ELECTRICAL 
 CHARAGTERISTICS 
 
 of 
 
 Transmission Circuits 
 
<r . I 
 
 THE LIBRARY 
 
 OF 
 
 THE UNIVERSITY 
 
 OF CALIFORNIA 
 
 DAVIS 
 
 FROM THE LIBRARY 
 
 OF 
 
 SOPHIA L. MC DONALD 
 
Digitized by the Internet Archive 
 
 in 2007 with funding from 
 
 IVIicrosoft Corporation 
 
 http://www.archive.org/details/electricalcharacOOnesbrich 
 
ELECTRICAL 
 CHARACTERISTICS OF 
 TRANSMISSION CIRCUITS 
 
 REPRINT 82 
 FEBRUARY. 1922 
 
 w 
 
 (WCSTINCHOUSCA 
 ELECTRIC 
 
 Reprinted from artidea originally appearing in the Electric Journal 
 
 WESTINGHOUSE ELECTRIC & MANUFACTURING CO. 
 
 EAST PITTSBURGH, PA. 
 
 LIBRARY 
 
Copyrighted, 1922, 
 
 BY THE 
 
 Westinghouse Electric & Manufacturing Co. 
 East Pittsburgh, Pa. 
 
PREFACE 
 
 THE rapid expansion in distributing and transmission systems will 
 continue unabated until the natural power resources will have been 
 fully developed. This expansion will necessitate a tremendous 
 amount of arithmetical labor m connection with the proper solution and 
 calculation of performance of projected transmission and distributing 
 circuits. It will demand much valuable time and energy in the educa- 
 tion of the younger engineers now going thru the technical schools and 
 others who will follow them. It was primarily to assist these younger 
 engineers by making their work more easy and less liable to error, and 
 providing them with all necessary tools that the data in this book have 
 been compiled. 
 
 Many articles each pertaining to some particular method of solution 
 of transmission circuits have been published from time to time. This book 
 constitutes a review of each of numerous methods perviously proposed by 
 different authors with examples illustrating each method of solution and 
 the accuracy which may be expected by its use. Thus by permission of 
 various authors the reader of this book is provided with a choice of nu- 
 merous methods ranging between the most simplified graphical forms of 
 solutions and complete mathematical solutions. He is also provided with 
 ntunerous and extensive tables of circviit and other constants making it 
 uimecessary for him to lose time and risk making mistakes in calculating 
 constants for each case in question. Much effort has been expended with 
 a view of simplifying explanations by the aid of supplementary diagrams 
 and tabulations. The engineer upon whose lot it only occasionally falls 
 to determine the size of conductors and performances of circuits appreciates 
 how easy it is to make errors in calculations which may prove very serious 
 and should find the quick estimating tables very useful particularly for 
 short line solutions. 
 
 For those preferring to avoid the more mathematical solutions the all 
 graphical methods for solving long line problems including the Wilkinson 
 & Kennelly charts for obtaining graphically the auxiliary constants should 
 prove helpful. 
 
 When borrowed material has been used in this book full credit has 
 been given the author at the place the material is used. It is desired, 
 however, at this place to mention the high appreciation of assistance given 
 by Ralph W. Atkinson, Herbert B. Dwight, Dr. A. E. Kennelly, Dr. A. S. 
 McAllister, Ralph D. Mershon, F. W. Peak Jr., J. F. Peters, Charles R. 
 Riker and T. A. Wilkinson. 
 
 y^ nuox 
 
CONTENTS 
 
 Page 
 
 Chapter I Resistance, Skin Effect, Inductance 1 
 
 Chapter II Reactance, Capacitance, Charging Current 10 
 
 Chapter III Quick Estimating Tables 23 
 
 Chapter IV Corona Effect 35 
 
 Chapter V Speed of Electric Propagation, Resonance, Parallel- 
 ing Transmission Circuits, Heating of Bare Con- 
 ductors out of Doors 40 
 
 Chapter VI Determination of Frequency and Voltage 45 
 
 Chapter VII Performance of Short Transmission Lines 49 
 
 Chapter VIII Performance of Long Transmission Lines (graphical) 61 
 
 Chapter IX Performance of Long Transmission Lines (mathe- 
 matical) 77 
 
 Chapter X Hyperbolic Functions 88 
 
 Chapter XI Performance of Long Transmission Lines (by hyper- 
 bolic functions) 95 
 
 Chapter XII Comparison of Various Methods Ill 
 
 Chapter XIII Cable Characteristics 121 
 
 Chapter XIV Synchronous Motors and Condensers for Power 
 
 Factor Improvement 129 
 
 Chapter XV Phase Modifiers for Voltage Control 138 
 
 Chapter XVI A Typical 220 Kv. Problem 145 
 
ELECTRICAL CHARACTERISTICS 
 of TRANSMISSION CIRCUITS 
 
 CHAPTER I 
 RESISTANCE-SKIN EFFECT-INDUCTANCE 
 
 THE transmission of alternating-current power in- 
 volves three separate circuits, one of which is 
 composed of the wires forming the transmission 
 line, while the others lie in the medium surrounding the 
 wires. The constants of these circuits are interde- 
 pendent; although any one may vary greatly from the 
 others in magnitude.* There is first the electric cir- 
 cuit through the conductors. Then since all magnetic 
 and dielectric lines of force are closed upon themselves 
 forming complete circuits there is a magnetic and a 
 dielectric circuit. The magnetic circuit consists of 
 magnetic lines of force encircling the current carrying 
 conductors and the dielectric circuit the dielectric lines 
 of force terminating in the current carrying conductors. 
 The close analogy of these is given in Table A, a care- 
 ful study of which will help those not familiar with the 
 subject to a clearer understanding of what happens in 
 an alternating-current transmission circtiit. 
 
 » For a unidirectional constant current the magnetic 
 field remains constant, and similarly for a unidirectional 
 constant voltage the dielectric field is constant. With 
 both the current and the voltage unidirectional and con- 
 stant, the electric circuit alone enters into the calcula- 
 tions. A changing magnetic flux introduces a voltage 
 into the electric circuit which modifies the initial or 
 impressed voltage. This eflfect of the magnetic circuit, 
 which is measured by the inductance L, storing the 
 energy o.^i^L, is a function of the current, and hence 
 is of most importance in dealing with heavy current 
 circuits. Similarly a changing electrostatic flux adds 
 
 *For a further description of these circuits see "Alteniat- 
 iiig Currents" by Prof. Carl E. Magnusson, from which Figs. 
 I to 5 are reproduced with the permission of the author. 
 
 (vectorially) a current to the main power current. 
 This effect of the dielectric circuit, which is measured 
 by the capacitance, storing the energy o.^e^C, is a func- 
 tion of the voltage, and hence is of most importance In 
 dealing with high-voltage circuits. 
 
 In an alternating-current circuit, both the voltage 
 and the current are continually varying in magnitude, 
 and morever, reversing in direction for each successive 
 half cycle. Therefore, with alternating currents, 
 energy changes occur continuously and simultaneously 
 in the interlinked magnetic, dielectric and electric cir- 
 cuits. 
 
 Figs. I to 5 inclusive illustrate the magnetic and 
 dielectric field surrounding conductors carrying current. 
 Figs. I and 3 represent respectively the magnetic and 
 dielectric circuits when the conductors are far apa'-t 
 and Figs. 2 and 4 when they are close together. Fig. 
 5 represents the resultant of the superimposed mag- 
 netic and dielectric fields. 
 
 The magnetic field surrounding a conductor which 
 is not influenced by any other field is represented by 
 concentric circles. This field is strongest at the surface 
 of the conductors and rapidly decreases with increasing 
 distance from the conductor as indicated by the spac- 
 ing of the lines of Figs, i and 2. 
 
 The dielectric stresses surrounding conductors are 
 represented by lines drawn radially from the con- 
 ductor. The strength of the dielectric field likewise 
 decreases with the distance from the conductor as is 
 indicated by the widening of the space between the 
 lines. The mignetic and the dielectric lines of force 
 always cross each other at right angles, as shown in 
 Fig- 5- 
 
RESISTANCE—SKIN EFFECT— INDUCTANCE 
 
 RESISTANCE OF COPPER CONDUCTORS 
 In Table I the resistance per thousand feet is listed 
 and in Table II per mile of single conductor. Values 
 are given for both solid and stranded copper conductors 
 at both loo and 97.3 percent conductivity and corres- 
 ponding to various temperatures between zero and 75 
 degrees C. The foot notes with these tables cover all 
 of the pertinent data upon which the values are based. 
 The resistance values in Table I corresponding to 
 temperatures of 25 and 65 degrees C. were taken from 
 
 FIG I f IG '' 
 
 MAGNETIC FIELD OF SINGLE CONDUCTOR [ MAGNETIC FIELD Of CIRCUIT 
 
 THE MAGNETIC CIRCUIT 
 
 FIG. 3 
 DIELECTRIC FIELD OF SINGLE CONDUCTOR DIELECTRIC FIELD OF CmCUIT 
 
 THE DIELECTRIC CIRCUIT' 
 
 FIG. 6, COMBINED DIELECTRIC AND MAGNETIC CIRCUITP 
 
 THE ELECTRIC CIRCUIT 
 
 Bulletin 31 of the Bureau of Standards issued April ist, 
 
 1912. The resistance values (taking into account the 
 
 expansion of the metal with rise in temperature) for the 
 
 other temperatures were calculated in accordance with 
 
 the following rule from page 10 of Bulletin No. 31. 
 
 The change of resistivity of copper per degree C. is 
 a constant, independent of the temperature of reference 
 and of the sample of copper. This resistivity-temperature 
 constant may be taken for general purposes as 0.0409 ohm 
 (mil foot). 
 
 As an illustration : — A 2 000 000 circ. mil stranded 
 
 copper conductor at 100 percent conductivity, has .1 
 
 resistance of 0.00623 ohm per 1000 feet at 65 degrees C. 
 
 Required to calculate its resistance at zero degrees C. 
 
 65X0.0409 = 2.6585 ohms (mil-foot) temper- 
 ature correction or 2658.5 ohms (mil, 1000 feet). 
 
 2658.5 
 
 = 0.0013^ ohm change m resistance. 0.00623 
 
 2000000 
 
 — 0.00133 = 0.0049 ohm resistance at zero degrees C. 
 It has been customary to publish tables of resist- 
 ance values based upon a temperature of 20 degrees C. 
 and 100 percent conductivity. The operating tempera- 
 tures of conductors carrying current is usually con- 
 siderably higher than 20 degrees C. and therefore cal- 
 culations based upon this temperature do not often re- 
 present operating conditions. Neither does copper of 
 100 percent conductivity represent the usual condition 
 for transmission circuit copper, whose average conduct- 
 ivity is probably nearer 97.3 percent. The values in 
 Tables I and II furnish a comparison of resistance for 
 annealed and hard drawn copper of stranded and solid 
 conductors at various temperatures based upon the new 
 "Annealed Copper Standard". 
 
 SKIN EFFECT 
 
 A solid conductor may be considered as made up 
 of separate filaments, just as a piece of wood is made 
 up of separate fibres. As a stranded conductor is 
 actually made up of a number of separate wires, such 
 a conductor will be considered in the following ex- 
 planation. The inductance of the various wires of the 
 cable will be different, due to the fact that those wires 
 near the center of the cable will be linked by more flux 
 lines than are the wires near the outer surface. The 
 self-induced back e.m.f. will therefore be greater in the 
 wires located near the center of the cable. The higher 
 reactance of the inner wires causes the current to dis- 
 tribute in such a manner that the current density will 
 be less in the interior than at the surface. This crowd- 
 ing of the current to the surface or "skin" of the wire 
 is known as "skin effect". 
 
 Since the self-induced e.m.f. is proportional to the 
 frequency as well as to the total flux linked, the skin 
 effect becomes more pronounced at higher frequencies 
 of the impressed e.m.f. It also becomes greater the 
 larger the cross-section, the greater the conductivity 
 and the greater the permeability of the conductor. 
 
 As a result the effective resistance of a conductor 
 to alternating current is greater than to direct current. 
 The effective resistance of nonmagnetic conductors to 
 alternating current may be obtained by increasing their 
 direct-current resistances by the percentages in Table 
 B, which were derived by the formulas in Pender's 
 Handbook. Thus the ohmic resistance of a 1000 000 
 circ. mil cable is approximately 8.4 percent greater at 
 60 cycles than its resistance to direct current at a tem- 
 perature of 25°C. If the temperature of the conductor 
 is 65 °C, its 60 cycle ohmic resistance will be approxi- 
 mately 6.4 percent greater than its direct-current re- 
 sistance. The practical result of skin effect is to re- 
 duce the carrying capacity of large cables. As in- 
 dicated by the values in Table B, skin effect may be 
 neglected when employing non-magnetic conductors ex- 
 
RESISTANCE—SKIN EFFECT— INDUCTANCE 
 
 cept in the use ot very large diameters. It is usual to viently large, a thousandth part of it, called the miUi- 
 
 manufacture cables of very large diameter, especially henry, is the usual practical unit. This unit is the co- 
 
 for service at high frequencies, with a non-conducting efficient of self-induction and is represented by the 
 
 core. In case of magnetic conductors, such as steel wire letter L. 
 
 or cable, as is some times used for long spans or short 
 high voltage feeders, skin effect must be carefully con- 
 sidered.* 
 
 DISTRIBUTION OF FLUX 
 
 When current flows through a conductor, a mag- 
 netomotive force (m.m.f.) is established of a value 
 
 TABLE A— COMPARISON OF THE THREE CIRCUITS 
 
 The Electric Circuit 
 
 The Magnetic Circuit 
 
 The Dielectric Circuit 
 
 Current / 
 Voltage E=RI 
 Electric Power 
 
 Resistivity 
 Resistance R—W/P 
 
 Magnetic Flux <t> 
 Magnetomotive Force /•"=»! 
 Magnetic Energy 
 
 Dielectric Flux V 
 Electromotive Force E-=.QIC 
 Dielectric Energy 
 
 Reluctivity 
 Reluctance R 
 Inductance Ij^^li 
 
 
 Reactance .r= wL — \l uiC 
 
 Elastivity \IK 
 Elastance 5" 
 Capacitance C=V IE 
 
 - Impedance z = 1/ r' -J- .^^ 
 
 Conductivity -y 
 
 Conductance 
 
 I g=W/E 
 \ 9=rlc 
 
 Permeability ix = BIH 
 Permeance jW= <^ /4JrF 
 
 Admittance y=i / :s:ig^^ J b = \/ g' -f- b'-- 
 
 INDUCTANCE 
 
 Any moving mass, for instance a flywheel in 
 motion, will resist a change in velocity. That is, the 
 inertia of the moving mass will tend to keep the mass 
 moving when disconnected from the source of power. 
 On the other hand the inertia will oppose any effort 
 to speed up the movement of the mass. 
 
 In a similar manner, the inductance of an electric 
 circuit resists a change in current. The cause of in- 
 ductance in an electric circuit is the magnetic field which 
 surrounds the circuit. When the current changes this 
 magnetic field changes correspond- 
 ingly, and in effect cuts the con- TABLE B— INCREASE OF EFFECTIVE RESISTANCE DUE TO SKIN EFFECT. 
 
 proportional to the current. 
 This m.nxf. is of zero 
 value at the center of the 
 conductor and increases as 
 the square of the distance 
 from the center until the 
 surface is reached, i (This 
 statement as well as those 
 following is based upon the 
 assumption of a uniform 
 distribution of current 
 throughout the conductor, 
 the conductor being of 
 non-magnetic material and 
 located i n non-magnetic 
 surroundings, such as air). At the surface it becomes 
 maximum for a given current and remains at this maxi- 
 mum value for all distances beyond the surface. It is 
 customary to think of the magnetic field surrounding 
 conductors as concentric circles of lines of force. 
 
 A physical picture of the magnetic field density 
 surrounding a current carrying conductor A is shown 
 by Chart I. The magnetic density due to the return 
 circuit (conductor B) is indicated in outline by broken 
 lines. The horizontal divisions represent the distance 
 from the center of conductor A and the height of the 
 
 I Permittivity A' 
 
 Permittance 
 I (Capacitance) C 
 Susceptance b=x/:r 
 
 ductor, producing an e.m.f. in it. 
 This e.m.f. of self induction has 
 such a direction as to resist the 
 change in current. While the cur- 
 rent is increasing, energy is stored 
 in the magnetic field and while the 
 current decreases, the magnetic 
 stored energy is returned to the 
 electric circuit. This effect of the 
 electric current on the surrounding 
 space is termed magnetic induction. 
 Unit of Inductance — When a 
 rate of change of current of one 
 ampere per second produces an 
 e.m.f. of one volt, the circuit is said 
 to have a unit of inductance called 
 a henry. The henry being incon- 
 
 *Rcfercnccs: — For a bihliography on the subject of skin 
 effect see article "Experimental Researches on Skin Effect in 
 Conductors" by A. E. Kennellv, F. A. Laws, and P. H. Pierce 
 in A. J. E. E. Trans., Vol. 34, Part II of Sept. 1915. .This 
 article ends with a bibliography on the subject embracing a 
 very complete list of articles. 
 
 "Calculation of Skin Effect in Strap Conductors" by H. 
 B. Dwight in Electrical World, March II, 1916. 
 
 "Skin Effect in Tubular and Flat Conductors" by H. B. 
 Dwight m A. I. E. E. Trans, for 1918. 
 
 For various sizes of solid copper rods. For stranded conductors of equivalent cross 
 sectional area the skin effect is practically the same as for the solid conductor. 
 
 1! 
 
 
 
 hi 
 
 = S " 
 
 *« 
 
 .5 
 
 h 
 
 si 
 
 P 
 
 J'ert-ent Increase of Copper Wires Above the Direct- Current 
 
 Resistance Due to Alternating -Currents of 
 
 Different Frequencies 
 
 Hased Upon Direct Current Re- 
 sistance at 25 Degrees 0. 
 (77 Degrees P.) 
 
 Baaed Upon Direct Current Re- 
 sistance at 65 Degrees C. 
 (149 Degrees F.) 
 
 V 
 
 
 u 
 
 si 
 
 U 
 
 U 
 
 >> 
 
 U 
 
 (A 
 U 
 
 Si 
 
 u 
 
 4 
 
 4 
 
 n 
 
 u 
 
 ■2 000 000 
 1 800 000 
 1 600 000 
 
 1.631 
 1.548 
 1.459 
 
 1.414 
 1.342 
 1.265 
 
 2.2 
 1.8 
 1.4 
 
 6.0 
 4.9 
 3.9 
 
 14.1 
 
 11.7 
 
 9.4 
 
 28.0 
 23.7 
 19.4 
 
 78.6 
 70.4 
 61.4 
 
 1.7 
 1.3 
 1.1 
 
 4.S 
 3.7 
 3.0 
 
 10.9 
 9.0 
 7.8 
 
 22.1 
 18.5 
 15.0 
 
 67.0 
 60.0 
 51.8 
 
 1 500 000 
 1 200 000 
 1 000 000 
 
 1.412 
 1.263 
 1.152 
 
 1.225 
 1.096 
 1.000 
 
 1.3 
 0.8 
 0.6 
 
 3.4 
 2.1 
 1.5 
 
 8.4 
 5.5 
 3.8 
 
 17.4 
 11.7 
 
 8.4 
 
 67.3 
 42.7 
 33.8 
 
 0.9 
 0.6 
 0.4 
 
 2.6 
 1.7 
 1.1 
 
 6.4 
 4.1 
 3.0 
 
 13.5 
 9.0 
 6.4 
 
 47.4 
 34.8 
 26.2 
 
 •. 50 000 
 500 000 
 250 000 
 
 0.998 
 0.815 
 0.575 
 
 0.866 
 0.707 
 0.500 
 
 0.3 
 0.1 
 
 0.0 
 
 0.9 
 0.4 
 0.1 
 
 2.2 
 1.0 
 0.3 
 
 4.9 
 2.2 
 0.6 
 
 20.6 
 
 10.1 
 
 2.7 
 
 0.2 
 0.1 
 0.0 
 
 0.7 
 0.8 
 0.1 
 
 1.7 3.7 
 0.7 1.7 
 0.2 0.4 
 
 16.4 
 7.7 
 2.0 
 
 curve rneasured vertically the intensity of the field at the 
 corresponding distance. The radius of the conductor 
 has been assumed as unity, and maximum field density 
 (always at the surface of the conductor) as 100 percent. 
 The intensity of the magnetic field starts at zero at 
 the conductor center, and increases (with uniform dis- 
 tribution of current in the conductor) directly as the 
 
4 RESISTANCE—SKIN EFFECT— INDUCTANCE 
 
 distance from its center until its surface is reached, The intensity of the magnetic field at any point is 
 
 where it becomes maximum. For distances beyond the proportional to the m.m.f. acting at that point and in- 
 
 surface of the conductor, the field intensity varies in- versely proportional to the length of its circular path 
 
 versely as the distance from its center. (magnetic reluctance). Thus at the surface of the 
 
 TABLE I— RESISTANCE PER 1000 FEET 
 
 OF COPPER CONDUCTORS AT VARIOUS TEMPERATURES 
 
 STRANDED CONDUCTORS 
 
 d 
 
 z 
 
 CO 
 
 '^ 
 
 00 
 
 AREA 
 
 CIRCULAR 
 MILS 
 
 OHMS PER 1000 FEET OF SINGLE CONDUCTOR 
 
 ANNEALED COPPER 
 
 100% CONDUCTIVITY 
 
 HARD DRAWN COPPER 
 
 97.3^ CONDUCTIVITY 
 
 CO 
 
 32° F 
 
 I5°C 
 
 59° F 
 
 20''C 
 
 68°F 
 
 25°C 
 
 77° F 
 
 35°C 
 
 95°F 
 
 50° C 
 
 I22°F 
 
 65°C 
 
 149° F 
 
 75°C 
 I67°F 
 
 0°C 
 32° F 
 
 I5°C 
 
 59° F 
 
 20°C 
 68° F 
 
 25°C 
 77° F 
 
 35°C 
 95°F 
 
 50°C 
 
 I22°F 
 
 65°C 
 I49°F 
 
 75°0 
 I67°F 
 
 oooo 
 
 Zooo ooo 
 1 fooooa 
 1 8O0 ooo 
 
 .oo.^tl 
 .005/2 
 
 00SI8 
 
 .00S4(. 
 .OOS77 
 
 OOS7.B 
 
 OOSSi 
 
 .OOS98 
 
 OO.S3<? 
 0OS6S 
 
 OOS99 
 
 OOSSf 
 
 00S90 
 
 006Z2 
 
 .0QS9I 
 
 .00623 
 
 OOtSl 
 
 00623 
 006S6 
 00692 
 
 00643 
 00&78 
 .007/6 
 
 .OO-SOO 
 OO.S26 
 
 OOSS6 
 
 0^-3: 
 .00S61 
 00S93 
 
 .00^-44 
 00S70 
 .0060S 
 
 aasS4 
 
 OOS74 
 0061s 
 
 OOS74 
 oo6oh 
 00640 
 
 .00607 
 .00640 
 00 67.J 
 
 0O640 
 00674 
 007// 
 
 00660 
 
 00697 
 .0073S 
 
 1 TOO ooo 
 
 0,00 000 
 
 / Soa ooo 
 
 OOS73 
 .00407 
 .006SO 
 
 .006/0 
 00647 
 .00690 
 
 006Z2 
 OO66O 
 .00704 
 
 .006ZS 
 00674 
 007/9 
 
 O06S9 
 
 .00700 
 .00746 
 
 oo69S 
 .00740 
 .00787 
 
 00733 
 00779 
 
 00830 
 
 007s 8 
 00 SOS 
 0O9S8 
 
 OOS90 
 .O0626 
 00668 
 
 .006ZI 
 0066S 
 .00709 
 
 0O640 
 O0678 
 00724 
 
 oo6S2 
 00693 
 .00739 
 
 ooblf 
 00720 
 .00766 
 
 007/4 
 .00760 
 008O8 
 
 O07-S'3 
 OOSOO 
 
 008.^3 
 
 0078O 
 .00827 
 00882 
 
 1 AGO ooo 
 / 300 ooo 
 /Zoo ooo 
 
 oai.9i 
 
 007^^ 
 
 .008/2 
 
 .00741 
 0079S 
 ooSfc'f 
 
 .007SS 
 
 .008/3 
 
 00 880 
 
 0077/ 
 
 00830 
 
 00899 
 
 00 Soa 
 00S6Z 
 00933 
 
 008-W 
 oolio 
 oo98.i 
 
 00890 
 009S8 
 0104 
 
 0093.0 
 00990 
 0107 
 
 0071s 
 00770 
 00^3S 
 
 00761 
 00820 
 
 .ooS-W 
 
 0077S 
 .00836 
 0090S 
 
 ■00793 
 ooS^SS 
 
 00724 
 
 00822 
 0088.S' 
 009^8 
 
 00868 
 093-S 
 0101 
 
 009IS 
 oa98S 
 .0107 
 
 0094s 
 O/OX 
 .0/ /O 
 
 1 loo ooo 
 
 1 000 ooo 
 
 9SOOOO 
 
 ooesi 
 .00974 
 .0/02 
 
 .0094Z 
 
 .0104 
 
 .0109 
 
 00960 
 .0/06 
 01 1 1 
 
 O09SI 
 0108 
 0/14 
 
 0107. 
 0/12 
 0//8 
 
 0/08 
 0/ /8 
 0/24 
 
 0//3 
 0/2S 
 OI3I 
 
 ■ 0117 
 0/29 
 .0/3^" 
 
 00910 
 
 100 
 o/oS 
 
 oo?6ff 
 0/07 
 01 IX 
 
 00986 
 0109 
 0/ /4 
 
 .0/0/ 
 
 ■ 01 1 1 
 .0117 
 
 o/os 
 
 0/IS 
 OI21 
 
 0/1/ 
 0/2/ 
 .0/27 
 
 .01/6 
 0/Z8 
 0/34 
 
 0/20 
 0/32 
 0/38 
 
 900 000 
 esoooo 
 900000 
 
 .0/08 
 
 .01 IS 
 
 OIZZ 
 
 . 01 IS 
 
 ■ OIZZ 
 
 .0130 
 
 .0117 
 0/Z4 
 0132 
 
 .0/20 
 .0 127 
 0/3S 
 
 0/24 
 
 0/32 
 0/40 
 
 0/31 
 0/39 
 0148 
 
 .0/38 
 
 0/47 
 
 OISi 
 
 0/42 
 OiSZ 
 .0/6/ 
 
 oil/ 
 0/18 
 012s 
 
 0118 
 
 OIZS 
 0/33 
 
 0/20 
 .0127 
 0136 
 
 .0/23 
 O/30 
 .0139 
 
 .0/27 
 .0I3S 
 ■ OI44 
 
 0/34 
 0/43 
 .OIS2 
 
 0/4^ 
 ■ O/SI 
 0/60 
 
 0/46 
 O/Sl 
 .0/6S 
 
 7SO 000 
 700000 
 6S0OO0 
 
 .0130 
 .0139 
 
 .OlSO 
 
 OI38 
 OI48 
 0I6O 
 
 .0140 
 
 .O/SI 
 
 OI63 
 
 0/44 
 0/S4: 
 .0/66 
 
 0/49 
 0/60 
 ■ 0/7Z 
 
 01S7 
 OI69 
 0182 
 
 0/66 
 .0178 
 
 0/92 
 
 0/7/ 
 OI84 
 .0199 
 
 0134 
 
 143 
 .0/S4 
 
 0/42 
 0/^2 
 .0/64 
 
 ol4f 
 DISS 
 0/67 
 
 .OI48 
 0IS8 
 0170 
 
 1 S3 
 
 .0/64 
 
 0/76 
 
 .0/6/ 
 .0/73 
 ■ 0187 
 
 0/70 
 .0183 
 ■ OI97 
 
 0.7S 
 ■ 0189 
 0204 
 
 600 000 
 ssoooo 
 
 SOOOOO 
 
 OlbT. 
 
 .0/77 
 .OI9S 
 
 .0173 
 .0/88 
 0207 
 
 0/76 
 
 .019/ 
 
 021 1 
 
 0/80 
 .0196 
 02/6 
 
 0/87 
 .0203 
 0224 
 
 .0197 
 
 . 02/4 
 
 0236 
 
 0208 
 
 0226 
 024f 
 
 021 S 
 0234 
 02S8 
 
 0/66 
 0/82 
 ozoo 
 
 0/78 
 0/93 
 02/3 
 
 .018/ 
 0196 
 02/7 
 
 ■ OI8S 
 
 0202 
 
 .0222 
 
 0/92 
 0209 
 O230 
 
 020 2 
 0220 
 .024Z 
 
 02/4 
 0232 
 ■ OZS6 
 
 022/ 
 
 0240 
 
 .026<r 
 
 -fSo 000 
 400000 
 3 so 000 
 
 .oa.l(, 
 .0243 
 ■ OZ78 
 
 .0230 
 Oi.S9 
 ■ 0297 
 
 .0234 
 .0264 
 0303 
 
 0240 
 0270 
 0308 
 
 0249 
 0280 
 03/9 
 
 0263 
 0296 
 0337 
 
 0277 
 03/1 
 .03S6 
 
 0286 
 0322 
 0368 
 
 0222 
 
 02S0 
 0286 
 
 0236 
 02 66 
 030s 
 
 .0240 
 .027/ 
 03/2 
 
 02.47 
 OX77 
 0316 
 
 o2.sr6 
 
 0288 
 .0328 
 
 O270 
 03<t4 
 .0 34^ 
 
 0Z8S 
 .0319 
 .036t 
 
 0x94 
 033/ 
 037* 
 
 300 000 
 ZSo 000 
 Zl 1 too 
 
 .0324 
 
 0390 
 
 .0460 
 
 ■ 0346 
 041S 
 
 ■ 0490 
 
 .03S3 
 
 .0423 
 
 O^SOO 
 
 .03 60 
 0432 
 OSIO 
 
 0373 
 0448 
 OS29 
 
 039+ 
 
 0+73 
 asS9 
 
 04/S 
 049e 
 OS89 
 
 04 2 8 
 OSIS 
 O609 
 
 0333 
 
 0400 
 
 o-?73 
 
 03S6 
 .0426 
 0S03 
 
 0363 
 043S 
 OSI4 
 
 .0370 
 0444 
 O^SZS 
 
 0383 
 0460 
 0S44 
 
 040i" 
 .0487 
 OS73 
 
 0427 
 OSIZ 
 .060S 
 
 0440 
 0S30 
 0626 
 
 ooo 
 oo 
 
 
 
 1 67 77Z 
 133 079 
 /OSS60 
 
 osso 
 
 .0732 
 0922 
 
 0&I8 
 0778 
 o98Z 
 
 06Z0 
 079S 
 100 
 
 o644 
 , 081/ 
 
 /OX 
 
 .0668 
 .084/ 
 . /Ob 
 
 0706 
 
 0888 
 1/2 
 
 0742 
 
 .0936 
 
 //8 
 
 0767 
 0967 
 122 
 
 QS96 
 07S2 
 0948 
 
 063S 
 OSOO 
 101 
 
 .0648 
 OS'S 
 /03 
 
 0662 
 
 .0834 
 
 /OS 
 
 0687 
 086S 
 ./09 
 
 072s 
 
 0900 
 //S 
 
 0762 
 09<2 
 /2/ 
 
 0788 
 099s 
 /2S 
 
 1 
 z 
 
 3 
 
 83 694 
 6i,3S8 
 SZ4,24 
 
 //6 
 ./47 
 /8S 
 
 .124 
 IS6 
 197 
 
 126 
 
 .lS9 
 201 
 
 ./29 
 ./63 
 20S 
 
 .'/It 
 2/3 
 
 /4I 
 /78 
 2ZS 
 
 /49 
 /88 
 237 
 
 IS4 
 194 
 2 4>f 
 
 1/9 
 
 ./s/ 
 
 ./90 
 
 127 
 160 
 202 
 
 IZ9 
 ./63 
 207 
 
 132 
 /67 
 Z/ / 
 
 ■ /3t 
 /73 
 2/9 
 
 /4S 
 /83 
 23/ 
 
 /S3 
 /93 
 Z44 
 
 IS8 
 ■ '2V2. 
 
 4 
 S 
 
 6 
 
 41 73S 
 33 079 
 Z&. Z44 
 
 233 
 
 .Z94 
 ■ 371 
 
 .248 
 
 ■ 314 
 39J 
 
 Z53 
 3ZO 
 .403 
 
 .ZS9 
 327 
 4/2 
 
 269 
 .339 
 427 
 
 .294 
 3S8 
 4S2 
 
 298 
 376 
 47S 
 
 30ff 
 388 
 
 49/ 
 
 239 
 30Z 
 3SI 
 
 2SS 
 323 
 
 406 
 
 260 
 328 
 
 41S 
 
 266 
 336 
 423 
 
 276 
 348 
 438 
 
 292 
 368 
 
 464 
 
 .306 
 3?^ 
 488 
 
 .3/6 
 
 ■ 399 
 
 ■ S04 
 
 7 
 
 8 
 
 ZO 822 
 
 /6 S/Z 
 
 ■ 468 
 .S90 
 
 • 497 
 .628 
 
 .>S-07 
 640 
 
 ■S-/9 
 6S4 
 
 S38 
 
 678 
 
 S69 
 7/6 
 
 S98 
 7SS 
 
 6/8 
 78/ 
 
 .482 
 607 
 
 S/Z 
 646 
 
 ■ SZO 
 6S8 
 
 S33 
 
 672 
 
 SS3 
 697 
 
 .S8S 
 736 
 
 ■ 6/S 
 • 770- 
 
 63S 
 80 2 
 
 S( 
 
 OLID CONDU 
 
 ICT( 
 
 DRS 
 
 oooo 
 ooo 
 
 00 
 
 zn too 
 
 H.7 77Z 
 133 079 
 
 04SI 
 .os(,9 
 07/8 
 
 0490 
 06O6 
 0764 
 
 0490 
 06I8 
 0779 
 
 OSOO 
 0630 
 079S 
 
 0S19 
 
 06S4 
 0826 
 
 OS48 
 691 
 .087/ 
 
 OS77 
 0727 
 09/7 
 
 OS96 
 07S2 
 0948 
 
 0463 
 OS8S 
 0738 
 
 0493 
 
 0623 
 a78S 
 
 0S03 
 063S 
 08OO 
 
 OSI4 
 
 .0647 
 
 08/7 
 
 0^33 
 0672 
 OSSO 
 
 .OS63 
 
 .07/0 
 .0894 
 
 0S9Z 
 0746 
 0942 
 
 .06/2 
 0772 
 0974 
 
 o 
 1 
 z 
 
 lOSSiO 
 93i,94 
 4,^358 
 
 .0 90S 
 
 .114 
 
 .144 
 
 09 63 
 
 .121 
 
 .IS3 
 
 0983 
 
 124 
 
 /S6 
 
 . /oo 
 
 ./26 
 
 ■ /S9 
 
 /04 
 
 ./3; 
 
 ./6S 
 
 / /O 
 139 
 /7S 
 
 //6 
 /46 
 /84 
 
 /20 
 
 ./SI 
 
 /to 
 
 0930 
 
 /17 
 
 /48 
 
 0988 
 
 /24 
 
 /S7 
 
 /Ol 
 /27 
 /60 
 
 /03 
 /29 
 /63 
 
 /07 
 ./34 
 ./70 
 
 1/3 
 ./43 
 ■ 180 
 
 ./I9 
 ./SO 
 ■ 189 
 
 /2 3 
 
 ■ /SS 
 
 ■ /9S 
 
 3 
 S 
 
 SZ ^24 
 •f/ 738 
 33 088 
 
 181 
 
 2Z9 
 
 289 
 
 193 
 Z44 
 .307 
 
 197 
 Z48 
 ■ 313 
 
 ZOI 
 2S3 
 
 ■ 319 
 
 209 
 2«3 
 33 / 
 
 220 
 2 7? 
 ZSO 
 
 232 
 293 
 368 
 
 240 
 302 
 38/ 
 
 /86 
 23X 
 297 
 
 ■ 198 
 ZSO 
 3 IS 
 
 ZoZ 
 2SS 
 
 32/ 
 
 207 
 260 
 328 
 
 .2/S 
 270 
 340 
 
 4226 
 .286 
 .360 
 
 Z38 
 
 .30/ 
 .37<' 
 
 • 246 
 310 
 391 
 
 7 
 9 
 
 Z6Z4-f 
 ZO 8ZZ 
 /CSJZ 
 
 ■ 364 
 4S9 
 S79 
 
 .387 
 488 
 .616 
 
 .3y>i- 
 498 
 .628 
 
 403 
 
 ■ SO 8 
 .640 
 
 418 
 S28 
 66S 
 
 442 
 SS7 
 70X 
 
 4<^ 
 S86 
 739 
 
 49' 
 606 
 764 
 
 374 
 47 Z 
 .S9S 
 
 398 
 
 SOX 
 
 633 
 
 +07 
 
 S/Z 
 64S 
 
 41S 
 S23 
 6S7 
 
 430 
 ,S-43 
 .68-S- 
 
 4X4 
 S7Z 
 7ZZ 
 
 •477 
 
 ■ 602 
 
 ■ 7S9 
 
 4^4 
 623 
 7?^ 
 
 These resistance values do not take into account skin effect. This should be considered when the larger conductors are used, particu- 
 larly at the higher frequencies. No allowance has been made for increased length due to sag when the conductors are suspended.. The resist- 
 ance values for the stranded conductors are two percent greater than for a solid rod of cross -section equal to the total cross -section of 
 the wires of the cable. ^ 
 
 The change of resistivity of copper per degree C. is a constant independent of the temperature of reference and of the sample of cop- 
 per. This resistivity -temperature constant is 0.0409 ohm (mil, foot). The fundamental resistivity used in calculating this table is the an- 
 nealed copper standard, viz. 0.15328 ohm (meter, gram) at '20 degrees C. 
 
 For sizes not given in the table computations may be made by the following formulas which were used in calculating the above table: — 
 
 Ohms per 1000 feet of annealed copper at 25 degrees C =- 
 
 10787 
 
 Circ. mils 
 
 at 65 degrees C — 
 
 12457 
 
 Circ. mili 
 
RESISTANCE—SKIN EFFECT— INDUCTANCE 
 
 conductor the m.m.f. reaches its maximum because all 
 of the current of the conductor is acting to produi^e 
 m.m.f. at this and all points beyond. On the other 
 hand the circular path subject to this maximum m.m.f. 
 is shortest at the surface, the reluctance a minimum 
 
 and consequently the field intensity is greatest. For 
 points beyond the surface the length of the circular 
 path through air is proportional to the distance from 
 the center of the conductor. Thus at a distance of ^ 
 from the center the circular path is twice as long as at 
 
 TABLE II— RESISTANCE PER MILE 
 
 OF COPPER CONDUCTORS AT VARIOUS TEMPERATURES 
 
 STRANDED CONDUCTORS 
 
 d 
 
 z 
 
 CO 
 
 AREA 
 
 CIRCULAR 
 MILS 
 
 OHMS PER MILE OF SINGLE CONDUCTOR 1 
 
 ANNEALED COPPER 
 
 100% CONDUCTIVITY 
 
 HARD DRAWN COPPER 
 
 97.3% CONDUCTIVITY 
 
 0°C 
 32° F 
 
 I5°C 
 59° F 
 
 20°C 
 68°F 
 
 25°C 
 
 77° F 
 
 35°C 
 95°F 
 
 50°C 
 I22°F 
 
 65°C 
 I49°F 
 
 75°C 
 I67°F 
 
 0°C 
 32°F 
 
 I5°C 
 59° F 
 
 20°C 
 68°F 
 
 25°C 
 77° F 
 
 35°C 
 
 95°F 
 
 50°C 
 I22°F 
 
 65''C 
 I49°F 
 
 75''0 
 I67°F 
 
 0000 
 
 Zooo 000 
 1 cfoo 000 
 
 1 800 000 
 
 .ozsa 
 .02,7/ 
 
 ,02.»* 
 
 0x74 
 
 OZSi 
 .030,1 
 
 .02.79 
 .0Z94 
 .0311 
 
 .03.iS 
 03OI 
 03/7 
 
 .OZ<iS 
 03I2 
 .0329 
 
 03/2 
 0330 
 0347 
 
 032f 
 
 .0347 
 .0346 
 
 0340 
 
 03S9 
 037? 
 
 026S 
 .0278 
 0x94 
 
 .0282 
 02?6 
 03/4 
 
 .029S 
 .030/ 
 0320 
 
 .0293 
 .0304 
 
 .032.S 
 
 0304 
 0320 
 
 0338 
 
 .032/ 
 0338 
 .03.17 
 
 0337 
 
 3J-6 
 037J 
 
 0349 
 
 .0368 
 03»9 
 
 1 700000 
 1 iOO 000 
 1 SOO 000 
 
 .O303 
 032.2 
 .0344 
 
 0323 
 ■ 034Z 
 03«sf 
 
 .0329 
 034? 
 .0373 
 
 0336 
 .03,S7 
 O380 
 
 .0348 
 .0370 
 ■ 0394 
 
 OZhS 
 
 .0391 
 
 0417 
 
 0388 
 o4'2 
 0438 
 
 0400 
 
 042.f 
 04.S4 
 
 .03/2 
 .0331 
 .0353 
 
 .033/ 
 
 .03.S-2 
 .0 37J- 
 
 0339 
 03.J8 
 0382 
 
 .0344 
 0367 
 039/ 
 
 03S8 
 038/ 
 040.J 
 
 0377 
 .0402 
 .0427 
 
 0398 
 
 0422 
 045/ 
 
 04/2 
 04 3* 
 04*7 
 
 1 400000 
 / 300000 
 
 tii^oooo 
 
 .03t8 
 .03»* 
 .04i.1 
 
 .0391 
 ■ 0422 
 .04S7 
 
 .0399 
 043C 
 + 6i 
 
 0408 
 .0439 
 
 .047S 
 
 .0423 
 04S6 
 0413 
 
 04 47 
 0482 
 .OS20 
 
 0470 
 OSO7 
 OSSC 
 
 04-87 
 0.S23 
 0S6S 
 
 .0378 
 •0407 
 04+2 
 
 .0402 
 .0433 
 0470 
 
 04/O 
 0442 
 047f 
 
 04/8 
 04.S"/ 
 0489 
 
 04 3.r 
 
 04*8 
 ojro7 
 
 o4.r9 
 
 041S 
 o.f34 
 
 0484 
 
 OJ-2/ 
 0S6S 
 
 OSOO 
 OJ39 
 o.r82 
 
 t' 100 000 
 
 / 600 boo 
 
 9So 000 
 
 OSlS 
 .0S3S 
 
 ■ 04 98 
 OSSO 
 .OJ77 
 
 .OS07 
 
 osio 
 .OS97 
 
 OSlS 
 OS70 
 0603 
 
 0539 
 
 OS9Z 
 0624 
 
 OS7Z 
 0623 
 
 o6se, 
 
 OS'n 
 066O 
 0613 
 
 06I8 
 0682 
 07/3 
 
 04 82 
 0.S28 
 OSSS 
 
 OSIZ 
 .OS6-i 
 .0S13 
 
 0S2I 
 0S77 
 Oi03 
 
 .0,S"33 
 .0~f87 
 OilS 
 
 osss 
 .0608 
 .0*40 
 
 0,S 87 
 O640 
 0472 
 
 06IS 
 067S 
 07IO 
 
 0«34 
 
 069/ 
 0730 
 
 ^00 000 
 9S0OOO 
 900000 
 
 ■ OS7I 
 .OiOS 
 
 ■ 0h4i 
 
 .0608 
 ■ Ot,4S 
 .0687 
 
 06I8 
 06SS 
 0C98 
 
 0^3S 
 o67Z 
 07/3 
 
 06SS 
 0698 
 .0740 
 
 .0693 
 .073S 
 0783 
 
 0730 
 0778 
 
 oszs 
 
 07SI 
 0803 
 08SI 
 
 OS87 
 0623 
 .0660 
 
 0623 
 .0660 
 0703 
 
 OiSS 
 672 
 07/8 
 
 0t,S0 
 0698 
 073-J 
 
 067Z 
 
 07/3 
 
 .0762 
 
 .0708 
 07SS 
 803 
 
 07SO 
 079S 
 094S 
 
 0772 
 o82.r 
 0873 
 
 7SO 000 
 700 000 
 6S0000 
 
 .okSS 
 
 ■ 073S 
 
 ■ 0793 
 
 ■ 07Zf 
 
 0783 
 
 .0946 
 
 .0740 
 0798 
 OSil 
 
 .076I 
 0814 
 0273 
 
 0788 
 .OS46 
 .09/0 
 
 0830 
 08^4 
 
 .0878 
 
 .0940 
 
 /02 
 
 090s 
 0973 
 lOS 
 
 0708 
 07S6 
 .OSlS 
 
 07SI 
 0803 
 .0847 
 
 0762 
 
 08lf 
 
 .0»83 
 
 7 82 
 
 .083-S- 
 
 0900 
 
 O80S 
 0866 
 0930 
 
 09/s 
 .0910 
 
 0900 
 096S 
 
 o92.f 
 . /oo 
 
 /or 
 
 600000 
 SSO 000 
 SOO 000 
 
 .0SS7 
 . /03 
 
 .09/S 
 .099S 
 ./lO 
 
 0930 
 .101 
 .IIZ 
 
 09 sz 
 
 .0988 
 
 ./04 
 
 .113 
 
 I2S 
 
 .110 
 /Zl 
 I3Z 
 
 / 14 
 /2A 
 136 
 
 .0878 
 0963 
 106 
 
 .0»-fO 
 /OZ 
 .//3 
 
 09S7 
 
 .104 
 
 ./IS 
 
 097^ 
 ./07 
 . 117 
 
 /OZ 
 
 .III 
 
 .IZZ 
 
 ■.'ill 
 
 IZ8 
 
 1/3 
 122 
 I3S 
 
 in 
 
 1X1 
 
 ./40 
 
 4SOOOO 
 
 ^00000 
 350000 
 
 ■ //-f 
 
 .JZ9 
 
 M7 
 
 .IZZ 
 .137 
 .IS7 
 
 IZ4 
 .140 
 .160 
 
 .163 
 
 I3Z 
 .148 
 .169 
 
 139 
 .IS7 
 .178 
 
 ./46 
 /6S^ 
 .188 
 
 /SI 
 /70 
 I9S 
 
 //8 
 I3Z 
 /SI 
 
 /ZS 
 141 
 I6Z 
 
 .127 
 
 ./44 
 
 I6S 
 
 /3I 
 
 /47 
 
 ./67 
 
 .136 
 /SZ 
 .174 
 
 143 
 ./6I 
 .183 
 
 ISO 
 /68 
 
 ./93 
 
 /S6 
 
 I7S 
 
 ZOO 
 
 300 000 
 ZSo 000 
 Zi 1 600 
 
 .171 
 ZOi, 
 ■ Z43 
 
 .183 
 .ZI9 
 ZS9 
 
 .187 
 
 .2Z4 
 
 264 
 
 .190 
 ZZ8 
 2t9 
 
 117 
 .237 
 .Z80 
 
 Z08 
 2 so 
 Z9i 
 
 .220 
 .2*3 
 .311 
 
 216 
 272 
 322 
 
 . 176 
 .Zll 
 Z49 
 
 188 
 22s 
 366 
 
 /9Z 
 230 
 .272 
 
 .196 
 Z3S 
 Z77 
 
 .203 
 .24 3 
 .Z88 
 
 2/4 
 ZS8 
 303 
 
 226 
 270 
 320 
 
 233 
 
 2 80 
 
 330 
 
 000 
 
 
 l(>7 77Z 
 13-3079 
 /OSSlO 
 
 .3oi 
 
 .387 
 
 488 
 
 .326 
 412 
 .S20 
 
 .333 
 
 .42.0 
 
 szs 
 
 .341 
 
 .4Z8 
 
 S40 
 
 .3S3 
 444 
 S60 
 
 .372 
 .470 
 .S9Z 
 
 39Z 
 
 49s 
 624 
 
 40.S- 
 SI2 
 64S 
 
 3IS 
 398 
 SOZ 
 
 .335 
 423 
 J3>J 
 
 342 
 
 .4-32 
 
 S4S 
 
 3,J-0 
 
 .4-42 
 
 ..sss 
 
 363 
 4S7, 
 .S76 
 
 383 
 
 608 
 
 402 
 SIO 
 64 
 
 4/6 
 S27 
 66/ 
 
 
 93 694 
 £a3S9 
 SX4,i-f 
 
 i,/Z 
 
 .777 
 
 9 7^ 
 
 6SS 
 .82S 
 
 / 04 
 
 . 66S 
 .840 
 1 07 
 
 682 
 /.09 
 
 708 
 89S 
 
 1 13 
 
 74s 
 942 
 119 
 
 787 
 
 99S 
 
 / 2S 
 
 8IS 
 103 
 
 / 30 
 
 630 
 798 
 
 / 01 
 
 672 
 84S 
 107 
 
 .68Z 
 .862 
 J /O 
 
 697 
 883 
 / /Z 
 
 730 
 .91S 
 //6 
 
 766 
 
 968 
 
 / 22 
 
 8IO 
 / OZ 
 
 / Z9 
 
 8as 
 
 /OS 
 / 33 
 
 4 
 S 
 
 417-38 
 33 079 
 
 1 ZZ 
 / Si, 
 1 9{, 
 
 1 31 
 /.i.6 
 Z09 
 
 1.34 
 J 69 
 Z 13 
 
 1 37 
 173 
 Z.17 
 
 I.AZ 
 1 90 
 2Z6 
 
 LSI 
 2 3-? 
 
 / S8 
 / 99 
 Z SI 
 
 /63 
 2 OS 
 2 S9 
 
 / 27 
 / 60 
 ZOI 
 
 / 3S 
 / 7/ 
 2/4 
 
 / 38 
 /73 
 ^,2o 
 
 /.4I 
 /78 
 Z.Z4 
 
 /.46 
 / 84 
 232 
 
 /.SS 
 / 9S 
 Z4S 
 
 /.6/ 
 Z04 
 Z S8 
 
 / 67 
 2 // 
 266 
 
 7 
 
 ZotiZ 
 fi 5IZ 
 
 Z.49 
 3tZ 
 
 Z.63 
 3 32 
 
 Z 68 
 3 38 
 
 Z74 
 3.46 
 
 <284 
 
 3-r8 
 
 3 01 
 3.78 
 
 3 16 
 399 
 
 3 27 
 
 4 13 
 
 Z SS 
 32/ 
 
 Z.7I 
 3.41 
 
 Z7S 
 3 48 
 
 Z8Z 
 
 3 SS 
 
 %l% 
 
 3 09 
 3. 89 
 
 3 2S 
 4/0 
 
 33S 
 
 4 24 
 
 SOLID CON 
 
 DUCTORS 
 
 oeeo 
 000 
 00 
 
 Zll 1,00 
 H,7 77Z 
 /33 079 
 
 Z38 
 30 1 
 3 SO 
 
 ZS4 
 
 320 
 .40'f 
 
 .ZS9 
 
 327 
 
 4IZ 
 
 264 
 
 333 
 
 .420 
 
 .274 
 .346 
 43i 
 
 28^ 
 3hS 
 460 
 
 30s 
 384 
 48S 
 
 3 IS 
 397 
 SOI 
 
 24,r 
 309 
 
 39(3 
 
 261 
 
 3Z9 
 
 .4 IS 
 
 .266 
 336 
 
 42-a 
 
 ■2.72. 
 .342 
 .432 
 
 2.82. 
 
 3.f,r 
 
 4SO 
 
 .298 
 
 .37S 
 
 473 
 
 3IZ 
 39s 
 
 497 
 
 323 
 408 
 
 SIS 
 
 
 
 / 
 Z 
 
 /OSSiO 
 93 4,94 
 (,i.3S9 
 
 478 
 
 .603 
 
 760 
 
 S09 
 
 .640 
 
 808 
 
 szo 
 
 .6SS 
 
 azs 
 
 SZ8 
 666 
 840 
 
 SSO 
 693 
 S7Z 
 
 S8Z 
 
 73S 
 
 .9ZS 
 
 6/3 
 77Z 
 97Z 
 
 63S 
 798 
 
 1 01 
 
 492 
 6I8 
 783 
 
 S22 
 6SS 
 830 
 
 .S3S 
 672 
 
 .»4.r 
 
 .S4S 
 .A80 
 .862 
 
 S6S 
 
 708 
 
 .900 
 
 ..r97 
 
 .7SS 
 ■ 9SO 
 
 628 
 .793 
 ) 00 
 
 6S0 
 820 
 
 / OS 
 
 3 
 5 
 
 SZi,Z4 
 41 739 
 33 ogg 
 
 9SS 
 1 Zl 
 1 S3 
 
 / oz 
 
 1 04 
 
 /3I 
 / 66 
 
 / 06 
 1.34 
 1 69 
 
 1 II 
 1 39 
 / 7S 
 
 1 16 
 lA7 
 I8S 
 
 / Z3 
 / SS 
 I9S 
 
 / 27 
 
 2 oz 
 
 983 
 
 /OS 
 / 32 
 / 67 
 
 I.07 
 1 3S 
 / 70 
 
 / /O 
 1 38 
 / 73 
 
 /./4 
 /43 
 
 / 80 
 
 J /9 
 /SI 
 
 / 90 
 
 /24 
 
 1 S9 
 
 2 00 
 
 / 30 
 
 /64 
 207 
 
 7 
 
 s 
 
 Zi,Z44 
 ZO 8Z7. 
 
 1 93 
 3 Ot, 
 
 zos 
 
 XSB 
 326 
 
 z 09 
 
 Z 63 
 3 33 
 
 2 /4 
 2 6f 
 33? 
 
 2 21 
 Z79 
 
 3 SI 
 
 Z 33 
 Z94 
 3 7/ 
 
 2 46 
 3/0 
 390 
 
 2 S4 
 
 3 20 
 
 4 04 
 
 'z'A 
 3 '4 
 
 Z 10 
 Z 6S 
 3 35- 
 
 z /s 
 
 a 7/ 
 
 341 
 
 Z ZO 
 
 Z 77 
 347 
 
 ■Z-Z7 
 
 2 87 
 
 3 6Z 
 
 2 40 
 
 3 OZ 
 38Z 
 
 X sz 
 
 3/8 
 40Z 
 
 3 29 
 
 4 If 
 
 These resistance values do not take into account skin effect. This should be considered when the larger conductor, are used partica- 
 larly at the higher frequencies. No allowance has been made for increased length due to sag when '*« '^""''"'^'^^ " ^ »P«°''~-„..\''» T"'": 
 anc6 values for the stranded conductors are two percent greater than for a solid rod of cross -section equal to the total crosi • section of 
 the wires of the cable. 
 
 The change of resistivity of copper per degree C. is a constant independent of the temperature »' "'«"°", J»°^,"' '»'VmT''u th.'^." 
 per. This resistivity temperature constant is 0.0409 ohm (mil. foot). The fundamental re.i.tivity used m calculating this Ubl* is Us u- 
 nealed copper standard, viz. 0.15328 ohm (meter, gram) at 20 degrees 0. 
 
RESISTANCE— SKIN EFFECT— INDUCTANCE 
 
 a distance of i (its surface) and consequently, although 
 the m.m.f. is the same the reluctance is double, per- 
 mitting only one-half as great a flux to flow as at the 
 surface. For a similar reason the density of the field 
 at a distance of lo is one-tenth the surface density; at 
 50 it is one-fiftieth, etc. The curve of field density be- 
 yond the surface of the conductor therefore assumes 
 the form of a hyperbola. 
 
 Inside conductor A the field density is represented 
 by a straight line joining the center of the conductor 
 to the apex of the density curve, represented as 100 per- 
 cent. Suppose it is desired to determine the field den- 
 
 CHART l-INDUCTANCE 
 
 The m.m.f. resulting from equal currents is the 
 same for all sizes of conductors. Thus the field den- 
 sity at points equally distant from the center of dif- 
 lerent sizes of conductors carrying equal currents is 
 equal provided these points lie beyond the surface of 
 the larger conductor. For points equally distant from 
 the center of different size conductors which lie inside 
 the conductors the density will be different. Thus if 
 the conductor diameter carrying equal current be re- 
 duced to one half, the m.m.f. at its surface will remain 
 the same, but since the flux path at the surface is now 
 only one-half as long, the flux density at the surface 
 
 rioo% 
 
 S « 
 
 PHYSICAL PICTURE ILLUSTRATING THE 
 MAGNETIC FLUX DENSITY SURROUND- 
 ING A CONDUCTOR CARRYING DIRECT 
 CURRENT (SKIN EFFECT ABSENT! WITH 
 EFFECT OF RETURN CIRCUIT IGNORED. 
 
 DOTTED AREA REPRESENTS THE MAGNETIC FLUX FIELD 
 RESULTING FROM CURRENT IN CONDUCTOR A - THE 
 PORTION OF THIS AREA CONTAINING CROSS SECTION 
 LINES REPRESENT THE FLUX DUE TO CURRENT IN CON- 
 DUCTOR A WHICH IS EFFECTIVE IN PRODUCING INDUCT- 
 ANCE IN CONDUCTOR A. THIS EFFECTIVE FLUX AREA 
 IS DIVIDED INTO THREE SECTIONS: TO THE LEFT THAT 
 EFFECTIVE WITHIN CONDUCTOR A. IN THE MIDDLE THAT 
 EFFECTIVE BETWEEN THE TWO CONDUCTORS AND TO 
 THE RIGHT THAT WHICH IS EFFECTIVE THRU CON- 
 DUCTOR B, 
 
 FORMULA 
 
 TOTAL INDUCTANCE iN MILLIHENRIES (L) PER 1000 FEET OF EACH CONDUCTOR L-0.0I6S4- 
 
 sity at a point midway between the center and surface 
 of the conductor. At this point the length of the cir- 
 cular path is one half its length at the conductor sur- 
 face. Since the current distributes uniformly through- 
 out the cross-section of the conductor, at a point mid- 
 way between the center and its surface, one-fourth of 
 the total current would be embraced by the circle. The 
 m.m.f. corresponding to this point would therefore be 
 one-fourth its value at the surface. With one-fourth 
 m.m.f. and one-half the surface reluctance the result- 
 ing density will be one-half of its surface density as 
 shown by this value falling on the straight line at this 
 distance from the center. 
 
 3.1403 LOG 
 10 R 
 
 FIG. 6 
 
 will be twice as great. In other words, the magnetic 
 field density at the surface of conductors having differ- 
 ent diameters but carrying the same currents is in- 
 versely proportional to their radii. 
 
 The area indicated by cross-sectional lines on the 
 inductance chart represents the amount of inductance 
 effective in conductor A resulting from current in this 
 conductor. It will be seen that the total area between 
 the adjacent surfaces of the conductors / to 9 below 
 the flux density line is effective. This part of the in- 
 ductance follows a logarithmic curve as illustrated on 
 the chart and is represented by the formula. 
 D-R 
 '~R' 
 
 L = 0.1403J logn 
 
 .(/) 
 
RESISTANCE—SKIN EFFECT— INDUCTANCE 
 
 Where L is in millihenries per looo feet of single 
 conductor. 
 
 The effective flux area departs from the flux den- 
 sity line at E dropping down in the form of a reverse 
 curve and terminating in zero at //. All flux to the 
 right of // cuts the whole of both conductors producing 
 the same amount of inductance in both of them in such 
 a direction as to oppose or neutralize each other. 
 
 The flux cutting conductor B from p to ii has its 
 full value of effectiveness in producing inductance in 
 conductor A. On the other hand it also produces to 
 a less extent inductance in conductor B but in a direc- 
 tion to oppose that which it produces in conductor A. 
 The difference between that produced in conductors A 
 and B is the effective flux producmg inductance in the 
 circuit and is represented by the shaded portion through 
 conductor B within the area E-p-ii-T-E. To illustrate 
 how the effective flux curved line E-T-ji was deter- 
 mined, suppose it is required to determine the effective 
 flux at the distance lo (center of conductor B). At 
 this point the flux density is ten percent, but since 
 these flux density lines are actually concentric circles, 
 having their center at the middle of conductor A this 
 flux density curve cuts conductor B in the form of an 
 arc (see lower right hand corner of inductance chart). 
 The area of the shaded portion between the two arcs is 
 a measure of the inductance in conductor B at its cen- 
 ter. The difference between this shaded area, and the 
 whole area of B, or the clear part to the right of the 
 shaded portion, is a measure of the difference in induct- 
 ance of the two conductors. In other words, for the 
 spacings shown, approximately 55 percent of ten or 
 5.5 percent is the value of the effective flux at distance 
 of 10 from conductor A. 
 
 ,r ■ , , , , ^-■^ 
 
 If in place of L = o./^oj^ logw = — ^ (/) 
 
 we take /, = o.r^o^y logm —^ (^) 
 
 we include all of the inductance area out to the vertical 
 Hne O-io. This would include the area E-O-T but not 
 the area T-io-ii. Since these two areas are equal, the 
 omission of one is balanced by including the other and 
 therefore formula (2) correctly takes into account all 
 of the effective inductance beyond the surface of con- 
 ductor A. 
 
 The inductance within conductor A is determined 
 as follows : — At a point midway between the center and 
 its surface the flux density is 50 percent as indicated by 
 the straight flux density line of the chart. However at 
 this point only one-fourth of the conductor area is en- 
 closed, so that, measured in terms of its effect if outside 
 the conductor, its effectiveness would be only one-fourth 
 of 50 or 12.5 percent. This is the reason that the so- 
 called effective flux line is curved and falls to the right 
 of the straight flux density line. The area of the trian- 
 gular section O-i-ioo is a measure of the effective in- 
 ductance within conductor A. This is a constant for 
 all sizes of solid conductors and is represented by the 
 
 constant 0.01524 of the inductance formula based upon 
 1000 feet of conductor. 
 
 The fundamental formula for the total effective 
 inductance (within and external to conductor A) ol a. 
 single solid non-magnetic conductor suspended in air >5 
 therefore : 
 
 /. = ox)i524 + 0.14037 logm -j^periooo/l. 
 or 
 
 D 
 
 (J) 
 
 L = 0.08047 + 0.7411S logw-/^ per mile (^) 
 
 It may be interesting to note here that the above 
 described graphical solution for inductance produces re- 
 sults in close agreement with these obtained by the fun- 
 damental formula for inductance. That is, lay out such 
 a chart on cross section paper to a large scale and count 
 the number of squares or area representing the internal 
 and the external inductance due to current in conductor 
 A. It will be seen that the relative values of the ex- 
 ternal and internal flux areas conform with the relative 
 values as determined by the formula. This will also 
 be true in the case of the conductors when so placed as 
 to give zero separation, as illustrated by Fig. 6. 
 
 VARIATIONS FROM THE FUNDAMENTAL INDUCTANCE 
 FORMULA 
 
 It has been proven mathematically by the Bureau 
 of Standards and others that the fundamental formula 
 (3) for determining inductance will give e.xact results 
 for solid, round, straight, parallel conductors, provided 
 skin and proximity effects are absent. Proximity ef- 
 fect is the crowding of the current to one side of a 
 conductor, due to the proximity of another current 
 carrying conductor. It is similar to skin effect in that 
 it increases the resistance and decreases the inductance. 
 Proximity effect as well as skin effect changes only the 
 inductance due to the flux inside the conductor. Proxi- 
 mity effect is more pronounced for large conductors, 
 high frequencies and close proximity. 
 
 For No. 0000 solid conductors at zero separation 
 and 60 cycles, the error in the results (as determined by 
 the fundamental inductance formula) due to skin ef- 
 fect is less than one-tenth of one percent. This error, 
 however, increases rapidly as the size of the conductor 
 increases. Proximity effect cannot be calculated but it 
 is believed to be less than two percent in the above case. 
 
 Should skin and proximity effect combined, be 
 sufficient to force all of the current out to within a verv 
 thin annulus at the surface of the conductor (a condi- 
 tion obviously never obtained at commercial frequen- 
 cies) their combined effect would be a maximum. In 
 such a case there would be no inductance within the 
 conductors and the first constant 0.01524 would dis- 
 appear from formula (3). 
 
 Skin and proximity effect are so small in the case 
 of the greater spacings of conductors required for high- 
 tension aerial transmission circuits that they may in 
 such cases be ignored. Even in the case of the close 
 
RESISTANCE—SKIN EFFECT— INDUCTANCE 
 
 O 
 
 h 
 
 o 
 
 D 
 O 
 
 z 
 o 
 o 
 
 LJ 
 
 o 
 
 z 
 
 (0 
 
 o 
 
 h 
 
 LJ 
 U 
 U. 
 
 O 
 O 
 O 
 
 cc 
 
 U 
 Q. 
 
 U 
 O 
 
 z 
 < 
 
 H 
 O 
 
 U 
 
 m 
 
 < 
 
 UJ >( 
 
 3 "^ 
 
 < 0> 
 
 I- 
 o 
 
 < 3 
 t- O 
 
 z 
 
 ^ 
 
 8 
 
 5 ^ 
 o 
 
 UJ 
 - ^ 
 
 < CD 
 
 O UJ 
 
 O 
 
 z 
 < 
 1- 
 
 O 
 
 00 Ql 
 
 O 
 H 
 O 
 
 O 
 
 z 
 o 
 o 
 
 CO CO 
 
 ; i 
 
 o 
 
 UJ 
 
 t 
 
 s 
 < 
 o 
 
 UJ 
 
 X 
 
 CI 
 
 z 
 
 X 
 
 X 
 
 o 
 
 U Q|CC 
 
 CO 
 
 
 
 Ol 
 CO 
 
 u. o cc 
 
 o o- o 
 •-So 
 
 UJ -J i< 
 
 8 o o 
 
 Z ^. 9. 
 
 -' '., 
 
 = o 
 
 < 
 
 5 o (d 
 H UJ ' 
 
 UJ 
 
 X 
 
 I- 
 
 S 
 
 o 
 
 K 
 
 < 
 
 (0 
 
 QC 
 O 
 
 I- 
 O 
 
 z> 
 
 Q 
 
 Z 
 
 o 
 o 
 
 muj 
 nui 
 
 I- 
 nuj 
 c«uj 
 
 »- 
 
 — UJ 
 CIUJ 
 
 nnn 
 
 mi 
 
 
 oa«oflo 
 
 r-t 
 
 I- 
 
 H 
 -UJ 
 
 -UJ 
 
 0& i^ 
 
 
 I- 
 
 
 ^A O— 
 
 «3 000« 
 
 c<Mp( 
 
 
 om'o 
 
 rl'o'- 
 
 < 3 CO 
 Ulz 3 -J 
 
 < oc 
 O 
 
 •ON s ^ a 
 
 S3H0NI Nl 
 
 3dAl 
 
 TVlaiLVW 
 
 
 
 
 
 *:oo 
 
 
 
 
 
 cjK •-^^ 
 
 
 ■♦■■et 
 
 ■♦•Oo*- 
 •inn 
 
 L,t-0 
 
 p)n n 
 
 00- 
 
 pinn 
 
 
 Mo- 
 no^ 
 
 
 oo—n 
 too-o- 
 
 Io>qOo 
 
 
 ^c — 
 
 
 
 ooo 
 
 ooo 
 
 ooo 
 ooo 
 ooo 
 ooo 
 
 Ooo 
 
 0«rv 
 
 
 
 
 
 
 i-O,0o 
 
 nsaflo 
 ooo 
 
 t-looo 
 
 
 
 ooot- 
 
 on** 
 
 •ato- 
 
 o«,^ 
 
 m><l-» 
 
 -nio 
 
 OoON 
 
 0«^ 
 
 ooo 
 
 ooo 
 ooo 
 ooo 
 ooo 
 ooo 
 ■•Tt- 
 
 
 nnn 
 
 
 nnn 
 
 Kko 
 oo- 
 
 ooo 
 
 nnn 
 
 
 
 
 
 oo~ 
 
 ooo 
 t-t-t 
 
 nnn 
 
 Oey 
 
 nnn 
 
 nnn nnn 
 
 Ooitj NAOOO 
 ■olO^J lots'* 
 
 nnn nnn 
 
 6oO + 
 
 nnn 
 
 on** 
 nnn 
 nf^n 
 
 nnn 
 
 ooo 
 nnn 
 
 
 r-o+ 
 c*nn 
 
 lo-s* 
 nnn 
 
 NNN 
 
 n*»- 
 ooo 
 N'X'y 
 
 
 'O'O'O 
 
 sS»-N 
 
 -N« 
 
 't-t^O 
 *-»-o 
 oo- 
 
 ooo 
 ooo 
 ooo 
 ooo 
 ooo 
 
 ry— O 
 
 n»-^ 
 
 -«0>n 
 
 ■»•+■<- 
 
 nnn 
 
 oott*- 
 
 nn-t 
 
 nnn 
 
 o^on 
 Nnn 
 nnn 
 
 -NN 
 
 nnn 
 
 N 
 o — 
 nnn 
 
 NN« 
 
 NNN 
 
 N'0-» 
 NNN 
 
 OooN 
 NNN 
 
 NNN 
 
 OOtOOD 
 
 tOON 
 **SN 
 
 vji-i-o 
 
 nior* 
 
 nn* <hf* 
 
 <*teo 
 NNn 
 
 ooo 
 
 ooo 
 ooo 
 ooo 
 Ooo 
 
 ••on 
 nnn 
 
 nM» 
 
 •T'tT 
 
 Nnr- 
 
 DOO 
 
 Wno 
 *-«*o 
 
 nnt 
 
 NO<^ 
 
 nnn 
 
 Ooon 
 nnn 
 
 ««N 
 
 V^O^* 
 
 doo 
 
 *»»-o 
 nn\^ 
 
 •^fy.n 
 nnn 
 
 
 not- 
 
 
 p»n 
 
 oT^N 
 NNn 
 
 ^ 
 ■♦•1-+ 
 
 >«o^ 
 o^— 
 
 o-oo 
 nrt 
 
 
 
 
 
 NNn 
 
 Nt--o 
 nnn 
 
 nnn 
 
 nnn 
 
 ■♦-«-■*- 
 nnn 
 
 nnn 
 nnn 
 
 Nnn 
 nnn 
 
 nnn 
 
 ON^ 
 
 o oo 
 
 nnn 
 
 *-^n 
 NNN 
 
 n+ 
 nNn 
 
 NNN 
 
 OINN 
 NNN 
 
 n<o 
 
 
 n-«-N 
 
 — n 
 f'o'o 
 
 nnn 
 
 oo— 
 
 oo 
 
 ooo 
 ooo 
 ooo 
 ooo 
 oioo 
 
 ttoKN 
 
 o».v 
 
 toon 
 nnn 
 
 nnn 
 
 ■«-^i■ 
 
 nnn 
 
 •<-r- 
 nnn 
 nnn 
 
 «-N*» 
 n^T 
 nnn 
 
 NNN 
 
 nnn 
 
 to-t, 
 Nnn 
 nnn 
 
 -«<^-N 
 oO~ 
 nnn 
 
 NOT 
 
 »-oo 
 Nnn 
 
 NNN 
 
 NNN 
 NNN 
 
 "ON- 
 
 lOlo-* 
 
 NNN 
 
 onN 
 nnn 
 
 NNN 
 
 SON 
 
 »-oo 
 
 <\N 
 
 ON>0 
 OMsCo 
 
 »-on 
 
 •»Nn 
 
 ioooo 
 
 "Olo-" 
 
 Hon 
 
 n^«o 
 
 NNN 
 
 ooo 
 
 ooo 
 ooo 
 ooo 
 ooo 
 
 V)0'0 
 
 ^•o'o 
 
 Vfct. 
 
 ^•on 
 
 NCoto 
 
 nnn 
 
 Non 
 nnn 
 
 N — * 
 nnn 
 
 Nnn 
 nnn 
 
 nnn 
 
 n*>'*- 
 
 NNN 
 
 sSO^ 
 «ONN 
 NNN 
 
 NNN 
 
 ^ton 
 
 oo 
 
 NNN 
 
 
 ^»on 
 
 KOoO) 
 
 n-«o 
 
 v«»-n 
 
 Ntttto 
 OOO 
 
 OOO 
 OOO 
 OOO 
 OpO 
 
 o>oo 
 "ot-t 
 
 ->N« 
 %»NN 
 
 r-JXN 
 nnn 
 
 i,o>i 
 
 NftJte 
 
 nnn 
 
 CdNOo 
 
 N»NN 
 
 nnn 
 
 nnn 
 
 oi,- 
 
 »-o- 
 Nnn 
 
 sftn— 
 
 "O-IN 
 NNN 
 
 jOX> 
 
 .mN 
 
 NNN 
 
 ^nfr- 
 «.oo 
 
 NN 
 
 NNf~ 
 4)0-0- 
 
 NN* 
 
 N-^N 
 
 nNN 
 nn-^ 
 
 ot-o 
 >-«-o 
 oo- 
 
 OOo 
 
 ooo 
 ooo 
 poo 
 "oow, 
 
 oW 
 nh 
 
 rt-1- 
 
 NO." 
 
 »-f-N 
 ^Nl» 
 
 Nl-^ 
 
 e-o-o 
 
 no N 
 N»N^ 
 
 ♦tt 
 
 t^N 
 10-*^- 
 
 ?5^ 
 
 "ol-- 
 
 +•♦■'»■ 
 
 oo- 
 
 »->«n 
 
 «0fr-O 
 
 nn't 
 
 noN 
 
 Ooft-»- 
 
 nnn 
 
 NOOOO 
 
 nnn 
 
 *ano 
 
 *«N0o 
 
 nnn 
 
 rN«- 
 
 lO-A-S 
 
 nnn 
 
 -00 lo 
 
 nnn 
 
 nooo 
 f4nn 
 nnn 
 
 10N«^ 
 
 nnn 
 
 -ooio 
 
 000,0- 
 
 nnN 
 
 ooio 
 nn^ 
 NNN 
 
 .l<sN 
 NctN 
 
 noN 
 
 0[>«-° 
 
 N»^ 
 
 NNOO 
 
 Nf- 
 t->ov« 
 
 "0<-<«. 
 
 o— -, 
 
 ON*- 
 
 ONN 
 n«NO 
 
 sn 
 >.-«n 
 
 6 o- 
 »ogo- 
 
 
 rtt-t 
 
 
 -^ — d 
 
 WO-.-* 
 
 
 ni-^ 
 
 NT- 
 
 ooo-o 
 nn^ 
 
 >»no 
 
 Noo*- 
 
 nnn 
 
 *s^o 
 o-o- 
 n<-^ 
 
 Novs 
 
 -ONK 
 
 nnn 
 
 NO^» 
 
 nnn 
 
 or-^ 
 NNn 
 nnn 
 
 NO--B 
 
 oo- 
 
 nnn 
 
 OolON 
 NOOO- 
 NNN 
 
 N*>-6 
 Ujlo-S 
 
 NNN 
 
 "ONO- 
 n-^-t- 
 NNN 
 
 "♦-— 00 
 
 Nnn 
 NNN 
 
 ON"*- 
 — — N 
 NNN 
 
 nON 
 
 o-oo 
 
 NN 
 
 Oo"oN 
 *«N0o 
 
 ^«no 
 ^vn"*- 
 
 ON J--* 
 NN flOfr* 
 OO " 
 
 oto^* 
 
 nNN 
 Nn»- 
 nnfi 
 
 ^cto 
 •-0- 
 
 Ooo-Q- 
 
 fc>y)N 
 
 Nfc*. 
 
 NN» 
 
 t-t-T 
 
 n— N 
 
 N»NN 
 
 
 NO^^ 
 --N 
 
 1--0O 
 
 n-T 
 
 i^no- 
 r<nn 
 TTt 
 
 1-tt 
 
 N>1 
 
 o-« 
 
 no>- 
 
 NOlto 
 
 0o<5« 
 
 *-<0-N 
 
 
 WON 
 
 v^N^ 
 
 ■iNO 
 
 nt-*o 
 
 ».v»n 
 NnT 
 
 NNO 
 
 OO — 
 
 o-^n 
 
 toO-O 
 
 nnT 
 
 noN 
 
 0OO-9* 
 
 nnn 
 
 ^-Oo 
 nOoOo 
 nnn 
 
 -sno 
 
 ^N«> 
 
 nnn 
 
 nnn 
 
 \*n o 
 noo*- 
 nnn 
 
 nn^ 
 
 lon^ 
 
 -»NN 
 
 nnn 
 
 nnn 
 
 n-N 
 einn 
 nnn 
 
 «-o- 
 Nnn 
 
 now 
 
 MOOD 
 
 NNN 
 
 sAno 
 "o^N 
 NNN 
 
 V<oio 
 NNN 
 
 —flow 
 nn't- 
 SNN 
 
 NN 
 NNN 
 
 »-\en 
 
 tOtt-0 
 -N 
 
 "ONd- 
 
 o— — 
 
 ■tfcto 
 «nte 
 
 >4K0 
 
 N-n 
 
 rn^ 
 
 oN-« 
 -•no 
 
 ftN<< 
 
 oaaNvuJjs 
 
 Nnn 
 nnn 
 
 •ON*- 
 
 i-'o'o 
 nnn 
 
 ^-ssn 
 o-o- 
 N^n 
 
 -OoW) 
 090BO- 
 
 NNN 
 
 -«n — 
 io-»N 
 
 NNN 
 
 OOW 
 
 nnt- 
 NNN 
 
 NN 
 NNN 
 
 noN 
 o— 
 NNN 
 
 ••N«n 
 oo»-o 
 
 -N 
 
 N»-^ 
 
 NN* 
 
 NV- 
 ■*■''>■* 
 
 "ON*" 
 O— — 
 
 0«»- 
 OKK 
 >«NO 
 
 ■vKn 
 — *n 
 
 N^^ 
 
 8?' 
 
 o o*^ 
 
 t-i-n 
 
 NV>N 
 W^N 
 
 oto-*- 
 
 t-tt 
 
 noN 
 
 no 
 
 no N 
 Ptfon 
 
 -N 
 
 NO--** 
 
 nn"t- 
 
 i--oo 
 o — ■ 
 ■♦-tt 
 
 Nnn 
 ■»■■»•■«■ 
 
 t>N»^ 
 O-OO 
 
 Nt-- 
 o»o--o 
 nnT 
 
 NJt-O 
 fr-O- 
 
 nT"*- 
 
 Noss 
 
 sSNN 
 
 nnn 
 
 No'-a 
 
 OoO-*- 
 
 nnn 
 
 ONf 
 
 NNn 
 nnn 
 
 N»-.» 
 
 oo- 
 
 nin^ 
 
 OoVoN 
 NOoo- 
 NNN 
 
 N»--« 
 loto^ 
 NNN 
 
 ioN«- 
 n-^t 
 NNN 
 
 ■<t-~0o 
 Nnn 
 NNN 
 
 NNN 
 
 noN 
 «-oo 
 
 NN 
 
 N«NCo 
 
 *»no 
 
 NON <--* ><>««- 
 •ANN 0B*-»* O — 
 '-00 
 
 Ov-Oo 
 
 *n\»n 
 ok>>« 
 
 nNN 
 
 «-o;~ 
 
 MNl, 
 
 «-»-o 
 
 ODft-«- 
 
 N»>»- 
 
 N0»>9 
 nnO» 
 
 n-xK 
 
 -©NN 
 
 -»--^0o 
 
 
 _n- 
 -Nn 
 
 O^N 
 
 ■«-rr 
 
 "on*^ 
 
 -»NN 
 
 nnn 
 
 O00+- 
 
 ■♦"♦-lo 
 nnn 
 
 n— '^ 
 Nnn 
 
 nnn 
 
 to-4N 
 Nnn 
 
 nOoOO 
 NNN 
 
 -•no 
 
 ■OstN 
 NNN 
 
 ioN»- 
 
 NNN 
 
 ~Cd>o 
 nn^ 
 NNN 
 
 ^»--<09 
 io(N 
 NNN 
 
 «-son 
 
 0o«-O 
 
 N-V- 
 
 t*5 
 Nn* 
 
 v»SO 
 
 N-n 
 v,<^n 
 
 n+io 
 
 »-*N 
 •*oto 
 NN- 
 
 anos 
 
 a3ddOO 
 
 o«»o 
 nNT 
 
 t,«<n 
 
 -NO 
 
 oL,«. 
 ---N 
 
 TJow 
 
 O— N 
 
 Nnij 
 t-o-- 
 ni-t 
 
 •>»-o 
 nnT 
 
 \40- 
 
 eoteo- 
 nnn 
 
 F7 
 
 o— »< 
 
 ■t-tt 
 
 -V)0o 
 
 NNto 
 
 qz^o 
 
 n-rt 
 
 TOT 
 
 W-aN 
 nnn 
 
 ■♦-ON 
 OoO-O- 
 
 nnn 
 
 N»- 
 
 nnn 
 
 L,no 
 Nfct- 
 nnn 
 
 n nn 
 
 N^oN 
 sSNOO 
 
 nnn 
 
 no— 
 nr^o 
 nnn 
 
 o-t-jn 
 
 *-7>«N 
 
 nnn 
 
 n»-o 
 NNt- 
 nnn 
 
 Nti:r 
 
 nnn 
 
 0-.N 
 nnn 
 
 N»- 
 
 «-«-o 
 NNn 
 
 _N'<1 
 ^NOD 
 NNN 
 
 Won 
 NNN 
 
 >ooN 
 Nni- 
 Nnn 
 
 oo- 
 nnn 
 
 toOgO 
 ^— N 
 
 Otooo 
 
 nnCo 
 
 Nnk) 
 
 10«BN 
 
 OV,00 
 
 ■«-r"o 
 
 lo-n 
 Nn 
 
 Nt**- 
 ooo 
 
 Ooo 
 
 OON 
 
 OOf 
 
 Wo"* 
 oon 
 v."on 
 
 •ooi- t)-- 
 
 r-no 
 
 or-^ 
 
 V»N»~ 
 BYT 
 
 ♦-Nn 
 Nnr 
 
 rrr 
 
 NON 
 
 Nnn 
 
 UN«- 
 -NN 
 
 I-N 
 
 ITT* 
 
 N*-5 
 «-o — 
 nTT 
 
 nnr 
 
 fr-s»n 
 
 N0o«- 
 
 nnn 
 
 -aNOo 
 nnn 
 
 ^QO N 
 
 nnn 
 
 ssn— 
 -Nn 
 nnn 
 
 Nft-w 
 
 O-tt-O 
 
 NNn 
 
 t:-s 
 
 NNN 
 
 NNN 
 
 io-*~ 
 Nnn 
 
 NNN 
 
 r^T-~, 
 
 NNN 
 
 W)NO 
 
 a-o- 
 
 NN 
 
 NO-tt 
 00»-O- 
 
 ^NN 
 
 TTio 
 
 »)>0N 
 
 •-0-; 
 
 O^- 
 
 Ooo 
 
 owo 
 
 «XN- 
 
 lyi 
 
 nV>o 
 
 -«N0o 
 
 TTT 
 
 N<»-V» 
 
 s9N«N 
 
 TTT TTt 
 
 NfO 
 
 TTt 
 
 O^L, 
 
 TTt 
 
 >^pNV)-to 
 
 T>^io 
 TTT 
 
 v-jno 
 nT4 
 TTT 
 
 t-*nT 
 TTT 
 
 o:t;N 
 
 TTT 
 
 'To 
 nn 
 
 TTt 
 
 C^no' 
 NnT 
 TTT 
 
 ooion 
 
 — Nf» 
 TTT 
 
 (NNT 
 OoO-o 
 
 nnT 
 
 TNn 
 nnn 
 
 NT- 
 nT"o 
 nnn 
 
 nON 
 -nN 
 nnn 
 
 lONlN 
 a-oo 
 Nnn 
 
 -B'O 
 NNto 
 NNN 
 
 -8 No 
 
 Tio-« 
 NNN 
 
 OoioN 
 NnT 
 
 NNN 
 
 -a^oN 
 — Nn 
 
 NNN 
 
 Ton 
 o— 
 
 NNN 
 
 -«NO 
 
 .<>->o 
 
 ♦--an 
 Nn 
 
 N0»«- 
 
 ooo 
 
 Ooo 
 onT 
 N^i-i 
 n4n 
 noO» 
 
 
 o r»-Vj -f«»i» 
 Tnn nNN 
 
 *»*-o 
 
 '»o^ 
 
 »7«I 
 
 N««- 
 
 TTt 
 
 nnS? 
 nno» 
 TTT 
 
 -aNN 
 TTT 
 
 P7 
 
 V>-*N 
 
 TT^ 
 
 TTJN 
 nTV) 
 TTT 
 
 JT 
 
 --N 
 TTT 
 
 ONS 
 
 OO 
 
 TT'W 
 
 i^no- 
 NOoto 
 nnn 
 
 oown 
 
 WsjN 
 
 nnn 
 
 1,-N 
 
 nTT 
 (nnn 
 
 T- 
 -;t<n 
 nnn 
 
 l^o■o 
 M^n 
 
 NT- 
 
 -»N" 
 NNN 
 
 0-^T 
 ■fl-vr-s 
 
 NNN 
 
 »Hon 
 
 NN« 
 
 T- 
 Nnn 
 
 NNN 
 
 NO)- 
 0-N 
 NNN 
 
 N»-N 
 
 ONT 
 
 TTV) 
 
 o-o 
 
 0- 
 
 ooo 
 
 NBT 
 
 nv>^ 
 
 viN- 
 
 a30ilOdU\3til331S 
 
 lAinNiiAimv 
 
 « 
 
 oj 
 
 «i 
 
 T 
 
 «y 
 
 
 
 "0 
 
 T 
 
 -- 
 
 « 
 
 
 
 »« 
 
 <^ 
 
 n 
 
 ^ 
 
 o 
 
 
 n 
 
 
 
 
 N» 
 
 Oo 
 
 n 
 
 s 
 
 
 
 ^ 
 
 T 
 
 N 
 
 n 
 
 s 
 
 
 
 
 
 N 
 
 N 
 
 n 
 
 -s 
 
 
 
 ^ 
 
 o 
 
 •^ 
 
 n 
 
 •< 
 
 
 
 o 
 va 
 
 o( 
 
 
 
 
 •o 
 
 N 
 
 In 
 
 N 
 
 
 O 
 
 o 
 
 1^ 
 
 ■o 
 
 N 
 O 
 
 "■ 
 
 
 •o 
 
 m 
 
 T 
 
 N 
 
 --^ 
 
 o 
 
 o 
 
 
 
 T 
 
 N 
 
 
 
 ^ 
 
 » 
 
 n 
 
 o 
 
 
 
 
 
 -s 
 
 0^ 
 
 ^ 
 
 
 
 
 lo 
 
 T 
 
 N 
 
 
 < 
 
 <J 
 
 
 
 
 N 
 
 — 
 
 
 
 
 ■O 
 
 oo 
 
 
 
 ■^ 
 
 
 
 
 
 •s 
 
 •v 
 
 
 
 ■N. 
 
 
 
 >o 
 
 n 
 
 o 
 
 
 
 o 
 
 
 
 < 
 
 m 
 
 o 2 
 
 a ;2 
 
 <! -.3 
 
 o a 
 
 3 S 
 
 » s 
 
 a a 
 
 5 a 
 
 :i II 
 
 Eg « 
 
 H » a 
 
 
 2 a 
 
 3 .a 
 
 o a o 
 
 •°-g ft 
 
 a V 
 
 o; o « 
 
 «S o 
 
 ga - 
 
 •G-g II 
 
 O *^ a) 
 so 
 
 RM 3 
 
 So.2i 
 
 " s s < 
 .=1 &-S 
 
 03-^ ».. n 
 
 IS *^«w 
 
 O -CO 
 
 s * ft*; 
 
 O o 
 
 5 s 
 
RESISTANCE— SKIN EFFECT— INDUCTANCE 
 
 spacings required for three conductor cables these com- 
 bined effects are usually less than four percent. 
 
 EFFECT OF STRANDING ON INDUCTANCE 
 
 The fundamental formula (3) for determining in- 
 ductance is based upon a solid conductor, R being taken 
 as the radius of the conductor. In stranded cables the 
 effective value for R lies between the actual radius and 
 that of a solid rod having an equivalent cross-section 10 
 that of the cable. The effective value for R varies with 
 the stranding of the cable employed. 
 
 Formulas for use in determining the inductance of 
 stranded cables when used for high-tension aerial trans- 
 mission have been calculated by Mr. H. B. Dwight as 
 follows : — 
 
 For a 7-wire cable, L = 0.741 logjo -^-^ — (5) 
 
 _ . , . , 2.640 D 
 
 For a 19-wire cable, L — 0.741 logw -3 {6) 
 
 T. ■ r , 2-605 D 
 
 For a 37-wire cable, L = 0.J41 logw — ^ (7) 
 
 2590 D 
 For a 6i-wire cable, L = o.jtr logya — j {S) 
 
 where L is in millihenries per mile of a single conductor, 
 D is the spacing between centers of cables, and d is the 
 outside diameter of the cables measured in same units 
 as D. 
 
 SPIRALING EFFECT UPON INDUCTANCE 
 
 Spiraling of the strands of a cable and spiraling of 
 the conductors of a three-conductor cable tend to in- 
 crease the inductance. It is difficult to calculate the 
 viffect of spiraling for the various cases, but it may be 
 considered negligible for high-tension aerial transmis- 
 sion circuits using non-magnetic conductors. For 
 three-conductor cables the effect of spiraling is probably 
 in the neighborhood of two percent. 
 
 Values for inductance per thousand feet of single 
 conductor are given in Table III, for commercial sizes 
 of copper and steel reinforced aluminum conductors. 
 The formula by which the values were derived are: — 
 
 L = oj3r524 -f- 0.1403 logv) -^ 0) 
 
 where L = Millihenries per 1000 feet of single conductor of a 
 single phase, or of a symetrical three-phase circuit. 
 
 D = Distance between centers of conductors. 
 
 R = Radius (to be measured in same units as D) of 
 solid conductor. In the case of stranded con- 
 ductors, A' was taken as the radius of a solid rod 
 of equivalent cross-section to that of the stranded 
 conductors. 
 
 Table III has been carried out to three figures only. 
 This would seem sufificiently accurate for working 
 values when it is considered that there are numerous 
 sources of variation from the calculated values. In 
 the first place formulas are based upon a uniform dis- 
 tribution of current throughout the cross section of the 
 •onductors, whereas the current is seldom uniform and 
 n the larger conductors, especially at 60 cycles, may be 
 to a large extent crowded to the outer strands as a 
 result of skin effect. This condition is further modi- 
 
 fied when the conductors are placed close together, by 
 the pro.ximity effect. Stranded conductors made up of 
 various stranding combinations result in variation of 
 inductance of several percent. In practice the length 
 and spacing of conductors will vary more or less from 
 those assumed when determining the calculated values. 
 
 The values for inductance of stranded conductors 
 in Table III, as stated above, were derived by taking R 
 as the radius of a solid rod having an equivalent cross- 
 section area to that of the stranded conductors. Thus 
 for 1 000 000 circ. mil cable the outside diameter is 
 1. 152 in. and that of an equivalent solid iron is i.o in. 
 R was therefore in this case taken as 0.5 in. The ef- 
 fective radius is really slightly greater than that of the 
 solid rod and less than that of the cable, varying with 
 the stranding employed. The actual inductance of 
 cables will therefore be slightly less (usually two or 
 three percent) than those indicated in the table for 
 solid rods. The table values are therefore conservative. 
 
 The steel core of steel reinforced aluminum cables 
 carries so little current on account of its relatively 
 greater resistance that for practical purposes it has been 
 customary to ignore its presence and to consider such 
 conductors as solid rods of same area as that of the 
 aluminum strands. In the absence of accurate data 
 this practice was followed in determining the values for 
 inductance of such cables in Table III. 
 
 The minimum value for inductance occurs when 
 
 the conductors have zero separation -„■ = z, (Fig. 6). 
 In this case the inductance in millihenries is independent 
 of the size of the conductor. As given by formula (3) 
 it is L = 0.05124 -\- O.J403 log^a 2 = 0.0575 millihen- 
 ries per 1000 feet of each conductor. Obviously in- 
 sulation requirements will not permit of such a low 
 value for inductance although it will be closely ap- 
 proached in low voltage cables. 
 
 Any given percentage difference in distance be- 
 tween centers of conductors represents a definite and 
 constant value in inductance regardless of their siz*?. 
 These values are given in column B at the bottom of the 
 table for various percentages increase in spacings. 
 Thus if the distance between conductor centers is in- 
 creased 50 percent the corresponding increase in in- 
 ductance is 0.025 as indicated in column B, under the A 
 values of 1.50. Likewise doubling the distance in- 
 creases the inductance by an amount of 0.042. For in- 
 stance the table value for inductance of No. o solid 
 copper conductor is for one-half inch spacing 0.084, and 
 for one inch spacing 0.126 (an increa.se of 0.042.) For 
 four foot spacing the table value is 0.362, and for eight 
 foot spacing 0.404, also a difference of 0.042. 
 
 References: — An article by Prof. Giarles F. Scott, "In- 
 ductance in Transmission Circuits" in The Electric Journ.\l 
 for Feb. 1906 very clearly covers the field of self and mutual 
 inductance external to the conductors. 
 
 H. B. Dwight, "Transmission Line Formulas." 
 V. Karapetoff, "The Magnetic Circuit" p. 189. 
 
CHAPTER II 
 REACTANCE-CAPACITANCE-CHARGING CURRENT 
 
 REACTANCE 
 
 A CONDUCTOR carrying an electric current is 
 surrounded by a magnetic flux, whose value is 
 proportional to the current. If the current 
 varies, this flux also changes, thereby inducing an elec- 
 tromotive force in a direction which opposes the change. 
 This counter e.m.f. is proportional to the rate of change 
 and hence in alternating current is proportional to the 
 frequency. It can be expressed in ohms per mile of 
 each conductor of a single-phase or of a symmetrical 
 three-phase circuit as follows: — 
 
 Ohms Reactance = ^ tt / L (9) 
 
 When / = Frequency in cycles per second 
 
 L = Henries per mile of single conductor. 
 
 The value for 2 -n- f are as follows : — 
 
 Frequency 2 v f 
 
 I 6.28 
 
 15 9425 
 
 25 157.1 
 
 40 251.3 
 
 60 3770 
 
 133 835-7 
 
 Tables IV and V indicate the reactance in ohms 
 per mile, of a single conductor at 25 and 60 cycles re- 
 spectively for various spacings of conductors. The 
 foot notes to these tables cover the pertinent points re- 
 lating to them. The resistance per 1000 feet, and per 
 mile of single conductor at 25 degrees C. {"jj degrees 
 F) is given in parallel columns as a convenience for 
 comparison of the resistance and reactance values. The 
 resistance corresponding to other temperatures when 
 desired may be taken from Tables I and II. 
 
 Tables VI and VII indicate the relative importance 
 of reactance and resistance. In some cases of short 
 lines and large single conductors, the reactance and not 
 the resistance may determine the size and number of 
 cables necessary. In other words, it may be necessary 
 to keep the resistance abnormally low so that the react- 
 ance will not be so high as to result in an abnormal volt- 
 age drop in the circuit. In such cases the values in 
 Tables VI and VII may be used for determining the 
 permissible resistance in order not to exceed the desired 
 reactance. 
 
 Example: — It is desired to use 1000 000 circ. mil single 
 conductor cables at 60 cycles, spaced two feet apart; from 
 Table VII it is seen that the reactance drop under these condi- 
 tions is 8.52 times the ohmic drop at 25 degrees C. If an ohmic 
 drop of five percent at 25 degrees C is suggested the corre- 
 sponding reactive drop would be 5 X 8.52 or 42.6 percent which 
 would be excessive. If it is desired to limit the reactive drop 
 to ID percent in this case, the ohmic drop at 25 degrees C must 
 be 10 H- 8.52 or 1. 18 percent. 
 
 Probably a more important use for Tables VI and 
 VII is for determining the reactance of a conductor di- 
 rectly from its resistance. To do this it is only neces- 
 sary to multiply its resistance (at 25 degrees C) by the 
 
 ratio value in table VI or VII corresponding to the con- 
 ductor and spacing desired. 
 
 UNSYMMETRICAL SPACING 
 
 The inductance and capacitance per conductor of 
 a three-phase circuit for symmetrical spacing of con- 
 ductors is the same as the inductance and capacitance 
 per conductor of a single-phase circuit for the same size 
 conductor and the same spacing. For irregular spac- 
 ing of conductors, the inductance and capacitance will 
 be different. When the three conductors are placed in 
 the same plane (flat spacing), the inductance of each of 
 the outside conductors is greater than that of the middle 
 conductor. By properly transposing the conductors, 
 the inductance and capacitance may be equalized in all 
 three conductors. However, the effect of flat spacing 
 
 ^— A,--Y- e -» 
 
 9 • 1 
 
 I 
 
 --Ci- 
 FIG. 7 
 
 J 
 
 
 c 
 
 
 \, 
 
 
 
 FIG. 8 
 
 
 
 1 
 
 ^-K 
 
 1 
 
 e^- 
 
 i 
 
 t 
 V- 
 
 -■>•> 
 
 — c -■ 
 
 .. _.^. 
 
 1 
 
 Conductor Spacings. 
 
 For three-phase irregula r flat or triangular spacing (Figs. 
 7 and 8) use D = ^' A B C. 
 
 For three-phase regular flat spacing Fig. 9 use D ^ 1.26 A. 
 
 For two-phase line the spacing is the average distance be- 
 tween centers of conductors of the same phase. It makes no 
 difference whether the plane of the conductors with flat spacing 
 is horizontal, vertical or inclined. 
 
 is equivalent to that of a symmetrical arrangement of 
 greater spacing. 
 
 Various arrangements of conductors are indicated 
 in Figs. 7, 8 and 9. Many three-phase high tension 
 circuits have the three conductors regularly spaced in 
 a common plane (regular flat spacing) Fig. 9. Beneath 
 these figures are placed statements indicating the de- 
 termination of "effective spacings" for any arrange- 
 ment of conductors. 
 
 Since the so called "effective spacing" correspond- 
 ing to unsymmetrical arrangements of conductors is 
 usually a fractional number, the line constants for such 
 effective spacing can usually not be taken directly from 
 
REACTANCE— CAPACITANCE— CHARGING CURRENT 
 
 II 
 
 the tables but may be obtained by the use of the values 
 in columns A and B at the foot of these tables. 
 
 Example : — It is desired to determine the 60 cycle reactance 
 per mile of a single conductor for flat spacing of 11 ft. between 
 adjacent 0000 solid copper conductors. The effective spacing 
 is 1.26 X II or 13.8 feet. The reactance (Table V) for this 
 conductor at 13 feet symmetrical spacing is 0.820 ohm. The 
 value for A, (bottom of Table V) = 13.8 -^ 13 = 1.06. The 
 value of B corresponding to the value for A of 1.06 is approxi- 
 mately 0.006 which, added to 0.820 gives a reactance of 0.826 
 ohm for the effective spacing of 13.8 feet. The values of re- 
 actance for all effective spacings not included in the Table 
 may be determined in a similar manner. 
 
 With an unsymmetrical arrangement of conductors 
 there must be a sufficient number of transpositions of 
 conductors to obtain balanced electrical conditions along 
 the circuit. 
 
 CAPACITANCE 
 
 When mechanical force is exerted against a liquid 
 or a solid mass, a displacement takes place proportional 
 to the force exerted and inversely proportional to the 
 resistance offered by the liquid or solid mass subjected 
 to the force. If the mass consists of some elastic ma- 
 terial, such as rubber, the displacement will be greater 
 than if it consists of a more solid material, such as 
 metal. 
 
 In a similar manner when an e.m.f. is applied to a 
 condenser, a certain quantity of electricity will flow into 
 it until it is charged to the same pressure as that of ihe 
 applied circuit. A condenser consists of plates of con- 
 ducting material separated by insulating material known 
 as the dielectric. All electric circuits consist of conduc- 
 tors separated by a dielectric (usually air) and there- 
 fore act to a greater or less extent as condensers. The 
 ability of a condenser or any electric circuit to receive 
 the charge is a measure of its "capacity" more properly 
 known as its "capacitance". Just as the rubber mass 
 referred to above will, for a given force, permit of 
 greater displacement so will circuits of greater capaci- 
 tance permit more current to flow into them for a given 
 e.m.f. impressed. 
 
 The process of charging a dielectric consists of set- 
 ting up an electric strain in it similar to the mechanical 
 strain in a liquid or mass referred to above. If an al- 
 ternating voltage is impressed upon the terminals of a 
 circuit containing capacitance, the charging current will 
 vary directly with the impressed e.m.f. There is cur- 
 rent to the condenser during rising and from the con- 
 denser during decreasing e.m.f. Thus the condenser is 
 charged and then discharged in the opposite direction 
 during the next alternation, making two complete 
 charges and discharges for each cycle of impressed 
 e.m.f. (Fig. 10). As long as the e.m.f. at the terminals 
 is changing, the condenser will continue to receive or 
 give out current. The current flowing to and from the 
 condenser, assuming negligible resistance, leads the im- 
 pressed e.m.f. by 90 electrical degrees. 
 
 DEFINIITION 
 
 The capacitance of a circuit or condenser is said to 
 be one farad when a rate of change in pressure of one 
 volt per second at the terminals produces a current of 
 
 one ampere. Stated another way, its capacitance in 
 farads is numerically equal to the quantity of electricity 
 in coulombs which it will hold under a pressure of one 
 volt. The farad being an inconveniently large unit, one 
 millionth part of it, the microfarad, is the usual prac- 
 tical unit. 
 
 CAPACITANCE TOnUULA 
 
 An exact formula for the capacitance between 
 parallel conductors must take into account the nonuni- 
 formity of the distribution of charge around the con- 
 ductors. Such a formula* is formed by considering the 
 charges as concentrated at the inverse points of the 
 conductors; thus, — 
 0.00S46/ 
 
 C = 
 
 COih'^-J 
 
 (10) 
 
 Where C equals the microfarads per 1000 feet of 
 conductor between two parallel bare conductors in air, 
 D, the distance between centers of the conductors and 
 d, the diameter and R the radius of the conductors, 
 measured in the same units as D. 
 
 Since Cosk-^ X = loge {X -f- 1 ' A'--/) (//) 
 
 0.008462 
 
 C = '- 
 
 (") 
 
 If— CHAROe- 
 FIG. 10 — CH.^HGINC CURBENT 
 
 -CNSOHARQEh' 
 
 Reducing to common logarithms and capitancc to 
 neutral, — 
 
 0.00735* 
 
 
 (/i) 
 
 Microfarads 'per 1000 feet of single conductor to neutral. 
 
 or 
 
 C = 
 
 o.o^SSiC) 
 
 '■"(4+v(^)"-') 
 
 U4) 
 
 Microfarads per mile of single conductor to neutral. 
 
 When D is greater than 10 d, which is always the 
 case in high-tension transmission lines employing bare 
 conductors, the following simplified formulas may be 
 used with negligible error. — 
 
 0fiO7354 
 
 D 
 
 Ingvi -^ 
 
 as) 
 
 ♦See article by Pender & Osborne in Electrical World of 
 Sept. 22, 1910, Vol. 56. 
 
REACTANCE— CAPACITANCE— CHARGING CURRENT 
 
 bJ 
 O 
 
 Z 
 < 
 I- 
 
 o 
 
 < 
 
 LI 
 
 cr 
 u 
 
 o 
 
 >- 
 
 o 
 
 CM 
 
 o 
 
 z 
 < 
 
 IJ u 
 
 52s 
 
 u 
 
 CD 
 
 < 
 
 e 
 
 og 
 
 UJ O 
 
 OS 
 
 coZoc O 
 
 UJ 
 
 <oo to 
 < K w 
 
 10 Lj 
 C4UJ 
 
 4. 
 
 nui 
 
 C4UJ 
 
 
 ■OUl 
 
 t- 
 
 
 
 
 ffiUJ 
 
 UJ ^ 
 0-S 
 
 IE O K 
 
 5 5 e» 
 Z3 -• 
 
 5-05 
 O 
 
 ON 8 T a 
 
 83H9NI Nr 
 U3X3NVia 
 
 BdAJ. 
 
 TVniBXVM 
 
 nil 
 
 
 c 
 
 
 ♦■00 
 
 r«<rt 
 
 
 
 
 t-SOO 
 
 •to 
 
 >O0DO 
 
 ~1^ 
 
 
 .p)^ 
 •-«•.•- 
 
 ^»sO 
 
 
 
 ^0- 
 
 
 t<'»->i 
 
 QOO 
 
 000 
 
 c,too 
 ooo 
 
 000 
 
 000 
 000 
 
 000 
 
 Ooo 
 000 
 000 
 000 
 
 OMN 
 
 
 
 
 S^*. 
 
 
 000 
 
 
 
 
 
 
 
 
 
 
 
 
 000 
 
 
 
 
 
 000 
 
 000 
 
 -OQ — 
 000 
 
 000 
 
 000 
 
 000 
 
 000 
 
 000 
 000 
 000 
 
 
 
 
 
 
 
 4^^ 
 00- 
 
 ^oc* 
 
 ff-OO 
 
 
 
 
 OsObOb 
 
 
 0-(^ 
 
 
 
 
 
 000^ 
 fntt 
 
 
 
 
 
 *-oo 
 
 ftjO<^ 
 000 
 
 CMnoo 
 oomOD 
 
 000 
 
 (So- 
 
 0-- 
 
 000 
 
 
 rtfjr; 
 
 
 00 
 
 Pint, 
 
 o««^o 
 
 
 
 tooooo 
 
 
 
 
 
 
 
 ^r- 
 «-%»«• 
 
 4^N 
 
 
 
 
 
 
 000 
 
 000 
 
 ^"2 
 
 000 000 
 000 000 
 ooo 
 000 
 
 
 T + t 
 
 r^»'0 
 
 N + ^ 
 
 
 
 
 Obo^ 
 o*»-^ 
 
 o 00 
 
 0--N 
 
 000 
 
 
 Coo 
 
 
 
 
 ^o<< 
 
 000 
 
 00 
 
 NNOo 
 000 
 
 »-toto 
 
 fftro 
 000 
 000 
 000 
 
 5M 
 
 
 
 
 
 
 
 
 o 00 
 
 
 
 WfHN 
 
 t-«o 
 NNfc 
 
 
 
 
 
 
 000 
 
 OotoOo 
 
 
 *4S 
 
 
 ft**-*- 
 000 
 
 000 
 
 0«D 
 
 OnN 
 (SO-O 
 0^ 
 
 O»ori 
 NteO 
 
 000 
 
 ooo 
 000 
 
 000 
 
 000 
 
 •00 »o 
 
 - V, 
 
 55^ 
 
 
 
 +■♦■♦■ 
 
 
 nnp) 
 
 
 on* 
 
 
 
 000 
 
 %^-«-o 
 
 
 *>oo 
 
 flodDA» 
 
 c4 
 
 
 
 
 
 
 Tf4 
 
 
 
 or»» 
 000 
 
 000 
 
 
 000 
 
 OtftJ 
 
 000 
 000 
 000 
 ,.o*>o 
 'OTT 
 
 5Sf 
 
 
 4.HW 
 
 roP)«n 
 
 
 
 
 or"* 
 
 
 
 00 
 
 r7r7n 
 
 TO- 
 
 TO 
 
 
 htof 
 
 ot*- 
 
 o"o- 
 
 OOo 
 
 
 
 
 Toor) 
 
 ■M-4 
 
 ODte 
 
 NMJo 
 000 
 
 
 ~NT 
 
 000 
 
 §00 
 00 
 000 
 
 
 
 
 (own 
 
 
 
 (y«-T 
 Iron 
 
 
 to<-o 
 
 -co 
 
 0^»-■ 
 
 00- 
 
 DO" 
 
 0-00 
 ■SIT) 
 
 
 »T0 
 
 00 T*- 
 
 not- 
 
 
 Nt-,n 
 
 
 
 "«)T 
 *T4 
 
 
 000 
 
 COT 
 
 *o>o 
 
 00 
 
 N0« 
 NioT 
 «r)T 
 
 000 
 
 o«J- 
 
 sWvO 
 
 KoH 
 •OTT 
 
 *-«-o 
 
 "1 + 
 
 
 fco-o- 
 
 nno 
 
 noT 
 
 tstofc 
 
 •*^^ 
 
 ♦•To 
 ^•jS 
 
 
 0^« 
 
 o-»r/ 
 TTl 
 
 ,•--1- 
 i^TT 
 OfOP] 
 
 no**) 
 
 oipjn 
 rtnn 
 
 
 o-»N 
 00- 
 
 •«W0o 
 
 00 »* 
 <•(«<•( 
 
 4 
 
 ««c( 
 
 10>9M 
 
 ON»r( 
 <T^P)T 
 
 o-« 
 
 L,0<« 
 0.0- 
 
 
 TO-» 
 
 0>«r< 
 
 
 To* 
 
 •«r-09 
 OOo 
 
 
 Oi*>>8 
 
 OT*> 
 
 35 
 
 Ot-M 
 
 O-- 
 
 (Vr-p) 
 00-- 
 
 »-oo 
 
 »^o 
 
 
 nBNOo 
 TOO 
 
 0(>'(-O 
 
 
 t - 
 
 DPjP) 
 
 
 (yfon 
 
 onf^ 
 
 00- 
 
 
 
 
 ciPJrt 
 
 
 00-- 
 
 Oort 
 ^»-o 
 
 relate 
 
 000 
 
 
 
 *«KO 
 
 ««« 
 
 oaoNvaxs- 
 
 
 onn 
 
 nnn 
 
 nnn 
 
 
 not- 
 
 rvOBOo 
 
 PJfOO 
 
 nn" 
 
 
 
 nnt 
 
 <vc(n 
 nnn 
 
 ^>i-o 
 
 p>'«-+ 
 
 r-n»- 
 
 ntnn 
 
 p>»->^ 
 00- 
 nnn 
 
 •■00 
 
 o(«on 
 
 nn'*J 
 
 
 00- 
 
 OO't-O 
 >«N0o 
 
 
 00 f*^ 
 
 
 n»^ 
 
 nv.t 
 
 *-»-0 
 
 NP>0 
 
 
 r*nft> 
 
 n«ot- 
 
 00>.«> 
 
 000 
 
 000 
 
 K-fn 
 
 000 
 
 o«^> 
 
 OKN 
 NO 
 
 —sn 
 
 So, 
 
 *-»-o 
 
 OP>^ 
 
 p)n*^ 
 
 OoO-«- 
 
 nnr^ 
 
 60V70 
 
 O'^cM 
 
 nnn 
 
 0^»N 
 (••JDO 
 
 
 -»T«> 
 (TO**) 
 
 c(0-T 
 00- 
 
 rtntn 
 
 
 0>4N 
 
 (onT 
 
 ft-»-o 
 o-n 
 
 ViO>» 
 voo 
 
 cd%-«- 
 
 To-« 
 
 NQqOo 
 
 
 TO-0 
 
 O<0- 
 
 000 
 
 'OOM 
 T0o-» 
 
 
 OT^O 
 
 ok>« 
 
 5»^ 
 
 
 
 00 
 TTt 
 
 «-QO 
 
 riTT 
 
 tVo 
 
 OMT 
 
 T-v4 
 
 kOdOs 
 
 &>TO 
 
 »jTo 
 
 i-(o-T 
 
 T-M 
 TIOl^ 
 
 nn*7 
 
 f-T»- 
 OTT 
 niVO 
 
 nnn 
 
 c<».T 
 
 CoTO- 
 
 t-TV 
 T>04 
 
 rJnpv 
 
 ^«iT 
 
 rtftJT 
 00- 
 
 to" 
 «-«.o 
 
 NOV 
 N<0«0 
 
 NfjOa 
 
 rtNtO 
 
 000 
 
 oOon 
 
 (N«" 
 
 TegW 
 
 ~«lsO 
 *>l-n 
 "ot") 
 
 nr'o 
 
 
 anos 
 
 UBddOa 
 
 
 l»l»T 
 T^-» 
 nnn 
 
 Toto 
 
 rjpjn 
 
 OTT 
 
 nnn 
 
 i^oo- 
 iTt 
 non 
 
 <HIOT 
 
 M«n 
 nnn 
 
 TTO 
 
 <vNn 
 
 T"»- 
 
 TOO 
 
 Non 
 
 
 
 — Na^3 
 
 TT'O 
 
 0-J^ 
 
 ^-5 
 titifl 
 
 ^•00 
 teTO 
 
 
 
 TT'O 
 
 
 000 
 
 loOoN 
 
 000 
 
 Ooo 
 
 00»( 
 OQ<f 
 "flON* 
 
 oon 
 
 n , 
 
 >OOT 
 
 oi, 
 
 nnn 
 
 •AptTs 
 
 «>»-»■ 
 
 T0-» 
 nnn 
 
 nnn 
 
 nnn 
 
 o-»t>l 
 
 nn'o 
 
 T«-^ 
 
 TT^ 
 
 nnn 
 
 ^OCo 
 
 nTT 
 nnn 
 
 r^ojo 
 «nT 
 nnn 
 
 T~- 
 o~ 
 nnn 
 
 «iT«^ 
 toa-T 
 
 ft(iyf* 
 
 
 ri*>T 
 
 TT" 
 
 NPlOs 
 
 Knn 
 
 
 Nrvto 
 
 
 Kin 
 
 NN«o 
 nTT 
 
 
 •-NP> 
 OoOv9> 
 000 
 
 0«lo 
 
 nf^ 
 
 «T 
 
 •»»o 
 00- 
 
 OOO 
 
 51,0 
 
 "JT- 
 
 f<«-IO 
 
 nnn 
 
 oo- 
 
 fcTO 
 NfcT 
 
 nnn 
 
 00^ 
 
 Kfcto 
 
 nnn 
 
 09P)»s 
 
 nnn 
 
 o sn 
 nnn 
 
 Oo^o 
 
 nnn 
 
 nnn 
 
 ^:;jr 
 
 nnn 
 
 cl<»T 
 nnn 
 
 lOOOo 
 
 nTT 
 
 nnn 
 
 nnn 
 
 «nT 
 nnn 
 
 rIOoT 
 wcvn 
 nnn 
 
 o-«W 
 
 TT'O 
 
 nnn 
 
 i.O-» 
 
 1^0^ 
 0-- 
 
 nnn 
 
 ToT 
 fr^OO 
 
 Nnn 
 
 NOb*- 
 
 ^no 
 
 »-o 
 
 t^nn 
 
 v>^»^ 
 
 c(f<N 
 
 TON* 
 
 TkllO 
 
 T0.» 
 wnn 
 
 0«VN 
 
 T*%s 
 
 00- 
 
 1,0 
 Oo«-0 
 
 ♦.■or* 
 
 »-TO 
 
 'Oh-* 
 
 nrt 
 
 ^To 
 
 Nn 
 
 «N- 
 
 bOo 
 
 oWt 
 fi'oi 
 
 TOn 
 
 OOL( 
 o- — 
 
 %^oo 
 ntt 
 
 oVx^ 
 
 ♦■TO 
 
 nnt 
 
 nnn 
 
 T'oio 
 
 -NT 
 
 nnT 
 nnn 
 
 <s»-T 
 nnn 
 
 nnn 
 
 f^iNN 
 
 «»>T 
 
 >«>IN 
 
 TT4 
 
 — Nn 
 rtrtn 
 
 Oono 
 «-o- 
 
 «>».T 
 
 
 •sNto 
 
 TOW 
 000 
 
 ^»n(< 
 
 NnT 
 
 000 
 
 Nf»T 
 
 n>»N 
 
 nany^y 
 
 ^§ 
 
 a30tlOdNiaU133l% 
 
 MnNivunnv 
 
 
 
 
 
 '1 
 
 •< 
 
 T 
 
 n 
 
 
 ■ 
 
 
 
 
 
 to 
 
 
 
 
 
 
 
 (0 
 
 
 
 <0 
 N 
 
 to 
 
 
 
 
 
 S 
 
 •^ 
 
 N 
 « 
 
 
 NO 
 
 <o 
 cx 
 
 
 
 
 T 
 
 « 
 
 
 lo 
 
 N 
 
 
 
 
 N 
 
 
 T 
 
 
 
 
 
 T 
 
 
 
 4 
 
 (0 
 
 
 
 
 n 
 
 
 
 
 
 
 
 
 •< 
 
 •n 
 
 
 
 
 K 
 
 
 
 
 
 "9 
 
 
 
 
 "o 
 
 
 
 
 
 
 •t 
 
 « 
 
 , 
 
 
 ^ « 
 
 e 
 
 
 
 oja 
 
 « 
 
 
 
 <» 
 
 
 .a 
 
 O* 
 
 If 
 
 a 
 
 
 
 
 
 -a « 
 
 9 
 
 
 
 
 
 
 
 
 
 2 
 
 
 
 1 
 
 ^ 
 
 bS 
 
 
 
 
 
 
 
 
 
 N 
 
 
 
 S 
 
 »a 
 
 fi** 
 
 ■0 
 
 « 
 
 
 
 
 
 
 
 
 
 
 
 
 a 
 
 
 fed-- 
 
 9> 
 
 « 
 
 «s 
 
 
 ,13 
 
 B^ 
 
 .s* 
 
 <S 
 
 » 
 
 =i 
 
 wo 
 a a 
 
 .2 
 
 .9 
 
 II 
 
 2S 
 
 CO OS 
 Oi 
 
 ^■° 
 
 o 
 «w as 
 
 
 5w 
 
 &« s 
 
 'J 
 
 ,3» 
 
 i-g 
 
 V 
 
 S 5 
 
 a ta 
 
 p. 
 ta 
 
 ® s 
 
 liaS 
 
 Ho -S 
 "S fee o 
 
 5 to M 
 
 ^ O t>A 
 
 Q J3 en 
 
 3 o es 
 O cs ,d 
 
 a* 
 
 25 
 
 3 U 
 
 o 
 
 "a 
 
 o • ® 5 
 
 O BO 'O'— 
 
 .2 .ri a S 
 «-3 » a 3 
 
 a 0) oj « 
 ■a (-2 
 
 00 V O 2 
 
 MpC >> . 
 
 .0 ben ^ 
 
 »-° a o- 
 a»> "So 
 
 "■-■9 j> 
 •r 5 fc. 
 
 •^Sg ° 
 §■3-1 " , 
 
 ^ ^ oj o r 
 
 0> " BO ^ 
 
 ►>" o 9 
 
 BO a Q k 
 a », »> ^ 
 
 £"•5 «S 
 
 .9 w 2 5 
 "" 3 " Cm 
 
 ? ai -^ O ,i 
 
 ; sb 
 
 J aj ©M 
 
 
 t3 &. 
 
 01 *** 
 
 s -35, 
 
 0; = <! 
 
 g| II 
 
 .t ft 
 
 bti*3 
 
 .i o> 
 
 O'S 3 
 
 
 
 l« .9 oi 
 
 w •" 
 
 CS BO 
 
 CO p« 
 
 oj "^ o S j; 
 
 .1-3 -as a 
 
 » c o S 3 
 
 ffl « M eo o 
 
 ©BO'— ^ 
 
 s oo o> £ 
 "«■« «2 
 
 r~j= I. " » 
 
 » 2 ° g- 
 01 3 a ^ » 
 
 •»f O ee 03 *3 
 
 -ij: j: •" « 
 
 5 S 
 
 
 a " S e , 
 01.9 —ja ' 
 
 o 5 0)~ 
 v^ O h O 
 
 * > • 
 S o» S i? 
 
 g E s2 
 
 'S a a 
 
 CO rr 
 
 ^ «2 
 
 o ■= " 
 
 o o fta 
 
 « « it's 
 
 0) 2 » 9 
 
 .a-o ■- 
 
 H 9 >. 
 
 wVm ta 
 
 j2-S O'a 
 
 )»g s- 
 
 t. "O „ U"^ 
 
 0.-S > -o . 
 
 EB ** » to 
 2i£ B3 
 
 •u 2 " 
 X 9 " IP'S 
 
 4) "3 
 
 .38 5 
 
REACTANCE— CAPACITANCE— CHARGING CURRENT 
 
 13 
 
14 
 
 REACTANCE— CAPACITANCE— CHARGING CURRENT 
 
 o 
 
 o 
 10 
 CM 
 
 h 
 < 
 
 LJ 
 
 o 
 
 z 
 < 
 I- 
 
 (0 
 
 (0 
 LJ 
 GC 
 
 O 
 h 
 
 UJ 
 
 o 
 
 z 
 < 
 h 
 o 
 
 < 
 
 UJ 
 
 u 
 
 -I 
 
 o 
 
 >- 
 o 
 
 CM 
 
 L. 
 O 
 
 o 
 
 h 
 < 
 
 Ld 
 
 -I 
 
 m 
 
 < 
 
 uJ 
 
 UJ ^ 
 
 o s 
 
 > K 
 
 m o 
 
 "^ HI 
 
 O pC 
 
 Z ll- 
 
 >• o: 
 
 I 
 a. 
 O 
 
 I- 
 
 Q CO 
 < 
 
 O 03 
 
 UJ O 
 
 ^ 2 
 
 I- Q 
 
 _ z 
 
 Q o 
 
 = o 
 
 CQ 
 
 o 
 
 > 
 
 UJ UJ 
 
 o 
 
 m 
 S9 z 
 
 < 2 
 
 UJ O c 
 
 g 5 I 
 
 z ^ "^ 
 
 £ CO o 
 
 UJ UJ ° 
 
 m X "o 
 
 ^ £ "o 
 
 > 
 o 
 
 03 
 
 H 
 Z 
 
 Z 
 
 o 
 o 
 
 15. 
 
 UJ O CO w 
 
 < t> o s 
 
 i_ lu ^ h- 
 
 >^ -» z f^ 
 
 CO t3 = -^ 
 
 coZccO 
 
 UJ CO O to 
 
 CC < H <N 
 
 
 f- 
 — UJ 
 CJUJ 
 
 
 
 6(^46 
 
 QbboN 
 
 CdOoN 
 
 so—"* 
 
 06h.S 
 
 
 
 
 
 OoOoN 
 
 
 6oSN 
 
 
 
 
 
 UJ d 
 
 a- 5 
 
 Q- O 1- 
 
 S O liJ 
 
 < 3 CO 
 UJzO ll 
 
 < cc 
 Q 
 
 ON & V a 
 
 S3H0NI Nl 
 8313 W Via 
 
 3dAJ. 
 
 iviasivw 
 
 V,-^r^ 
 
 06KK 
 
 
 OWrx 
 
 
 
 
 OOOO 
 
 
 o^w 
 
 
 
 
 
 
 
 ^l-'T 
 
 •ott 
 
 ^^W TYH 
 
 
 
 ^•^t 
 
 
 
 
 
 
 ^Y^O 
 
 
 
 
 
 
 000 
 
 ^55 
 
 OOO 
 OOO 
 
 000 
 OOO 
 
 ooo 
 
 OOO 
 
 ooo 
 
 OWN 
 
 001-5CX 
 
 
 t-«o 
 
 •*)*OCX 
 
 to(DN 
 
 000 
 
 000 
 000 
 
 o 
 o 
 600 
 ooo 
 000 
 
 
 
 
 
 5g 
 
 OtoN 
 
 OdHCD 
 
 000 
 
 000 
 
 ooo 
 000 
 
 000 
 
 000 
 00 o 
 
 CX-vO 
 
 ■>«0<o 
 
 
 -o* 
 
 •-jOtO 
 
 
 
 '-OON 
 
 Tf^ 
 
 
 OOOsS 
 
 
 njftjf^ 
 
 
 loOoO 
 
 
 tor<ss 
 
 
 NO O 
 
 00*8 + 
 
 
 00 f^^ 
 
 
 C<-.0 
 
 
 
 COOCo 
 >Oto 
 
 000 
 
 000 
 000 
 000 
 00,0 
 
 00 
 
 
 (»^i- 
 -^•^P) 
 
 NNflD 
 000 
 
 
 000 
 000 
 000 
 
 80NN 
 
 00- (h 
 
 
 
 
 03 low 
 
 
 teO'+ 
 
 
 
 
 
 
 
 
 
 r^NNS ^^^0'^ 
 
 nooo ^'^"i" f^rt*^ 
 
 fOooo 
 
 '-?NO 
 
 ONt 
 
 (SsO-t- 
 
 
 flo'on 
 
 O + Q 
 
 
 «0(j. 
 
 Oo-^>f- — cnN 
 
 
 
 ooa 
 
 ONN 
 
 oV)« 
 
 N(0O 
 
 000 
 
 000 
 
 000 
 000 
 ooO 
 
 
 
 
 Oot^K 
 
 000 
 
 ooo 
 
 000 
 000 
 
 000 
 
 0*90 
 
 OONN 
 
 
 ^"i' 
 
 
 
 N'0 + 
 
 
 
 n Nof "-jMW 
 
 sS + (^ 
 
 OoNCo 
 
 v^OvlO 
 
 lOoOo t^^-t (Ortr( 
 
 
 
 tJt^to 
 
 
 
 
 -«-jO ^tYt- 
 
 t1-^ Oomoo 
 
 ^w^ 
 
 K'fi'^t^'^D 
 
 
 '^OT 
 -NT 
 
 000 
 
 000 
 000 
 000 
 000 
 
 rnroot 
 
 
 
 
 
 N>«0O 
 
 -oC'O 
 
 NOP* 
 
 000 
 
 NfO 
 
 ogo 
 
 •ooto 
 
 
 
 
 
 
 
 o>oo 
 
 OOON© 
 
 
 
 tONO ,*»-t 
 
 
 
 
 ■+Oocy 
 
 >0 ^rr) 
 
 
 KKN 
 
 
 <tot-*^ r(«-5to woo^ 
 
 
 
 
 nSnO 
 
 o->-r< (Tl-'o 
 
 
 •*no 
 
 G3aNvaxs 
 
 
 ooo 
 
 ONK 
 
 0*0 
 
 NO+- 
 
 
 
 
 
 
 
 
 
 
 + (tosS 
 
 
 0-^6j 
 
 OflONS 
 
 O^t^ 
 
 Note 
 
 OvjN 
 
 MINO 
 
 
 
 n^^o 
 
 
 anos 
 
 UBddOO 
 
 
 Wtori 
 
 1.00 r^ 
 
 
 
 
 
 "-JlOto 
 
 
 mooo 
 
 
 
 qCDnS 
 
 O0o'>« 
 
 oOo-« 
 
 O0o^9 
 
 00^ 
 
 
 
 QtO 
 OonO*0 
 
 
 
 'O + t 
 
 0(h-fi 
 
 
 000 
 
 000 
 
 oocx 
 
 OO't- 
 
 oor> 
 N*ion 
 
 
 
 
 
 
 
 
 
 
 ^i-fo 
 
 
 to*— 
 
 
 
 
 MM 
 
 
 
 
 
 Tft-^ 
 
 «to'4- 
 
 
 ^r(0 
 
 
 
 NY. 
 
 oo>. 
 
 JOY- 
 
 000 
 
 
 to-'*' 
 
 (qOO 
 
 Ynm 
 
 sonSO 
 (yp)Y 
 
 ooo 
 
 0«Y 
 (^oto 
 
 000 
 
 NP)Y 
 
 io>*o 
 
 -fto'^sS 
 
 Yfr-^O 
 YfO(^ 
 
 
 [Q30UOdNI3U13318 
 
 lAinNiiAimv 
 
 §5 
 
 
 t*. ^ d 
 
 ! 5-0 
 
 4J 0,0 
 
 ^,2 ° ' - 
 ■a £ ?o 
 
 is-" " 
 
 I g ° SI 
 
 5 u o ^ *• 
 ^ .5 ■;:; "^ 
 
 -■ ^ »-l OJ 
 
 '••° a, * B 
 
 i to 
 
 ^^ ' 
 
 " © ""^ .2 ^ 
 
 c= "^ i!^ E"^ 
 
 >^ s -5 
 
 .£ « o ^ « 
 
 y c *> o 
 
 o - M ».9 
 
 O to M ~ rt 
 
 _ O i3 CO 
 
 a a><^ CO 
 
 5 = o - t3 
 
 to OJ ^ w 
 
 
 
 ^ « ^ > 
 
 ^- 1: ■ 
 
 j^ M w J3 2 
 
 .2 " ^ 
 
 S to C bo 
 
 ,- S ^ °^ 
 
 g a; a o ^ 
 
 ^^T3 to 
 
 ^ « 5 « 
 
 ® i M o ■-; 
 
 o fe « t, S 
 
 
 -•>«•-. 9 9 
 
 (U 4) o « o 
 
 to P. 
 
 fe © ^ 
 
 r; !« no 
 
 c£» 5« 
 
 a" 
 
 So c 
 
 ■*< _ OJ 
 
 "•^ ft OfH 
 
 K " " ^ 
 
 «S O A 
 
REACTANCE— CAPACITANCE— CHARGING CURRENT 
 
 15 
 
 o 
 
 o 
 U) 
 CM 
 
 I- 
 < 
 
 111 
 
 o 
 
 z 
 < 
 
 (0 
 (0 
 
 u 
 cc 
 
 o 
 
 h 
 
 UJ 
 
 o 
 
 z 
 < 
 
 O 
 
 < 
 bJ 
 GC 
 
 LJ 
 
 -I 
 
 O 
 
 >• 
 
 o 
 o 
 
 (0 
 
 Ll 
 O 
 
 GC 
 
 LJ 
 
 m 
 
 < 
 
 to 0) 
 
 H ID 
 §? 
 
 fe 2 
 
 0> O 
 
 a 
 
 I 
 I- 
 o 
 
 O 
 
 z 
 O S 
 
 2 o 
 
 f g 
 
 ? 8 
 
 to li. 
 
 2 C 
 
 X 
 
 I- 
 
 q: 
 O 
 u. 
 
 s 
 
 o 
 
 O to 
 
 > UJ 
 
 o 5 
 
 z 2 
 
 > 
 
 5 05 
 
 z 
 < 
 
 H 
 eo 
 
 z 
 
 o 
 o 
 
 ■^ 6 •- 
 
 UJ Q (C X 
 zOir 
 
 < O" 
 . UJ r- 
 t^ -< Z fc 
 05 O - 
 «ZccO 
 
 UJ CO O to 
 cr < H tN 
 
 I- 
 
 (OUJ 
 CIUJ 
 
 Do+n 
 
 I- 
 — UJ 
 CMuj 
 
 lOUJ 
 
 
 O\»*0 
 
 
 •^tnn 
 
 
 
 ti-V> 
 
 
 
 cc Uj 
 UJ — 
 
 e- o H 
 
 < 3o> 
 o 
 
 "ON S ?? 8 
 
 S3H0NI Nl 
 H3131^Via 
 
 3dAl 
 
 ivia3ivw 
 
 OOQN 
 
 
 
 Co**"*- NtH^ 
 
 
 
 
 
 
 
 
 
 
 ^■^N 
 «^(^I< 
 
 
 
 
 -«ioT tow 
 
 
 
 
 
 
 wo- 
 
 nooo 
 
 
 
 ^ rt rv 
 
 
 
 now 
 
 ■TOT 
 
 000 
 
 000 
 
 000 
 
 000 
 
 000 
 000 
 000 
 oftr- 
 
 (Of o 
 
 00 
 
 
 
 
 
 
 
 
 ooo 
 
 NNNft 
 
 KOO 
 
 
 OOQ 
 
 000 
 
 000 
 000 
 
 ooo 
 000 
 
 
 
 
 
 
 
 
 ^fO 
 
 
 ^00 
 
 000 
 
 
 Ho"-) 
 
 
 
 V.ON 
 
 000 
 
 OOO 
 000 
 000 
 
 odo 
 ooo 
 
 r<^0 
 
 
 -V l) o 
 
 
 
 OoNsS 
 
 obtt- 
 
 
 oOoiy 
 to** fv 
 
 
 
 
 
 
 
 
 1- 
 
 
 otoi- 
 
 OoO%> 
 000 
 
 000 
 
 000 
 000 
 000 
 
 
 ^00 
 
 00*00 
 
 
 O^Q 
 
 ^^^c*o 
 
 OOOOK 
 
 
 (Oft)'-) 
 
 p^oi- 
 
 OoOON 
 
 toNK 
 
 aoNK 
 
 OOMN 
 
 Oofnte 
 
 OotoN 
 
 
 
 
 
 
 00 v^ 
 
 NBoto 
 
 O0D>» 
 f POO 
 
 ^0— 
 
 knOd 
 
 000 
 
 000 
 
 ooO 
 ooo 
 00 o 
 o-oo 
 
 0*-%. 
 
 
 
 QDN 
 
 ••^'o 
 
 <isCl'0 
 
 
 sfll-)!^ 
 
 
 
 
 ^'-30 
 
 + 0-+ 
 
 toto 
 
 
 t-TT 
 
 t-'T'O 
 
 
 '^N'(■ 
 
 
 
 
 sS60- 
 
 O ^vN 
 
 000 
 
 000 
 000 
 
 000 
 00 
 
 Ooooo 
 
 
 ^tn 
 
 
 oto 
 
 1-*-n 
 
 
 ^f»JP) 
 
 c(^»0 
 
 
 
 kO« 
 
 
 OfcO 
 
 
 ■ttn 
 
 >-JOf 
 
 
 
 Ootxo 
 
 
 ^^T^ 
 000 
 
 000 
 
 000 
 000 
 000 
 
 , r>^ "or**© 
 
 tt-toWeoNK 
 
 
 torjfc 
 
 
 
 
 
 PC'S 
 
 
 to--* 
 
 
 eooo 
 
 
 000 
 
 000 
 000 
 
 000 
 
 fnnct 
 
 
 
 Oo-*»to 
 •©no 
 
 
 
 
 
 4d<o*0 
 
 
 f tl'-* 
 
 
 
 
 Oct 
 
 
 
 O 
 
 OP*ft- 
 ON^- 
 
 >eNO 
 
 .fvfO 
 
 OvOto 
 
 ooto 
 
 "-jrop* NN^ ^J^(^c 
 
 ^•'0•*- ntf^ rioi 
 
 h'r'2 
 
 tooc< 
 
 ♦ on 
 
 NB'Ot 
 
 
 
 
 
 OV*a 
 
 
 on;* 
 
 
 np(^ 
 
 
 ^^0(^ 
 
 *kW 
 
 
 OSK'O 
 
 \»n>.. 
 
 teK'o 
 
 ^sno 
 
 
 
 op(Sd 
 
 <toO< K»<0 
 
 
 r^-**- 
 
 nS^ 
 
 to«>o 
 
 
 -Sty 
 
 
 
 
 mnf^ 
 
 
 
 
 
 
 
 
 03aNVU18 
 
 toiD^ 
 
 o^•n 
 
 
 )>.*-n 
 
 
 2?^"^ 
 
 •^■^Q- 
 
 
 t-Tn 
 
 st-n 
 
 000 
 
 o«<>^ 
 
 -•NO 
 
 00*0 
 
 ^'^^ 
 
 oto>© 
 
 
 0-O(0 
 
 
 s»^* 
 
 
 
 
 OTbo 
 0•0^« 
 
 n«« 
 
 
 
 <4inn 
 
 ^ 00 
 
 
 
 
 
 noio 
 rtftn 
 
 Ttoto 
 
 rente 
 
 -•NO 
 
 (0't-*O 
 
 »p» 
 fxOte 
 
 C1« 
 
 ano9 
 
 d3dd03 
 
 
 
 oioki 
 
 ■of) 
 
 '.no- 
 's ■♦•> 
 
 ON*- 
 
 
 fOto«o 
 
 + T^ 
 
 ■i■•v^ 
 
 
 
 n>» 
 
 
 own 
 
 ■00* 
 
 
 
 0>K 
 
 
 t-ON 
 «vtof 
 
 
 to^^n 
 
 
 o-cn 
 
 
 on* 
 
 N.,« Oof 
 
 t»-o 
 
 
 
 nof 
 
 
 0^% 
 
 f «•- 
 
 00 -•^ 
 
 
 f<K 
 
 i,tos 
 
 000 
 
 oon 
 
 OOY 
 
 loose 
 oon 
 
 
 ^"-90 ^<y«» 
 >on V ooN'o 
 
 note 
 
 
 
 OBnN 
 
 rrOOB 
 
 ••IT 
 
 
 toNO 
 0»>^- 
 
 O0os» 
 
 ■•'0+ 
 
 snn 
 
 f«« 
 •.■of 
 
 noo 
 
 00 
 
 009 »- 
 
 ••fn 
 
 f»)^n j-K»s 
 
 'rrn 
 
 'of') 
 
 ^f>« 
 
 "^f n 
 
 
 ~4^ 
 f - 
 
 flof<^ 
 
 nN»~ 
 
 
 •An 
 
 TfllJ 
 
 o«<o 
 4f>^ 
 
 mT 
 ,,nv, 
 « too 
 00- 
 
 000 
 
 QVjO 
 
 >0f~ 
 
 n^fiO 
 
 \^fin 
 
 nV4»-f». 
 
 
 
 OOO 
 
 onf 
 
 5q 
 
 f nr 
 
 f n/l 
 
 n«trt 
 
 fr.w 
 
 nrx-. 
 
 -•OT 
 
 f KW 
 
 4>on 
 ^nf 
 
 000 
 r^nt- 
 
 T 
 
 M-0 
 
 n«« 
 
 030aOJNI3ai33±S 
 
 iAinNiMn-iv 
 
 : I 
 
 
 a 
 
 
 •0 • • 
 
 ^ 
 
 
 
 V 5 
 
 
 
 s 
 
 .a 
 
 ^o^** 
 
 -0 
 
 C-- ; 
 
 
 
 fao 
 
 
 « 
 
 ■S5« -s 
 
 
 
 ja 
 
 iF-;i 
 
 .2 
 
 
 
 a i- a - !• 
 
 0. 
 
 ^^::i 
 
 
 
 A 
 
 ilsiJS 
 
 
 
 sB-t: 
 
 
 
 
 
 
 
 ^■""ta 
 
 
 |s- = s 
 
 fX 
 
 &io -2 
 
 
 
 " ^ .: * 
 
 ft 
 
 
 
 
 
 
 « 
 
 iJo" "CS 
 
 »4 
 
 &4 
 
 Piin 
 
 
 
 
 s^S ''a 
 
 u 
 
 ^ "-=1 
 
 < 
 
 "is •Si 
 
 II 
 
 §,§° -•s 
 
 
 
 P 
 
 
 g 
 
 Iss'^l 
 
 
 5^=2 Sxo 
 
 
 .0 MO = » 
 
 a 
 
 2=:=^ 
 
 
 
 a 
 
 ►.5°-^- 
 
 
 S" « 
 
 
 ^"3 -'5 
 
 
 >>* 2 
 
 a 
 
 -ir-i 
 
 
 
 0) 9 M a ^ 
 " = S 
 
 
 
 
 ta""* 
 
 
 
 
 m « 
 
 a 
 
 I'l 5S 
 
 *** 
 
 ^"g-ga 
 
 
 •S£?-8 
 
 g 
 
 e 
 
 k 
 
 .b 
 
 S t " a " 
 
 
 3 N crta 
 
 £ 
 
 
 
 8 " « 
 
 
 p. 
 
 £ to k a 
 
 <9 
 
 hi 
 
 
 
 
 M 
 
 S~^?5 
 
 %> 
 
 S5-3 - 
 
 6. 
 
 ->i 
 
 ... OS 
 
 S * » fc » 
 
 <0 
 
 I 
 
 2 -3-5 
 
 ■si: la 
 
 
 
 II 
 
 k2~-„ 
 
 
 
 
 2 = 3 •= 
 
 
 
 
 
 
 
 3 
 
 "jo a ■ 
 
 
 ^_S =3 
 
 
 
 
 
 ^S 
 
 ■» =? " a 
 
 
 ^* 5 "• 
 
 
 
 
 
 
 S:;^^: 
 
 
 
 ■n-o 
 
 
 a,- 
 
 
 SE 
 
 
 
 
 J= tJ 
 
 
 
 i§ = -S8 
 
 
 £ = 
 
 5 " 
 
 >a H' 
 
 ^° 
 
 ;=?:-• 
 
 
 s-2s£5 
 
 
 •■= 
 
 a 
 
 
 
 
 2S 3 
 
i6 
 
 REACTANCE— CAPACITANCE— CHARGING CURRENT 
 
 O 
 
 u 
 u 
 
 o 
 o 
 o 
 
 UJ 
 
 GC 
 
 Q. 
 
 O 
 
 
 h 
 
 -J 
 
 (,) 
 
 < 
 
 h 
 D 
 UJ 
 
 D 
 O 
 
 z 
 o 
 
 z 
 
 o 
 
 o 
 
 h 
 
 u 
 
 < 
 
 u m 
 
 o 
 
 z 
 
 UJ 
 
 < 
 
 (3 
 
 z 
 
 < 
 
 CO 
 
 Q. 
 
 
 < 
 
 
 O 
 
 
 UJ 
 
 m 
 
 < 
 
 u! 
 
 z 
 _i 
 u 
 
 I 
 
 3; 
 
 tu 
 c 
 
 X 
 
 t- 
 
 < 
 o 
 q: 
 
 K 
 UJ 
 
 S 
 > 
 CO 
 
 < 
 
 s 
 
 UJ 
 
 eo 
 
 Q. 
 
 1 
 UJ 
 
 -1 
 
 C3 
 
 Z 
 
 CO 
 
 < 
 
 o< 
 
 o 
 
 o 
 
 z 
 
 8 
 
 I 
 
 o 
 
 < 
 
 UJ 
 
 u. 
 O 
 
 t- 
 
 UJ 
 UJ 
 
 u. 
 
 o 
 o 
 o 
 
 a 
 
 § 
 
 < 
 u. 
 O 
 cc 
 o 
 
 5 
 
 2 
 
 < 
 
 H 
 
 UJ 
 
 Z 
 
 O 
 
 t- 
 
 Ol 
 
 UJ 
 
 O 
 z 
 < 
 
 o 
 < 
 
 z 
 
 4s 
 
 X 
 
 CO 
 
 a. 
 O 
 
 1- 
 
 o 
 
 o 
 
 z 
 o 
 o 
 
 u. 
 
 o 
 
 CO 
 
 cc 
 
 UJ 
 
 1- 
 z 
 
 UJ 
 
 o 
 
 z 
 
 UI 
 
 UJ 
 03 
 
 S 
 
 UJ 
 
 O 
 
 z 
 < 
 t- 
 co 
 
 Q 
 
 1- 
 lOUJ 
 ClUI 
 
 u. 
 
 
 ooo 
 Ooo 
 
 n H r4 
 
 000 
 
 OQO 
 
 000 
 
 00 
 
 ■*^-* 
 
 nttrt 
 
 000 
 
 Ooo 
 
 rt 00 
 
 tiMn 
 000 
 
 000 
 
 nrtrt 
 000 
 000 
 
 o-»* 
 
 000 
 000 
 
 OK^ 
 
 + o« 
 
 000 
 
 000 
 
 000 
 
 Q 
 
 ooo 
 000 
 
 « «•< 
 
 00 
 000 
 
 000 
 000 
 
 000 
 000 
 
 
 00 
 
 QOO 
 
 =7^1 
 
 000 
 ooo 
 
 oo« 
 
 000 
 000 
 
 000 
 
 000 
 
 
 0> 
 
 UJ 
 
 £ 
 
 X 
 
 UI 
 
 O 
 
 i 
 
 z 
 
 8 
 
 H 
 fe 
 
 0) 
 
 o 
 
 UJ 
 
 I 
 (- 
 
 a 
 
 z 
 
 UJ 
 
 m 
 
 1- 
 
 OUJ 
 CMUI 
 
 u. 
 
 ooo 
 
 « OO 
 
 ooo 
 ooo 
 
 OOO 
 
 000 
 
 Ooo 
 
 000 
 
 o^J 
 000 
 
 Q 00 
 
 Sri 
 
 000 
 OO 
 
 ttt 
 
 Ooo 
 000 
 
 ♦ 0^ 
 
 000 
 00 
 
 n n n 
 « w« 
 ooo 
 OOO 
 
 000 
 000 
 
 « rtH 
 OoO 
 
 000 
 
 00 
 
 
 Ooo 
 
 000 
 
 ON«(y 
 
 000 
 000 
 
 On n 
 »<'x«x 
 
 OQO 
 000 
 
 ■*■*■« 
 
 OOO 
 OQO 
 
 
 1- 
 
 — UI 
 
 04 UJ 
 
 U. 
 
 ooo 
 ooo 
 
 Ooo 
 coo 
 
 fOO 
 
 000 
 000 
 
 000 
 000 
 
 4-«*o 
 
 •(^•♦^ 
 000 
 000 
 
 253 
 
 «-o 
 
 000 
 000 
 
 000 
 OoO 
 
 ■»--«■« 
 
 0° 
 000 
 
 00« 
 
 ««« 
 000 
 000 
 
 N«tM 
 000 
 000 
 
 t-mn 
 
 000 
 000 
 
 •( MM 
 
 00 
 000 
 
 5?5 
 
 
 
 Ooo 
 
 «55 
 
 000 
 000 
 
 t5? 
 
 000 
 000 
 
 f on 
 
 000 
 QOO 
 
 0-«N 
 ««« 
 
 «rtt< 
 
 OOQ 
 
 000 
 
 
 000 
 00 
 
 
 »- 
 
 u. 
 
 Ooo 
 
 OO 
 
 %«"»-o 
 
 ooo 
 
 ooo 
 
 00 
 000 
 
 **o 
 
 «> WW 
 
 000 
 Ooo 
 
 *l rt « 
 
 000 
 000 
 
 ^MO 
 
 OO 
 000 
 
 OOo 
 
 
 
 o-»+- 
 
 00 
 00 
 
 000. 
 
 000 
 
 • ■*-o 
 
 o« 
 
 00 
 000 
 
 <HM«X 
 
 Ooo 
 
 OQO 
 
 000 
 000 
 
 00 
 00 
 
 OQO 
 000 
 
 nrtw 
 000 
 000 
 
 OQO 
 
 000 
 
 OQO 
 000 
 
 •to-* 
 
 on" 
 
 000 
 000 
 
 
 
 
 
 
 
 h- 
 h-UJ 
 -UJ 
 
 u. 
 
 O OO 
 
 ooo 
 
 o o o 
 O CO 
 
 000 
 Op 
 
 00 
 00 
 
 000 
 
 000 
 
 00 
 
 000 
 
 ■*-00» 
 
 ««« 
 
 000 
 
 OQO 
 
 <XWM 
 
 000 
 OoO 
 
 :;S5 
 
 000 
 000 
 
 ♦•oo 
 
 «4r(C4 
 000 
 000 
 
 ««« 
 
 OQO 
 000 
 
 •.■4-0 
 
 00 
 
 Ooo 
 
 ^■WO 
 
 onn 
 
 OQO 
 
 000 
 
 QOO 
 
 00 
 
 000 
 000 
 
 •« w« 
 OoO 
 
 OQO 
 
 000 
 000 
 
 now 
 
 000 
 000 
 
 
 t- 
 
 iqUJ 
 
 = UJ 
 
 b. 
 
 ^oO 
 
 ooo 
 ooo 
 
 ooo 
 ooo 
 
 O O Q 
 
 •-•- 
 «'«■« 
 000 
 Ooo 
 
 ■•-wo 
 
 QO 
 
 
 
 o*»j 
 
 000 
 
 000 
 
 oo 
 
 00 
 
 000 
 
 000 
 
 o»*o 
 
 ooo 
 000 
 
 I-,— 
 
 00 
 000 
 
 «WCt 
 000 
 000 
 
 *-o-* 
 oo« 
 NnM 
 000 
 000 
 
 Ooo 
 Ooo 
 
 ••■00 
 
 000 
 000 
 
 o«w 
 
 000 
 000 
 
 000 
 000 
 
 3S5 
 
 oO 
 000 
 
 000 
 000 
 
 f'«0 
 
 OQO 
 QOO 
 
 
 =5 
 
 OoO 
 
 OQO 
 
 OOO 
 OOO 
 
 :ooo 
 000 
 
 Ooo 
 
 »■•- 
 
 000 
 000 
 
 000 
 
 000 
 
 C4 O'^ 
 
 OOO 
 00 
 
 Sis 
 
 OQO 
 OOO 
 
 000 
 000 
 
 00 
 
 OQO 
 
 000 
 000 
 
 oo» 
 00 
 
 000 
 
 2^ 
 ««« 
 
 000 
 
 OoO 
 
 ♦ +« 
 ««« 
 
 000 
 000 
 
 o« 
 
 
 Q 00 
 
 OQO 
 00 
 
 Ooo 
 000 
 
 4-0 
 
 Ooo 
 
 OQ 
 
 000 
 
 000 
 
 
 — UJ 
 
 (1. 
 
 28* 
 
 ooo 
 ooo 
 
 ooo 
 
 OOO 
 
 000 
 
 Ooo 
 
 own 
 000 
 000 
 
 000 
 
 oo<^ 
 000 
 
 00 
 
 OoasA 
 o*.»- 
 
 OoO 
 
 000 
 
 KOfc 
 
 000 
 ooo 
 
 to BoK 
 oO 
 
 Ooo 
 
 OQO 
 
 000 
 
 000 
 
 OOO 
 
 -•-0-* 
 
 00 
 
 Q 00 
 
 ooo 
 
 000 
 
 oo 
 
 000 
 
 0>flN 
 
 ^o«n 
 
 000 
 
 000 
 
 «4fr.fc 
 
 o»-* 
 
 ow<^ 
 
 OQO 
 OOO 
 
 on« 
 
 fc f^ f^ 
 
 000 
 000 
 
 ::5^ 
 
 000 
 Ooo 
 
 NCVN 
 
 OQO 
 
 000 
 
 
 u. 
 
 OQO 
 OOO 
 
 ooo 
 
 ooo 
 
 o>,^ 
 
 Oq 
 OoO 
 
 000 
 000 
 
 000 
 
 0^0' 
 000 
 
 OQO 
 
 PJOO 
 
 000 
 00 
 
 ^ob 
 
 OOfr- 
 
 oo*( 
 000 
 
 OQO 
 
 ooo 
 
 OOO 
 
 •^ rtc4 
 Ooo 
 
 000 
 
 ^Os» 
 
 000 
 000 
 
 000 
 000 
 
 •^o-J 
 
 000 
 
 000 
 
 QOO 
 
 000 
 
 ;t9 
 
 000 
 000 
 
 NOW 
 — .«. 
 
 000 
 000 
 
 o-ro 
 
 000 
 00 
 
 000 
 
 Ooo 
 
 J-O 
 OOO 
 
 00 
 
 
 
 
 t 
 o 
 o 
 
 D 
 
 c 
 r 
 
 c 
 
 1- 
 
 «*UJ 
 CO UI 
 
 U. 
 
 ion 
 Ooo 
 
 ooo 
 
 ooo 
 
 ooo 
 
 
 
 000 
 
 ofc^ 
 
 OOO 
 000 
 
 000 
 
 •4-wo 
 
 rt oo 
 ooo 
 
 000 
 
 oo*» 
 000 
 00 
 
 000 
 000 
 
 000 
 
 o»*- 
 
 00 
 00 
 
 otearv 
 
 000 
 
 Ooo 
 
 
 
 OQO 
 
 000 
 
 OQO 
 
 -L,o 
 
 000 
 000 
 
 000 
 QOO 
 
 000 
 
 QOO 
 
 rt-Q 
 00*1 
 00 
 
 ooo 
 
 000 
 
 OQO 
 
 000 
 000 
 
 IS*- 
 
 000 
 
 OOo 
 
 
 t- 
 
 i^UJ 
 
 f^ui 
 u. 
 
 523 
 
 ooo 
 
 ooo 
 
 ooo 
 00 
 
 ?4:? 
 
 n on 
 000 
 000 
 
 00 
 00 
 OoO 
 
 oo 
 
 oow 
 000 
 000 
 ooo 
 
 noo 
 
 OoO 
 
 000 
 
 00 
 
 Q 
 
 OOo 
 
 00 
 
 ■*.0oc< 
 
 000 
 000 
 
 OQO 
 
 000 
 
 Oo 
 
 000 
 
 
 OQO 
 
 oo 
 
 090 
 
 000 
 000 
 
 5it< 
 
 onn 
 000 
 00c 
 
 ♦■>0« 
 
 o»^»» 
 
 000 
 
 Ooo 
 
 2St 
 
 Ooo 
 
 Ooo 
 
 o*-o 
 
 (XNOt 
 
 00 
 000 
 
 
 1- 
 
 u. 
 
 loMO 
 
 ooo 
 
 oe o 
 
 O«0P4 
 
 o oo 
 
 O OO 
 
 ?55 
 
 00"» 
 000 
 
 OQO 
 
 oo" 
 
 
 000 
 
 00 
 
 000 
 
 «oo 
 Ooo 
 Ooo 
 
 OQO 
 
 ^-•^o 
 
 « WW 
 
 "no 
 Ooo 
 
 000 
 
 o<^0 
 oOo 
 ooo 
 
 000 
 000 
 
 000 
 000 
 
 teOO 
 Q 00 
 
 00 
 
 000 
 
 OQ 
 
 •oo*. 
 
 000 
 
 QOO 
 
 OQO 
 
 QOO 
 
 ^c-t-o 
 
 ooo 
 00 
 00 
 
 -00 
 
 onn 
 000 
 000 
 
 '''5 si 
 000 
 
 
 
 000 
 00 
 
 
 
 
 
 
 
 
 u. 
 
 OoO 
 
 ooo 
 
 -j-t-o 
 
 O oO 
 
 ooo 
 
 000 
 
 OoO 
 
 O*^ 
 oO 
 OoO 
 
 ■0*0 «> 
 
 000 
 000 
 
 OQO 
 
 •*-*- + 
 000 
 000 
 000 
 
 rj ciO 
 
 
 
 
 nno 
 000 
 ooo 
 
 2tt: 
 
 OOP* 
 000 
 000 
 
 
 oO 
 
 000 
 
 000 
 
 o<^« 
 
 000 
 
 000 
 
 2S^ 
 
 00 
 00 
 
 0-*Q 
 Wrt« 
 
 OQ 
 
 QOO 
 
 *0+0 
 
 noo 
 
 OqO 
 
 oo 
 
 « — -«■ 
 
 i^o 
 "QOO 
 
 
 0*^ c< 
 
 
 
 
 o 
 
 2 
 
 o 
 
 & 
 
 E 
 
 z 
 o 
 o 
 
 z 
 
 UJ 
 UJ 
 
 lU 
 CO 
 
 z 
 
 i3 
 
 UJ r^ 
 
 fS 
 
 Q u. 
 
 -J 
 CO < 
 
 < X 
 
 CO UJ 
 
 i§ 
 ^< 
 
 < o 
 
 CO UJ 
 
 Q :ooo 
 
 
 ^Ul 
 
 u. 
 
 5!i; 
 
 ooo 
 
 ooo 
 
 ^^o 
 ooo 
 
 Ooo 
 
 O OO 
 
 f-OO 
 
 
 000 
 
 TToST 
 
 teO>N 
 000 
 000 
 000 
 
 ■^ — .0 
 
 oon 
 000 
 
 
 
 000 
 000 
 
 000 
 
 000 
 00 
 000 
 
 ♦ 00 
 000 
 000 
 
 000 
 
 -s>Q>CJ 
 
 oon 
 
 OQO 
 000 
 
 OP< w 
 000 
 Ooo 
 
 006* r- 
 
 coo 
 00 
 
 KO+- 
 
 non 
 000 
 
 
 000 
 000 
 
 00° 
 
 oo 
 
 000 
 000 
 000 
 
 oo^ 
 oon 
 000 
 Ooo 
 
 •<o*- 
 000 
 000 
 
 QOO 
 
 OQO 
 
 000 
 
 
 O 
 
 z 
 o 
 
 a 
 
 UJ 
 UJ 
 
 I 
 
 H 
 S 
 
 u. 
 Q 
 
 UI 
 
 > 
 
 IT 
 UJ 
 Q 
 
 UJ 
 IT 
 
 UJ 
 
 O 
 
 z 
 < 
 t- 
 
 i 
 
 < 
 
 o 
 o 
 
 u. 
 
 CO 
 
 UJ 
 
 < 
 
 > 
 
 UJ 
 
 I 
 
 H 
 
 u. 
 
 ooo 
 ooo 
 
 nnt* 
 TTT 
 OOO 
 
 Ooo 
 
 VinO 
 
 000 
 000 
 
 Ooo 
 
 Ooo 
 000 
 
 00 
 oo 
 
 000 
 
 on" 
 000 
 oO 
 
 W o<r- 
 OoO 
 
 000 
 
 on 
 
 OoO 
 
 000 
 
 •t-ow 
 00 
 000 
 000 
 
 t^ on 
 
 OoO 
 oO 
 
 Oq^O) 
 
 n«cx 
 
 Ooo 
 ooo 
 
 o«w 
 
 00 
 
 000 
 ooo 
 
 t&5 
 
 oon 
 
 qC 
 
 000 
 
 »-0o0o 
 
 QOO 
 OOO 
 
 01«J 
 
 000 
 000 
 
 «M»-0 
 
 001 
 oo 
 
 OQO 
 
 f^6 
 oww 
 no 
 000 
 
 OQO 
 
 ooo 
 
 OOQ 
 
 
 H 
 
 CmUJ 
 
 "uj 
 u. 
 
 -«o 
 o*-o- 
 
 ooo 
 
 ooo 
 
 OOO 
 
 ooo 
 
 ■»-■♦- 1- 
 
 000 
 
 Ooo 
 
 OoO 
 OoO 
 
 ■*-00 
 
 ■'■^-^ 
 
 Ooo 
 Ooo 
 
 ow« 
 000 
 
 OQ 
 
 000 
 00 
 
 (non 
 00 
 000 
 
 000 
 000 
 00 
 
 0^- « 
 
 ooo 
 
 000 
 
 000 
 
 oon 
 00 
 00 
 
 noo 
 
 QOO 
 
 000 
 
 9^- + 
 
 o(^« 
 ono 
 000 
 000 
 
 p,o« 
 oO 
 OoO 
 
 or~->« 
 oNo 
 
 Tt-t- 
 
 OOO 
 
 ooo 
 
 ?5^ 
 nno 
 000 
 
 ooo 
 
 ooo 
 000 
 000 
 
 nO« 
 000 
 OOO 
 
 
 00 
 
 ooo 
 Ooo 
 
 ooo 
 
 Ooo 
 
 So.^ 
 »• 
 
 000 
 000 
 
 te 00 
 
 000 
 
 000 
 
 r + r 
 000 
 000 
 
 00 
 
 Q 00 
 
 ^o^ 
 ■r-f-o 
 
 Ooo 
 Ooo 
 
 000 
 
 000 
 
 o»-* 
 000 
 
 OQ 
 
 P,oo 
 oo 
 
 000 
 
 n 
 000 
 000 
 
 
 -0 
 
 ooo 
 ooo 
 000 
 
 o«« 
 MO 
 OoO 
 
 00 
 
 ^0^ 
 
 -««o 
 
 000 
 
 000 
 
 wOtt- 
 
 000 
 000 
 
 -jno 
 00 
 
 000 
 
 rote 
 4'oo 
 oon 
 000 
 000 
 
 
 CJ 
 
 OootO 
 
 ooo 
 O OO 
 
 I960 
 
 ooo* 
 
 000 
 000 
 
 17 to 10 
 000 
 OQO 
 
 
 
 OQO 
 
 >d«0 
 ooo 
 
 OQ 
 
 N — 
 
 000 
 000 
 
 00- 09 
 *9 ■*-■«- 
 000 
 000 
 
 00 
 00 
 
 OQO 
 OOO 
 
 O«^0b 
 ■*-00 
 
 000 
 000 
 
 ^^'^ 
 
 Ono 
 000 
 000 
 
 00 g 
 
 ooS 
 
 "•-on- 
 
 OoO 
 
 000 
 000 
 
 sow* 
 
 « 00 
 000 
 
 OQO 
 
 Ototo 
 
 M ^ U 
 
 ooo 
 000 
 
 <L,0 
 
 
 
 000 
 
 000 
 OQ 
 
 J25 
 
 nno 
 000 
 000 
 
 
 & 
 
 000 
 OOO 
 
 t^oo 
 
 Q 0«- 
 
 000 
 000 
 
 sOvSNO 
 000 
 000 
 
 Oo»< 
 
 
 000 
 
 OOO 
 
 000 
 
 1^0. b. 
 
 000 
 00 
 
 oofe 
 
 OQO 
 000 
 
 ort- 
 
 oO 
 oo 
 
 000 
 
 000 
 
 L,0« 
 
 00 
 
 Q 
 
 2SS 
 
 000 
 000 
 
 000 
 
 000 
 
 w -» 
 
 00 
 000 
 
 OtoiC 
 
 <-oo 
 Ooo 
 
 000 
 
 o«.i^ 
 000 
 
 OQO 
 
 *X00> 
 00 
 
 ooo 
 
 000 
 000 
 
 ^roTT 
 oc«o 
 
 000 
 000 
 
 
 «> 
 
 OOO 
 
 ooo 
 
 000 
 000 
 
 IS?:;: 
 
 OoO 
 
 Ooo 
 
 1^ oO 
 
 000 
 000 
 
 oo 
 
 OO 
 
 00 
 
 000 
 
 S5S 
 
 000 
 
 000 
 
 oO 
 
 00 
 
 ^«<o 
 
 00 
 000 
 
 6C^€*« s« 
 *J h. <0 
 
 000 
 OQO 
 
 000 
 
 000 
 
 5S? 
 
 000 
 
 000 
 
 000 
 
 000" 
 
 r-.aO 
 CM-Zo 
 
 000 
 000 
 
 ooo 
 000 
 
 000 
 000 
 
 000 
 000 
 
 000 
 000 
 
 
 u> 
 
 WW o 
 
 Ooo 
 Coo 
 
 Oo 
 0.00 
 
 WCDO ;eo 00 
 oj 0*- r- K*« 
 
 1 oO 
 
 0000 00 
 
 4o« 
 000 
 
 OOO 
 
 000 
 000 
 
 Ooo 
 Ooo 
 
 %«'«>0 
 
 000 
 
 000 
 
 1^- *00 
 
 Ooo 
 000 
 
 000 
 Q 
 
 'O'^O 
 
 000 
 000 
 
 ■oo? 
 
 000 
 000 
 
 K<^ 
 
 OoO 
 OoO 
 
 Ooo 
 
 000 
 
 f^ O^l 
 
 Q Q 
 
 00 
 
 5Si 
 
 000 
 
 000 
 
 000 
 000 
 
 o-^o 
 
 000 
 OQO 
 
 
 V 
 
 -OO 
 
 Ooo 
 
 •■•■ 
 -00 
 000 
 
 000 
 000 
 
 gate Ob 
 000 
 Ooo 
 
 wot- 
 
 NO 
 
 K«ote 
 
 000 
 
 00 
 
 OQO 
 000 
 
 ♦ Wo 
 
 00 
 
 000 
 
 tooH» 
 
 «■*■»■ 
 
 00 
 oc 
 
 355 
 
 000 
 000 
 
 000 
 
 ooo 
 
 000 
 000 
 
 <i 
 000 
 
 000 
 000 
 
 000 
 000 
 
 5 St 
 
 000 
 
 00 
 
 0DOr< 
 
 ooo 
 000 
 
 4.L,o nl,i' 
 «»>»'^ .f^ o«« 
 
 OoO '000 
 000 1 oOo 
 
 
 
 
 
 - Q 
 
 f «HO 
 
 000 
 
 000 
 000 
 
 fr-fr.00 
 
 oonv 
 
 60 lo*^ 
 
 000 
 000 
 
 J^r* 
 o^n 
 
 000 
 ooo 
 
 ^•0 
 
 M lO 'fl 
 000 
 000 
 
 owo 
 
 OQO 
 OQO 
 
 5S!^ 
 
 00 
 
 QOO 
 
 oOo 
 
 000 
 
 ool, 
 "to* 
 
 00 
 
 00 
 
 O-fr.* 
 
 Q qO 
 
 oo 
 
 000 
 000 
 
 ^S kuri 
 4«0 |ts2rt 
 
 000 10 00 
 00 000 
 
 
 ooo 
 
 000 
 
 000 
 
 000 
 
 1 00 
 
 
 sOI 
 
 1 — 
 
 OQO 
 
 0*-«» 
 
 
 OOvJ 
 
 nun 
 
 «< ^^ Ooo 
 
 t-on 
 
 000 
 1^0 
 
 000 
 000 
 
 000 
 000 
 
 OOo 
 
 Q 00 
 
 Oor-.O 
 
 
 000 
 
 s;5 
 
 000 
 
 ooo 
 
 000 
 000 
 
 c<-.0 
 
 000 
 
 nt^n rv»)0 
 
 000 QOO 
 000 00 
 
 OQO 
 
 00 Ci 
 
 
 000 
 
 000 
 
 000 
 
 000 
 
 000-000 
 
 
 
 
 
 1 
 
 «0 V) 
 
 no«" 
 •*-Oct 
 000 
 
 "^ DO 
 
 OQO 
 
 00 
 
 fit*--. 
 000 
 
 -•0^ 
 •^00 
 ■-00 
 000 
 
 OOo 
 
 ooo 
 
 OH 
 
 «<~.o 
 
 000 
 
 000 
 
 Q 
 
 ••t- 
 K r^ ^ 
 
 
 
 000 
 
 ^.o 
 
 000 
 
 000 
 
 -- -- Q 
 OQO 
 
 •SO)* 
 
 000 
 000 
 
 
 — |(N 
 
 
 
 
 
 
 
 
 <; 
 
 o^o 
 000 
 
 0^^ 
 
 ON- 
 
 000 
 
 «oo 
 000 
 
 O»o« 
 
 000 
 
 N-- 
 
 000 
 
 
 
 OtOO 
 000 
 
 000 
 
 
 q: 
 O 
 
 \- 
 o 
 
 D 
 O 
 
 z 
 o 
 o 
 
 
 < * 
 
 UJ z : 
 
 < a 
 c 
 
 ) 
 
 o« 
 
 ooo 
 000 
 OO 
 000 
 
 0»K 
 
 000 
 OQO 
 000 
 
 000 
 000 
 
 Ooo 
 ooo 
 000 
 
 OQO 
 
 000 
 
 N-^O 
 
 oOo 
 oOo 
 000 
 000 
 
 •oo'o 
 
 000 
 000 
 000 
 
 o'lo 
 
 ooo 
 ooo 
 000 
 000 
 
 000 
 
 00 
 
 00 
 000 
 0*0O 
 
 000 
 000 
 000 
 000 
 40'^ 
 
 o<y*- 
 
 Kf^ 
 s»KO 
 
 >o>»o 
 
 Oto>« 
 
 
 P^O^ 
 SK 
 -.,?sO 
 
 000 >« 
 
 r<oto 
 
 ^»K0 
 
 «-o 
 
 oO 
 ooc< 
 0'^ 
 
 000 
 
 000 
 
 i^o 
 
 000 
 
 onr 
 
 OQO 
 NO* 
 O-^N 
 '•^^ 
 ^*3t 
 
 
 ON s ^ a 
 
 
 
 
 
 
 
 
 
 
 0-fy 
 
 ni-S 
 
 |l« 
 
 O^t^ 
 
 oi-'o 
 
 
 
 
 
 
 S3H0NI Nl 
 
 3?? 
 
 
 >.o.o 
 
 ^00 
 
 •ifr--* 
 
 
 
 
 6woto 
 
 No»- 
 
 On« 
 
 Or<-« 
 
 •«o 
 
 o 0^ 
 t-t-o 
 
 0«P( 
 
 o-*c^ 
 noto 
 
 «oOT 
 
 
 f no 
 
 
 
 3dAJ. 
 
 aaoNVius 
 
 anos 1 
 
 a30HOdNI3a"I33XS 
 
 
 Tviaaivw 
 
 a3ddOD 
 
 lAinNIIAimV 
 
 
 F 
 
 k 
 
 4^ 
 
 5. 
 
 hi 
 
 s a 
 
 &. ^ 
 
REACTANCE— CAPACITANCE— CHARGING CURRENT 
 
 17 
 
 ll. 
 
 o 
 
 oc 
 
 u 
 
 0. 
 
 -J , 
 
 < 
 
 DC 
 
 h 
 
 D 
 
 U 
 
 z 
 
 o£ 
 
 Hp 
 
 h 
 
 LJ O 
 
 Od 
 
 zo 
 
 ?z 
 
 S8 
 
 O LJ 
 
 CO oc 
 
 D < 
 
 in m 
 
 >- 
 h 
 
 o 
 
 < 
 
 Q. 
 
 < 
 O 
 
 111 
 
 o 
 o 
 
 CM 
 
 U 
 
 -I 
 CD 
 
 < 
 
 U 
 
 o 
 
 z 
 
 
 oc 
 u. 
 
 X 
 
 CO 
 
 cc 
 
 o 
 
 1- 
 o 
 z> 
 o 
 z 
 o 
 o 
 
 u. 
 
 o 
 
 en 
 
 a: 
 
 UJ 
 
 1- 
 z 
 
 UJ 
 
 O 
 
 z 
 
 UJ 
 UJ 
 
 ? 
 
 (- 
 
 UJ 
 OD 
 
 5 
 
 UJ 
 
 O 
 
 z 
 
 4 
 t- 
 CO 
 
 o 
 
 1 
 
 1 
 
 1- 
 
 lOUJ 
 
 am 
 u. 
 
 
 
 rtHH 
 
 
 
 »<«••< 
 
 
 ooo 
 
 
 
 
 
 2U 
 
 
 
 
 0» 
 
 
 
 EACH CONDUCTOR OF A SINGLE-PHASE OR OF A SYMMETRICAL THREE-PHASE LINE-THE SUSCEPTANCE VALUES WERE DERIVED FRO 
 
 E CHARGING CURRENT in amperes PER MILE of single conductor to neutral - the (Susceptanoe 
 
 TRAU X \0~^ J THE SUSCEPTANCE BETWEEN CONDUCTORS EQUALS ONE HALF THE TABLE VALUES 
 
 COUJ 
 
 tNUI 
 
 U. 
 
 ♦ **« 
 
 
 
 
 rtrtH 
 
 
 
 ooo 
 
 1 •*-' 
 
 
 
 
 t:c 
 
 s^ 
 
 
 ooo 
 
 
 
 1- 
 ~ UJ 
 W Ui 
 
 u. 
 
 
 
 
 
 
 
 
 
 oo»- 
 
 ♦-»<•- 
 V'*'** 
 
 
 •1 "^-^ 
 
 ^1^* 
 
 ••%•• 
 
 
 
 *oo 
 
 • •■•I 
 
 iSl 
 
 u. 
 
 
 •* + - 
 ♦ ♦ t 
 
 
 
 
 
 
 
 qQ 
 
 
 
 
 
 ♦«♦■ 
 
 
 tt3 
 
 
 tr* 
 
 b. 
 
 
 
 ttrt 
 
 
 
 
 
 
 -.00 
 
 0«-» 
 
 
 
 
 ^1-»« 
 
 •>«»•• 
 
 Ss5 
 1 «it 
 
 
 
 jtt 
 
 1- 
 
 -UJ 
 
 u. 
 
 r*^'^ 
 
 
 
 
 
 
 
 
 
 oo». 
 1««l - 
 
 
 N. 
 
 
 
 in" 
 
 •.•.X 
 
 
 000 
 
 
 u 
 
 
 
 ^ t-o 
 
 
 
 
 4*N 
 
 
 
 000 
 
 ^♦•n 
 %-•'*■ 
 
 
 oo*. 
 
 -. - ^ 
 
 
 •in*! 
 
 
 o..» 
 
 1-- 
 
 1- 
 
 -UJ 
 - UJ 
 
 u. 
 
 
 
 
 rsWr 
 
 «-HO 
 
 
 
 
 
 t:3 
 
 
 
 000 
 
 u: 
 
 
 •IOC 
 
 •tut 
 
 000 
 
 1- 
 
 u. 
 
 
 
 r40O 
 
 r-r- J 
 
 
 
 
 
 
 
 
 ooO 
 
 • to 
 
 
 oo*- 
 
 
 ol<i«< 
 
 
 
 1- 
 
 <X>UJ 
 
 u. 
 
 
 HtotO 
 
 
 
 
 
 
 O^W 
 
 
 
 -00 
 
 
 
 000 
 
 »*•*« 
 <<>» 
 •(«l«1 
 
 ♦ 1-n 
 
 n«rt 
 •>(«f •« 
 
 •HO 
 
 1- 
 
 u. 
 
 
 
 
 0«is> 
 
 
 otor^ 
 
 rt 0»- 
 
 
 "in 
 
 oV^o 
 
 
 
 
 
 
 
 
 
 u. 
 
 5S3 
 
 ooo 
 
 
 
 n« o 
 
 00 -*r) 
 
 
 
 
 Si--* 
 
 mrtrt 
 
 
 r» f* "^ 
 
 
 
 
 
 
 •iWO 
 
 -B 
 
 u. 
 
 
 
 -O o 
 
 0*)-« 
 
 o»^» 
 
 
 
 Co MS 
 
 
 
 fmrt 
 
 
 
 
 
 
 
 0^0 
 
 
 1- 
 
 a 
 
 
 in*) 
 
 "IT) 
 
 
 of*-* 
 ^ oo 
 
 •no 
 
 nor* 
 o» 
 
 
 t-OVl 
 
 
 
 
 
 IS, *rs 
 
 
 ooi. 
 
 
 •XOCWt 
 
 
 t- 
 u. 
 
 
 
 
 NT'' 
 
 
 
 Too 
 
 K'>*0 
 
 o»-» 
 
 
 
 
 
 o*«- 
 
 
 fo.» 
 
 •(NO 
 
 
 
 
 1- 
 u. 
 
 t,<»JI 
 
 -oo 
 
 
 nTo 
 
 
 
 
 t-ot 
 
 o+N 
 
 ^ o» 
 
 OBOtN 
 
 
 
 
 
 
 WvO 
 
 
 2f- 
 •tt** 
 
 s 
 
 
 
 OL9«0 
 
 oo5 
 
 tt") 
 
 •TT) 
 
 no*» 
 
 oaOoN 
 
 
 
 
 teOW 
 
 oo«- 
 
 
 <<« -.0 
 
 
 
 urn'* 
 
 ♦ «*< 
 
 •inn 
 
 no* 
 
 ^ 
 
 Wo- 
 rt -o 
 
 o».te 
 
 •or* 
 
 
 
 
 
 
 
 
 
 
 
 
 o«.to 
 
 nno 
 tit 
 
 
 i.t" 
 
 ■•kh 
 
 CO 
 
 
 L»o« 
 
 
 
 
 1 1* 
 
 
 
 
 *«p 
 
 ON" 
 
 
 5?? 
 
 o«^ 
 
 ttt 
 
 ^o^ 
 
 nt<o 
 
 125 
 
 «7l-n 
 
 •inn 
 
 b 
 
 
 
 
 »-o»o 
 
 'O'O'O 
 
 
 o« 
 
 
 OKI 
 
 *tt 
 
 
 
 
 
 
 ot-l. 
 
 ■•♦0 
 
 oro.44 
 
 tt-) 
 
 »nW 
 nnn 
 
 ^ 
 
 235 
 
 f. r- N 
 
 
 
 
 « - 
 
 
 
 no*. 
 
 •o Mi- 
 
 
 
 
 
 ot^fti 
 
 '♦•in 
 
 •ill 
 
 oto 
 
 Ofct 
 
 2s: 
 
 t-tt 
 
 rvn-" 
 o»«. 
 
 tnn 
 
 ?» 
 
 oo»« 
 
 + «o 
 
 wooto 
 
 
 TOM© 
 W-^O 
 
 oo»^ 
 
 K«0 
 
 Od'^'4 
 
 ?ofc 
 
 osis 
 
 
 MOt- 
 
 2?: 
 
 
 rtNo 
 
 
 
 ♦ 0»i 
 
 4no 
 
 s;5 
 ttt 
 
 t-tr 
 
 
 
 
 
 
 MICROMHOS PER MILE OF 
 
 EQUATION b-27rFC TH 
 
 TABLE) X (VOLTS TO NEU 
 
 ■" 
 
 
 «oo^ 
 
 Wno 
 
 r< o »*■ 
 
 
 00 kn' 
 
 
 nO 
 
 
 
 
 
 + <^o 
 
 
 0-0 
 
 
 
 t^»» 
 
 ^•ot 
 
 s^t 
 
 t»t 
 
 Ot 
 
 
 w*-'- 
 
 -^^ 
 
 
 
 vOO 
 
 t,OW 
 
 
 NO«^ 
 ON" 
 
 OoNN 
 
 
 
 •O + t 
 
 5tr 
 
 or>*t- 
 
 t-Tt 
 
 Bist 
 
 Ot-«i 
 
 
 4«« 
 
 n^r. 
 HNO 
 
 - 
 
 
 
 
 
 w 
 ^ 
 
 »of ♦ 
 
 ^.. 
 
 ^t:: 
 
 O'oaH 
 
 
 
 
 
 ton*- 
 
 
 
 St** 
 
 
 n«^ 
 
 — KN 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^(io 
 
 
 
 
 
 5=; 
 
 10 
 QC 
 O 
 
 o 
 
 O 
 
 z 
 o 
 o 
 
 gz^3 
 o 
 
 000 
 Og 
 OO 
 000 
 
 ooo 
 
 0»N 
 
 oo 
 
 000 
 00 
 
 ooo 
 Opo 
 
 00 
 
 ooo 
 ooo 
 
 ooo 
 ooo 
 «<^o 
 
 O 00 
 OOO 
 
 ooo 
 
 000 
 
 »oo»o 
 
 000 
 
 000 
 
 
 0»00 
 
 OOO 
 
 000 
 
 
 
 000 
 v»o*o 
 
 QOO 
 000 
 
 000 
 
 gOO 
 
 pho 
 
 ooo 
 
 000 
 000 
 000 
 
 •JON 
 
 0«^^ 
 
 Kf^ 
 ^KO 
 
 1-* 
 
 oto>« 
 
 
 ?f%- 
 
 0T*> 
 
 
 000 
 ON 
 0» 
 
 on 
 
 000 
 
 Oh 
 
 000 
 o.nt 
 
 000 
 
 2?; 
 
 
 
 
 ON s V a 
 
 
 
 
 
 
 
 
 
 
 o-*»x 
 
 nt^ 
 
 • a 
 
 0^C( 
 
 1+4 
 
 
 
 
 
 S3H0NI Nl 
 
 a3X3wvia 
 
 5|l 
 
 
 5^3 
 
 ■< oo 
 
 35J 
 
 
 
 ^oS 
 
 en fv- 
 
 
 o.^.. 
 '**)« 
 «•<« 
 
 
 
 woto 
 
 
 
 
 
 3dAJ. 
 
 oaaNvaxs 
 
 anos 
 
 
 a30aOdNI3ai33XS 
 
 nviaiLvw 
 
 U3dd03 
 
 
 i/\inNii/\imv 
 
 
 5^ 
 
 
 -a 
 
i8 
 
 REACTANCE— CAPACITANCE— CHARGING CURRENT 
 
 Li. 
 O 
 
 UJ 
 
 LJ 
 Q. 
 
 -J 
 < 
 
 h 
 D 
 UJ 
 
 h 
 
 o 
 
 z 
 < 
 h 
 
 Q. 
 UJ 
 
 D 
 Q 
 
 Z 
 
 o 
 o 
 
 (/) 
 
 D 
 0) 
 
 >- 
 
 h 
 
 o 
 
 < 
 
 < 
 O 
 
 UJ 
 
 -J 
 
 o 
 
 >- 
 
 o 
 o 
 
 (0 
 
 UJ 
 
 -J 
 
 m 
 
 < 
 
 GC 
 < 
 ffl 
 
 UJ 
 
 -I 
 
 o 
 
 z 
 
 (0 
 
 UJ 
 
 I 
 
 Z 
 
 o 
 
 cr 
 
 u. 
 
 O 
 UJ 
 
 > 
 
 UJ 
 
 o 
 
 UJ 
 
 oc 
 
 UJ 
 
 ? 
 
 CO 
 Ul 
 
 t 
 
 UJ 
 
 o 
 
 z 
 < 
 
 a 
 
 UJ 
 
 o 
 CO 
 
 CO 
 
 1 UJ 
 
 I 
 
 (7 
 1 
 
 UJ 
 
 z 
 
 UJ 
 CO 
 
 < 
 I 
 
 Q. 
 
 U 
 UJ 
 CC 
 
 I 
 h- 
 
 § 
 
 a: 
 
 H 
 Ul 
 
 2 
 S 
 
 < 
 u. 
 O 
 q: 
 O 
 >^ 
 
 UJ 
 
 _i 
 C 
 z 
 CO 
 <t 
 u. 
 O 
 a: 
 O 
 H 
 
 o 
 
 Z> 
 Q 
 
 Z 
 
 o 
 o 
 
 I 
 o 
 < 
 
 UJ 
 
 u. 
 O 
 
 UJ 
 
 _i 
 a. 
 
 UJ 
 
 a. 
 « 
 
 O 
 
 i 
 
 o 
 
 c: 
 
 o 
 
 5 
 
 GC 
 li. 
 
 i 
 
 UJ 
 
 O 
 
 CO 
 CO 
 
 UJ 
 
 I 
 
 H 
 1 
 
 < 
 
 Z) 
 Ul 
 
 z 
 o 
 
 CE 
 
 o 
 
 13 
 O 
 
 z 
 
 8 
 
 o 
 
 z 
 
 CO 
 u. 
 
 o 
 UJ 
 
 _l 
 
 DC 
 U 
 Q. 
 
 CO 
 
 UJ 
 
 cc 
 
 Ul 
 Q. 
 
 s 
 < 
 
 2 
 
 1- 
 z 
 
 LJ 
 (£ 
 QC 
 
 D 
 O 
 
 o 
 
 z 
 
 o 
 (r 
 
 < 
 
 o 
 
 LJ 
 
 I 
 
 H 
 
 d 
 
 u. 
 
 n 
 z 
 O 
 h- 
 < 
 
 o 
 
 UJ 
 
 CO 
 
 UJ 
 
 3 
 
 < 
 
 > 
 
 Ul 
 
 _J 
 CD 
 < 
 
 1- 
 UJ 
 
 I 
 1- 
 
 u. 
 
 —1 
 
 < 
 
 I 
 
 Ul 
 
 z 
 O 
 CO 
 
 < 
 
 a 
 
 UJ 
 
 CO 
 
 cr 
 O 
 
 1- 
 
 o 
 
 Q 
 
 Z 
 
 o 
 o 
 
 z 
 
 UJ 
 UJ 
 
 CO 
 UJ 
 
 O 
 
 z 
 
 £ 
 
 UJ 
 
 O 
 CO 
 
 CO 
 
 UJ 
 
 I 
 
 H 
 CD 
 
 'o 
 
 X 
 
 < 
 
 o: 
 
 1- 
 
 UJ 
 
 z 
 
 O 
 
 1- 
 co 
 
 h- 
 _l 
 
 o 
 
 > 
 
 X 
 
 UJ 
 
 -J 
 
 CD 
 < 
 
 X 
 
 CO 
 
 DC 
 
 o 
 
 1- 
 o 
 
 3 
 
 o 
 
 Z 
 
 o 
 o 
 
 u. 
 
 o 
 
 CO 
 DC 
 UJ 
 
 1- 
 z 
 
 UJ 
 
 O 
 
 z 
 
 Ul 
 Ul 
 
 1- 
 
 ut 
 
 S 
 Ul 
 
 O 
 
 z 
 < 
 
 co 
 o 
 
 lOUJ 
 CIUJ ' 
 
 u. 
 
 
 
 T + r* 
 
 »o«o'o 
 
 
 3^ 
 
 OOO 
 
 
 
 5T? 
 
 
 lO"-* 
 
 
 
 
 
 
 OtVPJ 
 
 ■s»rfn 
 
 1- 
 
 eoui 
 
 C<UJ 
 
 u. 
 
 ^-l^l^ 
 
 
 
 
 
 
 oonM 
 -.^ O 
 
 wr-o 
 
 
 •^■r'^ 
 
 
 
 
 
 ocxo 
 
 r-o<^ 
 
 
 
 H 
 
 _UJ 
 
 CNUI 
 
 U. 
 
 1,1,- 
 • <•• 
 
 
 
 
 
 
 
 Loo- 
 
 
 
 
 
 
 0»O«^ 
 
 
 L,Is.O 
 
 oft-ft- 
 
 
 
 1- 
 
 2uJ 
 M. 
 
 
 
 
 
 
 
 »o^ 
 
 
 
 
 
 
 i,r~o 
 
 
 
 -■ Oft. 
 
 ».r<0 
 
 
 — Ul 
 
 u. 
 
 qoo 
 
 
 
 
 
 1,1^1.) 
 
 
 
 ol^o 
 
 
 NO-* 
 
 
 
 i^r" 
 
 
 
 
 
 H 
 
 Suj 
 u. 
 
 
 -o o 
 
 
 
 
 
 
 cof<" 
 
 
 
 N»00 
 
 00- CO 
 
 
 
 
 o--(« 
 
 Nno 
 o»-.ft- 
 •OTV 
 
 
 <2a 
 
 u. 
 
 
 
 — OO 
 
 OQ^«- 
 
 »-<j — 
 aooooo 
 
 - I^fr~ 
 
 
 toc<m 
 
 00 0, do 
 
 
 
 -,0*- 
 
 o^<•^ 
 
 
 
 'o'o'o 
 
 ^o«- 
 
 ft-r--N 
 
 — UJ 
 
 — UJ 
 
 u. 
 
 
 rrr 
 
 
 
 9-VoO 
 o o o 
 
 
 CVObH 
 
 
 
 
 Ool^h- 
 
 wn 
 
 oe-» 
 
 o.o<* 
 
 Oft-N 
 
 teOtiD 
 *oH»n 
 
 OfloO 
 
 'o'o'o 
 
 OftJto 
 
 1- 
 
 o.|;j 
 
 u. 
 
 
 
 
 
 
 
 Oo-O* 
 
 l/jOoO 
 
 
 OOoO 
 
 
 OOoO 
 
 Co o'* 
 ^-.0 
 1,1,., 
 
 
 «~.ft- 
 
 nS'>S'0 
 
 
 •*-n«^ 
 
 -o»- 
 io*ot 
 
 -*UJ 
 
 u. 
 
 too <^ 
 oo<r 
 
 
 «o o 
 
 
 
 nc<c< 
 
 
 
 nNO 
 
 
 So3: 
 
 
 5tS 
 
 
 ort 
 
 ft~r-'3 
 
 l^lOlo 
 
 
 CH--.0 
 
 1- 
 
 K-UJ 
 
 I^UJ 
 
 u. 
 
 
 -.00 
 
 o«oto 
 
 
 
 q*-jo 
 
 ^ + t 
 
 
 ...Oft. 
 
 
 fON O 
 
 N-.0 
 
 nN 
 
 ooo« 
 
 ON" 
 
 
 i,oN 
 o»-» 
 
 OtoO 
 
 
 1- 
 
 COS 
 
 u. 
 
 
 
 
 «0 0-^ 
 ♦-00 6. 
 
 
 
 t)*n 
 
 
 O «or- 
 
 
 i^+*> 
 
 •o^^ 
 
 Co I-.S* 
 
 lo'o'o 
 
 1-1 PI 10 
 v,tn 
 
 
 
 «<tor- 
 « Oft- 
 
 
 ow*- 
 
 
 6e O rt 
 
 NKN 
 
 
 o r^rt 
 
 ooo 
 
 
 ^«so 
 
 
 
 'o'o'o 
 
 o«-o 
 
 Oft-* 
 
 
 
 NO-- 
 
 Ort<to 
 
 ♦ n-- 
 
 Nor' 
 
 Oft-r- 
 
 ON^ 
 
 V)'o^ 
 
 1- 
 u. 
 
 roo« 
 
 lOC< - 
 
 o o»- 
 
 
 o»-.N 
 
 
 *-0« 
 
 
 
 
 O%.0» 
 
 Otoo 
 
 -00 L, 
 0^- 
 
 
 •0',', 
 
 rt*-n 
 »)(^ft> 
 
 
 M-O 
 
 'o'o'o 
 
 "a 
 
 u. 
 
 
 
 10 s*^ 
 
 O*0O 
 -;00 
 
 toooto 
 
 nSn 
 
 
 
 
 
 
 oOor- 
 
 
 
 
 
 ^1^^ 
 
 K**^ 
 
 I^OOd 
 
 ^*on 
 
 ^Jo^. 
 
 
 1- 
 
 u. 
 
 
 
 
 
 
 
 P(-«0 
 
 
 
 
 
 W Oft- 
 
 
 iOo«'« 
 
 noto 
 
 OoOoto 
 
 
 o*-f' 
 nOQB 
 
 K«OT 
 
 CO 
 
 
 QOO 
 
 
 
 
 
 
 
 
 nixo 
 
 
 
 
 
 5:!!? 
 
 «-<*;.•• 
 
 
 so, tJ 
 
 4Krl 
 
 CM 
 
 
 
 ^+- 
 
 
 
 bo« 
 
 
 
 
 0«'« 
 
 OaSS 
 
 00*0 
 
 •^rt 
 OofcCO 
 
 
 
 OOft- 
 
 >^*»o 
 ft^QoOa 
 
 hrto 
 oJtoOij 
 
 •)0«J 
 
 «i-«n 
 
 C3 
 
 
 
 
 
 
 •-kV 
 
 — ■i « 
 
 0«a 
 
 
 
 
 1^ 
 
 
 
 ONO 
 
 Oft. 
 
 
 toh-o 
 dQ«o4i> 
 
 CO 
 
 -.■* "♦• 
 
 
 4t^ 
 
 
 -inn 
 
 HO<» 
 
 
 Ool,-. 
 
 (b-i-o 
 
 Oo*J 
 
 
 o-«o 
 
 rtO.1, 
 
 
 o»»n 
 
 Qatote 
 
 
 < ■- 
 
 
 ^«.o 
 
 ft^floflg 
 
 "'. 
 
 o n o 
 
 
 
 
 
 t^OOO 
 
 1-rt*: 
 
 nnn 
 
 
 
 Oil- ft- 
 
 *.ObOB 
 
 < 
 
 "S2 
 
 
 •t-00 
 
 
 ft--*--^ 
 tab 
 
 
 ^ 
 
 «-:o 
 
 rttofO 
 
 
 
 ^"■^ 
 
 S4^ 
 
 
 
 
 
 
 .^< 
 
 
 Ot< 
 
 ft->«r* 
 
 ^4T 
 
 nflot* 
 nrt«i 
 
 -.-.0 
 
 b bft- 
 
 o 
 
 
 
 C( -o 
 
 
 ^^~ 
 »;*•* 
 
 rt*-"^ 
 
 o«»^ 
 
 ,.0* 
 
 ».nr. 
 
 -~. >»rt 
 
 00b 
 
 j-.^ 
 
 -•-,1^ 
 
 
 
 
 oV»». 
 
 n<<:^ 
 
 
 CJ 
 
 Vj T t- 
 
 
 "inn 
 
 
 ..-- 
 ^•^I- 
 ^»ii 
 
 
 IxT — ; 
 -. OO 
 
 
 00 to — 
 
 
 
 
 ♦ «-+ 
 
 -l^O 
 
 ♦•nd 
 
 ft.K.a. 
 
 
 nts-W 
 
 ■- 
 
 
 
 
 
 o 
 
 ooir- 
 
 b f 6 
 
 nn»< 
 
 on« 
 
 b %-f- 
 ft "^-^ 
 
 
 
 
 
 00^ 
 o^-'r 
 
 ft-r-r- 
 n*x»y 
 
 
 
 — lOJ 
 
 
 
 
 
 
 
 
 
 
 
 
 "4 + 
 
 ctwo 
 
 »4* 
 
 
 o)-^ b 
 
 
 
 n»1- 
 ••Tn 
 
 nt^t 
 
 cc 
 o 
 
 O 
 
 D 
 Q 
 
 Z 
 
 o 
 o 
 
 
 ir 
 
 < 3 
 
 < c: 
 
 o 
 
 CO 
 
 -J 
 
 5 
 
 OO 
 00 
 00 
 OO 
 00 
 0»N 
 
 000 
 
 00 o 
 00 
 
 ooo 
 
 oO 
 
 oOo 
 ooo 
 ooo 
 
 00 
 rc^-o 
 
 OOO 
 O OO 
 OOO 
 00 
 ■00*9 
 
 000 
 
 ooo 
 
 qOO 
 OOo 
 ©•oo 
 
 OoNN 
 
 ooo 
 
 ooo 
 
 000 
 
 ooo 
 tooyj 
 
 ooo 
 ooo 
 ooo 
 ooo 
 oi^o 
 
 ooo 
 ooo 
 ooo 
 ooO 
 
 ^#^^0 
 
 -sn 
 
 01-60 
 
 DCON» 
 
 MKO 
 
 ONty- 
 
 NN 
 N«NO 
 
 
 ♦ »*• 
 MNO 
 
 000 
 OON 
 Pt- 
 ^ 0^ 
 on 
 
 00 
 Q i^o 
 too.* 
 
 000 
 
 «<^n 
 nooD 
 
 000 
 
 ON s ? a 
 
 
 
 
 
 
 
 
 
 8oo 
 
 o ° o 
 
 O-fy 
 
 •>t4 
 
 ogo 
 
 0^0< 
 
 <^i-^ 
 
 
 
 
 
 S3H0N 
 B3X3VN 
 
 Nl 
 
 
 ?^5 
 
 >«OVJ 
 
 c^*;-* 
 
 ^«« 
 
 <-i^ 
 
 S55 
 
 Oft.*. 
 
 
 
 
 
 nN«v 
 
 •*no 
 
 
 
 «o« 
 
 0,»-Ni 
 
 
 ▼ft-'^ 
 n-nn 
 
 •«-o 
 
 n«<r( 
 
 3dAX 
 
 aaoNvais 
 
 anos 
 
 a3oaodNi3ai33iS 
 
 
 ivia3i 
 
 vw . 
 
 iJ3ddOO 
 
 lAinNiiAimv 
 
 
 
 
 
REACTANCE— CAPACITANCE— CHARGING CURRENT 
 
 19 
 
 UJ 
 
 oc 
 
 UJ 
 0) 
 
 h 
 
 D 
 
 o 
 
 cr 
 
 o 
 
 u 
 
 < 
 
 X 
 
 Q. 
 
 I 
 
 UJ 
 
 Ul 
 
 oc 
 
 I 
 h 
 
 oc 
 
 o 
 
 I- 
 o 
 
 o 
 
 z 
 o 
 o 
 
 UJ 
 
 oc 
 
 I 
 
 *=^. UJ 
 
 > oc 
 
 o 
 oc 
 < 
 
 I 
 
 V 
 
 UJ 
 
 -I 
 OQ 
 
 < 
 
 fe 
 
 ^5 
 
 •* 
 
 I 
 1- 
 co 
 
 < 
 
 -1 
 _l 
 
 UJ 
 
 5 
 
 Q 
 UJ 
 Q 
 
 _J 
 
 
 
 z 
 
 UJ 
 OQ 
 
 1- 
 co 
 
 CO 
 
 1- 
 
 D 
 
 
 ir 
 
 
 I 
 
 z 
 < 
 a: 
 
 OQ 
 
 -1 
 _i 
 < 
 
 
 
 I 
 1- 
 
 a 
 
 z 
 
 UJ 
 
 _i 
 
 UJ 
 
 I 
 1- 
 
 S 
 
 UI 
 
 z 
 
 
 CO 
 
 z 
 
 UJ 
 
 1- 
 I 
 
 I 
 
 UJ 
 
 oc 
 
 z 
 
 UJ 
 Ul 
 
 I 
 
 H 
 
 a 
 
 
 u. 
 
 < 
 > 
 
 
 
 z 
 
 
 oc 
 
 < 
 I 
 
 
 Ul 
 
 I 
 
 i?i 
 
 Z 3 
 
 i £ 
 
 Ul 
 
 uJ Z 
 
 Ul 
 
 I 
 
 H 
 
 w 
 
 LJ 
 
 J 
 
 >- 
 
 
 
 <0 
 
 It 
 
 
 f»clM 
 
 
 
 
 
 --0O 
 
 ;5t 
 
 ?W 
 
 
 
 
 
 
 
 
 
 c« 
 
 floK 
 
 
 OLTAQES AND SPACINGS 
 3 WILL VARY DIRECTLY AS 
 RAD X (VOLTS TO NEUTR 
 
 it 
 
 
 
 
 
 
 
 ?!c 
 
 
 
 »^». 
 
 t;*^^ 
 
 K4*; 
 
 
 OtoN 
 
 t?CtJ 
 
 
 5c5 
 
 3« 
 
 
 
 it 
 
 
 
 5"* 
 
 tt^ 
 
 4W* 
 
 ♦nt* 
 
 ^°s 
 
 oo« 
 
 <JOO 
 
 
 n-^o 
 
 
 »■«-•■ 
 
 WWO 
 
 5?^ 
 
 000 
 
 ton 
 
 OW>* 
 
 
 It 
 
 
 
 
 
 0.00. 
 
 
 
 •titiK 
 
 ON 
 
 Ote«4 
 
 
 NKK 
 
 noo 
 
 
 
 4H4 
 
 
 
 
 St 
 9 
 
 00 0005 
 
 
 o*^ 
 
 
 
 
 
 
 
 NtV) 
 
 
 OOos 
 
 
 5W 
 
 rirJo 
 
 
 
 
 
 > 
 
 UJ 
 
 i 
 
 < 
 
 Iz 
 
 =•0 
 
 5 CO 
 
 a 
 
 
 
 ^0^ 
 
 
 333 
 
 
 
 
 Ho- 
 
 0^0 
 
 0»^«S 
 
 •0+t 
 
 'O'O'O 
 
 
 ♦ ft- 
 
 rtK»> 
 
 2i^ 
 
 
 S^4 
 
 
 
 00 <» 
 
 
 ■♦•"JO 
 
 + ■♦'«■ 
 
 
 
 
 
 OK** 
 
 
 
 
 to-* 
 
 
 
 KWK 
 
 KO«* 
 
 
 
 
 CHARGING CURRENT PER MILE (EXPRESSED IN KVA 3 PHASE) FOR A SYMETRICAL 3 PHASE CIRCUIT AT THE 
 CONDUCTORS STATED. FOR OTHER ARRANGEMENTS OF CONDUCTORS SEE FOOT NOTES. Xf FOR OTHER SPACINGS T 
 SUSCEPTANCEiVALUES OF THE SPACINGS COMPARED. THE CHARGING K.V.A. 3 PHASE - (CHARGING CURRENT IN AMPEF 
 
 it 
 
 
 
 
 If) 
 
 
 
 
 o^»- 
 
 •(•OK 
 
 
 Mo^ 
 
 
 rto'>« 
 
 
 to^♦ 
 
 -« -Q 
 
 
 rt*^t 
 
 ••■NT 
 
 
 it: 
 
 
 
 
 
 
 
 
 v,rto 
 
 
 o».». 
 
 
 
 
 «.<»<< 
 ?*'*' 
 
 
 
 
 •.'•»o 
 
 i 
 
 i^ 
 
 10 <o 
 
 
 ■t-ioo 
 
 to'**- 
 
 OH- 
 
 
 
 
 N^m 
 ^^^ 
 
 
 
 
 J + T 
 
 •On" 
 
 
 
 
 
 
 
 
 
 10 ef- 
 
 *.I^-» 
 
 ^^■^ 
 
 rtr)t< 
 
 < <, 1 
 
 tj«a 
 
 0^0 
 
 1>.K»4 
 
 o-^»- 
 
 
 onN 
 
 
 
 
 ^^0 
 
 OQ^ 
 
 rtr»c< 
 
 
 
 is 
 
 
 
 
 «0 K 
 
 tofc^ 
 
 
 
 6ao rt 
 
 nnf4 
 
 
 n o9- 
 
 
 
 
 
 o«-0b 
 
 
 J.N 
 
 
 o^«. 
 
 i 
 
 i^ 
 
 
 10 N«a 
 
 
 000 
 
 inn 
 
 0.^0 + 
 
 
 ■f ON 
 
 k^»M 
 
 
 
 tol-O 
 
 •» + ♦ 
 
 
 i;^:- 
 
 «^'i*< 
 
 It** 
 
 
 
 
 
 u 
 
 _l 
 
 
 >- 
 
 
 U) 
 CM 
 
 it 
 
 
 
 1060 »o 
 
 ^^^ 
 1^-^ 
 
 v^^ 
 
 
 
 tetods 
 
 (feto*o 
 
 0,4,K 
 
 N»teK 
 
 
 09NK 
 
 nto h- 
 KK.K 
 
 *(>-o 
 
 KKK 
 
 
 
 Sfcr. 
 
 
 
 
 
 
 
 
 
 
 -.(00» 
 
 
 "Toy) 
 
 < 00- 
 
 to«toK 
 
 
 pvto 
 
 40WW 
 
 kN-« 
 
 
 
 
 o«-*» 
 
 
 j' u. 
 
 
 - - a 
 
 
 
 tetok.. 
 
 
 0to«-) 
 
 ■t-tt 
 
 + >♦-« 
 
 
 OO't- 
 
 ^•06. 
 
 
 Oi(->» 
 
 0*-«B 
 
 o>«o 
 
 •iKN 
 
 
 ••"OH 
 
 
 oo-* 
 
 
 ii 
 
 
 N-t 
 
 won 
 
 
 h--* 
 
 ^ no 
 
 
 ■••N+ 
 
 + 
 
 'O'n'o 
 
 "i"jn 
 
 Oo + K 
 00*- 
 
 Nrvt>l 
 
 ».rtfc 
 
 o«^*- 
 
 
 
 
 
 "J -It 
 
 
 St 
 
 r-^ 
 
 to "in 
 
 
 
 '^ "in 
 
 TO"! 
 
 000 
 
 (0(0") 
 
 0(Sfr- 
 
 OsJW 
 
 Kcjfcl 
 
 nD>0»0 
 
 
 
 
 
 
 06«6K 
 
 0100 
 
 SO-* 
 *0>0* 
 
 
 St 
 00 
 
 
 
 no 00 
 
 
 
 
 
 
 
 o^t< 
 
 
 to-t-O 
 
 ■~^ 
 
 oo*< 
 
 
 f "On 
 
 •Mil 
 
 >#OK 
 
 
 it 
 
 
 
 
 it" 
 
 
 00 «0^ 
 
 
 
 
 
 
 
 
 
 
 Ni-o 
 
 
 oi^l- 
 
 
 i£ 
 
 
 
 
 
 
 T>n 
 
 
 
 *^N«<^ 
 
 ^ -N 
 
 
 ~»*M 
 
 oo<a 
 
 obo 
 
 
 -S s* "^^ 
 
 '•Trt 
 
 OK^ 
 <oo 
 
 
 |t 
 
 in--o 
 
 
 
 i-f»n 
 
 060 
 
 
 
 
 n-l- 
 w>« 
 
 Ori«0 
 
 OoWto 
 
 Qtok 
 OoNS 
 
 fco-o 
 
 tono 
 
 r«-.0 
 
 ooW 
 
 OoNVt 
 
 Ott-i, 
 00-%. 
 
 Wo»» 
 
 
 00*^ 
 
 
 it 
 
 
 4o 
 
 
 
 qOO 
 
 
 
 
 Kr>o 
 
 o4.Vo 
 
 
 S5t 
 
 
 
 
 «>r)0 
 
 
 o«o 
 
 
 it 
 
 ftrt - 
 
 -00 
 
 
 
 
 
 o.*^o 
 
 
 ^o«^ 
 
 
 >»o*< 
 
 
 
 
 
 KOo 
 
 -1-11 
 
 
 
 is' 
 
 CO " 
 
 mi 
 
 Tit 
 
 
 
 
 n OK 
 
 
 
 
 
 
 
 
 000 
 
 
 o<*o 
 
 -in.* 
 
 0..W 
 
 
 > , ■ 
 
 St 
 
 
 ♦"♦IT 
 
 
 -••<«-» 
 *<«« 
 
 *Flr< 
 
 sst 
 
 ?^^ 
 
 JKO 
 
 00 
 
 
 
 
 
 
 f* « ■«. 
 
 -*oo 
 
 55; 
 
 «* 
 
 
 ^'~"' 
 
 
 
 
 
 
 
 
 
 
 
 
 
 CO 
 
 tc 
 
 
 f- 
 
 
 D 
 Q 
 Z 
 
 
 
 < 
 
 Ul 2 
 
 a: - 
 
 < 
 
 r * 
 
 
 
 000 
 000 
 00 
 
 QO 
 0»K 
 
 ooo 
 
 000 
 000 
 000 
 000 
 
 000 
 
 000 
 ooo 
 
 00 
 000 
 
 000 
 
 ooo 
 000 
 000 
 
 OOO 
 000 
 000 
 
 00 «) 
 "o 
 
 tosN 
 
 000 
 
 00 
 000 
 
 OQO 
 
 OQO 
 000 
 000 
 000 
 0*00 
 
 000 
 000 
 000 
 
 owv 
 
 K^ 
 
 o*>>« 
 
 
 ONK 
 — ••) 
 
 0+5 
 
 0•>^• 
 
 N#KO 
 •nt"i 
 
 
 000 
 
 oor* 
 DOn 
 
 •♦'on 
 
 OQO 
 Ot<)0 
 
 ont- 
 r>o*» 
 
 J 
 
 
 
 
 
 
 ON s Tj a 
 
 
 
 
 
 
 
 
 
 li^ 
 
 0.., 
 
 I*'! 
 
 
 0-^N 
 
 .ni-^ 
 
 
 
 
 
 
 
 S3H0NI NI 
 
 a3i3wvia 
 
 zH 
 
 
 5S5 
 
 «S3 
 
 
 
 
 tof9K 
 
 -Cob 
 
 ""4 
 
 
 
 U»- to 
 
 Koto 
 
 
 
 
 
 ;:5 
 
 
 3dAJ. 
 
 03aNvai8 
 
 anos 
 
 
 D3oaojNi3a n33is 
 
 
 TViasivw 
 
 U3ddOO 
 
 
 iMnNiiAinnv 
 
 
 Pi 
 
 s 
 
 hi 
 
REACTANCE— CAPACITANCE— CHARGING CURRENT 
 
 Microfarads per icxx) feet of single conductor to neutral, 
 or 
 
 0.03SSJ 
 
 r = 
 
 (i<5) 
 
 logK -jr 
 Microfarads per mile of single conductor to neutral. 
 
 The above formulas are only applicable to ordinary 
 overhead circuits when the distance from the conductor 
 to other conductors, particularly the earth, is large 
 compared to their distance apart. However, since the 
 effect of the earth is usually small in most practical 
 cases, the formulas give a very close approximation to 
 the actual capacitance of overhead circuits. 
 
 The values of capacitance in Table VIII were de- 
 rived by using formula (13). For calculating the 
 capacitance for the stranded conductors, the actual 
 overall diameter of the cable was taken. This intro- 
 duces a small error which is negligible except for very 
 close spacings not used in high tension transmission 
 lines employing bare conductors. 
 
 CHARGING CURRENT 
 
 RELATION OF CHARGING CURRENTS OF SINGLE AND 
 THREE-PHASE SYSTEMS 
 
 The diagrams (Fig. 11) may assist in forming a 
 clear understanding of the relation of charging current 
 
 < <?■ 
 
 0.281 AMPERE 
 CHARGING 
 CURRENT 
 
 50 000 
 VOLTS 
 
 100 000 
 VOLTS 
 
 ■I- - 
 
 50 000 
 
 VOLTS r: 
 
 — 7^ — '^ 
 1 00 000 
 
 VOLTS 
 
 1 00 000 
 VOLTS 
 
 1 00 000 
 VOLTS 
 
 system is 15.5 percent greater than in the single-phase 
 system, and the resulting charging k.v.a. is just double 
 that of the single-phase system. The charge on any 
 particular conductor is in phase with the voltage be- 
 tween that conductor and the neutral and the charging 
 current for that conductor is 90 degrees ahead of the 
 voltage drop from that conductor to neutral. 
 
 Grounding of the neutral point of a system has no 
 effect upon the charging current when the system is in 
 static balance. In determining the total charging cur- 
 rent to be supplied by a given generating station, it 
 should be remembered that in cases of duplicate trans- 
 mission circuits, when both circuits are excited, the 
 charging current will be approximately double what it 
 would be if only one of the circuits were in use. 
 
 Tables IX and X contain values for capacitance 
 susceptance to neutral in micromhos per mile of con- 
 ductor. As indicated, the charging current in amperes 
 per mile of single conductor to neutral = the (suscep- 
 tance from table) X (volts to neutral) X io-*.Thus 
 in a three-phase, 60 cycle, 100 000 volt, (57 740 volts to 
 neutral), symmetrical circuit, the No. 0000 stranded 
 conductors being arranged at the corners of an equi- 
 lateral triangle spaced nine feet apart, the charging cur- 
 rent per mile would be determined as follows: — 
 
 S.62 y. 57 740 X /o-« = 0.3245 
 
 amperes to neutral 
 
 or 0.3245 X 5774" = ^S.737 
 
 k.v.a. to neutral 
 
 ^S.737 X ^ = 5i5.^ K.v.a. 
 
 total three phase 
 
 Table XI is an extens- 
 ion of Tables IX and X 
 from which values 
 
 T 
 
 0.324 AMPEPE CHARGING CURRENT 
 
 z 
 
 0.32A AMPERE CHARGING CURRENT 
 
 in 
 
 SINGLE PHASE CIRCUIT THREE PHASE CIRCUIT 
 
 FIG. II — CHARGING CURRENT IN SINGLE AND THREE-PHASE CIRCUITS 
 
 to susceptance for single and three-phase circuits. In 
 the following consideration No. 0000 stranded copper 
 conductors will be assumed as spaced nine feet between 
 any two conductors, frequency 60 cycles, vol'.age 
 100 000 volts between conductors. Voltage to neutral 
 will therefore be, for single phase circuit, 50000 volts 
 and for three-phase circuit 57 740 volts. Distance of 
 transmission one mile. From Table VIII, a capaci- 
 tance to neutral of 0.00282 microfarads per 1000 feet is 
 obtained which is equivalent to 0.0149 microfarads per 
 conductor to neutral for this one mile of circuit. The 
 susceptance will therefore be as follows: — 
 
 Per conductor to neutral 2 ir i Cn ^ 5.62 microhms 
 Between conductors 2 tt f Qj = 2.81 microhms 
 
 For Single-Phase Circuit— To neutral 5.62 X 5° 
 000 X 10° '^= 0.281 amperes or between conductors 
 2.81 X -too 000 X 10' = 0.281 amperes therefore charg- 
 ing k.v.a. is 0.281 X 50000 X 2 = 28.1 k.v.a. single 
 phase or 0.281 X 100 000 ^ 28.1 k.v.a. single phase. 
 
 For a Three-Phase Circuit — To neutral 5.62 X 57 
 740 X 10* = 0.324 amperes. Therefore charging k.v.a. 
 is 0.324 X 57 740 X 3 = 56.2 k.v.a. three-phase. 
 
 It will be seen from the above that the charging 
 current per conductor in the three-phase symmetrical 
 
 k.v.a., three-phase for 
 charging current have 
 been calculated for certain assumed spacings and aver- 
 age voltages. In the case cited above it was found that 
 the charging current would be 56.2 k.v.a., three-phase 
 per mile. Table XI gives this value directly for the 
 conditions specitied. 
 
 CHARGING CURRENT AT ZERO LOAD 
 
 The term charging current of a transmission circuit 
 refers to the amount of current which flows into the 
 circuit at the supply end with normal voltage held at 
 the receiver end at aero load. If the circuit is long, its 
 capacitance will be high and therefore the voltage at 
 the supply end may be considerably less than at the re- 
 ceiver end. For instance a 60 cycle circuit 300 miles 
 long, having certain constants will, with 100 000 volts 
 maintained at the receiver end, have a voltage of only 
 80000 volts at the supply end at zero load. This same 
 circuit will at full load and lOO 000 volts maintained at 
 the receiver end, require 120000 volts at the supply 
 end. It is evident therefore that, since the charging 
 current varies with the voltage, if the circuit has much 
 capacitance the voltage along the circuit, and particu- 
 larly near the supply end, will vary to a large extent 
 
REACTANCE— CAPACITANCE— CHARGING CURRENT 
 
 31 
 
 and consequently the charging current of the circuit 
 will be different for different loads. 
 
 In case of the 300 mile circuit referred to above, 
 the charging current at zero load will be very much less 
 than it is at full load, because the average voltage at 
 zero load is less than the average voltage at full load. 
 At zero load the average voltage is less and at full load 
 it is greater than the receiver end voltage. 
 
 It is customary to calculate the total charging cur- 
 rent for the circuit by multiplying the total susceptance 
 by the receiver end voltage. This would be correct if 
 the voltage throughout the length of the circuit were 
 held constant and of the same value as at the receiver 
 end. This condition is approximately met within 
 commercial lines and this method of determining the 
 
 70,- 
 
 susceptance by the receiver voltage. For a circuit 300 
 miles long the error in charging current is only two 
 percent for 25 cycles and seven percent for 60 cycle cir- 
 cuits. The error in charging k.v.a. is four percent for 
 25 cycle and 32 percent for 60 cycle circuits. 
 
 RELATION OF INDUCTANCE TO CAPACITANCE 
 
 As conductors are brought closer together, the in- 
 ductance decreases and the capacitance increases. 
 These values change with changes in spacings between 
 conductors in such a mannar that their product L X C" 
 is practically a constant for all spacings (except very 
 close spacings such as used in low-voltage service and 
 lead-covered cables) and for all sizes of conductors. 
 If there were no losses encountered by the electric 
 
 '' propagation in the 
 
 conductors themselves 
 
 the product of L and 
 
 C would be a constant 
 
 55 5 for all spacings and 
 
 i sizes of conductors. 
 
 ="> I In Table C is in- 
 
 o dicated the relation 
 
 g of the total induct- 
 
 *°g ance and capacitance, 
 
 j6 o and their product, 
 
 ~ in two bare par- 
 
 '' 8 allel conductors in air 
 < 
 
 25 a for a circuit one mile 
 
 < 
 
 = long. The values for 
 
 2C y 
 
 ^ 
 
 SUPPLV END 
 
 DISTANCE IN MILES FROM SUPPLY END 
 12 — CHARGING CURRENT AT ZERO LOAD FOR VARIOUS LENGTHS 
 
 At zero load the voltage (on account of the effect of capacitance) decreases as 
 the supply end of the circuit is approached. The charging current at points along 
 the circuit decreases directly as the voltage. If the charging current for zero load is 
 estiinated by the approximate method based upon the receiver voltage being main- 
 tained throughout the length of the circuit the result will be too high. The error will 
 increase as the length of the circuit is increased; it will also increase rapidly as the 
 frequency is raised. The error in the resulting K.V.A. required to charge the circuit 
 will therefore increase very rapidly with an increase in distance or frequency. The 
 curves below represent an approximation of this error. 
 
 total charging current is therefore sufficiently accurate 
 for most practical purposes. 
 
 For the purpose of making exact calculation of the 
 total current at the supply end of long circuits, the 
 charging current must be calculated by mathematical 
 formulas which accurately take into account the change 
 in voltage along the circuit at zero load. This will be 
 taken up in a later article. It may be interesting to 
 note approximately, however, how the charging cur- 
 rent and charging k.v.a., as determined by the above 
 method, varies from what it would be if calculated by 
 the rigorous formula. The curves in Fig. 12 represent 
 an approximation to the error when calculating the 
 charging current at zero load by multiplying the total 
 
 L are in millihenries 
 and for C in micro- 
 farads. Since the 
 formulas by which L 
 and C were calculated 
 account for the flux 
 within the conductors 
 themselves, the prod- 
 uct LC is not a con- 
 stant, as will be seen 
 by the tabulated 
 values, although for 
 the larger spacings 
 such as used in high- 
 
 tension transmission the product is nearly a constant. 
 TABLE C— PRODUCT OF (TOTAL) L AND (TOTAL) C 
 
 Solid Conductors 
 
 a" 
 
 Induct- 
 ance /. 
 F'ormula 
 
 (4) 
 
 C.apac tanc« 
 
 
 Size 
 
 Diam. 
 Inches 
 
 Formula 
 
 Product 
 
 1000 000 
 
 I 000 000 
 
 I 000 000 
 
 0000 
 
 uooo 
 
 0000 
 
 1. 00 
 1. 00 
 1. 00 
 0.46 
 0.46 
 0.46 
 
 2 
 
 24 
 300 
 
 2 
 
 24 
 300 
 
 1053 
 2.653 
 4.279 
 
 1-553 
 3153 
 4-779 
 
 0.0339S 
 0.01 1 55 
 0.00695 
 0.02079 
 0.00961 
 0.00623 
 
 0.03S7S 
 0.03064 
 
 0.02974 
 0.03228 
 0.03030 
 a02977 
 
 RELATION OF INDUCTANCE AND CAPACITANCE TO LIGHT 
 VELOCITY 
 
 The propagation of the electric and the magnetic 
 
22 
 
 REACT A NCE~CA PA CI TA .V CE—CHA RGING C URREN T 
 
 fields in a dielectric, such as air, is the same as that of 
 light. Along a transmission line it is retarded only 
 slightly due to losses or the fact that the current is not 
 confined to the surface of the conductors. If the induc- 
 tance inside the conductors is negligible, then the ve- 
 locity of the electric and the magnetic fields is the same 
 as light, that is approximately i86 ooo miles per second 
 or approximately 3 X io^° cm. per second. For high- 
 tension transmission lines of large spacings, the induc- 
 tance inside the conductor is relatively small, so that the 
 speed of the electric field is practically that of light. 
 
 The following relation exists between inductance L 
 in henries, capacity C in farads and velocity of light V 
 per second: — 
 
 LC (in air) = -jTT.or, F = 
 
 ]' LC 
 
 (17) 
 
 Thus it will be seen that if either L or C is known, 
 the other may be determined since the velocity of light 
 V is known. If values for L and C are taken which in- 
 clude the inductance inside the conductors, particularly 
 if the conductors are very close together, it would be 
 necessary to assume a velocity of electric propagation 
 
 somewhat less than that of light. If, on the other 
 hand, the values for L and C external to the conductors 
 are taken, then the above equation is rigidly correct. 
 
 In Table C, it was shown that for No. 0000 con- 
 ductors, 300 inch spacing, the total values of L and C 
 were for a single-phase line, — 
 
 L = 0.004779 henries per mile of circuit. 
 C = 0.000 000 006 23 farads per mile of cir- 
 cuit. 
 
 therefore, 1 = .. , = 
 
 y o.oo^j/g X 0.000 000 006 2 J 
 
 183 000 miles per second (18) 
 
 which is less than the speed of light. 
 
 If we take the inductance in the air space between 
 the conductors. Formula (2) ; we arrive at the values, — 
 /- = 0.0046179 henries per mile of circuit. 
 C = 0.000 000 006 23 farads per mile of cir- 
 cuit. 
 
 therefore T'' = ~ , 
 
 V ooo,? 6// 9 X 0.000C0000O2J 
 
 186 000 miles per second ( 19) 
 
 which is approximately the spee<.l of light. 
 
CHAPTER III 
 QUICK ESTIMATING TABLES 
 
 FOR every occasion where a complete calculation 
 of a long distance transmission line is made, there 
 are many where the size of wire needed to trans- 
 mit a given amount of power economically is required 
 quickly. This knowledge is, moreover, the basis for 
 all transmission line calculations, as all methods of cal- 
 culating regulation presuppose that the size of wire is 
 known. To determine quickly and with the least pos- 
 sible calculation the approximate size of conductor cor- 
 responding to a given I^R transmission loss for any 
 ordinary voltage or distance, is the function of Tables 
 XII to XXI inclusive. By includmg so many trans- 
 mission voltages it is not intended to indicate that any 
 of them might equally well be selected for a new in- 
 stallation. On the contrary it is very desirable in the 
 consideration of a new installation, to eliminate con- 
 sideration of some of the voltages now in use. This 
 point will be considered later. 
 
 Since both the power-factor of the load, and the 
 charging current of the circuit, as well as any change in 
 the resistance of the conductors, will alter the PR loss, 
 it is evident that it is impractical to present tables which 
 will take into account the effect of all of these variables. 
 The accompanying tables do, however, give the per- 
 centage I^R loss corresponding to the two temperatures 
 (25 and 65 degrees C) ordinarily encountered in prac- 
 tice and the usual load power-factors of unity and 80 
 percent lagging, upon which the k.v.a. values of the 
 tables are based. The effect, however, of charging cur- 
 rent, corona or leakage loss is not taken into account In 
 these table values. The latter two (corona and leak- 
 age) are usually small and need not be considered here. 
 The effect of charging current, may, however, with 
 long circuits be material and will be discussed. 
 
 The values of k.v.a. in these tables are based upon 
 
 the following percentage PR loss in transmission 
 
 (neglecting the effect of charging current) : — 
 
 Percent Percent 
 
 Loss Loss 
 
 At 25°C At 6s°C 
 
 Load at 100 percent P-F. 8.66 lo.o 
 
 Load at 80 percent P-F. 10.8 12.5 
 
 These loss values are based upon the power de- 
 livered at the end of the circuit as 100 percent, and 
 not upon the power at the supply end. If raising or 
 lowering transformers are employed, the loss and vol- 
 tage drop in them will, of course, be in addition to the 
 above. 
 
 At first glance, some of these tables may appear 
 to have been carried to extremes of k.v.a. values for 
 the conductor sizes. This is because the tables are cal- 
 culated for ten percent loss, (at 100 percent power- 
 
 factor and 65 degrees C) whereas tne permissible loss 
 is frequently much less than ten percent. As the loss, 
 is directly proportional to the load, the permissible loads 
 for a given size wire and distance can be read almost 
 directly for any loss. Thus for a two percent loss the 
 permissible k.v.a. will be two-tenths the table values. 
 Conversely, the size of wire to carry a given k.v.a. 
 load at two percent loss will be the same as will carry 
 five (1CK-2) times the k.v.a. at ten percent loss. In 
 other words to find the size of wire to carry a given 
 k.v.a. load at any desired percent loss, find the ratio of 
 the desired PR loss to the PR loss upon which the table 
 values are based (corresponding of course to the tem- 
 perature and the load power- factor). Divide this ratio 
 into the k.v.a. to be transmitted. The result will be the 
 table k.v.a. value corresponding to the desired PR loss. 
 
 For example : — Assume 400 k.v.a. is to be delivered 
 a distance of 14 miles at 6000 volts, three-phase, and 
 80 percent power-factor lagging, at an assumed temper- 
 ature of 25 degrees C. Table XV indicates that this con- 
 dition will be met with an PR loss of 10.8 percent if 
 No. O copper or 167800 circ. mil aluminum conductors 
 are used. 
 
 Now assume that the PR loss should not exceed 
 5.4 percent, in place of 10.8 percent (upon which the 
 table values are based) . 5.4 -;- 10.8 = 0.5 and 400 -i- 0.5 
 = 800 k.v.a. as the table value corresponding to an PR 
 loss of 5.4 percent. The conductors corresponding to 
 800 k.v.a. table value (5.4 percent PR loss) will be 
 seen to be No. 0000 copper or 336420 circ. mil alum- 
 inum. 
 
 If conductors corresponding to 15 percent PR 
 loss are desired the same procedure will be followed : — 
 i5-f-io.8 =1.39 and 400-^-I.39 =287 k.v.a. table value. 
 This table value corresponds to approximately No. i 
 copper or 133 220 circ. mil aluminum conductors. 
 
 The table k.v.a. values have been tabulated for var- 
 ious distances. Should the actual distance be different 
 from the table values and it is desired to obtain k.v.a. 
 values corresponding to the losses upon which the table 
 k.v.a. values have been calculated, the following pro- 
 cedure may be followed :— 
 
 For a given PR loss in a given conductor (effect 
 of charging current neglected) the k.v.a. X feet or the 
 k.v.a. X miles is a constant. Thus Table XII indicates 
 that for 2 000 000 circ. mil cable, 756 coo k.v.a. X f^^t 
 is the constant; that is 756 k.v.a. may be transmitted 
 1000 feet ; 378 k.v.a., 2000 feet, and so on. If the actu;'.l 
 distance to be transmitted is 1300 feet the correspond- 
 ing k.v.a. value will be 756000-7-1300 or 581 k.v.a. 
 Usually the k.v.a. value can readily be approximated 
 
24 
 
 QUICK ESTIMATING TABLES 
 
 for any distance with sufficient accuracy for the pur- 
 pose for which these quick estimating tables are pre- 
 
 sented. One way of dong this would be as follows: — 
 The k.v.a. value corresponding to 2500 ft. is 302 k.v.a. 
 
 TABLE XII-QUICK ESTIMATING TABLE 
 
 CONDUCTORS 
 
 KILOVOLT- AMPERES. 3 PHASE. WHICH MAY BE DELIVERED AT THE FOLLOWING VOLTAGES OVER THE VARIOUS 
 CONDUCTORS FOR THE DISTANCES STATED. BASED UPON THE FOLLOWING |2r LOSS (EFFECT OF CHARGING CURRENT 
 NEGLECTED) ^T^g.^ ^^^^.^ 
 FOR LOAD POWER-FACTOR OF IOO%-8.66% LOSS- 10 0% LOSS 
 
 d 
 
 z 
 
 a> 
 «( 
 m 
 
 COPPER 
 
 MEA 
 
 IN 
 OIROULAfr 
 
 MIL* 
 
 ALUMINUIM 
 
 UEA 
 
 IN 
 
 CIRCULAR 
 
 MILS 
 
 FOR LOAD POWER-FACTOR OF 8(Wfr-l0.8% LOSS- 12.5 LOSS 
 
 220 VOLTS DELIVERED 
 
 50 
 FEET 
 
 100 
 FEET 
 
 150 
 FEET 
 
 200 
 
 FEET 
 
 250 
 FEET 
 
 300 
 FEET 
 
 400 
 FEET 
 
 500 
 FEET 
 
 600 
 FEET 
 
 750 
 FEET 
 
 1000 
 FEET 
 
 1500 
 FEET 
 
 2000 
 FEET 
 
 2500 
 FEET 
 
 3500 
 FEET 
 
 6000 
 FEET 
 
 1 
 MILE 
 
 
 X, eoo 000 
 1 BOO 000 
 1 7O0OO0 
 
 
 IS/ZS 
 13 730 
 IZ8ZI 
 
 7 Sit 
 
 6 8iS 
 i^lo 
 
 S09Z 
 9S77 
 9X79 
 
 3 781 
 3-t32 
 3ZOS 
 
 3 02.J 
 Z796 
 2S69 
 
 ZSZI 
 
 zzas 
 3/37 
 
 / «fo 
 
 17 16 
 I60Z 
 
 /SIZ 
 
 'iViz 
 
 1 Z60 
 
 //4-1 
 1 068 
 
 / oo« 
 
 91S 
 
 kss 
 
 Hi 
 
 641 
 
 SO 9 
 
 4 si 
 
 9Z7 
 
 378 
 343 
 
 jIo 
 
 302 
 
 274 
 ZS6 
 
 183 
 
 ISI 
 /37 
 IZ8 
 
 /43 
 /30 
 
 121 
 
 
 tt>oo 000 
 /soo 000 
 /■foo 000 
 
 
 12 loo 
 II3ZI 
 /OS 79 
 
 i oso 
 
 S-iiO 
 SZ99 
 
 ■9033 
 3 779 
 3SZ6 
 
 3 02.S 
 2 830 
 
 2 44-f 
 
 2 920 
 
 2 on 
 
 1 887 
 1763 
 
 /SIZ 
 I9IS 
 1322 
 
 /ZIO 
 
 1 I3Z 
 loss 
 
 /OO8 
 99 3 
 
 806 
 
 7SS 
 70s 
 
 60s 
 Sbi 
 SZ9 
 
 403 
 377 
 3 S3 
 
 30Z 
 Z83 
 
 Z69 
 
 24 Z 
 226 
 
 211 
 
 173 
 I6Z 
 ISI 
 
 121 
 113 
 106 
 
 119 
 
 lei 
 100 
 
 
 /30O 000 
 
 1 100 000 
 /ooo 000 
 
 / S9O 000 
 
 9 097 
 B3-tS 
 7Si2 
 
 9SZ3 
 9 172 
 3 781 
 
 3 OIS 
 2 782. 
 ZSZI 
 
 2 2*2 
 Z086 
 1 890 
 
 '/in 
 /S/Z 
 
 /S07 
 1391 
 1260 
 
 1/3 1 
 
 /093 
 
 I 99S 
 
 SS5 
 
 7^* 
 
 7S3 
 69S 
 
 603 
 
 SS6 
 so 9 
 
 4SZ 
 417 
 37 S 
 
 30/ 
 Z7S 
 ZSZ 
 
 ZZi. 
 
 209 
 
 189 
 
 181 
 167 
 ISI 
 
 IZ 
 //■ 
 /a 
 
 fj 
 
 If 
 
 
 9 so 000 
 900 00c 
 aso 000 
 
 /SIS •oo 
 /\3I 000 
 1 3SI SOO 
 
 ^9IO 
 
 3 ilZ 
 3 908 
 3 20S 
 
 2 408 
 a 272 
 2 136 
 
 / sot 
 /709 
 / 6oi. 
 
 /99S 
 /363 
 
 /zaz 
 
 /209, 
 
 '/'oli 
 
 Vs\ 
 
 BOI 
 
 722 
 
 60Z 
 
 sta 
 
 S39 
 
 98Z 
 4S9 
 9Z7 
 
 36/ 
 34' 
 320 
 
 Z9I 
 227 
 219 
 
 181 
 170 
 /60 
 
 'lit 
 
 128 
 
 If 
 
 72 
 V4 
 
 ^5 
 
 
 900 000 
 
 7SO 00a 
 
 700000 
 
 1 Z72 000 
 1 192. Soo 
 1/13 000 
 
 6 0S0 
 Si.78 
 S290 
 
 3 0ZS 
 3 939 
 Zi-*.S 
 
 zon 
 
 1893 
 1763 
 
 / SIZ 
 
 /920 
 /izz 
 
 /ZIO 
 / /3S 
 /OSS 
 
 zoos 
 
 117 
 
 7S6 
 
 710 
 66/ 
 
 60S 
 
 S6g 
 SZ9 
 
 S09 
 
 -f73 
 -♦40 
 
 40 3 
 37» 
 3S3 
 
 m 
 
 ZOZ 
 
 ISI 
 I9Z 
 132 
 
 12/ 86 
 1/3 81 
 lOS 7S 
 
 io 
 
 S7 
 S3 
 
 S7 
 S4 
 SO 
 
 
 isoooo 
 *ooooo 
 
 S50 000 
 
 1 033 SOO 
 
 9S9 OOO 
 87-tSOO 
 
 9 119 
 9S2^ 
 
 9 173 
 
 Z-*S7 
 ZZkZ 
 ZoSi 
 
 /638 
 /S07 
 /39I 
 
 1 2X4 
 / 131 
 /093 
 
 983 
 90S 
 834 
 
 %'/3 
 69S 
 
 6/9 
 S6S 
 SZZ 
 
 491 
 9SZ 
 ■9 17 
 
 347 
 
 3Z8 
 278 
 
 Z9i 
 Z09 
 
 /69 
 '1V9 
 
 /23 
 113 
 109 
 
 Vo 
 83 
 
 70 
 64 
 S9 
 
 4», 4S 
 
 rz t? 
 
 soo 000 
 ■i- so 000 
 400 000 
 
 79S 000 
 7IS~SOO 
 <i,3i 000 
 
 3 7SI 
 3 39i 
 3 039 
 
 1 890 
 /i98 
 
 /sn 
 
 / Z60 
 //3Z 
 /oil 
 
 99S 
 B4* 
 7Sa 
 
 7S6 
 679 
 6^7 
 
 630 
 Si 6 
 SOS 
 
 472 
 ■9ZS 
 379 
 
 378 
 390 
 
 303 
 
 3/S 
 
 ZS2 
 ZZ6 
 ZOZ 
 
 /89 
 170 
 ISZ 
 
 '/n 
 
 /Ol 
 
 99 7S S9 
 8S 68 49 
 76 60 .43 
 
 38 
 
 34 
 30 
 
 s 
 
 
 3SO 000 
 
 300 000 
 
 ZSO 000 
 
 SS^ SOO 
 ■977 000 
 397SOO 
 
 2i*S 
 ZZtl 
 I89S 
 
 1 3ZZ 
 
 1133 
 
 999 
 
 88Z 
 7SS 
 633 
 
 661 
 S67 
 979 
 
 S29 
 -♦.S3 
 379 
 
 991 
 
 378 
 
 3/6 
 
 330 
 2 83 
 237 
 
 24-f 
 ZZ7 
 /9b 
 
 220 
 189 
 /SS 
 
 176 
 'Al 
 
 132 
 113 
 9S 
 
 86 
 
 66 
 S7 
 47 
 
 S3 38 
 
 4S 32 
 
 38 27 
 
 26 
 
 « 
 
 ooot 
 
 000 
 00 
 
 Zl 1 600 
 limz 
 
 /33 079 
 
 334,920 
 Zi.t>800 
 Zl 1 9SO 
 
 / iOO 
 /271 
 /ooft 
 
 800 
 
 S09 
 
 S33 
 9ZS 
 336 
 
 900 
 319 
 2SZ 
 
 320 
 2SS 
 20Z 
 
 ^^1 
 
 J200 
 
 'lit 
 
 /60 
 /Z7 
 /o/ 
 
 /33 
 
 /ol 
 
 89 
 
 "hi 
 67 
 
 80 
 so 
 
 S-i 
 
 55 
 
 fz 
 2S 
 
 32 23 
 
 Zi IS 
 20 14 
 
 16 
 13 
 /O 
 
 ? 
 
 / 
 
 Z 
 
 /0SS60 
 83C9t 
 it,3S8 
 
 167 900 
 J33ZZO 
 /O^S^SO 
 
 SCO 
 63Z 
 SOI 
 
 900 
 3lt 
 zso 
 
 Z61.* 
 211 
 167 
 
 200 
 /S8 
 IZS 
 
 /60 
 /Z6 
 /oo 
 
 '/U 
 
 83 
 
 /oo 
 Vz 
 
 90 
 63 
 SO 
 
 66 
 S3 
 
 9 1 
 
 S3 
 11 
 
 — 40 
 
 27 
 Zl 
 17 
 
 20 
 16 
 12 
 
 /6 
 
 IZ. 
 10 
 
 II 
 
 9 
 7 
 
 8 
 
 6 
 s 
 
 ! 
 
 3 
 
 S2 £24 
 .33 OSS 
 
 S3 6-fO 
 £« 370 
 SZ630 
 
 39i 
 316 
 ZSI 
 
 IfS 
 /S8 
 /2S 
 
 I3Z 
 
 lOS 
 
 99 
 
 if 
 
 11 
 
 so 
 
 66 
 SZ 
 
 92 
 
 99 
 
 fl 
 
 Vz 
 
 ZS 
 
 33 
 26 
 2/ 
 
 26 
 
 20 
 
 13 
 
 lo 
 
 8 
 
 /O 
 1 
 
 8 
 
 6 
 
 s 
 
 £ 
 s 
 4 
 
 1, 
 
 2 
 
 4. 
 
 1 
 
 /iS/Z 
 
 •91 79>o 
 
 m 
 
 11 
 
 it 
 
 if 
 
 90 
 31 
 2S 
 
 i! 
 
 21 
 
 ZS 
 ZO 
 IS 
 
 ZO 
 /6 
 /Z 
 
 16 
 13 
 /o 
 
 13 
 
 /o 
 8 
 
 /o 
 
 1 
 
 6 
 S 
 
 4 
 
 4" 
 
 3 
 
 4 
 
 i 
 
 3 
 
 3. 
 1 
 1 
 
 2 
 1 
 
 1 
 
 
 440 VOLTS DELIVERED 
 
 50 
 FEET 
 
 100 
 FEET 
 
 150 
 FEET 
 
 200 
 FEET 
 
 250 
 FEET 
 
 300 
 FEET 
 
 400 
 FEET 
 
 500 
 FEET 
 
 600 
 FEET 
 
 750 
 FEET 
 
 1000 
 FEET 
 
 1500 
 FEET 
 
 2000 
 FEET 
 
 2500 
 FEET 
 
 3500 
 FEET 
 
 5000 
 FEET 
 
 1 
 
 MILE 
 
 
 Z 000 000 
 J 900 000 
 / 700 000 
 
 
 to sot 
 
 S992' 
 
 30 a,so 
 27961 
 2Si*Z 
 
 ZO li,6 
 IS307 
 1709s 
 
 ISIZS 
 1373 
 I28ZI 
 
 / z/00 
 /0988 
 /02S7 
 
 10 083 
 
 9 /S3 
 
 es-^7 
 
 7S62 
 6 86S 
 6910 
 
 6osro 
 S99Z 
 S/28 
 
 S042 
 ■9S76 
 
 4273 
 
 4 033 
 
 3 6-t-S 
 
 3419 
 
 3 02J- 
 Z746 
 2S69 
 
 Z017 
 / 831 
 /7/0 
 
 /SIZ 
 
 / 373 
 1 Z82 
 
 / ZIO 
 
 'llVi 
 
 864 
 73 Z 
 
 60s 
 
 S49 
 S/3 
 
 48S 
 
 
 / AOO 000 
 
 / •soo 000 
 /-♦oo 000 
 
 
 9a4oa 
 9SZ?i 
 
 9Z3I7 
 
 2-f 200 
 Z2i93 
 21 ISS 
 
 16133 
 /S09S 
 /4I0S 
 
 12100 
 1/322 
 IOS7f 
 
 9680 
 90S7 
 
 e-tts 
 
 8066 
 7S47 
 70S2 
 
 4, oso 
 S66I 
 SZ89 
 
 9S40 
 9SZ8 
 9Z3I 
 
 4033 
 3 773 
 
 sszt 
 
 3Z26 
 30/9 
 ZSZ/ 
 
 2420 
 Z269 
 2 //S 
 
 /6/3 
 
 / s/o 
 
 1410 
 
 12/0 
 '/iV7 
 
 III 
 
 646 
 604 
 
 489 
 4S3 
 423 
 
 4SS 
 4Zf 
 400 
 
 
 / zoo 000 
 1 loo 000 
 /ooo 000 
 
 iS9o oao 
 
 3tlS7 
 3-3 37f 
 
 18093 
 l6tS7 
 ISIZS 
 
 IZOiZ 
 II IZ6 
 /0 083 
 
 9097 
 S3 4^ 
 7S6Z 
 
 7237 
 6676 
 6OS0 
 
 603I 
 SSi.^ 
 S092 
 
 9SZ3 
 4 /7Z 
 3 78/ 
 
 3 6/8 
 3 337 
 3 0ZS 
 
 30/S 
 
 zjaz 
 
 2SZ/ 
 
 24/2 
 222.r 
 ZOli 
 
 1 ^°t 
 / 668 
 /SIZ 
 
 1 206 
 1/13 
 1008 
 
 7si 
 
 724 
 66& 
 60s 
 
 S/7 
 477 
 432 
 
 3tZ 
 334 
 302 
 
 34 3 
 
 
 ?So 000 
 900 000 
 aso 000 
 
 / SIS 000 
 1931 000 
 /3SI Soa 
 
 ZSS-fi 
 27 2 68 
 25*12 
 
 I999S 
 1363', 
 12921 
 
 9632 
 90S9 
 8S-*7 
 
 722-t 
 6817 
 69IO 
 
 S779 
 S9S3 
 SI28 
 
 ■48/6 
 
 -4 273J 
 
 36/Z 
 3908 
 320i 
 
 2 889 
 27Z6 
 ZS69 
 
 Z90S 
 ZZ7Z 
 Z/36 
 
 I9Z6 
 IS/7 
 /709 
 
 1 499 
 I3t,3 
 1 28Z 
 
 96 3 
 909 
 8SS 
 
 7ZZ 
 682 
 
 641 
 
 S78 
 S9S 
 SI 3 
 
 9/Z 
 3 89, 
 366 
 
 ZS9 
 
 2 73 
 ZS6 
 
 243 
 
 BOO 000 
 7SO 000 
 
 700 000 
 
 1 Z7Z 000 
 / /9Z soo 
 / / 13 000 
 
 ^-♦200 
 227/0 
 Zl ISS 
 
 12 ZOO 
 I/3SS 
 I0S79 
 
 'soii 
 
 7S70 
 70S3 
 
 6 OSO 
 S677 
 SZ89 
 
 1890 
 ■*S9Z 
 423/ 
 
 4033 
 
 37as 
 
 3S27i 
 
 302s 
 2838 
 2i-»-» 
 
 Z9ZO 
 227/ 
 2 / /S 
 
 ZO/i 
 1 89Z 
 /763 
 
 / 613 
 IS/9 
 /4/0 
 
 IZ/O 
 II3S 
 /OS7 
 
 807 
 7S7 
 70S- 
 
 60s 
 S67 
 
 Szi 
 
 484 
 ■4S9 
 423 
 
 346 
 324 
 302 
 
 242 
 
 227 
 Zl 1 
 
 2Z6 
 
 z/s 
 20 1 
 
 
 600 000 
 ■sso 000 
 
 1 033 JOO 
 9S9 000 
 879 SOO 
 
 19 ASS 
 IS093 
 It, (.90 
 
 9827 
 9096 
 839S 
 
 6SSZ 
 603I 
 SS63 
 
 9913 
 9SZZ 
 9I7Z 
 
 3931 
 3 618 
 J 3 3? 
 
 3Z76 
 30 /S 
 Z78/ 
 
 29S6 
 ZZCI 
 20 86 
 
 I96S 
 I809 
 /i6f 
 
 /63S 
 
 '/i%l 
 
 1 3IO 
 IZ06 
 I//3 
 
 98Z 
 
 tiss 
 
 603 
 SS6 
 
 491 
 4SZ 
 
 417 
 
 iVz 
 334 
 
 281 
 2SS 
 238 
 
 196 
 /S/ 
 167 
 
 ',^8 
 
 
 •soo 000 
 ■^ so 000 
 
 ■900 000 
 
 79s 000 
 7/,s^oo 
 636 000 
 
 IS/ZS 
 
 13 sat 
 
 I2I3S 
 
 7SiZ 
 6793 
 t.0t>9 
 
 S09Z 
 
 ■*SZ9 
 9046 
 
 378/ 
 3oi9 
 
 302s 
 27/7 
 ZtZJ 
 
 ZSZI 
 2Z&9 
 2023 
 
 1 890 
 1698 
 /S17 
 
 is/z 
 
 I3S8 
 1213 
 
 /ZiO 
 II3X 
 lOIZ 
 
 1008 
 
 607 
 
 SO 9 
 4S3 
 40s 
 
 3 78 
 340 
 Jo3 
 
 302 
 272 
 24 3 
 
 Z/i 
 /94 
 173 
 
 /S/ 
 139 
 12 1 
 
 '/t% 
 
 /'S 
 
 
 3SO 000 
 •3 00 000 
 ZSO 000 
 
 s^s-6 soo 
 977 000 
 397 SOO 
 
 IOS79 
 
 9oti 
 7S92 
 
 SZ89 
 9S39 
 3796 
 
 3SZi 
 30Z3 
 2S3I 
 
 24-»4 
 22^7 
 1898 
 
 2 //S 
 1 813 
 / S/8 
 
 1763 
 /SIZ 
 /ZhS 
 
 /322 
 1 133 
 999 
 
 /OSS 
 906 
 7S9 
 
 881 
 7S6 
 632 
 
 Voi 
 
 sol 
 
 SZ9 
 4 si 
 379 
 
 3S3 
 302 
 2 S3 
 
 ikl 
 
 189 
 
 Z/l 
 18/ 
 /SZ 
 
 IS/ 
 
 IZ9 
 /at 
 
 lOi 
 
 n 
 
 '%t 
 
 72 
 
 000 
 00 
 
 3.1 1 <>oo 
 'V77Z 
 /33079 
 
 3369ZO 
 26isoo 
 Zl 1 9S'0 
 
 i900 
 
 S09S 
 903-3 
 
 3200 
 ^547 
 Zolt 
 
 213-3 
 1698 
 l3-*9 
 
 /too 
 IZ73 
 1008 
 
 izao 
 
 /oi9 
 
 eo6 
 
 /067 
 899 
 67 Z 
 
 800 
 636 
 ^04 
 
 £90 
 sol 
 
 S3 3 
 92s 
 336 
 
 424 
 3i? 
 
 3ZO 
 ZS9 
 ZOZ 
 
 ZI3 
 170 
 139 
 
 —7TS 
 'iVi 
 
 128 
 
 lOZ 
 
 81 
 
 91 
 73 
 S8 
 
 69 
 SI 
 40 
 
 38 
 
 1 
 Z 
 
 /OJ-S40 
 83699 
 
 1^7 eoo 
 
 /33ZZO 
 /OSS30 
 
 3 ZOO 
 2S30 
 zooi 
 
 H.OO 
 IZ6S 
 I003 
 
 /066 
 
 800 
 632 
 SOI 
 
 690 
 so 6 
 
 .901 
 
 S3Z 
 922 
 339 
 
 ■AOO 
 316 
 2 SO 
 
 3ZO 
 2S3 
 200 
 
 • Z6i 
 
 /34 
 
 /60 
 /Z7 
 /OO 
 
 107 
 84 
 67 
 
 1% 
 
 so 
 
 64 
 S/ 
 40 
 
 46 
 36 
 28 
 
 32 
 
 ii 
 
 30 
 
 n 
 
 ^ 
 
 •sz tz-» 
 
 •9 1 736 
 
 ■3 3oee 
 
 83 6-»0 
 
 IS Si 
 iZiS 
 
 I003 
 
 SOZ 
 
 ill 
 
 339 
 
 396 
 3/6 
 ZS3 
 
 3/7 
 2S3 
 Zpl 
 
 269 
 ZIZ 
 167 
 
 /?8 
 /S8 
 
 /ZS 
 
 /ss 
 
 /Zi 
 
 too 
 
 /3Z 
 '%% 
 
 /OS 
 V7 
 
 so 
 
 S3 
 If 
 
 Tz 
 
 ZS 
 
 32 
 
 2^ 
 20 
 
 ZZ 
 It 
 14 
 
 /i 
 It 
 /« 
 
 
 6 
 7 
 
 3.<>299 
 ^ogiz 
 
 ■91 790 
 
 S.oo 
 
 39S 
 319 
 ZSO 
 
 2 69 
 
 '/ii 
 
 197 
 157 
 IZS 
 
 /S8 
 IZ6 
 /oo 
 
 /3d 
 
 /OS 
 
 83 
 
 98 
 78 
 6Z 
 
 11 
 
 so 
 
 66 
 SZ 
 9Z 
 
 ■ S3 
 
 ■91 
 33 
 
 — s? 
 
 3Z 
 ZS 
 
 zi 
 
 21 
 17 
 
 20 
 16 
 
 IZ 
 
 /6 
 It 
 10 
 
 '4 
 
 z. 
 
 ■ «• 
 
 6 
 
 •r 
 
 %■ 
 
 The heating limitations may, for the shorter distances, particularly if insulated or concealed conductors are employed, necessitate the 
 nae of larger conductors, resulting in a correspondingly less transmission loss. In the case of insulated or concealed conductors, should the 
 It.T.a. values fall near or to the left of the heavy line, consult Table XXV" for insulated or Table XXIII for bare conductors. The reactance for 
 the larger tonductors may be excessive, particularly for eOcycIe service, producing excessive voltage drop. This may be obviated by installing 
 two or more parallel circuits or using three-conductor cables. For single-phase circuits the k.v.a. will be one-half the table values. 
 
QUICK ESTIMATING TABLES 
 
 ^5 
 
 Hence the value corresponding to half this distance 
 (1250 ft.) is 604 k.v.a., which is sufficiently accurate 
 for practical purposes. 
 
 REACTANCE LIMITATIONS 
 
 The k.v.a. value of the tables naturally do not take 
 into account the reactance of the circuit. It will be 
 
 TABLE XIII-QUICK ESTIMATING TABLE 
 
 CONDUCTORS 
 
 KILOVOLT- AMPERES. 3 PHASE WHICH MAY BE DELIVERED AT THE FOLLOWING VOLTAGES OVER THE VARIOUS 
 CONDUCTORS FOR THE DISTANCES STATED. BASED UPON THE FOLLOWINQ |2r LOSS (EFFECT OF CHARGING CURRENT 
 NEGLECTED) AT25"C 
 
 FOR LOAD POWEB-FACTOR OF IOO%-8 66% LOSS- 10.0% LOSS 
 
 i 
 
 05 
 
 m 
 
 COPPER 
 
 AREA 
 
 IN 
 
 CIRCULAR 
 
 MII.S 
 
 ALUMINUM 
 
 AREA 
 
 IN 
 
 CIRCULAR 
 
 MILS 
 
 FOR LOAD POWER-FACTOR OF 80%-l0.8% LOSS- 12 5 LOSr 
 
 550 VOLTS DELIVERED 
 
 50 
 
 FEET 
 
 100 
 FEET 
 
 150 
 
 FEET 
 
 200 
 
 FEET 
 
 250 
 
 FEET 
 
 300 
 
 FEET 
 
 400 
 FEET 
 
 500 
 
 FEET 
 
 600 
 FEET 
 
 750 
 FEET 
 
 1000 
 FEET 
 
 1500 
 
 FEET 
 
 2000 
 FEET 
 
 2500 
 FEET 
 
 3500 6000 
 FEET FEET 
 
 1 
 
 MILE 
 
 
 Z 000 000 
 
 / aoo 000 
 
 / 700 000 
 
 
 94S3I 
 8oJ3i 
 
 47266 
 42907 
 A.006t 
 
 31S10 
 Z96oi 
 
 3 67/1 
 
 X3631 
 2I4S3 
 20033 
 
 I8906 
 
 17/63 
 /602 6 
 
 irytt 
 
 I4302 
 l33Si 
 
 II 816 
 /0727 
 
 looi 6 
 
 So 13 
 
 7877 
 
 7ISI 
 
 4677 
 
 I 302 
 S7ZI 
 S342 
 
 -♦727 
 4Z9I 
 4007 
 
 3/SI 
 2 860 
 2671 
 
 23*3 
 
 21+S 
 2003 
 
 189/ 
 
 1716 
 
 1603 
 
 I3S0 94S 
 izai gs» 
 
 //4 4 got 
 
 »/S 
 
 7^» 
 
 
 /<oo 000 
 / soo 000 
 /400 000 
 
 
 7S62S 
 70760 
 iiliO 
 
 37812 
 3 S3 SO 
 3306O 
 
 2S20i 
 23S»7 
 Z204O 
 
 18906 
 /7690 
 /6S30 
 
 /SI2S 
 l-tlSZ 
 13324 
 
 IZ 604 
 11793 
 
 II 020 
 
 94S3 
 g84S 
 82 64) 
 
 7S62 
 7074 
 66IZ 
 
 6302 
 S896 
 SSIO 
 
 J-O-4 2 
 -47/7 
 
 -♦-♦or 
 
 3781 
 3S38 
 3306 
 
 2SZI 
 33S3 
 230+ 
 
 1891 
 17 69 
 I6S3 
 
 /SIZ 
 I41S 
 1322 
 
 /09I 
 
 /oil 
 
 94s 
 
 7J-6 
 70 8 
 661 
 
 7/f 
 470 
 62S 
 
 
 1 zoo 000 
 / 100 000 
 t 000 000 
 
 / S90 000 
 
 S6S-12 
 S2/SS 
 
 472iS 
 
 2S27I 
 26077 
 i3*32 
 
 18 S41 
 
 n3SS 
 
 IS7SS 
 
 I4I3S 
 /303S 
 /I 816 
 
 II 30S 
 
 /O't.31 
 
 94S3 
 
 9+23 
 869 z 
 7#77 
 
 70 67 
 6SI9 
 S908 
 
 S6S4 
 SZlS 
 47Z7 
 
 4711 
 -»34* 
 393S 
 
 3769 
 3477 
 3ISI 
 
 ilii 
 
 J363 
 
 188-S 
 /73S 
 /S76 
 
 1413 
 /3 04 
 1182 
 
 1131 90s S6S 
 
 10+3 74 S SZZ 
 
 9+S 6 7S 473 
 
 S3i 
 
 +9Z 
 + 49 
 
 
 ^^0 000 
 
 '900 000 
 
 iso 000 
 
 /S/S OOQ 
 /■f3/ 000 
 / 3SI SOO 
 
 43i04, 
 
 ■4-006^ 
 
 22S74 
 21 303. 
 200^3 
 
 /SO SO 
 14202 
 I33SS 
 
 II 287 
 I06S2 
 10016 
 
 9030 
 
 8S2I 
 
 gal3 
 
 7J2X' 
 
 7/01 
 
 <i77 
 
 S643 
 S326 
 SOO 9 
 
 42 60 
 4007 
 
 3762 
 3SSO 
 3338 
 
 3010 
 2840 
 2671 
 
 Z2S7 
 ZI30 
 Z003 
 
 /SOS 
 I4Z0 
 1336 
 
 1129 
 
 /a6S 
 
 /002 
 
 *03 
 8S2 
 
 80 1 
 
 i+S 
 «>8 
 .5 72 
 
 4SZ 
 426 
 401 
 
 403 
 390 
 
 
 BOO OOQ 
 7S0 000 
 700 000 
 
 / Z7Z 000 
 J /9Z Soo 
 / 1 13 000 
 
 378^3 
 3S4S-t 
 33060 
 
 IS9C7 
 177*2 
 I6S30 
 
 IZ60* 
 II 82t 
 
 11020 
 
 9*S3 
 8S7' 
 
 azis 
 
 7S62 
 7097 
 6612 
 
 6302 
 S9I4 
 SS/O 
 
 -•72:5 
 443S 
 
 4132 
 
 3781 
 3S48 
 3306 
 
 3 ISI 
 Z9S7 
 Z7SS 
 
 3 S2I 
 2366 
 2204 
 
 189/ 
 1774 
 /6J-3 
 
 IZtO 
 //83 
 fIOZ 
 
 94S 
 
 15; 
 
 7S6 S40 
 708 SO 7 
 661 -472 
 
 378 
 3SS 
 
 331 
 
 3sa 
 
 33J 
 
 3/3 
 
 
 6 SO 000 
 ioo 000 
 
 yS^OOOO 
 
 / 033 soo 
 9S4 000 
 S74 Soo 
 
 307/0 
 a»27/ 
 
 3.t07S 
 
 IS3Si 
 I4I3S 
 I3039 
 
 /0 2 37 
 9424 
 8693 
 
 7677 
 70*7 
 6S20 
 
 6/42 
 S6S4 
 S2IS 
 
 SI IS 
 47/2 
 -♦3-»« 
 
 3S3S 
 3S32 
 3260 
 
 3 07 J 
 
 Z8Z7 
 3607 
 
 ZSi9 
 .J3.r6 
 .J/73 
 
 20 + 7 
 /8S4 
 1738 
 
 /S3S 
 /4I4 
 /3a4 
 
 /OZ4 
 %V9 
 
 868 
 707 
 6S2 
 
 6/4 4 3» 
 S6S 404 
 S22 372 
 
 307 
 
 zsi 
 
 26/ 
 
 2»/ 
 248 
 247 
 
 
 *soo 000 
 ^SO 000 
 4oO OOQ 
 
 79SOOO 
 7/SSOO 
 636 000 
 
 ZI 22s 
 
 I891S 
 
 /IS/6 
 
 /06I4 
 
 94S2 
 
 7878 
 7076 
 6322 
 
 S90S 
 S307 
 
 4741 
 
 .472* 
 42-K 
 3 7^3 
 
 3»3f 
 3S3S 
 3/il 
 
 2fJ-+ 
 .2 6.^3 
 2370 
 
 2363 
 Z123 
 /S96 
 
 1969 
 1769 
 /S80 
 
 /S7S 
 /4/S 
 /2 64 
 
 //82 
 
 /06/ 
 
 948 
 
 788 
 708 
 632 
 
 S9I 
 S3I 
 -474 
 
 473 
 42s 
 379 
 
 337 
 
 29i 
 2/2 
 /90 
 
 224 
 ZOI 
 
 /to 
 
 
 ^so 000 
 300 obo 
 SSO 000 
 
 SS*> SOO 
 ■477 000 
 3 97 S-OO 
 
 /&S30 
 
 /-f /48 
 //giS 
 
 SZ6S 
 7aS4 
 S932 
 
 SSIO 
 4723 
 
 3 9S4 
 
 4132 
 3S42 
 2f<< 
 
 3306 
 2823 
 2 3 72 
 
 Z7SS 
 Z36I 
 1977 
 
 2066 
 177/ 
 1483 
 
 /6S3 
 7416 
 1 186 
 
 1377 
 /ISO 
 98S 
 
 //03 
 791 
 
 826 
 708 
 S93 
 
 SS' 
 473 
 39S 
 
 4/3 
 3S4 
 297 
 
 331 
 283 
 337 
 
 236 
 
 ZoZ 
 170 
 
 /6S 
 //9 
 
 /si 
 
 734 
 
 //a 
 
 ooot 
 000 
 00 
 
 Zll 60O 
 /<,7 77Z 
 /33 079 
 
 33i>4-ZO 
 
 Zbbgoo 
 
 ZI 1 9SO 
 
 JO 000 
 796O 
 i3o2 
 
 Sooo 
 39SO 
 3ISI 
 
 3333 
 Z6S3 
 XIO 1 
 
 ZSOO 
 1990 
 IS7.r 
 
 2000 
 IS92 
 1260 
 
 7666 
 /327 
 
 /O.SO 
 
 /2S0 
 99S 
 787 
 
 /OOO 
 796 
 ^30 
 
 833 
 ^63 
 SZS 
 
 ^i6 
 S3I 
 420 
 
 SOO 
 398 
 ■3IS 
 
 333 
 
 3tS 
 
 3 /O 
 
 3sa 
 
 199 
 /SS 
 
 200 
 
 IS9 
 126 
 
 1 + 3 
 //3 
 90 
 
 /OO 
 
 15 
 
 ri 
 
 
 
 1 
 i. 
 
 /o.s'Sio 
 
 it 358 
 
 /67 800 
 /33 ZZO 
 /0SS30 
 
 Sooo 
 3 9S4 
 3/34 
 
 ZSoo 
 1977 
 /S67 
 
 /666 
 1318 
 /04S 
 
 IZSO 
 
 9»S 
 
 783 
 
 /OOO 
 7«/ 
 <27 
 
 !5? 
 
 S22 
 
 62i 
 3 9i. 
 
 SOO 
 
 IV3 
 
 416 
 ill 
 
 333 
 2 6-4 
 Z09 
 
 J'So 
 /IS 
 IS7 
 
 /66 
 73 2 
 /OS 
 
 /2S 
 
 Vs 
 
 /OO 
 
 11 
 
 71 
 S6 
 4S 
 
 so 
 
 40 
 
 31 
 
 ■47 
 
 i* 
 30 
 
 1 
 
 oS-Z^Z4 
 41 735 
 J3088 
 
 93i-fO 
 6(.370 
 SZ 6 30 
 
 Z479 
 1977 
 /S67 
 
 1239 
 
 98S 
 
 7f3 
 
 826 
 6S9 
 S22 
 
 tze 
 
 494 
 392 
 
 496 
 39S 
 3/3 
 
 413 
 3Z9 
 Z6/ 
 
 310 
 297 
 196 
 
 Z48 
 197 
 /■S6 
 
 ^^4 
 /30 
 
 /6S 
 /32 
 /o4 
 
 /Z4 
 1? 
 
 u 
 
 S2 
 
 i3 
 49 
 39 
 
 so 
 
 40 
 
 31 
 
 3S 
 
 28 
 32 
 
 2S 
 ZO 
 /6 
 
 24 
 
 '/J 
 
 i. 
 
 I 
 
 zo si a 
 
 41 740 
 
 I23J 
 9S4 
 7 SO 
 
 619 
 492 
 390 
 
 413 
 328 
 2AO 
 
 310 
 Z46 
 I9S 
 
 ^■^7 
 1*7 
 /S6 
 
 Z06 
 164 
 /30 
 
 /sj- 
 
 /as 
 
 97 
 
 123 
 
 J03 
 SZ 
 6S 
 
 ^2 
 6S 
 S2 
 
 il 
 11 
 
 41 
 33 
 26 
 
 Vs 
 20 
 
 zs 
 
 ZO 
 
 16 
 
 li 
 14 
 II 
 
 /I 
 
 /o 
 8 
 
 '4 
 
 7 
 
 
 1100 VOLTS DELIVERED 
 
 100 
 
 FEET 
 
 200 
 FEET 
 
 300 
 FEET 
 
 500 
 FEET 
 
 750 
 FEET 
 
 1000 
 FEET 
 
 2500 
 FEET 
 
 4000 
 FEET 
 
 1 
 
 MILE 
 
 \ + 
 
 MILES 
 
 1 jr 
 MILES 
 
 2 
 MILES 
 
 2i 
 
 MILES 
 
 3 
 
 MILES 
 
 3i 
 MILES 
 
 4 
 MILES 
 
 6 
 
 MILES 
 
 
 2000000 
 / Soo 000 
 1 700 000 
 
 
 /tfCiZ 
 I7lt3l 
 /i02I.S 
 
 44S3I 
 8SS/4 
 
 80132 
 
 630ZI 
 
 S7ZI0 
 X3-»il 
 
 37812 
 34326 
 3^oi^ 
 
 2S208 
 
 32gg-f 
 21368 
 
 /S906 
 /7 163 
 /602i 
 
 7S62 
 6S6S 
 6410 
 
 + 736 
 +291 
 4006 
 
 3S80 
 32SO 
 303S 
 
 286.f 
 3600 
 2*X% 
 
 23gg 
 2/66 
 2 02 3 
 
 /740 
 
 /62S 
 
 ISI? 
 
 /432 
 /300 
 1214 
 
 1194 
 I083 
 101 2 
 
 /02I 
 8fc7 
 
 89 S 
 
 812 
 7St 
 
 7/fc 
 
 6.ro 
 
 407 
 
 
 /too 000 
 / .5-00 000 
 l-^oo oao 
 
 
 /SI2S0 
 141 SZO 
 I372V> 
 
 7S62.S' 
 70760 
 66120 
 
 S04I6 
 47173 
 
 302S0 
 2 6 304 
 2.614S 
 
 30lkl, 
 iSSiS 
 1 7631 
 
 /SI2S 
 I4IS2 
 13224 
 
 60.5-0 
 .f66/ 
 S2>9 
 
 3781 
 3S3S 
 3306 
 
 3S6S 
 2680 
 ZSOS 
 
 2291 
 2144 
 2003 
 
 / 9/0 
 /786 
 7664 
 
 /432 
 /340 
 /2S2 
 
 II4S 
 1072 
 1002 
 
 ISS 
 893 
 834 
 
 766 
 7/6 
 
 716 
 67 
 
 «26 
 
 .572 
 .534 
 >50/ 
 
 
 / zoo 000 
 / joo 000 
 / 000000 
 
 1 Sfo 000 
 
 /I3CS4 
 104310 
 94S3I 
 
 S6S42 
 SZIS-C 
 472i'S^ 
 
 37694 
 34770 
 31-^10 
 
 22617 
 ZOS62 
 l89ol 
 
 /■soy 8 
 1390S 
 12604 
 
 1130a 
 
 7043I 
 
 •I4S3. 
 
 f i23 
 
 4172 
 37SI 
 
 2S37 
 2608 
 2363 
 
 Zl + l 
 1976 
 /790 
 
 /7/3 
 ISSO 
 1+32 
 
 /■42« 
 
 /3I7 
 /19 + 
 
 /O70 
 9SS 
 S9S 
 
 gS6 
 
 790 
 7/6 
 
 714 
 6si 
 S9 7 
 
 4/2 
 S6S 
 SI2 
 
 S3S 
 
 499 
 
 447 
 
 428 
 19S 
 3St 
 
 
 9SO 000 
 900 000 
 6S0000 
 
 / S/S 000 
 l43t 000 
 / 3SI SOO 
 
 90:1.9s 
 8 S3. II 
 ao/32 
 
 4SI49 
 42t,3S 
 j^oo6t 
 
 30019 
 28404 
 26711 
 
 ISOSf 
 I7042 
 16026 
 
 12 040 
 
 Il3lpl 
 
 IOiS4 
 
 9030 
 SSZI 
 
 800 
 
 36/2 
 340t 
 320s 
 
 J2.S7 
 .3/30 
 2003 
 
 1 710 
 16/3 
 /SI 8 
 
 /368 
 1291 
 121 + 
 
 1140 
 /076 
 /0I2 
 
 SSS 
 807 
 
 7S9 
 
 684 
 6+S 
 
 <07 
 
 S70 
 S3S 
 SO 6 
 
 434 
 
 427 
 
 3*2 
 321 
 
 303 
 
 
 900 000 
 7SO 000 
 700 000 
 
 / Z7Z 000 
 J I9Z SOO 
 / J 13 000 
 
 7Si2i 
 709tS 
 ii,l20 
 
 378/4 
 3S4S4 
 33061 
 
 2S209 
 Z3iS6 
 22040 
 
 ISI26 
 14193 
 133 24 
 
 /0083 
 9462 
 8Sli 
 
 7J63 
 7097 
 6612 
 
 303.f 
 3t3f 
 26-M 
 
 /S9I 
 177 + 
 /6S3 
 
 1 + 32 
 /3*+ 
 /2S2 
 
 11 + 6 
 /07S 
 1002 
 
 83i 
 
 7/6 
 672 
 626 
 
 .573 
 .5-37 
 
 SOI 
 
 477 
 ■448 
 4/7 
 
 3St 
 
 313 
 
 286 
 
 268 
 
 a SO 
 
 
 4tSo 000 
 AOO 000 
 SSo 000 
 
 1 033 SOO 
 9S4-000 
 974 SOO 
 
 i,l42l 
 StS42 
 SZI.S.S 
 
 3 07/0 
 
 29271 
 26017 
 
 .t047+ 
 18 847 
 
 n3SS 
 
 I2 2S^ 
 11308 
 /043I 
 
 8189 
 7J'3f 
 6TS4 
 
 6142 
 S6^S4 
 S2tS 
 
 Z4S7 
 3.2(.l 
 20 si 
 
 /S3.S\ 
 1413 
 1304 
 
 //63 
 
 7 071 
 
 988 
 
 790 
 
 77 S 
 l's% 
 
 S82 
 S3S 
 
 -♦»♦ 
 
 46S 
 428 
 3 9S 
 
 3 88 
 
 3S7 
 329 
 
 JS2 
 J04 
 
 291 
 267 
 
 247 
 
 232 
 
 ZI+ 
 197 
 
 
 soo 000 
 
 4SOOOO 
 JfOOOOO 
 
 79s 000 
 7/SSOO 
 636000 
 
 3793! 
 
 .23633 
 
 XI228 
 1 S9iS 
 
 I.S7SS 
 I-HS2 
 12 6+4 
 
 94SZ 
 8411 
 7SS6 
 
 6302 
 S66I 
 S0.i7 
 
 -f72<i 
 4246 
 3793 
 
 /Sf 
 7698 
 /S/7 
 
 1181 
 /06I 
 948 
 
 89S 
 80+ 
 718 
 
 7/6 
 6-»3 
 S7S 
 
 ill 
 
 479 
 
 -♦■47 
 402 
 3S9 
 
 3S8 
 323 
 287 
 
 398 
 268 
 339 
 
 zsi 
 
 224 
 ZOS 
 
 223 
 
 201 
 179 
 
 1 19 
 /*/ 
 /4» 
 
 
 3 SO 000 
 300 000 
 ZSo 000 
 
 SS6 SOO 
 477000 
 39ySOO 
 
 330t>0 
 ^S337 
 Z372S 
 
 HS30 
 14168 
 II S62 
 
 11020 
 9446 
 
 790 s 
 
 i6/2 
 J-t47 
 -«7+i 
 
 440s 
 
 3306 
 2t33 
 2372 
 
 7332 
 //33 
 9+9 
 
 az6 
 
 70s 
 .r»3 
 
 626 
 3^36 
 
 4*^ 
 
 SOI 
 430 
 3S9 
 
 417 
 JSS 
 300 
 
 3/3 
 26* 
 22-4 
 
 3 SO 
 3 IS 
 179 
 
 'iff 
 /SO 
 
 179 
 /J-3 
 /2» 
 
 /S6 
 J34 
 112 
 
 /as 
 
 /»7 
 
 8» 
 
 ooot 
 
 000 
 00 
 
 zi 1 too 
 
 li7 77Z 
 / 33 07f 
 
 336420 
 Z66 900 
 2 / / 9SO 
 
 30000 
 IS9ZI 
 t2i04 
 
 /000a 
 6302 
 
 6666 
 S307 
 4201 
 
 400a 
 
 3I>4 
 
 2666 
 ZIX3 
 /<80 
 
 Zooo 
 IS9Z 
 IZ60 
 
 SOO 
 637 
 .5-04 
 
 soo 
 
 3»S 
 3 IS 
 
 378 
 JO/ 
 238 
 
 303 
 
 Z4I 
 191 
 
 ZS2 
 Zol 
 IS9 
 
 /*9 
 /SO 
 119 
 
 IS2 
 /ZO 
 
 4S 
 
 IZ6 
 
 /OO 
 
 7f 
 
 /a9 
 86 
 68 
 
 94 
 7S 
 S9 
 
 76 
 40 
 47 
 
 
 
 1 
 
 Z 
 
 / os^to 
 
 /67 800 
 /33200 
 / ,5" J-30 
 
 J aooo 
 
 IV4 
 
 sooo 
 39S4 
 3134 
 
 3333 
 263£ 
 Z0S9 
 
 .2 000 
 ISSI 
 I3S4 
 
 /333 
 
 /OS4 
 
 83S 
 
 /OOO 
 7 9/ 
 
 627 
 
 -400 
 3/6 
 2^-/ 
 
 i-io 
 I9S 
 /S7 
 
 19 
 /SO 
 
 118 
 
 /SI 
 
 /20 
 
 9S 
 
 126 
 
 /OO 
 
 79 
 
 %'s 
 
 J9 
 
 7S 
 6e 
 -47 
 
 63 
 
 S4 
 42 
 34 
 
 47 
 
 Is 
 
 ■s 
 
 ■■ SZi>ZA 
 -*/ 736 
 ■33oSff 
 
 g3 640 
 66 370 
 SZ630 
 
 49iS 
 3fi4 
 3134 
 
 2-^79 
 1977 
 /S67 
 
 /iS3 
 /3I 8 
 
 991 
 791 
 637 
 
 661 
 418 
 
 4 96 
 39S 
 3/3 
 
 /S8 
 /2i 
 
 134 
 
 S9 
 
 7S 
 60 
 47 
 
 62 
 SO 
 40 
 
 -47 
 J? 
 
 37 
 
 31 
 
 ii 
 
 26 
 
 21 
 /7 
 
 23 
 IS 
 1 + 
 
 It 
 
 IS 
 
 /z 
 
 J_ 
 
 3 6 2-14 
 
 zoazz 
 
 fiSIZ 
 
 ■4 1740 
 
 2479 
 I9i7 
 /St' 
 
 /Z39 
 9S3 
 7 So 
 
 826 
 6SS 
 SZO 
 
 496 
 
 330 
 Z62 
 ZOS 
 
 197 
 /S6 
 
 ft 
 78 
 62 
 
 62 
 
 -♦» 
 39 
 
 47 
 37 
 30 
 
 fo 
 Z4 
 
 31 
 
 zs 
 
 20 
 
 1i 
 IS 
 
 '/J 
 
 IZ 
 
 /S 
 
 IZ 
 
 /a 
 
 /4- 
 /O 
 
 8 
 
 12 
 
 9 
 7 
 
 9 
 I 
 
 The heating limitations may. tor the shorter distances, particularly if insulated or concealed conductors are employed, necessitate the 
 use of larger conductors, resulting in a correspondingly less transmission loss. In the case of insulated or concealed conductors, should the 
 k.v.a. values fall near or lo the left of the heavy line, consult Table XX\' for insulated or Table XXTII for bare conductors The reactance (or 
 the larger conductors may be excessive, particularly for 60-cycle service, producing excessive voltage drop. This may be obviated by inatalling 
 two or more parallel circuits or using three-conductor cables. For single-phase circuit* the k.v.a. will be one-half the table valuai. 
 
26 
 
 QUICK ESTIMATING TABLES 
 
 necessary in some cases of low voltage and single con- 
 ductors^ (where the reactance is high) to use lower 
 values of k.v.a. or even in some cases to multiple cir- 
 
 cuits in order to keep the reactance within satisfactory 
 operating limits. This will be considered later by ex- 
 amples on voltage regulation. 
 
 TABLE XIV-QUICK ESTIMATING TABLE 
 
 CONDUCTORS 
 
 KILOVOLT-AMPERES. 3 PHASE. WHICH MAY BE DELIVERED AT THE FOLLOWING VOLTAGES OVER THE VARIOUS 
 CONDUCTORS FOR THE DISTANCES STATED. BASED UPON THE FOLLOWING |2r LOSS (EFFECT OF CHARGING CURRENT 
 NEGLECTED) AT26"C AT 65- O 
 FOR LOAD POWER-FACTOR OF IOO%-8 66% LOSS- 10.0% LOSS 
 
 i 
 
 CO 
 
 «1 
 
 CO 
 
 COPPER 
 
 AAEA 
 
 IN- 
 
 OIROUUUt ' 
 
 MILS 
 
 ALUMINUM 
 
 AREA 
 
 IN 
 
 CIRCULAR 
 
 MIL* 
 
 FOR LOAD POWER-FACTOR OF 80%- 10.8% LOSS- 12.5 LOSS 
 
 2200 VOLTS DELIVERED 
 
 100 
 
 FEET 
 
 200 
 FEET 
 
 300 
 FEET 
 
 500 
 FEET 
 
 750 
 FEET 
 
 1000 
 FEET 
 
 2500 
 FEET 
 
 4000 
 FEET 
 
 MILE 
 
 MILES 
 
 MILES 
 
 2 
 
 MILES 
 
 22 
 MILES 
 
 3 
 
 MILES 
 
 3i 
 
 MILES 
 
 4 
 
 MILES 
 
 5 
 
 MILES 
 
 
 2 OOO OOO 
 I900 ooo 
 
 /700 ooo 
 
 
 7S6 000 
 
 6S6000 
 641 00c 
 
 a79ooa 
 
 34'3 000 
 32000c 
 
 3S1000 
 
 329000 
 
 ZI4000 
 
 ISIOOO 
 
 137000 
 129000 
 
 101000 
 91 soo 
 SSSoo 
 
 7S600 
 69600 
 6>* 100 
 
 30200 
 27400 
 2S6ao 
 
 ;»Too 
 17 100 
 16000 
 
 /4300 
 /3000 
 /z 100 
 
 // 460 
 10400 
 97IO 
 
 9 sso 
 
 S670 
 8100 
 
 7/*0 
 
 «^oo 
 
 *070 
 
 J-730 
 S20O 
 
 4 ejo 
 
 4 770 
 4330 
 4ojro 
 
 4 o?o 
 37/0 
 3470 
 
 3SSC 
 
 3 2SC 
 3 04c 
 
 2 8*0 
 2*00 
 
 2430 
 
 
 / 600 ooo 
 /^oo ooo 
 /■foo ooo 
 
 
 iOSOOO 
 
 Sttooo 
 SZ^ooa 
 
 302000 
 293000 
 
 202000 
 199000 
 
 176000 
 
 121000 
 113000 
 
 106000 
 
 80 700 
 7S.sao 
 70 .soo 
 
 6 a SOO 
 S6 600 
 S2 900 
 
 Z4 200 
 
 22600 
 21 10a 
 
 IS 100 
 I4IOO 
 
 I3300 
 
 II 400 
 10700 
 10 000 
 
 9 170 
 8S80 
 8 010 
 
 7640 
 7 /SO 
 6,680 
 
 S730 
 S3k0 
 SOlO 
 
 4S8a 
 4290 
 4010 
 
 3 82 
 
 3.5^70 
 
 J340 
 
 3 2 70 
 3 0*0 
 2 8*0 
 
 Z 860 
 2&SC 
 2 SOO 
 
 2290 
 Z140 
 
 3 oao 
 
 
 /zoo ooo 
 1 /oo ooo 
 / ooo coo 
 
 / S9Q ooo 
 
 4S200O 
 
 417000 
 37900a 
 
 226000 
 iOfOOC^ 
 
 19^000 
 
 ISIOOO 
 131000 
 126000 
 
 ^0 SOO 
 93400 
 
 7S6ao 
 
 60 300 
 JS600 
 S0400 
 
 4S2O0 
 4170a 
 378O0 
 
 I8100 
 
 /6700 
 /J/OO 
 
 II 300 
 
 /0400 
 
 9 4S0 
 
 8S60 
 7900 
 7160 
 
 6 SSO 
 6320 
 S730 
 
 S7/0 
 SZ70 
 4 770 
 
 42B0 
 3 9S0 
 3SZ0 
 
 3430 
 
 3 160 
 2860 
 
 2 sso 
 2630 
 2 390 
 
 2 4 SO 
 2 2^0 
 
 2 040 
 
 2I40 
 1970 
 1 790 
 
 mo 
 /SSO 
 / 430 
 
 
 9SO ooo 
 900 ooo 
 SSO ooo 
 
 / S/SOOO 
 
 /43I ooo 
 /3SI soo 
 
 361 000 
 341000 
 3 2000c 
 
 ISO 000 
 170000 
 
 l&OOOO 
 
 120000 
 
 114 000 
 
 107000 
 
 72200 
 
 i>9 100 
 
 <-» too 
 
 4SI00 
 4S4oa 
 
 42700 
 
 36ioa 
 
 34100 
 32 000 
 
 14400 
 /3 600 
 /2SOO 
 
 9030 
 
 8S30 
 SOlO 
 
 6 840 
 &4S0 
 6070 
 
 S470 
 SI60 
 4«*0 
 
 4 £60 
 4 300 
 4 O.S-0 
 
 3420 
 3230 
 3030 
 
 274° 
 ZSSO 
 2430 
 
 Z 280 
 2 /SO 
 2020 
 
 1 9SO 
 / 840 
 1 73 
 
 17 10 
 /610 
 IS2C 
 
 / 370 
 I290 
 12IQ 
 
 
 ooo OOO 
 
 7SO ooo 
 700 ooo 
 
 /Z7Z ooo 
 / I9Z soo 
 / 113 ooo 
 
 joa 000 
 ZS^^ooo 
 264000 
 
 ISIOOO 
 
 142000 
 132000 
 
 101 000 
 94 600 
 99 100 
 
 ^osoo 
 S6SOO 
 S290O 
 
 40300 
 37 SOO 
 3S200 
 
 30200 
 29400 
 2t400 
 
 /2100 
 I13O0 
 /06OO 
 
 7S60 
 7 100 
 6 610 
 
 S730 
 S370 
 S 010 
 
 4SBO 
 4300 
 4010 
 
 3»20 
 3 SSO 
 3340 
 
 2»*0 
 2 TOO 
 ZSOO 
 
 3290 
 ZISO 
 2 000 
 
 1910 
 /79a 
 
 / 670 
 
 /640 
 /S40 
 /430 
 
 1430 
 /340 
 
 1 2Sa 
 
 /I4i- 
 1 070 
 /OOO 
 
 
 isoooo 
 too ooo 
 ssoooo 
 
 /033 soo 
 
 9S+ ooo 
 
 S74 soo 
 
 2A60O0 
 2 26000 
 
 20900c 
 
 12300C 
 113 000 
 
 10400a 
 
 91900 
 7S400 
 69 SOO 
 
 49100 
 
 4i200 
 4/700 
 
 33700 
 30 100 
 2 7»O0 
 
 24 SOO 
 Z26aa 
 20 Sao 
 
 9 800 
 
 90SO 
 
 8 3SO 
 
 6 140 
 S6SO 
 S 210 
 
 4 6S0 
 4 ZSO 
 
 3 9SO 
 
 3720 
 343a 
 3 I60 
 
 3100 
 2 SSO 
 2 630 
 
 2 3 30 
 2/40 
 
 / y7o 
 
 IS6I 
 1 7/0 
 /SSO 
 
 /SSO 
 /430 
 /3I0 
 
 /330 
 /220 
 //30 
 
 / 160 
 
 1070 
 
 987 
 
 931 
 
 as7 
 790 
 
 
 so oooo 
 ssoooo 
 4-00 ooo 
 
 79S ooo 
 7/SSOo 
 63i> ooo 
 
 IS1 000 
 
 170000 
 /S2000 
 
 99 soo 
 
 8490a 
 
 7SSO0 
 
 630O0 
 S6 600 
 
 So 600 
 
 3 7 SOO 
 33900 
 30 30c 
 
 2S200 
 22*00 
 20200 
 
 1S900 
 17000 
 iszoa 
 
 7S60 
 6,800 
 6 070 
 
 4730 
 4 240 
 3 7?0 
 
 3 SSO 
 
 3220 
 2S70 
 
 2S60 
 2S70 
 Z 300 
 
 Z3SO 
 Z 140 
 1 910 
 
 / 790 
 1 610 
 /440 
 
 /430 
 1290 
 1 ISO 
 
 1 190 
 
 1070 
 
 9SS 
 
 /020 
 ^20 
 
 824 
 
 89S 
 80 4 
 7/8 
 
 7/* 
 *43 
 
 .S7>r 
 
 
 •3SO ooo 
 
 Jooaoo 
 zso ooo 
 
 SS£ soo 
 
 4.77000 
 
 397SOO 
 
 132000 
 
 113000 
 
 94100 
 
 66 100 
 S6JOO 
 -♦7-»00 
 
 44 100 
 37 SOO 
 31 600 
 
 24+00 
 22700 
 /»ooo 
 
 17600 
 
 /s 100 
 
 I2 600 
 
 13Z00 
 
 n 300 
 
 9 soo 
 
 S290 
 4S30 
 3 800 
 
 33/0 
 Z930 
 
 z 370 
 
 ZSOO 
 ZlSO 
 / 800 
 
 ZOOO 
 1 720 
 1 440 
 
 1 670 
 I430 
 IZOQ 
 
 /2.ro 
 
 /073 
 900 
 
 1000 
 »40 
 720 
 
 83S 
 71s 
 600 
 
 7/6 
 */3 
 SI3 
 
 *2* 
 .5-3* 
 4 SO 
 
 Sol 
 429 
 3S9 
 
 oooo 
 ooo 
 
 zi / too 
 /i7 7'72 
 /33 079 
 
 336 4ZO 
 ZtL soo 
 ZI 1 9 so 
 
 80000 
 
 63700 
 so 400 
 
 40000 
 31 900 
 ZSiao 
 
 2*700 
 21 200 
 I69O0 
 
 /4000 
 12.700 
 /OIOO 
 
 10700 
 9490 
 6720 
 
 8 oao 
 6 370 
 S040 
 
 3 ZOO 
 2 SSO 
 Zooo 
 
 Zooo 
 / S90 
 / 260 
 
 / SIO 
 
 / 200 
 9SS 
 
 / ZIO 
 lbs 
 760 
 
 /OIO 
 804 
 636 
 
 7S7 
 6oi 
 477 
 
 606 
 483 
 392 
 
 SOS 
 
 403 
 
 318 
 
 433 378 
 3+i 301 
 273 238 
 
 303 
 24/ 
 191 
 
 o 
 / 
 2 
 
 /OSSiO 
 i&3SS 
 
 /67 soo 
 y33 ZIO 
 /OS S30 
 
 40000 
 3/ 60O 
 ZSIOO 
 
 Zoooo 
 tSSOO 
 12 SOO 
 
 /3 300 
 
 /osoa 
 
 »3iO 
 
 8000 
 6Z30 
 Saio 
 
 S330 
 4-2ZO 
 33-40 
 
 4000 
 SII60 
 ZSIO 
 
 /600 
 /Z60 
 1 000 
 
 /OOO 
 7,90 
 i,30 
 
 Vol 
 47i 
 
 606 
 480 
 3 80 
 
 400 
 317 
 
 3 80 
 300 
 .237 
 
 303 
 23? 
 190 
 
 2S3 
 ZOO 
 /S8 
 
 .2/7 
 171 
 /36 
 
 189 
 
 , /SO 
 
 /IS 
 
 ISI 
 
 IZO 
 
 4S 
 
 3 
 -» 
 
 41 73» 
 33 OSS 
 
 *3 640 
 66 370 
 SZ 6 30 
 
 19 SOO 
 IS Soo 
 12 SOO 
 
 9900 
 7910 
 6270 
 
 6 610 
 
 s 270 
 
 4 190 
 
 3970 
 
 3 I6O 
 ZSIO 
 
 Z640 
 ZIIO 
 / 670 
 
 / f so 
 /sso 
 1 2.SO 
 
 790 
 63a 
 soo 
 
 SOO 
 39S 
 
 313 
 
 37s 
 300 
 237 
 
 300 
 Z40 
 I90 
 
 ZSO 
 ZOO 
 IS9 
 
 ISS 
 /SO 
 
 /IS 
 
 /SO 
 
 I30 
 
 9S 
 
 ISS 
 
 ]0a 
 
 79 
 
 /07 
 8* 
 69 
 
 94 
 7S 
 S9 
 
 60 
 47 
 
 i. 
 
 7 
 
 Z&Z4-1 
 i0 822 
 
 ■91 740 
 
 9900 
 7900 
 4 2 JO 
 
 4960 
 3»3<I 
 3 120 
 
 3310 
 
 Z620 
 Z090 
 
 I980 
 /S70 
 /iso 
 
 / 320 
 
 / oso 
 
 8 30 
 
 990 
 
 790 
 6 20 
 
 400 
 3IO 
 zso 
 
 ZSO 
 /97 
 /Sb 
 
 188 
 
 '/Vs 
 
 /so 
 
 /I9 
 9S 
 
 /2S 
 99 
 79 
 
 II 
 
 47 
 
 63 
 
 49 
 39 
 
 S4 
 42 
 
 34 
 
 47 
 37 
 30 
 
 37 
 30 
 24 
 
 
 4000 VOLTS 
 
 DELIVERED 
 
 
 
 
 
 
 
 
 
 
 1 
 
 MILE 
 
 li 
 
 MILES 
 
 2 
 
 MILES 
 
 22 
 MILES 
 
 3 
 MILES 
 
 3i 
 MILES 
 
 4 
 
 MILES 
 
 6 
 
 MILES 
 
 6 
 
 MILES 
 
 7 
 MILES 
 
 8 
 MILES 
 
 9 
 
 MILES 
 
 10 
 
 MILES 
 
 MILES 
 
 12 
 MILES 
 
 13 
 MILES 
 
 14 
 
 MILES 
 
 
 A so OOO 
 
 Aoo ooo 
 
 sso ooo 
 
 / 033 000 
 
 9S4 000 
 S74 soo 
 
 1S400 
 lA 200 
 /3 100 
 
 10 2SO 
 9460 
 S7S0 
 
 770a 
 7 100 
 
 6, sso 
 
 6 ISO 
 S670 
 S2SO 
 
 S/zo 
 
 47 so 
 4370 
 
 4-»0O 
 40JO 
 37 JO 
 
 3S.J-0 
 3SS0 
 3280 
 
 3080 
 ZS40 
 262a 
 
 ZS60 
 2370 
 21 so 
 
 3 200 
 Z030 
 
 /S70 
 
 /920 
 
 /77S 
 /640 
 
 /7/0 
 /sso 
 /460 
 
 /S40 
 /420 
 /3IO 
 
 l-^oa 
 /290 
 /190 
 
 /280 
 /1 80 
 /a 90 
 
 //so 
 /090 
 
 /O/O 
 
 II 00 
 /02a 
 940 
 
 
 soo ooo 
 ^so ooo 
 ■^oo ooo 
 
 79s 000 
 
 7/S Soo 
 63600O 
 
 /1 900 
 
 /0 700 
 
 9490 
 
 7 9 SO 
 7/20 
 6330 
 
 S9SO 
 
 »r3<ro 
 
 4 7sa 
 
 4 7-ro 
 4170 
 3 800 
 
 3 960 
 3S60 
 316S 
 
 3400 
 3060 
 2713 
 
 Z980 
 Z670 
 Z373 
 
 Z390 
 2/40 
 1900 
 
 I9SO 
 I7SO 
 /S83 
 
 /70a 
 /S30 
 
 1 3S7 
 
 /490 
 /33a 
 1 186 
 
 /320 
 /190 
 /OSS 
 
 //90 
 /0 70 
 
 9-t9 
 
 /080 
 970 
 8*3 
 
 9fo 
 
 890 
 
 791 
 
 9/s 
 
 822 
 736 
 
 sso 
 
 763 
 6,78 
 
 
 ■3 so ooo 
 300 ooo 
 zso ooo 
 
 ssi. soo 
 
 -}77 000 
 397 soo 
 
 S270 
 7090 
 S92a 
 
 SSIO 
 
 4 72a 
 
 3 9S0 
 
 4 130 
 3S4° 
 2960 
 
 3307 
 2S32 
 237a 
 
 Z7S6 
 Z3S9 
 I97S 
 
 Z36Z 
 Z023 
 /S9 3 
 
 Z067 
 /769 
 148 1 
 
 /6S3 
 14! 6 
 IISS 
 
 /378 
 /I79 
 987 
 
 /ISI 
 
 /O / 1 
 
 ^46 
 
 /033 
 8 84 
 740 
 
 919 
 
 Vs% 
 
 827 
 708 
 S93 
 
 7S2 
 644 
 .S3? 
 
 699 
 
 SS9 
 493 
 
 63* 
 S4S 
 4S6 
 
 S90 
 S06 
 
 42 3 
 
 oooo 
 ooo 
 
 oo 
 
 ZI 1 boo 
 /i,7 772 
 /33 079 
 
 33^ 420 
 Zi6 soo 
 
 2 1 1 9SO 
 
 Soao 
 3990 
 
 3 140 
 
 3 330 
 Z6SO 
 Z090 
 
 3 soo 
 ,990 
 IS70 
 
 2000 
 ./S92 
 12SS 
 
 1666 
 1326 
 /OA.6 
 
 /42S 
 /I37 
 896 
 
 /ZSO 
 
 /OOO 
 796 
 
 627 
 
 S33 
 6*3 
 S23 
 
 7/4 
 S68 
 448 
 
 ^3S 
 4^7 
 
 391 
 
 sss 
 
 ''•'2 
 34? 
 
 Soo 
 39s 
 313 
 
 454 
 3*2 
 
 2S.r 
 
 416 
 33/ 
 261 
 
 38S 
 306 
 Z4I 
 
 3S7 
 Z84 
 224 
 
 o 
 
 'z 
 
 /OSS60 
 93 (,94 
 66 3SS 
 
 /67 soo 
 /33 Z20 
 /0SS30 
 
 zsoo 
 
 1970 
 /S70 
 
 1 670 
 / 320 
 1 04s 
 
 / 2Sa 
 ■9S7 
 7»3 
 
 1000 
 790 
 627 
 
 S\3 
 6i9 
 S22 
 
 7/3 
 ^^3 
 4-fS 
 
 62S 
 493 
 392 
 
 SOO 
 
 39s 
 
 3/3 
 
 416 
 329 
 261 
 
 3S7 
 ZZ4 
 
 3/2 
 
 247 
 I9li 
 
 277 
 2/f 
 174 
 
 2 SO 
 /97 
 /S7 
 
 179 
 /42 
 
 208 
 164 
 /30 
 
 192 
 IS2 
 130 
 
 na 
 
 '/n 
 
 ■3 
 
 SZ t,34 
 -»; 738 
 
 3 3 OSS 
 
 83 640 
 6i370 
 S2 4 30 
 
 /2+0 
 %V6 
 
 i3.6 
 661 
 SZ4 
 
 620 
 4-9S 
 393 
 
 4?6 
 396 
 314 
 
 4/3 
 J30 
 262 
 
 3S4 
 ZS3 
 Z2S 
 
 310 
 
 in 
 
 Z48 
 199 
 /S7 
 
 Z07 
 167 
 /3l 
 
 177 
 /4I 
 /IZ 
 
 /SS 
 /2 3 
 98 
 
 I3S 
 /lO 
 87 
 
 /24 
 99 
 78 
 
 //3 
 9o 
 7/ 
 
 ^11 
 6S 
 
 ^0 
 
 S6 
 
 
 4-400 VOLTS DELIVERED 
 
 1 
 
 MILE 
 
 MILES 
 
 2 
 
 MILES 
 
 2i 
 MILES 
 
 3 
 
 MILES 
 
 3i 
 
 MILES 
 
 4 
 
 MILES 
 
 5 
 MILES 
 
 6 
 MILES 
 
 7 
 MILES 
 
 8 
 MILES 
 
 9 
 
 MILES 
 
 10 
 
 MILES 
 
 M 
 
 MILES 
 
 12 
 MILES 
 
 13 
 MILES 
 
 14 
 MILES 
 
 
 6 so ooo 
 
 Aoo ooo 
 sso ooo 
 
 / 033 000 
 
 9S4 000 
 
 874 Soo 
 
 IS 600 
 
 n 100 
 IS 900 
 
 /2-»00 
 // 400 
 
 /o 600 
 
 9300 
 8 sso 
 
 7 9 so 
 
 7 4 So 
 
 AS so 
 
 637a 
 
 i,2O0 
 S700 
 S300 
 
 ^3k30 
 4 880 
 4SSO 
 
 4i4.ro 
 4270 
 3 9 So 
 
 3720 
 3420 
 3/ 80 
 
 31 00 
 ZSSO 
 Z6SO 
 
 2660 
 Z440 
 2270 
 
 2330 
 2140 
 1990 
 
 2070 
 1900 
 
 1770 
 
 1860 
 /7/0 
 /S9a 
 
 / 690 
 /sso 
 /ASo 
 
 ISSO 
 
 /t3a 
 /33a 
 
 / 430 
 132a 
 /330 
 
 /330 
 12 20 
 //40 
 
 
 soo ooo 
 ^so ooo 
 40O ooo 
 
 79S 000 
 7/S soo 
 636 000 
 
 14300 
 /Z900 
 
 11 soo 
 
 9 SSO 
 S600 
 7 6SO 
 
 71 so 
 64SO 
 S7S0 
 
 S7Z0 
 SISO 
 4«00 
 
 4770 
 4300 
 
 3S30 
 
 40 90 
 3690 
 3290 
 
 JS70 
 3230 
 2S80 
 
 2970 
 
 2S80 
 2300 
 
 2380 
 ZISO 
 19Za 
 
 20S0 
 /940 
 /640 
 
 17 BO 
 /620 
 /440 
 
 /S90 
 /430 
 /ZSO 
 
 /430 
 /290 
 //SO 
 
 /300 
 /ISO 
 
 /oso 
 
 1 190 
 
 /080 
 
 96a 
 
 /loo 
 99S 
 887 
 
 /020 
 
 irs 
 
 
 3S0 ooo 
 300 ooo 
 zso ooo 
 
 ssi soo 
 
 477 000 
 397 Soo 
 
 10000 
 
 SS90 
 7 no 
 
 6 670 
 S720 
 ■*7SO 
 
 SOOO 
 4290 
 3SSS 
 
 4000 
 3490 
 3970 
 
 3330 
 ZSto 
 2390 
 
 Z860 
 24 SO 
 20S0 
 
 2SOO 
 ZISO 
 /SOO 
 
 Zooo 
 
 /720 
 /430 
 
 /&70 
 1430 
 /20Q 
 
 /430 
 /Z30 
 /030 
 
 /2SO 
 /070 
 900 
 
 // 10 
 
 9SO 
 800 
 
 /OOO 
 SSS 
 7/7 
 
 9/0 
 
 780 
 6,S3 
 
 832 
 7/S 
 600 
 
 770 
 460 
 SS2 
 
 I'/i 
 
 S/3 
 
 oooo 
 coo 
 oo 
 
 ZI 1 4.00 
 
 lirr 77Z 
 
 /J3 079 
 
 336420 
 Z66eoo 
 ZI 1 9SO 
 
 60S0 
 
 4 820 
 
 3790 
 
 4030 
 3220 
 2 SSO 
 
 ^02S 
 24IO 
 
 /srs 
 
 Z420 
 /930 
 /S20 
 
 20X0 
 /6IO 
 I260 
 
 1730 
 /38o 
 /oeo 
 
 /SIO 
 
 1210 
 
 9SO 
 
 'U°s 
 
 7SS 
 
 /O/O 
 SOS 
 633 
 
 96S 
 &S8 
 S4 3 
 
 7SS 
 <o3 
 47S 
 
 *73 
 S3S 
 422 
 
 S.OS 
 482 
 37f 
 
 SSO 
 438 
 34.S- 
 
 SOS 
 402 
 317 
 
 467 
 
 IVz 
 
 43-3 
 34S 
 27/ 
 
 o 
 
 1 
 
 3. 
 
 /OS S60 
 S3 694 
 i(,3SS 
 
 / 67 800 
 /33 220 
 /OS S30 
 
 3030 
 Z390 
 
 /90a 
 
 2020 
 /S90 
 /270 
 
 /S/S 
 //9S 
 9 so 
 
 /220 
 9SS 
 7 60 
 
 /O/O 
 79s 
 633 
 
 8<.7 
 683 
 S4 2 
 
 7S8 
 S9S 
 47S 
 
 607 
 
 478 
 3 SO 
 
 soi, 
 398 
 3/7 
 
 433 
 34/ 
 27/ 
 
 379 
 399 
 237 
 
 337 
 266 
 21/ 
 
 303 
 23^ 
 
 /9a 
 
 276 
 2/7 
 173 
 
 2S3 
 /99 
 /SS 
 
 233 
 /83 
 /46 
 
 2/7 
 /7/ 
 /36 
 
 3 
 ■S 
 
 SZ&Z4 
 
 S3 640 
 6^ 370 
 S-2 630 
 
 ISoo 
 
 1200 
 
 9SO 
 
 / 000 
 Soo 
 03 
 
 7SO 
 Aoo 
 47S 
 
 600 
 4-SO 
 3»o 
 
 Soo 
 
 400 
 318 
 
 42» 
 
 34X 
 
 i-rz 
 
 37 S 
 .300 
 237 
 
 300 
 Z40 
 190 
 
 ZSO 
 ZOO 
 
 /as 
 
 214 
 172 
 /3* 
 
 188 
 
 /SO 
 /19 
 
 /67 
 
 /SO 
 
 /36 
 
 I/Li 
 
 l/<, 
 
 51 
 
 11 
 
 The heating limitations may, for the shorter distances, particularly if insulated or concealed eonductore »re employed, necasstttte the 
 see of larger conductors, resulting in a correspondingly less transmission loss. In the case of insulated or concealed conductors, should the 
 k.T.a. Talues fall near or to the left of the heavy line, consult Table XXV for insulated or Table XXIII for bare conductors. The reactance for 
 the larger conductors may be excessive, particularly for 60-cycle service, producing excessive voltage drop. ' This may be obviated by installin( 
 two or more parallel circuits or using three-conductor cables. For ■iscle-phase circuits the k.y.k. will be one-halt the table value*. 
 
QUICK ESTIMATING TABLES 
 
 TABLE XV-QUICK ESTIMATING TABLE 
 
 27 
 
 CONDUCTORS 
 
 KILOVOLT- AMPERES. 3 PHASE. WHICH MAY BE DELIVERED AT THE F 
 CONDUCTORS FOR THE DISTANCES STATED. BASED UPON THE FOLLOWING 
 NEGLECTED) 
 
 FOR LOAD POWER-FACTOR OF I00*-856*r7§a_ 
 
 OLLOW 
 
 i'rlc 
 
 AT 66 
 10.0% L 
 12.6 L 
 
 D 
 
 NO VOLTAGES OVER THE VARIOUS 
 ISS (EFFECT OF CHARGING CURRENT 
 
 •c 
 
 d 
 
 z 
 
 CO 
 
 COPPER 
 
 AREA 
 
 IN 
 
 CIRCULAR 
 
 MIL« 
 
 ALUMINUM 
 
 AREA 
 
 tN 
 
 CIRCULAR 
 
 MIL* 
 
 FOR LOAD POWER-^^ACTOR OF 80%- 10.8% LOSS- 
 
 OSS 
 3SS 
 
 
 
 6000 VOLTS DELIVERE 
 
 
 1 
 MILE 
 
 MILES 
 
 2 
 
 MILES 
 
 2x 
 
 MILES 
 
 3 
 
 MILES 
 
 3i 
 
 MILES 
 
 4 
 
 MILES 
 
 5 
 
 MILES 
 
 6 
 
 MILES 
 
 7 
 MILES 
 
 8 
 
 MILES 
 
 9 
 
 MILES 
 
 10 II 12 13 14 
 MILES MILES MILES MILES MILES 
 
 
 eoQooa 
 j-soooo 
 
 ^.r4 000 
 
 »74 J-OO 
 
 3 1 eoo 
 
 2 9S0C 
 
 33100 
 31 300 
 
 19 700 
 
 /7300 
 
 'S900 
 /* 700 
 
 I3SOO 
 IX700 
 "SOO 
 
 1 1 soo 
 
 'OkOO 
 9 9SO 
 
 9900 
 9'oo 
 S930 
 
 86S0 
 
 7 9 so 
 
 7370 
 
 <i93o 
 6 370 
 
 S900 
 
 S7SO 
 S30O 
 ■9920 
 
 4 9SO 
 
 ■9 SSO 
 
 4220 
 
 ■♦320 3».ro 3460 
 3980 3S90 3180 
 3690 3X80 39SC 
 
 3 'SO 2890 2470 
 i^.VL -24.50 24SC 
 
 12470 
 \Z270 
 
 
 ♦ i'oooo 
 
 .*0 ODO 
 
 7^^000 
 7/^J-OO 
 ^ 3£ 000 
 
 3390c 
 3 I300 
 
 moo 
 /^ooo 
 
 l-t300 
 
 " 900 
 10700 
 
 'O^OO 
 9SSO 
 9S10 
 
 »8iC 
 Sooo 
 
 7'3o 
 
 7600 
 6 9SO 
 
 6100 
 
 6 6 70 
 .5-400 
 .5-340 
 
 S320 
 4 770 
 4270 
 
 4 4.50 
 -4 00a 
 3S6I 
 
 3 800 
 3490 
 30S0 
 
 %%%°r, itiS ^m ^■*^° ^"'° ^O^O '90O 
 39S0 2660 2390 2/70 2000 / »40 /72o 
 X670 3Z70 Xl30 1940 naa '64.a /.c-*-. 
 
 
 300 000 
 
 2SO 000 
 
 -*77 000 
 397 SOO 
 
 /960O 
 IS900 
 /33O0 
 
 IXAOO 
 
 10 boo 
 SS40 
 
 9300 
 
 79i>0 
 6&70 
 
 7440 
 *370 
 -5-330 
 
 4,300 
 S3I0 
 4440 
 
 S3I0 
 ■9SSO 
 3910 
 
 4 6SO 
 3 9 90 
 3330 
 
 3 720 
 3190 
 
 3 670 
 
 3 100 
 2iSO 
 Z310 
 
 2270 
 
 / 9ao 
 
 2 320 
 
 I990 
 ' 67o 
 
 2070 
 
 1770 
 I980 
 
 I860 
 'S90 
 '330 
 
 I690 
 I9SO 
 /2/0 
 
 'SSO 
 /330 
 'no 
 
 /43o 
 '2X0 
 'OXO 
 
 /33o 
 // 30 
 
 4 JO 
 
 000 
 
 /«7772 
 /J3 079 
 
 3 66 SOO 
 2 1 1 <i.SO 
 
 a9<ro 
 
 70t,0 
 
 7^00 
 
 S970 
 47/0 
 
 Sb30 
 ■44 SO 
 3J30 
 
 4>S"oo 
 
 3 sea 
 3S3a 
 
 3 7 JO 
 39*0 
 23iO 
 
 3310 
 
 2 360 
 202c 
 
 28IO 
 2240 
 / 76o 
 
 3XS0 
 1791 
 '9 10 
 
 ' 970 
 149a 
 " ko 
 
 '610 
 I390 
 ' 01 
 
 /4lO 
 IIIO 
 883 
 
 13 SO 
 
 99S 
 7*4 
 
 "30 
 89S 
 706 
 
 '030 
 
 814 
 442 
 
 937 »44 
 744 6S9 
 SSS J-4 3 
 
 goo 
 
 690 
 
 1 
 Z 
 
 e3 6*-f 
 
 '33 230 
 '0^S3 
 
 -4440 
 3 J3o 
 
 3 7.ro 
 
 2460 
 33SO 
 
 2S/0 
 2220 
 /760 
 
 2 ISO 
 I780 
 ' 9 1 
 
 IS70 
 l9iO 
 1 170 
 
 IblO 
 /2 70 
 '010 
 
 /4/0 
 1 / 10 
 
 sex 
 
 1 IZO 
 990 
 706 
 
 937 
 791 
 SS9 
 
 80-3 
 634 
 SO't 
 
 703 
 
 sss 
 
 441 
 
 42^- 
 444 
 342 
 
 S62 
 499 
 3J3 
 
 SIS 
 404 
 3x1 
 
 441 
 
 3 70 
 
 291 
 
 ■433 
 34a 
 27/ 
 
 400 
 
 317 
 2S0 
 
 
 ■»' 73 8- 
 33 088- 
 
 &i. 370 
 J-2 630 
 
 2230 
 /770 
 
 ' Sbo 
 
 i-tio 
 
 '390 
 
 1 no 
 8»4 
 
 ' "0 
 
 S9I 
 
 70? 
 
 930 
 743 
 S9o 
 
 797 
 637 
 SOS 
 
 6?7 
 SSh 
 442 
 
 Sis 
 -»44 
 3S4 
 
 ■96S 
 3 7/ 
 39 S 
 
 399 
 318 
 3SX 
 
 349 
 278 
 231 
 
 3'o 
 
 27f 
 223 
 '77 
 
 253 
 203 
 
 I61 
 
 233 XlS 
 ISS '71 
 I4i /3» 
 
 3 00 
 /«« 
 
 /24 
 
 
 
 
 6600 VOLTS DELIVERED 
 
 
 
 
 
 1 
 
 MILE 
 
 li 
 
 MILES 
 
 2 
 
 MILES 
 
 2i 
 MILES 
 
 3 
 
 MILES 
 
 3i 
 MILES 
 
 4 
 
 MILES 
 
 5 
 
 MILES 
 
 e 
 
 MILES 
 
 7 
 MILES 
 
 8 
 
 MILES 
 
 9 
 
 MILES 
 
 10 
 
 MILES 
 
 MILES 
 
 12 
 
 MILES 
 
 13 
 
 MILES 
 
 14 
 
 MILES 
 
 
 600 000 
 
 / 033 000 
 9s-tooo 
 974 SOO 
 
 4/4<JO 
 39 SOO 
 35700 
 
 2SO0O 
 3S700 
 23 S-OO 
 
 30 90Q 
 19 300 
 n eoo 
 
 /6S00 
 
 /-S-400 
 /4 300 
 
 /4000 
 
 /2gOO 
 
 // 400 
 
 13 000 
 II 000 
 10000 
 
 /o.roo 
 
 9630 
 
 8930 
 
 S>400 
 7700 
 7/40 
 
 7000 
 6400 
 S9SO 
 
 6000 
 .5-.roo 
 S'l 00 
 
 .5-2.5-0 
 48/0 
 
 4 4 60 
 
 4470 
 
 4270 
 3960 
 
 4140 
 
 39 so 
 
 3S70 
 
 3830 
 3JO0 
 33SO 
 
 3saa 
 
 32/0 
 
 2480 
 
 3230 
 2 9iO 
 37 SO 
 
 30OO 
 
 2 7J-0 
 
 3 SSO 
 
 
 4SO 000 
 •400 000 
 
 7'SSOO 
 &3(,00O 
 
 32300 
 X9 lao 
 XStoo 
 
 31 SOO 
 /*4<J0 
 17300 
 
 19-SOO 
 
 1x900 
 
 /2400 
 1 1 &00 
 /O30O 
 
 1 0900 
 
 9 700 
 
 S6.ro 
 
 92SO 
 SB30 
 7400 
 
 807S 
 
 7370 
 
 6970 
 
 6460 
 SS30 
 
 SI eo 
 
 S4O0 
 4SS0 
 
 4330 
 
 4620 
 4 1 SO 
 3700 
 
 4030 
 3 640 
 3240 
 
 3400 
 3230 
 2 8 80 
 
 3330 
 24/0 
 3 S90 
 
 2990 
 2 440 
 2340 
 
 2640 
 2420 
 2/40 
 
 24 SO 
 3X40 
 '990 
 
 23/0 
 20 80 
 / SSO 
 
 
 3S0 000 
 300 000 
 2 SO 000 
 
 -477 000 
 
 397-S-ao 
 
 21600 
 /4300 
 /6 200 
 
 'Sooo 
 13.900 
 10 eoo 
 
 1 I30O 
 
 9 is a 
 Siaa 
 
 ^040 
 7720 
 6480 
 
 7.r2o 
 1^990 
 
 S900 
 
 6 9 SO 
 6S20 
 4 6 20 
 
 S6SO 
 4820 
 
 4o.ro 
 
 4S20 
 3860 
 33-90 
 
 3770 
 3220 
 3700 
 
 3330 
 2760 
 23/0 
 
 3830 
 24X0 
 2030 
 
 2SIO 
 XISO 
 ISOO 
 
 2240 
 I930 
 '63a 
 
 2o.ro 
 
 1760 
 1470 
 
 /880 
 /4/0 
 I3S0 
 
 1740 
 19 SO 
 1340 
 
 '6'0 
 '3 So 
 "60 
 
 000 
 00 
 
 XI' iOO 
 ' i.7 772 
 /33 079 
 
 33 if 3.0 
 266 SOO 
 
 3.1 , 9SO 
 
 /3600 
 
 /o eoo 
 gsso 
 
 9oso 
 7200 
 
 S7O0 
 
 ^800 
 S-*oo 
 -4270 
 
 J'4'fo 
 4320 
 3420 
 
 9-S30 
 3600 
 
 2s.ro 
 
 3990 
 3090 
 
 3440 
 
 3400 
 
 X700 
 
 3 190 
 
 2720 
 
 2/60 
 1710 
 
 2240 
 
 '800 
 '93a 
 
 /440 
 'S-90 
 
 'xJo 
 
 I700 
 '3S0 
 '070 
 
 ' S'O 
 
 '300 
 
 9S0 
 
 I360 
 
 'OSO 
 
 8SS 
 
 /230 
 4»o 
 778 
 
 1130 
 900 
 7/3 
 
 10*0 
 830 
 447 
 
 470 
 770 
 4/0 
 
 / 
 z 
 
 »3 <f-? 
 
 '&7 foo 
 '33 220 
 
 /o,rj^3o 
 
 S3S0 
 4-270 
 
 4S30 
 3S90 
 3fS0 
 
 3400 
 
 2640 
 2/30 
 
 272 
 
 XI SO 
 17 1 
 
 Xlio 
 
 17 90 
 
 1*30 
 
 1990 
 'S3 
 
 1310 
 
 1 700 
 /340 
 1070 
 
 I360 
 I070 
 
 SS4 
 
 'I30 
 89S 
 710 
 
 970 
 767 
 4/0 
 
 SSO 
 672 
 .533 
 
 7SS 
 S97 
 477 
 
 480 
 S3t 
 
 -427 
 
 4/7 
 490 
 390 
 
 S6S 
 448 
 
 3i6 
 
 J2X 
 
 4/3 
 324 
 
 4 84 
 3tS 
 
 306 
 
 s 
 
 SX 424 
 -»/ 73P 
 330»S 
 
 »3 6-K) 
 i6 370 
 .^2 ^30 
 
 3 3»o 
 
 2.7/0 
 2/40 
 
 ZZSO 
 1 SOO 
 /420 
 
 Ii90 
 '3 so 
 1070 
 
 13 SO 
 
 ' 080 
 
 Sio 
 
 II30 
 900 
 7/2 
 
 9&S 
 77S 
 6/2 
 
 477 
 
 S3S 
 
 676 
 S43 
 ■928 
 
 S6S 
 ■9S0 
 3S6 
 
 483 
 
 -»27 
 334 
 248 
 
 374 
 300 
 23» 
 
 338 
 27/ 
 2/4 
 
 307 
 246 
 'tS 
 
 281 
 234 
 '7S 
 
 2 60 24/ 
 204 /fj 
 'iS 'Si 
 
 
 10 000 VOLTS DELIVERED 
 
 
 
 6 
 MILES 
 
 7 
 MILES 
 
 8 
 
 MILES 
 
 9 
 MILES 
 
 10 
 MILES 
 
 II 
 
 MILES 
 
 12 
 MILES 
 
 13 
 MILES 
 
 14 
 MILES 
 
 16 
 MILES 
 
 16 
 
 MILES 
 
 18 
 MILES 
 
 20 
 
 MILES 
 
 22 
 MILES 
 
 24 
 MILES 
 
 26 
 
 MILES 
 
 28 
 
 MILES 
 
 
 fciO 000 
 <6oo 000 
 ■ssooao 
 
 / 033 000 
 9s-^ooo 
 S74 J-00 
 
 /6 000 
 
 '■teoo 
 
 '3 700 
 
 13 SOO 
 
 I3700 
 II700 
 
 ix'oo 
 
 II 100 
 /0200 
 
 /o 700 
 
 9S30 
 
 9'00 
 
 9b30 
 
 89SO 
 S300 
 
 ? 7.5-0 
 90S0 
 79 so 
 
 S030 
 73 SO 
 6930 
 
 7400 
 
 6ffoo 
 
 6300 
 
 6870 
 6320 
 S8S0 
 
 6,920 
 
 S900 
 S9SO 
 
 6000 
 
 SS30 
 S'30 
 
 S3SO 
 4930 
 4 SSO 
 
 4830 
 4930 
 ■9 100 
 
 4 3 80 
 4020 
 3720 
 
 4020 
 3680 
 
 3910 
 
 3700 
 
 3910 
 31S0 
 
 34-fO 
 3/Jo 
 2420 
 
 
 600 000 
 -450000 
 .^00 000 
 
 79s OOq 
 7'S SOO 
 6>3^ 000 
 
 /1300 
 
 // too 
 IVIo 
 
 'O&OO 
 9SSO 
 S4gO 
 
 93SO 
 S9SO 
 7-43Q 
 
 83 50 
 7930 
 
 iiS9-l 
 
 7900 
 6690 
 S930 
 
 67 30 
 
 soao 
 
 S390 
 
 6, ISO 
 
 jS6a 
 4440 
 
 S'30 
 4S60 
 
 S380 
 ■477a 
 4 240 
 
 ■9920 
 
 ■94 so 
 39SO 
 
 ■96,20 
 ■41 70 
 37/0 
 
 ■4-1 00 
 3700 
 3300 
 
 3700 
 3390 
 2470 
 
 3360 
 3040 
 
 2700 
 
 27S0 
 3470 
 
 3SSO 
 3S70 
 3 3Sa 
 
 2440 
 2 340 
 2l30 
 
 
 3^0000 
 300 000 
 2^0 000 
 
 •rsi SOO 
 ■^77 000 
 
 397 SOO 
 
 9&IO 
 7 370 
 6 / 70 
 
 73»0 
 6320 
 sx-!o 
 
 6460 
 .f.S30 
 4430 
 
 S792 
 44/0 
 4 //O 
 
 SI 7 
 9930 
 3700 
 
 ■97O0 
 4030 
 
 337a 
 
 43 10 
 3690 
 3040 
 
 3970 
 3+00 
 3SSO 
 
 3640 
 3/60 
 2640 
 
 3440 
 24J^O 
 
 2470 
 
 3230 
 2740 
 23/0 
 
 2 8 70 
 X960 
 30 60 
 
 3 SSO 
 23IO 
 1 SSO 
 
 23.f0 
 
 Xaio 
 
 '6*0 
 
 2 'SO 
 I840 
 'S40 
 
 1990 
 1700 
 '9XO 
 
 IS40 
 'SSO 
 
 '3xe 
 
 oooo 
 000 
 00 
 
 ;2 / / 600 
 /67 77a 
 / 3 3 79 
 
 33&-f20 
 266 S'OO 
 2 / / 9SC 
 
 SZIO 
 ^lAO 
 3370 
 
 ■^9i,0 
 
 3ssa 
 3900 
 
 34/0 
 3//0 
 
 2-».ro 
 
 34.70 
 2760 
 
 3 I30 
 X990 
 1 460 
 
 2 99 
 23&0 
 
 neo 
 
 .260O 
 2O70 
 /630 
 
 2400 
 1910 
 ISIO 
 
 2230 
 17 SO 
 '400 
 
 xaso 
 '66a 
 '31 a 
 
 /y.50 
 'SSO 
 '33a 
 
 1740 
 /JffO 
 '090 
 
 'S60 
 
 '34a 
 
 9 SO 
 
 /420 
 
 "s^f 
 
 ' 300 
 /040 
 
 8/7 
 
 '300 
 9S7 
 7J4 
 
 "to 
 890 
 
 700 
 
 / 
 
 2 
 
 /ossiio 
 
 '<i7 SoO 
 '33200 
 
 /o.r.r3o 
 
 3 &00 
 3oeo 
 /633 
 
 3330 
 
 /760 
 /4O0 
 
 /4,ro 
 /J-40 
 
 /220 
 
 1790 
 I370 
 '090 
 
 IS60 
 
 1330 
 
 990 
 
 1430 
 
 1 1 2a 
 991 
 
 '300 
 
 '030 
 
 8/4 
 
 ' 3ao 
 
 444 
 7.J4 
 
 'no 
 es2 
 
 7oa 
 
 /040 
 833 
 6S3 
 
 976 
 77/ 
 4/2 
 
 848 
 4 84 
 .544 
 
 781 
 4/7 
 440 
 
 7'0 
 
 S6I 
 44S 
 
 6SI 
 J/4 
 408 
 
 40/ 
 474 
 377 
 
 SSI 
 441 
 3SO 
 
 3 
 
 J-2&Z4 
 -»/ 73 J 
 3308S- 
 
 S'3640 
 66370 
 S3. 630 
 
 'Z9o 
 
 /030 
 
 SI') 
 
 1 1 10 
 8S4 
 701 
 
 91,9 
 774 
 4/4 
 
 i>'6/ 
 6»» 
 S9b 
 
 7 7S 
 619 
 ■99 1 
 
 70s 
 563 
 
 447 
 
 444 
 
 SI 6 
 4-04 
 
 S96 
 476 
 37y 
 
 .SS9 
 ■993 
 3SI 
 
 S17 
 
 ■913 
 337 
 
 4 84 
 387 
 307 
 
 43/ 
 344 
 273 
 
 3 87 
 304 
 241- 
 
 3S2 
 291 
 2X3 
 
 32 3 
 3SS 
 
 304 
 
 248 
 
 277 
 22/ 
 /7J 
 
 
 11000 VOLTS DELIVERED 
 
 6 
 MILES 
 
 7 
 MILES 
 
 8 
 MILES 
 
 9 
 
 MILES 
 
 10 
 MILES 
 
 II 
 
 MILES 
 
 12 
 MILES 
 
 13 
 MILES 
 
 14 
 
 MILES 
 
 16 
 MILES 
 
 16 
 
 MILES 
 
 18 
 
 MILES 
 
 20 
 
 MILES 
 
 22 
 
 MILES 
 
 24 
 MILES 
 
 26 
 MILES 
 
 28 
 
 MILES 
 
 
 6 ^<3 O0£) 
 ^<30 «>00 
 ^f"^0 000 
 
 / 032 000 
 9S4O0O 
 S7-fSOO 
 
 / y 400 
 
 /7*oO 
 'iSOO 
 
 /6 60O 
 /J-300 
 /4 300 
 
 /4 JOO 
 /3400 
 '3 400 
 
 13 900 
 II 900 
 II 100 
 
 1 1 600 
 
 /07O0 
 99x0 
 
 '060Q 
 
 9 7SO 
 9000 
 
 9700 
 9900 
 
 S3i0 
 
 S42 
 *230 
 7620 
 
 S300 
 76SO 
 7OS0 
 
 7 7SO 
 7'SO 
 6600 
 
 73SC 
 6 6 to 
 6X00 
 
 ^■fSO 
 S9SO 
 SSOO 
 
 SSOO 
 S3SO 
 4440 
 
 SX7a 
 4 8 70 
 4 SOO 
 
 4 SSO 
 44SO 
 
 4 1X0 
 
 44 Jo 
 4/20 
 
 3 SlO 
 
 4/J-o 
 38ZO 
 JJ4« 
 
 
 sS'OO 000 
 -^Jo 000 
 ■<oo 000 
 
 79S 000 
 7'SSOO 
 4i3COOO 
 
 14900 
 
 I3SO0 
 '3 000 
 
 I3SOO 
 
 lisao 
 '0300 
 
 1 1 300 
 
 JOIOO 
 
 9000 
 
 10 000 
 
 S9S0 
 
 e 000 
 
 9970 
 S07O 
 7300 
 
 9 170 
 73Sa 
 6SS0 
 
 74 J-O 
 4 7io 
 9000 
 
 69O0 
 6200 
 S.SSO 
 
 6400 
 S770 
 SI So 
 
 S9SO 
 S3go 
 4 SOO 
 
 S600 
 So SO 
 4- SOO 
 
 soao 
 4470 
 4000 
 
 4480 
 4030 
 3400 
 
 40S0 
 3470 
 3270 
 
 3 730 
 3370 
 3000 
 
 3 4 JO 
 3 100 
 17S0 
 
 3200 
 
 2880 
 
 XS70 
 
 
 3 .5~0 000 
 300 000 
 3 SO 000 
 
 •rs & ■s-00 
 
 ■477 000 
 397 SOO 
 
 'OSOO 
 S930 
 7470 
 
 8 9SO 
 
 7iS0 
 «4oo 
 
 7S30 
 i7O0 
 S (,00 
 
 A9io 
 S9S0 
 4 4SO 
 
 i>370 
 S3i0 
 •44»0 
 
 SiSo 
 
 -9970 
 4070 
 
 S3SO 
 4460 
 3730 
 
 ■9S30 
 4/20 
 3440 
 
 497c 
 3*30 
 
 3xaa 
 
 4 ISO 
 3S60 
 3910 
 
 34/0 
 33iO 
 2 800 
 
 3480 
 2440 
 2440 
 
 3/30 
 2 470 
 
 2340 
 
 2 840 
 3930 
 X030 
 
 2630 
 3x30 
 1 8 70 
 
 241 
 3060 
 1 730 
 
 2230 
 1910 
 '600 
 
 tfOOO 
 CO 
 
 2 1 / ^00 
 767772 
 /330T9 
 
 336 -tSO 
 266 900 
 2/ / 9SO 
 
 6300 
 So'^o 
 3960 
 
 .5"400 
 4 300 
 
 3340 
 
 ■9 730 
 3 760 
 39il> 
 
 4 200 
 3 340 
 2 640 
 
 3 7»0 
 30IO 
 
 3440 
 2 740 
 ;»/JO 
 
 3/J^O 
 Ji/O 
 19 SO 
 
 24/0 
 23/0 
 '»30 
 
 370a 
 XiSO 
 noo 
 
 XS30 
 30 10 
 'StO 
 
 2360 
 'SSO 
 '4So 
 
 2 1 oa 
 
 ' 670 
 '330 
 
 IS90 
 
 'SOO 
 "SO 
 
 I7XO 
 /370 
 '070 
 
 JS70 
 
 '3SO 
 
 990 
 
 '4 SO 
 'ISO 
 9'0 
 
 /J JO 
 
 layo 
 »se 
 
 
 
 2^ 
 
 /OSS&O 
 
 /67 i-OO 
 /33 210 
 7«'JJ-30 
 
 3 ISO 
 2440 
 
 1 Ho 
 
 2 700 
 
 3 19-0 
 1700 
 
 3 370 
 1 S70 
 /48o 
 
 3 100 
 
 two 
 
 '320 
 
 i»90 
 
 '■990 
 
 "to 
 
 '730 
 '360 
 
 /oto 
 
 'S70 
 
 '390 
 
 990 
 
 '*SO 
 'ISO 
 910 
 
 '3SO 
 
 '07a 
 
 89 S 
 
 /260 
 
 '000 
 
 790 
 
 ' 'So 
 93S 
 7-*o 
 
 'OSO 
 83S 
 660 
 
 99S 
 747 
 J91. 
 
 840 
 480 
 S40 
 
 787 
 422 
 4fJ- 
 
 72 J 
 S7S 
 ■4tl 
 
 67 S 
 S3S 
 •42 2 
 
 s 
 
 S3&34 
 4/ 73S 
 
 3 3orf- 
 
 r3 640 
 <i6 370 
 ^■2 630 
 
 ' S70 
 '3SO 
 
 990 
 
 1390 
 
 '070 
 
 SSO 
 
 / /70 
 440 
 740 
 
 '040 
 i3i 
 640 
 
 990 
 7S0 
 J90 
 
 eso 
 
 6tS 
 S90 
 
 7rs 
 
 63S 
 
 ^9S 
 
 710 
 S7t 
 ■9S7 
 
 670 427 
 J37 JO 3 
 414 3 4J- 
 
 SSS 
 47/ 
 37/ 
 
 S30 
 
 ■418 
 330 
 
 ■9 69 
 376 
 396 
 
 4XS 
 391 
 
 270 
 
 393 
 
 3/1 
 247 
 
 340 
 28f 
 iXt 
 
 33 J 
 
 248 
 -r/J, 
 
 The heating limitations may. for the shorter distances, particularly if insulated or concealed conductors are employed, necessitate tha 
 ase of larger conductors resulting id a correspondingly less transmission lo8$ In the case of insulated or concealed conductors should the 
 k.v.a valueir fal' near or to the left of the heavy line consult Table XXV for insulated or Table XXIII for bare conductors The reactance lor 
 the larger conductors may be excessive particularly for 60-cycle service, producing excessive voltage drop This may be obviated by installinf 
 two or more parallel circuits or using three-conductor cables. For single-phase circuits the k.v.a. will be one-half the table values 
 
QUICK ESTIMATING TABLES 
 
 TABLE XVI-QUICK ESTIMATING TABLE 
 
 CONDUCTORS 
 
 KILOVOLT- AMPERES. 3 PHASE. WHICH MAY BE DELIVERED AT THE FOLLOWING VOLTAGES OVER THE VARIOUS 
 CONDUCTORS FOR THE DISTANCES STATED. BASED UPON THE FOLLOWING i2r LOSS (EFFECT OF CHARGING CURRENT 
 
 FOR LOAD POWER-FACTOR OF I00%-8.66% LOSS- 10.0% LOSS 
 
 i 
 
 CO 
 
 ffi 
 
 COPPER 
 
 AREA 
 
 IN 
 CIRCULAR 
 
 MILS 
 
 ALUMINUM 
 
 AREA 
 
 IN 
 
 CIRCULAR 
 
 MILS 
 
 
 
 
 
 12 
 
 000 VOLTS DELIVERED 
 
 6 
 MILES 
 
 7 
 MILES 
 
 8 
 
 MILES 
 
 9 
 
 MILES 
 
 10 
 
 MILES 
 
 II 
 
 MILES 
 
 12 
 
 MILES 
 
 13 
 
 MILES 
 
 14 
 MILES 
 
 15 
 MILES 
 
 16 
 
 MILES 
 
 18 
 
 MILES 
 
 20 
 MILES 
 
 22 
 
 MILES 
 
 24 
 MILES 
 
 26 
 
 MILES 
 
 28 1 
 MILES 
 
 
 ^ooooo 
 SSO ooo 
 
 '' 033 000 
 
 9.5:9 000 
 
 87<SOO 
 
 23/00 
 2/200 
 /9700 
 
 '9 eoo 
 / Szoo 
 
 /4>900 
 
 /S90O 
 /990O 
 
 /4300 
 /3 300 
 
 /a 700 
 
 // 800 
 
 13 Lao 
 II LOO 
 
 10 800 
 
 II Loo 
 
 loLOO 
 9 8SO 
 
 10700 
 9 too 
 
 9 / 00 
 
 f9ao 
 9/00 
 
 84SO 
 
 93SO 
 gsao 
 7900 
 
 8LS0 
 
 7 9 so 
 7400 
 
 7 700 
 7/00 
 6L00 
 
 L9SO 
 L3SO 
 S900 
 
 6>300 
 S900 
 
 S400 
 
 S800 
 S300 
 4920 
 
 S3S0 49SO 
 9%00 4 SSO 
 
 
 Soo ooo 
 4 so ooo 
 
 •400000 
 
 79SOOO 
 7/SSOO 
 636 OOO 
 
 liooc 
 
 /•»200 
 
 /37OQ 
 
 m 3.00 
 
 /3300 
 /300a 
 /0 700 
 
 /O LOO 
 9490 
 
 /O 700 
 9LO0 
 
 es4o 
 
 g6oo 
 
 7 770 
 
 890O 
 SOOO 
 7 130 
 
 9300 
 7380 
 LS70 
 
 7iOO 
 68Sa 
 
 LlOO 
 
 L400 
 S700 
 
 6iio 
 LO 00 
 S340 
 
 S900 
 S300 
 9 7SO 
 
 S3S0 
 4 800 
 
 4370 
 
 48SO 
 4300 
 38»o 
 
 94SO 
 9000 
 3S6a 
 
 4100 
 3690 
 
 J39a 
 
 3 800 
 3420 
 3 0S'0 
 
 
 <3SOOOO 
 
 3oo ooo 
 
 3.SO ooo 
 
 SS^SOO 
 47 7 000 
 39/ SOO 
 
 i-x^oo 
 
 loLoo 
 
 8900 
 
 104,00 
 
 9/00 
 
 7im30 
 
 79L0 
 iiTO 
 
 8370 
 7080 
 
 S930 
 
 74-40 
 L370 
 J-330 
 
 S790 
 4 8S0 
 
 L3O0 
 
 S3IO 
 *440 
 
 S730 
 4foo 
 4100 
 
 S300 
 9SS0 
 38IO 
 
 4 9LO 
 43SO 
 3SS0 
 
 ALSO 
 3980 
 3330 
 
 4I30 
 3S40 
 396a 
 
 373a 
 
 3190 
 21,70 
 
 J3S0 
 3S90 
 3430 
 
 3 too 
 
 3 6 so 
 2220 
 
 3 86a 
 34S0 
 20S0 
 
 366a 
 3370 
 1 900 
 
 CO 
 
 /tT77Z 
 /33 079 
 
 336-^30 
 ZAi8O0 
 2 // 9SO 
 
 S9T 
 41IO 
 
 SI3Q 
 -4033 
 
 SL30 
 4 980 
 3f30 
 
 sooo 
 3980 
 
 3140 
 
 -^J-oo 
 
 3S90 
 3 9 30 
 
 4 090 
 33S0 
 XSLO 
 
 3 7S0 
 3990 
 3.3S0 
 
 3 460 
 3~7S0 
 2/70 
 
 32lO 
 3S60 
 3010 
 
 3 000 
 3390 
 / 8 80 
 
 2S/0 
 22+0 
 / 740 
 
 J.S0O 
 
 1990 
 
 /S70 
 
 2 3 so 
 179/ 
 /4 10 
 
 3040 
 
 /63a 
 
 13 90 
 
 / 970 
 / 490 
 1 1 90 
 
 1730 
 13 90 
 / 04o 
 
 /6/0 
 128 
 
 / 
 
 /OSStO 
 i 1,358 
 
 /t,7 800 
 /33 220 
 /OSS30 
 
 3 7 JO 
 2 3.SO 
 
 SS4Q 
 3. 0/0 
 
 38IO 
 3330 
 I7I0O 
 
 3 soo 
 
 1 970 
 
 / S70 
 
 3 3 SO 
 1780 
 /4-I0 
 
 mo 
 1 390 
 
 1 870 
 I480 
 
 1170 
 
 I730 
 1370 
 1090 
 
 I6/0, 
 /370 
 / Q/O 
 
 /SOO 
 
 // 80 
 
 940 
 
 1410 
 
 /no 
 
 880 
 
 /2SO 
 990 
 7 80 
 
 I/30 
 990 
 700 
 
 /03O 
 SIO 
 &40 
 
 930 
 740 
 
 86S 
 a93 
 
 go 3 
 
 t. 3S 
 
 3 
 •5 
 
 41 738 
 33088 
 
 83 t40 
 it, 370 
 S3 &3o 
 
 / »«0 
 /490 
 
 // »o 
 
 / S9o 
 /X70 
 / 0/ 
 
 /39O 
 IIIO 
 884 
 
 / 340 
 990 
 790 
 
 /no 
 89 
 7/0 
 
 / 010 
 810 
 
 <-«o 
 
 930 
 740 
 S90 
 
 Sio 
 
 L80 
 S4 
 
 790 
 L40 
 Soo 
 
 740 
 
 sro 
 
 472 
 
 700 
 S60 
 
 440 
 
 &20 
 490 
 3fO 
 
 SbO 
 
 440 
 3S0 
 
 SIO 
 400 
 330 
 
 460 430 
 370 343 
 39S 373 
 
 900 
 3'f 
 2SZ 
 
 
 13 200 VOLTS DELIVERED 
 
 
 
 6 
 
 MILES 
 
 7 
 
 MILES 
 
 8 
 
 MILES 
 
 9 
 
 MILES 
 
 10 
 
 MILES 
 
 II 
 
 MILES 
 
 12 
 
 MILES 
 
 13 
 
 MILES 
 
 14 
 
 MILES 
 
 15 
 MILES 
 
 16 
 MILES 
 
 18 
 
 MILES 
 
 20 
 MILES 
 
 22 
 
 MILES 
 
 24 
 
 MILES 
 
 26 
 MILES 
 
 28 
 MILES 
 
 
 iSOOOO 
 600 OOO 
 SSOOOO 
 
 / 03-3 000 
 9SAOOO 
 87-» SOO 
 
 27^00 
 
 2 5 700 
 Z3 9O0 
 
 3AO00 
 13000 
 30.400 
 
 3090a 
 /f 300 
 17900 
 
 i»ioe 
 17 too 
 /S900 
 
 /47O0 
 /S400 
 /4 300 
 
 IS300 
 14000 
 13 000 
 
 14000 
 13 800 
 x/ 900 
 
 I3900 
 II 800 
 II 000 
 
 13 000 
 II 000 
 /0300 
 
 n zoo 
 10300 
 
 9SS0 
 
 i09oa 
 9 4J-0 
 8 9SO 
 
 9 300 
 8SS0 
 
 7 9S0 
 
 8 370 
 7700 
 7 /SO 
 
 7 600 
 7000 
 
 LSOO 
 
 7 000 
 
 L400 
 S9SO 
 
 6 4SO 
 S930 
 SSOO 
 
 (3 OOO 
 SSOO 
 Si 00 
 
 
 ■soo ooo 
 ■fSo ooo 
 •400 ooo 
 
 79SOOO 
 7/ S SOO 
 
 asiooo 
 
 X/ Soo 
 /^3QO 
 
 /7300 
 
 J8SOO 
 
 /Ltoa 
 14 800 
 
 / i/00 
 
 P9SOO 
 
 '3 900 
 
 /4 300 
 /3 900 
 
 / / .TOO 
 
 /3 900 
 // 600 
 /0300 
 
 II 70a 
 /O LOO 
 f t30 
 
 I0700 
 9 700 
 SLSO 
 
 9930 
 
 S930 
 9000 
 
 9330 
 8380 
 7400 
 
 SLoa 
 7 7SO 
 L 900 
 
 90S0 
 
 73S0 
 6 4S0 
 
 7 ISO 
 L4S0 
 S7SO 
 
 6 4S0 
 SSOO 
 
 S 170 
 
 SSSO 
 S300 
 
 4 7/0 
 
 S3SO 
 4 SSO 
 
 -4 3ao 
 
 4960 
 9460 
 
 4ooa 
 
 9 6IO 
 4140 
 3 70a 
 
 
 3 so ooo 
 
 .300000 
 :iSoooo 
 
 SS& SOO 
 477000 
 3 9 7 SOO 
 
 /S/OQ 
 /O900 
 
 /Z900 
 //ooo 
 9a.ro 
 
 1 1300 
 9 QSO 
 
 SOSO 
 
 /oooo 
 SS70 
 
 7IS0 
 
 9 030 
 
 7730 
 
 C4SO 
 
 s 300 
 
 70i0 
 S870 
 
 7S2a 
 
 L430 
 S3S0 
 
 Lfso 
 S930 
 19<o 
 
 L4S0 
 SS30 
 -4620 
 
 6 03Q 
 S /SO 
 
 .4300 
 
 SLSo 
 4 830 
 4040 
 
 SOOO 
 4 380 
 3 too 
 
 4SIO 
 3 9 60 
 3 320 
 
 4 100 
 3 SIO 
 3 93o 
 
 3 7t^O 
 3 Z'O 
 3 700 
 
 3 470 
 3960 
 34 90 
 
 3 330 
 3 760 
 
 3 3iO 
 
 oooo 
 ooo 
 oo 
 
 3.1 1 «00 
 m772 
 
 /33079 
 
 336-420 
 36t, too 
 
 2>/9SO 
 
 ^ /OO 
 
 yzio 
 
 S70Q 
 
 7F00 
 i, 300 
 ^9oc 
 
 i SSO 
 S930 
 4XtO 
 
 L070 
 4830 
 3 SOO 
 
 S4SO 
 -♦330 
 
 3430 
 
 4970 
 3990 
 3 1 10 
 
 4 SSO 
 3620 
 3 9SO 
 
 4300 
 
 3330 
 2*30 
 
 3900 
 3 100 
 34SO 
 
 3 640 
 3 890 
 
 3 3 80 
 
 3430 
 271 
 3 140 
 
 3 030 
 
 2 4'0 
 1 9oa 
 
 3730 
 3160 
 
 1710 
 
 3 4 SO 
 1 9 70 
 /SSO 
 
 3370 
 1 910 
 1 420 
 
 3 lao 
 
 1 660 
 
 1310 
 
 ' 9SO 
 isso 
 / 230 
 
 o 
 
 / 
 
 /OS SiO 
 
 /b7 eoo 
 
 /33230 
 /OSS30 
 
 4SSO 
 3S9C 
 2»^0 
 
 3-/OC 
 
 3070 
 
 3440 
 
 3410 
 31,90 
 3 I4Q 
 
 3030 
 
 3390 
 / 900 
 
 3 730 
 3 ISO 
 / 710 
 
 34 to 
 1 9 bo 
 
 /SLO 
 
 3370 
 1 900 
 1430 
 
 2 /OO 
 1 L60 
 IZ30 
 
 1 9S0 
 ' S90 
 1330 
 
 1 830 
 1430 
 1 140 
 
 1 7O0 
 1340 
 / 070 
 
 / SIO 
 
 1 300 
 9 so 
 
 /360 
 
 / 070 
 
 8sa 
 
 1340 
 
 fsa 
 
 710 
 
 1 1 3a 
 9oo 
 7/0 
 
 losa 
 
 830 
 
 66a 
 
 9 7S 
 770 
 
 6 10 
 
 1 
 
 J2 «2-» 
 4/73g 
 33099 
 
 93i-to 
 
 i * 370 
 
 sa i3o 
 
 2 3-50 
 / ffOO 
 
 /■9%a 
 
 /ISO 
 
 /sjo 
 /aao 
 
 / iSO 
 /340 
 / 070 
 
 / soo 
 
 /3O0 
 
 9 SO 
 
 1 3S0 
 
 / ogo 
 
 esc 
 
 1 330 
 9SS 
 
 77» 
 
 J 1 3a 
 900 
 7/0 
 
 1030 
 830 
 (to 
 
 9*0 
 770 
 S/0 
 
 900 
 73 
 J^70 
 
 840 
 680 
 S30 
 
 7S0 
 600 
 470 
 
 670 
 -J'-fO 
 -(SO 
 
 6/0 
 490 
 390 
 
 S60 
 4SO 
 360 
 
 SIO 
 430 
 330 
 
 -»»o 
 390 
 300 
 
 
 15 000 VOLTS DELIVERED 
 
 6 
 MILES 
 
 7 
 MILES 
 
 8 
 
 MILES 
 
 9 
 MILES 
 
 10 
 MILES 
 
 II 
 
 MILES 
 
 12 
 
 MILES 
 
 13 
 MILES 
 
 14 
 MILES 
 
 16 
 MILES 
 
 16 
 MILES 
 
 18 
 MILES 
 
 20 
 MILES 
 
 22 
 MILES 
 
 24 
 
 MILES 
 
 26 
 MILES 
 
 28 
 
 MILES 
 
 
 i-SO 000 
 600000 
 SSOOOO 
 
 / 033 000 
 9S4 000 
 87-* soo 
 
 di&OOO 
 33200 
 3OLO0 
 
 3/ 000 
 19SOO 
 2*300 
 
 %7 000 
 X'i'fOO 
 23000 
 
 24 OOO 
 32200 
 30S00 
 
 2/ LOO 
 19900 
 I8400 
 
 /9 Soo 
 
 18300 
 /L700 
 
 18000 
 ILioo 
 /S300 
 
 /6L00 
 
 /S300 
 I4IOO 
 
 IS400 
 14200 
 
 /3IOO 
 
 14900 
 
 /3300 
 /2 300 
 
 /3SOO 
 /2400 
 /ISOO 
 
 /3 000 
 II 1 00 
 
 /ozoo 
 
 10900 
 9 9SO 
 9200 
 
 9900 
 
 9 100 
 8 3 SO 
 
 9 000 
 8 3 00 
 7 6SO 
 
 S300 
 7 6SO 
 7 OSO 
 
 7700 
 7100 
 SSSO 
 
 
 SOOOOO 
 4SOOOO 
 
 •qoo 000 
 
 7 9SOOO 
 7/SSoo 
 ^36 000 
 
 37 800 
 3SOO0 
 
 22 2 00 
 
 23*00 
 2/-»oo 
 
 19/00 
 
 30 eoo 
 /8 700 
 
 / 4,700 
 
 I8LO0 
 /L700 
 14 800 
 
 IL700 
 /SOOO 
 13 300 
 
 /S300 
 /3LOO 
 I3IOO 
 
 /3900 
 /3 soa 
 II 100 
 
 13 goo 
 /I soo 
 
 I0300 
 
 // 900 
 /0 7oa 
 
 9 S40 
 
 11 /OO 
 /OOOO 
 8 900 
 
 /0900 
 93S0 
 9340 
 
 9300 
 
 83S0 
 
 7 430 
 
 83SO 
 7SO0 
 L L70 
 
 7 LOO 
 6 800 
 L 07a 
 
 69SO 
 L3SO 
 
 SS60 
 
 6400 
 
 S 7S0 
 SI30 
 
 J9SO 
 S3S0 
 
 4 770 
 
 
 3 SO 000 
 300 000 
 3.SO 000 
 
 SSL Soo 
 477 000 
 39 7 soo 
 
 /94O0 
 /6600 
 13900 
 
 /6L00 
 19300 
 // 900 
 
 t4SOO 
 /X-^OO 
 
 /OAOO 
 
 13900 
 
 9 3 to 
 7 810 
 
 II LOO 
 9960 
 8 330 
 
 lo Loo 
 9 0S0 
 7S70 
 
 9L90 
 8 390 
 L940 
 
 9940 
 
 7660 
 
 6410 
 
 S-ioo 
 7 100 
 S9SO 
 
 7 7SO 
 6 4->o 
 
 7 360 
 6220 
 ^2/0 
 
 6460 
 SS30 
 4630 
 
 S8IO 
 SOOO 
 4 1 60 
 
 S380 
 4 S30 
 3 790 
 
 4890 
 4 ISO 
 3 470 
 
 4 470 
 3 S30 
 3 3 00 
 
 -* !SO 
 
 3 SSO 
 3990 
 
 oooo 
 ooo 
 oo 
 
 3.1 / too 
 /t777Z 
 / 33 079 
 
 33<, 430 
 3i6 soo 
 
 3 1 1 9SO 
 
 // 700 
 9330 
 73sa 
 
 / 000 
 
 9000 
 £300 
 
 9790 
 
 7000 
 
 ■SS/o 
 
 L330 
 4900 
 39fO 
 
 7O30 
 
 J-*oo 
 
 4410 
 
 i 39o 
 S090 
 4010 
 
 S Sio 
 4 LLO 
 .? 6 70 
 
 S4IO 
 -J 300 
 339a 
 
 S030 
 4 OOO 
 3 ISO 
 
 4 6 80 
 3 730 
 29-»0 
 
 3 SOO 
 
 3 740 
 
 3 9 10 
 3 no 
 
 2 4S0 
 
 3 SIO 
 Z 800 
 3 3 to 
 
 3300 
 
 3S40 
 3 OOO 
 
 3 930 
 2330 
 / 840 
 
 3 700 
 2 ISO 
 1 700 
 
 2 SIO 
 2 000 
 / S70 
 
 o 
 
 / 
 
 2 
 
 'OS SiO 
 83<»-f 
 <it 3S8 
 
 /67 800 
 /33 220 
 /ojr.S3o 
 
 Sato 
 
 4L30 
 3t70 
 
 Sooo 
 3t7t> 
 3 ISO 
 
 ■* 3 90 
 
 3 ■4-70 
 27S0 
 
 3090 
 3 9Sa 
 
 / 940 
 
 3SIO 
 3 7 SO 
 33IO 
 
 3 l9o 
 3S30 
 2000 
 
 23/0 
 / S90 
 
 3 700 
 3 190 
 / 690 
 
 19 80 
 / S70 
 
 2 3-tO 
 / 8S0 
 1 470 
 
 2 300 
 1 740 
 1 380 
 
 1 9SO 
 / S40 
 1 330 
 
 I760 
 1 390 
 / 1 00 
 
 / 600 
 I360 
 /OOO 
 
 I4LO 
 / / 60 
 920 
 
 1 3SO 
 
 1070 
 
 SSO 
 
 / 3 SO 
 990 
 7 SO 
 
 3 
 
 S3i^4 
 -♦( 73* 
 3308r 
 
 S3*-»0 
 <S« 370 
 .S3&30 
 
 39/ 
 2 320 
 / 94a 
 
 3990 
 / 9fO 
 / S9o 
 
 XtBo 
 t 740 
 /3»o 
 
 "777? 
 
 /330 
 
 970 
 
 1 740 
 1390 
 
 1 no 
 
 1 S90 
 / 370 
 / 000 
 
 / 4S0 
 // 60 
 930 
 
 / 340 
 
 1 070 
 
 8 SO 
 
 / Z40 
 990 
 790 
 
 / 1 60 
 930 
 730 
 
 / 090 
 8 70 
 69 
 
 9 70 
 
 774 
 614 
 
 870 
 L90 
 -«30 
 
 790 
 L30 
 SOO 
 
 730 
 J^ffO 
 460 
 
 470 
 
 330 
 
 •420 
 
 L30 
 490 
 390 
 
 
 16 f500 VOLTS DELIVERED 
 
 6 
 MILES 
 
 7 
 MILES 
 
 8 
 
 MILES 
 
 9 
 
 MILES 
 
 10 
 MILES 
 
 II 
 
 MILES 
 
 12 
 MILES 
 
 13 
 MILES 
 
 14 
 
 MILES 
 
 15 
 MILES 
 
 16 
 MILES 
 
 18 
 MILES 
 
 20 
 
 MILES 
 
 22 
 
 MILES 
 
 24 
 
 MILES 
 
 26 
 MILES 
 
 28 
 MILES 
 
 
 ASO 000 
 
 600 000 
 
 sso 000 
 
 / 033 000 
 9S.i 000 
 974 SOO 
 
 37SO0 
 
 38900 
 39 Soo 
 31 900 
 
 3-^000 
 
 30ZOO 
 37 900 
 
 30-300 
 2A»00 
 3-^900 
 
 37300 
 34100 
 33 300 
 
 24700 
 
 31 900 
 30 300 
 
 33700 
 xo /oo 
 18600 
 
 2/ 000 
 
 18600 
 1 7300 
 
 l9SOO^ 
 I7100 
 /S9ao 
 
 If 300 
 /6100 
 14 Soo 
 
 I7000 
 IS 100 
 13 900 
 
 IS 100 
 
 /3+00 
 13400 
 
 13 Loo 
 IZOOO 
 II 1 00 
 
 13 300 
 
 I0900 
 /o 100 
 
 II 300 
 /o 000 
 
 9 300 
 
 /o»roo 
 9300 
 
 8600 
 
 9 7Sa 
 8600 
 
 79SO 
 
 
 .SOO 000 
 
 ^so 000 
 ^00 000 
 
 79s 000 
 7/.S- SOO 
 L3L 000 
 
 3 3 700 
 
 30 -^OO 
 
 27000 
 
 39900 
 24000 
 23/00 
 
 3S300 
 ZX900 
 30X00 
 
 3 3 SOO 
 30 300 
 
 /y 000 
 
 30300 
 1 8300 
 /L 300 
 
 18300 
 /LLOO 
 /4 700 
 
 1 L 800 
 
 /s 300 
 
 /3 .5"00 
 
 /SSOO 
 14000 
 
 13 400 
 
 /4 400 
 /3 000 
 // 600 
 
 13 soo 
 13100 
 10 800 
 
 /3LOO 
 // 900 
 
 /o 100 
 
 1 1 300 
 
 10 100 
 
 9 000 
 
 /o /oo 
 9 100 
 8 100 
 
 9 ISO 
 8300 
 7 3SO 
 
 940a 
 7 600 
 6 7SO 
 
 7 7SO 
 7000 
 6 3O0 
 
 LSOO 
 SSOO 
 
 
 3S0 000 
 300 000 
 
 aso 000 
 
 SSL SOO 
 477000 
 
 2 3 7O0 
 ZOXOO 
 
 /6 Soa 
 
 3 300 
 I73O0 
 /9900 
 
 /77OO 
 JS /OO 
 
 fS 800 
 
 I 3-i-OO 
 
 II 3Q0 
 
 14200 
 13 100 
 10/00 
 
 I3900 
 1/000 
 
 // too 
 /o too 
 8 4SO 
 
 /090O 
 9300 
 7 7 80 
 
 /0300 
 8 6SO 
 7330 
 
 9480 
 806O 
 673a 
 
 9 8 SO 
 
 7JJ-0 
 6 3ao 
 
 7900 
 6 700 
 S600 
 
 7 /OO 
 
 60S0 
 
 so so 
 
 6 4SO 
 SSOO 
 4600 
 
 S9oo 
 Soso 
 
 -»220 
 
 S4SO 
 4LSO 
 3 890 
 
 4330 
 36IO 
 
 oooo 
 ooo 
 oo 
 
 2 t 1 Loo 
 / L7 772 
 /33 079 
 
 33«-»20 
 2«i 900 
 
 3 1 1 9SO 
 
 /■*3oo 
 
 // 300 
 
 9920 
 
 /3 300 
 9700 
 
 7 (.SO 
 
 II 7 00 
 
 e^oo 
 
 «7O0 
 
 9 Soo 
 7SSO 
 S9SO 
 
 8S30 
 L7 90 
 3370 
 
 7 770 
 
 Lno 
 
 4 870 
 
 7 100 
 
 s 670 
 
 4 460 
 
 6S60 
 
 s 330 
 
 4 I30 
 
 6 080 
 4S6 
 
 S700 
 4S30 
 3 S80 
 
 SSSO 
 43SO 
 3 3 JO 
 
 4 7J-0 
 3770 
 2970 
 
 •f 240 
 
 3 3 90 
 3 L70 
 
 3 880 
 3 080 
 
 3430 
 
 3 SSO 
 3930 
 3330 
 
 3 3 80 
 3610 
 30LO 
 
 3 040 
 2 9 30 
 
 1910 
 
 o 
 1 
 
 2 
 
 /OS SAO 
 93 <»■» 
 ^i3S8 
 
 /*7 800 
 /33330 
 /0SS30 
 
 7/00 
 
 S&20 
 
 ^^so 
 
 Lioo 
 4%30 
 3830 
 
 ^320 
 
 -^230 
 33-10 
 
 -4720 
 J 7^0 
 2.570 
 
 A3LO 
 3370 
 3 LIU 
 
 3 8L0 
 3 070 
 3430 
 
 3S90 
 3 810 
 3 330 
 
 3370 
 3S90 
 3060 
 
 3040 
 2-f/o 
 / 910 
 
 2 840 
 33SO 
 
 1 780 
 
 3L60 
 3110 
 
 1 670 
 
 2340 
 / *70 
 / 48o 
 
 XI30 
 / LtO 
 / 330 
 
 /930 
 /S3o 
 / 3/0 
 
 1 770 
 /900 
 ///O 
 
 / L30 
 1 Z90 
 / 030 
 
 /300 
 •930 
 
 3 
 
 S3 i34 
 
 <S4370 
 0^2430 
 
 3S'fO 
 J 930 
 3 Q30 
 
 3030 
 3430 
 /9oo 
 
 3i.S0 
 3/30 
 /*7C 
 
 33ifl 
 /AHO 
 
 3 130 
 /i,90 
 /330 
 
 1930 
 /S30 
 /3IO 
 
 1 7 60 
 /410 
 / I30 
 
 / 630 
 1 yoo 
 / 030 
 
 /.S/o 
 
 / 310 
 
 9S0 
 
 1 410 
 
 1 13a 
 
 890 
 
 / 330 
 
 / 060 
 
 »30 
 
 1 1 80 
 940 
 
 740 
 
 / OLO 
 84 
 L70 
 
 960 
 7LO 
 600 
 
 8 90 
 700 
 SLO 
 
 810 
 6 SO 
 .SIO 
 
 400 
 
 ->»o 
 
 ,^ 5 ''""*a''oi8 may, for the shorter distances, particularly if insulated or concealed conductors are employed, necessitate the 
 
 ase 01 larirer cou/luctors. resultin? in a correspondingly less transmission loss. Tn the case of insulated or concealed conductors should ihfl 
 Kv.a values fall near or to the left of the heavy line, consult Table XXV for insulated or Table XXIII for bare conductors. The reactance for 
 ine larcer conductors may he excessive, particularly for 60-cycle service, producing excessive voltage drop. This may be obviated by installing 
 two or more parallel circuits or usine ihreecondurtor cables. For sinrle-phRse circuits the k.v.a. will be one-half the table values. 
 
QUICK ESTIMATING TABLES 
 
 i9 
 
 TABLE XVII-QUICK ESTIMATING TABLE 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 CONDUCTORS 
 
 KILOVOLT- AMPERES. 3 PHASE. WHICH MAY BE DELIVERED AT THE FOLLOWING VOLTAGES OVER THE VARIOUS 
 CONDUCTORS FOR THE DISTANCES STATED. BASED UPON THE FOLLOWING |2rlOSS (EFFECT OF CHARGING CURRENT 
 NEGLECTED) 
 
 FOR LOAD POWER-FACTOR OF IOO*-8.66% LOSS- lOOTb^^ 
 FOR LOAD POWER-FACTOR OF 80«^I0,8% LOSS- 12 5 LOSS 
 
 
 i 
 
 0} 
 oil 
 CD 
 
 COPPER 
 
 ARIA 
 
 IN 
 
 aROULAR 
 
 MILS 
 
 ALUMINUM 
 
 ARIA 
 
 IN 
 
 CIRCULAR 
 
 MIL* 
 
 
 
 20 000 VOLTS DELIVERED 
 
 
 7 
 MILES 
 
 8 
 
 MILES 
 
 9 
 
 MILES 
 
 10 
 
 MILES 
 
 II 
 
 MILES 
 
 12 
 MILES 
 
 13 
 MILES 
 
 14 
 MILES 
 
 16 
 MILES 
 
 16 
 MILES 
 
 18 
 MILES 
 
 20 
 MILES 
 
 22 
 
 MILES 
 
 24 
 
 MILES 
 
 26 
 MILES 
 
 28 
 MILES 
 
 30 
 MILES 
 
 
 
 itso ooo 
 600 ooo 
 
 i^SOOOO 
 
 / 033 OOO 
 fS-iOOO 
 874 soo 
 
 SS ooo 
 •So soo 
 
 16 900 
 
 ■♦7200 
 -<4 200 
 
 11 OOO 
 
 ■»2 70O 
 39100 
 3 b SOO 
 
 39 SOO 
 3S100 
 32800 
 
 31900 
 32 IOC 
 29 SOO 
 
 3ZOO0 
 29 SOO 
 
 27 300 
 
 29 boo 
 27x00 
 
 2S200 
 
 27 soo 2S7O0 
 2S 300 23bOO 
 23 100121 goo 
 
 2 3 600 21 300 
 22100 19 700 
 2OS0O I8ZQO 
 
 19200 171-OC 
 I7700 /b/OO 
 /61O0 11900 
 
 /«ooc 
 
 I970C 
 /3 60C 
 
 /4»oo i3iadi/3Soc 
 /3boa 12 7oo\// 80a 
 /3toO /I 70olio9ao 
 
 
 
 .soo ooo 
 •^soooo 
 ^oo ooo 
 
 79SOOO 
 7/S SOO 
 b3bOOO 
 
 12 SOO 
 3»2oO 
 3 3 9O0 
 
 37000 
 3 3 loo 
 
 2f boo 
 
 33 OOO 
 39600 
 Ztloo 
 
 Z9700 
 
 Zb^oo 
 
 23 700 
 
 27 000 
 21300 
 
 XI boo 
 
 21 700 
 2X 200 
 
 19900 
 
 2 2 900 
 20 SOO 
 19200 
 
 XI 200 
 19000 
 16 900 
 
 19 Soo 
 
 17 800 
 IS 900 
 
 ItSOO 
 16700 
 I1800 
 
 lb soo 
 11800 
 13 200 
 
 II8OO I3SOC 
 I3300 12 1 OQ 
 
 II 800 la Sec 
 
 IX 3aJfTiToo loioe 
 /1 1001/0200 9 soa 
 
 9 890t 9/30 fitta 
 
 9900 
 8 900 
 
 
 
 ■3SO ooo 
 
 3ao ooo 
 2SO ooo 
 
 SS6 SOO 
 ■^ 77 OOO 
 3 9 7 SOO 
 
 19 SOO 
 2S30C 
 2 1 loo 
 
 2S900 
 22IOO 
 / 8SO0 
 
 Z3 OOO 
 
 19 70O 
 /blOO 
 
 X0700 
 17 700 
 11 800 
 
 19 900 
 IblOO 
 /3100 
 
 1 7 XOO 
 /I 700 
 
 /2 3iao 
 
 IS900 
 /3 6O0 
 // 100 
 
 11700 
 12 boo 
 /ObOO 
 
 /3900 
 / 1 800 
 9990 
 
 12900 
 II 000 
 9260 
 
 II SOO 
 9830 
 9330 
 
 /0 300 
 9 SSO 
 
 7110 
 
 9400 
 80S0 
 
 b730 
 
 S6/0 
 7 370 
 
 b 170 
 
 7 9S0 
 6 SIC 
 S70C 
 
 7 380 
 6 32C 
 S290 
 
 6 990 
 S900 
 1910 
 
 
 cooo 
 ooo 
 oo 
 
 /4,7-r7Z 
 133 079 
 
 33b4XO 
 3 bb too 
 2/ / 9SO 
 
 moo 
 
 t1 2O0 
 / 1 2O0 
 
 ISbOO 
 
 /2100 
 
 9 800 
 
 /3 9O0 
 
 // OOO 
 
 971 o 
 
 /3 SOO 
 
 9 9 SO 
 7 910 
 
 II 300 
 9010 
 7/30 
 
 /0100 
 
 82BO 
 
 bS3a 
 
 9 b/O 
 7bS0 
 b030 
 
 8 930 
 7/10 
 SbOO 
 
 9 330 
 bb30 
 SZ30 
 
 7 8/0 
 6220 
 
 4900 
 
 691 
 
 SS 30 
 
 4 3*0 
 
 bZSO 
 1970 
 3 920 
 
 ilto 
 
 1S2C 
 3SbC 
 
 szia 
 IISO 
 
 3 270 
 
 4 8/0 
 3 83C 
 3010 
 
 4 4 40 
 
 3 SSO 
 
 1 170 
 3 310 
 
 
 1 
 
 2 
 
 / OJSiO 
 663Sf 
 
 /b7 800 
 /333XO 
 /OSS30 
 
 893C 
 
 7osa 
 
 SbOC 
 
 7 8io 
 b I70 
 19oa 
 
 b91o 
 S190 
 13bo 
 
 ^zso 
 1910 
 39X0 
 
 SbtO 
 
 1190 
 3Sbo 
 
 S2IO 
 
 1110 
 
 3 270 
 
 ISIO 
 38IO 
 
 1160 
 3S30 
 2 800 
 
 1 1 70 
 3290 
 ZblO 
 
 3 9/0 
 3090 
 21 so 
 
 3470 
 2 740 
 2180 
 
 3120 
 2 470 
 1960 
 
 ZSIO 2 boo 
 2240 20*0 
 /780 Ib30 
 
 2100 
 1900 
 1 SIO 
 
 2330 
 
 /7*a 
 /loo 
 
 zote 
 /bio 
 / 310 
 
 
 3 
 ■♦ 
 
 -♦/ 738- 
 33 off* 
 
 83 bio 
 bb370 
 SZb30 
 
 1130 
 3S10 
 2W/0 
 
 3 870 
 3IOO 
 21b0 
 
 3110 
 27S0 
 Xiao 
 
 3100 
 
 2190 
 /9bO 
 
 2920 
 22SO 
 / 790 
 
 ZSfo 
 Zabo 
 /bio 
 
 Z390 
 / 900 
 /SIO 
 
 22/0 
 / 770 
 /IOC 
 
 3 070 
 /bSO 
 / 310 
 
 1910 
 ISSO 
 1230 
 
 1 720 
 /3»0 
 1090 
 
 ISSO 
 
 1X10 
 
 990 
 
 1110 
 
 IIXO 
 
 S90 
 
 1x90 
 
 /O30 
 
 930 
 
 1190 
 9S0 
 7S0 
 
 /I/O 
 891 
 7O0 
 
 /03a 
 sxo 
 bsa 
 
 
 
 22 000 VOLTS DELIVERED 
 
 
 
 
 7 
 MILES 
 
 8 
 MILES 
 
 9 
 
 MILES 
 
 10 
 
 MILES 
 
 II 
 
 MILES 
 
 12 
 MILES 
 
 13 
 
 MILES 
 
 14 
 MILES 
 
 16 
 
 MILES 
 
 16 
 
 MILES 
 
 18 
 MILES 
 
 20 
 MILES 
 
 22 
 MILES 
 
 24 
 MILES 
 
 26 
 MILES 
 
 28 
 MILES 
 
 30 
 MILES 
 
 
 
 6S0 ooo 
 too ooo 
 sso ooo 
 
 / 033 OOO 
 
 9 SI- OOO 
 
 87-fSOO 
 
 a SOO 
 
 i/200 
 sbSoo 
 
 S9 aoo 
 .53 soo 
 19600 
 
 -5"/ 800 
 17 boo 
 
 11 zoo 
 
 lb boo 
 1Z800 
 
 39700 
 
 1x200 
 
 38900 
 3bloo 
 
 38 900 
 3S6O0 
 33 100 
 
 3SB00 
 3Z9O0 
 30S00 
 
 33 300 
 30 600 
 
 29 100 
 
 31 000 
 27 soo 
 26 600 
 
 29100 
 
 2 b 700 
 
 21800 
 
 2S900 
 2 3 800 
 22 1 00 
 
 23300 
 2/400 
 19 900 
 
 XI 100 
 19100 
 19 000 
 
 /4400 
 17900 
 16S00 
 
 /7 fOO 
 /»400 
 
 IS 200 
 
 '6 600 
 
 IS 300 
 
 11200 
 
 /SSOO 
 13700 
 I3200 
 
 
 
 •soo ooo 
 ■tso ooo 
 <oo ooo 
 
 79s ooo 
 
 7/SSOO 
 63b OOO 
 
 S/3O0 
 
 lb ooo 
 
 11 300 
 
 11 soo 
 
 10 200 
 3 b ooo 
 
 10 000 
 3S800 
 32 OOO 
 
 3S900 
 32 ZOO 
 29900 
 
 3Z 600 
 29 300 
 Zizoo 
 
 29 900 
 26900 
 XIOOO 
 
 27600 
 2180O 
 21XOQ 
 
 3S700 
 23000 
 20 boo 
 
 23900 
 2 1 SOO 
 19 ZOO 
 
 2x100 
 
 Zo 100 
 
 18 000 
 
 30 000 
 17900 
 /6 000 
 
 17900 
 /6I00 
 11100 
 
 16 300 
 /1600 
 /3IOO 
 
 11900 
 
 /3100 
 IZOOO 
 
 /3 800 
 
 /24ao 
 
 // too 
 
 /3800 
 II SOO 
 10300 
 
 II 900 
 /0700 
 9600 
 
 
 
 3SO ooo 
 3oo ooo 
 sso ooo 
 
 SSb SOO 
 -♦77 OOO 
 397 SOO 
 
 3S900 
 30b00 
 2SbOO 
 
 31 loo 
 2b700 
 2Z100 
 
 27 900 
 23 900 
 
 /9 9ao 
 
 2S/00 
 2/100 
 17900 
 
 ZX8O0 
 /9100 
 /6 300 
 
 20900 
 
 n aoo 
 11900 
 
 /9 300 
 /6 SOO 
 13 900 
 
 n 900 
 
 IS300 
 IZ8O0 
 
 /b700 
 /120a 
 /XOOO 
 
 IS 700 
 
 /3300 
 II 200 
 
 /3900 
 /I 900 
 9 9 SO 
 
 12 SSO 
 
 10 700 
 
 9 9S0 
 
 /1 100 
 9700 
 9 ISO 
 
 loioai 9 bSO 
 aioM 82SO 
 7isei 6900 
 
 99SO 
 
 7bSO 
 blOO 
 
 8 SSO 
 
 7iao 
 4ooa 
 
 
 oooo 
 ooo 
 oo 
 
 2/ / 600 
 /bT!72 
 
 / 3-iaT! 
 
 3:3b -»20 
 
 Xbb 800 
 3.1 1 9 SO 
 
 2/ SOO 
 /7300 
 '3bOO 
 
 /9 9O0 
 
 /s /oo 
 
 /I 900 
 
 /beoo 
 
 /3100 
 /O boo 
 
 IS /OO 
 
 12100 
 
 9 Soo 
 
 /3700 
 II 000 
 9bSO 
 
 12 too 
 
 /o /oo 
 
 7920 
 
 II 600 
 9300 
 7 300 
 
 10 Soo 
 8bS0 
 6790 
 
 lo 100 
 9090 
 b330 
 
 9 ISO 
 7 SSO 
 
 S9ao 
 
 8100 
 6700 
 
 7ss(a 6 SSO 
 
 6oSd\ SSOO 
 
 nsof 1320 
 
 baoo 
 
 SOSO 
 3 960 
 
 SSOO 
 IbSO 
 3 6SO 
 
 S10O 
 1 320 
 339a 
 
 SOSO 
 
 V'tt 
 
 
 o 
 
 1 
 z 
 
 /ossto 
 
 66 3i^ff 
 
 / b7 fOO 
 /33 Z30 
 /0SS30 
 
 /o 9oo 
 SS20 
 b790 
 
 9 ISO 
 7190 
 S9SO 
 
 8 3 80 
 
 ibso 
 
 S290 
 
 7 SSO 
 S970 
 17 SO 
 
 68bo 
 S1XO 
 1320 
 
 &3O0 
 -»*80 
 3 960 
 
 S820 
 1600 
 3b60 
 
 S100 
 1270 
 3100 
 
 S030 
 3990 
 3 ISO 
 
 1720 
 3710 
 
 2 970 
 
 1 190 
 3320 
 
 2 bio 
 
 3 770 
 2980 
 2370 
 
 3 430 
 2710 
 2160 
 
 3 ISO 
 
 2190 
 1990 
 
 3910 
 
 2 300 
 
 1 830 
 
 2 700 
 Z13O 
 1700 
 
 ISIO 
 /9S0 
 / S90 
 
 
 3 
 
 J2 624 
 •»/ 73 8 
 3308? 
 
 93 bio 
 66 370 
 sr2630 
 
 S3bO 
 1300 
 3 390 
 
 IbSO 
 3770 
 2970 
 
 11 70 
 3310 
 ZblO 
 
 37 SO 
 30I0 
 2370 
 
 31IO 
 2710 
 
 2 160 
 
 3 I30 
 ZSIO 
 
 1 980 
 
 2990 
 2 320 
 1 920 
 
 2 690 
 2 ISO 
 1 b9o 
 
 2 SOO 
 2010 
 IS80 
 
 2310 
 1 890 
 1980 
 
 209C 
 / 670 
 / 320 
 
 / 870 
 /SOO 
 /ISO 
 
 1 700 
 I370 
 /OSO 
 
 /S60 
 
 izso 
 
 990 
 
 I140 
 II60 
 910 
 
 1 310 
 
 I070 
 
 SIO 
 
 / zso 
 /ooo 
 
 790 
 
 
 
 30 000 VOLTS DELIVERED 
 
 
 12 
 MILES 
 
 14 
 
 MILES 
 
 16 
 
 MILES 
 
 18 
 
 MILES 
 
 20 
 MILES 
 
 22 
 
 MILES 
 
 24 
 MILES 
 
 26 
 MILES 
 
 28 
 MILES 
 
 30 
 MILES 
 
 32 
 MILES 
 
 36 
 MILES 
 
 40 
 
 MILES 
 
 kllLES 
 
 48 
 MILES 
 
 S2 
 MILES 
 
 66 
 MILES 
 
 
 
 &SO ooo 
 (•oo OOO 
 
 sso ooo 
 
 / 03 3 OOO 
 
 9 SI OOO 
 
 871 soo 
 
 7Z200 
 66200 
 bl SOO 
 
 bxooo 
 SSfoo 
 
 S2800 
 
 SI 200 
 19 700 
 1b200 
 
 •^8200 
 11200 
 
 n 000 
 
 43200 
 
 39 800 
 3 b 900 
 
 39100 
 
 3 b zoo 
 33 SOO 
 
 3bl00 
 33 200 
 30 800 
 
 33300 
 30 soo 
 
 2810O 
 
 30 900 
 
 Z81O0 
 2*400 
 
 28 900 
 2 b SOO 
 
 21700 
 
 27100 
 21 soo 
 
 23 100 
 
 24/00 
 22/00 
 20J00 
 
 2/bOO 
 19 900 
 IS10O 
 
 I9 700 
 IS 100 
 /6700 
 
 19000 
 
 IbbOO 
 /S100 
 
 lb 600 
 IJ300 
 /IZOO 
 
 /.f400 
 /4200 
 /320O 
 
 
 
 soo ooo 
 ■4 so ooo 
 
 400000 
 
 79SOOO 
 7/SSOO 
 63b OOO 
 
 ssboa 
 so ooo 
 11 soo 
 
 n8oo 
 
 -t2 800 
 39 lao 
 
 11 800 
 37 soo 
 33-400 
 
 37 100 
 33100 
 29 70a 
 
 33100 
 30 000 
 2 b 700 
 
 30I00 
 27 300 
 21 300 
 
 27900 
 2S loa 
 22200 
 
 2S7O0 
 23100 
 20 SOO 
 
 23 900 
 2 1 SOO 
 19 loo 
 
 22300 
 Xoooo- 
 17 900 
 
 Z0900 
 19700 
 Ib700 
 
 IB SOO 
 16 70O 
 
 il ioo 
 
 /670O 
 
 /soao 
 
 I330O 
 
 /szoo 
 /3600 
 12100 
 
 I3900 
 IZSOO 
 It 100 
 
 12 900 
 II Soo 
 10 2do 
 
 // 90a 
 
 /0700 
 9S10 
 
 
 
 3SO ooo 
 3 00 000 
 XSOOOO 
 
 SSbSOO 
 .♦77000 
 J 97SOO 
 
 38 7O0 
 33200 
 2.7900 
 
 3320O 
 Z9100 
 23 900 
 
 29 /oo 
 21900 
 20 too 
 
 xseoo 
 
 22 100 
 19 SOO 
 
 23x00 
 
 1" 900 
 /b700 
 
 21 100 
 19/00 
 /S /OO 
 
 19 100 
 IbbOO 
 /3 90O 
 
 I7900 
 IS300 
 IZ900 
 
 IbbOO 
 11200 
 II 900 
 
 ISSOO 
 13 300 
 1 1 100 
 
 11 Soo 
 12100 
 10100 
 
 1X900 
 
 It 000 
 
 9360 
 
 II boo 
 9 9 so 
 9330 
 
 /otoo 
 90S0 
 
 7S70 
 
 9&90 
 
 az9o 
 
 b910 
 
 8940 
 7660 
 610O 
 
 szoo 
 
 7/00 
 
 S9SO 
 
 
 oooo 
 ooo 
 oo 
 
 X.U iOO 
 1 67 772 
 /33 07f 
 
 33b130 
 3bb 900 
 XI 1 9SO 
 
 2310O 
 ISbOO 
 /1700 
 
 ZO lOO 
 /bOOO 
 /ZbOO 
 
 17 boo 
 11000 
 /I 030 
 
 ISbOO 
 
 12 100 
 
 9 800 
 
 11 100 
 IIZOO 
 89XO 
 
 /Z900 
 /O2O0 
 8020 
 
 II 700 
 9330 
 73SO 
 
 10 800 
 8&IO 
 b790 
 
 10 OOO 
 8000 
 *300 
 
 9370 
 7160 
 SSSO 
 
 8 7 90 
 7000 
 SSIO 
 
 7810 
 6220 
 A900 
 
 70 30 
 .5"* 00 
 
 44/0 
 
 6 390 
 SO90 
 10/0 
 
 S860 
 IbbO 
 3b70 
 
 S1I0 
 4300 
 3390 
 
 S020 
 
 1000 
 
 3 ISO 
 
 
 o 
 
 1 
 2 
 
 /OSSbO 
 
 »3&9-f 
 6b 3S« 
 
 / 67SOO 
 
 /'33zao 
 
 /0SS30 
 
 // 700 
 93bO 
 73S0 
 
 /OOOO 
 
 7910 
 «>3oo 
 
 8BOO 
 b9lO 
 
 SSI a 
 
 79/0 
 bl70 
 1900 
 
 7O30 
 SSSO 
 1110 
 
 6 390 
 SOSO 
 1010 
 
 SSbo 
 4b30 
 3b70 
 
 S1O0 
 4270 
 3 3?0 
 
 Sozo 
 3 970 
 3 ISO 
 
 1b90 
 3 700 
 29*0 
 
 1390 
 3470 
 27«0 
 
 3910 
 3090 
 21S0 
 
 asio 
 
 X790 
 
 3x10 
 
 3190 
 2S30 
 3000 
 
 2930 
 2310 
 I910 
 
 27 00 
 2110 
 I690 
 
 ZSIO 
 I9SO 
 IS70 
 
 
 3 
 
 ■s 
 
 SX. iS2-» 
 -*' 738 
 
 33offy 
 
 S3e.io 
 
 *b370 
 SXb30 
 
 S9IO 
 1610 
 3b9i 
 
 -t^ffO 
 3 980 
 3 Ibo 
 
 13iO 
 31SO 
 Z760 
 
 3 990 
 3 100 
 21S0 
 
 3 190 
 
 z 790 
 
 Z2IO 
 
 3I70 
 ZS30 
 2010 
 
 29 10 
 2320 
 
 1 910 
 
 2b90 
 
 2/40 
 / 700 
 
 2490 
 
 /990 
 
 /S90 
 
 Z310 
 i860 
 I170 
 
 2190 
 I710 
 I3 90 
 
 I910 
 ISSO 
 
 1x30 
 
 1710 
 1390 
 
 IIOO 
 
 IS90 
 /Z70 
 /OOO 
 
 1IS0 
 II 60 
 920 
 
 1310 
 
 /070 
 
 SSO 
 
 l210 
 990 
 7*0 
 
 
 
 33000 VOLTS 
 
 DELIVERED 
 
 
 
 
 
 12 
 MILES 
 
 14 
 
 MILES 
 
 IS 
 
 MILES 
 
 18 
 MILES 
 
 20 
 MILES 
 
 22 
 MILES 
 
 24 
 MILES 
 
 26 , 
 MILES 
 
 28 
 MILES 
 
 30 
 MILES 
 
 32 
 
 MILES 
 
 36 
 MILES 
 
 40 
 MILES 
 
 44 
 MILES 
 
 48 
 MILES 
 
 62 
 MILES 
 
 66 
 MILES 
 
 
 
 (>So ooo 
 
 600Q00 
 
 sso ooo 
 
 / 033000 
 9S1000 
 
 871SOO 
 
 87200 
 9O2O0 
 
 7Sseo 
 
 7'» 700 
 68 800 
 b1800 
 
 bS300 
 bO ZOO 
 Sb700 
 
 SSzoo 
 
 S3 boo 
 S01Q0 
 
 Sz zoo 
 IS zoo 
 
 1 S 300 
 
 17600 
 43 900 
 11 zoo 
 
 4 3 700 
 40200 
 37700 
 
 10 200 
 37100 
 
 31 900 
 
 37 400 
 34400 
 
 3240O 
 
 31800 
 32100 
 30200 
 
 J2 6S0 
 
 30100 
 
 29 300 
 
 2-?/J<I 
 2 6 900 
 ZSZOO 
 
 ZblOO 
 X1IOO 
 
 xxboa 
 
 2 3 800 
 ZI900 
 30 boo 
 
 XI 800 
 
 30100 
 ISSOO 
 
 J / 
 
 I9SOO 
 '7100 
 
 9 700 
 ,■7 ZOO 
 >6 300 
 
 
 
 soo ooo 
 ^soooo 
 '^OO ooo 
 
 79SOQO 
 7'SSOO 
 &3&OO0 
 
 6730O 
 boioo 
 SI ooo 
 
 S7 800 
 
 SI Soo 
 
 1b3O0 
 
 SO boo 
 1S300 
 
 lObOO 
 
 1SOOQ 
 10 ZOO 
 3bOOO 
 
 loiao 
 
 3b300 
 3Z100 
 
 3 b eoo 
 33 000 
 29 soo 
 
 3 3 700 
 JO 200 
 27000 
 
 31100 
 Z7 900 
 2SOOO 
 
 Z9 9ao 
 3S900 
 2 3 200 
 
 27000 
 
 24/ 00 
 ZI 600 
 
 2 5 300 
 22600 
 2O300 
 
 2 2 SOO 
 20100 
 1 9000 
 
 xoxoo 
 
 1 9 100 
 IbXOO 
 
 I9100 
 I6S00 
 in 00 
 
 /bSoo 
 /tlOO 
 /3SOO 
 
 'SSOO 
 13900 
 
 iz soo 
 
 moo 
 
 2?f 
 // 6 10 
 
 
 
 3SO ooo 
 
 300000 
 2 so OOO 
 
 SSbSOO 
 ■177000 
 397 SOO 
 
 nooo 
 
 10200 
 3 3 700 
 
 10 300 
 
 31 soo 
 
 2 9 800 
 
 3S300 
 30 XOO 
 
 zszoo 
 
 31 100 
 
 zb9co 
 22100 
 
 28 Zoo 
 Z1IOO 
 
 zoxoo 
 
 2S700 
 ZI900 
 
 1 9300 
 
 23 SOO 
 20 100 
 16 SOO 
 
 zi 700 
 
 19600 
 IS SOO 
 
 20200 
 17300 
 
 moo 
 
 19 900 
 IblOO 
 13 100 
 
 17 boo 
 IS 100 
 
 IX boo 
 
 IS 700 
 
 13 100 
 
 II ZOO 
 
 11100 
 12000 
 
 10 100 
 
 IZSOO 
 
 /o9oa 
 9 /so 
 
 II TOO 
 
 /OOOO 
 
 81O0 
 
 10 Soa 
 9300 
 
 7 7SO 
 
 10 100 
 
 S6SO 
 
 7x00 
 
 
 oooo 
 ooo 
 oo 
 
 3.1 1 boo 
 /67 772 
 /3-iOTf 
 
 33b130 
 
 a.bb8oo 
 a / / 9SO 
 
 283O0 
 2-2.b00 
 17 900 
 
 21300 
 
 I93O0 
 /S200 
 
 2/200 
 16900 
 
 13 300 
 
 18 900 
 
 IS 000 
 /I 900 
 
 1 7000 
 
 ,3 soo 
 lO 7O0 
 
 isloo 
 
 IZ300 
 
 9 700 
 
 11 200 
 11300 
 S900 
 
 13 100 
 10 SOO 
 8220 
 
 IZ200 
 9t9o 
 
 7b20 
 
 I1 100 
 90S0 
 7120 
 
 /O60O 
 9*SO 
 
 i 6sa 
 
 9 ISO 
 7 SOO 
 S9oa 
 
 a soo 
 
 67 so 
 S3SO 
 
 7700 
 biso 
 
 19SO 
 
 7 100 
 
 S6Sa 
 
 11S0 
 
 6 SSO 
 S ZOO 
 
 1 1 10 
 
 6 loa 
 
 1940 
 3 »/0 
 
 
 o 
 
 1 
 
 2 
 
 /OSSbO 
 bb3Si 
 
 /b780O 
 /33220 
 /0SS30 
 
 11 ZOO 
 II XOO 
 9900 
 
 /Z200 
 9 boo 
 7b20 
 
 /O 600 
 9100 
 6 blO 
 
 91bo 
 7190 
 S9Z0 
 
 ffSXO 
 b 720 
 S330 
 
 7 7 so 
 bl20 
 19SO 
 
 7100 
 SbOO 
 11SO 
 
 iSSo 
 Sl90 
 
 1 100 
 
 bo90 
 19ZO 
 
 38 10 
 
 S 690 
 1SOO 
 3S60 
 
 S300 
 4200 
 3340 
 
 1730 
 3710 
 Z960 
 
 12bO 
 3360 
 2 6 70 
 
 3»70 
 3a 60 
 X130 
 
 3SS0 
 2900 
 32 30 
 
 3 270 
 2 S90 
 2 OSO 
 
 3010 
 2110 
 
 1 910 
 
 
 ^3" 
 
 SXbXI 
 ■91739 
 33 08f 
 
 S>3 bio 
 6b370 
 
 7OS0 
 SbSO 
 -»f 30 
 
 ioso 
 
 18 SO 
 39O0 
 
 S300 
 1230 
 3 330 
 
 1700 
 3760 
 29*0 
 
 1230 
 3290 
 
 X b70 
 
 39S0 
 3090 
 2*20 
 
 3S30 
 2920 
 ZX20 
 
 32fO 
 ZblO 
 20S0 
 
 3020 
 2120 
 1900 
 
 2930 
 2260 
 1770 
 
 2 6SO 
 
 XI 10 
 
 1 660 
 
 2 ISO 
 
 1990 
 I1SO 
 
 21 10 
 I690 
 1330 
 
 1920 
 /SIO 
 / 210 
 
 1 760 
 1110 
 1110 
 
 1 620 
 
 1 yoo 
 1 020 
 
 1 SiO 
 1 XIO 
 
 rso 
 
 
 The heating limitations may, for the shorter distances, particularly if insulated or concealed conductors are employed, necessitate th» 
 ase of larger conductors, resulting in a correspondingly less transmission loss. In the case of insulated or concealed conductors, should the 
 k.v.B. values fall near or to the left of the heavy line, consult Table XXV for insulated or Table XXIII for bare conductors. The reactance for 
 the larger conductors may bo excessive, particularly for eo-cycle service, producing excessive voltage drop. This may be obviated by instaUiii( 
 two or more parallel circuits or using three-conductor cables. For Bin;le-phase circuits the k.v.a. will be one-half the table values. 
 
30 
 
 QUICK ESTIMATI.W. TABLES 
 
 TABLE XVIM-QUICK ESTIMATING TABLE 
 
 CONDUCTORS 
 
 KILOVOLT- AMPERES. 3 PHASE. WHICH MAY BE DELIVERED AT THE FOLLOWING VOLTAGES OVER THE VARIOUS 
 CONDUCTORS FOR THE DISTANCES STATED. BASED UPON THE FOLLOWING |2r LOSS (EFFECT OF CHARGING CURRENT 
 NEGLECTED) Ajjg.c AT66°0 
 
 FOR LOAD POWER-FACTOR OF IOO%-8.66% LOSS- 10.0% LOSS 
 
 FOR LOAD POWER-FACTOR OF 80%- 10.8% LOSS- 12.6 LOSS 
 
 CO 
 
 m 
 
 COPPER 
 
 MCA 
 
 IN 
 
 cinouLjM 
 
 MIL* 
 
 ALUMINUM 
 
 AREA 
 
 IN 
 
 OlltCULAII 
 
 mxA 
 
 40 000 VOLTS 
 
 DELIVERED 
 
 
 
 
 14 
 MILES 
 
 le 
 
 MILES 
 
 IB 
 MILES 
 
 20 
 MILES 
 
 11 
 
 MILES 
 
 24 
 
 MILES 
 
 26 
 
 MILES 
 
 28 
 
 MILES 
 
 30 
 MILES 
 
 32 
 MILES 
 
 36 
 MILES 
 
 40 
 
 MILES 
 
 44 
 MILES 
 
 48 
 MILES 
 
 62 
 MILES 
 
 66 
 MILES 
 
 60 
 
 MILES 
 
 
 iSoooo 
 Aooooa 
 
 ' 033 ooo 
 
 ^s^ ooo 
 
 ST-t SOO 
 
 //oooo 
 loioao 
 93foa 
 
 99 700 
 
 SXOOO 
 
 SS300 
 
 7 SSOO 
 73 OOQ 
 
 74 Soo 
 70 soo 
 
 6 SSOO 
 
 70 000 
 
 64 Soo 
 S9700 
 
 6 4 300 
 S900O 
 S4-700 
 
 S9000 
 S4SOO 
 SOSOO 
 
 SSOOO 
 S07OO 
 4 6 SOO 
 
 s/3ao 
 
 + 7300 
 + 3 700 
 
 4 8 000 
 ++300 
 + / 000 
 
 + 2400 
 
 39+00 
 34.500 
 
 38400 
 3S400 
 33700 
 
 3S0OO 
 33300 
 29800 
 
 33 /OO 
 39S00 
 27 300 
 
 2 9^00 
 27200 
 2.S2O0 
 
 2 7 .TOO 
 25300 
 23+00 
 
 2.r40O 
 2 3 400 
 2/ 800 
 
 
 J-OOOOO 
 •4-SO ooo 
 
 400 ooo 
 
 79S ooo 
 7/J'SOO 
 «3»000 
 
 S-*6oa 
 
 76300 
 67SOO 
 
 74 ooo 
 i^700 
 S93Q0 
 
 6 6 OOO 
 S9300 
 S3.700 
 
 S9200 
 S3 soo 
 47 Sao 
 
 S4aoa 
 4SS00 
 43/00 
 
 49400 
 
 44 Sao 
 
 3 9 soo 
 
 4S600 
 41 000 
 3 4J-00 
 
 43 300 
 38/00 
 33900 
 
 39 £00 
 3S60O 
 3/600 
 
 e7ooo 
 33300 
 
 29 700 
 
 33 000 
 29400 
 2 6400 
 
 39600 
 26700 
 33700 
 
 37000 
 2 + 200 
 2/400 
 
 2+700 
 22200 
 /9 800 
 
 2 2 80c 
 20^00 
 /8200 
 
 21 /oa 
 /9oao 
 '6900 
 
 /9 700 
 /78O0 
 /SSOO 
 
 
 3S0 ooo 
 
 300 ooo 
 
 zso coc 
 
 SS6 -TOO 
 H77 ooc 
 3*>7S'oo 
 
 S9000 
 SO60C 
 4 3 300 
 
 SI 700 
 37 OOO 
 
 4S900 
 3930O 
 3 3 9O0 
 
 ■41 300 
 3S40O 
 39 600 
 
 3 7 400 
 
 32aoo 
 
 2 4 900 
 
 34+00 
 
 39sao 
 
 34700 
 
 31 SOO 
 27200 
 2 2 800 
 
 39 SOO 
 3S300 
 3 1 /OO 
 
 37 SOO 
 23400 
 /9 700 
 
 ZSSOO 
 22/00 
 /SSOO 
 
 33000 
 /9 700 
 /&4O0 
 
 3 700 
 /7 700 
 /4SOO 
 
 /SSOO 
 /6/0O 
 /3SOO 
 
 /7 300 
 /+700 
 /2 300 
 
 /S90O 
 /3 6O0 
 //+00 
 
 /4700 
 
 /3 600 
 /O600 
 
 /3 Soo 
 
 /I soo 
 9 SSO 
 
 oooo 
 
 OOQ 
 OO 
 
 XI I 400 
 /*777J 
 /33079 
 
 336-fao 
 2*& eoo 
 
 a / / fs'o 
 
 3S7O0 
 XS^OO 
 
 3a-*oo 
 
 31 S.OO 
 ZASOO 
 /9<»0O 
 
 27 SOO 
 3-X/OO 
 /74O0 
 
 2SOO0 
 /9 900 
 /S7O0 
 
 21700 
 
 /8/00 
 
 / + 2O0 
 
 20 SOO 
 /6 600 
 /3 /OO 
 
 /9200 
 /S300 
 /X/OO 
 
 /7S00 
 
 /4200 
 / / 200 
 
 /li700 
 /3200 
 /0400 
 
 /S600 
 /Z400 
 
 tsoo 
 
 /3900 
 
 // 000 
 
 8 7/0 
 
 /3SOO 
 9 9 SO 
 7 840 
 
 //400 
 9040 
 7/30 
 
 /O+OO 
 8230 
 4 .S30 
 
 96/0 
 76S0 
 6030 
 
 8 930 
 
 7 / /O 
 S60O 
 
 8330 
 4430 
 .5'230 
 
 o 
 
 / 
 2 
 
 /OSJ^O 
 6b3Sg 
 
 /*7 »00 
 /33 2JO 
 /oss3a 
 
 17 SOO 
 
 /■^ too 
 
 //30Q 
 
 /S^OO 
 
 /2.300 
 
 9 900 
 
 /3900 
 
 / o 900 
 
 g 7/0 
 
 72 SOO 
 9 SSO 
 7?-»0 
 
 // 300 
 
 S9SO 
 7/30 
 
 70400 
 S330 
 6S40 
 
 9 620 
 7S90 
 6030 
 
 8930 
 7QSO 
 S600 
 
 8 330 
 6SS0 
 
 S33a 
 
 7 8/0 
 6 /70 
 4900 
 
 6 940 
 
 S49a 
 
 4 360 
 
 63SO 
 + 9 + 
 3920 
 
 S6S0 
 
 4490 
 3S60 
 
 S3/a 
 4 //o 
 3370 
 
 4 8/0 
 
 3 soa 
 
 3O30 
 
 + + 40 
 3i30 
 2 800 
 
 4170 
 
 3 290 
 
 3 6/0 
 
 3 
 
 ^a4S4 
 
 S3A40 
 
 0-370 
 
 .SX&30 
 
 7P40 
 7070 
 ,5-4/o 
 
 7 7 SO 
 6190 
 
 4»»0 
 SSOO 
 + 370 
 
 csoo 
 ■49SO 
 3^30 
 
 S640 
 4SO0 
 3S70 
 
 ■sno 
 
 4 /3Q 
 J280 
 
 4770 
 3 SIO 
 3020 
 
 4430 
 3S40 
 3SIO 
 
 + /30 
 3300 
 2420 
 
 3 970 
 3090 
 2460 
 
 344a 
 
 2 7 JO 
 3 /So 
 
 3 /OO 
 2+80 
 / 940 
 
 2S20 
 
 2 2Sa 
 
 / 790 
 
 3 SSo 
 3 0&0 
 /640 
 
 3 380 
 / 9/0 
 7S/0 
 
 22/0 
 / 770 
 /+00 
 
 2 070 
 / 6SO 
 /3/0 
 
 
 44 OOO VOLTS DELIVERED 
 
 14 
 MILES 
 
 16 
 
 MILES 
 
 18 
 
 MILES 
 
 20 
 MILES 
 
 22 
 
 MILES 
 
 24 
 MILES 
 
 26 
 MILES 
 
 28 
 MILES 
 
 30 
 MILES 
 
 32 
 MILES 
 
 36 
 MILES 
 
 40 
 
 MILES 
 
 44 
 MILES 
 
 48 
 MILES 
 
 61 
 MILES 
 
 66 
 MILES 
 
 60 
 
 MILES 
 
 
 « ^o ooo 
 <oo ooo 
 
 JSO OOO 
 
 1 033 ooo 
 
 ^s-tooo 
 
 S7t SOO 
 
 /3300a 
 /i^ooa 
 //300C 
 
 //^OOO 
 I07000 
 
 99000 
 
 /03o»a 
 
 9SOO0 
 SSoao 
 
 93000 
 
 sssoo 
 
 79300 
 
 S4SOO 
 77 BOO 
 73 000 
 
 77 soo 
 
 7/SOO 
 66000 
 
 7/SOO 
 44000 
 6/000 
 
 66 SOO 
 
 6/300 
 Si SOO 
 
 42000 
 
 J7200 
 S3 800 
 
 J-S-ooo 
 S3 soo 
 49SOO 
 
 S/SOO 
 47 soo 
 44000 
 
 + 4 .TOO 
 43700 
 39 600 
 
 43 soa 
 38900 
 36000 
 
 3 8 700 
 3S700 
 33000 
 
 3S7ao 
 3300a 
 3 osoo 
 
 3 3 200 
 30 600 
 SSSOO 
 
 31 oao 
 
 28400 
 
 24+00 
 
 
 soo ooo 
 4-so ooo 
 ^oo ooo 
 
 /^.S-ooo 
 7/S'S'oo 
 ^3i ooo 
 
 /oa.ooo 
 S3^oo 
 
 99 700 
 So7oa 
 
 72 ooo 
 
 79 700 
 
 7/600 
 44 ooc 
 
 7/SOO 
 64 SOO 
 S7 7O0 
 
 6S300 
 SS700 
 S3300 
 
 S9S0O 
 S3SO0 
 
 4Soao 
 
 SS200 
 49700 
 44 Zoo 
 
 S/ 300 
 + 4200 
 + /200 
 
 47 SOO 
 + 3000 
 3S400 
 
 + 4800 
 + 300 
 
 34 OOO 
 
 39 soo 
 3 SSOO 
 33000 
 
 3S900 
 3Z2O0 
 28800 
 
 32 600 
 
 39300 
 36/00 
 
 39 f 00 
 26900 
 34000 
 
 27400 
 2+ 800 
 22/00 
 
 3S600 
 23 /oa 
 30 60a 
 
 23900 
 3/ SOO 
 
 /93oa 
 
 
 ^•so ooo 
 ^oo ooo 
 x.-^o ooo 
 
 ssc Soo 
 
 -477 ooo 
 397^00 
 
 7/SOO 
 6/300 
 
 ^/3oe 
 
 62 Foe 
 S3 ^00 
 -*4 8oo 
 
 sssoo 
 
 47 70O 
 3 9 SOO 
 
 SO 300 
 4.3 900 
 3 SSOO 
 
 4S600 
 39 000 
 32 400 
 
 4/ Soo 
 3S700 
 
 39900 
 
 38700 
 33000 
 37 600 
 
 3i-800 
 30 400 
 25400 
 
 3 3 SOO 
 28400 
 23900 
 
 3/ 400 
 3 6 SOO 
 22+00 
 
 27 900 
 2 3 800 
 / 9 900 
 
 2S/O0 
 2/+0O 
 /T900 
 
 33SOO 
 /9SOO 
 /6300 
 
 3O9O0 
 /7SS0 
 /490a 
 
 / 9 3ao 
 /6Soa 
 
 /3 soo 
 
 /7 9ao 
 /S3oa 
 
 /3 800 
 
 / 6 700 
 /4 300 
 
 // 900 
 
 OOOO 
 
 ooo 
 
 • OO 
 
 S. 1 1 4>oo 
 /«7 77a 
 /33 07f 
 
 336 -fao 
 
 366800 
 311 9SO 
 
 ■i3xoo 
 
 3-»-»0o 
 27 /OO 
 
 37900 
 30/00 
 2 3 7O0 
 
 33700 
 2 4»00 
 2 / /OO 
 
 30 300 
 34/00 
 /9000 
 
 37 SOO 
 3/ 900 
 /7300 
 
 3S3(^ 
 30 /OO 
 /SSOO 
 
 33 300 
 /SSOO 
 /4600 
 
 2/400 
 /73O0 
 /3S00 
 
 20 200 
 /6 000 
 /3 6O0 
 
 /S9SO 
 /SOSO 
 // SSO 
 
 /6S00 
 /3/4O0 
 /OSOO 
 
 /S/oa 
 /3 000 
 9 Soo 
 
 /370a 
 
 /090O 
 
 S6S0 
 
 /3 600 
 
 /O 000 
 
 790a 
 
 // 600 
 
 92SO 
 
 7300 
 
 /a soo 
 8 400 
 
 6 7S0 
 
 / / 00 
 800a 
 
 6300 
 
 o 
 
 / 
 
 2 
 
 /OS J to 
 S3 *f-f 
 
 / 67 800 
 733S.O0 
 
 /ossao 
 
 2/700 
 /7/00 
 /3S00 
 
 I9 90O 
 /■*9oo 
 
 1 1 900 
 
 /< VOO 
 /3 300 
 
 /<J.roo 
 
 /S /OO 
 // 900 
 9470 
 
 /3 SOO 
 
 /O SOO 
 
 S400 
 
 /2 40O 
 9970 
 7 SSO 
 
 / /700 
 9300 
 7280 
 
 /OSOC 
 SSSO 
 6 770 
 
 /O /OO 
 7970 
 6 330 
 
 94S0 
 74SO 
 S900 
 
 8 400 
 iiSO 
 S2S0 
 
 7SS0 
 S90O 
 + 730 
 
 69O0 
 S400 
 4 300 
 
 6 300 
 49S0 
 3940 
 
 SSSO 
 
 + 400 
 
 3 4+0 
 
 S4O0 
 + 270 
 3 3 80 
 
 SOSO 
 3 9SO 
 3/60 
 
 3 
 
 -«/ 73? 
 33 OSS 
 
 S3 6-tO 
 £4370 
 ■5-2 430 
 
 /0 700 
 S60O 
 4 7P0 
 
 9400 
 
 7 soo 
 S930 
 
 S3S0 
 4 4*0 
 SX70 
 
 7S20 
 6000 
 ■4 7S0 
 
 6S30 
 S470 
 4330 
 
 6 3S0 
 SOOO 
 3960 
 
 S7S0 
 4630 
 3 660 
 
 S360 
 4300 
 3400 
 
 SOOO 
 + 000 
 3 /70 
 
 4700 
 3 7SO 
 3960 
 
 4 170 
 3340 
 2 430 
 
 3 740 
 3000 
 2 370 
 
 34/0 
 2 730 
 2/40 
 
 3 /SO 
 3 SOO 
 /9S0 
 
 3S90 
 23/e 
 
 / 830 
 
 2 480 
 2/.S-0 
 / 700 
 
 2 SOO 
 2000 
 
 /sao 
 
 
 50 000 VOLTS DELIVERED 
 
 20 
 MILES 
 
 24 
 MILES 
 
 28 
 MILES 
 
 32 
 
 MILES 
 
 36 
 
 MILES 
 
 40 
 
 MILES 
 
 44 
 
 MILES 
 
 48 
 
 MILES 
 
 62 
 MILES 
 
 66 
 
 MILES 
 
 60 
 
 MILES 
 
 64 
 MILES 
 
 72 
 
 MILES 
 
 80 
 MILES 
 
 88 
 MILES 
 
 96 
 MILES 
 
 104 
 MILES 
 
 
 ^So ooo 
 Aoo ooo 
 
 ^SO O OO 
 
 / 033 OOO 
 
 9.r-*ooo 
 
 »7-f JOO 
 
 /xeooe 
 ///ooo 
 
 /0 3 0O0 
 
 /OO ooo 
 
 93 Soo 
 assoc 
 
 S6000 
 79200 
 73 soo 
 
 7SOOO 
 69 300 
 4* a 00 
 
 66S00 
 
 6/SOO 
 S7 000 
 
 60 000 
 
 sssoo 
 
 S/ 300 
 
 ■5-+ 800 
 so 300 
 
 46600 
 
 so 300 
 44200 
 
 + 2 800 
 
 + 4200 
 + 2^roo 
 
 39S00 
 
 43000 
 
 39600 
 
 3 6700 
 
 40000 
 37000 
 
 J430O 
 
 37.roo 
 
 3+600 
 32/00 
 
 J34O0 
 30 700 
 
 2 8.yoo 
 
 3oooa 
 27700 
 
 2 5 400 
 
 27+00 
 2S/00 
 2 3 300 
 
 2S/00 
 33/00 
 3/ 400 
 
 23/00 
 3/200 
 /9700 
 
 
 •S'OO OOO 
 •^SO OOO 
 
 ■400 ooo 
 
 79SOOO 
 7/S^S-OO 
 434 Oao 
 
 93 goo 
 ^3 800 
 
 74 soo 
 
 77300 
 69SOO 
 6/ Soo 
 
 6 6 ZOO 
 S9 Soo 
 S3 000 
 
 ss 000 
 
 S3 300 
 46300 
 
 S/SOO 
 
 46 soa 
 41 300 
 
 4 6 400 
 4/ soa 
 37/00 
 
 43300 
 38 000 
 33700 
 
 J 8 700 
 3Sooa 
 3o 900 
 
 3S600 
 
 33300 
 
 3SSOO 
 
 33 /OO 
 29 900 
 36 soo 
 
 3090a 
 27900 
 34700 
 
 29000 
 24/0O 
 33200 
 
 2.S700 
 23200 
 20400 
 
 23200 
 20900 
 
 /SSOO 
 
 X/ /oa 
 
 /9000 
 /6 9O0 
 
 /9300 
 /7 SOO 
 /S400 
 
 /7 800 
 
 /6/ao 
 
 /+200 
 
 
 3 so OO'O 
 
 joo ooo 
 xso ooo 
 
 ^S& SOO 
 -t77000 
 3 97J-00 
 
 6-t6oa 
 
 SS300 
 4 6 300 
 
 S3 soo 
 46/00 
 3S60O 
 
 A^/00 
 39 SOO 
 33 /OO 
 
 40400 
 3+400 
 
 2S9oa 
 
 3S9O0 
 
 30700 
 
 3S7O0 
 
 33 300 
 
 27400 
 
 23/00 
 
 39300 
 3S/00 
 
 2 / ooa 
 
 24900 
 3 3000 
 / 9 300 
 
 24 soo 
 Z/ 300 
 /7SO0 
 
 2 3 /OO 
 /9 TOO 
 /6SOO 
 
 3/ soo 
 /S4oa 
 /S400 
 
 3 300 
 /7 300 
 /4 SOO 
 
 /7900 
 /S400 
 /Z9O0 
 
 /6 /OO 
 
 /3Sao 
 
 / / 600 
 
 /+ 700 
 
 /2400 
 
 /osoa 
 
 /3400 
 
 / / Soa 
 
 94+0 
 
 /2+00 
 /04OO 
 8 900 
 
 Oooo 
 ooo 
 OO 
 
 2.1 t &0O 
 /<7 772 
 /33 07y 
 
 334+20 
 2 44 900 
 2 / / 9SO 
 
 3 9 /OO 
 3/ /OO 
 29SOO 
 
 33 SOO 
 3S900 
 30 400 
 
 27900 
 32200 
 /7SOO 
 
 34400 
 /9400 
 
 /S300 
 
 3/ 700 
 /7300 
 /3400 
 
 /9SOO 
 /SSOO 
 /2200 
 
 /7SOO 
 /4 /OO 
 // /OO 
 
 /6200 
 /300a 
 /O ZOO 
 
 /SOOO 
 
 // 900 
 
 9430 
 
 /390O 
 // /OO 
 8 7S0 
 
 /3OO0 
 
 /04OO 
 
 8/70 
 
 /2300 
 9 720 
 7440 
 
 /O SOO 
 8 4+0 
 4 8/0 
 
 9 760 
 
 7770 
 
 6 730 
 
 8 8 80 
 7 070 
 SS70 
 
 8/40 
 
 6 480 
 S'/OO 
 
 7S/0 
 
 S9Sa 
 
 47/0 
 
 c 
 / 
 2 
 
 /os-s&o 
 93 t9f 
 *t,3sr 
 
 7 67 SOO 
 /33220 
 /0^S■J30 
 
 /9S00 
 /<S40Q 
 
 /aaoo 
 
 /6 300 
 /X900 
 
 /o soo 
 
 73900 
 // 030 
 S7S/ 
 
 /2 300 
 
 9 64a 
 
 7660 
 
 /O SOO 
 SS70 
 49/0 
 
 9740 
 77/0 
 4/30 
 
 ssto 
 
 70/0 
 SS70 
 
 S/40 
 £430 
 S/00 
 
 7S/0 
 S930 
 + 7/0 
 
 6 970 
 j-s/o 
 
 4 370 
 
 6S/0 
 sf/+0 
 
 ■9aao 
 
 6/00 
 + 822 
 3833 
 
 ,r+2o 
 + 290 
 3+00 
 
 4 SSO 
 3 840 
 3040 
 
 4440 
 
 3S/0 
 X 7S0 
 
 4070 
 
 33/0 
 3 SSO 
 
 3 740 
 2 970 
 2 350 
 
 3 
 S 
 
 ^l 738 
 J3 0SS 
 
 83 6-t^O 
 4 4 370 
 ■S-3 630 
 
 9 690 
 
 7740 
 6/40 
 
 S070 
 6 4SO 
 S/AO 
 
 6 930 
 SS3a 
 ■4370 
 
 6 O60 
 ■4S40 
 3S40 
 
 J"3»o 
 + 300 
 
 3 + /0 
 
 4S4 
 3S70 
 3070 
 
 4400 
 3S30 
 3 790 
 
 4040 
 3330 
 
 3S6a 
 
 3 730 
 2 980 
 2340 
 
 3460 
 2 740 
 3/90 
 
 3230 
 2.5-80 
 3 0SO 
 
 3030 
 2+20 
 / 930 
 
 3 69a 
 2 /SO 
 / 7/0 
 
 3420 
 /930 
 /S30 
 
 2200 
 1760 
 1400 
 
 2020, 
 /4/0 
 /2 80 
 
 /840 
 J 490 
 //SO 
 
 
 60 000 VOLTS DELIVERED 
 
 20 
 MILES 
 
 24 
 
 MILES 
 
 28 
 
 MILES 
 
 32 
 
 MILES 
 
 36 
 MILES 
 
 40 
 
 MILES 
 
 44 
 
 MILES 
 
 48 
 MILES 
 
 62 
 
 MILES 
 
 66 
 MILES 
 
 60 
 
 MILES 
 
 64 
 MILES 
 
 72 
 MILES 
 
 80 
 
 MILES 
 
 88 
 MILES 
 
 96 
 MILES 
 
 104 
 
 MILES 
 
 
 4, so ooo 
 Aoo ooo 
 SSo ooo 
 
 I033 OOO 
 <>S-9 0aO 
 S7-tSOO 
 
 / 73O00 
 /•Sfooo 
 1^^000 
 
 /44OO0 
 /33000 
 /33000 
 
 >2-f000 
 
 //4ooe 
 /06000 
 
 /osooo 
 99SOO 
 93 soo 
 
 f 4000 
 »».roo 
 82200 
 
 944<W 
 
 7't«ol 
 74000 
 
 78800 
 72J-00 
 47300 
 
 73 200 
 
 6 6 soo 
 6/SOO 
 
 66 soo 
 6/30O 
 S7000 
 
 6/ Soo 
 S7000 
 sssoo 
 
 S7 7O0 
 S3300 
 49 + 00 
 
 S4 000 
 
 4970a 
 
 + 4200 
 
 48000 
 44300 
 
 4/ /OO 
 
 + 3 300 
 39 800 
 
 37000 
 
 3f4ao 
 
 362O0 
 3 3 600 
 
 34/00 
 33200 
 30900 
 
 33200 
 30700 
 2SSao 
 
 
 Soo ooo 
 ■t-so ooo 
 •^oo ooo 
 
 79SOOO 
 
 7/ssao 
 *36ooa 
 
 /3300C 
 / 30 OOO 
 /O700O 
 
 / // OOO 
 
 /OO ooo 
 S9ooe 
 
 9Soeo 
 
 as7oc 
 76300 
 
 S3 soo 
 7S300 
 44 7<?0 
 
 7+000 
 
 4 4 700 
 S9 3O0 
 
 6660a 
 6OO0O 
 S3 400 
 
 40 .TOO 
 S4 7O0 
 
 4 3 soa 
 
 SSSOO 
 
 SO 200 
 
 44 SOO 
 
 S/3oa 
 46300 
 41 /OO 
 
 47S00 
 
 43800 
 
 38/oa 
 
 + 4+00 
 +■0 000 
 3J400 
 
 + / 700 
 37600 
 33400 
 
 3700a 
 33300 
 39700 
 
 3330O 
 30000 
 3 6700 
 
 30300 
 37300 
 2+200 
 
 27700 
 
 3S /OO 
 2 2 200 
 
 25400 
 23/00 
 3 Sao 
 
 
 3So ooo 
 
 3 00 000 
 
 xso oae 
 
 ■SS6 soo 
 
 -♦77 OOO 
 3»7^O0 
 
 9300C 
 79^00 
 ^t,7oo 
 
 77 soo 
 66300 
 SSSOO 
 
 66400 
 S6900 
 47600 
 
 SS /OO 
 
 49 SOO 
 41 700 
 
 s/ 700 
 
 + +200 
 37000 
 
 4-6S00 
 39 SOO 
 33300 
 
 43300 
 3 6 300 
 30 300 
 
 39700 
 33/00 
 27900 
 
 3 SSOO 
 30600 
 3S600 
 
 33300 
 39400 
 
 33 soo 
 
 31 000 
 26SOO 
 22200 
 
 29/00 
 2+900 
 20S00 
 
 3SSao 
 33/00 
 /SSOO 
 
 23 200 
 /9 9O0 
 /67OO 
 
 21 100 
 /S 100 
 
 /s/00 
 
 /94ao 
 /6600 
 
 /3 90a 
 
 17 900 
 /SSOO 
 /3 Soa 
 
 oooo 
 ooo 
 
 OO 
 
 3.1 1 coo 
 /330Tf 
 
 J34-f 20 
 3 66 SOO 
 2 / / 9SO 
 
 ■4^*900 
 3 4 300 
 
 4 6900 
 37300 
 
 .J^-too 
 
 40 300 
 32000 
 1S200 
 
 3S/OC 
 3SOOO 
 33000 
 
 3; 200 
 2+800 
 
 /9 too 
 
 2S/00 
 22+00 
 /7 600 
 
 2 SSOO 
 20 300 
 / 6 000 
 
 23 400 
 1 840O 
 /470Q 
 
 2/600 
 /7300 
 /36QO 
 
 20/00 
 /600O 
 /3600 
 
 /S700 
 /4900 
 /I 700 
 
 17 600 
 /+000 
 // 000 
 
 /S600 
 
 /340a 
 
 9 SOO 
 
 /4/ao 
 
 // 300 
 SSSO 
 
 /2900 
 
 /O2O0 
 
 8020 
 
 / 1 70a 
 
 9330 
 7 3S0 
 
 /a Soo 
 86/0 
 6 790 
 
 c 
 1 
 
 JOSSiO 
 «3«1i-f 
 *t3SS 
 
 / 67 Soo 
 733 23a 
 
 /OSS30 
 
 29/00 
 2 27.O0 
 /760O 
 
 2 3-»00 
 /SSOO 
 /-t700 
 
 30 /oa 
 /s soo 
 
 /3 600 
 
 /7 600 
 /390O 
 //OOO 
 
 /S600 
 /2 30O 
 9800 
 
 /4 0OO 
 // /OO 
 
 SS30 
 
 /2 8oo 
 
 /o /OO 
 
 80 20 
 
 // 700 
 9240 
 73 .yo 
 
 /osoo 
 
 SSSO 
 
 679a 
 
 /OOOO 
 7930 
 4300 
 
 9370 
 7410 
 SSSo 
 
 8 79a 
 49+0 
 ss/o 
 
 78/0 
 6/70 
 4900 
 
 7030 
 
 SSSO 
 
 44/0 
 
 4390 
 SOSO 
 
 40/0 
 
 SS60 
 4630 
 3 470 
 
 S4IO 
 4370 
 
 339a 
 
 ■ 3 
 
 SXI.Z'^ 
 4} 73ff 
 
 S3 «f O 
 44370 
 
 /3900 
 u too 
 
 / 1 600 
 
 fa»o 
 
 9970 
 
 7 960 
 
 S730 
 ^970 
 
 7 7SO 
 6I90 
 
 6 9SO 
 fS70 
 
 4 3+0 
 S070 
 
 S8/0 
 + 4 + 
 
 •S370 
 4390 
 
 + 980 
 3fSo 
 
 4 6SO 
 3 7/0 
 
 + 340 
 3+80 
 
 3 SSO 
 3 100 
 
 3490 
 2790 
 
 3 /70 
 2 SSO 
 
 29/0 
 2 320 
 
 36SO 
 3/40 
 
 The loss due to corona will not be excessive with any of the above conductors used at sea level for the voltages stated. For elevations 
 above sea level, check the values with Table XXII, especially for the smaller conductors. On long circuits of high voltage, the effect of 
 charging current (also corona and leakage losses) will be to increase or decrease the I'R loss, depending on the amount of load and its power- 
 factor. See Pig. 18 
 
QUICK ESTIMATING TABLES 
 
 TABLE XIX-QUICK ESTIMATING TABLE 
 
 31 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 CONDUCTORS 
 
 KILOVOLT- AMPERES. 3 PHASE, WHICH MAY BE DELIVERED AT THE FOLLOWING VOLTAGES OVER THE VARIOUS 
 CONDUCTORS FOR THE DISTANCES STATED. BASED UPON THE FOLLOWING |2r LOSS (EFFECT OF CHAROINO CURRENT 
 NEGLECTED) AT26'C 
 
 FOR LOAD POWER-FACTOR OF I00%^8.66% LOSS- IO.O%^LOSS 
 FOR LOAD POWER-FACTOR OF 80%^I0.8% LOSS- 12 6 LOSS 
 
 
 6 
 
 2 
 
 CO 
 
 <4 
 
 CD 
 
 COPPER 
 
 AREA 
 
 IN 
 
 CIROUIAR 
 
 MILS 
 
 ALUMINUM 
 
 AftCA 
 
 IN 
 
 OIHOULAH 
 
 MIL* 
 
 
 
 66 000 VOLTS DELIVERED 
 
 
 20 
 MILES 
 
 24 
 MILES 
 
 28 
 MILES 
 
 32 
 MILES 
 
 38 
 MILES 
 
 40 
 MILES 
 
 44 
 MILES 
 
 48 
 MILES 
 
 62 
 MILES 
 
 66 
 MILES 
 
 60 
 MILES 
 
 64 
 
 MILES 
 
 72 
 
 MILES 
 
 80 
 
 MILES 
 
 88 86 
 
 MILES MILES 
 
 104 
 MILES 
 
 
 4>00 000 
 
 sso 000 
 
 9S'9 000 
 87-* .5-00 
 
 XlOOOO 
 
 I93OO0 
 178 OQO 
 
 17* 00c 
 
 /6/ 000 
 
 /48QO0 
 
 /So 00c 
 / 39 000 
 /X70QQ 
 
 /3/QOO 
 
 txoooo 
 / / / 000 
 
 //&OO0 
 
 /07000 
 99000 
 
 /osaoa 
 
 94 200 
 9920C 
 
 9S300 
 3 7 soo 
 9/ aoQ 
 
 sysoo 90 soo 
 80X00 7-f aoo 
 
 74 zoo 69 700 
 
 7.5'0O0 
 
 *ff»oo 
 
 iZSOC 
 
 «»»oo tssoa SBooAszsoc 47too 43700I40400 
 
 t4300 tOOOO S3 SOO\4»IOO 43 7O0 40/00137/00 
 S9SOO SSSOO 49 Sao\44iOO 40S0i 37/oo\il-w,° 
 
 
 ^So 000 
 
 ^00 000 
 
 79S 000 
 
 7/ SSOO 
 4>3t>000 
 
 f 6/000 
 J4SOO0 
 /3OO00 
 
 / 34 OQO 
 
 /xi 000 
 /08000 
 
 / J SOOO 
 /040O0 
 9X700 
 
 /o/ 000 
 9/ oao 
 8/000 
 
 89 soo 
 
 90700 
 
 7X000 
 
 90 soa 
 
 72 700 
 ASOOO 
 
 73 300 
 4 6 000 
 
 ^9000 
 
 syaoo 
 
 iOSOC 
 
 SI 00a 
 
 izooo 
 s&ooo 
 so 000 
 
 S7t00 
 
 SI 900 
 9&3oa 
 
 S370C 
 4* Joe 
 433O0 
 
 ^0^00 44 700 40200 3ttOO 3 3 SOO 31000 
 4SS00 4O300 3&30O 33000 3OZ00 ZtOOO 
 40 SOO JtOOO 3ZSOO X9.^oo ZTOae J rnnn 
 
 
 
 3SO 000 
 300 000 
 
 zso 000 
 
 SS4, Soo 
 -*77 OQO 
 39 7 SOO 
 
 naooo 
 94000 
 80 soo 
 
 94000 
 80000 
 
 4>7XO0 
 
 90 Soo 
 
 49700 
 
 S7400 
 
 70 Soo 
 40000 
 so soo 
 
 ^2 soo 
 S3 soo 
 44 700 
 
 S4S0O 
 
 4 9 000 
 *O3O0 
 
 s-/ soo 
 
 43700 
 3 47O0 
 
 47200 
 ■40000 
 33ioO 
 
 43S00 
 37000 
 3/000 
 
 40 soa 
 
 34 300 
 XSSOO 
 
 3 7 700 
 3ZOO0 
 Zt9oo 
 
 3SZ0C 3/ ZOO 
 30 000 Z^7O0 
 ZSZOO 22300 
 
 ZSZOO ZS700 Z3t06 
 24000 Zl Soo ZOOOO 
 
 Zl 700 
 / 8JOO 
 
 
 000 
 
 00 
 
 /67 77Z 
 /33 079 
 
 33h -^ZO 
 
 a. €6 s-oo 
 
 a. f 1 9S0 
 
 48O00 
 S4OO0 
 42 soo 
 
 S(,7oo 
 4SO00 
 3SSO0 
 
 49700 
 
 39^00 
 30400 
 
 4ZS00 
 33 900 
 a.4itOQ 
 
 37700 
 30 too 
 3.3700 
 
 34000 
 27 /OO 
 2/300 
 
 31 000 
 24600 
 /930O 
 
 3»3ao 
 ZISOO 
 17700 
 
 Z 6 IQO 
 Zofoo 
 Ii300 
 
 Z43O0 
 I9300 
 IS^OO 
 
 ZX700 
 la/oa 
 
 /4ZO0 
 
 Zl ZOO 
 lt9oo 
 /3 300 
 
 /y»oo 
 /sooo 
 
 17000 
 13 SOO 
 
 /o too 
 
 ISSOO 
 /2300 
 9tS0 
 
 /4/00 
 
 i/zoa 
 
 SSSO 
 
 13000 
 
 /O4O0 
 S ISO 
 
 
 3 
 
 /OS-SSO 
 
 /67 Soa 
 
 /33 33.0 
 
 /ass3o 
 
 34 000 
 Z7000 
 
 XI 300 
 
 2 9400 
 
 xasoo 
 
 /7 8C0 
 
 X4300 
 
 /9300 
 /szoo 
 
 2/ 300 
 /6 800 
 /3 30O 
 
 /8 900 
 
 /sooo 
 //too 
 
 /7000 
 /3S00 
 
 /a 400 
 
 /SSOO 
 /ZXOO 
 *f700 
 
 /4a.oo 
 //zoo 
 S900 
 
 /3/QO 
 
 /0400 
 
 9 ZOO 
 
 /zzoo 
 
 9 too 
 7 too 
 
 1 1300 
 g9oo 
 
 7IOO 
 
 /otoo 
 
 S400 
 4.tSO 
 
 94SO 
 7 soo 
 S900 
 
 SSOO 
 i7S0 
 S300 
 
 7 7S0 
 4/00 
 4 SSO 
 
 7/00 
 stoo 
 
 4 4S0 
 
 tsso 
 
 SZOO 
 AlOO 
 
 
 -4 
 
 4/ 73ff 
 
 »3 640 
 4^ 370 
 
 /i,8oo 
 /3SQO 
 
 /400Q 
 // 200 
 
 /2 000 
 
 9400 
 
 /osoo 
 
 9400 
 
 9 300 
 7 soo 
 
 84QO 
 
 6 700 
 
 7400 
 4/00 
 
 7000 
 S600 
 
 6400 
 .5500 
 
 tooo 
 ■ffoo 
 
 Stoo 
 4SOO 
 
 SXSO 
 43.10 
 
 4 47o| 4200 3tZ0 
 3 7SA 337S 3 070 
 
 3 soo 
 38/0 
 
 3 2*0 
 z too 
 
 
 
 70 000 VOLTS DELIVERED 
 
 
 
 
 36 
 MILES 
 
 40 
 MILES 
 
 44 
 MILES 
 
 48 
 MILES 
 
 62 
 MILES 
 
 66 
 MILES 
 
 80 
 
 MILES 
 
 64 
 MILES 
 
 72 
 MILES 
 
 eo 
 
 MILES 
 
 88 
 MILES 
 
 86 
 MILES 
 
 104 
 MILES 
 
 112 
 MILES 
 
 120 
 
 MILES 
 
 128 
 MILES 
 
 144 
 MILES 
 
 
 
 ^00 000 
 
 1 033 000 
 fS'f OQO 
 974 SOO 
 
 /3000Q 
 
 IZQQQQ 
 i/XOOO 
 
 //8OO0 
 /0 8000 
 
 / 00 000 
 
 /07000 
 
 9 9 soo 
 *f/XOO 
 
 99000 
 90 soo 
 
 93 soo 
 
 9O40O 
 83 soo 
 
 77 soo 
 
 84 000 
 
 77S0O 
 
 7/ 70^ 
 
 7 940Q 
 7230O 
 4? 000 
 
 73.100 
 67 700 
 
 *i4oo 
 
 a SSOO 
 
 &0300 
 
 SSSOO 
 
 S9000 
 S4000 
 
 so 000 
 
 S3 soo 
 4-9 ZOO 
 4 SiOO 
 
 49000 
 
 4S10O 
 
 4-/ 700 
 
 4 J 300 
 4/ 700 
 37700 
 
 42000 
 37700 
 
 3SSOO 
 
 3»3o<: 
 3^/00 
 3 3 ZOO 
 
 3 4 700 
 33 SOO 
 31 300 
 
 3Z700 
 30100 
 
 Z7900 
 
 
 
 soo 000 
 
 ■4 so 000 
 
 ^00 000 
 
 79S 000 
 7/S SOO 
 63* 000 
 
 /O/OOO 
 
 90800 
 ^0 roo 
 
 90 soo 
 
 B'6oo 
 
 7Z 700 
 
 82 soo 
 
 74 2.00 
 
 44/00 
 
 7SS00 
 <i,900Q 
 40&00 
 
 49800 
 
 4290a 
 
 SS900 
 
 44900 
 
 S9S0Q 
 
 S/ 900 
 
 40 SOO 
 S4SOa 
 4940Q 
 
 St700 
 
 s/oaa 
 •^s-^oo 
 
 so soo 
 
 4SS00 
 
 40400 
 
 4SZ0O 
 4 1 SOO 
 3 4 300 
 
 41 Zoo 
 37IOO 
 33 000 
 
 3 7 700 
 J400O 
 3030a 
 
 34900 
 31400 
 ZtOOO 
 
 32400I30200 
 29Zoo\ Z7ZO0 
 ZS900\Z4 ZOO 
 
 ZS300 
 ZSSOO 
 22 700 
 
 ZSZOO 
 
 ZX 700 
 xozoa 
 
 
 
 3SO 000 
 
 300 000 
 zsc 000 
 
 S^S6 SOO 
 -477 000 
 397 SOO 
 
 7Q30Q 
 
 60x00 
 
 so 400 
 
 43300 
 
 S4XOO 
 
 4S30Q 
 
 S7SOi> 
 
 49x00 
 
 4/ XOO 
 
 S2700 
 
 4S/00 
 
 37900 
 
 49700 
 
 4 1 700 
 
 34900 
 
 4S2O0 
 
 39700 
 
 3Z400 
 
 4X200 
 34/00 
 30200 
 
 39 soo 
 
 33900 
 ze-aoo 
 
 ~3SIO0 
 JO 100 
 
 zszoo 
 
 3/600 
 57/00 
 Z17O0 
 
 Z^soo 
 Z4too 
 
 2 400 
 
 .24400 
 
 Z7ioo 
 IS900 
 
 Z4300 
 20 800 
 /74O0 
 
 22 too 
 19300 
 It ZOO 
 
 Zl 100 
 
 /SOOO 
 /SIOO 
 
 19 TOO 
 
 /t90o 
 /I Zoo 
 
 /- too 
 /sooo 
 /ztoo 
 
 
 0000 
 000 
 
 
 21/ 4>00 
 /A7 77a 
 /33079 
 
 33<» 4-ZO 
 
 a / 1 9SO 
 
 -^xsoo 
 
 33 900 
 X47oa 
 
 383O0 
 
 304OO 
 X4 000 
 
 34900 
 
 X7 700 
 2/ soo 
 
 3/900 
 2S400 
 
 zoooo 
 
 29400 
 
 23400 
 
 /8400 
 
 27300 
 
 XI 700 
 
 / 7 /OO 
 
 2 SSOO 
 2O3O0 
 /4 000 
 
 ;i3'?oo 
 /9 000 
 /sooo 
 
 Zl ZOO 
 /t900 
 13 300 
 
 19 100 
 /SZOO 
 12 000 
 
 /74O0 
 /3S00 
 /090O 
 
 IS900 
 iz 700 
 zoooo 
 
 14700 
 /I 700 
 fZ40 
 
 13 too 
 
 /0900 
 
 SSSO 
 
 /ztoo 
 
 10 100 
 
 sooo 
 
 1 / 900 
 
 9SZO 
 7SO0 
 
 / 4,00 
 g460 
 <«70 
 
 
 / 
 
 2 
 
 /o€fS^O 
 9Z 69'f 
 «A 3S-9 
 
 /67 9oa 
 /33 220 
 /0^S30 
 
 2/200 
 /6 800 
 
 /3300 
 
 /f/oo 
 
 /SJOO 
 /XOOO 
 
 /7400 
 /3700 
 /0900 
 
 /S90Q 
 /2 4oo 
 
 /QOOO 
 
 /4 700 
 //600 
 9240 
 
 / 3 6,00 
 
 /O8O0 
 
 8Sto 
 
 /Z700 
 
 /Q /OO 
 
 Sooo 
 
 /3O0O 
 9-tSo 
 7S0O 
 
 /O ^00 
 8400 
 4*70 
 
 9S70 
 7S60 
 
 <ooo 
 
 9700 
 4 870 
 S4to 
 
 7 97a 
 t 300 
 sooo 
 
 73io 
 SSZO 
 4 420 
 
 tsso 
 
 S400 
 4 290 
 
 63S0 tOOO 
 S040 4720 
 4 000 3 7SO 
 
 S3IO 
 4 ZOO 
 3330 
 
 
 
 
 93 A-fo 
 
 /osoo 
 
 9 soo 
 
 9430 
 
 7 930 
 
 7300 
 
 4 790 
 
 6330 
 
 S900 
 
 ^•370 
 
 4 7S0 
 
 4320 
 
 39to 
 
 3 479 
 
 3 3»« 
 
 3/40 
 
 X970 
 
 Xi40 
 
 
 
 80 000 VOLTS 
 
 DELIVERED 
 
 
 
 
 
 
 
 
 
 
 
 
 36 
 
 MILES 
 
 40 
 MILES 
 
 44 
 
 MILES 
 
 48 
 MILES 
 
 62 
 MILES 
 
 66 
 
 MILES 
 
 60 
 
 MILES 
 
 84 
 
 MILES 
 
 72 
 
 MILES 
 
 80 
 MILES 
 
 88 
 
 MILES 
 
 86 
 MILES 
 
 104 
 MILES 
 
 112 
 MILES 
 
 120 
 MILES 
 
 128 
 MILES 
 
 144 
 MILES 
 
 
 
 A SO 000 
 i,oo 000 
 sso 000 
 
 / 033 000 
 9S4^OO0 
 97-4 S-OO 
 
 17/000 
 
 /57000 
 /■*i,ooo 
 
 /S400i 
 
 / 4 XOOO 
 /3/ QOO 
 
 /40000 
 /X900O 
 
 //9000 
 
 /28000 
 //800Q 
 J 1 QOOO 
 
 /I80QQ 
 
 / 09000 
 
 /O/OOO 
 
 //QOOO 
 /O/OOO 
 
 93i.oo 
 
 'OXOQO 
 94S00 
 87600 
 
 <=1i>OQO 
 88.500 
 ^XOOO 
 
 8 SSOO 
 78SQO 
 72900 
 
 77000 
 7/000 
 6SS00 
 
 70000 
 
 t4O00 
 S9S00 
 
 ^4000 
 S9000 
 
 SSOOO 
 
 S9 000 
 S4-SOO 
 SO SOO 
 
 SSOOO 
 
 so soo 
 44900 
 
 SI 000 
 
 47700 
 
 •+3*00 
 
 4 8 000 
 44 200 
 4/ 000 
 
 42700 
 
 3f 200 
 34400 
 
 
 
 Soo 000 
 •4 so 000 
 ^00 000 
 
 79SOOO 
 7fSSOO 
 434000 
 
 / 3 3.000 
 
 /iSooo 
 1 OS 000 
 
 / /9000 
 
 JO7O0O 
 
 94900 
 
 J 08 QOO 
 
 97000 
 
 8^300 
 
 99000 
 
 88 800 
 
 79/00 
 
 9/S00 
 32000 
 73000 
 
 &SOOO 
 76ZQO 
 47 800 
 
 79000 
 7/000 
 4 3 300 
 
 74200 
 
 6i7ao 
 S9300 
 
 6 4 000 
 S920O 
 SX700 
 
 S9SOO 
 S3 soa 
 47400 
 
 S4 000 
 
 4SSOO 
 43/00 
 
 47 soo 
 44400 
 39 soo 
 
 45700 
 
 ■41 000 
 3iSO0 
 
 42 JOO 
 38/00 
 33900 
 
 39SOC 
 3SS0C 
 
 3/ too 
 
 37/00 
 33300 
 Z9tOO 
 
 33000 
 2»400 
 2 4 300 
 
 
 
 3SO 000 
 
 300 000 
 
 2SO QOO 
 
 SS4 SOO 
 ^77000 
 39 7 SOO 
 
 91 Boo 
 
 ys&oo 
 
 ASBOO 
 
 9X400 
 70 900 
 S9200 
 
 7S/00 
 
 44300 
 
 S3 900 
 
 68 9oa 
 S9O00 
 
 4 ^400 
 
 43400 
 S4400 
 4S400 
 
 S9000 
 so soo 
 
 42300 
 
 SS/OO 
 •47X00 
 39 SOO 
 
 S/600 
 
 44 zoo 
 
 37 000 
 
 4S900 
 393Q0 
 32900 
 
 41 300 
 3S400 
 Z9 too 
 
 37 soo 
 3ZIO0 
 ZC900 
 
 34400 
 Z9SOO 
 24700 
 
 3/ Soo 
 Z7ZO0 
 2 2 800 
 
 29 Soo 
 ZS30O 
 
 Zt 1 00 
 
 27 SOO 
 2 3400 
 11700 
 
 zo soo 
 
 ZX 100 
 1 SSOO 
 
 2J»O0 
 /»40O 
 
 It400 
 
 
 0000 
 000 
 00 
 
 2 1 > ^00 
 /J3 079 
 
 33^430 
 X 1 1 9 SO 
 
 SSSOO 
 -♦■»J00 
 
 So OQO 
 
 39900 
 3/300 
 
 4S4O0 
 36XOO 
 Z8SOQ 
 
 'i-l 700 
 
 33 100 
 X4,/OQ 
 
 33400 
 
 30 600 
 24 /OQ 
 
 3S7ao 
 2840O 
 ZX400 
 
 3 3 300 
 Z4 SOO 
 
 xo 9oo 
 
 31 ZOO 
 Z4800 
 
 1 9 iOO 
 
 Z78O0 
 2x100 
 /7400 
 
 ZSOOO 
 19 900 
 /SiOO 
 
 2 2 700 
 
 IS 100 
 
 14 ZOO 
 
 20 SOO 
 I i^tOO 
 /3 /OO 
 
 /f 200 
 /S300 
 
 /z 100 
 
 1 7 soo 
 14 ZOO 
 /I 200 
 
 It 700 
 /3Z00 
 70 400 
 
 istoo 
 IZ 400 
 9 soo 
 
 13 900 
 1/000 
 8 700 
 
 
 
 1 
 
 /OSS^O 
 S3 69-^ 
 
 f47 SOO 
 /33 ZZO 
 
 /o*r^3o 
 
 Z7B00 
 
 3.1900 
 
 2SOOO 
 
 /4700 
 /S7oa 
 
 zi 700 
 
 /7 900 
 /4XOO 
 
 Z0800 
 /4400 
 
 /3 /OO 
 
 /9200 
 /SXOO 
 /X /OO 
 
 J? 900 
 
 /4 1 00 
 
 / / zoo 
 
 /4 70a 
 /3 /OO 
 /040a 
 
 IS 600 
 
 /Z 300 
 9»oo 
 
 /3900 
 
 /a 900 
 
 9 7/0 
 
 IZSOO 
 9Sto 
 7»40 
 
 /I 300 
 8990 
 7/30 
 
 /0400 
 8230 
 4 J30 
 
 9tio 
 7 too 
 6 030 
 
 8930 
 7OS0 
 SiOO 
 
 8330 
 
 tsso 
 
 S330 
 
 7 tlO 
 4 /70 
 
 4900 
 
 4T40 
 J-4f0 
 4340 
 
 
 
 88 000 VOLTS DELIVERED 
 
 
 36 
 MILES 
 
 40 
 
 MILES 
 
 44 
 MILES 
 
 48 
 MILES 
 
 62 
 MILES 
 
 66 
 
 MILES 
 
 80 
 MILES 
 
 64 
 MILES 
 
 72 
 MILES 
 
 60 
 
 MILES 
 
 88 
 
 MILES 
 
 96 
 MILES 
 
 104 
 MILES 
 
 112 
 MILES 
 
 120 
 MILES 
 
 128 
 MILES 
 
 144 
 MILES 
 
 
 
 4S0 000 
 400 000 
 %SSO 000 
 
 / 033 000 
 9S4 000 
 
 87'* SOO 
 
 X07000 
 
 /91 OQO 
 I7700Q 
 
 /86O00 
 
 17/ OQQ 
 /S9000 
 
 / 49 000 
 
 IS400C 
 
 /■*-*000 
 
 /SSOOO 
 /43000 
 /3Z000 
 
 /*3000 
 /3XO0O 
 /2-2.O0O 
 
 /33000 
 
 12X000 
 
 //3000 
 
 124000 
 
 //4000 
 /0 4 000 
 
 uiooo 
 / 07 000 
 It soo 
 
 /03000 
 9SS0Q 
 88 200 
 
 93O00 
 SSSOO 
 79 soo 
 
 S4S00 
 7 a 000 
 72 000 
 
 77 soo 
 7/S00 
 ^0000 
 
 7/ soo 
 tiooo 
 t/000 
 
 tiSOO 
 
 tiooo 
 St soo 
 
 4 2 000 
 S7000 
 S3 000 
 
 Sfoao 
 S3 Sao 
 49 700 
 
 Si Soe 
 
 47700 
 
 44100 
 
 
 
 Soo 000 
 
 ■9 so 000 
 4OQ ODO 
 
 79s QOO 
 7/S SOO 
 
 434 OQ 
 
 /S9000 
 /*300Q 
 /Z8O00 
 
 /43000 
 /X90O0 
 //SOOO 
 
 131 000 
 //7C00 
 /05COO 
 
 /zoooo 
 /07000 
 94'r>Q0 
 
 / 1 1 coo 
 99000 
 
 99700 
 
 /0 2O00 
 
 ^2000 
 
 82 soo 
 
 94000 
 8^000 
 
 76 900 
 
 90000 
 Saioo 
 
 7ZOOO 
 
 90000 
 7/ SOO 
 44 000 
 
 7/ SOO 
 i4SOO 
 S7SOO 
 
 issoo 
 
 SSSOO 
 
 szsoo 
 
 to 000 
 S3iOO 
 4 SOOO 
 
 SSSOO 
 49 soo 
 
 44300 
 
 51 000 
 4teoo 
 4 ' 200 
 
 49000 
 43 000 
 3S400 
 
 4SOOO 
 903OO 
 3tOOO 
 
 40000 
 
 3S700 
 
 3x000 
 
 
 
 3S0 000 
 300 000 
 2SO 000 
 
 s-s^ Soo 
 
 •477 000 
 397SOO 
 
 1/ 1000 
 
 9SOO0 
 79SOO 
 
 lOQOOO 
 8SSQQ 
 7 /soo 
 
 9/ 200 
 
 79000 
 
 iSOOO 
 
 93 soo 
 
 7/SQO 
 S9 9O0 
 
 77200 
 
 460OO 
 SSOOO 
 
 7/40Q 
 4/XOQ 
 Si QOO 
 
 k,7000 
 S72OO 
 4790a 
 
 6X700 
 ■J3i00 
 
 -»■» soo 
 
 SS800 
 47700 
 39 Soo 
 
 SO 000 
 
 42 700 
 3S 700 
 
 4Stoo 
 
 39000 
 
 3XS00 
 
 41 7SO 
 3S 700 
 2?fOO 
 
 3 stoo 
 33000 
 Z7SOO 
 
 3 SSOO 
 
 30 too 
 zssoo 
 
 33SOO 
 
 zsteo 
 
 Z390O 
 
 3/ 300 
 24800 
 J 2400 
 
 Z7 90O 
 Z3S00 
 
 19900 
 
 
 0000 
 000 
 00 
 
 Zi 1 AOO 
 /67 7 7Z 
 /33 079 
 
 3244ZO 
 2 6£ SOO 
 
 X / 1 9SO 
 
 47X00 
 S3 Soo 
 •4Z30Q 
 
 6 osoo 
 
 49200 
 39000 
 
 SSOOO 
 
 43900 
 
 3-t^OO 
 
 sSOSao 
 
 40 300 
 J/ 700 
 
 4iX00 
 
 37000 
 
 29300 
 
 43 zoo 
 
 34400 
 
 27200 
 
 40300 
 32X00 
 2S400 
 
 37 Soo 
 30Z00 
 xseoo 
 
 3 3 400 
 X490C 
 2.1 ZOO 
 
 30 ZOO 
 24/00 
 19100 
 
 Z7SOO 
 2/ 900 
 /7300 
 
 ZSZOO 
 20 100 
 IS Soo 
 
 23200 
 ISSOO 
 /4 too 
 
 Zl too 
 
 I7ZOO 
 /3 too 
 
 ZOIOO 
 
 li 100 
 IX 700 
 
 / 8 900 
 IS 100 
 II 900 
 
 itsoo 
 1340a 
 
 to too 
 
 
 
 1 
 
 / os-s^o 
 
 83 694 
 
 / 47 »oo 
 /33 2ZO 
 /OS S30 
 
 33700 
 
 XCSOO 
 Xl /oo 
 
 30 300 
 
 X39O0 
 /9000 
 
 X7400 
 XI700 
 /7200 
 
 2S30O 
 
 /99QO 
 /S80O 
 
 23300 
 / 9 300 
 1*600 
 
 2I700 
 /7 000 
 /3SOO 
 
 ZQ200 
 
 /s 90Q 
 iz 400 
 
 /9000 
 14900 
 II 900 
 
 14900 
 
 /J 200 
 / SOO 
 
 /SIOO 
 
 It 900 
 
 9 soo 
 
 13 SOO 
 'OS JO 
 9KiO 
 
 IZ too 
 9 too 
 7900 
 
 II too 
 9 100 
 7 300 
 
 10 SOO 
 SSOO 
 
 4700 
 
 10 100 
 7900 
 
 t300 
 
 9 SOO 
 
 7400 
 
 S900 
 
 8400 
 
 t too 
 
 SZOO 
 
 
 The loss due to corona will not be excessive with any of the above conductors used at nea level for the voltaRes slated. For elevatioDi 
 
 above ses level, checlc the values with Table XXII. especially for the smaller conductors. On lonK circuits of high voltaje, the eflFecl of 
 
 charging current (also corona and leakage losses) will be to increase or decrease the I R loss, depending on the amount of load and its power- 
 factor. See Fig. 13 
 
32 
 
 QUICK ESTIMATING TABLES 
 
 TABLE XX-QUICK ESTIMATING TABLE 
 
 CONDUCTORS 
 
 KILOVOLT- AMPERES. 3 PHASE. WHICH MAY BE DELIVERED AT THE FOLLOWING VOLTAGES OVER THE VARIOUS 
 CONDUCTORS FOR THE DISTANCES STATED. BASED UPON THE FOLLOWINQ (Sr lqsS (EFFECT OF CHARGING CURRENT 
 
 f^E^^EC™ AT25»C ATefi«C 
 FOR LOAD POWER-FACTOR OF IOO%-8.66% LOSS- 10 0% LOSS 
 FOR LOAD POWER-FACTOR OF 80?fr- 1 8% 1 n<?<? 19 r 1 noo 
 
 i 
 
 4 
 m 
 
 COPPER 
 
 AREA 
 
 IN 
 
 CtROUUM 
 
 MILS 
 
 ALUMINUM 
 
 AREA 
 
 IN 
 
 CIRCULAR 
 
 MILS 
 
 
 100 000 VOLTS DELIVERED 
 
 52 
 MILES 
 
 56 
 
 MILES 
 
 60 
 
 MILES 
 
 64 
 MILES 
 
 72 
 MILES 
 
 80 
 
 MILES 
 
 88 
 
 MILES 
 
 96 
 
 MILES 
 
 104 
 MILES 
 
 112 
 MILES 
 
 120 
 MILES 
 
 128 
 
 MILES 
 
 144 
 MILES 
 
 160 
 MILES 
 
 176 
 MILES 
 
 192 
 MILES 
 
 208 
 MILES 
 
 
 6SO OOO 
 too OOO 
 
 ssoooo 
 
 / 033 OOO 
 9S-t OOO 
 S7-* ^oo 
 
 l»Sooc 
 
 /70000 
 
 /ssooo 
 
 / 7X 000 
 /S9OO0 
 
 IA7000 
 
 /AOOOO 
 /48O00 
 
 /37000 
 
 /so 000 
 /39000 
 
 /290OO 
 
 ■'33004 /ZOOOO 
 123000 / /OOOO 
 i I40Q0 103000 
 
 /09000 
 /a /ooo 
 93 SOO 
 
 /OOOOO 
 
 93 SOO 
 8,5 SOO 
 
 fZSOO 
 
 sssoo 
 79000 
 
 86000 
 
 79000 
 
 73SOO 
 
 80000 
 
 74000 
 essoQ 
 
 7SOOO 
 69000 
 
 6fOOO 
 
 6 6 SOO 
 6/ SOO 
 S7 000 
 
 60000 
 SSOOO 
 Si SOO 
 
 ^■4S00 
 So SOO 
 ■46700 
 
 SOOOO 
 ■96200 
 ■4 3 700 
 
 46 200 
 4 2 700 
 
 39SOO 
 
 
 J-OO OOO 
 ^SO OOO 
 400 OOO 
 
 79S OOO 
 7'S SOO 
 t3£i OOO 
 
 / -4- 300c 
 /J»<JOO 
 
 / / + 000 
 
 /3-XOOO 
 //*fOOO 
 
 1 Ob 060 
 
 /34000 
 
 ///ooo 
 99900 
 
 /J^OOO 
 /04OOO 
 
 92700 
 
 /O^ooe 
 93000 
 92 400 
 
 93000 
 
 83SO0 
 
 74 200 
 
 94 SOO 
 
 7&OOQ 
 
 <i74ao 
 
 7 7 SOO 
 69SOO 
 A/8-00 
 
 7/J^OO 
 (i-f 200 
 >5'7 000 
 
 6 6 000 
 S9SOQ 
 S3 000 
 
 (,2 ooo 
 sssoo 
 49400 
 
 S8000 
 S2000 
 •46 300 
 
 S/SOO 
 ■4(0600 
 ■41200 
 
 ^(oSOO 
 ■4 1 700 
 37 100 
 
 -J22O0 
 
 3»ooo 
 
 33700 
 
 39700 
 3-»700 
 30900 
 
 3S700 
 33/00 
 
 2 9 SOO 
 
 
 ■3S0 OOO 
 300 OOO 
 2SO OOO 
 
 ■SS6 -SOO 
 -♦77000 
 397J-00 
 
 9f-f00 
 S'S/OO 
 
 7/200 
 
 9Z300 
 
 7fooo 
 4.S/00 
 
 8h/oo 
 
 73700 
 6/ 700 
 
 90700 
 
 69/00 
 S7900 
 
 7/ 900 
 ^/ 400 
 
 s/ 400 
 
 ^9S00 
 SS300 
 ■46300 
 
 S9 700 
 S0 300 
 •42 / 00 
 
 S3 900 
 46 /OO 
 38600 
 
 *9700 
 .4 2 SOO 
 3S 600 
 
 46/CO 
 3 9 SOO 
 3 3 000 
 
 43000 
 
 3 4>9oo 
 30800 
 
 ■40300 
 JfSOO 
 29900 
 
 3S900 
 
 30 700 
 2S700 
 
 32 300 
 27600 
 33/00 
 
 39300 
 3S/00 
 31 000 
 
 2«foO 
 23000 
 /*300 
 
 34 SOO 
 Zl ZOO 
 17 SOO 
 
 oooo 
 
 OOO 
 OO 
 
 311 too 
 
 /«7 77» 
 
 . /33 079 
 
 336 -?20 
 Zii SOO 
 3/1 9SO 
 
 60/00 
 47S00 
 37700 
 
 SSSao 
 
 3SOOQ 
 
 sz/00 
 41400 
 32700 
 
 ■49 eoo 
 
 38900 
 30 6>00 
 
 43400 
 34 SOO 
 27ZOO 
 
 39000 
 3 / /OO 
 2-4 SOO 
 
 SSSOO 
 Z8ZOO 
 2 2 300 
 
 32 SOO 
 ZS9oa 
 ::o4oo 
 
 30000 
 23900 
 /»goo 
 
 27 900 
 22 200 
 /7 SOO 
 
 2 6 000 
 20700 
 /6 3O0 
 
 24-too 
 I9-400 
 
 /s 300 
 
 21 700 
 17 200 
 /3 600 
 
 /9SO0 
 /SSOO 
 /2 300 
 
 n 700 
 
 /♦/OO 
 
 // 100 
 
 16 200 
 /2 900 
 /0 200 
 
 /SOOO 
 
 II 900 
 9 4oe 
 
 o 
 
 /oS'fto 
 
 it7 eoo 
 
 /33 Z30 
 /0SS3O 
 
 30000 
 33700 
 /esoo 
 
 Z7900 
 ZXOOQ 
 J7SOO 
 
 2 4>QQO 
 3.0^00 
 /&3O0 
 
 24-400 
 
 /9300 
 
 /S300 
 
 21 700 
 /7/00 
 /3^0O 
 
 I9SOO 
 /S400 
 12 200 
 
 / 7 700 
 /■9OOO 
 // /OO 
 
 /6>300 
 / X 900 
 /0200 
 
 /S 000 
 
 / / 900 
 
 ■?-)00 
 
 /3foo 
 // 000 
 9700 
 
 /3000 
 /0300 
 9200 
 
 / 3300 
 9600 
 7600 
 
 /OSOO 
 8600 
 6 SOO 
 
 9 SOO 
 7700 
 6 /OO 
 
 9900 
 
 7 000 
 S600 
 
 S/oo 
 6-400 
 S/oo 
 
 7 SOO 
 S900 
 
 4 7oa 
 
 
 110 000 VOLTS DELIVERED 
 
 52 
 MILES 
 
 56 
 MILES 
 
 60 
 MILES 
 
 64 
 MILES 
 
 72 
 MILES 
 
 80 
 
 MILES 
 
 88 
 MILES 
 
 96 
 
 MILES 
 
 104 
 MILES 
 
 112 
 MILES 
 
 120 
 MILES 
 
 128 
 MILES 
 
 144 
 MILES 
 
 160 
 MILES 
 
 176 
 MILES 
 
 192 
 MILES 
 
 208 
 MILES 
 
 
 tSO OOO 
 too OOO 
 
 ^s^o OOO 
 
 / 033 OOO 
 9S^ OOO 
 V7-t SOO 
 
 aa-4000 
 
 2.04>000 
 /<f/000 
 
 3O9000 
 f^/OOQ 
 / 77 000 
 
 /94000 
 /7SOQO 
 /6SOO0 
 
 / 92000 
 /hyooo 
 
 /SSQOO 
 
 /fozooo 
 
 148000 
 / 39000 
 
 /4^000 
 /3^0QQ 
 
 /z*iooo 
 
 / 33000 
 /22000 
 
 //3000 
 
 /2 tOOC 
 /I /OOO 
 7 03 000 
 
 //3000 
 /O3 0O0 
 ■?^^oo 
 
 /04000 
 9SSO0 
 ^^Soo 
 
 97000 
 
 99000 
 
 9 2 SOO 
 
 ^/ooo 
 
 B3SO0 
 77S00 
 
 Si ooo 
 
 7-4 ooo 
 
 69000 
 
 7 3 000 
 67OO0 
 62 000 
 
 66S00 
 
 61 000 
 S6SOO 
 
 <tOJ00 
 
 sssoo 
 
 i4S00 
 
 S6000 
 
 S/SOO 
 
 47700 
 
 
 ^OO OOO 
 -f fO OOO 
 •400 OOO 
 
 79S OOO 
 7/.S SOO 
 t3t OOO 
 
 /73OOO 
 
 /ssooo 
 
 / 4,0000 
 /4400O 
 129 000 
 
 /SO 000 
 /34OQO 
 
 / / 9 000 
 
 /^oooo 
 
 /ZAOCO 
 / /Z'OOO 
 
 / 2.5 000 
 //2Q0O 
 99700 
 
 //ZOQO 
 /O/OOO 
 
 89 700 
 
 JOZOOQ 
 
 9/600 
 9/600 
 
 9 3 SOO 
 8-fOOO 
 74 900 
 
 »6000 
 77600 
 69OO0 
 
 90000 
 72000 
 A4/00 
 
 7SOOO 
 6>7OO0 
 Sf9oO 
 
 70000 
 &3 000 
 S^/OO 
 
 62SQ0 
 S600O 
 ■49800 
 
 S6O00 
 So SOO 
 ■4490a 
 
 SI 000 
 ■4S900 
 40800 
 
 4 6 700 
 ■42000 
 37 400 
 
 43000 
 3»»00 
 3 4 SOO 
 
 
 3SO o6o 
 
 300 OOO 
 ZSO OOO 
 
 S'St SOO 
 ^77 OOO 
 3 97 ^OO 
 
 JXQQOO 
 
 /0300C 
 96, /OO 
 
 III OOO 
 
 fs^oo 
 so 000 
 
 /04000 
 89Z00 
 
 74700 
 
 97700 
 
 83600 
 
 70000 
 
 9L900 
 
 74-30Q 
 62 SOO 
 
 79/00 
 66900 
 
 S6 00O 
 
 7/000 
 
 60 800 
 
 so 600 
 
 6S/O0 
 
 ss 700 
 
 ■4b600 
 
 60/00 
 
 S/Soo 
 ■^3 /OO 
 
 SS900 
 ■f 7 900 
 40 000 
 
 SZ/QO 
 44 600 
 37300 
 
 ■48 800 
 
 ■41 eoo 
 
 3SOOO 
 
 ■43*00 
 373/0 
 3 1 1 00 
 
 39/00 
 33-fOO 
 
 ZSooo 
 
 3SSOO 
 30^400 
 3S300 
 
 33 SOO 
 
 31900 
 2JJJJ 
 
 30000 
 3S7O0 
 3 1 SOO 
 
 oooo 
 
 OOO 
 OO 
 
 211 t,oo 
 /(,7 772 
 /33 07f 
 
 336-J20 
 
 z&i, eoo 
 
 311 9so 
 
 73700 
 S7 900 
 ■4SA00 
 
 ^7 SOO 
 •S3 700 
 42300 
 
 63OO0 
 
 ySO /OO 
 
 39 SOO 
 
 S9 /OO 
 
 ■^7000 
 37000 
 
 S2Soa 
 ■4/ 900 
 32900 
 
 ■47200 
 37600 
 29^00 
 
 4-2900 
 
 34-200 
 26900 
 
 39400 
 3 / 300 
 24 70O 
 
 3h300 
 2S9O0 
 3 2 too 
 
 33700 
 Z(^9oo 
 2/ /OO 
 
 3 /SOO 
 ZS /OO 
 If 900 
 
 39 SOO 
 2 3 SOO 
 /SSOO 
 
 2 6 200 
 30900 
 / 6400 
 
 2 3 600 
 19 SOO 
 /■4SOO 
 
 2 1 SOO 
 17 1 00 
 /3SOO 
 
 19 700 
 /S600 
 1 2300 
 
 IS300 
 14400 
 /I 400 
 
 
 
 /67 ffOO 
 
 /33a»o 
 
 3i30fi 
 29 700 
 
 2(,ioO 
 
 3 /SOO 
 Ji49oo 
 
 29 SOO 
 Z3 3QO 
 
 ZhZOO 
 
 2070a 
 
 2S600 
 /9600 
 
 2/^900 
 /h900 
 
 /9700 
 
 /ssoo 
 
 /% / 00 
 /-f 300 
 
 J6900 
 /3300 
 
 /S700 
 /Z400 
 
 /I 700 
 II 670 
 
 13129 
 /0300 
 
 // 800 
 9300 
 
 / 700 
 SSOO 
 
 9 SOO 
 7 SOO 
 
 9/00 
 7200 
 
 
 120 000 VOLTS DELIVERED 
 
 64 
 
 MILES 
 
 72 
 MILES 
 
 80 
 
 MILES 
 
 88 
 MILES 
 
 96 
 MILES 
 
 104 
 MILES 
 
 112 
 
 MILES 
 
 120 
 
 MILES 
 
 128 
 MILES 
 
 144 
 MILES 
 
 160 
 MILES 
 
 176 
 
 MILES 
 
 192 
 
 MILES 
 
 208 
 MILES 
 
 224 
 MILES 
 
 240 
 MILES 
 
 256 
 MILES 
 
 
 tSO OOO 
 600 OOO 
 SSO OOO 
 
 / 033 ooo 
 
 'iS^ ooo 
 S7-9 SOO 
 
 /*?9ooa 
 
 /S-^ooo 
 
 /93.00c 
 177000 
 
 /&4000 
 
 /73OO0 
 
 /4>oooe 
 
 /48O0O 
 
 IS700Q 
 I4SOO0 
 / 34 000 
 
 144000 
 
 /33OO0 
 /23000 
 
 /3300c 
 J 23000 
 //3000 
 
 123 000 
 //4OOO 
 /OSOOO 
 
 /JSOOO 
 
 706000 
 9S,foo 
 
 /OSooo 
 
 /OOOOO 
 9 2 SOO 
 
 9^000 
 
 ^9 SOO 
 82000 
 
 9^ SOO 
 9QQQQ 
 74000 
 
 7SSO0 
 72 SOO 
 (s7O00 
 
 72000 
 &6S00 
 
 61 SOO 
 
 t6SO0 
 6/SOO 
 S6SO0 
 
 6/SOO 
 S7000 
 S3 SOO 
 
 -r7^oo 
 ^■3000 
 
 ■f9 200 
 
 S4 000 
 SOOOO 
 46 200 
 
 
 ■SOO OOO 
 4SO OOO 
 400 OOO 
 
 7?S OOO 
 7/S SOO 
 63hOOO 
 
 /4,7O0O 
 /SO 000 
 /^3000 
 
 /4HOOO 
 /33O0C 
 1/9000 
 
 / 3 3000 
 /ZOOOO 
 
 / 0700a 
 
 /2/O0O 
 /ofooa 
 *f7/00 
 
 // / 000 
 /OOOOO 
 89 000 
 
 /02OOO 
 fzsoo 
 
 S3 100 
 
 9SOOO 
 
 8^000 
 
 7^300 
 
 S9000 
 JO 000 
 7/300 
 
 Z3SO0 
 
 7Saoo 
 
 6 6 700 
 
 74000 
 4><,SOO 
 S9300 
 
 ^4, SOO 
 60000 
 S3 ^00 
 
 60 SOO 
 S4SO0 
 9-SSOO 
 
 sssoo 
 
 SOOOO 
 ■4ASOO 
 
 S/000 
 46200 
 ■4 1 /OO 
 
 47SO0 
 4ZOOO 
 38/00 
 
 4 4 SOO 
 40O00 
 3S60O 
 
 41 700 
 3 7 SOO 
 33300 
 
 
 JSO OOO 
 300 OOO 
 ZSO OOO 
 
 SStSOO 
 ■^77 OOO 
 397 SOO 
 
 //60OO 
 
 f9ooo 
 
 93000 
 
 i 03000 
 8&SOO 
 74/00 
 
 9300a 
 
 6 6 7O0 
 
 9^ SCO 
 7Z-^0O 
 40AOO 
 
 77 SOO 
 
 4,4,-ioo 
 
 sssoo 
 
 7/ SOO 
 6/ZOQ 
 S/ZOO 
 
 6S-40O 
 Sh9O0 
 47^00 
 
 63 OOO 
 S3/00 
 
 ■*1fao 
 
 ss/00 
 ■49SOO 
 ■Hiao 
 
 SI^OO 
 4-43-00 
 37 000 
 
 46>SOO 
 39 900 
 33300 
 
 ■42 20a 
 36 300 
 30300 
 
 39 700 
 33IOO 
 27 700 
 
 3SSoa 
 30600 
 
 2S6O0 
 
 33200 
 
 2S-40 
 2 3 800 
 
 31 000 
 26 SOO 
 32200 
 
 39/00 
 24 800 
 20 Sao 
 
 oooo 
 
 OOO 
 
 Zl 1 *oo 
 
 /6777i 
 
 J34420 
 
 7O300 
 SiOQO 
 
 ■4-^ /OO 
 
 &ZSOO 
 ^9700 
 39 ZOO 
 
 S6 2oa 
 
 3.530C 
 
 •s/ too 
 
 4-0 700 
 32 too 
 
 46 900 
 37300 
 29400 
 
 43 2QO 
 3440Q 
 27 100 
 
 40/00 
 32000 
 
 ZSZOO 
 
 3 7S00 
 39SO0 
 33SOO 
 
 3S/00 
 
 29000 
 
 33 000 
 
 3/ ZOO 
 Z-9 800 
 /_3^0Q 
 
 38/OQ 
 3Z3Q0 
 /7 60Q 
 
 2 SSOO 
 30300 
 /6000 
 
 23-4->0 
 
 /Shoo 
 
 I-4700 
 
 31 600 
 17 300 
 /3SOO 
 
 20 100 
 /6 000 
 /2600 
 
 /S 700 
 /4 900 
 /I 700 
 
 17 Sao 
 
 I4O00 
 
 II 000 
 
 
 
 /t7 eoo 
 
 JS/09 
 
 3 tZOO 
 
 2.9/ 00 
 
 zsi.00 
 
 2 3400 
 
 Zl 600 
 
 2a /OO 
 
 /8 700 
 
 /7SOO 
 
 /Shoo 
 
 /4OO0 
 
 /3 700 
 
 // 700 
 
 /o 900 
 
 /OOOO 
 
 9iao 
 
 SSOO 
 
 
 132 000 VOLTS DELIVERED 
 
 64 
 
 MtLES 
 
 72 
 
 MILES 
 
 80 
 MILES 
 
 88 
 MILES 
 
 96 
 MILES 
 
 104 
 
 MILES 
 
 112 
 MILES 
 
 120 
 MILES 
 
 128 
 MILES 
 
 144 
 MILES 
 
 160 
 MILES 
 
 176 
 MILES 
 
 192 
 MILES 
 
 208 
 MILES 
 
 224 
 
 MILES 
 
 240 
 MILES 
 
 256 
 
 MILES 
 
 
 ^SO OOO 
 too OOO 
 ^S^SO OOO 
 
 1 033 OOO 
 fS^OOO 
 97-^ SOO 
 
 %i.2ooa 
 s.^ 2 000 
 
 323000 
 
 Z33O00 
 
 z/sooa 
 
 /98OO0 
 
 z/0000 
 
 /93O0O 
 /78O00 
 
 /9/ 000 
 1 74, 000 
 /^2ooo 
 
 /7SOOO 
 /&/ 000 
 /490OO 
 
 /b/000 
 
 /48000 
 ,3700a 
 
 /SOOOO 
 
 /■a 8 000 
 
 /28O00 
 
 /■joooo 
 
 /Z1OO0 
 
 jo'fooo 
 
 /3/ 000 
 /2/ 000 
 / 12 ooo 
 
 i/4>ooo 
 /07 000 
 99000 
 
 /OSOOO 
 96 SOO 
 S90OO 
 
 9SS00 
 88000 
 8/ 000 
 
 »7J^O0 
 SO SOO 
 7-4 SOO 
 
 ?o^oo 
 7^+000 
 
 69 SOO 
 
 7SOOO 
 
 69000 
 
 64 00a 
 
 70 000 
 
 A 4 SOO 
 
 S4SOO 
 
 6SS00 
 
 6 OSOO 
 S6 000 
 
 
 SOO OOO 
 
 ■9-Soooo 
 
 •400 OOO 
 
 79s OOO 
 7/S SOO 
 t36000 
 
 Zozooo 
 /8SOOO 
 
 /h'iOOO 
 
 / 79 000 
 
 /&JOO0 
 /44000 
 
 /&/ 000 
 /4SOO0 
 / 29 000 
 
 / 47 000 
 /320O0 
 //^OOO 
 
 /3400c 
 
 /z 1 000 
 /08000 
 
 /240OQ 
 /1 2000 
 /OOOOO 
 
 i JSOOO 
 /0-4000 
 
 /07000 
 9 A SOO 
 
 fi-soe 
 
 /OlOOQ 
 
 9o9qo 
 9/000 
 
 99 SOO 
 90 SCO 
 7ZOOO 
 
 ^OSOO 
 73 SOO 
 t'^Soa 
 
 73 SOO 
 66000 
 S9OO0 
 
 67000 
 
 iosoo 
 
 S'fOOO 
 
 62QOO 
 S6 000 
 SOOOO 
 
 S7SOO 
 S2 000 
 46 300 
 
 S3SOO 
 48 400 
 43300 
 
 SSO SO 
 4S400 
 40 SOO 
 
 
 3SO OOO 
 3QOOOO 
 ZSO OOO 
 
 SSt SOO 
 ■977 OOO 
 377v5'O0 
 
 / 40 000 
 JZQQOO 
 /OOOOO 
 
 fZSOOQ 
 
 J0700Q 
 
 S9 300 
 
 J/XOOO 
 9i>000 
 SO SOO 
 
 /ozooo 
 37SOO 
 
 73000 
 
 9 3 SOO 
 90000 
 67000 
 
 96 SOO 
 74000 
 
 <i>zooo 
 
 wo SOO 
 6S700 
 ■S7S0 
 
 7S0OO 
 
 £-f 000 
 
 .53700 
 
 70 300 
 
 6,0000 
 so 200 
 
 &ZS00 
 
 S3 SOO 
 -44 600 
 
 S6000 
 •i-Sooo 
 ■i-oaoo 
 
 SI 000 
 ■4-3700 
 36S00 
 
 ■4t.7SO 
 ■40000 
 33 SOO 
 
 ■43 200 
 37000 
 31000 
 
 ■^0200 
 343O0 
 2 8 700 
 
 37Soe 
 
 32000 
 36S00 
 
 3S/00 
 30000 
 3S/00 
 
 OOOO 
 
 z/ / too 
 
 336-»20 
 zttsoo 
 
 Zf 1 9SO 
 
 ssooo 
 
 7SSOO 
 60 200 
 ■47,500 
 
 A9000 
 S-fZOO 
 -42700 
 
 S/80O 
 
 49300 
 3W8oa 
 
 S6 7O0 
 4S200 
 3S400 
 
 S230Q 
 41 700 
 3 2900 
 
 -4 9 SOO 
 3 9 70O 
 30 SOO 
 
 -».f300 
 36100 
 
 3tsao 
 
 42 SOO 
 
 33900 
 
 Z6 800 
 
 37 700 
 30/00 
 
 Z370O 
 
 3-4O0O 
 
 27/00 
 2/300 
 
 30900 
 2-f *00 
 /9-400 
 
 2y3oo 
 
 2260a 
 
 I7800 
 
 3 6 /OO 
 20900 
 /6^400 
 
 24 ZOO 
 /9300 
 /S200 
 
 23 600 
 
 /Soao 
 
 /4300 
 
 21200 
 /69O0 
 /3 400 
 
 
 140 000 VOLTS DELIVERED 
 
 64 
 MILES 
 
 72 
 MILES 
 
 80 
 
 MILES 
 
 88 
 
 MILES 
 
 96 
 
 MILES 
 
 104 
 MILES 
 
 112 
 
 MILES 
 
 120 
 MILES 
 
 128 
 MILES 
 
 144 
 MILES 
 
 160 
 MILES 
 
 176 
 MILES 
 
 192 
 MILES 
 
 208 
 
 MILES 
 
 224 
 MILES 
 
 240 
 MILES 
 
 266 
 MILES 
 
 
 6 so OOO 
 too OOO 
 SSO OOO 
 
 / 033 OOO 
 9S-f OOO 
 B7-tSOO 
 
 29SOO0 
 
 Z73000 
 ZS/OOO 
 
 zezooo 
 
 242000 
 3-23000 
 
 23 AOOO 
 
 xnooo 
 
 2QIO0O 
 
 2/SOOC 
 /470OO 
 /9ZOQO 
 
 ^9^000 
 /81OQO 
 /A 7000 
 
 /n/000 
 
 tia7000 
 ISSOQC 
 
 /6900C 
 /S6000 
 /43Q00 
 
 /S7000 
 
 /■fSOOO 
 /3-4000 
 
 /47000 
 
 136000 
 /26OO0 
 
 /3JOOO 
 /2/000 
 *// 000 
 
 //Sooo 
 /oSooo 
 
 /OOOOO 
 
 / 07000 
 9 SSOO 
 ■f/000 
 
 9 6000 
 90SOO 
 S3SOO 
 
 90 SOO 
 8-iSOO 
 77S00 
 
 9-i-ooo 
 79000 
 
 7/SOC 
 
 7SSO0 
 
 72 SOO 
 67O0O 
 
 73S00 
 69 000 
 63 000 
 
 
 SOO OOO 
 ■t-SO OOO 
 ■900 OOO 
 
 79S ooo 
 7/S SOO 
 ^3t OOO 
 
 303000 
 
 /ejooo 
 
 201 000 
 /»/ 000 
 i4tlO00 
 
 / 9/ Ooo 
 /A3000 
 /4SOO0 
 
 /^sooc 
 
 /49 OOa 
 /32 OQC 
 
 /Si 000 
 /3SOOC 
 12/ OOO 
 
 I4000C 
 ISSOOO 
 .'/2 000 
 
 /30000 
 / /600Q 
 J04OQQ 
 
 /3IOOO 
 /0 9000 
 969O0 
 
 / /3aoo 
 /02000 
 90900 
 
 / 1 000 
 90400 
 8Q800 
 
 90 700 
 91-^00 
 
 72700 
 
 Si SOO 
 7-»ooo 
 66/00 
 
 7S6CO 
 67 SOO 
 
 60 600 
 
 69 SOO 
 63 600 
 
 SS900 
 
 ^4 eoo 
 
 S8/00 
 
 Sl 900 
 
 60 SOO 
 S4 300 
 49400 
 
 S6700 
 SO SOO 
 
 4S400 
 
 
 3S0 OOO 
 300 OOO 
 ZSO OOO 
 
 SS6SOO 
 ■477 OOO 
 397SOO 
 
 /58-OO0 
 /3S-000 
 //3000 
 
 / -40000 
 
 120000 
 
 /o / 000 
 
 /270OO 
 /oSooo 
 
 90700 
 
 1 /Soao 
 9 9SOO 
 9 2 SOO 
 
 /OSQOO 
 903O0 
 7SAO0 
 
 97400 
 9340c 
 &9^oo 
 
 90 400 
 77400 
 ^4900 
 
 e--*-4oo 
 72200 
 
 60 SOO 
 
 79/00 
 4>77O0 
 S6 700 
 
 70 300 
 60200 
 ■So 400 
 
 63300 
 S420O 
 
 ■ts^too 
 
 S7SOO 
 ■4930a 
 
 ■41 :ioo 
 
 S3700 
 ■4S/00 
 37 SOO 
 
 ■49700 
 
 ■41 700 
 3A900 
 
 4S2O0 
 39700 
 32400 
 
 42200 
 3 6 /OO 
 3O20O 
 
 39iOO 
 3 3 900 
 2S3O0 
 
 oooo 
 
 2// too 
 
 336-*20 
 Ztt ^OO 
 
 3 1 / 9SO 
 
 'r<S7oa 
 
 7*»ZOO 
 ^OOCO 
 
 9^000 
 
 S7 700 
 %S-3400 
 
 7^SOO 
 
 ^/ 000 
 
 4»000 
 
 *96o<> 
 
 SS4ao 
 ■43700 
 
 C38oa 
 
 So 9oo 
 
 '^OOOC 
 
 S9 90C 
 -4 6900 
 3 6900 
 
 S4700 
 4 3 €00 
 34 300 
 
 s/000 
 ■*o6oa 
 33000 
 
 ■47900 
 39/00 
 OOOOO 
 
 42 SOO 
 
 33 900 
 3 4,700 
 
 3 9 300 
 3OSO0 
 3-*ooo 
 
 3-4SOO 
 X7 700 
 31 SOO 
 
 31900 
 3S-4oa 
 30000 
 
 39-400 
 33400 
 /SSOO 
 
 27300 
 
 z/9oa 
 /7/00 
 
 3 SSOO 
 20 300 
 / ioOO 
 
 3390O 
 
 /9ooe 
 
 /SOOO 
 
 The loss due to corona will not be excessive with any of the above conductors used at sea level for the voltages stated. For elevations 
 above sea level, check the values with Table XXII, especially for the smaller conductors. On long circuits of high voltage, the effect of 
 charging current (also corona and leakage losses) will be to increase or decrease the I'B loss, depending on the amount of load and its power- 
 factor. See Fin. 13 
 
QUICK ESTIMATING TABLES 
 
 33 
 
 HEATING LIMITATIONS 
 
 The k.v.a. values given in these tables do not take 
 into account the heating and consequently carrying cap- 
 acity of the conductors. This may be ignored in the 
 case of the longer overhead high-voltage transmission 
 circuits. For very short circuits (especially for the 
 lower voltages and particularly for insulated or con- 
 cealed conductors) the carying capacity (safe heating 
 limits) of the conductors must be carefully considered. 
 
 approximately the point at which the carrying capacity 
 of that particular conductor is reached if insulated and 
 installed in a fully loaded four duct line. If the con- 
 ductor is to be installed in a duct line having more than 
 four ducts its Opacity will be still further reduced. 
 The position of this line is based upon the use of lead 
 covered, paper insulated, three conductor, copper cables 
 for sizes up to 700000 circ. mils and of lead covered, 
 paper insulated, single conductor, copper cables for the 
 larger sizes. In other words, the position of this heavy 
 
 TABLE XXI-QUICK ESTIMATING TABLE 
 
 CONDUCTORS 
 
 KILOVOLT. AMPERES. 3 PHASE WHICH MAY BE DELIVERED AT THE FOLLOWING VOLTAGES OVER THE VARIOUS 
 CONDUCTORS FOR THE DISTANCES STATED. BASED UPON THE FOLLOWING |2r LOSS (EFFECT OF CHARGING CURRENT 
 
 FOR LOAD POWER-FACTOR OF IOO%-8 66% LOSS- 10 0% LOSS 
 FOR LOAD POWER-FACTOR OF 8CWfr- 10.8% LOSS- 12.5 LOSS 
 
 
 COPPER 
 
 AREA 
 IN 
 
 CIRCULAR 
 
 MILS 
 
 ALUMINUM 
 
 AREA 
 
 IN 
 
 CIRCULAR 
 
 MILS 
 
 154,000 VOLTS DELIVERED 
 
 
 98 
 MILES 
 
 104 
 MILES 
 
 112 
 MILES 
 
 120 
 MILES 
 
 128 
 MILES 
 
 144 
 MILES 
 
 160 
 MILES 
 
 176 192 
 MILES MILES 
 
 208 
 MILES 
 
 7U 
 
 MILES 
 
 240 
 
 MILES 
 
 256 
 MILES 
 
 288 
 MILES 
 
 320 
 MILES 
 
 352 
 MILES 
 
 384 
 MILES, 
 
 
 6S0 000 
 600 000 
 SSO 000 
 
 / 033 000 
 954 000 
 8 74 SOO 
 
 237 SOC 
 ZI9 OOO 
 202 000 
 
 2I9SOO 
 202 OOO 
 1^7000 
 
 203 .SOO 
 IS7i00 
 1 73 000 
 
 190 000 
 175000 
 161 SOO 
 
 I7SS00 
 I6i000 
 IS 1 SOO 
 
 iMsoo 
 
 I4^.sr:c 
 134 SOO 
 
 142 son 
 
 :Zi Skiu 
 
 121 300 
 
 139^0.- 
 119 SOO 
 110 200 
 
 IISSOO 
 I09S0O 
 101 000 
 
 JO9S00 
 lot 000 
 
 93 30O 
 
 102000 
 
 93 SOO 
 
 96 600 
 
 9 J 000 
 
 S7J00 
 80 800 
 
 89 000 
 92000 
 7S700 
 
 7930O 
 73 000 
 67300 
 
 7/300 
 
 <J700 
 
 60700 
 
 49 700 
 S9700 
 SS300 
 
 SI 300 
 59 700 
 5o.rao 
 
 
 500 00 
 46 000 
 400 000 
 
 795 000 
 7/S SOO 
 63i 000 
 
 /83J00 
 /it SOO 
 147 000 
 
 IbtOOO 
 ISIOOO 
 131,000 
 
 r 37 000 
 141 000 
 12 1 SOO 
 
 I46S00 
 I3H00 
 117 SOO 
 
 l3.'Son 
 I23SOO 
 //OSOO 
 
 I29nnrt 
 I09S00 
 93 000 
 
 109 Sf^^ 
 98800 
 88300 
 
 99 SOO 
 99 900 
 so 200 
 
 91 SOO 
 «3 300 
 73 SOO 
 
 94300 
 76000 
 67 900 
 
 79 SOO 
 70 600 
 63 000 
 
 73 «0 
 45 800 
 S3 SOO 
 
 69 700 
 6/ 700 
 SS 300 
 
 6/000 
 S4900 
 49000 
 
 ssooo 
 
 49300 
 
 44200 
 
 SO 000 
 4 4 900 
 40200 
 
 4S700 
 4/30O 
 34 800 
 
 
 3S0 000 
 300 000 
 250 000 
 
 S5(, SOO 
 477 000 
 3?7 SOO 
 
 12^200 
 I09S00 
 91 Soo 
 
 IIS SOO 
 102000 
 S4 600 
 
 IO9S00 
 94 000 
 78 50O 
 
 102 SOO 
 S7.S00 
 73200 
 
 96 000 
 92 000 
 6t600 
 
 83 300 
 73000 
 61 000 
 
 74800 
 6S70O 
 S4800 
 
 69 900 
 
 S9 70O 
 49 900 
 
 64 000 
 S470O 
 4S700 
 
 ■S92O0 
 SO SOO 
 42200 
 
 S4 300 
 
 4690a 
 
 39200 
 
 SI 200 
 
 43700 
 36iOO 
 
 49000 
 41000 
 34 300 
 
 42700 
 36 SOO 
 30 SOO 
 
 3».£00 SSOOO 
 32 900 39 900 
 37400 24 900 
 
 33.IOO 
 27 300 
 22900 
 
 
 
 336420 
 Zbt, SOO 
 
 77 200 
 
 tisoa 
 
 7/XOO 
 St SOO 
 
 6(200 
 S2700 
 
 61 iOO 
 49200 
 
 S7700 
 46100 
 
 SI SOO 
 41000 
 
 46200 
 36900 
 
 ■9 3 000 
 33 600 
 
 3SS00 
 30700 
 
 3S60O 
 29 400 
 
 33 000 
 
 2640a 
 
 30 SOO 
 34 600 
 
 29 900 
 23/00 
 
 2S70t\ 23200 
 30S0C\ /S'fOO 
 
 ZIOOO 
 
 /tsoo 
 
 19 200 
 
 /.r9oo 
 
 
 
 187,000 VOLTS DELIVERED 
 
 
 06 
 MILES 
 
 104 
 
 MILES 
 
 112 
 
 MILES 
 
 120 
 MILES 
 
 128 
 MILES 
 
 144 
 
 MILES 
 
 160 
 
 MILES 
 
 176 
 
 MILES 
 
 192 
 MILES 
 
 208 
 MILES 
 
 224 
 MILES 
 
 240 
 
 MILES 
 
 266 
 MILES 
 
 288 
 MILES 
 
 320 
 
 MILES 
 
 352 
 
 MILES 
 
 384 
 
 MILES 
 
 
 6SO 000 
 
 SCO 000 
 
 SSO 000 
 
 1 033 000 
 954 000 
 87-f 500 
 
 J4"0 00O 
 
 aasooo 
 
 399 000 
 
 32 Z ooo 
 397000 
 27S0O0 
 
 300 000 
 27700a 
 
 2Sioao 
 
 z^oooo 
 
 2S9 00C 
 239000 
 
 2 63 000 
 2420CO 
 224000 
 
 334 000 
 ZlSOOO 
 199000 
 
 zio 000 
 
 193 SOO 
 179 000 
 
 191 000 
 1 7« 000 
 162 SOO 
 
 I7SOOO 
 161 SOO 
 149000 
 
 1 61 SOO 
 149 000 
 I37SOO 
 
 /SO ooo 
 
 /38 0OO 
 (28000 
 
 /40 000 
 /29 000 
 / 1 9 SOO 
 
 /3/ SOO 
 121 000 
 1/2000 
 
 / 16 SOO 
 107 SOO 
 99 SOO 
 
 /OS 000 
 91000 
 99 SOO 
 
 IS 30a 
 99000 
 
 9/ SOO 
 
 %7 SOO 
 80 700 
 
 79400 
 
 
 SOO 000 
 4S0 000 
 40c 000 
 
 79S 000 
 7 IS SOO 
 636 000 
 
 270 ooo 
 2^7000 
 
 2 SO ooo 
 32SOO0 
 200 000 
 
 232000 
 209000 
 
 lessoo 
 
 2)6000 
 194 SOO 
 173 SOO 
 
 203000 
 1 92000 
 I62SOO 
 
 1 80000 
 / 62 000 
 /44SO0 
 
 I6Z000 
 l*S SOO 
 130 OOO 
 
 147 SOO 
 /32S0O 
 IISSOO 
 
 /3SOOO 
 12/ SOO 
 
 lotsoo 
 
 I2SO00 
 112000 
 100 000 
 
 //SSOO 
 104 000 
 93 000 
 
 109000 
 97200 
 8*700 
 
 10 1000 
 91 000 
 
 Si 300 
 
 90 000 
 Si 000 
 12200 
 
 9 1000 
 13 000 
 6SO00 
 
 13 700 
 44 200 
 S9000 
 
 67400 
 60 700 
 .59 200 
 
 
 
 SS6 SOO 
 477 000 
 397 SOO 
 
 1^1 OOO 
 
 /3-fvyoo 
 
 '74S0a 
 
 149 ooe 
 
 114 SOO 
 
 162 000 
 /JSOOO 
 IISSOO 
 
 /SI OOO 
 179000 
 I07S00 
 
 141 Soo 
 121 000 
 101 000 
 
 / 26000 
 107 SOO 
 
 ?9aoo 
 
 / 13 SOO 
 96S00 
 
 SO 800 
 
 103 000 
 
 tfooo 
 
 73 SOO 
 
 94 SOO 
 SO SOO 
 67 300 
 
 87000 
 7-* 300 
 43200 
 
 SI 000 
 690O0 
 S7SO0 
 
 7SS0O 
 64 SOO 
 .53 800 
 
 70 800 
 60 SOO 
 
 So SOO 
 
 63 000 
 
 S3 700 
 44 Soo 
 
 Si TOO 
 -18 300 
 ♦ 0+00 
 
 SI SOO 
 ■44 000 
 36 900 
 
 47300 
 -to SOO 
 33 700 
 
 
 
 336 4ZO 
 
 /l-f^OOO 
 
 /osooo 
 
 9 7 SOO 
 
 91000 
 
 %S200 
 
 7S9O0 
 
 ««JOO 
 
 63 000 
 
 S7000 
 
 52 400 
 
 4S800 
 
 4SSO0 
 
 .♦2 700 
 
 J»ooo 
 
 39 200 
 
 3/ 000 
 
 xtioo 
 
 
 
 220,000 VOLTS DELIVERED 
 
 
 86 
 MILES 
 
 104 
 MILES 
 
 112 
 MILES 
 
 120 
 MILES 
 
 128 
 MILES 
 
 144 
 
 MILES 
 
 160 
 MILES 
 
 176 
 
 MILES 
 
 192 
 MILES 
 
 208 
 
 MILES 
 
 224 
 MILES 
 
 240 
 
 MILES 
 
 256 
 MILES 
 
 288 
 MILES 
 
 320 
 MILES 
 
 352 
 MILES 
 
 384 
 MILES 
 
 
 6S0 000 
 
 1 033 000 
 9S4 000 
 874 SOO 
 
 ■^vsooo 
 
 4-i7ooO 
 413 000 
 
 447000 
 ^13000 
 3^2000 
 
 417 ooo 
 3S3 OOO 
 3S4OO0 
 
 3SiO00 
 
 sstooo 
 
 331 000 
 
 364 ooo 
 336 000 
 310 000 
 
 323 000 
 399000 
 27^000 
 
 29' ooo 
 36S000 
 249000 
 
 26 S 000 
 24*000 
 236000 
 
 243 000 
 224 000 
 207 000 
 
 2 34 000 
 207000 
 19/ 000 
 
 2OJ000 
 192 000 
 /77 SOO 
 
 194000 
 / 79 000 
 /6SS00 
 
 192000 
 169000 
 ISSOOO 
 
 / 63 000 
 149000 
 139 000 
 
 143 SOO 
 /34S00 
 124000 
 
 132 SOO 
 /22 000 
 II2900 
 
 /2/ SOO 
 1/2 000 
 
 tOiSao 
 
 
 
 79s 000 
 71 S SOO 
 63<, 000 
 
 374 000 
 33iOOO 
 300000 
 
 34SOOO 
 3 10 OOO 
 277O0O 
 
 33.1 oaa 
 !SS ooo 
 2S7 000 
 
 299000 
 269000 
 240000 
 
 39/ 000 
 1S2OO0 
 33S0OO 
 
 Z49000 
 22.f 000 
 300000 
 
 234 000 
 202 000 
 
 ISO 000 
 
 204000 
 1 S3 SOO 
 163 SOO 
 
 187000 
 l6tOOO 
 
 'SO 000 
 
 173 000 
 
 fSS 000 
 
 133 SOO 
 
 160000 
 I44OO0 
 I29SOO 
 
 /.<7.roo 
 
 134 SOO 
 120 000 
 
 '4o ioo 
 / 24000 
 "2J0O 
 
 124 SOO 
 
 /•z 000 
 
 '00000 
 
 1/3 SOO 
 to- OOO 
 
 9000a 
 
 103 000 
 92000 
 92000 
 
 93 SOO 
 
 89 20O 
 
 7SOOO 
 
 
 The loss due to corona will not be excessive with any of the above condnctors used at sea level for the voltages stated. For elevatioui 
 
 above sea level, check the values witli Table XXII, especially for the smaller conductors. On long circuits of hijrh yoitatre, the effect of 
 
 charging current (also corona and leakage losses) will be to increase or decrease the I'R loss, depending on the amount of load and its power- 
 factor. See Fig. 13 
 
 For circuits of short length the carrying capacity of 
 conductors will frequently determine these sizes and 
 not the economic transmission loss. The carrying cap- 
 acity of bare copper conductors suspended in air and 
 of insulated copper conductors in duct lines are given 
 in tables XXIII and XXIV, both of which are to appear 
 in subsequent articles. 
 
 Running diagonally across each table from XII to 
 XVII inclusive, is a heavy line. The point at which 
 this heavy line intersects the horizontal line containing 
 the k.v.a. values for a given size of conductor indicates 
 
 line is based upon the k.v.a. values for carrying capacity 
 given in Table XXIV and is placed upon the tables as a 
 warning that the heating limit capacity of the conductors 
 must be considered. To illustrate, suppose 220 volts is 
 to be delivered, over i 000 cxx) circ. mil, insulated, single 
 conductor, copper cables in a fully loaded four duct 
 conduit. Table XII indicates that 189 k.v.a. can be 
 transmitted over these conductors a distance of 2000 ft. 
 without overheating the cable. If it is desired to trans- 
 mit 378 k.v.a. a distance of 1000 feet, the fact that this 
 value occurs to the left of the heavy line, indicates that 
 
34 
 
 QUICK ESTIMATING TABLES 
 
 it is beyond the safe carrying capacity for this size con- 
 ductor in a four duct line. Reference to Table XXIV 
 will show that 297 k.v.a. is the maximum capacity of 
 this cable under the conditions stated. In this case, 
 either a larger conductor, or two or more smaller con- 
 ductors must be used to prevent overheating. This will 
 result in a less loss than those upon which the table 
 k.v.a. values are based, and in this case the heating of 
 the cable will probably determine the size to use. 
 
 EFFECT OF CHARGING CURRENT IN ABOVE I^R LOSS VALUES 
 
 As stated previously, the percent PR losses in the 
 quick estimating tables are based upon the load current 
 and therefore do not take into account the effect of the 
 charging current which is of a distributed nature and 
 superimposed upon the load current. The effect of the 
 charging current is to increase or decrease the current 
 in the circuit by an amount depending upon the relative 
 
 2800 
 
 SUPPLY END 
 
 200 300 
 
 DISTANCE IN MILES FROM SUPPLY END 
 
 FIG. 13 — EFFECT OF CHARGING CURRENT ON TR TRANSMISSION LOSS 
 
 The curves represent (for certain circuits) an approximation of the resultant I'R loss, compared to 
 what it would have been if there were no charging current present in the circuit. The effect of the 
 charging current superimposed upon the receiver current is either to increase or to decrease the PR 
 loss of the circuit depending principally upon the relative amount of the leading and lagging components 
 of the current in the circuit. 
 
 values of the lagging and leading quadrature compon- 
 ents of the current in the circuit. 
 
 For instance assume that the power-factor of the 
 load is unity. In such case there is no quadrature com- 
 ponent in the load current. If, however, the circuit is 
 of considerable length, and particularly if the frequency 
 is 60 cycles, there will be an appreciable amount of 
 charging current (quadrature leading component) 
 added vectorially to the load current. The sum of 
 these two currents in quadrature with each other wiil 
 result in an increase of current in the circuit with a 
 consequent increase in the I^R loss. Thus the effect 
 of charging current in a circuit delivering a load of 100 
 percent power-factor will always be to increase the PR 
 loss. 
 
 If, however, the power- factor of the load is laggin(». 
 
 there will be a lagging component in the load current. 
 The charging or leading current v,-ill be practically in 
 opposition to the lagging component of the load current 
 and will therefore tend to cancel or neutralize the lag- 
 ging component of the load current. The result will 
 be a reduction of the current in the circuit and con- 
 sequently in the PR loss. But if the circuit is very 
 long, particularly if the frequency is 60 cycles and the 
 load power-factor is near unity (lagging component in 
 load current small) the comparatively large leading 
 component (charging current) will not only neutralize 
 'the lagging component of the load current, but will 
 produce a leading power-factor at points along the cir- 
 cuit. If the charging current is sufficiently high it will 
 ncrease the current, causing an increase in the PR 
 OSS. Thus the effect of charging current in circuits 
 lelivering a lagging load is to decrease the PR loss up 
 
 to a certain 
 amount and then, 
 if the charging 
 current is s u ffi- 
 ciently large, to in- 
 crease PR loss. 
 
 The curves in 
 Fig. 13 show this 
 effect for 25 and 
 60 cycle circuits 
 delivering loads of 
 unity power-fac- 
 tor; also loads 
 of 80 percent lag- 
 ging power-factor 
 for circuits up to 
 500 miles long. It 
 will be seen that 
 for circuits 300 
 miles long the 
 effect of charging 
 current will be to 
 reduce the PR 
 loss by approxi- 
 mately 25 percent 
 if the load is 80 
 percent lagging. If the load power-factor is unity the 
 PR loss will be increased approximately 10 percent for 
 these particular problems if the frequency is 25, and 
 30 percent if the frequency is 60 cycles. 
 
 The curves in Fig. 13 show that for circuits 500 
 miles long, in which the entire charging current is fur- 
 nished from one end of the circuit, the effect of this 
 charging current is to increase the PR loss by 300 per- 
 cent if the frequency is 60 cycle and the load power- 
 factor 100 percent. In other words a large part of the 
 current in the circuit for such a long 60 cycle circuit 
 is charging current so that the effect of the load current 
 on the PR loss is comparatively small. Of course such 
 a long circuit, unless fed from two or more generating 
 stations located at widely separated points along the 
 transmission lin3, would not be commercially practical. 
 
CHAPTER IV 
 CORONA EFFECT 
 
 In 1898 Dr. Chas. F. Scoft presented a paper before the A.I.E.E. describing experimental tests (made 
 during several years previous) relating to the energy loss between conductors due to corona effect. These 
 investigations began at the Laboratory at Pittsburgh and were continued at Telluride, Colorado, in con- 
 junction with the engineers of the Telluride Power Company. Preliminary observations were made by 
 Mr. V. G. Converse and were continued in notable measurements by Mr. R. D. Mershon. These investiga- 
 tions were later followed by the work of Professor Ryan, by Mr. R. D. Mershon, Mr. F. W. Peek, Jr., Dr. 
 J. B. Whitehead, Mr. G. Faccioli and others. The electrical profession is particularly indebted to Mr. Peek 
 and Dr. Whitehead for the large amount of both practical and theoretical data which they have presented 
 to the electrical profession on the subject. Mr. Peek developed and presented the empirical formulas 
 which follow, for determining the disruptive critical voltage, the visual critical voltage and the power 
 loss due to corona effect. The close accuracy of Mr. Peek's formulas has been confirmed by various in- 
 vestigators in different sections of the country. The following deductions concerning corona have to a 
 large extent been previously presented by Mr. Peek. 
 
 CORONA, manifesting its presence usually by an 
 electrostatic glow or luminous discharges, and 
 audibly by a hissing sound, was clearly observed 
 and studied in connection with electrostatic machines. 
 It did not become a serious factor to be considered in 
 connection with the design of commercial electrical ap- 
 paratus until the increasing generator and transmission 
 voltage emphasized its important. 
 
 Although it is usual to think of corona effect only 
 in connection with high-voltage transmission lines, it 
 has received not a little thought of late by the designers 
 of high-voltage generators and motors, notably large, 
 high-voltage turbogenerators. By effectively insulating 
 the portion of the conductor embedded in the iron of 
 the armatures of alternating-current machines, particu- 
 larly with mica, punctures to ground due to corona ef- 
 fect are not likely to occur. However, at the ends of 
 the armature coils (where it is difficult to employ mica 
 for insulating), where air is partially depended upon 
 as an insulating medium between coils and ground, 
 corona may appear. The presence of these corona 
 stresses results in disintegrating and weakening some 
 kinds of insulating materials, causing them to break 
 down after a period of service. This deterioration of 
 insulation may be due to local heating, mechanical vi-. 
 bration or chemical formations in the overstressed air, 
 such as ozone, nitric acid, etc. 
 
 Higher voltages are being chosen as an economic 
 means for reducing loss in transmission. These higher 
 voltages may result in corona loss far in excess of the 
 saving in transmission loss due to the adaptation of the 
 higher voltages. It is, therefore, pertinent that par- 
 ticular consideration be given to the limitation of corona 
 loss when the choice of conductors is made. This con- 
 sideration will sometimes make it desirable to take ad- 
 vantage of the higher critical voltage limits of aluminum 
 conductors (with steel reinforced centers) of an equiva- 
 lent resistance, due to their greater diameter; or it may 
 be desirable to obtain the necessary larger diameter by 
 the use of copper conductors having some form of non- 
 conducting centers or, for still larger diameters, of 
 
 aluminum conductors having such centers, in order to 
 avoid skin effect. The use of copper conductors having 
 hemp centers has in some instances given mechanical 
 trouble. 
 
 The critical voltage at which corona becomes mani- 
 fest, is not constant for a given line, but is somewhat 
 dependent upon atmospheric conditions. Assuming a 
 Ime employing conductors just within the critical volt- 
 age limitations for the conditions to be met, the corona 
 loss in such a line would be almost negligible during fair 
 weather, but during stormy weather, (particularly dur- 
 ing snowstorms) this corona loss would be many times 
 what it is during fair weather. On the other hand, 
 since the storm will usually not appear over the whole 
 length of lines at the same time and since storms occur 
 only at intervals, it may often be economical to allow 
 this loss to reach fairly high values during storms. Fog, 
 sleet, rain and snowstorms lower the critical voltage 
 and increase the losses. The effect of snow is greater 
 than any other weather condition. Increase in tem- 
 perature or decrease in barometric pressure lowers the 
 voltage at which visual corona starts. 
 
 The critical voltage increases with both the dia- 
 meter of conductors and their distance apart. This 
 sometimes makes it desirable to use aluminum conduc- 
 tors as previously stated. It also increases with the 
 horizontal or vertical arrangement of conductors, due 
 to the fact that the two outside conductors considered 
 as a pair are twice as far apart as are the other pairs. 
 The same general rules apply to stranded conductors as 
 to solid conductors, the actual diameter of the former 
 being considered as the effective diameter of the con- 
 ductor. 
 
 The losses due to corona effect increase very 
 rapidly with increa.se in voltage after the critical voltage 
 has been reached. A long transmission line having 
 considerable capacitance may deliver a higher voltage 
 than appears at the generator end of the fine due to 
 capacitance effect. The corona loss would in this case 
 be greater per mile at the receiving end than at the 
 sending end of the line. 
 
36 
 
 CORONA EFFECT 
 
 The magnitude of the losses, as well as the critical 
 voltage, is affected by atmospheric conditions; — hence 
 tliey probably vary with the particular locality and the 
 season of the year. Therefore, for a given locality, a 
 voltage which is normally below the critical point, may 
 at times be above the critical voltage, depending upon 
 changes in the weather. 
 
 The material of the conductors does not seem to 
 affect the losses. Sometimes the conductors of. new 
 transmission lines, when first placed in service will show 
 visual corona, which may entirely disappear after a few 
 hours or weeks of service. This may be due to 
 scratches, particles of foreign substances, etc., on the 
 conductors which are eliminated after the voltage stress 
 has been kept on the conductors for a short time. Un- 
 der such conditions the corona loss will also become 
 less as the visual effect disappears. 
 
 The loss of power due to corona effect increases 
 with frequency and increases as the square of the ex- 
 cess voltage above a certain critical voltage referred to 
 as the "disruptive critical voltage" e^. This disruptive 
 critical voltage is that voltage, at which a certain definite 
 and constant potential gradient is reached at the con- 
 ductor surface. This gradient go is 30 kv maximum 
 (21. 1 kv effective) per centimeter, or 76.2 kv maximum 
 (53.6 kv effective) per inch. These values are based 
 upon an air density at sea level (25° C, 29.92 inches or 
 76 cm. barometer). This gradient is independent of 
 the size of conductors and thcr distance apart, but is 
 proportional to the air density, that is to the barometric 
 pressure and the absolute temperatures. It may be con- 
 sidered as the dielectric strength of air. The presence 
 of corona at a certain point of the system shows that a 
 critical electric stress has been exceeded at that point. 
 The corona loss is also proportional to the square root 
 of the conductor radius r and inversely proportional to 
 the square root of the conductor spacing. 
 
 The law by which corona losses increase with the 
 voltage does not give a very steep curve, but a rather 
 mild curve following the quadratic law at and above the 
 critical limit. In other words there is no sharp elbow 
 in the curve above which the losses increase very rapidly 
 with the voltage and which could be adopted as the 
 normal operating point of the circuit. 
 
 Table XXII, indicating the voltage limitations due 
 to corona effect, has been worked up from Mr. F. W. 
 Peek's formula as indicated at the bottom of the table. 
 The values in this table are conservative and may in 
 many cases be exceeded. They are the effective e^ dis- 
 ruptive critical voltage between conductors for fair 
 weather based upon S values for 25 degrees C. (^y 
 degrees F) and m^ values of 0.87 for cable and 0.93 
 for wire. With these table values, corona loss should 
 not be excessive during storms. If the values of 
 Table XXII indicate that the conductors contemplated 
 are close to the limit due to corona effect, a careful 
 check should be made by the formula to determine 
 definitely the corona loss for such conductors under 
 storm operating conditions. 
 
 e, = 2.302 7>H go hr 
 
 (-^) 
 
 logv 
 
 . {21) 
 
 (e — Co) ^ lo--' {22) 
 
 F. W. PEEK S CORONA FORMULAE 
 
 Disruptive Critical Volts, Fair Weather (parallel wires) 
 
 fo = 2.302 niogo 8 r logm— {20) 
 
 effective kv to neutral, — 
 Visual Critical Volts — Fair Weather (parallel wires) 
 
 s 
 r 
 
 effective kv to neutral 
 Power Loss (fair weather) — 
 
 P -'f {/ + ^5) ^T 
 
 kw per mile of each conductor 
 
 Power Loss {Storm) — Storm power loss is higher and can 
 generally be found with fair approximation by assuming <?<, ^= 
 0.80 times fair weather c„. It generally works out in practice 
 that the Co voltage is the highest that should be used on trans- 
 mission lines {22A ) 
 
 All of the above voltages are to neutral. To find voltages 
 between lines multiply by 1.73 for three-phase, and by 2 for 
 single phase. 
 
 Notation — 
 
 e = Effective applied voltage in kv to neutral. 
 
 (This will vary at different points of the circuit and 
 at different loads. At low loads and long lines of 
 high voltage it may be higher at the receiving end 
 than at the generator end due to inductive capaci- 
 tance) 
 
 fo = effective disruptive critical voltage in kv to neutral. 
 It is the voltage that gives a constant break down 
 gradient for air of 76 kv maximum per inch, the 
 "elastic limit" at which the air breaks down. 
 Visual corona does not start at the disruptive 
 critical voltage, but at a higher voltage Cv. 
 
 Cf = effective visual critical kv to neutral (voltage at 
 which visual corona starts) 
 
 P = power loss in fair weather in kw per mile of single 
 conductor, 
 
 J 7.9b 
 
 s = 
 
 This takes care of the effect of altitude 
 
 459 + t 
 
 and temperature, (air density). It is I at 25 de- 
 grees C. {77 degrees F.) and 29.92 inches (76 cm.), 
 barometric pressure. 
 (7o = 53.6 kv per inch effective (disruptive gradient of air) 
 b =: barometric pressure in inches. 
 t = maximum temperature in degrees F. 
 / = frequency in cycles per second, 
 m. = irregularity factor. 
 = I for polished wires. 
 
 =^ 0.98 to 0.93 for roughened or weathered wire. 
 = 0.87 to 0.83 for cables. 
 »ir = nto for wires (l to O.93) 
 
 = 0.72 for local corona all along cables (7 strands) 
 := 0.82 for decided corona all along cables (7 strands) 
 r r=z radius of conductor in inches. 
 
 J = spacing in inches between conductor centers, based 
 upon the assumption of a symmetrical triangular 
 arrangement. For three-phase irregular flat or tri- 
 angular spacing take .r := f Anc . For three- 
 phase regular flat spacing tako s = J.26A. 
 Theoretically, if the conductors were perfectly smooth, no 
 loss would occur until the critical voltage, Ct is reached, when 
 the loss should suddenly take a definite value, equal to that 
 calculated by quadratic law, with Cy as the applied voltage and 
 Co as the critical voltage in the equation. It should then follow 
 the quadratic law for all higher voltages. On the weathered 
 conductors used in practice, the quadratic law is followed over 
 the whole range of voltage, starting at eo. 
 
 Example :— In order to show the variation in corona loss 
 at different voltages and for different weather conditions, Table 
 E has been calculated for No. o stranded copper conductors 
 (105560 circ. mils, 0.373 in. diameter) and for steel reinforced 
 aluminum conductors (167800 circ. mils, 0.501 in. diameter) 
 having an quivalent resistance but greater diameter. F. W. 
 Peek's formulas were used and the following assumptions were 
 made : — 
 
 / =60 cycles, 
 mo ^ 0.87 
 m, = 0.72 
 go = 53-6 
 
CORONA EFFECT 
 
 37 
 
 r 
 s 
 b 
 t 
 s 
 r 
 
 4 
 
 = 0.186 in. for copper = 0.250 in. for aluminum. 
 =; 144 inches (delta arrangement of conductors). 
 = 28.9 corresponding to an altitude of 1000 feet 
 = 77 degrees F. S therefore = 0.967. 
 
 = 774 for copper = 576 for aluminum 
 logio 774 = 2.89 and logio 576 = 2.76 
 
 iL= 0.036 for copper and 0.0415 for aluminum. 
 s 
 
 DISRUPTIVE CRITICAL VOLTAGE— Fair Weather 
 
 eo = 2.302 nto gohr log\a — {20) 
 
 effective kv to neutral 
 For the Copper Conductors 
 
 (To = 2.302 X 0.S7 X 53.6 X 0.967 X 0.186 X 2.89 
 = 55.8 kv to neutral (96500 volts between conductors). 
 Table XXII gives, by interpolation, the limitation of e, for 
 above conditions, as 96500 volts between conductors. To find 
 Co to neutral for any other altitude or temperatures insert the 
 corresponding values of 8 for the altitude and temperature in 
 the formula. 
 
 TABLE D— WORKING TABLE- S (DENSITY) VALUES 
 
 Altitude and Temperature Correction Factors 
 
 17.9b 
 
 8 = 
 
 459 + t 
 t = temperature in degrees F. 
 
 where b = barometric pressure in inches and 
 
 .Mtitude ill 
 
 Barometer 
 
 Values for Different Temp. 
 
 
 
 
 
 
 Feet 
 
 III 
 
 
 o»C. 
 
 25° c. 
 
 see. 
 
 
 Inches 
 
 1 
 
 
 (S:!" F.) 
 
 (77° F.) 
 
 (I3i» F.) 
 
 Sea Level 
 
 30.0 
 
 76.2 
 
 1.09 
 
 * 
 1. 00 
 
 0.925 
 
 500 
 
 2945 
 
 74.8 
 
 1.07 
 
 0.985 
 
 0.910 
 
 1000 
 
 28.90 
 
 73-3 
 
 1.05 
 
 0.967 
 
 0.892 
 
 1500 
 
 28.30 
 
 71.8 
 
 1.03 
 
 0.947 
 
 0.873 
 
 2000 
 
 27,80 
 
 70.7 
 
 I.OI 
 
 0.932 
 
 0.860 
 
 2500 
 
 27.25 
 
 69.2 
 
 0.955 
 
 0.912 
 
 0.841 
 
 3000 
 
 26.80 
 
 68.0 
 
 0.980 
 
 0.897 
 
 0.827 
 
 4000 
 
 2575 
 
 65.3 
 
 0.940 
 
 0.860 
 
 0.793 
 
 5000 
 
 24.70 
 
 62.7 
 
 0.902 
 
 0.827 
 
 0.762 
 
 6000 
 
 23.90 
 
 60.7 
 
 0.875 
 
 0.800 
 
 0.738 
 
 7000 
 
 22.95 
 
 58.3 
 
 0.840 
 
 0.770 
 
 0.710 
 
 8000 
 
 22.05 
 
 56.0 
 
 0.805 
 
 0.738 
 
 0.682 
 
 9000 
 
 21.30 
 
 54-1 
 
 0.778 
 
 0.712 
 
 0.657 
 
 10 000 
 
 20.50 
 
 52.1 
 
 0.750 
 
 0.687 
 
 0.633 
 
 12000 
 
 19.00 
 
 48.3 
 
 0.697 
 
 0.637 
 
 0.588 
 
 14000 
 
 17.55 
 
 447 
 
 0.643 
 
 0.588 
 
 0.543 
 
 15000 
 
 16.90 
 
 42.9 
 
 0.618 
 
 0.566 
 
 0.522 
 
 ♦This column contains the values for 8 which were used 
 in determining the values of Co in Table XXII. That is, the 
 ■values for sea level in Table XXII multiplied by these 8 values 
 gives the c, values of the table for the higher altitudes. 
 
 For the Aluminum Conductors 
 
 eo = 2.302 X 0.8/ X 33-6 X 0.967 X 0.23 X 2.76 
 = 71.5 kv to neutral (123500 volts between conductors). 
 Table XXII gives (by interpolation) the limitation for 
 above conditions as 123500 volts between conductors. 
 
 To find Co to neutral for any other altitude or temperature 
 insert the corresponding value of 8 for that altitude and tem- 
 perature in the formula. 
 DISRUPTIVE CRITICAL VOLTAGE— Stormy Weather 
 
 Co during storm = approximately 80 percent eo during 
 fair weather. 
 
 For the Copper Conductors 
 
 eo for storm = 55.8 X O.80 = 44.6 kv to neutral or 
 77000 volts between conductors. 
 
 For the Aluminum Conductors 
 
 fo for storm = 715 X 0.80 = 57.2 kv to neutral or 
 98800 volts between conductors. 
 
 VISUAL CRITICAL VOLTAGE— Fair Weather 
 
 o./8g 
 
 e, = 2.302 Wv goSr 
 effective kv to neutral 
 
 ( 0./S9 y, s 
 
 (^/) 
 
 For Copper Conductors 
 
 e, - i^o2Xo.72Xs3.6Xo.967Xou86(i+-^^\ 2J9 
 
 = 66.4 kv to neutral (115 000 volts between conductors). 
 
 To find e, to neutral for any other altitude and temperature, 
 insert the corresponding values of 8 for that altitude and tem- 
 perature in the formula above. 
 
 For the Aluminum Conductors 
 
 Cr - 2^O2X0.72XS3-6Xo.<f67X0.2s(/+—-^\ 2.76 
 
 = 82 kv to neutral (141S00 volts between conductors). 
 
 To find e, to neutral for any other altitude and temperature, 
 msert the corresponding values of S for that altitude and tem- 
 Jerature in the formula above. 
 POWER LOSS _ 
 
 P=Y ^ + ^^^ J-T ^^ " ^°^' "^ ^"^ 
 
 kw per mile of each conductor 
 
 The corona power loss corresponding to various conditions 
 for the above circuit has been calculated by formulae (22) and 
 (22A). They are given in Table E. However, in order to 
 illustrate the application of the power loss formula the losses 
 for the following conditions are determined below. Assuming 
 that the No. stranded copper conductors will be operated at 
 105 kv between conductors (60.7 kv to neutral). 
 
 For Fair Weather— Max. Temp. 50 degrees C. (122 degrees 
 F.)— £„ = 51.3 kv. 
 
 P = ^^U + ^5) 
 
 AjL {.e- eoY-iar^ {ji2) 
 
 kw per mile of each conductor 
 
 ^ ° TSgT e^ + ^.^) X "•"3'^ (^-7 - Sr.3)^ /o"' 
 
 = 1.2 kw per mile of each conductor or 3.6 kw per mile 
 for three conductors. 
 
 For Stormy Weather— Max. Temp. 25 degrees C. {77 de- 
 grees F.)— £. = 55.8 X 0.8 = 44.6 kv. 
 3go 
 ^ = ~^:^ (^ + ^^) X °-°S^ (^-7 - 44.6Y 10-^ {22 A) 
 
 = 3.2 kw per mile of each conductor or 9.6 kw per mile 
 for three conductors. 
 
 By applying formula (20) to the ^.bove case it develops 
 that the fair weather values of e, are for 25 degrees C. {77 
 degrees F.) 96500 kv and for 50 degrees C. (122 degrees F.) 
 B8800 kv between conductors. Table XXII values for 25 de- 
 grees C. {77 degrees F.) confirm this. 
 
 Table E values for corona loss indicate that No. o 
 copper conductors can, with 144 inch delta arrangement 
 of conductors and looo ft. elevation be used at line volt- 
 ages as high as 100000 volts without excessive corona 
 loss during stormy weadier. At 100 000 volts and as- 
 suming a 25 degrees C. {yy degrees F) temperature dur- 
 ing fair weather and storm conditions, the corona losses 
 would be o.i kw per mile for fair weather and 6.5 kw 
 per mile for stormy weather. If the transmission is 
 single circuit 100 miles long and without branches, has 
 an average altitude of 1000 feet and the storm condition 
 existed throughout the length of the circuit, the power 
 loss due to corona would be 6.5 X 100 or 650 kw. The 
 capacity of such a circuit at 100 000 volts (see Table 
 XX) would be roughly 15000 kw at ten percent l^R 
 loss. The storm corona loss therefore would represent 
 
 650 
 or 4.3 percent. This, in addition to ten percent 
 
 PR loss, would represent approximately 14 percent loss 
 in transmission during the storm conditions. 
 
 In the above case it would probably be considered 
 good engineering (so far as corona loss is concerned) 
 
38 
 
 CORONA EFFECT 
 
 TABLE XXM-APPROXIMATE VOLTAGE LIMITATIONS 
 
 RESULTING FROM CORONA 
 
 STRANDED COPPER CONDUCTORS 
 
 AND 
 MILS 
 
 ii 
 
 i i 
 
 CLEVATlO^ 
 IN 
 
 LIMIT IN KILOVOLTSIBETWEENI CONDUCTORS 
 3 PHASE; FOR VARIOUS! SPACINQSI X 
 
 
 AND 
 
 OtHOULAB 
 
 MILS 
 
 SS 
 
 ii 
 
 ELEVATION 
 
 IN 
 
 FEET 
 
 LIMIT IN KILOVQLTS BETWEEN CONDUCTORS 
 3 PHASE iFOFI VARIOUS SPACINQS X 
 
 
 3 
 FT. 
 
 4 
 
 FT 
 
 5 
 
 FT. 
 
 6 
 
 FT. 
 
 7 
 F_T 
 
 8 
 FT. 
 
 9 
 
 FT 
 
 II 
 
 FT 
 
 13 
 
 FT 
 
 16 
 
 FT 
 
 19 
 
 FT 
 
 25 
 
 FT. 
 
 
 3 
 FT 
 
 4 
 
 FT 
 
 5 
 FT 
 
 6 
 
 FT 
 
 7 
 FT 
 
 8 
 FT. 
 
 9 
 FT 
 
 II 
 FT 
 
 13 
 
 FT 
 
 15 
 
 FT 
 
 19 
 FT 
 
 26 
 FT 
 
 
 4 
 
 .232 
 
 BEA LEVEL. 
 lOOO 
 BOOO 
 
 sz 
 ■so 
 
 Si 
 
 it 
 
 Sg 
 S6 
 S-9 
 
 io 
 SS 
 Si 
 
 42. 
 
 to 
 SS 
 
 43 
 
 tt 
 i>o 
 
 46 
 
 it 
 
 If 
 
 49 
 
 4% 
 
 44 
 
 73 
 
 260 000 
 
 676 
 
 SEA LEVEL 
 
 tooo 
 
 2000 
 
 1 IZ m 
 
 '09 ll'j 
 /04 "0 
 
 '2.4 /2« 
 /IO /2- 
 
 iii Hi 
 
 131 134 137 I4Z 14s /49!'55/6b 
 /27 'Jo I3Z 137 140 /44 '50 /SS 
 liZ IZS /ZS /3Z I3S '3f '45/49 
 
 6C00 
 SOOO 
 
 ■io 
 
 ■99 
 ■9S 
 ■91 
 
 SO 
 ■9b 
 43 
 
 SI 
 48 
 ^■9 
 
 S3 
 
 49 
 ■ft 
 
 ^•4 
 SO 
 44 
 
 SS 
 
 SI 
 ■47 
 
 57 
 
 SZ 
 49 
 
 58 
 
 S9 
 SS 
 SI 
 
 6/ 
 5-4 
 
 J-2 
 
 43 
 
 SS 
 Sf 
 
 
 4000 
 
 eooo 
 aooo 
 
 9i 
 89 
 83 
 
 /Oi '07 
 94 99 
 87 9' 
 
 no 1 13 1 IS 
 loz los 107 
 94 96 98 
 
 /'8 
 no 
 101 
 
 /ZZ IZS IZ8 '33 13a 
 113 1 H / 19 '24 '28 
 /05 '07 "0 114 1 IS 
 
 
 lOOOp . 
 ■SOOO 
 •4 000 
 
 37 
 3-t 
 
 3i. 
 
 3g 
 3S 
 33 
 
 -40 
 37 
 34 
 
 ■41 
 38 
 3S 
 
 ■9Z 
 39 
 
 34 
 
 43 
 40 
 37 
 
 44 
 40 
 37 
 
 45 
 1? 
 
 44 
 42 
 39 
 
 47 
 44 
 40 
 
 ^1 
 42 
 
 so 
 
 ft 
 
 ■93 
 
 
 10.000 
 12000 
 
 MOOO 
 
 77 
 
 IL 
 
 8/ 
 75 
 49 
 
 !l 
 
 98 
 SI 
 75 
 
 1% 
 
 71 
 
 92 
 
 94 
 Io' 
 
 97 
 90 
 83 
 
 99 
 92 
 8S 
 
 /oz lot 
 
 94 98 
 
 87 91 
 
 I/O 
 /ox 
 
 ff 
 
 3 
 
 26G 
 
 filA LEVEL 
 1000 
 3000 
 
 S7 
 SS 
 
 iZ 
 io 
 ss 
 
 64 
 6Z 
 to 
 
 it 
 it 
 iZ 
 
 48 
 44 
 43 
 
 49 
 
 47 
 44 
 
 70 
 
 15 
 
 72 
 70 
 47 
 
 74 
 
 7* 
 
 74 
 7/ 
 
 73 
 
 81 
 79 
 74 
 
 
 300 000 
 
 630 
 
 SEA LEVEL 
 ■ 000 
 2000 
 
 IZO /ZS '32 '38 
 /'6 /24 '28 /34 
 1 IZ 119 1 ^3 IZi 
 
 I4Z 145 I4« 
 /38 '40 /43 
 /32 /3S '38 
 
 /Sri 
 '48 
 /43 
 
 'SS 'tl li? '74 
 'S3 'St /tz Itf 
 I4S 'SO /Si /tX 
 
 *00O 
 
 eooo 
 
 80O0 
 
 SI 
 ■13 
 
 S3 
 ■99 
 9i 
 
 SS 
 SI 
 ■97 
 
 S7 
 
 i'9 
 
 S8 
 54 
 50 
 
 59 
 5-5 
 5' 
 
 40 
 
 J-4 
 SI 
 
 42 
 5-7 
 53 
 
 S9 
 
 iS 
 
 a/ 
 
 Si 
 
 47 
 
 42 
 S7 
 
 70 
 44 
 60 
 
 
 4000 
 6000 
 
 aooo 
 
 96 
 
 //o /I3 
 /OX /oi 
 94 97 
 
 I'? 
 
 no 
 
 '02 
 
 '12. '25 '27 '32 /3i 
 119 Hi "8 '22 /Zi 
 / 05/07 /09 /,3 Hi 
 
 139 '43 /5a 
 129 '3 3 /39 
 '7? IZ3/2M 
 
 to 000 
 12,000 
 ■4 000 
 
 ■to 
 37 
 J4 
 
 9Z 
 11 
 
 44 
 90 
 
 37 
 
 9S 
 
 AZ 
 
 39 
 
 44 
 43 
 •40 
 
 47 
 
 4? 
 44 
 4/ 
 
 49 
 44 
 42 
 
 S/ 
 47 
 43 
 
 sz 
 
 48 
 
 -»5 
 
 S3 
 
 49 
 
 44 
 
 SS 
 
 
 ■OOOO 
 
 "laooo 
 
 MOOO 
 
 sz 
 
 74 
 70 
 
 88 
 
 8/ 
 75 
 
 90 
 
 77 
 
 95" 
 88 
 
 81 
 
 97 
 
 99 
 92 
 85 
 
 /02 
 94 
 87 
 
 /OS 'OS 
 97 100 
 90 93 
 
 no / 14 
 
 lO'i. 'Oi 
 
 9S 99 
 
 /I9 
 
 /IO 
 
 loz 
 
 
 2 
 
 .2G2 
 
 SEA LEVEL 
 • 000 
 JOOO 
 
 iS 
 i3 
 Al 
 
 6S 
 it 
 <4 
 
 li 
 
 73 
 Ik 
 
 75 
 73 
 70 
 
 11 
 
 78- 
 V3 
 
 SO 
 7S 
 75 
 
 M 
 
 84 
 8/ 
 78 
 
 87 
 84 
 81 
 
 90 
 87 
 84 
 
 
 350 000 
 
 68t 
 
 SEA LEVEL 
 1000 
 2000 
 
 /27 
 /23 
 119 
 
 /35 '4/ 
 /3/ /3i 
 / 2i /3I 
 
 /■ft 
 141 
 /3t 
 
 'SI 
 /4i 
 141 
 
 iSS /SS 
 /SO /S3 
 I4S /4S 
 
 /63 /f 171 '78 
 
 /SS 'tz Its nz 
 
 'SZ ISi '40 /44 
 
 Iti 
 
 ISO 
 174 
 
 
 4000 
 
 eooo 
 
 SOOO 
 
 Si 
 
 sz 
 
 ■*» 
 
 Si 
 S9 
 SO 
 
 S7 
 SZ 
 
 43 
 SS 
 54 
 
 iS 
 40 
 
 55 
 
 it 
 il 
 
 sr 
 
 47 
 42 
 
 57 
 
 49 
 
 44 
 S9 
 
 70 
 
 is 
 ^0 
 
 72 
 47 
 42 
 
 75 
 49 
 44 
 
 77 
 72 
 44 
 
 
 4000 
 6000 
 
 aooo 
 
 /Ol 
 /O/ 
 93 
 
 //i 
 /Oi 
 /OO 
 
 IXI 
 113 
 /04 
 
 IZS /3C 
 117 IZI 
 /07 III 
 
 733 
 /Z4 
 /I4 
 
 /34 
 /2 4 
 II? 
 
 IfO IA4 '47 'S3 /io 
 130 I3f i-ij ifX 141 
 /ZO 123 /Zt 131 /37 
 
 
 •OOOO 
 ■2.000 
 t4O0O 
 
 14 
 
 9C. 
 ■93 
 ■90 
 
 ■99 
 ■9S 
 9Z 
 
 SO 
 44 
 43 
 
 5; 
 47 
 44 
 
 S3 
 
 49 
 
 ■4S 
 
 53 
 
 Vi 
 
 SS 
 S/ 
 47 
 
 Si 
 SZ 
 
 48 
 
 S7 
 51 
 
 60 
 55" 
 SI 
 
 42 
 57 
 S3 
 
 
 -OOOO 
 12 000 
 ■4D00 
 
 87 
 
 8/ 
 75 
 
 92 
 
 84 
 79 
 
 94 
 Vz 
 
 IOC 
 93 
 Si 
 
 '03 
 9i 
 89 
 
 lot 
 98 
 
 f / 
 
 109 
 
 /oa 
 
 93 
 
 IIZ 11^ 
 103 lat 
 9i 98 
 
 ' ' 17 
 "t 
 /Ol 
 
 '22 727 
 "3 "9 
 /as 'Of 
 
 
 1 
 
 332 
 
 SU LEVEL 
 ■ 000 
 20CX) 
 
 7i 
 
 70 
 47 
 
 7i 
 
 73 
 7 1 
 
 79 
 
 8' 
 78 
 74 
 
 ?3 
 
 80 
 77 
 
 SS 
 
 sz 
 
 79 
 
 87 
 
 84 
 8' 
 
 S9 
 Si 
 83 
 
 Vs 
 
 SS 
 
 93 
 90 
 87 
 
 9t 
 93 
 90 
 
 /OO 
 97 
 93 
 
 
 4t)0 000 
 
 728 
 
 SEA LEVEL 
 1000 
 2000 
 
 '35 '43 
 '3/ /39 
 '24 /J3 
 
 1*9 
 '44 
 '39 
 
 /SS'io /43 
 /so /SS /SS 
 /4S /49 '52 
 
 /<4 /72 
 HI lit 
 ISS 74/ 
 
 177 I8Z '89 
 17 1 '74 '83 
 7 45 / 70 /74 
 
 197 
 19/ 
 184 
 
 4000 
 
 8000 
 
 eooo 
 
 iZ 
 S7 
 S3 
 
 iS 
 io 
 
 Si 
 
 i8 
 
 i3 
 SS 
 
 it 
 
 iS 
 io 
 
 7' 
 44 
 4/ 
 
 73 
 48 
 42 
 
 If 
 44 
 
 77 
 7/ 
 45 
 
 78 
 73 
 47 
 
 80 
 
 It 
 
 82 
 
 '7? 
 
 84 
 80 
 
 74 
 
 4000 
 6000 
 
 eooo 
 
 //4 
 /08 
 99 
 
 IZI 
 II f 
 /OS 
 
 128 
 "9 
 "0 
 
 133 
 IZf 
 
 114 
 
 /3S 
 '2S 
 I'S 
 
 /40 
 /30 
 /20 
 
 '43 
 /33 
 '22 
 
 /4S 
 '3S 
 '^? 
 
 /SZ /S? '63 /4f 
 '42 '44 IS' /SJ 
 /30 134 / 39 /4S 
 
 10 000 
 12 000 
 MOOO 
 
 t1 
 *S 
 
 A-a. 
 
 sz 
 
 48 
 
 44 
 
 II 
 
 SS 
 Si 
 47 
 
 57 
 52 
 
 -♦9 
 
 SS 
 
 s^^ 
 so 
 
 io 
 SS 
 
 SI 
 
 4/ 
 54 
 52 
 
 42 
 SS 
 S3 
 
 44 
 J-9 
 
 J-4. 
 
 44 
 4/ 
 54 
 
 48 
 43 
 59 
 
 •OOOO 
 
 ■2000 
 
 ■4 000 
 
 9Z 
 Si 
 
 79 
 
 If 
 
 84 
 
 '02 
 9S 
 87 
 
 /06 
 
 V/ 
 
 / /O 
 '02 
 94 
 
 "Z 
 /03 
 96 
 
 '14 
 'Oi 
 97 
 
 ''S 
 '09 
 /O 1 
 
 / 22 
 /IZ 
 '04 
 
 'ZS izt 
 
 lit 'ZO 
 'O? Ill 
 
 '35 
 
 in 
 
 
 
 
 .373 
 
 SEA LEVEL 
 ■ 000 
 
 aooo 
 
 79 
 77 
 
 74 
 
 ?3 
 
 SO 
 
 77 
 
 87 
 84 
 8/ 
 
 89 
 84 
 83 
 
 92 
 89 
 
 84 
 
 9-9 
 
 91 
 
 88 
 
 9i 
 93 
 89 
 
 9? 
 95 
 9' 
 
 /Ol 
 
 9S 
 9^9 
 
 /03 
 /OO 
 94 
 
 /07 
 /04 
 /oo 
 
 / // 
 
 /°7 
 '04 
 
 460 000 
 
 772 
 
 SEA LEVEL 
 1000 
 2000 
 
 /fl 
 /3i 
 '32 
 
 '50 
 /fS 
 /fo 
 
 '57 
 /52 
 
 /44 
 
 /tz 
 IS? 
 
 ISI 
 
 /47 
 'it 
 /Si 
 
 /7/ 
 /46 
 /4o 
 
 '75 
 
 181 
 /75 
 
 /49 
 
 /S7 
 '8/ 
 '74 
 
 191 
 /SS 
 '73 
 
 '98 207 
 '92 200 
 '85 /93 
 
 
 4000 
 
 eooo 
 
 8000 
 
 i3 
 
 ss 
 
 7/ 
 It 
 
 i'9 
 
 77 
 7/ 
 
 44 
 
 79 
 73 
 48 
 
 SI 
 
 82 
 74 
 7/ 
 
 8* 
 78 
 72 
 
 Si 
 
 SI 
 74 
 
 88 
 82 
 74 
 
 9Z 
 95 
 79 
 
 9S 
 11 
 
 4000 
 
 eooo 
 
 6000 
 
 / 21 
 //2 
 /04 
 
 /29 
 
 /zo 
 
 III 
 
 /35 /39 
 /25 /29 
 "i "9 
 
 /43 
 /33 
 /23 
 
 /47 
 /37 
 /24 
 
 '5/ 
 /40 
 /29 
 
 /54 /i' 
 I4S 149 
 733 /3S 
 
 /Cf 
 '53 
 / 41 
 
 ' 70 
 'SS 
 
 '44 
 
 '78 
 '45 
 753 
 
 
 ■0000' 
 
 laooo 
 
 •4.000 
 
 12 
 
 S7 
 SZ 
 44 
 
 to 
 
 if 
 
 1^ 
 5i 
 
 43 
 i'9 
 
 if 
 
 S9 
 SS 
 
 ii 
 il 
 Si 
 
 47 
 42 
 5-7 
 
 i9 
 
 44 
 
 S9 
 
 70 
 iS 
 io 
 
 73 
 48 
 
 63 
 
 74 
 70 
 45 
 
 10 000 
 ■2 000 
 14 000 
 
 97 
 
 /03 
 95" 
 88 
 
 /OS 
 /OO 
 92 
 
 / / / 
 
 /03 
 95 
 
 "S 
 /oi 
 ?8 
 
 "7 
 '09 
 /o/ 
 
 /2 
 // / 
 /03 
 
 '24 
 "S 
 /Ot 
 
 'Z8 
 "9 
 "0 
 
 13 J 
 IZI 
 1 'Z 
 
 '34 
 '24 
 
 "6 
 
 '41 
 '32 
 '2/ 
 
 
 00 
 
 ^8 
 
 SEA LEVEl 
 lOOO 
 
 87 
 84 
 81 
 
 91 
 SS 
 
 9i 
 9Z 
 
 SB 
 
 98 
 9S 
 
 9/ 
 
 101 
 98 
 94 
 
 /03 
 /OO 
 9i 
 
 lOS 
 
 /oz 
 9S 
 
 /Of 
 
 /05 
 /02 
 
 / / 1 
 107 
 
 /Of 
 
 //■9 
 I/O 
 /07 
 
 l/S 
 /I-9 
 I/O 
 
 '22 
 
 "8 
 
 '/4 
 
 boo 000 
 
 816 
 
 SEA LEVEl 
 1000 
 2000 
 
 '44 
 if-t 
 '34 
 
 'Si 
 
 /s/ 
 '44 
 
 /43 
 /SS 
 'SZ 
 
 /70 
 /44 
 
 '58 
 
 174 
 
 /i9 
 /43 
 
 '79 
 /73 
 'i? 
 
 '83 
 '77 
 171 
 
 '89 
 '83 
 '71 
 
 /9S 
 
 '89 
 ISZ 
 
 I9t 
 
 i91 
 
 mi 
 
 X.0 7 
 zoo 
 
 193 
 
 2/4 
 209 
 Z02. 
 
 
 6000 
 
 SOOO 
 
 i-9 
 
 7g 
 73 
 i7 
 
 SZ 
 
 7i 
 70 
 
 84 
 7C 
 72 
 
 87 
 
 S9 
 
 Vi 
 
 Vz 
 
 77 
 
 94 
 ¥0 
 
 If 
 SZ 
 
 9S 
 9/ 
 Sf 
 
 101 
 94 
 87 
 
 /o5 
 97 
 90 
 
 4000 
 6000 
 8000 
 
 •lV7 
 'OS 
 
 /34 
 IZS 
 l/S 
 
 /fO 
 /30 
 /2 
 
 '46 
 /36 
 
 /ZS 
 
 'SO 
 
 'ii 
 
 'SI 
 143 
 /3Z 
 
 'SS 
 I4i 
 /3S 
 
 'iZ 
 /Si 
 '39 
 
 /48 
 '54 
 '44 
 
 1 7 ' 
 
 I'S? 
 
 '*? 
 
 ■11 
 
 'SI 
 
 /8t 
 /73 
 
 '59 
 
 
 lOOOO 
 •2000 
 14,000 
 
 St 
 SS 
 SI 
 
 iZ 
 SS 
 S3 
 
 iS 
 io 
 St. 
 
 47 
 42 
 J7 
 
 69 
 59 
 
 7/ 
 If 
 
 72 
 45 
 
 42 
 
 li 
 
 74 
 70 
 iS 
 
 78 
 72 
 47 
 
 8' 
 75 
 49 
 
 84 
 77 
 7/ 
 
 10 000 
 12000 
 14 000 
 
 /OO 
 93 
 SS 
 
 /07 
 1? 
 
 "2 
 /03 
 94 
 
 1/7 
 /OS 
 /OO 
 
 "9 
 //<? 
 /02 
 
 723 
 
 "4 
 /OS 
 
 /Zi 
 //i 
 /07 
 
 /29 
 '20 
 / / 1 
 
 734 
 
 '24 
 "4 
 
 137 
 
 'Zt 
 "7 
 
 '4Z 
 73 / 
 
 1 Zl 
 
 '48 
 '37 
 '27 
 
 
 000 
 
 A70 
 
 SEA LEVEL 
 lOOO 
 2000 
 
 fs 
 
 101 
 94 
 
 /OS 
 lOZ 
 98 
 
 /08 
 /04 
 /Ol 
 
 //2 
 
 /04. 
 
 I/-9 
 //O 
 /Ot 
 
 //t 
 /IZ 
 /OS 
 
 /zo 
 
 /It 
 
 /IZ 
 
 /Z3 
 
 //9 
 
 l/S 
 
 /ZS 
 /Zl 
 /li 
 
 /30 
 /24 
 IZI 
 
 '35" 
 '3/ 
 72* 
 
 760 000 
 
 998 
 
 SEA LEVEL 
 lOOO 
 2000 
 
 nz 
 
 Hi 
 
 /to 
 
 /S3 
 /77 
 17/ 
 
 /fZ 
 /Si 
 /go 
 
 200 
 '93 
 /S7 
 
 r9^ 
 
 7 9/ 
 
 Zl' 
 20f 
 '97 
 
 Zli 
 Z09 
 20 2 
 
 224 
 
 Zli 
 209 
 
 230 
 223 
 2'5 
 
 Z3i 
 
 2ZS 
 ZZO 
 
 24 4 
 238 
 230 
 
 257 
 248 
 240 
 
 
 4000 
 6000 
 
 eooo 
 
 8/ 
 7< 
 
 70 
 
 87 
 80 
 79 
 
 90 
 8-9 
 
 77 
 
 so 
 
 94 
 
 98 
 91 
 84 
 
 100 
 92 
 SS 
 
 'U 
 
 lot 
 
 It 
 
 9c 
 
 lOS 
 lOQ 
 92 
 
 IIX 
 '04 
 94 
 
 '/6 
 /08 
 99 
 
 4000 
 6000 
 8000 
 
 /9S 
 /37 
 /J7 
 
 '57 
 /44 
 /3 5 
 
 '45 
 /S3 
 /fl 
 
 '72 
 /40 
 /48 
 
 774 
 /44 
 /SI 
 
 '8/ 
 /it 
 /SS 
 
 /84 
 '73 
 /40 
 
 7 93 
 i79 
 /iS 
 
 /98 
 
 189 
 
 '70 
 
 203 
 788 
 / 74 
 
 2/2 
 794 
 '91 
 
 lo'4 
 '90 
 
 
 tO.OOO 
 
 13.000 
 MOOO 
 
 60 
 Si 
 
 n 
 
 2f 
 
 It 
 43 
 
 77 
 4'i 
 
 78 
 72 
 47 
 
 79 
 73 
 
 4» 
 
 sz 
 74 
 70 
 
 84 
 78 
 72 
 
 86 
 V3 
 
 11 
 
 93 
 85 
 
 79 
 
 •OOOO 
 .2 000 
 ■4 000 
 
 /It 
 /09 
 /O/ 
 
 /Zt 
 //i 
 /07 
 
 /3l 
 /ZZ 
 //3 
 
 /37 
 
 /27 
 //7 
 
 ifl 
 130 
 /ZO 
 
 I4S 
 134 
 124 
 
 /48 
 /37 
 '27 
 
 /S4 
 '4i 
 /3/ 
 
 /S8 
 744 
 'J5 
 
 742 
 'SC 
 /3t 
 
 'it 
 'Si 
 
 Its 
 
 '74 
 743 
 'SO 
 
 
 OQPO 
 
 628 
 
 SEA LEVEL 
 lOOO 
 .»Q00 
 
 /04 
 /o/ 
 *7 
 
 III 
 
 107 
 I03 
 
 /// 
 '07 
 
 1/9 
 //S 
 /Il 
 
 /23 
 //9 
 IIS 
 
 /IS 
 IZI 
 1/7 
 
 /2 8 
 
 /2+ 
 /20 
 
 /32 
 IZS 
 /Z3 
 
 /34 
 /32 
 /27 
 
 /39 
 /34 
 /3o 
 
 '44 
 '39 
 /34 
 
 ISO 
 
 '45 
 '40 
 
 U 000 000 
 
 i.t62 
 
 SEA LEVEL 
 ■ 000 
 3000 
 
 /9Z 
 ISi 
 179 
 
 ZOS 
 
 '99 
 
 /9I 
 
 2/4 
 X09 
 Zo2 
 
 224 
 
 •^'^ 
 2 09 
 
 23/ 
 223 
 2/5 
 
 137 
 it? 
 
 24 3 
 
 2 35 
 227 
 
 ZSZ 
 244 
 23.! 
 
 240 
 ZSZ 
 242 
 
 247 
 258 
 249 
 
 271 
 24f 
 258 
 
 2 to 
 Zto 
 27' 
 
 
 40O0 
 6000 
 SOOO 
 
 S9 
 93 
 77 
 
 9S 
 S8 
 8' 
 
 99 
 93 
 SS 
 
 loz 
 9S 
 SS 
 
 /ot 
 9S 
 9/ 
 
 /07 
 /OO 
 92 
 
 95 
 
 113 
 /OS 
 97 
 
 II? 
 
 /oe 
 
 loe 
 
 '/il 
 /03 
 
 '24- 
 //s 
 /oi 
 
 '29 
 /2o 
 '/O 
 
 4000 
 6000 
 
 eooo 
 
 liS 
 
 'S3 
 '42 
 
 '74 
 '44 
 
 /SI 
 
 /SS 
 /73 
 /S9 
 
 /93 
 '79 
 /45 
 
 /9S 
 /SS 
 /70 
 
 204 
 190 
 /?S 
 
 209 
 /94 
 /80 
 
 2/7 
 i0 2 
 /84 
 
 214 
 20 8 
 '92 
 
 230 
 2/4 
 '97 
 
 238 
 222 
 
 205 
 
 250 
 232 
 
 2/4 
 
 
 I0.0O0 
 I3 00O 
 ■4 000 
 
 7/ 
 ti 
 il 
 
 7i 
 70 
 6X 
 
 79 
 1'7 
 
 SI 
 75 
 70 
 
 84 
 
 Vz 
 
 84 
 
 88 
 
 »/ 
 75 
 
 9/ 
 84 
 77 
 
 li 
 80 
 
 If 
 82 
 
 99 
 9' 
 84 
 
 '03 
 95 
 St 
 
 
 to 000 
 13 000 
 
 '4 000 
 
 '3 2 
 IZZ 
 // 3 
 
 iVo 
 
 I2LC 
 
 /+S 
 /3 7 
 '27 
 
 /Sf 
 /4Z 
 /3Z 
 
 /St 
 /47 
 13 i 
 
 /is 
 
 /SI 
 /3 9 
 
 747 
 /Sf 
 /fl 
 
 /73 
 /60 
 /4S 
 
 '78 
 
 /is 
 
 'S3 
 
 '83 
 /il 
 IS7 
 
 >to 
 
 ni 
 
 /43 
 
 /9f 
 'ts 
 
 '7Q 
 
 
 
 
 SOLID COPPER CONDUCTORS 
 
 
 
 
 
 4' 
 
 204 
 
 SEA LEVEL 
 • 000 
 20OO 
 
 SI 
 *9 
 ■<7 
 
 ^4 
 SZ 
 SO 
 
 Si 
 ft 
 
 SS 
 Si 
 
 J4 
 
 Sf 
 
 If 
 
 40 
 58 
 
 5-4 
 
 4' 
 .59 
 57 
 
 43 
 il 
 S9 
 
 44 
 42 
 io 
 
 If 
 4 ' 
 
 48 
 44 
 i3 
 
 70 
 48 
 45 
 
 
 
 
 325 
 
 SEA LEVEL 
 
 lOOO 
 2000 
 
 75 
 72 
 70 
 
 74. 
 
 82 
 79 
 7i 
 
 79 
 
 87 
 It 
 
 II 
 
 83 
 
 91 
 88 
 85 
 
 %7 
 
 88 
 
 94 
 91 
 90 
 
 9S 
 9S 
 92 
 
 10%. 
 98 
 
 95 
 
 /OS 
 '?}■ 
 
 
 6000 
 
 eooo 
 
 44 
 4) 
 37 
 
 9(. 
 43 
 40 
 
 48 
 ■9S 
 4 ' 
 
 i2 
 
 42 
 
 5-' 
 -♦7 
 43 
 
 ■^■9 
 
 SZ 
 49 
 45 
 
 5"4 
 
 So 
 
 ft 
 
 SS 
 
 SI 
 
 47 
 
 J-4 
 52 
 48 
 
 S8 
 Sf 
 SO 
 
 40 
 Si 
 
 SI 
 
 4000 
 6000 
 BOOO 
 
 tt 
 SS 
 
 48 
 43 
 58 
 
 70 
 
 is 
 
 40 
 
 73 
 
 48 
 42 
 
 75 
 49 
 44 
 
 76 
 7/ 
 44 
 
 7? 
 if 
 
 8/ 
 75 
 49 
 
 82 
 74 
 7/ 
 
 ■P* 
 
 Vz 
 
 SS 
 92 
 
 75 
 
 90 
 
 84 
 77 
 
 
 K)JXX) 
 ■2 000 
 14 000 
 
 Ji 
 3Z 
 30 
 
 il 
 
 32 
 
 38 
 3J- 
 33 
 
 4.0 
 37 
 34 
 
 40 
 37 
 35 
 
 41 
 38 
 35" 
 
 11 
 34 
 
 43 
 40 
 37 
 
 44 
 40 
 37 
 
 44. 
 ■91 
 38 
 
 44 
 43 
 
 40 
 
 ft 
 
 44 
 
 fl 
 
 •OOOO 
 12 000 
 ■4 000 
 
 SI 
 
 Sf 
 SO 
 44 
 
 5"4 
 
 52 
 48 
 
 58 
 5^4 
 SO 
 
 40 
 55 
 
 .5/ 
 
 4/ 
 5-4 
 .92 
 
 iZ 
 SS 
 S3 
 
 44 
 40 
 55 
 
 44 
 4/ 
 511 
 
 47 
 
 49 
 
 is 
 
 40 
 
 72 
 47 
 il 
 
 
 3 
 
 229 
 
 SEA LEVEL 
 '000 
 20Q0 
 
 S7 
 
 Si 
 
 io 
 
 se 
 
 Si 
 
 i,3 
 il 
 
 S9 
 
 44 
 42 
 
 io 
 
 45" 
 43 
 4/ 
 
 47 
 
 a 
 
 48 
 44 
 43 
 
 70 
 48 
 45 
 
 7/ 
 
 il 
 
 73 
 7/ 
 48 
 
 75 
 73 
 70 
 
 78 
 
 75 
 73 
 
 "*7 
 42 
 
 57 
 
 
 00 
 
 365 
 
 SEA LEVEL 
 .000 
 SOOO 
 
 83 
 80 
 77 
 
 8? 
 
 85 
 82 
 
 9/ 
 88 
 85 
 
 94 
 |8 
 
 94 
 93 
 90 
 
 98 
 94 
 9Z 
 
 loo 
 9? 
 93 
 
 /04 
 100 
 97 
 
 /o4 
 /0 3 
 94 
 
 '09 
 
 705 
 'OX 
 
 " 2 
 '08 
 lOf 
 
 III 
 'i'o% 
 
 
 4000 
 
 eooo 
 
 8O0O 
 
 4y 
 
 IS 
 
 4a 
 
 SI 
 ■4S 
 44 
 
 S9 
 SO 
 44 
 
 SS 
 
 s/ 
 
 4? 
 
 5-4 
 52 
 
 4-8 
 
 57 
 
 58 
 S9 
 SO 
 
 40 
 54 
 5-/ 
 
 i/ 
 
 57 
 
 52 
 
 43 
 
 58 
 54 
 
 44 
 <o 
 
 55 
 
 4000 
 6000 
 
 eooo 
 
 7/ 
 ii 
 4/ 
 
 75 
 70 
 45 
 
 78 
 73 
 47 
 
 91 
 75 
 49 
 
 82 
 
 77 
 7/ 
 
 84 
 78 
 72 
 
 84 
 
 80 
 
 74 
 
 89 
 
 83 
 74 
 
 9' 
 85 
 7«' 
 
 94 
 87 
 80 
 
 94 
 
 at 
 
 82 
 
 /OO 
 
 92 
 85 
 
 
 I2 0OO 
 MOOO 
 
 it 
 
 33 
 
 3S 
 
 43 
 40 
 37 
 
 44 
 40 
 37 
 
 44 
 38 
 
 44 
 11 
 
 44 
 43 
 40 
 
 48 
 
 44 
 4/ 
 
 49 
 45 
 42 
 
 •^3 
 
 46 
 43 
 
 5-/ 
 
 .53 
 49 
 44 
 
 
 ■OOOO 
 12 000 
 14 000 
 
 S7 
 53 
 48 
 
 40 
 54 
 5 / 
 
 42 
 58 
 5-3 
 
 44 
 59 
 55 
 
 44 
 4/ 
 Si 
 
 47 
 42 
 .57 
 
 48 
 43 
 59 
 
 li 
 4/ 
 
 72 
 47 
 42 
 
 f.4 
 
 77 
 71 
 ii 
 
 79 
 
 u 
 
 
 a 
 
 
 SEA LEVEl 
 tOOO 
 2O0O 
 
 i2 
 i" 
 St 
 
 it 
 
 i9 
 
 il 
 
 48 
 44 
 43 
 
 70 
 48 
 45 
 
 72 
 70 
 47 
 
 73 
 
 I's 
 
 75 
 73 
 70 
 
 77 
 75 
 72 
 
 79 
 74 
 74 
 
 80 
 7? 
 75 
 
 83 
 80 
 78 
 
 84 
 83 
 80 
 
 
 000 
 
 
 SEA LEVEL 
 ■ 000 
 
 l'8 
 85 
 
 94 
 
 90 
 
 100 
 97 
 93 
 
 '03 
 00 
 94 
 
 loi 
 02 
 99 
 
 09 
 05 
 
 loz 
 
 /// 
 '07 
 /03 
 
 1/4 
 //o 
 /oi 
 
 "7 
 /'3 
 /09 
 
 IZQ 
 Hi, 
 112. 
 
 724 
 '20 
 
 Hi 
 
 iZ9 
 /ZO 
 
 
 268 
 
 4O0O 
 
 eooo 
 
 8000 
 
 S3 
 44 
 IS 
 
 S7 
 SZ 
 48 
 
 SV 
 S-9 
 SO 
 
 40 
 54 
 SI 
 
 42 
 57 
 53 
 
 43 
 58 
 J"4 
 
 44 
 40 
 SS 
 
 44 
 il 
 57 
 
 48 
 43 
 
 58 
 
 49 
 59 
 
 7/ 
 ii 
 4' 
 
 1^9 
 £3 
 
 410 
 
 4000 
 £000 
 
 eooo 
 
 78 
 
 73 
 47 
 
 sz 
 
 77 
 
 7/ 
 
 84 
 SO 
 74 
 
 88 
 82 
 74 
 
 ii 
 
 1? 
 
 80 
 
 9S 
 89 
 82 
 
 9S 
 
 9/ 
 84 
 
 /OO 
 93 
 84 
 
 '?i 
 
 St 
 
 10? 
 99 
 tl 
 
 no 
 
 'ti 
 
 
 
 10 000 
 •2O00 
 I4O00 
 
 41 
 3» 
 3i 
 
 4i 
 42 
 34 
 
 44 
 43 
 
 40 
 
 48 
 
 i1 
 
 49 
 45 
 42 
 
 SO 
 '>3 
 
 SI 
 
 Vt 
 
 53 
 
 5'4 
 5"o 
 44 
 
 SS 
 
 SI ■ 
 47 
 
 S7 
 
 S3 
 49 
 
 So 
 
 
 ■OOOO 
 
 12000 
 
 14 000 
 
 42 
 
 58 
 53 
 
 44 
 61 
 Si 
 
 49 
 43 
 59 
 
 I's 
 
 il 
 
 73 
 
 47 
 42 
 
 7S 
 
 74 
 70 
 45 
 
 7? 
 72 
 47 
 
 80 
 
 82 
 74 
 70 
 
 8S 
 79 
 73 
 
 88 
 
 8/ 
 75 
 
 
 ». 
 
 289 
 
 SEA LEVEL 
 
 • ooo 
 
 2O0O 
 
 if 
 ii 
 il 
 
 72 
 47 
 
 7S 
 72 
 70 
 
 77 
 74 
 
 72 
 
 79 
 74 
 74 
 
 8; 
 
 78 
 
 74 
 
 S3 
 
 80 
 77 
 
 SS 
 
 82 
 
 7-9 
 
 87 
 84 
 81 
 
 89 
 
 9Z 
 89 
 Si 
 
 9S- 
 9Z 
 9f 
 
 
 OOOO 
 
 
 SEA LEVEl 
 ■ 000 
 2000 
 
 'OO 
 
 Ot 
 02 
 99 
 
 no 
 oi 
 
 103 
 
 "4 
 
 'O 
 04 
 
 / 7 
 /3 
 09 
 
 120 
 
 Hi 
 
 HZ 
 
 22 
 
 "8 
 "4 
 
 24 
 '22 
 
 /IS 
 
 '29 
 'ZS 
 ZO 
 
 32 
 '27 
 23 
 
 128 
 
 4Z 
 137 
 '33 
 
 
 woo 
 
 eooo 
 aooo 
 
 S9 
 ss 
 
 SI 
 
 i Z 
 57 
 S3 
 
 44 
 
 io 
 
 SS 
 
 44 
 4/ 
 5-7 
 
 48 
 43 
 58 
 
 i's 
 
 io 
 
 7/ 
 ii 
 il 
 
 73 
 48 
 43 
 
 7S 
 
 74 
 
 45 
 
 79 
 73 
 48 
 
 St 
 7* 
 7o 
 
 A60 
 
 4000 
 
 6000 
 
 aooo 
 
 84 
 
 80 
 74 
 
 9/ 
 as 
 
 78 
 
 9S 
 SS 
 SI 
 
 If 
 84 
 
 93 
 84 
 
 03 
 9i 
 
 SS 
 
 OS 
 97 
 90 
 
 loS 
 101 
 93 
 
 " / 
 03 
 95 
 
 "3 
 /05 
 97 
 
 lis 
 109 
 101 
 
 XX 
 
 "3 
 /OS 
 
 
 
 ■OOOO 
 19 000 
 ■4 000 
 
 4 1 
 44 
 
 44 
 9S 
 4» 
 
 SI 
 47 
 44 
 
 53 
 
 54 
 
 SS 
 S/ 
 47 
 
 57 
 52 
 49 
 
 58 
 54 
 5-0 
 
 io 
 SS 
 5/ 
 
 4/ 
 5-4 
 52 
 
 43 
 58 
 5-4 
 
 6r 
 
 io 
 
 Sb 
 
 
 10 000 
 12000 
 
 t4 00O 
 
 4f 
 59 
 
 II 
 
 75 
 70 
 44 
 
 78 
 V7 
 
 80 
 
 82 
 
 74 
 70 
 
 84 
 77 
 
 r 1 
 
 84 
 to 
 74 
 
 88 
 82 
 74 
 
 1; 
 
 77 
 
 1? 
 80 
 
 il 
 
 
 X For single phase or 2 phase multiply the 3 phase values by 1.16. The above are the disruptive critical voltage values for fair weather based upon a tt;"»- 
 perature of 25° C. (77° F.) and values for M, of 0.S7 for stranded and 0.% for solid conductors. Derived by Peek's formula: Kilovolts to neutral ^ 2.302 M„ G„ S 
 
 T = Temperature in degiees P.; B =■ 
 
 S 
 
 R I-"K ,. -j^. where G„ — 53.6 Kilovolts per inch; S = Spacing in iiitlies; R = Radius of conductor in inches; S - 
 
 R 
 Barometer pressure in inches. 
 
 i7.q B 
 459+T' 
 
CORONA EFFECT 
 
 39 
 
 to operate the No. o copper conductors at as high a line 
 voltage as too coo volts. If, however, for other reasons, 
 ijoooo is selected as the desirable operating voltage, 
 then either a large diameter copper conductor or an 
 
 aluminum conductor having a greater diameter hut an 
 equivalent conductivity to that of the No. o copper con- 
 ductor should be selected. 
 
 TABLE E— COMPARISON OF CORONA LOSS 
 
 For 
 
 No. Stranded Copper Conductors 105 S60cir.mil (diameter 0.373 in.) and equivalent Aluminum Conductors 167 800 dr. mil (diametei 0.501 in.) 
 Conductor Spacing (s) Delta - 144 in. Altitude 1000 feet— Barometer 28.9 inches. Calculated from formula (22) 
 
 Kilovolts 
 
 Corona Loss in Kw. per Mile for Three Conductors at 60 CycUs 
 
 Fair Weather- 
 
 (Formula 22) 
 
 Stormy Weather— (Formula 22-A) 
 
 Between 
 Conduct- 
 ors 
 
 To 
 Neutral 
 
 No. Copper 
 Radius 0.186 in. 
 
 Aluminum 
 Radius 0.25 in. 
 
 No. Copper 
 Radius 0.186 in. 
 
 Alnroinuro 
 Radius 0.2S in. 
 
 0° C 
 32° F 
 6-1. OS 
 ««-60.5 
 
 25° C 
 77° F 
 5-0.967 
 eo-5S.7 
 
 50* C 
 
 122° F 
 
 5=0.892 
 
 eo-51.3 
 
 0° C 
 
 32° P 
 
 5=1.05 
 
 eo=77.S 
 
 25»C 
 
 77° F 
 
 5-0.967 
 
 eo-71.5 
 
 SO* c 
 
 122* F 
 
 5-0.892 
 
 eo-66.0 
 
 0* C 
 
 32* F 
 
 5 = 1.05 
 
 «o-48.4 
 
 25* C 
 
 77° F 
 
 5-0.967 
 
 eo-44.5 
 
 SO* C 
 
 122" F 
 
 5-0.892 
 
 eo-41.0 
 
 0* C 
 
 32* F 
 
 S-\.OS 
 
 (•-62. 
 
 2S* C 
 
 77* F 
 
 5-0.967 
 
 eo-S7.2 
 
 SO* C 
 
 122* F 
 
 5-0.892 
 
 <.-52 7 
 
 100 
 110 
 120 
 
 57.8 
 63.5 
 69.2 
 
 0.0 
 0.3 
 2.6 
 
 0.1 
 2.3 
 6.7 
 
 0.2 
 
 6.0 
 
 12.8 
 
 
 
 
 
 
 
 
 
 
 
 0.4 
 
 0.3 
 
 7.8 
 
 14.8 
 
 6.S 
 13 3 
 22 6 
 
 11.3 
 20.3 
 32.0 
 
 
 
 
 
 2.0 
 
 
 
 17 
 6.2 
 
 1.1 
 
 4.6 
 
 12.6 
 
 130 
 140 
 150 
 
 75.1 
 80.8 
 86.7 
 
 7.25 
 13.8 
 22.4 
 
 13.9 
 23.3 
 35. S 
 
 22.6 
 34.8 
 50.2 
 
 0.0 
 0.3 
 3.3 
 
 0.5 
 
 3.7 
 9.9 
 
 3.8 
 10.1 
 19.7 
 
 24.4 
 35. 8 
 SO. 2 
 
 34. 6 
 48.7 
 66. 
 
 46. S 
 63.7 
 84. 
 
 6.7 
 13 9 
 24. 
 
 13.7 
 23 8 
 37 2 
 
 23.2 
 36.4 
 S3. 3 
 
 160 
 180 
 
 92.4 
 104.8 
 
 35.0 
 66.0 
 
 49.8 
 89.0 
 
 67.7 
 115.0 
 
 8.7 
 29.3 
 
 18.7 
 47.3 
 
 32.2 
 69.5 
 
 66. 
 108. 
 
 85. 
 135. 
 
 106 
 163. 
 
 36. 
 
 72. 
 
 53. 
 96. 
 
 73. 
 I2S. 
 
 fl + 25 
 
 Note: At 25 cycles the losses would be 
 
 f + 25 
 
 25 + 25 
 
 50 
 
 ■ times the above table values. For conductors in a cow (flat spacing) the 
 
 60 + 25 85 
 
 corona loss would be reduced below the values for delta or triangular arrangement. For the higher voltages in the above table the conductor spacings would, in aa 
 actual installation, be greater than 144 in. (upon which basis the table values are given) thus giving actually less corona loss for the higher voltages thao indicated 
 by the table values. 
 
 The accompanying photograph illustrating corona 
 on an experimental line is published with the kind per- 
 mission of F. W. Peek, Jr. 
 
 Since the formulas pertaining to corona effect are to 
 some extent worked up from test data they may be 
 slightly changed from time to time. In case the problem 
 Jit hand seems vitally near fhe critical point it will be well 
 to consult the latest literature at that time as an additional 
 check on the work. 
 
 roROX.A .\T 230 Kv. t.ig cm. dt.wietf.r, 0.47" c.\ble, 310 cm. 
 
 10 FEF.T SPACING. 
 
CHAPTER V 
 
 SPEED OF ELECTRIC PROPOGATION RESONANCE 
 PARALLELING TRANSMISSION CIRCUITS 
 HEATING OF BARE CONDUCTORS 
 
 . SPEED OF ELECTRIC PROPAGATION 
 
 ASTRONOMERS and investigators by various 
 methods of determination have arrived at 
 sHghtly different values for the speed of light. 
 The Smithsonian Physical Tables give i86 347 miles per 
 second as a close average estimate. In electrical engi- 
 neering, the speed of light is usually stated as approxi- 
 mately 3 X io^° centimeters per second. This is the 
 equivalent of 186 451 miles per second. The speed of 
 electrical propogation (assuming zero losses) is the 
 same as .that of light. 
 
 ELECTRIC WAVE LENGTH 
 
 Suppose a frequency of 60 cycles per second is im- 
 pressed upon a circuit of infinite length. At the end 
 of one sixtieth of a second the first impulse (neglecting 
 retardation due to losses) will have traversed a distance 
 of 186 347 -f- 60 or 3106 miles. A section of such a cir- 
 cuit 3106 miles long would be designated as having a 
 full wave length for a frequency of 60 cycles per 
 second. 
 
 In Fig. 14, the dotted line or one cycle wave is 
 shown as extending over a circuit 3106 miles long. In 
 this case, when the first part of the wave arrives at a 
 point 3106 miles distant, the end of the same wave is 
 at the beginning of the circuit. For each half wave 
 length the current is of equal value but flowing in op- 
 posite directions in the conductor. Such a circuit is 
 designated as of full wave length. Since the velocity of 
 the electric propagation is slightly less than that of light, 
 being slightly retarded due to resistance and leakage 
 losses, the actual wave length will be slightly less than 
 3106 miles. Thus for a 300 mile, 60 cycle, three-phase 
 circuit consisting of No. 000 copper conductors having 
 10 ft. flat spacing, the wave length is calculated to be 
 2959 miles. The wave length of such a circuit is in- 
 dicated by the heavy line on the accompanying sketch. 
 In the case of this particular circuit the electric field has 
 been retarded approximately five percent, due to the 
 losses of the circuit, as indicated by the displacement of 
 the dotted and full line curves. 
 
 QUARTER WAVE RESONANCE 
 
 If the end of a long trough filled with water is 
 struck by a hammer, the impact will cause a wave in the 
 water to start in front of the point of impact and travel 
 to the far end of the tank. When this wave reaches the 
 far end of the tank it will be reflected, traveling back 
 toward the point of origin, but on account of resistance 
 encountered it will be of diminishing height or ampli- 
 tude. If, at the instant it gets back to the point of origin, 
 the end of the tank is again struck by the hammer, the 
 
 resulting impulse will be that due to the second hammer 
 blow plus that remaining from the first blow. The re- 
 sult will be that the second wave from the near end of 
 the tank will be of greater amplitude than the first wave. 
 If when the second wave arrives back at the near end, 
 the end of the tank is struck again with the hammer the 
 resulting third impulse will be of greater amplitude than 
 the second impulse. If at the instant of the return of 
 each succeeding impulse the end of the tank is struck, 
 the result will be cumulative and each succeeding wave 
 will be of greater magnitude than the one preceeding 
 until the point is reached where the losses due to resist- 
 ance become sufficient to prevent a further increase in 
 amplitude of the wave. 
 
 Under certain conditions a similar phenomenon may 
 occur in electric circuits and this is known as "quarter 
 wave resonance". If an electric impulse* is sent into a 
 
 1.— WAVE LENGTH lE LOSSES WERE NOT PRESENT TO RETARD TmE ELECTRIC PROPAGATION = 3 106 MILES — »* 
 
 FIG. 14 — WAVE LENGTH OF 6o CYCLE CIRCUIT 
 
 conductor, such as a transmission circuit, this impulse 
 travels along the conductor at the velocity of light. If 
 the circuit is open at the other end, the impulse is there 
 reflected and returns at the same velocity. If at the 
 moment when the impulse arrives at the starting point 
 a second impulse is sent into the circuit, the returned 
 first impulse adds itself to, and so increases the second 
 impulse; the return of this second impulse adds itself to 
 the third impulse, and so on ; that is, if alternating im- 
 pulses succeed each other at intervals equal to the lime 
 required by an impulse to travel over the circuit and 
 back, the effects of successive impulses add themselves, 
 and large currents and high e.m.f.'s may be produced 
 by small impulses. This condition is known as quarter 
 wave electric resonance. To produce this condition, it 
 is necessary that the alternating impulses occur at time 
 intervals equal to the time required for the impulses to 
 travel the length of the line and back. For example, 
 the time of one half wave or cycle of impressed e.m.f. 
 
 *For a complete study of this subject see "Transient Electric 
 Phenomena and Oscillations" by C. P. Steinmetz, from which 
 the above description of quarter wave resonance has largely 
 been taken. 
 
PROP AG A TION— RESONANCE— PARALLELING CIRCUITS 
 
 41 
 
 is the time required by light to travel twice the length 
 of the line, or the time of one complete cycle is the time 
 light requires to travel four times the length of the line. 
 Stated another way, the number of cycles or frequency 
 of the impressed alternating em.f.'s in resonance condi- 
 tion, is the velocity of light divided by four times the 
 length of the line; or to have free oscillation or reson- 
 ance condition, the length of the line is one quarter 
 wave length of light. The cycles at which this condi- 
 tion is reached (if there were no losses present) would 
 be determined as follows: — 
 
 Pre uen ———^^^l— 
 
 ' ■' Length in miles ^ '' 
 
 * Length in miles ^— f 
 
 Frequency 
 
 ('4) 
 
 RESONANCE LENGTHS OF CIRCUITS 
 
 Commercial frequencies are so low that to reach a 
 quarter wave resonance condition with them the circuit 
 would have to be of great length. The following values, 
 for the sake of simplicity, are based upon the assump- 
 tion that there are no losses in the circuit. 
 
 Fundamental Frequency Resonance Length Wave Length 
 
 IS c>xlcs 3106 miles 12434 miles 
 
 25 cycles 1863 miles 7452 miles 
 
 40 cycles 1165 miles 4660 miles 
 
 60 cycles 776 miles 3106 miles 
 
 The above lengths are based upon the impressed or 
 fundamental frequencies. If these impressed fre- 
 quencies contain appreciable higher harmonics, some of 
 the latter may approach resonance frequency and, if of 
 sufficient magnitude, may cause trouble. Thus the 
 length of circuit corresponding to resonance conditions 
 of various harmonics of the fundamental is given below. 
 
 Cycles 
 
 Harmonics | 
 
 3rd. 
 
 5th. 
 
 7th. 
 
 15 
 25 
 40 
 60 
 
 1035 miles 
 621 miles 
 388 miles 
 258 miles 
 
 631 miles 
 372 miles 
 233 miles 
 155 miles 
 
 444" miles 
 266 miles 
 166 miles 
 III miles 
 
 Thus an impressed frequency of 60 cycles will not 
 produce quarter wave electric resonance unless the cir- 
 cuit be approximately 776 miles long. If a third har- 
 monic, however, is present in the impressed wave, this 
 harmonic will develop quarter wave resonance in a cir- 
 cuit approximately 258 miles long, a 5th harmonic in a 
 circuit approximately 155 miles long, and a 7th har- 
 monic in a circuit approximately iii miles long. 
 
 The above values are based upon no losses being 
 encountered in transmission. Obviously this is an in- 
 correct assumption, as electric propagation is always 
 accompanied by more or less loss, depending upon the 
 fundamental constants (resistance and leakage) of the 
 circuit. The effect of such losses is to retard the ve- 
 locity of the electric propagation, usually by an amount 
 of five to ten percent below that of light. The above 
 values of circuit lengths representing a condition for re- 
 sonance may therefore be as much as ten percent above 
 the actual lengths. 
 
 An investigation of the effects of higher harmonics 
 
 of the impressed wave is of importance in connection 
 with very long distance transmission systems. 
 
 PARALLELING TRANSMISSION CIRCUITS 
 
 Transmission lines are frequently constructed with 
 duplicate circuits which are normally operated in paral- 
 lel. In other cases two circuits may lead from the gen- 
 erating station in divergent directions and at some dis- 
 tant point come together and be connected in parallel. 
 
 If the two circuits are fed from different genera- 
 tors, or sources of supply, the only condition necessary 
 for paralleling the circuits is that the phase rotation of 
 the two circuits be the same and that the regulation in 
 speed of the prime movers of the generators feeding 
 the two systems can be adjusted so as to bring the 
 phases of the two circuits together for paralleling. 
 
 If, however, the two circuits which are to be con- 
 nected in parallel are fed from the same source of sup- 
 ply, the case may become involved. There will be no 
 trouble in obtaining the correct phase rotation, for 
 should the circuits not rotate alike, it is only necessary 
 to transpose any two of the connections of either of the 
 circuits (assuming that the circuits are three-phase). 
 The other condition to be met is that the phases of both 
 circuits to be paralleled are the same, i. e., the volt- 
 ages in the phases to be paralleled must pass through 
 their zero and maximum values at the same instant. 
 
 If neither circuit has transformers between the 
 points where they are to be connected in parallel, their 
 phases will coincide and there will be no trouble about 
 connecting them in parallel. If one circuit has no 
 transformers and the other has transformers, the phase 
 relations of the two circuits will depend upon the kind 
 of transformer connections employed. Suppose it is 
 assumed that the raising transformers are connected 
 delta to star and the lowering transformers are con- 
 nected delta to delta. With these connections the 
 phases of the two circuits will be 30 electrical degrees 
 apart and it will be impossible to parallel the circuits. 
 In other words one delta-star or star-delta transformer 
 connection produces a phase displacement of 30 degrees. 
 It will be obvious that a second delta-star or star-delta 
 connection will restore the original phase relation. A 
 delta-delta connection or a star-star connection does 
 not affect the phase relations. If both circuits have an 
 even number of star and even number of delta wind- 
 ings, the equivalent resultant will be the same as if all 
 the connections were either delta-delta or star-star; 
 hence, there will be no resultant change in phase rela- 
 tions and the two circuits can be paralleled with each 
 other or with a circuit having no transformations. If, 
 however, both circuits have an odd number of delta and 
 an odd number of star windings, any attempt to re- 
 solve them into the equivalent number of delta-delta 
 and star-star connections will leave one star and one 
 delta; the effect is the same as if there was one star- 
 delta connection in the circuits. This will twist the 
 phase relations of the terminals 30 degrees out of phase 
 from the generators. Since both circuits will have an 
 
42 
 
 PROPAGATION— RESONANCE— PARALLELING CIRCUITS 
 
 equivalent phase displacement, they can be paralleled 
 with one another, but since both are 30 degrees out of 
 phase with the generators, they cannot be paralleled 
 with a line having no transformations; nor with a line 
 having an even number of star and delta connections. 
 
 When the phase angles of the two transmission cir- 
 cuits (receiving their power from a common source) are 
 known to be such as to permit of parallel operation it 
 is then necessary to phase them out before connecting 
 the circuits together. The phase rotation can be checked 
 most readily by means of a polyphase motor connected 
 first to one circuit and then to the other, being careful 
 to connect the leads in the same order in each case. If 
 the motor runs in the same direction from both circuits, 
 the phase rotation of the circuits will be the same. The 
 phase angle can be readily tested by means of a single- 
 phase synchroscope*. In case a polyphase motor and 
 synchroscope are not available, the phasing out of the 
 circuits may be accomplished by the use of a voltmeter 
 and transformer.** As an illustration, assume that 
 from a 4400 volt bus in a generating station a 4400 volt 
 transmission circuit extends for some distance from the 
 station. A second transmission circuit fed from the 
 same bus but containing both raising and lowering 
 transformers is to be paralleled at the farther end with 
 the 4400 volt circuit which contains no transformers. 
 The phase angles of the lines are assumed to be such 
 as to permit paralleling the two circuits, with proper 
 connections. 
 
 One of the transmission circuits is connected to 
 one side of the paralleling switch as in Fig. 15 and the 
 other circuit to the other side of the same switch. The 
 three terminals on one side of the switch may be tagged 
 1-2-^. Likewise the three terminals on the other side 
 of the switch may be tagged 4-5-6. Connect any two 
 terminals together (i and 4 in this case) by a jumper. 
 Take voltage readings across the corresponding ter- 
 minals 2 to 5, s to 6, and j to 5, ^ to 6. From these 
 voltage readings it is a simple matter to indicate by a 
 vector diagram the relative phase relations at the switch 
 contacts of the two circuits to be paralleled. In the case 
 illustrated, the readings indicate that the relative voltage 
 relations on the two sides of the paralleling switches are 
 as indicated by the full line delta 1-2-^, and the broken 
 line delta 4-5-6. It will be seen that phase j-j will 
 parallel with phase 4-5, that phase 1-2 will parallel with 
 phase 6-5 and phase .?-j will parallel with phase 4-6. 
 In order to bring about this phase relation it will be nec- 
 essary to change the transformer connections on the 
 low-tension side of the lowering transformers, inside 
 of the delta. That is the 6 end of the transformer 
 windings 5-6 will be connected to the 4 end of transfor- 
 
 *These tests are described in an article on "Phasing Out 
 High Tension Lines" by E. C. Stone in the Journal for Nov. 
 1917, p. 448- 
 
 **This method is described in an article on "Determination 
 of Polarity of Transformers for Parallel Operation" by W. 
 M. McConahey, in the Journal for July 1912, p. 613. See also 
 article on "Polarity of Transformers" by W. M. Dann in the 
 Journal for July 1916, p. 350. 
 
 mer 4-5. The 4 end of transformer 4-6 will be 
 connected to the 5 end of transformer, 5-6 and the 6 end 
 of transformer 4-6 will be connected to the 5 end of 
 transformer 4-5. These changes will shift the position 
 of the delta 4-5-6 so that it will coincide with delta 
 1-2-3. -^ further test of voltage between switch term- 
 inals 2X.0 5 and 5 to d should indicate zero voltage across 
 the switch terminals to be connected together, in which 
 case the paralleling switches may be closed. In order to 
 measure the voltage across the paralleling switch con- 
 tacts it will usually be necessary to employ a potential 
 transformer. This transformer and voltmeter should be 
 capable of withstanding 1.73 times the voltage of the 
 circuit for, with the connections given in Fig. 15, one 
 reading gave 7610 volts, whereas the voltage of the cir- 
 cuit was only 4400 volts. 
 
 In case there is a ground on both systems, the plac- 
 ing of a jumper across two of the switch contacts would 
 result in a short-circuit. This jumper should not be 
 placed across the switch until it b.as been shown by con- 
 necting a transformer across these two contacts that no 
 potential exists between them. 
 
 HEATING OF BARE CONDUCTORS IN AIR 
 
 If the circuit is long, the voltage will probably be 
 high and consequently the current to be transmitted 
 
 -4400 VOLT LINE DIRECT 
 FROM STATION BUS 
 
 !< 
 
 2 TO &-4400 VOLTS 
 
 3 TO 6-4400 VOLTS 
 
 SWITCH ^ 2 TO 6-7610 VOLTS 
 
 C3) (^ 3 TO 6-ZERO VOLTS 
 
 FIG. 15 — TEST FOR PHASE SEQUENCE 
 
 small. In this case, the heating effect of the current 
 will be small and unimportant. If, however, the circuit 
 is short and an unusually large amount of power is to be 
 transmitted, the current will be large. Since the PR 
 loss varies as the square of the current and directly as 
 the resistance, the heat generated, if the current is large, 
 may be sufficient to overheat or anneal the material of 
 the conductors. In some cases of unusually large 
 amounts of power being transmitted short distances, the 
 heating effect of the currents resulting may be sufficient 
 to limit the amount of power that can be transmitted 
 at a given voltage. 
 
 Table XXIII should be consulted in cases where 
 the circuit is short and the amount of power to be trans- 
 mitted large. In this table are columns containing cur- 
 rent values which have been calculated corresponding 
 to 10, 25 and 40 degrees C. rise in temperature for vari- 
 ous sizes of bare copper conductors suspended in still 
 air at a temperature of 25 degrees C. In other words 
 these current values are based upon absolute temper- 
 atures of 35, 50 and 65 degrees C. The current values 
 corresponding to a temperature rise of 40 degrees C. 
 
1 
 
 i 
 
 ERRATUM 
 
 The formula used in calculating the values for table XXIII, page 43, 
 embodied the only available information on this subject at the time the values 
 were calculated. Recent exhaustive and carefully conducted tests, made by 
 George E. Luke, indicate a wide difTerence in results from the table values, 
 especially in the larger size conductors. The table values corresponding to 40° C 
 rise should not, therefore, be used. 
 
 In the April, 1923 issue of the Electric Journal, page 127, appears an article 
 entitled "Current Capacity of Wires and Coils" in which Mr. Luke gives the results 
 of his tests and the empirical formula he developed as a result of the test. 
 
'Oi* o) -, 
 
 MUTAHfla 
 
 .ijj 
 
 
PROP AG A TION-RF.SONAXCE-PA HA U.Iil.lSG CIRCUl TS 
 
 TABLE XXIII-HEATING CAPACITY FOR 40o C. RISE 
 
 OF BARE COPPER CONDUCTORS SUSPENDED OUT OF DOORS 
 
 43 
 
 CONDUCTORS 
 
 AMPERES- 
 
 BARE 
 
 APPROXIMATE CARRYING CAPACITY IN 
 TO A TEMPERATURE RISF DP /in r (R/iOcn non 
 
 K V A 3 PHASE corresponding] 
 
 > 
 
 d 
 
 z 
 
 CO 
 CD 
 
 AREA 
 
 IN 
 
 CIRCULAR 
 
 MILS 
 
 a. CO 
 
 1- X 
 
 u 
 
 STILL AIR FOR 
 TEMPERATURE 
 RISES STATED 
 
 .~' '^ .-...r >_.>«, urvt nioo ur- nu o (bAbtU UPO™ MlwrtKtO IIM UULUMN MARKED 
 
 FOR 40 C RISE ) FOR BARE COPPER CONDUCTORS SUSPENDED IN STILL AlH 
 OUT OF DOORS. 
 
 220 
 VOLTS 
 
 440 
 VOLTS 
 
 550 
 VOLTS 
 
 MOO 
 VOLTS 
 
 2200 
 VOLTS 
 
 4000 
 VOLTS 
 
 4400 
 VOLTS 
 
 6000 
 VOLTS 
 
 6600 
 VOLTS 
 
 6900 
 VOLTS 
 
 FOR 
 IO°C 
 RISE 
 
 FOR 
 25° C 
 RISE 
 
 f6r 
 
 40°C 
 RISE 
 
 K.V A. 
 
 KVA. 
 
 KVA. 
 
 KVA 
 
 KVA 
 
 KVA 
 
 KVA 
 
 KVA 
 
 KVA 
 
 KVA 
 
 O 
 UJ 
 O 
 
 z 
 < 
 
 cc 
 
 H 
 CO 
 
 
 / 8o oooo 
 1 700 ooo 
 
 /.Sfi 
 /.sot 
 
 2/40 
 1980 
 / 89a 
 
 328C 
 302C 
 292c 
 
 4050 
 3760 
 3600 
 
 / 540 
 /430 
 /370 
 
 3 080 
 2 860 
 2 740 
 
 3 850 
 3 580 
 
 3420 
 
 7 700 
 
 7 ISO 
 6 SSO 
 
 '5 400 
 /4 JOO 
 13 TOO 
 
 2 T 900 
 26 000 
 24 800 
 
 JO 800 
 
 2 8 6O0 
 27400 
 
 42 000 
 39 100 
 37400 
 
 46200 
 42 800 
 
 48 300 
 44 800 
 4 3 000 
 
 
 / Soo ooo 
 /-foo ooo 
 
 lAii. 
 
 /■HIO 
 
 1 720 
 
 /i35 
 
 Z780 
 
 2520 
 
 3440 
 J300 
 3IOO 
 
 /3/0 
 
 /zso 
 
 II 80 
 
 2 620 
 2 Soo 
 X360 
 
 3 270 
 3 /40 
 2 950 
 
 6 SSO 
 6 2 80 
 S900 
 
 13100 
 13 560 
 II SOO 
 
 2 3 800 
 22 800 
 
 2 / 400 
 
 26200 
 25 '2o 
 23600 
 
 35 800 
 34 300 
 
 J9 200 
 37 too 
 
 41 000 
 39200 
 
 
 1 /oo ooo 
 / ooo ooo 
 
 /.2 0? 
 I.ISZ 
 
 I460 
 /360 
 
 12 70 
 
 2 1 CC 
 1 9S0 
 
 2760 
 3S8C 
 2420 
 
 / OSO 
 980 
 I730 
 
 2 100 
 / 960 
 1 840 
 
 2620 
 2460 
 2300 
 
 5250 
 4920 
 1 60O 
 
 10 SOO 
 9 840 
 9 200 
 
 / 9 '00 
 
 1 7 eoo 
 
 16 700 
 
 2 / 000 
 
 19 bgo 
 1 8400 
 
 28 700 
 26 800 
 25 100 
 
 3 1 SOO 
 39 400 
 
 2 7 6 00 
 
 3 3 000 
 JO 800 
 2 8 800 
 
 
 'too ooo 
 8SO ooo 
 
 /.093 
 /o&z 
 
 //7S 
 1/20 
 
 18 70 
 I800 
 / 720 
 
 2330 
 2220 
 2/30 
 
 8 80 
 840 
 8IO 
 
 1 760 
 / 680 
 1 630 
 
 2 200 
 
 X 1 1 
 
 3 030 
 
 4 400 
 
 4220 
 
 4 O.SO 
 
 8 800 
 8440 
 8 100 
 
 / 6 000 
 /5 300 
 14 TOO 
 
 / 7 boo 
 /6 880 
 / 6 200 
 
 24 100 
 23 000 
 
 2 b 400 
 25 300 
 
 27 700 
 26500 
 
 
 7 so OOO 
 700 ooo 
 
 /03I 
 .998 
 ?6i 
 
 /07S 
 
 /03S 
 
 980 
 
 /C-^O 
 /S80 
 IA90 
 
 2030 
 1 940 
 /8 30 
 
 770 
 740 
 69 S 
 
 / S40 
 /4 80 
 1 39u 
 
 / 930 
 / 840 
 
 1 740 
 
 3 870 
 3 690 
 J 4 80 
 
 7740 
 
 7380 
 69 to 
 
 14 000 
 13400 
 13 UOO 
 
 15480 
 14 760 
 13 920 
 
 Zl 100 
 20 200 
 19 000 
 
 23 200 
 22 100 
 Jo 90a 
 
 2 4 200 
 2 3200 
 2 / 800 
 
 
 4,00 ooo 
 «fvSO ooo 
 
 ■ 893 
 .8SJ, 
 
 8 70 
 8 10 
 
 I330 
 /3SO 
 
 1740 
 1630 
 /S30 
 
 660 
 62S 
 ctSO 
 
 1 320 
 1 2SO 
 
 1 / 60 
 
 1 660 
 / S60 
 I4S0 
 
 3 3ZO 
 3 IXQ 
 2 900 
 
 6640 
 6 240 
 5 800 
 
 12 000 
 1 1 300 
 1 a 600 
 
 13 380 
 12 480 
 
 / 1 &00 
 
 1 8 100 
 1 7 000 
 
 's 900 
 
 19 aoo 
 
 18 TOO 
 1 7 soo 
 
 30 eoo 
 
 /9600 
 
 
 ^OQ OJO 
 4 so ooo 
 ■400 OOO 
 
 8'S 
 
 772 
 
 .7Z8 
 
 7SS 
 700 
 640 
 
 //60 
 
 /06O 
 
 980 
 
 1430 
 /330 
 I3IO 
 
 54 5 
 SOO 
 460 
 
 1090 
 
 /OOO 
 
 930 
 
 1 360 
 1 3SO 
 1 / So 
 
 3 730 
 2 500 
 
 X 300 
 
 ,5440 
 5 000 
 4 600 
 
 9 86O 
 q /oo 
 8 350 
 
 10 eso 
 /o 000 
 
 9 300 
 
 14 900 
 '3 700 
 12 600 
 
 I6300 
 
 IS 100 
 
 n /oo 
 'S 700 
 
 I440O 
 
 
 3SO ooo 
 joo ooo 
 zso ooo 
 
 .68' 
 .630 
 .S7.S 
 
 S7S 
 
 s/s 
 4Sa 
 
 88S 
 78S 
 6 8S 
 
 1090 
 970 
 840 
 
 4 IS 
 370 
 330 
 
 830 
 740 
 640 
 
 1 O40 
 930 
 800 
 
 2 OSO 
 / 840 
 / li,00 
 
 4160 
 3 680 
 
 3 200 
 
 7 soo 
 
 6 TOO 
 5 800 
 
 8320 
 / 360 
 6 400 
 
 // 300 
 
 10 /OO 
 
 8 700 
 
 13 400 
 
 11/00 
 
 9 SSO 
 
 / 3 000 
 / / boo 
 10 000 
 
 OOO 
 
 oo 
 
 3, 1 / (,00 
 li,7 77Z 
 
 /33 or? 
 
 528 
 
 .-f 70 
 .418 
 
 3^S 
 33a 
 2%0 
 
 &OS 
 SOS 
 4ZS 
 
 7S0 
 625 
 527 
 
 28S 
 238 
 200 
 
 S70 
 4 75 
 ■400 
 
 71S 
 S9S 
 SOO 
 
 1 430 
 1 / 90 
 / 000 
 
 Z 860 
 2 3 80 
 2 000 
 
 S 1 70 
 
 4320 
 3 640 
 
 5 720 
 4 760 
 4 000 
 
 7 800 
 6 SOO 
 5470 
 
 8 550 
 
 7 / 30 
 6000 
 
 -8 9SQ 
 
 • 74S0 
 
 6Z80 
 
 'z 
 
 / OS StoO 
 S3 ^9'^ 
 t,(, 3S8 
 
 ■373 
 ■ 332 
 .29Z 
 
 Z3S 
 I9S 
 I6Z 
 
 360 
 300 
 2S0 
 
 444 
 370 
 307 
 
 /70 
 741 
 116 
 
 336 
 282 
 233 
 
 423 
 352 
 292 
 
 846 
 
 704 
 584 
 
 1 693 
 /40S 
 / i6>8 
 
 3 ObO 
 ZSSO 
 2 1 20 
 
 3384 
 28/6 
 2 336 
 
 4 600 
 3 840 
 J / 80 
 
 SOSO 
 
 4220 
 
 3SOO 
 
 S30O 
 4400 
 3 660 
 
 3 
 
 ■4- 
 
 £Z <,ZA 
 33 OS8 
 
 .Z6o 
 .232 
 .206 
 
 /36 
 
 7/4 
 
 zio 
 176 
 /4 7 
 
 258 
 2 35 
 
 I8Z 
 
 If 
 69 
 
 196 
 178 
 138 
 
 245 
 2 24 
 
 / 73 
 
 490 
 44? 
 346 
 
 980 
 896 
 692 
 
 1 770 
 / 620 
 / 3&0 
 
 / 960 
 / 792 
 /384 
 
 2 680 
 2440 
 / 890 
 
 Z940 
 2 680 
 208O 
 
 J 080 
 2 SOO 
 2 ISO 
 
 Q 
 
 -J 
 O 
 m 
 
 ooo 
 oo 
 
 :i 1 1 &0O 
 /6 7 77-2 
 /33 079 
 
 .-f60 
 .410 
 ■.3i,S 
 
 370 
 
 3IO 
 
 zse 
 
 S6S 
 47'S 
 420 
 
 7Z8 
 S88 
 
 49 s 
 
 Z7S 
 234 
 /88 
 
 SSO 
 448 
 376 
 
 690 
 56 
 470 
 
 /380 
 / /20 
 
 940 
 
 3760 
 Z Z4-0 
 1 8«0 
 
 S 030 
 406O 
 34x0 
 
 S S30 
 4 480 
 376O 
 
 7 SSO 
 6100 
 SISO 
 
 8 300 
 6,700 
 S650 
 
 8 TOO 
 7000 
 S9O0 
 
 1 
 Z 
 
 lOS s&o 
 ■S3 691 
 
 .325 
 
 2/S 
 /82 
 /54 
 
 335 
 Z80 
 235 
 
 41S 
 J48 
 Z9^ 
 
 'S8 
 ' 3X 
 / IZ 
 
 3/6 
 2 64 
 2 24 
 
 39S 
 330 
 280 
 
 790 
 
 660 
 S60 
 
 / SSO 
 / 320 
 / 120 
 
 Z860 
 Z400 
 3 040 
 
 3 I60 
 Z640 
 2240 
 
 4 300 
 3620 
 3 060 
 
 4 730 
 3 970 
 3 370 
 
 4 950 
 4 'SO 
 3S30 
 
 3 
 ■4 
 
 -? / 73S 
 
 33 oes- 
 
 .2 ay 
 .Z04 
 
 ./SZ 
 
 /2'S 
 
 /08 
 
 90 
 
 2 00 
 /67 
 /40 
 
 Z4S 
 
 Z07 
 
 174 
 
 93 
 78 
 66 
 
 /86 
 IS8 
 /3Z 
 
 233 
 19 7 
 /6S 
 
 465 
 39'4 
 330 
 
 930 
 788 
 660 
 
 / 690 
 I430 
 / 300 
 
 / 860 
 / S76 
 / J30 
 
 ZS40 
 Z ISO 
 I8IO 
 
 2 790 
 2 3 to 
 19 80 
 
 Z920 
 Z4TO 
 3 090 
 
 
 
 10 000 
 VOLTS 
 
 II 000 
 VOLTS 
 
 13200 
 VOLTS 
 
 15000 
 VOLTS 
 
 20000 
 VOLTS 
 
 22000 
 VOLTS 
 
 30 000 
 VOLTS 
 
 33 000 
 VOLTS 
 
 50 000 
 VOLTS 
 
 60 000 
 VOLTS 
 
 K.V.A. 
 
 K.V.A. 
 
 K.VA 
 
 K.V.A. 
 
 K.V.A. 
 
 K.V.A. 
 
 K.V.A. 
 
 KVA 
 
 KVA 
 
 KVA 
 
 a 
 
 UJ 
 
 O 
 
 z 
 < 
 
 
 z OOO ooo 
 1 800 ooo 
 
 1 700 ooo 
 
 /.63/ 
 /.5+S 
 A 50-? 
 
 2140 
 1980 
 1890 
 
 3 a SO 
 
 3030 
 Z920 
 
 40S0 
 3760 
 3600 
 
 70 000 
 65 000 
 62 300 
 
 7 7 000 
 7/ Soo 
 68 soo 
 
 92.200 
 8-5500 
 82 000 
 
 <!7 5O0 
 
 93500 
 
 
 1 
 
 
 
 
 
 
 / (,oo ooo 
 /soo ooo 
 / 400 ooo 
 
 /.-»5<? 
 /4/2 
 /.36-f 
 
 1810 
 1 rzo 
 /6 3 5 
 
 2780 
 2 640 
 2520 
 
 3440 
 3300 
 31 00 
 
 S9 soo 
 5 7 000 
 S3 600 
 
 6SSOO 
 6 3 000 
 S9 000 
 
 78SOO 
 75 300 
 70 600 
 
 S92O0 
 8 5 500 
 8 0500 
 
 
 
 
 
 
 
 
 /zoo ooo 
 / /oo ooo 
 / ooo ooo 
 
 /.263 
 /2 0? 
 /./SZ 
 
 lAhO 
 I3t0 
 1370 
 
 2 2 30 
 2/00 
 / 950 
 
 2760 
 2 5 80 
 2'420 
 
 47 700 
 44 600 
 4 / 800 
 
 SZ SOO 
 
 49 000 
 4 6 000 
 
 63 000 
 58 700 
 5 S 300 
 
 7/ 60O 
 6 7 000 
 6 3 000 
 
 '?5400 
 89 200 
 83 600 
 
 <f8 000 
 92 000 
 
 
 
 
 
 
 9SO ooo 
 <foo ooo 
 kso ooo 
 
 /.1 2 3 
 I.093 
 /.o62 
 
 IZIS 
 II 75 
 1 IZO 
 
 1870 
 
 leoo 
 
 I730 
 
 2 3 20 
 2220 
 2/30 
 
 40 000 
 38400 
 3 6 800 
 
 44 200 
 4 2 200 
 4 SOO 
 
 S3 000 
 SO SOO 
 48 6<50 
 
 60 200 
 57500 
 SSZOO 
 
 SO 000 
 77 800 
 73 600 
 
 88400 
 84400 
 81 000 
 
 
 
 
 . 
 
 
 800 ooo 
 7SO ooo 
 700 ooo 
 
 /.0 3I 
 .998 
 .964 
 
 lOVS 
 
 I03S 
 
 980 
 
 /i.40 
 'i80 
 1490 
 
 2030 
 1940 
 1830 
 
 3S 100 
 33 600 
 3 1 6O0 
 
 38600 
 36 900 
 34 800 
 
 46 300 
 44 200 
 4 / 600 
 
 52600 
 SO SOO 
 47 SOO 
 
 70200 
 67200 
 63 200 
 
 77200 
 73 800 
 69 600 
 
 95 000 
 
 
 
 
 
 6SO ooo 
 6 oo ooo 
 jso ooo 
 
 ■929 
 .893 
 
 .ess 
 
 920 
 870 
 810 
 
 1410 
 /330 
 I2S0 
 
 1740 
 16-40 
 /■S30 
 
 JO /OO 
 
 2840O 
 X6400 
 
 33 000 
 3 1 ZOO 
 29 000 
 
 39 700 
 37400 
 .14 900 
 
 45 000 
 42500 
 39 TOO 
 
 60 200 
 56 800 
 52 800 
 
 66 000 
 62400 
 S8OO0 
 
 90 000 
 85 000 
 79400 
 
 99 000 
 i)3 SOO 
 87 000 
 
 
 
 
 soo ooo 
 4SO ooo 
 400 ooo 
 
 ■ eis 
 
 .772 
 .728 
 
 755 
 700 
 640 
 
 1 160 
 
 /06O 
 
 980 
 
 14 30 
 /320 
 13 1 
 
 24 800 
 23 SOO 
 
 X 1 000 
 
 2.7200 
 3S 000 
 2 3 000 
 
 33 600 
 30 100 
 37 600 
 
 37 200 
 34 200 
 3 1 400 
 
 49 6 00 
 456O0 
 ■4X 000 
 
 5 4 400 
 SO 000 
 «46 000 
 
 74400 
 68400 
 62 SOO 
 
 81 SOO 
 7SOOO 
 69 000 
 
 
 
 
 3SO ooo 
 
 300 ooo 
 
 3 so ooo 
 
 .68/ 
 630 
 S7S 
 
 575 
 SIS 
 4 SO 
 
 88 S 
 78S 
 685 
 
 1090 
 970 
 840 
 
 18 900 
 
 /6 eoo 
 
 I4SOO 
 
 20 800 
 / 8400 
 16 000 
 
 24 800 
 22 100 
 
 19 /OO 
 
 28300 
 2 5 300 
 X 1 Soo 
 
 37 800 
 33 600 
 29 000 
 
 41 600 
 36 80O 
 3 2 000 
 
 56600 
 SO-^-OO 
 43600 
 
 62 000 
 55200 
 4 7»oo 
 
 94 000 
 83500 
 72 500 
 
 
 oooo 
 ooo 
 oo 
 
 Z 1 1 &0O 
 /67 772 
 /33 OTJ 
 
 52 6 
 470 
 
 418 
 
 38i 
 330 
 280 
 
 60s 
 S08 
 425 
 
 7SO 
 
 625 
 527 
 
 13 000 
 
 lO 800 
 
 9/00 
 
 14 300 
 // 900 
 10 000 
 
 1 7 /OO 
 14 200 
 13 000 
 
 I9SOO 
 /CXOO 
 /3 700 
 
 2 6 000 
 2/ 60O 
 / 8 200 
 
 286O0 
 2 3 800 
 20000 
 
 39 000 
 3x400 
 XT400 
 
 4 2 700 
 3S600 
 30 000 
 
 65 000 
 54 000 
 4S500 
 
 7S'OCC 
 
 4..1 900 
 
 54 ?0O 
 
 
 o 
 1 
 
 z 
 
 /OS s&o 
 
 373 
 332 
 292 
 
 235 
 I9S 
 1 6 3 
 
 360 
 300 
 3S0 
 
 444 
 370 
 307 
 
 7700 
 6400 
 
 s ^ipo 
 
 ■84SO 
 7050 
 5850 
 
 /O /OO 
 
 8420 
 7 000 
 
 //SOO 
 960O 
 7 9S0 
 
 /5400 
 IX 800 
 10 600 
 
 /6 900 
 /4 100 
 // TOO 
 
 23 000 
 1 9 xoo 
 1^ 900 
 
 2S400 
 2 / 100 
 '7 SOO 
 
 38 SOO 
 3x000 
 XbSOO 
 
 4 6 000 
 
 
 3 
 
 ■4 
 S 
 
 SZ 4,24 
 
 1^ lU 
 
 2(,0 
 232 
 20i 
 
 I3(> 
 1 l'^- 
 96 
 
 XlO 
 176 
 /47 
 
 ZiS 
 232. 
 /»2 
 
 4460 
 4 000 
 3 ISO 
 
 4900 
 4410 
 3470 
 
 S880 
 S300 
 4 ISO 
 
 6 700 
 6000 
 4 700 
 
 8930 
 8000 
 6300 
 
 9 80O 
 8 840 
 6 940 
 
 '340O 
 
 IX 000 
 
 9400 
 
 14 700 
 I33O0 
 10400 
 
 3XAOO 
 30 000 
 
 IS 300 
 
 26 Soo 
 
 2 4 000 
 18 800 
 
 
 oooo 
 ooo 
 oo 
 
 Zl 1 too 
 /h7 77Z 
 / 33 079 
 
 ■fto 
 
 410 
 ZiS 
 
 370 
 310 
 2S8 
 
 S6S 
 47S 
 4X0 
 
 73» 
 SS» 
 49S 
 
 1x600 
 
 /O ZOO 
 
 8 SSO 
 
 /3 800 
 
 // aoo 
 ?4oo 
 
 16 600 
 13400 
 II 300 
 
 /8 900 
 
 /.s 300 
 
 13 900 
 
 2 5-200 
 20400 
 / 7 / 00 
 
 27 600 
 2 2 400 
 /8?00 
 
 J78O0 
 30 600 
 ZS SOO 
 
 41 SOO 
 
 33500 
 
 2 8 200 
 
 63 OOO 
 SO 800 
 43 6O0 
 
 TS 600 
 6/ 300 
 Si 600 
 
 a ■ 
 
 -J 
 o 
 
 o 
 
 1 
 z 
 
 /OS S60 
 83 694 
 66 3SS 
 
 32S 
 289 
 2SS 
 
 218 
 182 
 /S4 
 
 335 
 230 
 298 
 
 4'S 
 34S 
 395 
 
 7 1 80 
 6 0X0 
 S /OO 
 
 7900 
 6630 
 
 scoo 
 
 94SO 
 7fSO 
 67-iO 
 
 /O BOO 
 
 9 o<ro 
 7 6 so 
 
 14360 
 13040 
 / OXOO 
 
 IS800 
 
 /3 600 
 
 // ZOO 
 
 XI 600 
 IS /OO 
 /s 300 
 
 23 600 
 /9 8O0 
 /6SO0 
 
 36 000 
 30 000 
 35SOO 
 
 4320a 
 3640O 
 306OO 
 
 
 3 
 
 ■4 
 
 S3 i>Z4 
 4 1 738 
 3 3 OS 8 
 
 239 
 30'^ 
 /82 
 
 I3S 
 
 108 
 
 90 
 
 2 00 
 /67 
 /40 
 
 3-4S 
 207 
 /74 
 
 4 3 SO 
 3 SSO 
 30^0 
 
 4 660 
 3 940 
 3 330 
 
 S 600 
 
 4 700 
 3970 
 
 6 3SO 
 
 s 370 
 4 500 
 
 8 SOO 
 7 1 60 
 6040 
 
 9330 
 
 7 880 
 6640 
 
 13 700 
 10 740 
 
 9 060 
 
 /4000 
 
 / / goo 
 
 9900 
 
 2/ 200 
 / 7 800 
 /SOOO 
 
 25400 
 2. 480 
 /f 000 
 
 Formula (from Foster's Hand Book) Amperes = r/M-yj — - for stranded conductors, and Amfirres-tiso-^ -rr- for solid conductors, where 7"=- trmperatare 
 
 rise is degrees C, D ^ diameter of conductor in inches, and ^ = resistance in ohms per mil foot at the final lcmp«ralure, Basetl on an air temperature of as 
 degrees C. 
 
44 
 
 PROPAGATION— RESONANCE—PARALLELING CIRCUITS 
 
 (absolute temperature of 65 degrees C.) have also been 
 expressed in the form of k.v.a., three-phase values cor- 
 responding to various transmission voltages. Thus No. 
 0000 stranded bare copper conductors suspended in still 
 air out doors at 25 degrees C. wrill carry 750 amperes 
 with a temperature rise of 40 degrees C. (absolute tem- 
 perature 65 degrees C). If the transmission voltage is 
 220 volts, the corresponding k.v.a. value will be 285 
 k.v.a. three-phase and if the transmission voltage is 
 10 000 volts, 13000 k.v.a. may be transmitted with the 
 same temperature rise. 
 
 As indicated by foot notes the values of the table 
 were calculated by formulas from Foster's Handbook 
 as follows: — 
 
 Amperes = Jioo -^ — -— for stranded conductor. 
 
 .(<5) 
 
 Amperes = 1250 . 
 
 -for solid conductor {26) 
 
 Where 
 
 T = Temperature rise in degrees C. 
 D = Diameter of conductors in inches. 
 R = Resistance of conductors in ohms per mil-foot at 
 final temperature. 
 
CHAPTER VI 
 
 DETERMINATION OF FREQUENCY & VOLTAGE 
 
 FREQUENCY DETERMINATION 
 
 Cost of Transformers — Sixty cycle transformers 
 cost approximately 30 to 40 percent less than 25 cycle 
 transformers ; or stated another way, 25 cycle trans- 
 formers cost approximately 40 to 66 percent more than 
 60 cycle transformers. The saving in first cost may 
 vary between $1.50 and $2.50 per kv-a. in favor of 
 60 cycles. Assuming that the total kv-a. of trans- 
 former capacity connected to a transmission circuit is 
 2.5 times the kv-a. transmitted over the circuit, the sav- 
 ing in favor of 60 cycle transformers would be $3.75 to 
 $6.25 or an average of $5.00 per kv-a. transmitted. As- 
 suming 20000 kv-a. to be transmitted, the saving in 
 cost at $5.00 per kv-a. will be $100000 in favor of 60 
 cycle transformers. The actual difference in cost will 
 depend upon the type of the transformers, that is, 
 whether water or self-cooled and also upon their aver- 
 age capacity. The difference in cost will be greater for 
 the self-cooled type and for the smaller capacities. 
 
 Weight and Space of Transformers — The less 
 weight of 60 cycle transformers makes them easier to 
 handle and they require less space for installation. 
 
 Higher Reactance — Inductive reactance at 60 
 cycles is 2.4 times its value at 25 cycles. This tends 
 to produce poorer voltage regulation of the circuit. 
 Higher reactance has one advantage for the larger sys- 
 tems in that it tends to limit short-circuit currents and 
 thus assists the circuit opening devices to function 
 properly. By virtue of the higher reactance it might 
 be possible in some cases to obtain sufficient reactance 
 in the transformers without the addition of current 
 limiting reactance coils. 
 
 Efficiency — The efficiency of 60 cycle transformers 
 is usually 0.25 to 0.50 percent higher than for 25 cycle 
 transformers. 
 
 Charging Current — At 25 cycles both the charging 
 current and the reactance are approximately 42 percent 
 of their values for 60 cycles. This tends to give better 
 regulation and usually higher efficiency in transmission. 
 On the other hand, the higher transmission efficiency 
 may be offset by the slightly lower eflficiency of 25 cycle 
 transformers. In cases of very long circuits (par- 
 ticularly if the circuits are in duplicate and both in ser- 
 vice) or of transmission systems embracing many miles 
 of high tension mains and feeders, the charging currents 
 may be so great as to limit the choice in transmission 
 voltage. On the other hand large charging currents 
 may be permitted, provided under excited synchronous 
 motors are used at various parts of the transmission 
 
 system for partially neutralizing this charging current 
 and for maintaining con.jtant voltage. 
 
 Inductive Disturbances— Lighimng, switching and 
 other phenomena cause disturbances on conductors of 
 transmission circuits. The frequency of these disturb- 
 ances is independent of that impressed on the system. 
 After the removal of the disturbing influence they 
 oscillate with the natural frequency of the line. 
 
 The natural frequency of the line is far above com- 
 mercial frequencies but, if the transmission line is long, 
 there may be some odd harmonic present in the funda- 
 mental impressed frequency which corresponds with the 
 natural period of the line. This might tend to produce 
 an unstable condition or resonance. This condition is 
 somewhat less likely to occur at 25 cycles. 
 
 Summary — Although there are a number of large 
 25 cycle transmission systems in operation, they were 
 mostly installed before the design of 60 cycle converting 
 apparatus and electric light systems had reached their 
 present state of perfection. Unless it is desirable to 
 parallel with an existing 25 cycle system located in ad- 
 joining territory without the introduction of frequency 
 changers, it is now quite general practice to choose the 
 frequency of 60 cycles.* 
 
 VOLTAGE DETERMINATION 
 
 From a purely economic consideration of the con- 
 ductors themselves, Kelvin's law for determining the 
 most economical size of conductors would apply. Kel- 
 vin's law may be expressed as follows : — 
 
 "The most economical section of a conductor is that 
 which makes the annual cost of the PR losses equal to 
 the annual interest on the capital cost of the conducting 
 material, plus the necessary annual allowance for depre- 
 ciation". That is, the economical size of conductor for 
 a given transmission will depend upon the cost of the 
 conducting material and the cost of power wasted in 
 transmission losses. The law of maximum economy 
 may be stated as follows : — "The annual cost of the en- 
 ergy wasted per mile of the transmission circuit added 
 to the annual allowance per mile for depreciation and 
 interest on first cost, shall be a minimum". 
 
 Attempts have been made to determine by mathe- 
 matical expression the most economical transmission 
 voltage, all factors having been taken into account. 
 There are so many diverse factors entering into such a 
 
 ♦For a complete discussion of this subject see a paper hy 
 D. B. Rushmore before the Schenectady section A. I. E. E., 
 May 17, IQI2, on "Frequency" and an article by B. G. Lamme 
 on "The Technical Story of the Frequencies" in the Journal. 
 for June, 1918, p. 23a 
 
46 
 
 DETERMINATION OF EREQUENCY AND VOLTAGE 
 
 treatment as to make such an expression complicated, 
 difficult and unsatisfactory. There are many points re- 
 quiring careful investigation, not embraced by Kelvin's 
 law, before the proper transmission voltage can be de- 
 termined. Some of these points are given below. 
 
 Cost of Conductors — For a given percentage energy 
 loss in transmission, the cross-section and consequently 
 the weight of conductors required by the lower and 
 medium voltage lines (up to approximately 30000 
 volts) to transmit a given block of power varies in- 
 versely as the square of the transmission voltage. Thus 
 if this voltage is doubled, the weight of the conductors 
 will be reduced to one fourth with approximately a cor- 
 responding reduction in their cost. This saving in con- 
 ducting material for a given energy loss in transmission 
 becomes less as the higher voltages are reached, becom- 
 
 TABLE E 1— WEIGHT OF BARE COPPER 
 CONDUCTORS 
 
 b 
 
 z 
 CO 
 
 ED 
 
 AREA IN 
 
 CIRCULAR 
 
 MILS 
 
 WEIGHT IN POUNDS 
 
 PER 1000 FEET 
 OF CIRCUIT 
 
 PER MILE 
 OF CIRCUIT 
 
 NUMBER OF 
 CONDUCTORS 
 
 NUMBER OF 
 CONDUCTORS 
 
 ONE 
 
 TWO 
 
 THREE 
 
 ONE 
 
 TWO 
 
 THREE 
 
 
 2 000 000 
 / 900 000 
 / 800 000 
 
 i, I<i0 
 3 8 70 
 SS(,n 
 
 /i 340 
 (/ 740 
 1 1 1 20 
 
 19 S40 
 17 <olO 
 Ik 6S0 
 
 32 630 
 Z0 994 
 293S7 
 
 hS 2hQ 
 bl 9S8 
 Si7l4 
 
 <?7 ?90 
 92 9S2 
 88 071 
 
 
 / 700 000 
 
 / feoo 000 
 / soo 000 
 
 S2S0 
 4 630 
 
 10 SOO 
 <?880 
 9 2h0 
 
 IS 7S0 
 I4S20 
 I3S90 
 
 27720 
 26083 
 2444t 
 
 55 440 
 52 lit 
 48 892 
 
 83'60 
 78 249 
 7?338 
 
 
 1 -»oo 000 
 1 zoo 000 
 1 zoo 000 
 
 4 320 
 ■4 010 
 3 7/0 
 
 S6-I0 
 8020 
 7420 
 
 12 9&0 
 12 030 
 // 130 
 
 22810 
 21 1 73 
 I9SS9 
 
 451,20 
 4234h 
 39 I7S 
 
 iS'430 
 635/9 
 58 767. 
 
 
 1 100 000 
 
 1 000 000 
 9S0 000 
 
 3 400 
 3 090 
 2 9-30 
 
 6 SOO 
 6 l%0 
 
 ssto 
 
 10 200 
 9 270 
 8 790 
 
 17 9S2 
 liSIS 
 15470 
 
 35 904 
 32 630 
 30940 
 
 53856 
 4 8 945 
 4i4IO 
 
 
 900 000 
 eso 000 
 eoo 000 
 
 2 780 
 2 <,20 
 2 470 
 
 ssto 
 
 S240 
 
 4 940 
 
 ?3-#0 
 7 860 
 7410 
 
 14 67S 
 13 83-) 
 13 047 
 
 793Si, 
 27668 
 26084 
 
 44 034 
 4/ 502 
 39 /26 
 
 
 7 50 000 
 TOO 000 
 
 eso 000 
 
 2320 
 2lt0 
 2010 
 
 4h40 
 4320 
 .4020 
 
 6 9i0 
 6480 
 (,0^0 
 
 12 2S0 
 II 40S 
 10 6/3 
 
 24 -500 
 228/0 
 2/226 
 
 36 750 
 342'X 
 31 S39 
 
 
 (,00 000 
 sso 000 
 soo 000 
 
 1 SSO 
 1 TOO 
 1 S40 
 
 3 700 
 3400 
 3 080 
 
 SiiO 
 S 100 
 4&20 
 
 9768 
 8976 
 8 131 
 
 /9336 
 17952 
 
 itztz 
 
 29304 
 26928 
 Z4393 
 
 
 4S0 000 
 400 000 
 3 50 000 
 
 1 390 
 / 240 
 1 O80 
 
 2780 
 24S0 
 2 ItO 
 
 4 1 70 
 3 72 
 3 240 
 
 7339 
 iS47 
 S 702 
 
 14 67? 
 13094 
 
 11404 
 
 2201 7 
 19 641 
 17 lOi 
 
 0000 
 
 300 000 
 2SO 000 
 2/2 000 
 
 ?26 
 77Z 
 HSZ 
 
 ISSZ 
 1 S44 
 
 1 306 
 
 2778 
 
 23l(, 
 1 9 59 
 
 4 889 
 ^076 
 344S 
 
 977S' 
 8/52 
 6S9<= 
 
 /4 6 67 
 12 228 
 10 344 
 
 000 
 00 
 
 
 /68 000 
 /33 000 
 /06 000 
 
 SIS 
 4 II 
 32(, 
 
 1036 
 822 
 (,SZ 
 
 1 SS4 
 
 1 233 
 978 
 
 2 73S 
 2 170 
 1721 
 
 5470 
 4 340 
 3442 
 
 8205 
 65/0 
 5/63 
 
 / 
 2 
 
 3 
 
 83 700 
 
 a 400 
 ^3 600 
 
 2S8 
 20s 
 /i3 
 
 Sli 
 4 10 
 32t, 
 
 774 
 US 
 4S9 
 
 I 362 
 
 /082 
 
 86' 
 
 2724 
 2 /64 
 / 72 2 
 
 40?i, 
 3 246 
 2.SS3 
 
 
 ■4 1 700 
 33 100 
 26300 
 
 129 
 
 102 
 
 SI 
 
 2SS 
 
 204 
 It2 
 
 387, 
 30i 
 243 
 
 68/ 
 ^39 
 -♦28 
 
 /362 
 
 I07S 
 
 SSi 
 
 2043 
 /6/7 
 /2 84 
 
 I 
 
 30 SOO 
 lb SOO 
 
 i1 
 
 IZS 
 101 
 
 19-2 
 /S3 
 
 338 
 26? 
 
 6 76 
 S3% 
 
 / 014 
 807 
 
 ing increasingly less as voltages go higher. This is for 
 the reason that for the higher voltages at least two other 
 sources of losses, leakage over insulators and the escape 
 of energy through the air between the conductors 
 (known as "corona") appear. In addition to these two 
 losses, the charging current, which increases as the 
 transmission voltage goes higher, may either increase or 
 decrease the current in the circuit depending upon the 
 power-factor of the load current and the relative 
 amount of the leading and lagging components of the 
 current in the circuit. Any change in the current of the 
 circuit will consequently be accompanied by a corre- 
 sponding change in the I^R loss. In fact, these sources 
 of additional losses may, in some cases of long circuits 
 or extensive systems, materially contribute toward limit- 
 ing the transmission voltage. The weight ol copper 
 
 conductors, from which their cost may readily be cal- 
 culated, is given in Table E-i. As an insurance 
 against breakdown, important lines frequently are built 
 with circuits in duplicate. In such cases the cost of 
 conductors for two circuits should not be overlooked. 
 
 Table E-1 contains the weights of bare stranded 
 copper cables per 1000 feet of circuit, also per mile of 
 circuit. For the purpose of facilitating rapid calcula- 
 tion for any given case, the weights are given corre- 
 sponding to one, two and three conductors for these two 
 lengths of circuit. 
 
 Reduced Electric Surges — The better insulation 
 necessitated by higher transmission voltages tends to 
 make the circuit more secure against ordinary disturb- 
 ances. Also the smaller currents resulting with the 
 higher voltages cause less disturbance in the circuit in 
 the case of grounds, short-circuits, switchings, light- 
 ning and other disturbances. 
 
 Less Reactance Volts Drop — Since the current cor- 
 responding to higher transmission voltages goes down 
 as the voltage goes up, the voltage necessary to over- 
 come the reactance of the circuit will be less, and the 
 percentage reactance volts much less for higher volt- 
 
 TABLE F— PRESENT RELATIVE COSTS OF 
 
 HIGH TENSION APPARATUS 
 
 Expressed in Percent (6600 Volt Costs Taken as 100%) 
 
 Trans- 
 formers 
 
 Switches 
 
 Electro- 
 lytic Ar- 
 resters 
 
 Insulators 
 
 too 
 100 
 
 s> 
 
 102 
 100 
 
 100 151 
 100 135 
 
 :> 
 
 104 
 100 
 
 160 
 185 
 
 106 
 100 
 
 19s 
 36s 
 
 s> 
 
 108 
 100 
 
 205 
 430 
 
 »> 
 
 5> 
 
 no 
 
 320 
 650 
 
 125 
 -15 
 
 430 
 1250 
 
 150 
 
 155 
 
 640 
 3S00 
 
 175 
 
 255 
 
 1600 
 5500 
 
 20c 
 420 
 
 1900 
 6500 
 
 225 
 
 2400 
 7700 
 
 ages. Thus, if the transmission voltage is doubled, the 
 current will be halved and for the same spacing of 
 conductors the reactance volts drop will be one half, re- 
 sulting in one fourth the percentage of the reactance- 
 volts drop. 
 
 Cost of Transformers — If the transmission voltage 
 exceeds 13 200 volts, banks of step-up transformers will, 
 be required of sufficient capacity to transform all of 
 the kv-a. to be transmitted. A still greater capacity of 
 step down transformers will be lequired to reduce the 
 voltage to that suitable for operating motors and lights. 
 In some cases two reductions from the transmission 
 circuit voltage may be required, the first usually re- 
 ducing to 22 000, 1 1 000 or 6600 volts for general dis- 
 tribution and the second reducing from the general dis- 
 tribution voltage to the proper voltage for motors and 
 lights. The net result is that the total capacity in trans- 
 formers connected to a transmission system employing 
 both step up and step down transformers may vary 
 from a minimum of two to a maximum of about four 
 times the kv-a. transmitted over the high-tension cir- 
 cuits. The average condition we will assume as 2.5 
 times the kv-a. to be transmitted. 
 
 The cost of power transformers at the present time 
 
DETERMINATION OF FREQUENCY AND VOLTAGE 
 
 47 
 
 for 66 coo volts service will vary between $1.25 to $3.00 
 for 60 cycle and $2 to $5 per kv-a. for 25 cycle ser- 
 vice, depending upon their type and capacity. The 
 total cost per kv-a. of transformers on a system would 
 therefore be represented by approximately 2.5 times the 
 above costs. The present relative costs of trans- 
 formers for different voltages are given in Table F. 
 For instance if the transmission voltage is increased 
 from 33 000 to 66 000 volts the transformers will cost in 
 the neighborhood of 150 -^ 115 or 31 percent more 
 than they would cost for 33 000 volts. Knowing the 
 amount of power to be transmitted, an approximate 
 estimate may be made as to the additional cost of the 
 necessary transformers for a higher voltage. 
 
 Cost of Insulators — Table F values indicate a wide 
 difference in the cost of insulators for the higher volt- 
 
 Efficiency — The efficiency of transformers will be 
 slightly higher for the lower voltages. 
 
 Small Customers — The furnishing of power to 
 small customers at points along the transmission cir- 
 cuits should receive careful consideration. The cost of 
 switching apparatus, lightning arresters and trans- 
 formers required to permit service being given to such 
 customers will be less for the lower voltage. 
 
 Charging Current — The amount of current re- 
 quired to charge the transmission circuits varies ap- 
 proximately as the transmission voltage. Therefore 
 the charging current, expressed in kv-a. varies ap- 
 proximately as the square of the voltage. Thus the 
 charging current required for a 33 000 volt circuit is 
 approximately one half and the charging kv-a. one 
 fourth that of a 66 000 volt circuit. 
 
 TABLE G— FORM OF TABULATION FOR DETERMINING VOLTAGES 
 
 AND CONDUCTORS 
 
 BASED ON THE TRANSMISSION OF 10 000 KV-A. FOR TEN MILES AT 80 PERCENT POWER-FACTOR LAQQINQ 
 
 60 CYCLES, THREE PHASE 
 
 VOLTAGE 
 
 a: 
 
 o< 
 
 "-> 
 co^ 
 
 UJO 
 
 < 
 
 
 CONDUCTORS 
 
 
 1 
 
 VOLTAGE 
 DROP AT 
 
 FIRST COST 
 
 ANNUAL 
 OPERATING COST 
 
 
 
 FULL LOAD 
 
 CONDUCTORS 
 
 AT 25 CTS. PER 
 
 POUND 
 
 CO 
 
 £< 
 
 Si 
 
 CO 
 
 t- 
 
 z 
 
 Oco 
 
 co^ 
 
 l-H 
 15 
 
 Oco 
 
 I 
 
 S UJ 
 ZH 
 H CO 
 
 -"< 
 
 CO 
 
 oc 
 
 
 
 CO 
 
 z 
 
 _l 
 2 
 
 
 
 
 si 
 
 UJ 
 
 Z 
 
 CO 
 
 _l 
 
 °i 
 
 CO < 
 
 *^ 
 
 00 
 cc 
 
 
 is 
 
 1- 
 
 Ml 
 
 
 1% 
 
 COI 
 
 03O 
 
 UJ 
 
 TOTAL |2rL0SS 
 
 •INTEREST ON 
 FIRST COST 
 ■ AT 6% 
 
 DEPRECIATION 
 
 ON FIRST COST 
 
 AT 10% 
 
 |2r LOSSES 
 
 AT 1 CT. PER 
 
 KW-HOUR 
 
 1 
 
 UJ 
 
 
 
 coir 
 
 UJ — 
 (E 
 
 UJ 
 
 UJ 
 CE 
 
 a. 
 
 
 tr 
 
 
 
 > 
 
 10.000 
 KVA 
 
 2500 
 KVA 
 
 TOTAL LOSS 
 PER YEAR IN 
 KW-HOURS 
 
 I2 
 
 2 
 
 lECO 
 go: 
 
 li.soo 
 
 VSZi 
 
 3 so 
 
 SOO 000 
 
 243 930 
 
 i-n 
 
 4 30 
 
 .5.3 
 
 27 
 
 / 707470 
 
 4.3 
 
 2/.7 
 
 17.S 
 
 >60 9SZ 
 
 *7SO0O 
 
 *3 000 
 
 '1 000 
 
 *foo 
 
 'nOiiZ 
 
 'S4i3 
 
 ■'/4OS8'/7075fJ9</<l 
 
 3 00 000 
 
 Itk L70 
 
 i.tt> 
 
 720 
 
 9.0 
 
 45 
 
 2 867y50 
 
 7.2 
 
 22.7 
 
 30 
 
 3* 470 
 
 7SO00 
 
 3000 
 
 1000 
 
 900 
 
 116 snd 
 
 6994 
 
 //6S7 
 
 28 580 
 
 47 331 
 
 "000 
 
 iioso 
 
 3.S0 
 
 /2S« 
 
 16.1 
 
 20 
 
 5 /02 700 
 
 /3.9 
 
 24.2 
 
 as 
 
 20 5/2 
 
 76 000 
 
 3 000 
 
 1000 
 
 too 
 
 100 4li 
 
 6 02s 
 
 10 041 
 
 5/027^ 
 
 67091 
 
 Z7 fiOO 
 
 n 702 
 
 ZiZ 
 
 iooooo 
 
 lAi 670 
 
 i.it 
 
 403 
 
 s.o 
 
 2S 
 
 1 S9S 700 
 
 ■^.0 
 
 /2.8 
 
 II 
 
 36 670 
 
 76 500 
 
 3 000 
 
 loSO 
 
 1300 
 
 /'« 420 
 
 7 lOS 
 
 //842 
 
 /5 987 
 
 3-»9* 
 
 itooo 
 
 82 030 
 
 3.50 
 
 720 
 
 9.0 
 
 -ts 
 
 2 SS7 9SC 
 
 7.2 
 
 /3.i 
 
 14 
 
 XOS12 
 
 76 SOO 
 
 3 000 
 
 loso 
 
 1 200 
 
 /02 362 
 
 i, 136 
 
 /0 226 
 
 28 580 
 
 44942 
 
 •0 
 
 .5/ t30 
 
 .j.ii 
 
 ;/43 
 
 14.3 
 
 7/ 
 
 + i34 7M 
 
 //.5 
 
 14-1 
 
 I7.S 
 
 12 910 
 
 76 500 
 
 3 000 
 
 1 050 
 
 1200 
 
 94 «0 
 
 S 6B0 
 
 9466 
 
 45 348 
 
 60 4f# 
 
 33000 
 
 I9 0S3 
 
 ns 
 
 »oo 
 
 ii 100 
 
 4.42 
 
 -f06 
 
 5.1 
 
 25 
 
 / &09 6SC 
 
 4.0 
 
 6.S 
 
 7.0 
 
 16 275 
 
 S3S00 
 
 3 300 
 
 /600 
 
 1990 
 
 /05 <« 
 
 6 340 
 
 10 565 
 
 16 0?733002 
 
 •2 
 
 32 -4^0 
 
 S.83 
 
 8" 
 
 /O.I 
 
 60 
 
 3 2/5 iSO 
 
 8.0 
 
 6.« 
 
 /0.S 
 
 8 117 
 
 82500 
 
 3 300 
 
 1600 
 
 /9S0 
 
 974V 
 
 5 850 
 
 9749 
 
 33 IS6 47 755 
 
 >»+ 
 
 20 430 
 
 H.I 
 
 IZfS 
 
 I6.Z 
 
 8/ 
 
 5 HO 660 
 
 /2.9 
 
 7.1 
 
 I4.S 
 
 5 I07 
 
 82 500 
 
 3 300 
 
 1 600 
 
 1920 
 
 94 4S7 
 
 5 470 
 
 944? 
 
 5/407 66525 
 
 44 000 
 
 zs'io-t 
 
 131 
 
 »2 
 
 32 460 
 
 8.83 
 
 A St 
 
 S.7 
 
 29 
 
 / SOS 790 
 
 4.6 
 
 3.9 
 
 6.0 
 
 z 117 
 
 90 000 
 
 34i0 
 
 2 300 
 
 3960 
 
 107721 
 
 6 463 
 
 /0 772 
 
 IS 053 
 
 3SZIS\ 
 
 "■S 
 
 /i no 
 
 /7.8 
 
 9/6 
 
 11.4 
 
 S3 
 
 3 639 780 
 
 9.1 
 
 ■4.0 
 
 9.S 
 
 4 040 
 
 90 000 
 
 3450 
 
 2 200 
 
 3960 
 
 10 3 tSO 
 
 6319 
 
 ;0 3t5 
 
 36 3»S529«| 
 
 ages : thus the increased cost of 66 000 volt insulators 
 above the cost of 33 000 volt insulators is stated as 
 3500 -f- 650 or 540 percent. 
 
 Cost of Other Apf>aratus — The cost of lightning 
 arresters, high-tension circuit breakers and general in- 
 sulation increase with the voltage. - The increased cost 
 of these items, however, may not have sufficient weight 
 to materially influence the selection of the transmission 
 voltage. 
 
 Cost of Buildings — Lower voltage transformers, 
 switching equipment and lightning arresters require 
 less space for insulation. If this apparatus is to be 
 placed indoors, the cost of necessary buildings may be 
 less. The amount of real estate required may also be 
 less in case of the lower voltage. 
 
 • Relative Cost Values- — Table F contains relative 
 co.st values for different transmission voltages. They 
 indicate approximately the variation, at the present 
 time, in cost of the principal material which is affected 
 by a change in transmission voltage. Cost values are 
 very unstable at present but the table will serve in a 
 general way to indicate comparative costs. 
 
 Summary — In deciding upon the transmission 
 voltage, careful and full consideration should be given 
 to the present (or probable future) voltage of any 
 neighboring or adjacent systems. There is an increas- 
 ing tendency to combine generating and transmission 
 systems for purposes of economy, and insurance against 
 breakdown in service. If a possible future consolida- 
 tion is not kept in mind when selecting the transmission 
 voltage, a voltage may be decided upon which would 
 render it impossible to parallel with a neiglAoring sys- 
 tem, except through connecting transformers. In this 
 case the transformers of the two systems would prob- 
 ably not be interchangeal)le for service on either system. 
 
 If the contemplated transmission system is remote 
 from any exi.sting system, a study of the initial and op- 
 erating costs should be made corresponding to various 
 sizes of conductors and to various assumed transmission 
 voltages. A suggested tabulation for such compari- 
 sons is shown in Table G. In this table, it is assumed 
 that 10 000 kv-a. (8000 kw at 80 percent power-factor 
 i:igging), is to be transmitted a distance of ten miles 
 at 60 cycles, three-phase for ten hours, followed by 
 
DETERMINATION OF FREQUENCY AND VOLTAGE 
 
 2500 kv-a. (2000 kw at 80 percent power-factor lag- 
 ging) for 14 hours. Delta spacing is assumed of three 
 feet for the lower two and four feet for the higher two 
 voltages. Raising and lowering transformers will be 
 required of an assumed total capacity of 2.5 X 10 000 
 or 25 000 kv-a. Conductors of hard drawn stranded 
 copper are employed, the resistance of the conductors 
 being taken at a temperature of 25 degrees C. from 
 Table II. 
 
 The cost of the pole or tower line, the righi of 
 way, buildings and real estate for buildings is not in- 
 cluded in this tabulation. Neither is the difference in 
 transformer efficiencies taken into account. The differ- 
 ence in these items will not be sufficient in this case 
 greatly to influence the choice of the transmission volt- 
 age, because all of the voltages compared are relatively 
 low. Because of the large amount of power to be 
 transmitted a comparatively short distance, the approxi- 
 mate rule of 1000 volts per mile for short lines does 
 not hold true for this problem. 
 
 Assuming for the sake of argument that the price 
 values given in this form of tabulation are approxi- 
 mately correct for this problem and that there are no 
 neighboring transmission systems, then the problem re- 
 duces to cost economics. 
 
 Since both the first and operating costs in Table G 
 are higher for 16 500 volts than they are for 22 000 
 volts, it is evident that 16 500 volts is economically too 
 low a voltage. 
 
 ■ In the consideration of 22 000 volts it will be seen 
 that, of the three sizes of conductors, the largest size 
 (300000 circ. mil.) will be the cheaper in the end. 
 Thus, if No. 000 were selected, the first cost would be 
 $16159 less than for 30000 circ. mil conductors, but 
 the operating cost (due to greater loss in transmission) 
 will be approximately $10000 a year more. For a 
 similar reason No. o conductors will be disqualified. 
 
 In the consideration of 33 000 volts, No. 00 con- 
 ductors will be the choice and in the consideration of 
 44000 volts. No. 2 conductors will be the choice. The 
 choice then comes down to the following: — 
 
 Voltage 
 Transmission 
 
 Conductors 
 
 Total 
 
 Cost 
 
 First 
 
 Annual 
 
 Operating 
 
 Cost 
 
 22000 
 33000 
 44000 
 
 300000 circ. mils 
 No. 00 
 No. 2 
 
 $118420 
 105655 
 107 727 
 
 $34934 
 33002 
 35288 
 
 It will thus be seen that a voltage of 33 000 volts 
 and No. 00 conductors are the most economical of those 
 tabulated. The transmission loss will be 5.1 percent, 
 the reactance 6.5 percent and the voltage drop seven 
 percent at full load. The value assigned as the cost per 
 
 kw-hour for power lost in transmission will obviously 
 have great influence in determining the proper economic 
 size of conductors for any given transmission voltage. 
 The cost of the copper will have a relatively greater im- 
 portance on longer lines. As a matter of fact, a larger 
 s'.ze than any of the conductors listed in Table G would 
 be still more economical, under the conditions given. 
 There have been numerous mistakes made in under-esti- 
 mating the ultimate demand for electrical power and 
 consequently adopting too low a transmission voltage. 
 When in doubt the higher voltage will, in the course of 
 time, most likely justify its adoption by reason of fu- 
 ture growth not apparent at the time the choice is made. 
 The design and construction of transformers, cir- 
 cuit breakers, lightning arresters, etc. for a multiplicity 
 of high-tension voltages is expensive. The manufac- 
 turers of such apparatus are endeavoring to standardize 
 transmission voltages for the purpose of minimizing the 
 number of designs of high-tension apparatus. This 
 point could with mutual profit be taken up with the 
 
 TABLE H-COMMON TRANSMISSION VOLTAGES 
 
 Length of Line 
 
 Voltages 
 
 I to 3 miles 
 
 550 or 2200 volts 
 
 3 to 5 miles 
 
 2200 or 6600 volts 
 
 5 to 10 miles 
 
 6600 or 13200 volts 
 
 10 to 15 miles 
 
 13200 or 22000 volts 
 
 15 to 20 miles 
 
 22000 or 33000 volts 
 
 20 to 30 miles 
 
 33000 or 44000 volts 
 
 30 to 50 miles 
 
 44000 or 66000 volts 
 
 SO to 75 miles 
 
 66000 or 88000 volts 
 
 75 to 100 miles 
 
 88000 or 1 10 000 volts 
 
 100 to 150 miles 
 
 no 000 or 132000 volts 
 
 150 to 250 miles 
 
 132000 or 154000 volts 
 
 250 to 350 miles 
 
 154000 or 220000 volts 
 
 manufacturers before any particular voltage is decided 
 upon. 
 
 The amount and cost of power to be transmitted 
 is a very important factor in determining the economic 
 transmission voltage. For average conditions isolated 
 from existing transmission lines the voltages shown in 
 Table H have been quite generally used. For excep- 
 tional cases, exceptional values will be used. For ex- 
 ample if 40000 kv-a. is to be transmitted 20 miles, 
 66000 volts or higher might be used. On the other 
 hand if a very small amount of power is to be trans- 
 mitted, lower voltages would probably be selected. 
 
 At the present time the prospects seem bright for 
 the standardization of the following "normal" system 
 voltages. 
 
 44000 
 
 132000 
 
 66000 
 
 154000 
 
 88000 
 
 ♦187000 
 
 110000 
 
 220 000 
 
 ♦The use of 187000 volts is likely to occur only in case it 
 i? found necessary to have a voltage between 154 000 and 220 000 
 volts. 
 
CHAPTER VII 
 PERFORMANCE OF SHORT TRANSMISSION LINES 
 
 (EFFECT OF CAPACITANCE NOT TAKEN INTO ACCOUNT) 
 
 THE PROBLEMS which come under the general 
 heading of short transmission lines are those in 
 which the capacitance of the circuit is so small 
 that its effect upon the performance of the circuit may, 
 for all practical purposes, be ignored. The effect of ca- 
 pacitance is to produce a current in leading quadrature 
 with the voltage, usually designated as charging current. 
 This leading component of current in the conductor 
 does not appear in the load current at the receiving end 
 of the circuit. It is zero at the receiving end of the 
 circuit but increases at nearly a uniform rate as the 
 sending end of the circuit is approached, at which point 
 it ordinarily becomes a maximum. 
 
 The effect of this charging current flowing through 
 the inductance of the circuit is to increase the receiv- 
 ing-end voltage and therefore to decrease the voltage 
 drop under load. Since the charging current is 2.4 
 times greater for a frequency of 60 cycles than it is for a 
 frequency of 25 cycles, its effect upon the voltage regu- 
 lation will be considerably greater at 60 cycles than at 
 25 cycles. The effect of charging current upon the 
 voltage regulation will also increase as the distance of 
 transmission is increased. 
 
 If the circuit were without capacitance, there would 
 be no charging current and consequently the mathe- 
 matical and the two graphical solutions (impedance 
 methods) which follow under the general heading of 
 "short transmission lines" would all produce accurate 
 results. All circuits, however, have some capacitance, 
 and as the length or the frequency of the circuit in- 
 creases, these three methods will therefore yield re- 
 sults of increasing inaccuracy. Some engineers con- 
 sider these impedance methods sufficiently accurate for 
 circuits 20 to 30 miles long while others use them for 
 still longer circuits. To act as a guide. Table J indicates 
 the error in the supply voltage as determined by these 
 impedance methods, for circuits of different lengths 
 corresponding to both 25 and 60 cycle frequencies. 
 These three impedance methods produce practically the 
 same results, and the sending end voltage, as determined 
 by any of these methods, is always slightly high. In 
 other words the effect of the charging current is to 
 reduce the voltage necessary at the sending end, for 
 maintaining a certain voltage at the receiving end of 
 the circuit. The error referred to below for the three 
 methods is expressed in percentage of the receiving end 
 voltage. Thus, for a 30 mile, 25 cycle circuit, the error 
 is 0.04 percent, and for a 30 mile, 60 cycle circuit the 
 error is 0.2 percent. If an error of 0.5 percent is con- 
 
 sidered permissible, then the Dwight or the Mershon 
 Chart methods, or the corresponding mathematical so- 
 lution, may be used for 25 cycle circuits up to approxi- 
 mately 125 miles, and for 60 cycles circuits up to ap- 
 proximately 50 miles. Of course these impedance 
 methods may be used for still longer circuits by making 
 proper allowance to compensate for the fundamental 
 error. 
 
 DIAGRAM ILLUSTRATING A SHORT TRANSMISSION CIRCUIT 
 
 Fig. 16 illustrates the relation between the various 
 elements in short transmission circuits, when the effect 
 of capacitance and leakage is not taken into account. 
 The current flowing in such a circuit meets two op- 
 posing e.m.f's. ; i.e. of resistance in phase with the cur- 
 rent and reactance in lagging quadrature with the cur- 
 rent. 
 
 The upper part of Fig. 16 illustrates such a circuit 
 schematically and the lower part vector ially. The volt- 
 
 TABLE J 
 
 Length of 
 Circuit (Miles) 
 
 Error in Percentage of 
 Receiver Voltage 
 
 cycles 
 
 60 
 cycles . 
 
 20 
 
 30 
 
 50 
 
 100 
 
 200 
 
 300 
 
 -H>.02 
 
 -f-0.04 
 
 +ai 
 
 -f04 
 
 -fM 
 +3-3 
 
 -fa 10 
 -fas 
 -1-0.5 
 +1.9 
 -fao 
 -fi8.o 
 
 age component required at the sending end to overcome 
 the resistance IR of the circuit is indicated in the vector 
 diagram by a short line parallel with the base line /, 
 representing the phase of the current. These lines are 
 drawn parallel, since the resistance voltage drop is in 
 phase with the current. The voltage component re- 
 quired at the sending end to overcome the reactance IX 
 of the circuit is indicated by a line in quadrature or at 
 right angles, to the phase of the current The reactance 
 is in quadrature with the current for the reason that the 
 rate of change in the magnetic field (consequently the 
 e.m.f. of self-induction or reactance) surrounding the 
 conductor is greatest when the cufrent is passing 
 through zero. The hypotenuse IZ of this small right 
 angle impedance triangle represents the impedance volt- 
 age of the circuit. It represents the direction and value 
 of the resulting voltage necessary to overcome the com- 
 bined effect of the resistance and the reactance of the 
 circuit. 
 The relative values and phases of the receiving and 
 
50 
 
 PERFORMANCE OF SHORT TRANSMISSION LINES 
 
 sending end voltages, and their phase relations with the 
 current /, are also indicated on the vector diagram. 
 This diagram is plotted for a receiving end load based 
 upon 80 percent power- factor lagging. E, represents the 
 value of the voltage required at the sending end of the 
 circuit to maintain the voltage £r at the receiving end, 
 when the impedance of the circuit is IZ and the receiv- 
 ing end power-factor is 80 percent lagging. The phase 
 angle *, indicates the amount by which the current lags 
 behind the voltage at the sending end; cos «, being the 
 power-factor of the load as measured at the sending 
 end. Likewise cos «, is the power-factor of the load at 
 the receiving end. 
 
 TAPS TAKEN OFF CIRCUIT 
 
 Usually the main transmission circuit is tapped and 
 power taken off at one or more points along the circuit. 
 The performance of such a circuit must be calculated 
 by steps thus: — Assume a circuit 200 miles long with 
 10 000 kw taken off at the middle and 10 000 kw at the 
 receiving end. From the conditions known or assumed 
 at the receiving end, calculate the corresponding send- 
 
 R f X 
 
 Er 
 
 NEUTRAL _£_ 
 
 P. F. OF 
 LOAD=Co> e. 
 
 EQUIVALENT TRANSMISSION CIRCUIT TO NEUTRAL 
 
 * ErCoie, ^^ 
 
 VECTOR DIAGRAM OF TRANSMISSION CIRCUIT 
 
 FIG. 16 — DIAGRAMS FOR SHORT TRANSMISSION LINES 
 
 Impedance method, capacitance effect not taken into account 
 
 ing end conditions, that is the voltage, power and 
 power-factor at the substation in the middle of the cir- 
 cuit. To the calculated value of the actual power in 
 kilowatts add the losses at the substation in the middle 
 of the circuit. Any leading or lagging component in 
 the substation load current must also be added alge- 
 braically, in order to determine the power-factor at the 
 sending side of the substation. This will then be the 
 receiving end conditions at the substation in the middle 
 of the circuit, from which the corresponding conditions 
 at the sending end of the circuit may be calculated. If 
 the sending end conditions are fixed, and the receiving 
 end conditions are to be determined, the substation 
 losses will in such case be subtracted in place of added. 
 
 CABLE AND AERIAL LINES IN SERIES COMPOSITE LINES 
 
 In some cases it is necessary to place part of a 
 transmission circuit underground, and in other cases it 
 may be desirable to use two or more sizes of conductors 
 in series. The result will be that the circuit constants 
 will be different for the various sections. If the effect 
 of capacitance be neglected, the combined circuit may 
 
 be treated as a single circuit having a certain total re- 
 sistance R and a total reactance X. 
 
 PROBLEMS 
 
 Later a table will be presented listing a large num- 
 ber of transmission circuits from 20 to 500 miles long, 
 at both 25 and 60 cycles operating at from 10 000 to 
 200 000 volts. These problems are numbered from i to 
 64. When a reference is made in the following to some 
 problem number it will refer to one of this list of prob- 
 lems. 
 
 SYMBOLS 
 
 The symbols which will be employed in the follow- 
 ing treatment are given below: — 
 
 FOR LOAD CONDITIONS 
 
 Kv-Or = (total) at receiving end. 
 Kv-ciro = (one conductor to neutral) at receiving end. 
 Kv-a, = (total) at sending end. 
 Kv-OsB = (one conductor to neutral) at sending end. 
 Kwr = Kw (total) at receiving end. 
 Kwra = Kw (one conductor to neutral) at receiving end. 
 Kw, = Kw (total) at sending end. 
 Kw,n = Kw (one conductor to neutral) at sending end. 
 E, = Voltage between conductors at receiving end. 
 £rii = Voltage from conductors to neutral at receiving 
 
 end. 
 E, := Voltage between conductors at sending end. 
 £»« = Voltage from conductors to neutral at sending 
 end. 
 I, = Current in amperes per conductor at receiving 
 
 end. 
 /. = Current in amperes per conductor at sending end. 
 Cos 6, = Power-factor at receiving end. 
 Cos 6, = Power-factor at sending end. 
 
 FOR ZERO LOAD CONDITIONS 
 
 The symbols corresponding to zero load conditions 
 are as indicated above for load conditions with the ad- 
 dition of a sub zero. 
 
 THE FUNDAMENTAL OR LINEAR CONSTANTS 
 
 The fundamental, or "linear constants" of the cir- 
 cuit for each conductor per unit length are represented 
 as follows: — 
 
 r := Linear resistance in ohms per conductor mile (taken 
 
 from Table II) 
 X = Linear reactance in ohms per conductor mile (taken 
 
 from Table IV or V) 
 b = Linear capacitance susceptance to neutral in mhos per 
 
 conductor mile (taken from Table IX or X> 
 g = Linear leakage conductance to neutral in mhos per 
 conductor mile. (This represents the direct escape 
 of active power through the air between conductors 
 and of active power leakage over the insulators. 
 These losses must be estimated for conditions 
 similar to these of the circuit under consideration. 
 For all lines except those of great length and high 
 voltage it is common practice to disregard the 
 effects of leakage or corona loss and to take g as 
 
 equal to zero. 
 
 z ^ Linear impedance = ]/ r* -|- x* 
 y = Linear admittance = \/ g' -j- b' 
 If the length of each conductor of the circuit in 
 unit length is designated as / we have 
 
 rl = Total resistance in ohms per conductor = R 
 xl := Total reactance in ohms per conductor = X 
 hi = Total susceptance in mhos per conductor to neutral 
 
 D 
 
 gl = Total conductance in mhos per conductor to neutral 
 = G 
 
PERFORMANCE OF SHORT TRANSMISSION LINES 
 
 51 
 
 then, 
 
 Z = y R' + X' ohms 
 and, Y = \/G' + B' mhos 
 
 IR = Voltage necessary to overcome the resistance. 
 IX = Voltage necessary to overcome the reactance. 
 IZ ^ Voltage necessary to overcome the impedance. 
 
 METHODS FOR DETERMINING THE CONSTANTS OF THE 
 CIRCUIT 
 
 Several different methods for determining the fim- 
 damental constants of the circuit are in use. These 
 methods are illustrated below. 
 
 Problem — Find the resistance volts IR and the re- 
 actance volts IX in percent of delivered volts £, for the 
 following conditions: — lOO kw active power to be de- 
 livered at looo volts, three-phase, 60 cycles, over three 
 No. 0000 stranded, hard drawn, copper conductors, cir- 
 cuit one mile long, with a symmetrical delta arrange- 
 ment of conductors, two foot spacing, the temperature 
 being taken as 25 degrees C. 
 
 Resistance of one mile of single conductor = 0.277 
 
 ohm (from Table II) 
 
 Reactance of one mile of single conductor = 0-595 
 
 ohm (from Table V) 
 
 Method No. i — When three-phase circuits first 
 came into use, it was customary (and correct), in de- 
 termining the loss and voltage regulation, to consider 
 them equivalent to two single-phase circuits, each single- 
 phase circuit transmitting one-half the power of the 
 three-phase system. This practice is still followed by 
 some engineers; thus: — 
 
 50000 
 
 1000 
 
 50 amp. per conductor for each single-phase cir- 
 cuit. 
 
 0.277 X 2 X SO ^ . ■ , J- 
 '-'-^ — £i-^_ ^ jQQ _ 2.77 % resistance volts drop of 
 
 '"^^^ single-phase circuit. 
 
 0-595 X 2 X 50 ^ jQQ _ 5 55% reactance volts drop of 
 
 '°'-'° single-phase circuit. 
 
 Method No. z consists of treating the case as a 
 straight three-phase problem. Thus: 
 
 —, • = ';7-7'? amperes per conductor of three- 
 
 1000 X 1-732 phase circuit. 
 
 0.277 X 1-732 X 57-73 y^ ^^ ^ ^.^.^^^ resistance volts 
 i°oo drop of three- 
 
 phase circuit. 
 0-595 X 1-732 X 57-73 ... y ^^ _ % reactance volts 
 
 1000 ^ ='"" drop of three- 
 
 phase circuit. 
 
 Method No. 3 consists in assuming one-third the 
 total power transmitted over one conductor with neutral 
 or ground return (resistance and reactance of return 
 being taken as zero). Such an equivalent circuit is 
 shown by diagram in the upper part of Fig. 16. Thus 
 the circuit constants for the above problem would be 
 determined as follows : — 
 
 100 000 
 Watts per phase = — - — — 33 333 waits. 
 
 Volts to neutral = 1000 X O-S774 or S77A volts. 
 
 33333. _ ^^^^ amperes per cotiducior; {smme as for 
 
 577-4 method No. 2) 
 
 0.277 X 57-74 . ^ 100 — 2.77% resistance volts drop of 
 
 577-4 three-phase circuit. 
 
 0-595 X 57-74 y ,00 _ 5.95% reactance volts drop of 
 577-4 three-phase circuit. 
 
 It will be seen that all three methods produce the 
 same results. Method No. j seems the most readily 
 adaptable to various kinds of transmission systems and 
 will be used exclusively in the treatment of the problems 
 which will follow. 
 
 APPLICATION OF THE TABLES 
 
 Numerous tables of constants, charts, etc., have 
 been presented, and a few more will follow. Chart II 
 plainly indicates the application of these tables, etc. to 
 the calculation of transmission circuits and the sequence 
 in which they should be consulted. 
 
 GRAPHICAL vs. MATHEMATICAL SOLUTIONS 
 
 At the time of the design of a transmission circtiit 
 the actual maximum load or power- factor of the load 
 that the circuit will be called upon to transmit is sel- 
 dom known. An unforseen development leading to an 
 increased demand for electrical energy may result in a 
 greatly increased load to be transmitted. The acttial 
 length of a circuit (especially when located in a hilly 
 or rolling country) is never known with mathematical 
 accuracy. Moreover, the actual resistance of the con- 
 ductors varies to a large extent with temperature varia- 
 tions along the circuit. 
 
 When it is considered that there are so many in- 
 determinate variables which vitally affect the per- 
 formance of a transmission circuit, it would seem that 
 a comparatively long and highly mathematical solution 
 for determining the exact performance, necessarily 
 based upon rigid assumptions, is hardly justified. In 
 many cases the economic loss in transmission will de- 
 termine the size of conductors and, if the circuit is very 
 long, synchronous machinery is likely to be employed 
 for controlling the voltage. 
 
 Mathematical solutions have one very important 
 virtue, in that they provide an entirely different but 
 parallel route in the solution of such problems, and 
 therefore are valuable as a check against serious errors 
 in the results obtained by the more simple graphical so- 
 lutions. 
 
 In the following treatment, simple but highly ac- 
 curate graphical solutions will be first presented, for 
 determining the performance not only of short trans- 
 mission lines, but also for long lines. For short lines 
 the Dwight and the Mershon charts will be used. For 
 long lines, where the effect of capacitance must be ac- 
 curately accounted for, the Wilkinson Charts, supple- 
 mented with vector diagrams will be used. These three 
 forms of graphical solutions will, when correctly ap- 
 plied to any power transmission problem, produce re- 
 sults in which the error will be much less than that due 
 to irregularities in line construction and inaccurate as- 
 sumptions of circuit constants. These three graphical 
 solutions will in each case be followed by mathematical 
 solutions. In the case of short lines the usual formulas 
 employing trigonometric functions will be employed, 
 and in the case of long lines the convergent series, and 
 two different forms of hyperbolic solutions will be em- 
 ployed. 
 
52 
 
 PERFORMANCE OF SHORT TRANSMISSION LINES 
 
 GRAPHICAL SOLUTION 
 
 When the receiving end load conditions, that is, 
 the voltage, the load and the power-factor are known, 
 the IR volts required to overcome the resistance and 
 the IX volts required to overcome the reactance of the 
 circuit, may be readily calculated. 
 
 On a piece of plain paper or cross-section paper 
 divided into tenths, a vector diagram of the current 
 and of the various voltage drops of the circuit may be 
 laid out to a convenient scale. Whichever kind of 
 paper is used, the procedure will be as in the following 
 example. 
 
 Single-Phase Problem — Find the voltage at the 
 sending end of a single-phase circuit i6 miles long, con- 
 sisting of two stranded, hard drawn No. ocxx> copper 
 conductors spaced three feet apart. Temperatures taken 
 as 25 degrees C. Load conditions at receiving end 
 assumed as 4000 kv-a (3200 kw at 80 percent power- 
 factor lagging) 20 000 volts, single-phase, 60 cycles. 
 4000 
 
 Kv-ar„ =- 
 
 Era 
 
 2 
 20 000 
 
 2000 kv-a to neutral. 
 
 10 000 volts to neutral. 
 
 2 000 000 
 Ir = — — ^ 200 amperes per conductor. 
 
 The fundamental constants per conductor are: — 
 i? = 16 X 0.277 'from Table II) = 4.432 ohms 
 X = 16 X 0.644 (from Table V) ^ 10.304 ohms 
 and IR = 200 X 4.432 ^ 886 volts resistance drop 
 886 
 = ,„,,^ X i«o = 8.86 percent 
 10000 "^ 
 
 IX = 200 X 10.304 = 2061 volts reactance drop 
 2061 ^ , 
 
 = ,„„^ X 100 — 20.61 percent 
 loooo ' 
 
 Having determined the above values a vector dia- 
 gram may be made as follows: — 
 
 Draw an arc quadrant having a radius of 10 000 
 (the receiving end voltage to neutral) to some con- 
 venient scale, as shown in Fig. 17. The radius which 
 represents the base, or horizontal line will be assumed 
 as representing the phase of the current at the receiv- 
 ing end of the circuit. Divide this base line into ten 
 equal parts. These ten divisions will then correspond 
 to loads of corresponding power-factors. Since a load 
 has been assumed having a power-factor of 80 percent 
 lagging, draw a vertical line from the 0.8 division on 
 the base line, until it intersects the arc of the circle. 
 From this point of intersection draw a line to the right 
 and parallel with the base line. To the same scale as 
 that plotted for the receiver voltage (10 000) measure 
 off to the right 886 volts to D. This is the voltage 
 which, as determined above is required to overcome the 
 resistance of one conductor of the circuit. It is some- 
 times stated as the voltage consumed by the line re- 
 sistance. It will be noted that this voltage drop is in 
 phase with the current at the receiving end. From this 
 point lay off vertically, and to the same scale, 2061 volts 
 which is, as determined above, the volts necessary to 
 overcome the reactance of one conductor of the circuit. 
 This is sometimes stated as the voltage consumed by the 
 line reactance. Connect this last point by a straight 
 
 CHART II.— APPLICATION OF TABLES TO 
 SHORT TRANSMISSION LINES 
 
 (EFFECT OF CAPACITANCE NOT TAKEN INTO AC- 
 COUNT) OVER HEAD BARE CONDUCTORS 
 
 Starting with the kv-a., voltage and power-factor at 
 the receiving end known. 
 
 QUICK ESTIMATING TABLES XII TO XXI INC. 
 
 From the quick estimating table corresponding to the 
 voltage to be delivered, determine the size of the con- 
 ductors corresponding to the permissible transmission loss. 
 
 HEATING LIMITATION— TABLE XXIII 
 
 If the distance of transmission is short and the amount 
 of power transmitted very large there is a possibility of 
 overheating the conductors — to guard against such over- 
 heating the carrying capacity of the conductors contem- 
 plated should be checked by this table. 
 
 CORONA LIMITATION— TABLE XXII 
 
 If the transmission is at 30000 volts, or higher, this 
 table should be consulted to avoid the employment of con- 
 ductors having diameters so small as to result in excessive 
 corona loss. 
 
 RESISTANCE— TABLES I AND II 
 From one of these tables obtain the resistance per 
 unit length of single conductor corresponding to the maxi- 
 mum operating temperature — calculate the total resistance 
 for one conductor of the circuit — if the conductor is large 
 (250000 circ. mils or more) the increase in resistance due 
 to skin effect should be added. 
 
 PR TRANSMISSION LOSS 
 
 Calculate the PR loss of one conductor by multiplying 
 its total resistance by the square of the current — to obtain 
 the total loss multiply this result by the number of con- 
 ductors of the circuit. 
 
 REACTANCE— TABLES IV AND V 
 
 From one of these tables obtain the reactance per unit 
 length of single conductor. Calculate the total reactance 
 for one conductor of the circuit. If the reactance is ex- 
 cessive (20 to 30 percent reactance volts will in many cases 
 be considered excessive) consult Table VI or VII. Hav- 
 ing decided upon the maximum permissible reactance the 
 corresponding resistance may be found by dividing this 
 reactance by the ratio value in Table VI or VII. When 
 the reactance is excessive, it may be reduced by instaljing 
 two or more circuits and connecting them in parallel, or 
 by the employment of three conductor cables. Using 
 larger conductors will not materially reduce the reactance. 
 The substitution of a higher transmission voltage, with its 
 correspondingly less current, will also result in less react- 
 ance. 
 
 GRAPHICAL SOLUTION 
 A simple graphical solution, as described in the text, 
 may be made by which the kv-a, the voltage and the power- 
 factor at the sending end of the circuit may be determined 
 graphically. Or the voltage at the sending end may be 
 determined graphically by the use of either the Dwight or 
 the Mershon chart. With the Mershon chart the power- 
 factor at the sending end may be read directly from the 
 chart. 
 
 MATHEMATICAL SOLUTION 
 As a precaution against errors the results obtained 
 graphically should be checked by a mathematical solution, 
 in cases where accuracy is essential. 
 
PERFORMANCE OF SHORT TRANSMISSION LINES 
 
 53 
 
 line with the center E of the arc. The length of this 
 line ES represents the voltage to neutral at the sending 
 end which, for this problem, is 1 1 998 volts. The dis- 
 tance this line extends beyond the arc represents the 
 drop in voltage for one conductor of the circuit. The 
 
 voltage drop for this problem is 
 
 1998 
 
 X 100 = 19.98 
 
 10000 
 percent of the receiving end voltage. 
 
 The phase difference between the current and the 
 voltage at the receiver end is e^ = 36° 52'. This is the 
 angle whose cosine is 0.8 corresponding to a power-fac- 
 tor at the receiving end of 80 percent. Likewise the 
 phase difference between the receiving end current and 
 the sending end voltage is *, ^ 42° 13' corresponding 
 to a power-factor at the supply end of 74.06 percent. 
 The difference in these two phase angles (5° 21') repre- 
 sents the difference in the phase of the voltages at the 
 sending and receiving ends of the circuit. The power- 
 
 I 
 
 I9»a VOLTS CONDUCTOR DROP 
 
 e206l VOLTS REACTANCE DROP 
 2S44 VOLTS IMPEDANCE DROP 
 
 eee volts resistance drop 
 
 POSITION FOR load 
 OF 80% UAQGING. 
 
 /i 
 
 /* 
 
 /.?" 
 
 $ 
 
 ey 
 
 0.a 0.4 06 0.6 0.7 0.8 
 POWER-FACTOR OF LOAD 
 
 > CURRENT 
 
 FIG. 17 — GRAPHICAL SOLUTION FOR A SHORT TRANSMISSION LINK 
 
 Capacitance effect not taken into account. 
 
 factor at the sending end of the circuit may be readily 
 obtained by dropping a vertical line down from the point 
 where the line representing the sending end voltage £5" 
 intersects the arc of the circle, to the base line 
 representing the phase of the receiving end current. 
 Such a line will correspond to a power-factor of 74.06 
 percent. This assumption that the vector representing 
 the direction of the receiving end current also repre- 
 sents the direction of the sending end current is upon 
 the basis that the circuit is without capacitance. It, 
 therefore, is permissible only with short lines. 
 
 In Fig. 17 the location of the impedance triangle 
 is also indicated (by broken lines) in positions corre- 
 sponding to a receiving end load of 100 percent power- 
 factor ; and also for a receiving end load of zero lagging 
 power- factor. It is interesting to note that in the case 
 of 100 percent power-factor the resistance drop (at 
 right angle to the arc) has a maximum effect upon the 
 voltage drop; whereas the reactance drop (nearly paral- 
 lel with the arc) has a minimum effect upon the volt- 
 
 age drop. At zero lagging power-factor load just the 
 reverse is true; namely the resistance drop is nearly 
 parallel with the arc and causes a minimum voltage 
 drop, while the reactance is at right angles and produces 
 a maximum effect upon the voltage drop. 
 
 VOLTAGE AT SENDING END AND LOAD AT RECEIVING 
 END FIXED 
 
 In cases of feeders to be tapped into main trans- 
 mission circuits, the voltage at the sending end is usually 
 fi.xed. It may be desired to determine what the volt- 
 age will be at the receiving end corresponding to a given 
 load. This may be obtained graphically as follows: — 
 
 Draw a horizontal line which will be assumed to 
 represent the phase of the current. (Fig. 17) Since 
 the power-factor of the load at the receiving end is 
 known, the angle whose cosine corresponds may be ob- 
 tained from Table K. This angle represents the 
 phase relation between the current and the voltage af 
 the receiving end of the circuit. For the problem illus- 
 trated by Fig. 17 this angle is 36° 52', corresponding to 
 a power-factor of 80 percent. Having determined this 
 angle, draw a second radial line intersecting the current 
 vector at the angle corresponding to the receiving end 
 load power-factor. This second line will then repre- 
 sent the direction of the voltage at the receiving end 
 of the circuit. If the load power- factor is lagging, this 
 line will be in the forward direction, and if the load 
 power-factor is leading it will be in the backward di- 
 rection from the current vector. Now with the inter- 
 section of the current and voltage vectors as a center, 
 draw an arc of a circle to some suitable scale, repre- 
 senting the voltage at the sending end. Calculate the 
 voltage necessary to overcome the resistance, and also 
 that necessary to overcome the reactance of the circuit 
 
 Draw a right angle impedance triangle to the same 
 scale, using the resistance volts as a base. Cut out the 
 impedance triangle to its exact size. Keeping the base 
 of the triangle (resistance voltage) in a horizontal posi- 
 tion (parallel with the current vector) move the triangle 
 over the diagram in such a manner that its apex follows 
 the arc of the circle representing the numerical value 
 of the voltage at the sending end. Move the triangle 
 up or down until a position is found where it makes 
 connection with the vector representing the voltage at 
 the receiving end. This is then the correct position for 
 the impedance triangle, and the receiving end voltage 
 may be scaled off. 
 
 GRAPHICAL SOLUTION BY THE MERSHON CHART 
 
 The above graphical solution is that employed in 
 the well known chart which Mr. Ralph D. Mershon 
 early presented to the electrical profession, and which 
 is reproduced as Chart III. The Mershon Chart is 
 simply a diagram on cross-section paper with vertical 
 and horizontal subdivisions each representing one per- 
 cent of receiving end voltage. On this chart a number 
 of concentric arcs are drawn, representing voltage drops 
 up to 40 percent. After the reactance and the resist- 
 ance volts have been calculated and expressed in per- 
 
54 
 
 PERFORMANCE OF SHORT TRANSMISSION LINES 
 
 CHART lll-MERSHON CHART 
 
 DIRECTIONS FOR USING CHART 
 
 BY MEANS OF THE TABLES CALCULATE THE RESISTANCE VOLTS AND THE REACTANCE. 
 ^VOLTS IN THE LINE, AND FIND WHAT PER CENT EACH IS OF THE E.M.F. DELIVERED 
 ■^AT THE END OF THE LINE. STARTING FROM THE POINT ON THE CHART WHERE 
 ^THE VERTICAL LINE CORRESPONDING WITH POWER FACTOR OF THE LOAD 
 JNTERSECTS THE SMALLEST CIRCLE. LAY OFF IN PER CENT THE RESIS- 
 iJ-ANCE E.M.F. HORIZONTALLY AND TO THE RIGHT^ FROM THE POINT 
 yTHUS OBTAINED LAY OFF UPWARD IN PER CENT THE REACTANCE- 
 THE CIRCLE ON WHICH THE LAST POINT FALLS GIVES 
 DROP IN PER CENT OF THE E.M.F. DELIVERED AT THE 
 THE LINE. 
 
 EXAMPLE 
 
 VTHUS A CIRCUIT HAVING 5 PER CENT RESISTANCE 
 
 D to PER CENT REACTANCE VOLTS HAS WITH 
 
 LOAD OF lOO PER CENT POWER FACTOR 5^ 
 
 PER CENT VOLTAGE DROP OR WITH A LOAD 
 
 OF ao PER CENT POWER FACTOR 10 PER 
 
 CENT VOLTAGE DROP. 
 
 RESISTANCE DROP_ 
 
 .6 .7 .8 
 
 LOAD POWER FACTORS 
 
 10 20 30 
 
 DROP IN PER CENT OF E.M.F. DELIVERED 
 
PERFORMANCE OF SHORT TRANSMISSION LINES 
 
 S5 
 
 cent of Ef the impedance triangle is traced upon the 
 chart and the voltage drop in percentage of £, is read 
 directly as indicated by the directions. All values on the 
 chart are expressed in percent of the receiving end volt- 
 age. 
 
 Single-Phase Problem — Taking the resistance volt- 
 age as 8.86 percent and the reactance voltages 20.61 per- 
 cent of the receiving end voltage, for the above single- 
 phase problem, (Fig. 17) and tracing these values upon 
 the Mershon Chart for a receiving end load of 80 per- 
 cent power-factor lagging, the voltage drop is de- 
 termined as 19.9 percent. The calculated value being 
 19.98 percent, the error by the chart is seen to be 
 negligible. 
 
 WHEN THE SENDING END CONDITIONS ARE FIXED 
 
 When the conditions at the sending end are fixed 
 and those at the receiving end are to be determined, 
 the solving of the problem by the Mershon Chart is 
 more complicated. In such cases, it is usual to estimate 
 what the probable receiving end condition will be. 
 From these estimated receiving end conditions, deter- 
 mine by the chart the corresponding sending end condi- 
 tions. If the conditions as determined by this assump- 
 tion are materially 'different from the known conditions, 
 another assumption should be made. The correspond- 
 ing sending end conditions should then be checked with 
 the known conditions. Several such trials will usually 
 be necessary to solve such problems. 
 
 GRAPHICAL SOLUTION BY THE DWIGHT CHART 
 
 Mr. H. B. D wight has worked up a straight line 
 chart, shown as Chart IV, in which the resistance and 
 the reactance of the circuit have been taken into ac- 
 count through the medium of spacing lines marked for 
 various sizes of conductors.* The use of this chart 
 does not, therefore, require the calculation of the re- 
 sistance and reactance or the use of tables of such con- 
 stants. The Dwight Chart is also constructed so as to 
 be applicable to loads of leading as well as to loads of 
 lagging power-factors, whereas the Mershon chart, as 
 generally constructed, is applicable to loads of lagging 
 power-factor only. However the Mershon Chart can 
 be made applicable for the solving of problems of lead- 
 ing as well as lagging power-factor loads by extending 
 it through the lower right-hand quadrant. The appli- 
 cation of synchronous condensers frequently gives rise 
 to loads of leading power-factor. The Dwight Chart 
 is well adapted to the solution of such circuits. Still 
 another feature of this chart is that formulas are given 
 which take capacitance effect into account with suffi- 
 cient accuracy for circuits with a length up to approxi- 
 mately 100 miles. 
 
 Single-Phase Problem — Find the voltage at the 
 sending end of a single-phase circuit 16 miles long, con- 
 sisting of two stranded, hard-drawn, No. 0000 copper 
 conductors, spaced three feet apart. Temperature 
 
 *The basis of the construction of this chart 13 described in 
 the Journal for July, 1915, p. 306. * 
 
 taken as 25 degrees C. Load condition at receiving end 
 assumed as 4000 kv-a (3200 kw at 80 percent power- 
 factor lagging) 20 000 volts single-phase, 60 cycles. 
 
 From Table II the resistance of No. 0000 stranded, 
 hard-drawn, copper conductors at 25 degrees C. is 
 fotmd to be 0.277 ohm per wire per mile. Lay a 
 straight edge across the Dwight Chart from the resist- 
 ance value per mile 0.277 (^s read on the lower half 
 of the vertical line to the extreme right) to the spacing 
 of three feet for copper conductors and 60 cycles at the 
 extreme left. Along this straight edge read factor 
 V = 0.62, corresponding to a lagging power-factor of 
 80 percent. This factor V is equivalent to the change 
 in receiving end voltage per total ampere per mile of 
 circuit, due to the line impedance. 
 
 It will be noted that opposite the resistance values 
 (extreme right vertical line) is placed the correspond- 
 ing sizes of copper and aluminum conductors on the 
 basis of a temperature of 20 degrees C. If the tempera- 
 ture is assumed to be 20 degrees C. it will not be neces- 
 sary to consult a table of resistance values. In such a 
 case, the straight edge would simply be placed over the 
 division of the vertical resistance line corresponding to 
 the size and material of conductors. Marking a resist- 
 ance value on this vertical line makes the chart adapt- 
 able to resistance values corresponding to conductors 
 at any temperature. Had the power factor been lead- 
 ing, in place of lagging, the corresponding resistance 
 point would have been located on the upper half of the 
 vertical resistance line. 
 
 Continuing following the directions on the chart for 
 short lines, we obtain the following. Since the circuit 
 is single-phase, use 2 V =^ 1.24 
 
 ,, ,, , ^ .. , X c lOOOOO X 4000 X 16 X 1.24 
 
 Voltage drop in percent of E, = 2ooocf 
 
 == 19.84 percent 
 
 The voltage drop, as calculated mathematically, is 
 19.98 percent representing an error of 0.14 percent by 
 the chart. 
 
 Three-Phase Problem (No. jj) — Find the voltage 
 at the sending end of a three-phase circuit, 20 miles 
 long, consisting of three No. ocoo stranded, hard-drawn, 
 copper conductors, spaced three feet apart in a delta 
 arrangement. Temperature taken as 25 degrees C. 
 Load conditions at receiving end assumed as 1300 kv-a 
 (1040 kw at 80 percent power-factor lagging) 10 000 
 volts, three-phase, 60 cycles. 
 
 From Table II, the resistance per wire per mile is 
 
 again found to be 0.277 ohm and since the spacing and 
 
 frequency are both the same as in the case of the above 
 
 single-phase problem, we again obtain V = 0.62. The 
 
 voltage drop in percent of £, is therefore 
 
 100 000 X 1300 X 20 X 0.62 
 
 , ^1 = 16.12 percent 
 
 10 000 
 
 The voltage drop as calculated mathematically is 
 
 16.16 percent, representing an error of 0.04 percent. 
 
 CAPACITANCE 
 
 In long circuits the effect of capitance is to de- 
 crease the voltage drop, or increase the voltage rise, as 
 
56 
 
 PERFORMANCE OF SHORT TRANSMISSION LINES 
 
 CHART-IV DWIGHT CHART 
 
 FOR DETERMINING THE VOLTAGE REGULATION OF TRANSMISSION CIRCUITS CONTAINING CAPACITANCE 
 
 CORRECT WITHIN APPROXIMATELY ONE HALF OF ONE PER CENT ^ ^0^^ o%^ 
 
 OF LINE VOLTAGE FOR LINES UP TO 100 MILES LONG AND FOR ,<^V-^ ® *" 
 
 LOADS GIVING NOT MORE THAN 15 PER CENT 
 RESISTANCE OR REACTANCE VOLTS , ->« 
 
 ^^NO. 3 ALUMINUM 
 7 1.7 -p- NO. 6 COPPER (J- 
 
 o 
 ^o 
 
 ub 
 
 ll'*J.4- NO. 2 ALUMINUM g i_ 
 ^NO. 4 COPPER < 
 
 q: 
 
 LJli. 
 
 u 
 
 Ot3 
 
 I ALUMINUM 
 NO. 3 COPPER 
 
 zz 
 
 ^NO. ALUMINUM Zi Ld 
 "^NO. 2 COPPER V) _l 
 
 u 
 q: 
 
 NO. 00 ALUMINUM 
 NO. I COPPER 
 
 NO. 000 ALUMINUM 
 Na COPPER 
 NO. 0000 ALUMINUM 
 NO. 00 COPPER 
 250.000 CM ALUMINUM 
 NO. 000 COPPER 
 ■300,000 CM ALUMINUM 
 NO. 0000 COPPER 
 ■350.000 CM ALUMINUM 
 '250.000 CM COPPER 
 
 CO 
 
 IT 
 
 o 
 
 Z 
 
 ^ O < 3 
 "S it! 3 
 
 5 o 
 o 
 
 ID 
 
 DIRECTIONS —Q, 
 
 LAY A STRAIGHT EDGE ACROSS T^i 
 THE CHART FROM THE SPACING POINT 
 TO THE RESISTANCE POINT. AND READ V 
 
 SHORT LI N ES 
 VOLTAGE DROP IN PER CENT OF E 
 
 100.000 KVA X LV 
 
 250.000 CM COPPER 
 1^350000 CM ALUMINUM 
 "^O. 0000 COPPER 
 ^ 300,000 CM ALUMINUM 
 ■ —NO. 000 COPPER 
 ^250,000 CM ALUMINUM 
 NO. 00 COPPER 
 NO. 0000 ALUMINUM 
 
 NO. COPPER 
 NO. 000 ALUMINUM 
 
 j^NO. I COPPER 
 
 ^NO. 00 ALUMINUM J 
 
 ^NO. 2 OOPPER 'q 
 
 "^NO. ALUMINUM ^. (yj 
 
 WHERE 
 
 DROP IN VOLTS - 1000 KVA X LV 
 
 "" E 
 
 KVA - KVA OF LOAD AT RECEIVER END 
 L - LENGTH OF LINE IN MILES 
 E — LINE VOLTAGE AT LOAD OR RECEIVER END 
 V- FACTOR READ FROM CHART. V REPRESENTS A DROP IN 
 VOLTAGE WHEN IT IS READ ON THE SAME SIDE OF THE ZERO 
 LINE AS THE RESISTANCE POINT. AND A RISE IN VOLTAGE WHEN 
 IT IS ON THE OPPOSITE SIDE. 
 
 LINES OVER 30 MILES LONG 
 FOR LONG LINES. THE LINE CAPACITANCE DECREASES THE DROP, OR INCREASES 
 THE RISE AS FOUND ABOVE. BY / , \2 / i \2 
 
 '°° ^ (iooo) ^^^ "^^N"^- °" ^'^(iooo) ^°'-"rS WHERE 
 
 \'°°"' V"00/ ^=2.16 FOR 60 CYCLES 
 
 K-0 375 FOR 25 CYCLES 
 THE EFFECTIVE SPACING OF 3 PHASE LINES IS S -^/^^ S - 1.26 A FOR FLAT SPACING. 
 
 THE ABOVE EQUATIONS ARE FOR 2 AND 3 PHASE LINES 
 
 FOR SINGLE PHASE USE 2 V1N PLACE OF V COPYRIGHT 1916 BY H. B. DWIGHT 
 
 ■ ^NO. 3 COPPER ^ . 
 
 - *^NO I ALUMINUM [J li. 
 
 < 
 
 q-q: 
 
 NO. 4 COPPER 2 _ 
 NO. 2 ALUMINUM ^ jj 
 
 l-O 
 (0 < 
 
 u 
 
 rNO. 6 COPPER 
 NO. 3 ALUMINUM 
 
 RESISTANCE-OHMS 
 ONE CONDUCTOR 
 
PERFORMANCE OF SHORT TRANSMISSION LINES 
 
 S7 
 
 will be explained later. The Dwight and Mershon 
 charts do not recognize the effect which capacitance has 
 upon the voltage drop. In the lower left hand corner 
 of the Dwight Chart, however, there is placed a formula 
 by which a correction may be applied to the voltage 
 drop as given by the chart This correction accounts 
 for the effect of the charging current (resulting from 
 capacitance) quite accurately, provided the circuit is 
 not too long or the frequency too high. The application 
 of this corrective factor will be evident from the follow- 
 ing problem. 
 
 TABLE K-COSINES, SINES AND TANGENTS 
 
 ANGLE 
 
 cos 
 
 (P F) 
 
 SIN 8 
 
 TAN 
 
 0° 00' 
 
 1. 000 
 
 0.0000 
 
 0.0000 
 
 8' o6' 
 
 0.990 
 
 0.1409 
 
 0.1423 
 
 11° 28' 
 
 0.980 
 
 0.1988 
 
 0.2028 
 
 '^o °4' 
 
 0.970 
 
 0.2430 
 
 0.2506 
 
 't '5 
 
 0.960 
 
 0.2798 
 
 O.29IS 
 
 i8° n' 
 
 0.950 
 
 0.3120 
 
 0.3285 
 
 '< 56; 
 
 0.940 
 
 0.3410 
 
 0.3627 
 
 21° 33 
 
 0.930 
 
 0.3673 
 
 0.3949 
 
 < 04 
 
 0.920 
 
 0.3918 
 
 0.4258 
 
 24: ^ 
 
 0.910 
 
 0.4144 
 
 0.4554 
 
 25„ 50' 
 
 0.900 
 
 0.4357 
 
 0.4841 
 
 ^L °7' 
 
 0.890 
 
 0.4558 
 
 O.512I 
 
 28° 21' 
 
 0.880 
 
 0.4748 
 
 0.5396 
 
 29^ 32' 
 
 0.870 
 
 0.4929 
 
 0.5665 
 
 30„ 41 
 
 0.860 
 
 0.5103 
 
 0.5934 
 
 31° 47' 
 
 0.850 
 
 0.5267 
 
 0.6196 
 
 3^0 si; 
 
 0.840 
 
 0.5424 
 
 0.6457 
 
 33 54 
 
 0.830 
 
 0.5577 
 
 0.6720 
 
 34° 54' 
 
 0.820 
 
 0.5721 
 
 0.6976 
 
 35 54' 
 
 0.810 
 
 0.5864 
 
 0.7239 
 
 36 52; 
 
 0.800 
 
 0.6000 
 
 0.7499 
 
 37° 48' 
 
 0.790 
 
 0.6129 
 
 0.7757 
 
 ^K 44' 
 
 0.780 
 
 0.6257 
 
 0.8021 
 
 39° 38 
 
 0.770 
 
 0.6379 
 
 0.8283 
 
 40 32 
 
 0.760 
 
 0.6499 
 
 0.8551 
 
 K ^ 
 
 0.750 
 
 0.6613 
 
 0.8816 
 
 42° 16' 
 
 0.740 
 
 0.6726 
 
 0.9089 
 
 43° 06' 
 
 0.730 
 
 0.6833 
 
 0.9358 
 
 43° 56' 
 
 0.720 
 
 0.6938 
 
 0.9634 
 
 44, 45; 
 
 0.710 
 
 0.7040 
 
 0.9913 
 
 "•1° ^ 
 
 0.700 
 
 0.7141 
 
 1.0199 
 
 46° 22' 
 
 0.690 
 
 0.7238 
 
 1.0489 
 
 47° 09' 
 
 0.680 
 
 0.7331 
 
 1.0780 
 
 47 ss; 
 
 0.670 
 
 0.7422 
 
 I.IO74 
 
 < 42 
 
 0.660 
 
 0.7513 
 
 1.1383 
 
 < 27 
 
 0.650 
 
 0.7598 
 
 1.1688 
 
 50° 12' 
 
 0.640 
 
 0.7683 
 
 1.2002 
 
 50° 57' 
 
 0.630 
 
 0.7766 
 
 1.2327 
 
 5'„° 41; 
 
 0.620 
 
 0.7846 
 
 J.26S5 
 
 52° 24' 
 
 0.610 
 
 0.7923 
 
 1.2985 
 
 53: °7' 
 
 0.600 
 
 0,8000 
 
 1.3327 
 
 S3 SO 
 
 0.590 
 
 0.8073 
 
 1.3680 
 
 54 32' 
 
 0.580 
 
 0.8145 
 
 '■4037 
 
 55 14' 
 
 0.570 
 
 0.8215 
 
 1.4406 
 
 55° 56' 
 
 0.560 
 
 0.8284 
 
 1.4788 
 
 56° 37' 
 
 0550 
 
 0.8350 
 
 1.5175 
 
 57 18' 
 
 0.540 
 
 0.8415 
 
 1.5577 
 
 57° 59' 
 
 0.530 
 
 0.8479 
 
 1.5993 
 
 58 40; 
 
 0.520 
 
 0.8542 
 
 1.6426 
 
 59 20' 
 
 0.510 
 
 0,8601 
 
 1.6864 
 
 60° 00' 
 
 0.500 
 
 0.8660 
 
 1.7320 
 
 60° 39' 
 
 0.490 
 
 0.8716 
 
 1.7783 
 
 61° 18' 
 
 0.480 
 
 0.8771 
 
 1.8265 
 
 61° 57' 
 
 0.470 
 
 0.8825 
 
 1.8768 
 
 Three-Phase Problem (No. 45) — Find the volt- 
 age at the sending end of a three-phase circuit, 100 
 miles long, consisting of three No. 0000, stranded, hard- 
 drawn copper conductors, spaced nine feet apart in a 
 delta arrangement. Temperature assumed as 25 de- 
 grees C. Load conditions at receiving end assumed as 
 22000 kv-a, 80 percent power-factor lagging, 88chx) 
 volts, 60 cycles. 
 
 load by the amount of 100 X ^- 
 
 .y 
 
 = 2.l6 
 
 From Table II the resistance is found to be 0.277 
 
 ohm per mile. From Dwight Chart read V = 0.70. 
 
 Then, the voltage drop in percent of £„ if the line were 
 
 short, would be, 
 
 looooo X 22000 X 100 X 0.70 
 
 83ooo» = 19.89 percent 
 
 From directions on the Dwight chart for circuits 
 over 30 miles long, the charging current of this circuit 
 is found to be such as to decrease the voltage drop un- 
 der load conditions or to increase the voltage at zero 
 
 \/OOOj 
 
 percent. Hence the voltage at the sending end, under 
 load conditions, will be ip.8p — 2.16 ^^^ 17-73 percent. 
 The actual result as calculated rigorously is 17.94 per- 
 cent. Thus the error by the Dwight graphical solution 
 is approximately 0.21 percent. 
 
 If the power-factor of the load is assumed as 100 
 percent (problem 46) in place of 80 percent lagging, we 
 get V = 0.3s and find the error for the Dwight graphi- 
 cal solution of this 100 mile, 60 cycle circuit to be ap- 
 proximately 0.75 percent. It should be noted, however, 
 that the reactance volts are in this case 22 percent of 
 the receiving end voltage. 
 
 SENDING END CONDITIONS FIXED 
 
 When the sending end conditions are fixed, a dif- 
 ferent form of solution must be employed to determine 
 the size of conductors corresponding to a given voltage 
 drop. In such cases, the Dwight Chart is particularly 
 applicable. To use the chart for the solution of such 
 problems proceed as follows. First V is calculated by 
 means of the formulas on the chart, and then a straight 
 edge is placed through V (on the line corresponding to 
 the power-factor of the load) and the point for the 
 spacing and frequency to be used, and the required size 
 of conductor can be seen at a glance on the resistance 
 scale at the right. To make this application of the 
 chart clear, the following is given, — 
 
 Voltage drop in percent of Er ■ 
 Hence 
 
 100 000 Kv-a y. L V 
 
 £,' 
 
 iiS) 
 
 V = 
 
 Voltage drop in percent of Et X E* 
 
 (^9) 
 
 100 000 Kv-a XL 
 
 Applying (29) to the above problem No. 33 we get 
 
 16.12 X 10 000' _ 
 ~iooooo X 1300 X 20 ~ **■ ^ 
 Following the above directions, the resistance per 
 mile is found to be 0.277 ohm and the corresponding 
 size of conductor No. 0000 copper. 
 
 MATHEMATICAL SOLUTION 
 
 In order to check any one, or all of the above de- 
 scribed graphical methods, a complete mathematical 
 solution may be made by applying the various trigono- 
 metrical formulas. Fig. 18, to the values of the problem 
 under consideration. These formulas have been ar- 
 ranged to meet the conditions of loads of either lagging 
 or leading power-factors, and for conditions fixed at 
 either the receiving or the senuing ends. 
 
 There are numerous problems requiring a solution 
 
S8 
 
 PERFORMANCE OF SHORT TRANSMISSION LINES 
 
 where the voltage at the sending end, and the kilowatts 
 and the power-factor of the load at the receiving end 
 are fixed. In such cases it is required to determine the 
 corresponding receiving end voltage. This determina- 
 tion can be made mathematically, but such a solution is 
 tedious, since the formulas applying to such cases are 
 cumbersome. Formulas are given at the bottom of Fig. 
 i8 which may be applied to such problems. Time and 
 labor may, however, be saved in solving such problems 
 by the employment of a cut-and-try method usually used 
 in such cases, as follows: — 
 
 Assume what the voltage drop will be, correspond- 
 ing to the size of conductors likely to be used. On the 
 basis of this assumption the receiving end voltage is 
 fixed; thus, all of the receiving end conditions are as- 
 sumed to be fixed. The corresponding sending end 
 voltage is then readily determined by one of the graphi- 
 cal methods described. If the sending end voltage thus 
 determined is found to be materially different from the 
 fixed sending end voltage, another trial, based upon a 
 different receiving end voltage, will probably suffice. 
 
 Single-Phase Problem — Find the characteristics of 
 the load at the sending end of a single-phase circuit, 
 16 miles long, consisting of two stranded, hard drawn, 
 copper conductors, spaced three feet apart ; temperature 
 taken as 25 degrees C ; load conditions at receiving end 
 assumed as 4000 kv-a (3200 kw at 80 percent power- 
 factor lagging) 20000 volts, 60 cycles; transmission 
 loss to be approximately ten percent. 
 
 Following the procedure given in Chart II, consult 
 Quick Estimating Table XVII for a delivered voltage 
 of 20000. Since the conditions of the above problem 
 are a power-factor of 80 percent, and a temperature 25 
 degrees C, the corresponding kv-a values are as in- 
 dicated at the head of the table on the basis of 10.8 per- 
 cent loss in transmission for a three-phase circuit. For 
 a single-phase circuit the corresponding values will be 
 one-half the table values. Thus the 4000 kv-a single 
 phase circuit of the problem is equivalent to 8000 kv-a, 
 three-phase on the table. From the table, it is seen that 
 for a distance of 16 miles 7810 kv-a, three-phase can 
 be transmitted over No. 0000 conductors with a loss of 
 10.8 percent. 7810 kv-a is near enough to 8000 kv-a, and 
 the loss of 10.8 percent is near enough to an assumed 
 loss of ten percent, so we decide that No. 0000 copper 
 • conductors come nearest to the proper size to meet the 
 conditions of the problem. The loss with No. 0000 
 
 8000 
 conductors will be-Q- X 10.8 =11.06 percent, as will 
 
 be shown later. 
 
 Table XXIII indicates that there will be no over- 
 heating of this size of conductor. 
 
 Table XXII indicates that 20 000 volts is too low to 
 
 result in corona loss with No. 0000 conductors, at any 
 
 reasonable altitude. Then, — 
 
 V 4000 
 
 Kv-a,n := — ^ 2000 kv-a to neutral. 
 
 3200 
 Kwrm = — — ^= 1600 kw to neutral. 
 
 Ei, = = 10 000 volts to neutral. 
 
 2000000 . . 
 
 ' ^ ' 10000 = 200 amperes per conductor. 
 
 The resistance per conductor is 
 
 i? = 16 X 0-277 (from Table II) = 4.432 ohms. 
 
 The reactance per conductor is 
 
 X = 16 X 0.644 (from Table V) = 10.304 ohms. 
 and IR = 200 X 4-432 = 866 volts, resistance drop 
 886 
 
 = X 100 =: 8.86 percent 
 
 10 000 ^ '^ 
 
 IX = 200 X 10.304 = 2061 volts, reactance drop 
 
 = X 100 = 20.61 percent 
 
 10 000 'j 
 
 E.n = 1/(10000 X 0.8 -I- 866)' 4- (loooo X 0.6 + 2061)' 
 
 = II 998 volts to neutral (30) 
 
 ,/ (10 000 X 0-6) +2061 \ 
 
 «• =''^''-'1(10 000X0.8) -f 886 J = ^2 13 (jO 
 
 Percent PF. = (Cos. 42° 13' ) X 100 = 7406 percent (3s) 
 
 Kv-a,a = ^°° 10^ = 2399.6 kv-a per conductor (33) 
 
 Kw,n = 2399.6 X 0.7406 = 1777-1 kw per conductor (34) 
 
 11998— lOOOO ^^ o .. . 
 
 Percent voltage drop = ^^^ X 100 = 19-98 percent 
 
 (46) 
 
 Transmission loss = ^^^^ "^^^ = '77-28 k^ Per conductor 
 (47) 
 
 177 28 ^^ 2 
 
 Percent transmission loss = " pp X loo = 11.08 percent 
 
 (4S) 
 
 Three-Phase Problem (No. 55)— Find the char- 
 acteristics of the load at the sending end of a three- 
 phase circuit 20 miles long, consisting of three stranded, 
 hard-drawn, copper conductors, spaced in a three foot 
 delta. Temperature taken as 25 degrees C. Load 
 conditions at receiving end assumed as 1300 kv-a. (1040 
 kw at 80 percent power-factor lagging) 10 000 volts, 60 
 cycles ; transmission loss not to exceed ten percent. 
 
 Following the procedure given in Chart II, the fol- 
 lowing results are obtained : — 
 
 Consult Table XV for a delivered voltage of 10 000 
 volts. Since the conditions of the above problems are, 
 power-factor of load 80 percent, temperature 25 degrees 
 C. the corresponding three-phase kv-a values of the 
 table are on the basis of 10.8 percent loss in transmis- 
 sion. From Table XV it is seen that 1240 kv-a, three- 
 phase can be transmitted over No. 000 conductors, or 
 15(30 kv-a., three-phase over No. 0000 conductors at 
 10.8 percent loss. Since the loss for the problem is 
 not to exceed ten percent and 1300 kv-a is to be tnms- 
 mitted, we will select No. 0000 conductors. The loss 
 
 j-jOO 
 
 for these conductors will therefore be — —- of 10.8. or 
 
 1560 
 
 nine percent as will be shown later. 
 
 Table XXIII indicates that there will be no over- 
 heating of this size of conductor when carrying 1300 
 kv-a, three-phase. 
 
 Table XXII indicates that 10000 volts Is too low 
 to result in corona loss with No. OOGG conductors at 
 any reasonable altitude. Then: — 
 
 J-3QQ 
 
 Kv-On = — I — = 433.33 kv-a to neutral. 
 Kwt, — — — = 346.6 kw to neutral. 
 
and 
 
 _ lOOOO 
 
 '■ ~ r.732 ~ 5774 volts to neutral. 
 
 , 433 333 
 
 'r = — = 75.05 amperes per conductor. 
 
 The resistance per conductor is, — 
 
 i? = 20 X 0.277 (from Table II) = 5.54 ohms. 
 
 The reactance per conductor is, — 
 
 X = 20 X 0.644 (from Table V) = 12.88 ohms. 
 
 IR = 75.05 X 5-54 = 415-8 volts, resistance drop. 
 4158 
 
 PERFORMANCE OF SHORT TRANSMISSION LINES 59 
 
 Transmission loss = ^^"^^ ^^ = 3120 *w per conductor 
 
 (^) 
 
 9.00 percent 
 
 %t 20 X 1 
 
 Percent transmission loss = -^^-^ — ^^"^X lOO = 
 
 1040 
 
 S774 
 
 X 100 = 7.20 percent. 
 
 OIAORAM 
 
 FOR 
 
 LAOQINQ POWEH-FAOTOR 
 
 MIXED SENDING AND RECEIVING END CONDITIONS FIXED 
 
 Brancn circuits are frequently run from the main 
 transmission trunk circuit to the center of some local 
 distribution. In such cases the voltage at the sending 
 end and the current or the power and power-factor at 
 
 -^8 
 
 ••CURIJENT 
 
 DIAGRAM 
 
 FOR 
 
 LEADING J>OWER-F ACTOR 
 
 P^ 
 
 • V i ' i 
 
 _i_i i h-jL* 
 
 K. ERCOsep — -»i ! 
 
 V EgCOSBg. »J 
 
 CURRENT 
 
 LOADS OF LAGGING POWER- FACTOR 
 
 ^^ 
 
 LOADS OF LEADING POWER -FACTOR 
 
 £3= y(ER cos Sr+IR)^ + (Er sin Sh + OCf (30) 
 (ErSIN9r)+1X) 
 
 WHEN es=TAN-l ,EoCOSeH^+IR) "'* 
 
 RECEIVING-END V " ") f 
 
 CONDITIONS % PFs=- COS Og X 100 (32) 
 ARE FIXED 
 
 KVAcu-LlrSN PER CONDUCTOR (33) 
 ^'^ "iooo 
 KW^K|- KV-Aqki X COS 9<~ PER CONDUCTOR (34) 
 
 Er- ^(EsCOS6s-IR/ + (EgSINeg-lX)^ (36) 
 
 (Eg SIN Sg)- IX) 
 
 WHEN Sr-TAN-I /EgCOS9s)-IR) "" 
 
 SENDING-ENO ^ '" "11 
 
 CONDITIONS % PFr- COS Or X 100 (37) 
 ARE FIXED ,^^ 
 
 KVAo.," BH PER CONDUCTOR (38) 
 
 "RN" 
 
 1000 
 
 KWrm-KVArnX COS Sr per CONDUCTOR (39) 
 
 Eg- "^CEr COS eR+ 1R)^ + (Er SIN 9r - IX) (40) 
 
 (ERS.NeR)-DC) 
 WHEN Sg-TAN-* (Er COS 9r)+1R) '*" 
 
 RECEIVING -END 
 CONDITIONS %PFg-OOS9sXl00 (32) 
 
 ARE FIXED ^u. l«EsN pER CONDUCTOR (33) 
 
 SN IOOO 
 KW;„- KVAgfi X COS 9^ PER CONDUCTOR (34) 
 
 Er- y(Es COS 98 - IR)^ + (Eg SIN 9s-t-lX)^ (4J) 
 (EgSINBs)-t-D<) 
 
 WHEN On-TAN-< /Vc C03Bc\-IR-> <*» 
 
 SENDING-ENO " ('^S'-^'S) '") 
 
 CONDITIONS « pp _ COS 9r X 100 (37) 
 ARE FIXED " " 
 
 in/.An,i— '"^RN PER CONDUCTOR (38) 
 "" IOOO 
 kWp,,- KVAnfi X COS Br PER CONDUCTOR (39) 
 
 GENERAL FORMULAS 
 
 WHEN THE VOLTAGE AT SENDING END AND THE AMPERES AND POWER^^ACTOR AT RECEIVINOEND ARE FIXED 
 
 Er = - 1 (r COS 9r ±X SIN eR)+ VeI-i' (r^ SIN^ Sr+X^ C0S° 9r)± 2 1^ R X COS 9r SIN 8r (44) 
 * USE + WHEN THE POWER-FACTOR OF THE LOAD IS LAGGING AND - WHEN THE POWER-FACTOR IS LEADING 
 
 WHEN THE VOLTAGE AT SENDING END AND THE POWER AND POWER-FACTOR AT RECEIVING END ARE FIXED 
 (POWER FACTOR LAGGING) 
 
 j = A Wit Vl- 
 
 (R nX')KWRN^XI08 
 A* COS* Or 
 
 AT, 1 000 KWru (R cos 9o + X SIN 9r) 
 WHEREA-Eg^yX eilL__a^ ! ,4« 
 
 i2r 
 
 % VOLTAGE DR0P= 
 
 -S~'-R 
 
 XIOO (46) 
 
 TRANSMISSION LOSS- j^ KW PER CONDUCTOR (47 > 
 % TRANSMISSION LOSS- - °tOTAl"kWb''*" ^ ' °° ***' 
 
 FIG. l8 — TRICWNOMETRICAL FORMULAS FOR SHORT TRANSMISSION UNES 
 
 Capacitance effect not taken into account. 
 
 IX = 75.05 X 12.88 = 966.6 volts, reactance drop. 
 966.6 ,, 
 = ■ - X 100 = 16.74 percent. 
 
 the receiving end are approximately fixed. In such 
 
 cases the calculation for the voltage at the receiving end 
 
 , — ^ -,^, , , , , ,^,,, , requires more arithmetical work than is required when 
 
 £.. = 1/ (5774 X 0.8 -f- 415-8)' -h (5744 X 0.6 -f966.6 "=6707 ,,'. ... ^ a t .x, ■ ■.• c a 
 
 volts to neutral (30) all the conditions at one end of the circuit are fi.xed. 
 
 ( 5774 X 0.6 -f 966.6 \ _ o , f ,, Such problems can be more readily solved graphically, 
 
 S774X0.8 4-41S-8 / ~ ^ " as previously explained, but may be solved mathe- 
 
 PF. = (Cos 41" 22') X 100 = 75-05 percent (3^) matically by applying formula (44) or (45), Fig. 18. 
 
 Kv-a„ = —j~ — ^^^ — S03.4 kv-a per conductor (33) To illustrate the application of formula (44) we 
 
 Kw,. = 503.4 X 0.7505 = 377-8 kw per conductor (3^) will apply the values of Problem jj to formula (44) and 
 
 Percent voltage drop =-^^^^r=^ X loo = 16.16 percent calculate the receiving end voltage. Thus we have as 
 
 S774 
 
 (.46) 
 
 fixed conditions: — 
 
6o 
 
 PERFORMANCE OF SHORT TRANSMISSION LINES 
 
 En = 6707 volts 
 
 It = 75-05 amperes 
 ■Cos 0, = 0.8 
 Sin S, =: 0.6 
 
 R ===^ 5-54 ohms 
 X = 12.88 ohms 
 IR = 415.8 volts 
 Then 
 
 £r = —7505 (5-54 X 0.8 + i2.{ 
 
 X 0.6) + 
 
 1/6707' — 75-05' (5-54' X 0.6 -' + 12.88= X 0.8') + 2 X 
 
 7505" X 5-54 X 12.88 X 0.8 X 0.6 U4) 
 
 = — 913 + F 44983849 — 660242 + 385831 
 = — 913 + 6637 = 5774 volts. 
 To illustrate the application of formula (45) we 
 v/ill apply the values of Problem 55 to formula (45) 
 
 TABLE L 
 ILLUSTRATING VARIATION IN REACTANCE 
 
 Resulting from Changes in the Conductors and Transmission Voltages 
 
 and calculate the receiving end voltage. Thus we have 
 as fixed conditions : — 
 E.n = 6707 volts ■ 
 Kw,„ = 346.6 kw 
 R = 5.54 ohms 
 X =^ 12.88 ohms 
 
 Cos 6, ■=: 0.8 
 
 Sin Br = 0.6 
 
 then 
 
 A = 6707 -y/ 0.5 
 
 loooX 346.6 (5.54 X 0.8 + 12.88 X 0.6) 
 
 6707' X 0.8 
 
 E,. = AJ^+y 
 
 i 
 
 A 
 
 (5-54' + 12.88') 346.6' X iC 
 A' X 0.8' 
 
 Us) 
 
 - 6707 j/'o.s — 0.1172 = 4152 
 
 = 41521/ I + 0.936 =: 5774 volts 
 
 Alternative to (44) and (^5)— The following 
 formulas have been proposed by Mr. H. B. Dwight to 
 meet the mixed conditions referred to, — 
 
 E,„ := 6707 volts 
 
 1000 X Kwr, = 346600 wafts 
 
 1000 X reactive Kv-a,^ = 346600 X -^= 260 000 v-a 
 
 R = 5.54 ohms 
 
 X = 12.88 ohms 
 
 L = 346600 X 5-54 + 260000 X 12.88 = 5270000 
 
 M = 346600 X 12.88 — 260000 X 5-54 = 3025000 
 
 E^ = 0.5 E^^ - L+ o.s VE,* - y E,^ I. - 4 M"- 
 •E = 5774 volts 
 
 R - R 1- £. J^ 
 
 ^ - ^'~ E,- E,'~ ^E,^' 
 
 Ei' 
 
 2 E,' 
 
 5L^ 
 E,- 
 
 ~ E,' - 
 E = 5779 volts 
 
 E,' 
 
 CONDUCTORS 
 
 Total 
 I' R 
 
 Loss 
 (KW) 
 
 IR 
 
 IX 
 
 Approximate 
 
 Voltage 
 Regulation at 
 
 Volts 
 
 Per 
 Cent. 
 
 Volts 
 
 Per 
 Cent. 
 
 100 
 
 Per 
 
 Cent. 
 
 Power 
 
 Factor 
 
 80 Per 
 Cent. 
 Power 
 Factor 
 (Lag.) 
 
 RECEIVING END VOLTAGE — 0600 
 
 Single Circuit of 
 three 500,000 circ.mil 
 bare overhead con- 
 ductors 
 
 129 
 
 123 
 
 3.22 
 
 622 
 
 16.32 
 
 4.5 
 
 12.8 
 
 Two circuits each 
 of three 250,000 circ. 
 mil bare overhead 
 conductors. 
 
 129 
 
 123 
 
 3.22 
 
 133 
 
 8.73 
 
 3.< 
 
 7.7 
 
 One Circuit of 
 500,000 circ. mil 
 three-conductor 
 cable. Insulation 
 thickness ij by a 
 inches. 
 
 129 
 
 1.23 
 
 3.22 
 
 172 
 
 4.52 
 
 3.2 
 
 5.0 
 
 RECEIVING END VOLTAGE — 13 200 
 
 Single circuit of 
 three 126,000 circ. 
 mil bare overhead 
 conductors. 
 
 129 
 
 247 
 
 3.22 
 
 354 
 
 4.M 
 
 3.2 
 
 5.1 
 
 CIRCUITS OF EXCESSIVE REACTANCE 
 
 If a large amount of power is to be transmitted at 
 comparatively low voltage, particularly if the frequency 
 is high, the reactance of the circuit will be high com- 
 pared with its resistance. If the reactance is excessive 
 (20 to 30 percent reactance volts may in some cases be 
 considered excessive), the voltage regulation of the cir- 
 cuit may be seriously impaired. 
 
 As will be seen by consulting Tables VI and VII, 
 there is a fixed relation between tlie resistance and the 
 reactance of a circuit for a given frequency, size and 
 spacing of conductors. This ratio is 2.4 times greater 
 for 60 cycle than it is for 25 cycle circuits. For a given 
 size of conductor the reactance can be varied only 
 slightly by changing the spacing of overhead bare con- 
 ductors. Substituting a larger or smaller conductor 
 may change the resistance materially, but this will have 
 little effect upon the reactance. 
 
 The reactance may be reduced by either or all of 
 the following methods. The circuit may be split up 
 into two or more circuits employing smaller conduc- 
 tors and these circuits connected in parallel. The volt- 
 age may be raised, if the installation is new, and smaller 
 conductors employed; or the overhead conductors may 
 be replaced by three conductor cables. To illustrate 
 the above methods, the following problem has been as- 
 sumed and the results tabulated. 
 
 A HIGH REACTANCE PROBLEM 
 
 Table L refers to the following problem — ^,}.ooo 
 kv-a, three-phase, 60 cycles, is to be delivered a dis- 
 tance of three miles over hard-drawn, stranded copper 
 conductors. The PR loss is to remain at 129 kw. The 
 spacing of the overhead conductors assumed as 3 by 3 
 by 3 ft. Temperature 25 degrees C. 
 
 It is evident from Table L that if two three-phase 
 circuits, each consisting of three 250000 circ. mil. con- 
 ductors are installed in place of one three-phase circuit, 
 consisting of three 500 000 circ. mil. conductors, the re- 
 actance will be reduced by nearly one half, and a cor- 
 responding improvement in the voltage drop or regula- 
 tion will occur, particularly if the load power-factor is 
 80 percent lagging. A further improvement along this 
 line will be obtained if a single three-conductor cable is 
 employed. Doubling the voltage for the overhead cir- 
 cuit and employing three 125 000 circ. mil. conductors 
 results in practically as good performance in voltage 
 regulation as for the 6600 volt three-conductor cable. 
 
 *See article by Mr. H. B. Dwight on "Effect of a Tie Line 
 between Two Substations" in the Electrical Review, Dec. 21, 
 1918, p. 966. The formulas given in this article make complete 
 allowance for the effect of capacitance and are very similar to 
 the above. 
 
CHAPTER VIII 
 PERFORMANCE OF LONG TRANSMISSION LINES 
 
 (GRAPHICAL SOLUTION) 
 
 THE E.M.F. of self-induction in a transmission 
 circuit may either add to or subtract from the 
 impressed voltage at the sending end, depend- 
 ing upon the relative phase relations between the cur- 
 rent and the voltage at the receiving end of the circuit. 
 This is illustrated by means of voltage vectors in Fig. 
 20, in which the phase of the current is assumed to be 
 constant in the horizontal direction indicated by the 
 arrow on the end of the current vector. The voltage at 
 the receiving end is also assumed as constant at 100 
 volts. The vector representing the receiving end volt- 
 age (Er = 100 volts) is shown in two positions corre- 
 sponding to leading current, two positions correspond- 
 ing to lagging current and in one position corresponding 
 to unity power-factor. The components IR and IX of 
 the supply voltage necessary to overcome the resistance 
 R and the reactance X (e.m.f. of self-induction) of the 
 circuit are assumed to be 10 volts and 20 volts respec- 
 tively. Since the current is assumed as constant. IX 
 and IR are also constant. The impedance triangle of 
 the voltage components required to overcome the com- 
 bined effect of the resistance and the reactance of this 
 circuit is therefore constant. It is shown in five differ- 
 ent positions about the semicircle, corresponding to five 
 different load power-factors. The voltage E, at the 
 sending-end required to maintain 100 volts at the re- 
 ceiving-end is indicated for each of the five positions of 
 the impedance triangle. 
 
 Counter-clockwise rotation of the vectors will be 
 considered as positive. This means that when th: cur- 
 rent is lagging behind the impressed e.m.f., the voltage 
 vector will be in the forward or leading direction from 
 the current vector as indicated by the arrow. When 
 the current leads the impressed voltage, the voltage 
 vector will be in the opposite, or clockwise direction 
 from the current vector. In other words, assuming the 
 vectors all rotating at the same speed about the poin, O 
 in a counter-clockwise direction, the current vector will 
 be behind the voltage vector when the current is lagging 
 and ahead of it when the current is leading. 
 
 The alternating magnetic flux surrounding the con- 
 ductors, resulting from current flowing through them, 
 generates in them a counter e.m.f. of self-induction. 
 This e.m.f. of self-induction has its maximum value 
 when the current is passing through zero and is there- 
 fore in lagging quadrature with the current. On the 
 c'iagrams an arrow in the line IX, indicates the direction 
 of the e.m.f. of self-induction. It will be seen that since 
 the direction of the current is assumed constant, 
 the e.m.f. of self-induction acts downward in all 
 
 five impedance diagrams. The sending-end voltage 
 is therefore opposed or favored by this self- 
 induced voltage (see arrows) to a greater or less 
 extent depending upon the power-factor of the load. 
 Thus at lagging loads of high power-factor, the self-in- 
 duced voltage acts approximately at right angles to the 
 5ending-end voltage, and therefore requires a small com- 
 ponent of the sending-end voltage to balance or neutral- 
 ize its effect. As the power-factor of the receiving-end 
 load decreases in the lagging direction (upper quadrant 
 of diagram) the sending-end voltage swings around 
 more nearly in line with the direction of the induced 
 voltage, thus requiring a greater component of the send- 
 ing-end voltage to counter-balance its effect. At zero 
 power-factor lagging, the direction of the sending-end 
 voltage and that of the induced e.m.f. are practically in 
 opposition, (as indicated by the arrows), so that the 
 component of the sending-end voltage required to over- 
 come the induced voltage is a maximum, or nearly as 
 much as the e.m.f. of self-induction. It is interesting to 
 note that at zero lagging power-factor, when the effect 
 of self-induction on line voltage drop reaches a maxi- 
 mum, the sending-end voltage component IR necessary 
 to overcome the resistance of the circuit, (now nearly 
 at right angles to the supply voltage), is a minimum. 
 The reverse of these conditions is true for receiv'ng- 
 end loads of power-factors near unity. 
 
 Now consider receiving-end loads of leading 
 power-factors, (lower quadrant of diagram). It will 
 be seen that the e.m.f. of self-induction does not now 
 oppose the sending-end voltage (indicated by direction 
 of the arrows) but has a direction more or less parallel 
 to that of the sending-end voltage. At high leading 
 power-factors, the e.m.f. of self-induction has little 
 effect on the sending-end voltage, but as zero leading 
 power-factor is approached these two e.m.f.'s more 
 rearly come in phase with each other. At zero power- 
 factor leading, the e.m.f. of self-induction adds almost 
 directly to the sending-end voltage. 
 
 It will be seen, therefore, that for receiving-end 
 loads of lagging power-factor, the sending-end voltage 
 is greater than the receiving-end voltage, by an amount 
 necessary to overcome the resistance and self-induction 
 of the circuit. For receiving-end loads of leading 
 power-factor, the sending-end voltage is less than the 
 receiving-end voltage, for the reason that the e.m.f. of 
 self-induction is in such a position as to assist the send- 
 ing-end voltage. 
 
 The following values from Fig. 20 illustrate these 
 conditions : 
 
6a 
 
 GRAPHICAL SOLUTION FOR LONG LIN EH 
 
 Power-Factor of 
 Receiving End Load 
 
 Supply Voltage 
 
 percent lagging 
 80 percent lagging 
 ICX) percent 
 80 percent leading 
 
 percent leading 
 
 120.4 
 120.4 
 111.8 
 
 98.5 
 80.6 
 
 The condition of leading power-factor at the re- 
 ceiving-end would be unusual in practice, since the 
 power-factor of receiving-end loads is usually lagging. 
 In cases, however, where condensers are used for 
 voltage or power-factor control, the power-factor at the 
 receiving-end may be leading. If the circuit were with- 
 out inductance, there could be no rise in voltage at the 
 
 '■—M'^OlSG^ 
 
 ■^P.F O^ LCtf D 80% LAGGING 
 
 P.F OF LOAD. 80% LEADING 
 
 BASIS OF DIAGRAM 
 tr HELD CONSTANT AT 100 VOLTS. 
 IR RESISTANCE VOLTS CONSTANT AT 10 VOLTS. 
 IX REACTANCE VOLTS CONSTANT AT 20 VOLTS. 
 .IZ IMPEDANCE VOLTS CONSTANT AT 22.4 VOLTS, 
 -P.F. OF LOAD. 0% LEADING 
 
 FIG. 20 — EFFECT OF SELF INDUCTION ON REGULATION 
 
 receiving-end, for in such a case, IX of the diagram 
 would disappear, and the voltage drop would be the 
 same as with direct current. All alternating-current 
 circuits are inductive, and the greater their inductance, 
 the greater will be the voltage drop, or the voltage lise 
 along the circuit. 
 
 Any alternating-current circuit may be looked upon 
 as containing three active e.m.f.'s out of phase with each 
 other. In addition to the impressed e.m.f. at the send- 
 ing-end, there are two e.m.f 's of self-induction, one as 
 the result of the receiving-end current and lagging 90 
 
 degrees behind it and the other as the result of the line 
 charging current and lagging 90 degrees behind it. 
 These two combine at an angle, with each other and 
 with the impressed e.m.f. at the sending-end. 
 
 CHARGING CURRENT 
 
 Conductors of a circuit, being separated by a di- 
 electric (such as air, in overhead circuits, or insulation 
 in cables), form a condenser. When alternating-cur- 
 rent flows through such a circuit, current (known as 
 charging current) virtually passes from one conductor 
 through the dielectric to the other conductors, which 
 are at a diflferent potential. This current is in shunt 
 with the circuit, and differs from the current 
 which passes between conductors over the insula- 
 tors etc. (leakage current) or through the air 
 (corona effect) only in that the charging current 
 leads the voltage by 90 degrees, whereas the leak- 
 age current is in phase with the voltage. 
 
 For a given spacing of conductors, the charg- 
 ing current increases with the voltage, the fre- 
 quency and the length of the circuit For long 
 high-voltage circuits, particularly at 60 cycles 
 per second, the charging current may be as much 
 as the full-load current of the circuit, or more. 
 In some cases of long 60 cycle circuits, where a 
 comparatively small amount of power is to be 
 transmitted, it is necessary to limit the voltage of 
 transmission, in order that the charging current 
 may not be so great as to overload the genera- 
 tors. This charging current, being in leading 
 quadrature with the voltage, represents nearly all 
 reactive power, but it is just as effective in heat- 
 ing the generator windings as if it represented 
 active power. On the other hand, it combines 
 with the receiving-end current at an angle (de- 
 pending upon the power-factor of the receiver 
 load) in such a manner that the addition of the 
 full-load receiving-end current, in extreme cases, 
 may not greatly increase the sending end current. 
 In other words (if the charging current is near 
 full-load current) the current at the generator 
 end may not increase much when full load at the 
 receiver end is added, over what it is when no 
 load is taken off at the receiving-end. 
 
 Since the e.m.f. of self-induction due to the 
 charging component is proportional to the charg- 
 ing current, its effect upon the voltage regulation 
 of the circuit will also be proportional to the charging 
 current. For a short low-voltage circuit, the charging 
 current is so small that its effect on voltage regulation 
 may be ignored. On the longer circuits, especially long 
 60 cycle circuits, such as will be considered later, its 
 effect must be given careful consideration. 
 
 VARIATION IN CURRENT AND VOLTAGE ALONG THE 
 CIRCUIT 
 
 It was explained above and illustrated in Fig. 20 
 that with a receiving-end load of leading power-factor. 
 
GRAPHICAL SOLUTION FOR LONG LINES 
 
 63 
 
 the voltage at the sending-end of the circuit might be 
 less than that at the receiving-end. It was shown that 
 the e.m.f. of self-induction, resulting from the leading 
 current, tends to raise the voltage along the circuit. 
 This boosting effect of the voltage is entirely due to the 
 leading component of the load current. 
 
 If, now, it is assumed that the power-factor of the 
 receiving-end load is 100 percent, there will be no lead- 
 ing component in the load current, and therefore there 
 can be no boosting of the voltage due to the load cur- 
 rent. Since, however, all circuits have capacitance, 
 and since the current is alternating, charging current 
 Vv'ill flow into the line and this being a leading current, 
 the same tendency to raise the voltage along the circuit 
 will take place as is illustrated by Fig. 20. 
 
 The upper part of Fig. 21 is intended to give a 
 physical conception of what takes place in an alternat- 
 ing-current circuit. As the load current starts out from 
 the sending-end, and travels along the conductor, it 
 meets with ohmic resistance. This is represented by r 
 in Fig. 21. It also meets with reactance in quadrature 
 to the current. This is represented by jx in the dia- 
 gram. Superimposed upon this load current is a cur- 
 rent flowing from one conductor to the others, in phase 
 with the voltage at that point and representing true 
 power. This current is the result of leakage over in- 
 sulators and of corona effect between the conductors. 
 It is represented by the letter g in the diagrams. Then 
 there is the charging current in leading quadrature with 
 the voltage. This current does not consume any active 
 power except that necessary to overcome the resistance 
 to its flow. 
 
 In Fig. 21 the four linear constants of the alternat- 
 ing-current circuit, r representing the resistance, jx re- 
 presenting the reactance, g representing the leakage and 
 b representing the susceptance, are shown as located, or 
 lumped, at six different points along the circuit. This 
 i., as they would appear in an artificial circuit divided 
 into six units. In any actual line, these four constants 
 are distributed quite evenly throughout the length of 
 the circuit. 
 
 VOLTAGE AND CURRENT DISTRIBUTION FOR PROBLEM X 
 
 The effect of the charging current flowing through 
 the inductance of the circuit gives rise to a very inter- 
 esting phenomenon. In order to illustrate this effect, 
 the current and voltage distribution for a 60 cycle, 1000 
 volt, three-phase circuit, 300 miles long, is plotted in 
 Fig. 21. This circuit will be referred to as problem X. 
 In such a long 60 cycle circuit, this phenomenon is quite 
 pronounced; so that such a problem serves well as an 
 illustration. The voltage and the current have been 
 determined for points 50 miles spart along the circuit. 
 Values for both the current and the voltage under zero 
 load, also under load conditions have been plotted. The 
 load conditions refer to a receiving-end load of i8ckx) 
 kv-a, at 90 percent power-factor, lagging, 60 cycle 
 three-phase. The voltage is assumed as being held 
 constant 104000 volts at the receiving-end, for both 
 ?:»'-o and full-load conditions. 
 
 Zero-Load Conditions — Without any load being 
 taken from the circuit, it will be seen that the charging 
 current at the sending-end approaches in value that 
 established when under full load; i.e., 94.75 amperes. 
 The charging current drops down to approximately 50 
 amperes at the middle, and to zero at the receiving-end 
 of the unloaded circuit. The lower full line curve 
 shows how this current is distributed along the circuit 
 Starting at zero, at the receiving-end of the circuit, it 
 increases as the sending-end of the circuit is ap- 
 proached, at which point it reaches its maximum value 
 cf 87.89 amperes. The voltage distribution under zero- 
 load conditions is some-what opposite to that of the cur- 
 rent distribution. That is the voltage ( 104 000 volts at 
 the receiving-end) keeps falling lower until it reached a 
 value of 84 676 at the sending-end. It should be noted 
 that the voltage curve for zero load condition drops 
 down rapidly as the sending-end is approached. The 
 reason for this is the large charging current flowing 
 through the inductance of the circuit at this end of the 
 circuit. The larger the charging current the greater 
 the resultant boosting of the receiving-end voltage. 
 
 Load Conditions — When 16 000 kv-a at 90 percent 
 power-factor lagging is taken from the circuit at the 
 receiving-end, the current at this end goes up to 99.92 
 amperes. As the supply end is approached the current 
 becomes less, reaching its lowest value (approximately 
 83 amperes) in the middle of the circuit. At the supply 
 end it is 94.75 amperes, which is less than it is at the 
 receiver end. Thus the full line representing the cur- 
 rent in amperes along the circuit assumes the form of an 
 arc, bending downward in the middle of the circuit. 
 The shape of this current curve is dependent upon the 
 relative values of the leading and lagging components 
 cf the current at points along the circuit. The reason 
 that the current is a minimum rear the middle of the 
 circuit, is because this is the point where the lagging 
 current of the load and the leading charging current of 
 the circuit balance or neutralize each other, and the 
 power-factor is therefore unity. Starting at the re- 
 ceiving-end, the power- factor is 90 percent lagging. As 
 the middle of the circuit is approached, the increasing 
 charging current neutralizes an increasing portion of the 
 lagging component of the load current. Near the 
 middle of the circuit, this lagging component is en- 
 tirely neutralized, and the power-factor therefore rises 
 to unity. Passing the middle and approaching the 
 sending-end there is no more lagging component to be 
 neutralized, and the increasing charging current 
 causes a decreasing leading power-factor which, when 
 the sending-end is reached, becomes 93.42 percent 
 leading. It will, therefore, be seen that the power- fac- 
 tor as well as the current and voltage varies through- 
 nut the length of the circuit. 
 
 The voltage distribution under load condition is 
 indicated by the top broken line. In order that the re- 
 ceiving-end voltage may be maintained constant at 
 104000 volts, the voltage at the sending-end will vary 
 
64 
 
 GRAPHICAL SOLUTION FOR LONG LINES 
 
 from 84 676 volts at zero load to 122 370 volts at the 
 pssumed load. 
 
 THE AUXILIARY CONSTANTS 
 
 With the impedance methods considered under the 
 general heading of "Short Transmission Lines" the cur- 
 rent vifas considered as of the same value throughout 
 the circuit, and the voltage drop along the circuit was 
 considered as proportional to the distance. These as- 
 sumptions, virhich are permissible in case of short Imes, 
 are satisfied by simple trigonometric formulas. 
 
 The rigorous solution for circuits of great elec- 
 trical length accurately takes into account the effect 
 produced by the non-uniform distribution of the cur- 
 rent and the voltage throughout the length of the circuit. 
 This effect wfill hereafter be referred to as the distribu- 
 tion effect of the circuit, and may be taken into account 
 
 SENDING 
 END E3 
 
 EQUIVALENT CIRCUIT TO NEUTRAL 
 r INDICATES RESISTANCE jx REACTANCE jb SUSCEPTANCE AND g LEAKANCE 
 
 (P.F. OF LOAD.90% LAGGING) 
 _VOLTAOE (LOAD CONDITION) 
 
 ti! 
 
 1 20s 
 
 SENDINO-END 
 
 100 l&O '^00 
 
 DISTANCE IN MILES FROM SENDING-END 
 
 FIG. 21 — DIAGRAMS OF TRANSMISSION CIRCUIT — PROBLEM X 
 
 300 miles long, 104000 volts delivered, 60 cycle. The upper diagram gives a 
 physical conception of the conditions along the line. The curves show the variation 
 in current and voltage along the circuit. 
 
 through the application of the so called auxiliary con- 
 stants of the circuit. 
 
 The auxiliary constants A, B and C of the circuit 
 are functions of its physical properties, and of the fre- 
 quency only. They are entirely independent of the 
 voltage or current of the circuit. The various solu- 
 tions for long transmission circuits are in effect 
 schemes for determining the values of these three 
 auxiliary constants. Mathematically they may be cal- 
 culated, by hyperbolic functions or by their equivalent 
 convergent series. Graphically they may be obtained 
 to a high degree of accuracy from the accompanying 
 Wilkinson Charts for overhead circuits not exceeding 
 300 miles in length. Having determined the values for 
 these three constants for a given circuit, the remainder 
 of the solution is just as simple as for short lines It 
 is only necessary to apply any desired load conditions 
 to these constants and plot the results by vector dia- 
 grams. 
 
 DIAGRAM OF THE AUXILIARY CONSTANTS 
 
 In Fig. 22 are shown voltage and current diagrams 
 lepresenting the application of the auxiliary constants 
 to the solution of transmission circuit problems. To 
 construct the voltage vector diagram, the two auxiliary 
 constants A and B are required, and to construct the 
 current vector diagram, constants A and C are required. 
 Since these diagrams are based upon one volt and 
 one ampere at the receiving-end, it is necessary to 
 multiply the values of the auxiliary constants by the 
 volts or the amperes at the receiving-end, in order to 
 apply the auxiliary constants to a specific problem. 
 Since the diagrams are shown corresponding to unity 
 power-factor, it will also be necessary to change the 
 position of the impedance and charging current triangles 
 in case the power-factor differs from unity. This will 
 be explained later. 
 
 Constants a■^ and a^ — Refer- 
 ring to the voltage diagram, Fig. 
 22, if the line is electrically short 
 the charging current, and conse- 
 quently its effect upon the volt- 
 age regulation is small. In such 
 a case the auxiliary constant o^ 
 would be unity, and the auxiliary 
 constant aj would be zero. In 
 other words, the impedance dia- 
 gram would (for a power- 
 factor of 100 percent) be 
 built upon the end of the 
 vector ER, the point coin- 
 ciding with the point R. In such 
 a case, the voltage at the send- 
 ing end, at zero load, would be 
 the same as that at the receiving- 
 end. If the circuit contains ap- 
 preciable capacitance, the e.m.f. 
 of self-induction, resulting from 
 the charging currents which will 
 flow, will result in a lower voltage at zero load 
 at the sending-end than at the receiving-end 
 of the line, as previously explained. Obviously, 
 the load impedance triangle must be attached 
 to the end of the vector representing the voltage at the 
 sending-end of the circuit at zero load. This is the 
 vector EO of the voltage diagram. Fig. 22. This volt- 
 age diagram corresponds to that of a 60 cycle circuit, 
 300 miles in length. In such a circuit, the effect of the 
 charging current is sufficiently great to cause the shift- 
 ing of the point from R (in a short line) to the posi- 
 tion shown in Fig. 22. In other words, the voltage at 
 zero load at the sending-end has shifted from ER for 
 circuits of short electrical length, to EO for this long 
 60 cycle circuit. The auxiliary constants a^ and a,, 
 therefore, determine the length and position of the vec- 
 tor representing the sending-end voltage at zero load. 
 Actually, the constant a^ represents the volts resistance 
 drop due to the charging current, for each volt at the 
 
 300 VllLES 
 RECEIVING-END 
 
GRAPHICAL SOLUTION FOR LONG LINES 
 
 65 
 
 receiving-end of the circuit. That is, the line OF equals 
 approximately one-half the charging current times the 
 resistance R, taking into account, of course, the dis- 
 tributed nature of the circuit. If the circuit is short, it 
 would be sufficiently accurate to assume that the total 
 charging current flows through one-half of the resist- 
 ance of the circuit. To make this clear, it will be shown 
 later that, for problem X, the resistance per conductor 
 R = 105 ohms and the auxiliary constant C, = 
 0.001463. Thus, this line will take 0.001463 ampere 
 charging current, at zero load, for each volt maintained 
 at the receiving-end, and since OF = approximately 
 
 R 105 
 /c X — \ve have OF (a^) = 0.001463 X = 
 
 0.0768075. The exact value of O2 as calculated rigor- 
 ously, taking into account the distributed nature of the 
 circuit, is 0.076831. Since the charging current is in 
 
 s 
 
 VOLTAGE DIAGRAM 
 
 <- VOLTAGE AT RECEIVING-EN0=ONE VOLT- 
 
 H 
 
 CURRENT DIAGRAM 
 
 -CURRENT AT RECEIVING -END=ONE AMPERE 
 
 FIG. 22 — DI.\GR.'\MMATIC REPRESENTATION OF AUXILIARY CONSTANTS 
 OF A TRANSMISSION CIRCUIT 
 
 The vectors are based upon one volt and one ampere being 
 delivered to the receiving end at unity power-factor. These 
 diagrams correspond to those of a long circuit, 
 leading quadrature with the voltage ER, the resistance 
 drop OF due to the charging current is also at right 
 angles to ER, as in Fig. 22. 
 
 The length of the line FR or (/ — Oj), represents 
 the voltage consumed by the charging current flowing 
 through the indi^ctance of the circuit. This may also 
 be expressed with small error if the circuit is not of 
 
 X 
 great electrical length as /o X — • The reactance per 
 
 conductor for problem X is 249 ohms. Therefore FR = 
 249 
 
 0.001463 X 
 
 0.182143 and Oi =: i.oooooo 
 
 0.182143 = 0.817857. The exact value for o, as cal- 
 culated rigorously, taking into account the distributed 
 nature of the circuit, is 0.810558. The vector FR, re- 
 presenting the voltage consumed by the charging cur- 
 rent flowing through the inductance, is naturally in 
 quadrature with the vector OF, representing the voltage 
 consumed by the charging current flowing through the 
 resistance of the circuit. 
 
 Constants b^ and b^ represent respectively the re- 
 sistance and the reactance in ohms, as modified by the 
 distributed nature of the circuit. The values for these 
 constants, multiplied by the current in amperes at the 
 receiver-end of the circuit, give the IR and IX volts 
 
 CONSTANT (A) 
 
 ~60 Too BO 200 555 5o5 350 400 460 600 
 TRANSMISSION DISTANCE— MILES 
 
 CONSTANT (B) 
 
 25 CYCLtS bj 
 J5 CVCLES^ 
 
 SO CYCLES b} 
 
 SO CYCLES b| 
 
 100 1 50 200 250 300 360 400 460 600 
 TRANSMISSION DISTANCE-MILES 
 
 CONSTANT (C) 
 
 100 150 MO 250 300 360 400 
 TRANSMISSION DISTANCE-MILES 
 
 FIG. 23 — VARIATION OF THE AUXILIARY CONSTANTS FOR CIRCUITS 
 OF DIFFERENT LENGTHS 
 
 drop consumed respectively by the resistance and the 
 reactance of the circuit. To illustrate this, the values 
 of R and X for problem X are i? = 105 ohms and X = 
 249 ohms per conductor. The distribution effect of the 
 circuit modifies these linear values of R and A' so that 
 
66 
 
 GRAPHICAL SOLUTION FOR LONG LINES 
 
 their effective values are b^ = 91.7486 and b^ = 
 235.868 ohms. The impedance triangle, as modified so 
 as to take into exact account the distributed nature of 
 the circuit, is therefore smaller than it would be if the 
 circuit were without capacitance. 
 
 Constants c^ and c^ represent respectively conduct- 
 ance and susceptance in mhos as modified by the dis- 
 tributed nature of the circuit. The values for these 
 constants, multiplied by the volts at the receiving-end 
 of the circuit, give the current consumed respectively 
 by the conductance and the susceptance of the circuit. 
 To illustrate, the value of B for problem X is 0.001563 
 mho per conductor. The distribution effect of the cir- 
 cuit modifies this fundamental value so that its effec- 
 
 CONSTANT (A)=(a|i-ja;) 
 
 V4TH wave 1/2 WAVE %TH WAVE FULL WAVE 
 
 FIG. 24 — VARIATION OF THE AUXILIARY CONSTANTS 
 
 For a 60 cycle circuit (problem X) up to full wave length, 
 tive value c^ = 0.001463. The value of Cj is so small 
 that its effect is negligible for all except very long cir- 
 cuits. For power circuits it will usually be sufficiently 
 accurate to neglect c^. The value c^ will in such cases 
 represent the charging current at zero load per volt at 
 the receiving-end. Thus c^, multiplied by the receiving- 
 end voltage, gives the charging current at zero load for 
 the circuit. For problem X, c^ = 0.001463, and this, 
 multiplied by the receiving-end voltage to neutral 
 60044 = 87.85 amperes charging current per con- 
 ductor. 
 
 VARIATION IN THE AUXILIARY CONSTANTS 
 
 The curves. Fig. 23, will serve to illustrate in a 
 general way how the auxiliary constants vary for t^oth 
 
 25 and 60 cycle circuits for lengths up to and includ- 
 ing 500 miles. In other words these curves have been 
 plotted from calculated values for these constants for 
 certain circuits. 
 
 When the circuit is short, these constants do not 
 vary materially from the linear constants of the cir- 
 cuit, but when the circuit becomes long, they depart 
 rapidly, particularly if the frequency is high. 
 
 AUXILIARY 
 OONSTANTS 
 
 WAVE LENGTH OF THE CIRCUIT AND TRANSMISSION DISTANCE-MILES [ 
 
 '/sTH 
 
 'A 
 
 . %TH 
 
 Vt 
 
 .VaTH 
 
 % 
 
 '/sTH, 
 
 FULL 
 
 369.9 
 MILES 
 
 1739.8 
 
 .MILES' 
 
 III09.7 
 MILES 
 
 1479.5 
 MILES. 
 
 1849.4 
 MILES, 
 
 2219.3 
 .MILES 
 
 2589.2 
 MILES 
 
 2959 1 
 MILES 
 
 ■a, 
 
 82 
 
 f .7/4 
 + ,//3 
 
 
 +.323 
 
 -.789 
 + .350 
 
 
 
 -■9-f2 
 -.622 
 
 
 -/. /O-f 
 
 + /./9/ 
 -.938 
 
 +1.922 
 
 
 
 * 105 
 
 ••26; 
 
 * 87 
 
 + 428 
 
 -77.5 
 *3S0 
 
 -27* 
 + SS.S 
 
 -330 
 -330 
 
 -/22 
 -60s 
 
 + 293 
 -S60 
 
 + &70 
 -/3S 
 
 0, 
 
 -.000075 
 +.00/7^ 
 
 -.00060 
 
 +.OOX.'*? 
 
 -oo/z 
 +.00169 
 
 'OPth 
 -.0003Z2 
 
 -,00101 
 -.00Z50 
 
 ■*-.ooo7/ 
 -.003S 
 
 +.O028 
 -.00233 
 
 +.0039 
 
 •^.ooo78 
 
 (A) 
 
 .72 5 
 
 .323 
 
 /9o°oo' 
 
 .843 
 //Si' OS' 
 
 I.Z09 
 
 //So'oo' 
 
 I.IZ9 
 
 /zi-fzC 
 
 /./04 
 /27d'00' 
 
 '■SZ8 
 /32l°ll' 
 
 I.9ZZ, 
 /3i<fOO 
 
 
 
 
 
 
 
 
 
 (B) 
 
 30I--* 
 
 .+ 37 
 
 3se.s 
 
 //02°2?' 
 
 ZB2.3 
 //(S°3l' 
 
 ■H9.S 
 /ZZS°07 
 
 */9.^ 
 /S5S°3V 
 
 ^35.7 
 /2 9 7=23 
 
 iSZ.t 
 /345:3.»' 
 
 
 
 
 
 
 
 
 
 (C) 
 
 .00'743 
 
 •002527 
 //o/°2«' 
 
 -OOZO7S 
 /IZi°ZI- 
 
 .00/633 
 //9/'Z(; 
 
 .0027/i" 
 /Z17'39' 
 
 .003J82 
 /zil'Zi 
 
 .O03677 
 /330'/S' 
 
 .003947 
 /37/''2«' 
 
 
 
 
 
 
 
 
 
 FIG. 25 — VAHIATION OF THE AUXILIARY CONSTANTS 
 
 For problem X up to full -wave length. 
 
 The auxiliary constants have been calculated for 
 problem A' up to and including a full wave length, 
 namely 2959 miles. Calculations were made only for 
 distances representing each i/8xh. wave, that is each 
 370 miles. The results are tabulated in Fig. 25, and 
 are plotted graphically in Fig. 24. It is interesting to 
 note how these auxiliary constants vary with increasing 
 negative and positive values as the circuit increases in 
 length. A polar diagram is plotted in Fig. 26, indicat- 
 ing the manner in which the auxiliary constant A and 
 its rectangular co-ordinates vary. Although these ex- 
 treme variations are instructive and interesting, ^hey 
 are not encountered in power transmission circuit:;, al- 
 though they will be in long distance telephone practice. 
 
 Vi WAVE LENGTH, 
 739.8 MILES 
 
 %TH WAVE LENGTH 
 1109.7 MILES, 
 
 V2 WAVE LENGTH 
 1479.6 MILES 
 
 '/4th wave LENGTH 
 
 369.9 MILES 
 
 i> ,FULL WAVE LENGTH 
 2959.1 MILES 
 
 T2.0 
 CONSTANT (a) 
 
 %TH WAVE LENGTH 
 I 849.4 MILES 
 
 %TH WAVE LENGTH 
 2589.2 MILES 
 
 3/4 WAVE LENGTH 
 2219.3 MILES 
 
 • •M . 6- 
 FIG. 26 — POLAR DIAGRAM 
 
 Showing the variation of the auxiliary constant A for 
 
 hlf*m y nr» *-n full timro 1at->n-fV. 
 
 .^..lu ,. »»'fe Vil^, Ytli lCH.i»_»ll Ul lilC 
 
 problem X, up to full wave length. 
 
 THE WILKINSON CHARTS 
 
 Mr. T. A. Wilkinson has prepared charts frum 
 v.hich the auxiliary constants may be read directly, thus 
 abridging a great amount of tedious mathematical cal- 
 culation. These charts, are plotted for circuits of 
 lengths up to and including 300 miles.* 
 
 ♦Similar Charts by Mr. Wilkinson were published in the 
 Electrical World for Mar. 16, igi8. 
 
GRAPHICAL SOLUTION FOR LONG LINES 
 
 67 
 
 CHART V— WILKINSON CHART A 
 
 (FOR DETERMININQ AUXILIARY CONSTANTS-ZERO LOAD VOLTAGE) 
 o 
 
 00 COOT en O (X O 
 
 CVCLES RESISTANCE (OHMS) X SUSCEPTANCE ( MlCROr." lOS)- PER MIUE 
 
66 
 
 GRAPHICAL SOLUTION FOR LONG LINES 
 
 their effective values are b^ = 91.7486 and b^ = 
 235.868 ohms. The impedance triangle, as modified so 
 as to take into exact account the distributed nature of 
 the circuit, is therefore smaller than it would be if the 
 circuit were without capacitance. 
 
 Constants c^ and c^ represent respectively conduct- 
 ance and susceptance in mhos as modified by the dis- 
 tributed nature of the circuit. The values for these 
 constants, multiplied by the volts at the receiving-end 
 of the circuit, give the current consumed respectively 
 by the conductance and the susceptance of the circuit. 
 To illustrate, the value of B for problem X is 0.001563 
 mho per conductor. The distribution effect of the cir- 
 cuit modifies this fundamental value so that its effec- 
 
 COrslSTANT (A) = (a| <-ia2) 
 
 '/4THWAVE 1/2 WAVE %THWAVE FULL WAVE 
 
 KIG. 24 — VARIATION OF THE AUXILIARY CONSTANTS 
 
 For a 60 cycle circuit (problem X) up to full wave length, 
 tive value c^ = 0.001463. The value of c^ is so small 
 that its effect is negligible for all except very long cir- 
 cuits. For power circuits it will usually be sufficiently 
 accurate to neglect c^. The value c^ will in such cases 
 represent the charging current at zero load per volt at 
 the receiving-end. Thus c^, multiplied by the receiving- 
 end voltage, gives the charging current at zero load for 
 the circuit. For problem X, c^ = 0.001463, and this, 
 multiplied by the receiving-end voltage to neutral 
 60044 ^= 87.85 amperes charging current per con- 
 ductor. 
 
 VARIATION IN THE AUXILIARY CONSTANTS 
 
 The curves. Fig. 23, will serve to illustrate in a 
 general way how the auxiliary constants vary for t^oth 
 
 25 and 60 cycle circuits for lengths up to and includ- 
 ing 500 miles. In other words these curves have been 
 plotted from calculated values for these constants for 
 certain circuits. 
 
 When the circuit is short, these constants do not 
 vary materially from the linear constants of the cir- 
 cuit, but when the circuit becomes long, they depart 
 rapidly, particularly if the frequency is high. 
 
 AUXILIARY 
 OONSTANTS 
 
 WAVE LENGTH OF THE CmOUIT AND TRANSMISSION DISTANCE-MILES | 
 
 '/sTH 
 
 'A 
 
 . %TH 
 
 '/2 
 
 .%TH 
 
 % 
 
 '/sTH, 
 
 FULL 
 
 369.9 
 
 MILES 
 
 1739 8 
 MILES' 
 
 III09.7 
 MILES 
 
 1479.6 
 MILES, 
 
 1849.4 
 MILES, 
 
 2219 3 
 .MILES 
 
 2589.2 
 
 MILES 
 
 2959.1 
 MILES 
 
 ■ai 
 
 + .7/i 
 
 
 +.323 
 
 -.789 
 + .350 
 
 -/. 20? 
 
 
 -■942 
 -.iZI 
 
 
 -I.I Of 
 
 *-l.l9l 
 -.953 
 
 + /.922 
 
 
 &.■ 
 
 *■ lOS 
 
 <-28; 
 
 + 87 
 + 428 
 
 -77.5 
 + 350 
 
 -Z7i 
 
 *ss.s 
 
 -330 
 -330 
 
 -IZi. 
 -60s 
 
 + Z92 
 -SiO 
 
 + &70 
 -I3S 
 
 0, 
 
 -.000075 
 +.00/7'* 
 
 -.00050 
 
 -oo/a 
 
 -onik 
 -.000322 
 
 -.OOIOI 
 
 -.00250 
 
 +.0007i 
 
 -.0035 
 
 +.O028 *:0039 
 -.00233 +.0007S 
 
 (A) 
 
 .7ZS 
 
 .3Z3 
 / 90^00 
 
 .8^3 
 
 lis(,° oi 
 
 1.2.09 
 
 //■ao'oo 
 
 I.IZ9 
 /2/3°24' 
 
 1.104 
 
 /Z7O°C0' 
 
 '■SZS 
 /32l°ll' 
 
 I.92Z, 
 /3t0°OO 
 
 
 
 
 
 
 
 
 
 (B) 
 
 301.* 
 
 1-37 
 /79°3^' 
 
 358.S 
 //02°2'7' 
 
 282.3 
 
 /ZUg'oT 
 
 il9.3 
 /sS8«3-f 
 
 i-iS.7, 
 
 /z^rz-i 
 
 i^2.4 
 /348-3-f' 
 
 
 
 
 
 
 
 
 
 (C) 
 
 ■ oon-t-i 
 
 ■002527. Ooa07i' 
 
 .00/433 
 //?/°26' 
 
 .0027/5 
 
 /217'rf 
 
 .0O3582 
 /2^l'2f 
 
 .003i77 
 /320'IS' 
 
 ■O03947 
 /37l°2i' 
 
 
 
 
 
 
 
 
 FIG. 25 — VARIATION OF THE AUXILIARY CONSTANTS 
 
 For problem X, up to full wave length. 
 
 The auxiliary constants have been calculated for 
 problem X up to and including a full wave length, 
 namely 2959 miles. Calculations were made only for 
 distances representing each i/d'th wave, that is each 
 370 miles. The results are tabulated in Fig. 25, and 
 are plotted graphically in Fig. 24. It is interesting to 
 note how these auxiliary constants vary with increasing 
 negative and positive values as the circuit increases in 
 length. A polar diagram is plotted in Fig. 26, indicat- 
 ing the manner in which the auxiliary constant A and 
 its rectangular co-ordinates vary. Although these ex- 
 treme variations are instructive and interesting, Ihey 
 are not encountered in power transmission circuits, al- 
 though they will be in long distance telephone practice. 
 
 '/4 WAVE LENGTH. 
 739.8 MILES 
 
 %TH WAVE LENGTH 
 MOB. 7 MILES. 
 
 Vl WAVE LENGTH 
 1479 5 MILES 
 
 %TH WAVE LENGTH 
 I 849.4 MILES 
 
 ■ •/gTH WAVE LENGTH 
 369.9 MILES 
 
 P /FULL WAVE LENGTH 
 2969.1 MILES 
 
 +2.0 
 CONSTANT (a) 
 
 %TH WAVE LENGTH 
 2589.2 MILES 
 
 3^ WAVE LENGTH 
 2219.3 MILES 
 
 FIG. 26 — POLAR DIAGRAM 
 
 Showing the variation of the auxiliary constant A for 
 problem A', up to full wave length. 
 
 THE WILKINSON CHARTS 
 
 Mr. T. A. Wilkinson has prepared charts frum 
 v.hich the auxiliary constants may be read directly, thus 
 abridging a great amount of tedious mathematical cal- 
 culation. These charts, are plotted for circuits of 
 lengths up to and including 300 miles.* 
 
 ♦Similar Charts by Mr. Wilkinson were published in the 
 Electrical World for Mar. 16, 1918. 
 
GRAPHICAL SOLUTION FOR LONG LINES 
 
 fi7 
 
 CHART V— WILKINSON CHART A 
 
 (FOR DETERMININQ AUXILIARY CONSTANTS-ZERO LOAD VOLTAGE> 
 
 CVCLES RESIST ANCe""(0HMS) X SUSCtPTANCE (MICR0K;M0S)-PER MILE 
 
68 GRAPHICAL SOLUTION FOR LONG LINES 
 
 CHART VI— WILKINSON CHART B 
 
 0.90 I op 
 
 
 (FOR DETERMINING AUXILIARY CONSTANTS-LINE IMPEDANCE) 
 2601 
 
 CHART B 
 
 PROBLEM 
 FIND THE AUXILIARY CONSTANTS FOR A TH^EE PHASE 80 CVCLE 
 TRANSMISSION CIRCUIT 300 MILES LONO GON3I8TINO OF THSEE 
 *000 STRANDED COPPER CONDUCTORS WITH lO' X Kl' X 20' FLAT SPACING 
 - ^10X10X20-13.6- EQUIVALENT DELTA SPACINGl-FROM WIRE 
 
 TABLES THE FOLLOWING VALUES PER MILE Of S'NGlE CONDUCTOR ARE 
 OBTAINED, r-0.3B OHM. X- 0.83 OHM 
 
 TO FIND THE CONSTANTS FROM THE CHART. 
 CONSTANT "b"- FROM THE DOT AT THE RIGHT AT THE INTERSECTION C 
 LINES FOfl 300 MILES AND 60 CYCLES FOLLOW HORIZONTALLY TO THE 
 RiCHT TO AN INTERSECTION WITH THE DIAGONAL FOR F- 35 
 VERTICALLY ABOVE THE LATTER POINT READ THE VALUE OF "b"- D I 7 
 CONSTANT "b^ -FROM THE DOT ABOVE AT THE INTERSECTION OF LINES 
 FOR 300 MILES AND 60 CYCLES FOLLOW VERTICALLY UPWARD TO AN 
 INTERSECTION WITH THE DIAGONAL FOR X-0 83 HORIZONTALLY TO 
 THE RIGHT OF THE LATTER PQINT READ THE VALUE OF "b'l ■■ 236. 
 CONSTANT (B)- THIS IS THE TQTAl IMPEDANCE PER CONDUCTOR TO 
 THE LOAD CURRENT IT MAY BE READ FROM THE CHART AS FOLLOWS 
 ON THE CIRCUUR ARC IN THE UPPER RIGHT HAND QUADRANT AT THE 
 INTERSECTION OFCOORDINATEa"bj-Bl 7 ANo"b'j- 336 HEAD THE 
 VALUE OF iai->2b2. 
 
 OOPVRIOHT I9ia BY T, K WILKINSON 
 
 CYCLES 
 
 0.15 0.20 0.25 0.30 
 RESISTANCE (OHMS)-PER MILE 
 
GRAPHICAL SOLUTION FOR LONG LINES 
 
 69 
 
 CHART VII— WILKINSON CHART C 
 
 (FOR DETERMINING AUXILIARY CONSTANTS-CHARGING CURRENT) 
 
 0019- 
 
 BOO 
 
 CHART C 
 
 wtoeiEM 
 
 riNO TMt 4UKH.IW* CONSTANTS FQI * TMftEE PMASC CO CClC 
 TR*f*SMiSSiON C'«CU'T 300 Mn.es LOhC consisting Of TMBCC 
 ♦000 8TBANDC0 COPPtC CON0UCTO«S WiTm pO'X iO« W'LAT s^fcC-MJ 
 
 1- .jToTrSTjo - I 3 8 couivw.cntoclt»sp*c«»«<-»»om-«c 
 
 TABLES THE FCXLQWtNO VALUES PC" M'LE OT SINGLE CONOUCTO« W>E 
 OBTAINED r-0.36OHiyl.b-» 2'"'C*'0*'**0S^0'«"'"*»- THe«tfO«C 
 
 rb'.oasKs 7>^ b w 
 
 TO f INO THE CONSTANTS raQM T>^ CWMT 
 qpNST ANT "c"- f WOM THE OCT AT TmC tCFT AT T*«C »iTEItS£CTK>«Of 
 LINE S FOft JOO MILES AND 80 CtClES f OLLOw mOBi20NTALL' TO '►< lE'T 
 TO»N iNTERSeCTONVWlTH Th| CHACON *L fO*» Ht - » *0 Vt»»TlC*H.* 
 ABOVE TmE LATTEB P0<NT ftCAO TmE VALUE O' "C, . - 0000« 
 gQfsiSTANT -C, -tnOU TNt DOT ABOvt AT TmC inTERSCCTONOT imtS 
 roe JOO MILES *N0 M CYCLE S 'OLLOW wCftTtCALLV U»^W*«0 TO *N INTER- 
 SECTION *nTM The DIAGONAL FQA b- S 3 1 mO«U'ONTA4.L» TO ThC tC' T 0* 
 THE LATTER POINT READ The VALUE Of "Cj-OOO'M 
 f^ONSTAtsiT (C) - THtS IS THE TOTAL ChARCnC CuMCNT RCR CONOUCO" 
 »»£RvOLr TONEuTRAl AT The RECEIVER END WiTntN PRACTCAL L«*«TS ThJ 
 VALUE Of 'C" WILL NEVER t «CECO s "^ VAluC Of "Cy TmC Cff ECT C 
 "CI'ON TMC NUMERICAL VALUE Of ' Ci iS TMERtf ORE NECl-CBlE AND 'C' M*» 
 
 BE TAKEN AS EQUAL TO"Cj -0 00<*«, 
 
 C0»tr>CmT )••» B» r A i*K " "NSON 
 
70 
 
 GRAPHICAL SOLUTION FOR LONG LINES 
 
 The reading of these charts is simplified by reason 
 of the fact that all three charts are somewhat similar. 
 In following any of therii, the start is made from the 
 intersection of the short arc representing length of cir- 
 cuit and the straight line representing the frequency. 
 From this intersection a straight line is followed tc a 
 diagonal line and thence at right angles to the constant 
 required. Thus in a few minutes the auxiliary con- 
 stants of the circuit may be obtained directly from the 
 chart, whereas by a mathematical solution from 15 
 minutes to an hour might be consumed in obtaining 
 them. It is not, however, the time saved in obtaining 
 these constants which is most important. The greatest 
 advantage in this graphical solution for the auxiliary 
 constants is that it not only abridges the use of a form 
 of mathematics which the average engineer is inefficient 
 in using, but it tends to prevent serious mistakes being 
 made. In calculating these auxiliary constants by 
 either convergent series or hyperbolic methods, an in- 
 correct algebraic sign assigned to a number may cause 
 a very serious error. Errors of magnitude are less 
 likely to occur when using a comparatively simple 
 graphical solution. 
 
 In order to determine the accuracy obtainable by a 
 complete graphical solution, using the Wilkinson Charts 
 for obtaining the auxiliary constants and vector dia- 
 grams for the remainder of the solutions, 48 problems 
 were solved both graphically and mathematically. 
 These problems consisted of circuits varying between 
 20 and 300 miles in length, and voltages varying be- 
 tween 10 000 and 200000 volts. Twenty- four prob- 
 lems were for 25 cycle, and the same number for 60 
 cycle circuits. The maximum error in supply end volt- 
 ='ge by the graphical solution employing a four times 
 magnifying glass was one-fourth of one percent. A 
 tabulation of the results as determined by various 
 methods for these circuits will follow later. 
 
 APPLICATION OF TABLES 
 
 The application of the tables to long transmission 
 lines follows, in general, the same plan as for short 
 lines, published as Chart II, with such modifications as 
 are produced by the effects of distributed capacitance 
 and reactance. The procedure best suited for long 
 transmission lines is shown in Chart VIII. 
 
 GRAPHICAL SOLUTION OF PROBLEM X 
 
 Problem X — Length of circuit 300 miles, conduc- 
 tors three No. 000 stranded copper spaced 10 by 10 by 
 20 feet (equivalent delta 12.6 feet) Temperature taken 
 as 25 degrees C. Load conditions at receiving-end 
 18000 kv-a, (16200 kw at 90 percent power- factor lag- 
 ging) 104 000 volts, three-phase, 60 cycles. 
 
 104000 
 £r. = = 60046 volts. 
 
 I, 
 
 1732 
 6000 X 1000 
 
 CHART VIM.— APPLICATION OF TABLES TO 
 LONG TRANSMISSION LINES 
 
 (EFFECT OF DISTRIBUTED CAPACITANCE TAKEN INTO 
 ACCOUNT) OVERHEAD BARE CONDUCTORS 
 
 Starting with the kv-a., voltage and power-factor at 
 the receiving end known. 
 
 QUICK ESTIMATING TABLES XII TO XXI INC. 
 
 From the quick estimating table corresponding to the 
 voltage to be delivered, determine the size of the con- 
 ductors corresponding to the permissible transmission loss. 
 
 CORONA LIMITATION— TABLE XXII 
 
 If the transmission is at 30000 volts, or higher, this 
 table should be consulted to avoid the employment of con- 
 ductors having diameters so small as to result in excessive 
 corona loss. 
 
 RESISTANCE— TABLE II 
 
 From this table obtain the resistance per unit length 
 of single conductor corresponding to the maximum operat- 
 ing temperature — calculate the total resistance for one con- 
 ductor of the circuit — if the conductor is large (250000 
 circ. mils or more) the increase in resistance due to skin 
 effect should be added. 
 
 REACTANCE— TABLES IV AND V 
 
 From one of these tables obtain the reactance per unit 
 length of single conductor. Calculate the total reactance 
 for one conductor of the circuit. If the reactance is ex- 
 cessive (20 to 30 percent reactance volts will in many cases 
 be considered excessive) consult Table VI or VII. Hav- 
 ing decided upon the maximum permissible reactance the 
 corresponding resistance may be found by dividing this 
 reactance by the ratio value in Table VI or VII. When 
 the reactance is excessive, it may be reduced by installing 
 two or more circuits and connecting them in parallel, or 
 by the employment of three conductor cables. Using 
 larger conductors will not materially reduce the reactance. 
 The substitution of a higher transmission voltage, with its 
 correspondingly less current, will also result in less react- 
 ance. 
 
 I 
 
 CAPACITANCE SUSCEPTANCE- 
 IX AND X 
 
 -TABLES 
 
 From one of these tables obtain the capacitance sus- 
 ccptance to neutral, per unit length of single conductor. 
 Calculate the total susceptance for one conductor of the 
 circuit to neutral. 
 
 GRAPHICAL SOLUTION 
 
 From the Wilkinson charts obtain the auxiliary con- 
 stants. Applying these auxiliary constants to the load con- 
 ditions of the problems, make a complete graphical solution 
 as explained in the text. Vector diagrams of the voltage 
 and the current at both ends of the circuit are then con- 
 structed, from which the complete performance can be 
 readily obtained graphically. 
 
 60046 
 
 = 99.92 amperes. 
 
 MATHEMATICAL SOLUTION 
 
 As a precaution against errors in those cases where 
 accuracy is essential, the result obtained graphically should 
 be checked by the convergent series or the hyperbolic 
 method. 
 
GRAPHICAL SOLUTION FOR LONG LINES 
 
 71 
 
 From tables the following linear constants per mile 
 
 are determined. — 
 
 »" = 0-35 ohm (Table No. II) 
 
 X = 0.83 ohm {Table No. V by interpolation) 
 
 b = 521 micromhos ( Table No. X by interpolation) 
 
 ' = {in this case taken as zero) 
 
 therefore, 
 
 rb 
 and, 
 
 rb'- — 
 
 = 0-35 X S-2I = 1.82 
 
 0.35 X 5-2r — 950 
 The auxiliary constants of the above circuits are 
 now taken directly from the Wilkinson Charts. This 
 problem is stated on the Wilkinson chart. Following 
 
 AUXILIARY CONSTANTS OF CIRCUIT 
 
 {CALCUUVTEO RIGOROUSLY BY CONVERGENT SERIES) " 
 (A)- + OS 1 006E ■> j 007883 1 (B)"= ♦ 91.7488 t j 238 8880 
 
 (ai + jaj) = (bi + jbj) 
 
 — 8142 / 6" 24' 53 ' -263 083 /e8'44' 41' OHM81 
 
 VOLTAGE DIAGRAM 
 
 (C) = 
 
 - CL00004 1 
 
 (o,. 
 
 j(u>oi4e3 
 
 iC2) 
 
 - 0001464 \ 91' 36' 36' MHO 
 
 RECEIVING-END LOAD OF 99.92 AMPERES 
 AT 90% P.F LAGGING 
 
 % 
 
 -48.871 VOLTS =Ern XSi 
 80.048 VOLTS' 
 
 CURRENT DIAGRAM 
 
 VECTOR. OF. reference; 
 
 S'*-POS1TION OF s 
 ^ FOR LOAD PFr 
 ',\ OF 80* leading 
 
 FOR ABOVE 
 
 PROBLEM 
 H 
 
 .^-<^ 
 .« 
 
 -(O < 
 
 
 u1^ 
 
 rent calculated rigorously which will appear in a later 
 section. The / terms preceding some of the numerical 
 values in Fig. 27 apply to the mathematical treatment, 
 and have no significance in connection with the 
 graphical solution. 
 
 VOLTAGE DIAGRAM 
 
 The vector ER, representing the constant voltage 
 at the receiving-end (for all loads) is first laid off to 
 some convenient scale. Along this vector, starting from 
 E, lay off a distance equal to the receiving-end voltage 
 multiplied by the constant a^ 
 (60046 X 0.810558 = 48,671 
 volts). This is EE of Fig. 27. 
 From F lay off vertically (to the 
 same scale) the line FO equal to 
 the receiving-end voltage multiplied 
 by the constant a^ (60046 X 
 0.076831 = 4613 volts). Connect 
 the points O and £ by a line. 
 This line EO represents the voltage 
 at the sending-end at zero load. 
 This voltage vector may, if desired, 
 +18.626 VOLTS bc locatcd by polar co-ordinates in 
 place of rectangular co-ordinates. 
 If it is desired to work with polar 
 co-ordinates lay off the line EO at 
 an angle of 5° 25' in the forward 
 direction from the receiving-end 
 voltage vector ER. (For the graphi- 
 cal solution it is not necessary to 
 take account of seconds in angles) 
 The length of the vector EO will 
 be found by multiplying the con- 
 stant A by the receiving-end volt- 
 age (0.8142 X 60044 = 48889 
 
 ^CORRESPONDING 
 TOLOAOpr„90% 
 LAGGING 
 
 
 A 
 
 too 
 ^°o volts). 
 
 o» 
 
 VECTOR OF REFERENCE 
 
 
 Fir.. 27 GRAPHIC SOLUTION OF PROBLEM X 
 
 the directions printed on the charts, we obtain for this 
 
 circuit the following values for the auxiliary constants. 
 
 o, = 0.81 bx = 91,7 fi = 0.00004 
 02 = 0.077 bs =^ 235 fs = 0.00146 
 
 From this point on, the solution is made graphically 
 
 as indicated in Fig. 27. It should be noted here that 
 
 the auxiliary constants obtained from the Wilkinson 
 
 Charts are practically the same as those stated at the 
 
 top of Fig. 27, which values were calculated rigorously 
 
 by convergent series. We will employ the rigorous 
 
 values in plotting the diagram so that the values on the 
 
 diagram will agree with the values of voltage and cur- 
 
 Having located the point 0, the 
 impedance triangle is built upon it 
 in the following manner. Since 
 the power-factor of the load is 90 
 percent lagging, determine from a 
 table of cosines what the angle is 
 whose cosine is 0.9. This is found 
 (from Table K) to be 25 degrees, 
 50 minutes. Lay off the line OD 
 an angle with the vector 
 25° 50' in the lagging di- 
 of OD will be determined 
 
 at 
 of reference ER of 
 rection. The length 
 by multiplying the current in amperes per conductor by 
 the auxiliary constant fc, (9992 X 91.7486 = 9167 
 volts). This represents the resistance drop per con- 
 ductor. From the point D thus found draw a line DS 
 .it right angles with OD. This line DS represents the 
 reactance volts per conductor; its length is found by 
 multiplying the current in amperes per conductor by the 
 auxiliary constant b^ (99-92 X 235.868 = 23 568 volts). 
 Connect the point 5 w'ith £, the length of which repre- 
 sents the voltage (70652 volts) at the sending-end for 
 
74 
 
 GRAPHICAL SOLUTION FOR LONG LINES 
 
 CHART IX-PETER'S EFFICIENCY CHART 
 
 FOR DETERMINING TRANSFORMER LOSSES AND EFFICIENCIES 
 
 >i 
 
 EFFICIENCY IN PERCENT AT LOADS STATED 
 
 3 I 1 
 
 I: 
 
 FULL 
 
 < 
 O 
 
 3 
 
 .i7o 
 
 .2 
 .3 
 .4 
 -.5 
 .6 
 .7 
 .8 
 .9 
 -1.0 
 
 -1.6- 
 
 -2 
 
 »- 
 Z 
 
 UJ 
 
 O 
 
 cc 
 
 UJ 
 Q. 
 
 CO 
 CO 
 
 o 
 
 _1 
 cc 
 
 UJ 
 
 a. 
 a. 
 o 
 o 
 
 -2.5 
 
 -3.5 
 
 -4 
 
 -4.5 
 
 5% 
 
 -99 
 
 ■98 
 
 -97 
 
 -96 
 
 -95 
 
 -94 
 
 -93 
 
 L92 
 
 -91 
 
 -99 
 
 r98 
 
 ^97 
 
 r 
 
 -96 
 |-95 
 7 94 
 -93 
 
 -92 
 
 -91 
 
 -99 
 
 -98 
 
 -97 
 
 -96 
 
 95 
 
 -94 
 
 -93 
 
 -92 
 
 ■99 
 
 -98 
 
 -97 
 
 -96 
 
 -95 
 
 •94 
 
 -93 
 
 -92 
 
 li |J- 
 
 '2 '4 
 
 r9i 
 FULL 
 
 -91 
 
 t 
 3 
 4 
 
 -99 
 
 r98 
 
 -97 
 
 -96 
 
 -95 
 
 -94 
 
 -93 
 
 -92 
 
 '91 
 
 -90 
 
 -89 
 
 99 
 r98 
 =-97 
 
 96 
 |-95 
 L94 
 
 93 
 f-92 
 i-91 
 1-90 
 i-89 
 1-88 
 ^87 
 
 i-86 
 1-85 
 -84 
 ■83 
 
 .1% 
 
 .2 
 
 .3 
 
 .4 
 
 .5 
 
 ,6 
 
 .7 
 
 .8 
 
 .9 
 
 ■I.Oq 
 
 < 
 
 O 
 
 _i 
 
 hl.5_^ 
 
 _i 
 
 u. 
 
 I— 2 
 
 - 2.5 
 
 — 3 
 
 z 
 
 UJ 
 
 O 
 
 UJ 
 Q. 
 
 Z 
 .3.5 - 
 
 CO 
 
 CO 
 
 O 
 
 — 4 
 
 i I 
 
 2 4 
 
 EFFICIENCY IN PERCENT AT LOADS STATED 
 
 — 4.5 
 
 5% 
 
 z 
 O 
 
 TO OBTAIN EFFICIENCY AT ANY LOAD LAY STRAIGHT EDGE AT GIVEN IRON AND COPPER LOSS POINTS 
 AND READ THE EFFICIENCY AT REQUIRED LOAD ON THEIR RESPECTIVE SCALES WHERE STRAIGHT EDGE 
 CROSSES THEM. 
 
 VICE VERSA. TO OBTAIN LOSSES. PLACE STRAIGHT EDGE ACROSS ANY TWO GIVEN EFFICIENCY POINTS 
 AND READ PER CENT IRON AND COPPER LOSS ON THEIR RESPECTIVE SCALES. 
 
GRAPHICAL SOLUTION FOR LONG LINES 
 
 75 
 
 CHART X-PETER'S REGULATION CHART 
 
 FOR DETERMINING TRANSFORMER REGULATION 
 
 POWER FACTOR OF LOAD IN PER GENT. 
 00 95 90 85 80 • 70 60 
 
 "T 
 
 Er~i 
 
 I— 
 
 — I 
 
 2— 
 
 »- H 
 
 z 
 
 UJ 
 
 O 
 
 cc' 
 
 UJ 
 Q. 
 
 3— 
 
 UJ 
 
 o 
 z 
 < 
 
 <o 
 
 CO 
 
 UJ 
 
 tr 
 
 4— 
 
 — 2 
 
 z 
 
 UJ 
 
 O 
 
 'en 
 
 UJ 
 0. 
 
 — z 
 
 — 3 
 
 z 
 
 O 
 
 \- 
 
 < 
 
 ■_l 
 
 D 
 O 
 
 UJ 
 
 cr 
 
 —4 
 
 l-z^ 
 
 13 
 
 I 
 
 J 
 
 -J 
 
 -J 
 
 I 
 
 — I 
 
 2-"-! 
 
 -I 
 
 ~3 
 
 -3 
 
 1 
 
 -I 
 
 -n 
 
 -D 
 
 3-1 
 -1 
 
 -I 
 --I 
 
 -| 
 ~J 
 —i 
 
 -I 
 — H 
 
 1 
 "H 
 
 -| 
 
 "D 
 _J 
 
 J 
 _j 
 
 J 
 4-1 
 
 Z "1 
 UJ— 1 
 
 o n 
 
 UJ ' 
 
 < -^ 
 H -I 
 CO--I 
 
 CO -1 
 UJ I 
 
 tr— I 
 
 6-:i 
 
 --1 
 
 H 
 
 --J 
 
 J 
 
 J 
 
 I 
 
 n 
 
 "1 
 —I 
 
 L 
 
 - 
 
 2— 
 
 2 — 
 
 ^3- 
 
 UJ 
 
 O 
 
 UJ 
 
 z 
 o 
 
 h4- 
 < 
 
 _i 
 
 o 
 
 UJ 
 QC - 
 
 I — 
 
 2— 
 
 3 — 
 
 -3— 
 
 4— 
 
 4— 
 
 2 — 
 
 3— 
 
 5— 
 
 5— 
 
 6 — 
 
 - 7— 
 
 H H^ 
 
 6— 
 
 I— 
 
 2 — 
 
 3 — 
 
 4— 
 
 7 — 
 
 I— 
 
 2 — 
 
 3— 
 
 4- 
 
 5— 
 
 6 — 
 
 7 — 
 
 t- 
 f-- 
 
 1^2 
 
 t: 
 
 (-- 
 
 P" 
 
 r-- 
 r- 
 
 F-3 
 h- 
 
 c: 
 I — 
 I" 
 r- 
 [". 
 
 U-4 
 
 F- 
 
 t- 
 
 h- 
 
 b" 
 
 t- 
 
 \- 
 
 F- 
 
 [-- 
 
 ^:^ 
 
 t. UJ 
 I- O 
 
 c 
 t 
 
 c. z 
 
 I- ~ 
 c- 
 
 1—6 
 
 f- 
 r 
 Tuj 
 
 :§ 
 
 U UJ 
 
 ^:| 
 
 r- 
 
 L- 
 
 C-7 
 
 h- 
 C. 
 
 ^ 
 r- 
 
 r 
 
 I— 
 
 L 
 
 L 
 
 L- 
 
 _k 
 
 REGULATION CHART 
 
 TO OBTAIN THE REGULATION FOR 
 A LOAD OF ANY POWER FACTOR. LAY 
 A STRAIGHT EDGE ACROSS THE 
 GIVEN RESISTANCE AND REACTANCE 
 VALUES AND READ THE REGULATION 
 AT REQUIRED POWER FACTOR ON 
 THEIR RESPECTIVE SCALES. 
 
 TO OBTAIN THE RESISTANCE AND 
 REACTIVE COMPONENTS WHEN THE 
 REGULATIONS AT ANY TWO POWER 
 FACTORS ARE GIVEN. PLACE THE 
 STRAIGHT EDGE ACROSS THE GIVEN 
 POINT AND READ THE REQUIRED 
 COMPONENTS ON THEIR RESPEC- 
 TIVE SCALES. 
 
 BROKEN LINES ARE FOR DETER- 
 MINING THE REACTANCE WHEN THE 
 IMPEDANCE DROP AND RESISTANCE 
 DROP ARE KNOWN. THIS IS DETER- 
 MINED WITH A STRAIGHT EDGE. 
 
 — o f'3 
 
 Z 
 ui 
 O 
 
 ■q: 
 
 UI 
 Q. 
 
 h z 
 
 cr 
 
 UJ 
 Q. 
 
 c- 
 1-- 
 
 I 
 
 c: 
 I— 
 
 L- 
 
 I- — 
 
 F- 
 
 (-"- 
 
 L". 
 
 f. 
 
 t.. 
 
 t- 
 
 C" 
 
 t- 
 
 \- 
 
 C- 
 
 h 
 
 C" 
 
 ^- 
 
 r- 
 d" 
 h- 
 CL 
 
 u 
 
 L_ UJ 
 
 r °- 
 r- z 
 
 z 
 
 UJ 
 
 o 
 
 I— 
 
 I- 
 I — 
 
 r 
 
 - UJ 
 
 o r- 
 
 r 
 
 < 
 
 -I- 
 
 o 
 
 ■< 
 
 r 
 
 L_ 
 
 L 
 
 P-uj L. 
 <r L 
 L.. 
 L 
 -4 !- 
 
 r 
 
 r- 
 
 I- 
 
 r~ 
 
 r 
 
 I— 
 
 h 
 
 I- 
 
 r 
 u- 
 
 u 
 
 UI 
 
 O 
 
 z 
 < 
 
 O 
 < 
 
 UJ 
 
 100 95 90 85 80 70 60 
 
 POWER FACTOR OF LOAD IN PER CENT. 
 
76 
 
 GRAPHICAL SOLUTION FOR LONG LINES 
 
 the resistance per conductor including an equivalent 
 value to correspond to the resistance in the high and 
 low tension windings of two transformers will be, — 
 /? + y^j = 105 + 6.25 + 6.2s = 1 17.5 ohms. 
 
 The percent reactance volts of a transformer hav- 
 ing 3.74 percent regulation at 80 percent lagging power- 
 factor and 1.04 percent resistance volts may be read 
 directly from Peter's Regulation Chart (Chart X) by 
 laying a straight edge along the points corresponding to 
 1.04 percent resistance and 3.74 on the 80 percent power- 
 factor line. The intersection of the straight edge with 
 the last solid line at the right will give the percent react- 
 ance, := 4.85 percent. 
 
 The percent reactance volts can also be read di- 
 rectly from the ]\Iershon Chart. To do this, follow 
 
 TABLE M— APPROXIMATION OF RESISTANCE~AND"rE- 
 
 ACTANCE VOLTS FOR TRANSFORMERS OF 
 
 VARIOUS CAPACITIES 
 
 ohms reactance will therefore be 
 
 2882 
 
 := 28.84 ohms 
 
 Transformer 
 Capacity 
 in Kv-a 
 
 Voltage Drop in Percent 
 
 Resistance 
 
 Reactance 
 
 25 cycles 
 
 60 cycles 
 
 25 cycles 
 
 60 cycles 
 
 300 
 500 
 
 750 
 
 2. IS 
 
 1.4 
 
 1.2 
 
 1-3 
 1.2 
 I.I 
 
 4.0 
 
 41 
 4.2 
 
 5-6 
 6.0 
 6.3 
 
 1000 
 1500 
 aono 
 
 1-7 
 1-4 
 1-3 
 
 I.I 
 o.g 
 0.8 
 
 6.0 
 6.2 
 6.4 
 
 6.5 
 7.0 
 7.0 
 
 3000 
 5000 
 7500 
 
 1.2 
 I.I 
 1.0 
 
 0.75 
 0.6s 
 0.6 
 
 6.8 
 
 7.2 
 7.8 
 
 7.0 
 7.0 
 8.0 
 
 10 000 
 15000 
 25000 
 
 I.O 
 
 0.9S 
 0.9 
 
 0.6 
 
 0.55 
 
 0.5 
 
 8.0 
 8.0 
 8.0 
 
 8.0 
 8.5 
 9.0 
 
 upward the vertical line in the Mershon Chart corre- 
 sponding to 80 percent power-factor until it intersects 
 the first arc. From this point of intersection follow the 
 horizontal line to the right a distance corresponding to 
 T.04 percent resistance volts. From this point thus ob- 
 tained follow the vertical line until the arc repre.senting 
 3.74 percent voltage drops is reached. The length of 
 this vertical line will be the percentage reactance volts 
 of the transformer, in this case 4.8 percent. Of course 
 the reactance may, if desired, be calculated by following 
 the general construction traced out as above described 
 upon the Mershon chart, but the chart will give sufiR- 
 ciently accurate values for practical purposes. 
 
 The volts necessary to overcome the reactance of 
 the windings of one of these transfonners is therefore 
 found to be 60 046 X 0.048 = 2882 volts to neutral. The 
 
 99.92 
 
 to neutral for each transformer. Since the reactance 
 of each line conductor is 249 ohms, the reactance per 
 conductor, including an equivalent value to correspond 
 to the reactance in the high and low tension windings of 
 two transformers will be, — 
 
 X + Xt = 24,g + 28.84 + 28.84 = 306.68 ohms. 
 The impedance of one conductor of the circuit of 
 problem X including the raising and lowering trans- 
 formers will be, — 
 
 2 = 117.S -|- y 306.68 ohms 
 and Y = (assumed to be the same as without the trans- 
 formers). 
 
 With the assumed values for the impedance, the 
 performance of the combined circuit may be calculated 
 as though there were no transformers in the circuit. 
 
 VOLTAGE AND CURRENT AT INTERMEDIATE POINTS ALONG 
 THE CIRCUIT 
 
 Thus far we have considered the electrical condi- 
 tion at the two ends of a transmission circuit only. Oc- 
 casionally it may be desired to determine the voltage or 
 the current at a point, or at various points along the 
 circuit. In Fig. 21, graphs of the voltage and of the 
 current are shown for points between the terminals of 
 a circuit corresponding to the condition of zero load, 
 and also of rated load. The graphs were plotted by 
 determining graphically the voltage and the current for 
 points at 50 mile intervals along this 300 mile circuit, 
 as follows: — 
 
 To determine the conditions 250 miles from the 
 sending-end, (50 miles from the receiving-end) the 
 three auxiliary constants were obtained from the 
 Wilkinson charts corresponding to a circuit 50 miles 
 long. In other words, it was assumed that the circuit 
 was only 50 miles long. By multiplying these auxiliary 
 constants by the known voltage and current at the re- 
 ceiving-end of the circuit, voltage and current diagrams 
 were constructed as in Fig. 27 and on these, the corre- 
 sponding values of voltage and current at the sending- 
 end of the 50 mile section were scaled off. This gives 
 the conditions, for the load assumed, at a point 250 
 miles from the sending-end. In a similar manner the 
 voltage and current at this point, corresponding to zero 
 load at the receiving-end, may be obtained. A similar 
 precedure will determine the electrical conditions for a 
 point 100 miles from the receiving-end (200 miles from 
 the sending-end). The auxiliary constants will this 
 time be read from the charts, corresponding to a 100 
 mile circuit, but the same receiving-end conditions will 
 be used, as before. The electrical condition for any 
 intermediate points along any smooth line, may thus be 
 readily determined. 
 
CHAPTER IX 
 PERFORMANCE OF LONG TRANSMISSION LINES 
 
 (RIGOROUS CONVERGENT SERIES SOLUTION) 
 
 THE APPROXIMATE electrical performance of 
 overhead circuits having a length not ex- 
 ceeding 300 miles, may readily be determined by 
 the use of the Wilkinson Charts for determining the 
 values of the auxiliary constants, supplemented by vec- 
 tor diagrams representing the current and voltages of 
 the circuits. In important cases, as a final check upon 
 the values obtained by the simple graphical solution, a 
 mathematical solution yielding rigorous results should 
 be made. If the circuit is more than 300 miles long, a 
 mathematical solution yielding rigorous values will 
 be required for determining the correct values of at 
 least the auxiliary constants. 
 
 FORMS OF RIGOROUS SOLUTIONS 
 
 The most direct method for determining mathe- 
 matically the exact performance of circuits of great 
 electrical length is by the employment of hyperbolic 
 functions, and the fundamental equations are usually 
 expressed in such terms. Many engineers have a 
 general aversion to the use of mathematical expressions 
 employing hyperbolic functions. One reason for this is 
 that the older engineers attended college before the 
 hyperbolic theory as applied to transmission circuits had 
 been developed, and tables of such functions were not at 
 that time available. 
 
 In 1893 Dr. A. E. Kennelly introduced vector 
 arithmetic into alternating-current computation for the 
 first time.* Although real hyperbolic functions had 
 well recognized uses in applied science, it was in 1894** 
 that he, for the first time, suggested and illustrated the 
 application of vector hyperbolic functions to the de- 
 terminations of the electrical performance of transmis- 
 sion circuits. Since ihat time Dr. Kennelly has been a 
 most persistent advocate of the employment of these 
 functions in electrical engineering problems. To ad- 
 vance their use, he has calculated and published numer- 
 ous tables and charts of such functions. Such tables 
 were, until recently, incomplete and the result was that 
 it was necegsary, in using these tables, to interpolate 
 values, thus introducing complications and inaccuracies 
 into the calculations. 
 
 Tables of hyperbolic functions and charts are now 
 sufficiently extensive and complete for accurate work. 
 The universities quite generally are encouraging instruc- 
 tion of students in the hyperbolic theory. It is there- 
 
 *Trans. Am. Inst. Elec. Engrs., Vol. X, page 175 "Im- 
 pedance." 
 
 ~ **"Elecfrical World", Vol. XXIII, No. i, page 17, January 
 1894, "The Fall of Pressure in Long-Distance Alternating- 
 Current Conductors." 
 
 fore to be expected that, in the future, the employment 
 of hyperbolic functions for the solution of long trans- 
 mission lines will come into general use. 
 
 The fundamental hyperbolic equations expressing 
 the electrical behavior of transmission circuits may be 
 expressed in the form of convergent series and, in such 
 form have, in some cases, certain advantages over the 
 hyperbolic form. The convergent series form of solu- 
 tion does not require the employment of tables or charts 
 of hyperbolic functions, whereas hyperbolic forms of so- 
 lutions do require such tables or charts. If, therefore, 
 such tables or charts are not available, hyperbolic solu- 
 tions cannot be employed. 
 
 While the amount of arithmetical work involved is 
 considerable, any degree of accuracy may readily be ob- 
 tained by the convergent series solution by working out 
 the terms for the auxiliary constants until they become 
 too small to have any effect upon the results. This can 
 also be done with hyperbolic functions, but exact inter- 
 polation of such functions from tabular values, may be 
 considered more difficult than the working out of an 
 extra term or two in the convergent series form of solu- 
 tion. The above remarks apply to cases where an un- 
 usual degree of accuracy is required. Later will be in- 
 cluded a tabulation of the performance of 64 different 
 electrical circuits, as determined by a rigorous, and also 
 by eight different approximate methods of calculation. 
 As the rigorous values are taken as 100 percent correct, 
 in determining the percent error by the approximate 
 methods, it was important that the so called "rigorous" 
 values be exact. To make them so, it was found con- 
 venient to employ the convergent series form of solu- 
 tion for these particular problems, covering circuits up 
 to 500 miles long and potentials up to 200000 volts. 
 For the calculation of the performance of practical 
 power transmission circuits, tables of hyperbolic func- 
 tions are now sufficiently complete to yield results well 
 within the errors due to variation in the assumed linear 
 constants of the circuits from their actual values. 
 
 The employment of convergent series requires a 
 working knowledge of complex quantities only, whereas 
 the employment of hyperbolic functions in addition 
 leads into hyperbolic trigonometry. As literature per- 
 taining to the hyperbolic theory becomes more generally 
 available, and as the younger engineers take up active 
 engineering work, the hyperbolic theory will become 
 more generally used. 
 
 For the purpose of providing a choice of rigorous 
 methods, both convergent series and two forms of hy- 
 
78 
 
 CONVERGENT SERIES SOLUTION FOR LONG LINES 
 
 perbolic solutions are given. The numerical values 
 employed in these solutions have been carried to what 
 may appear as an unnecessary degree of precision. The 
 reason for this is to demonstrate the fact that all of 
 these rigorous solutions yield the same results. For 
 practical problems less accuracy would be essential, 
 thus reducing the amount of arithmetical work. 
 
 Before taking up the rigorous solutions, it has been 
 thought desirable to review the rules regarding the use 
 of complex quantities and vector operations. 
 
 COMPLEX QUANTITIES 
 
 The calculation of the auxiliary constants of the 
 circuit by convergent series, and the further calculation 
 of the electrical performance of the circuit, involve the 
 use of complex numbers, that is, numbers containing / 
 terms. Thus A ^= a^ -\- ja^ is a complex quantity. To 
 the beginner, expressions containing / terms may seem 
 difificult to understand. It cannot be made too emphatic 
 that the rules governing the use of such terms are so 
 simple (embodying only the simple rules of algebra) 
 that the beginner will shortly be surprised with the ease 
 at which complex quantities are handled. 
 
 y Terms — In the complex notation Z =^ X -\- jY, 
 the prefix / indicates that the value Y is measured along 
 the axis perpendicular to that of X, or what is called 
 the imaginary axis. There need be no significance at- 
 tached to the symbol / other than that of a mere dis- 
 tinguishing mark, to designate a distance above or be- 
 low the reference axis in the vector diagram. How- 
 ever, great use is made of a further assigned signifi- 
 cance. It has a numerical significance in the form of 
 j = 1 w which enables all formal algebraic opera- 
 tions, multiplication, addition, extraction of roots, etc. 
 incident to computation involving complex quantities, to 
 be carried out rigorously. This numerical designation 
 for y does not prevent its use as a designating symbol 
 for the vertical direction in the vector diagram.* 
 
 PLANE VECTORS 
 
 Alternating voltages and currents which vary ac- 
 cording to the sine or cosine law, may be represented 
 graphically by directed straight lines, called plane vec- 
 tors. The length of the vector represents the effective 
 value of the alternating quantity, while the position of 
 the vector with respect to a selected reference vector, 
 base or axis, gives the phase displacement. The line 
 OP, of Fig. 29, represents a plane vector inclined at an 
 angle of 33° 41' with the base OS (the axis of refer- 
 ence). The length of the line OP is a measure, to some 
 assumed scale, of the effective value of the voltage or 
 current, while the angle SOP gives the phase displace- 
 ment. 
 
 Counter-clockwise rotation is considered positive. 
 Thus, in Fig. 29, if the line OS represents the instantan- 
 eous direction of the current and the line OP that of the 
 voltage at the same instant, the current is represented 
 
 ♦For an extended explanation of / terms, reference is made 
 to Dr. Charles P. Steinmetz's "Enginering Mathematics", and 
 Dr. A. E. Kennelly's "Artificial Electric Lines." 
 
 as lagging behind the voltage by the angle 33° 41', By 
 m.eans of vectors the relative phase position and value 
 of either currents or e.m.f.'s can be represented in the 
 same manner as forces in mechanics. 
 
 The position of P, with respect to O, is usually de- 
 fined in terms of rectangular or polar co-ordinates. In 
 rectangular co-ordinates there are two fixed mutually 
 perpendicular axes, ~XOX and -—YOY (Fig. 31) in 
 the plane of reference. The former, — XOX, is called 
 the real axis, or axis of real quantities. The latter, 
 — YOY, is called the imaginary axis, or axis of imagin- 
 ary quantities. The qualifying adjective "imaginary" 
 does not mean that there is anything indeterminate or 
 fictitious about this axis. The perpendicular projections 
 of P-i (Fig. 31) on the X and Y axes are respectively 
 the real component X, and the imaginary component Y. 
 
 The magnitude and sign of the rectangular com- 
 ponents X and Y completely determine the position of 
 the vector OP. Positive is indicated to right and up- 
 ward, negative to the left and downward as indicated in 
 Fig. 30. Thus, if X and Y are both positive, OP lies in 
 the first quadrant. If X and Y are both negative, OP 
 lies in the third quadrant. If Z is — and Y is +, OP 
 lies in the second quadrant. If X is + and Y is — , OP 
 lies in the fourth quadrant. Any plane vector may be 
 completely specified by its real and imaginary compon- 
 ents X and Y. Thus, beneath Fig. 31, is a table in 
 which the point P is located in the plane by co-ordinates 
 for all quadrants. 
 
 From Fig. 30 it is evident that, mathematically, the 
 quadrature numbers are just as real as the others. The 
 quadrature numbers represent the vertical, and the ordi- 
 nary numbers the horizontal directions. 
 
 VECTOR OPERATIONS 
 
 In general, in the handling of complex numbers in- 
 volving y terms, the simple rules of algebra are followed. 
 In Fig. 32 two vector quantities are shown. Vector A 
 has a magnitude of 5 units and is inclined in the positive 
 or leading direction at an angle of 36° 52' with the 
 horizontal reference vector, and vector B has a magni- 
 tude of 4.47 units, and is inclined in the positive or 
 leading direction at an angle of 63° 26' with the refer- 
 ence vector. These vector quantities are expressed in 
 rectangular co-ordinate rs A ^= -\- 4 -\- jj, B = -\- £ -\- 
 j/j or in polar co-ordinates as ^ = 5 /jd° 52', B = 4.47 
 /63° z6'. The prefix j simply means that the number 
 following it is measured along the vertical or Y axis. 
 The dot under the vector designation indicates that A is 
 expressed as a complex number, so that the absolute 
 value of A would be V i.4)^ -\' (j)* = 5 ^"d of B = 
 V {sY -\- {4Y ^= 4.47. The absolute value of a com- 
 plex number is called its "size" ; while the angle is called 
 its "slope". 
 
 In order to illustrate the handling of complex quan- 
 tities, the various operations of addition, subtraction, 
 multiplication, division, evolution and involution of 
 the vectors A and B in Fig. 32, will be performed. 
 
CONVERGENT SERIES SOLUTION FOR LONG LINES 
 
 » 
 
 Addition — Fig. 33 illustrates the addition of these units and is inclined in the forward direction at a slope 
 vectors expressed in rectangular co-ordinates. The re- of 49° 24' with reference to the initial line, OS. 
 suiting vector will have as its real component, the alge- Subtraction — Fig. 34 illustrates the subtraction A — 
 
 ■^«^ 
 
 >#^ 33- »^' 
 
 \ '; VECTOR OF . 
 """■"S" REFERENCE 
 
 FIG. 29 
 
 FIG. 
 
 31 
 
 
 V 
 
 
 
 
 
 
 
 ■ + j3 
 
 
 
 p 
 
 -a 
 
 
 
 +j2 
 
 
 P 
 
 -1 
 
 
 
 / 
 
 
 + jl 
 
 
 
 
 -3 
 
 
 -2 -1 
 
 
 
 + 1 
 
 1 
 
 +2 
 
 + 3 
 
 ->X 
 
 
 
 -J" 
 
 
 P-3 
 
 
 
 -J2 
 
 
 P-4 
 
 
 
 
 
 -is 
 
 
 
 
 -Y 
 
 p-1 5 x-*3, Y-+2 .•.P-I-+3+J2 
 
 P-2 : X--3. Y-*2 ,".P-2--3*j2 
 
 P-3 } X--3. Y--2 /. P-3--3-J2 
 
 p_4 ; X-+3 Y--2 .•.P-4-+3-J2 
 
 ADDITION A+B 
 
 
 + 
 
 ^i3 
 
 
 FIG. 30 
 
 
 + j2 
 
 \ 
 
 -3 -2 H 
 
 
 
 ♦1 
 
 •? *? ! 
 
 
 
 -jl 
 
 -j2 
 
 -J3 
 
 
 MULTIPLICATION 
 AXB 
 
 -J 
 
 FIG. 32 
 
 VECTORS A &, B 
 
 VECTOR (A) -(a| +182)= +4+] 3- S /if 62' 
 VECTOR (B) =(b| + j b2)"*-2 +j4= AA7 /S3i2Sl 
 
 SUBTRACTION A-B 
 
 FIG. 34 
 
 (A)= +4*j3 
 -(B)- - 2 - j 4 -CHANGE SIGNS AND ADO. 
 (C)'<-2-il 
 
 <C).lf(2)2+(l)2 - 2 J4 /-?y?4' 
 
 DIVISION 
 
 B 
 
 (A)= + 4 + j3 
 + (B)-+2 + i4 
 
 ((J)=«.e + j7 
 
 (C)=Y(6)*+m^ - 9.22 /4y24' 
 
 FIG. 35 "N, 
 
 O-VL-tlS 
 
 VECTOR (C) = I . I 2 -^^^f - 26* 34- 
 
 A 
 
 LENGTH VECTOR (C)—g-= I 12 
 ANGLE VECTOR (C) - " 
 
 36* 52 -63' 26' " " 26° 34' 
 
 {l^)-->t*j3 
 (g|= »2<-)4 
 
 -12-t-ie 
 
 (C)=-4+i22 
 
 FIG. 36 
 
 <C) 
 
 .YW 
 
 4.(22)' - 22.36 
 OR 
 
 /'go "If 
 
 LENGTH (0= & X 4 47 - 22.36 
 ANGLE (C)«=»3*26'*38'52'-I00-|8 
 
 FIGS. 29 TO 36 — EXAMPLES OF VECTOR SOLUTIONS 
 
 braic sum of the reals, and as its imaginary component, B. This is simply addition after the signs of both of 
 
 the algebraic siun of the imaginaries. Thus : the components of the vector to be subtracted have been 
 
 A = + 4 + J3 reversed. Thus,— 
 + B = + 2 + 14 
 
 A+B = C= + 6 + }7 
 
 C = V (6)' + (7)' = 9.22 absolute. 
 The resulting vector has, therefore, a size of 9.22 
 
 A = + 4 + J3 
 — B = — 2 — /4 
 A - B ='C = + 2 — ii 
 
 C — y (2)' + (i)' = 2.24 absolute. 
 
CONVERGENT SERIES SOLUTION FOR LONG LINES 
 
 The resulting vector C has therefore a size of 2.24 
 units and a slope of — 26° 34'. In polar co-ordinates, 
 
 C = 2.24\26° j/. 
 
 Division — To divide one plane vector by another, 
 divide their sizes and subtract their slopes. Fig. 35. 
 Thus, — 
 
 Absolute value of C : 
 
 1. 12 
 
 4-47 
 Angle of inclination of C = 36° 52' — 63° 2& = 
 
 — 26° 34' in the negative direction. In polar co-ordi- 
 nates C =^ i.i2\26° S4'- 
 
 Multiplication — Fig. 36 illustrates the multiplica- 
 tion of the vectors A and B. Here the rules of algebra 
 
 also apply, except that when two / terms are multiplied 
 signs are assigned opposite to those which would be used 
 in the ordinary solution of an algebraic problem. This 
 is for the reason that, — 
 
 hence, p ■= — / 
 
 Hence where p occurs it is replaced by its value 
 
 — I and therefore, — 
 
 - ;■ X ;■ = -f 7 
 ;■• = — ;■ 
 )* = + I 
 i' = -\- j, etc. 
 Thus, to get the product of A and B : — 
 
 ^ = + 4 -1- ;■ 3 
 B = + 2 -F y 4 
 
 + 8 -f y 6 
 — 12 -I- y 16 
 
 ^X5 = C = — 4-fy22 = 22.35 absolute 
 The resulting vector C has therefore a size of 22.35 
 units and is inclined in the positive direction at an angle 
 of 100° 18' to the vector of reference. The polar ex- 
 pression is C = 22.35 \ 100° 18' 
 
 The magnitude and position of the product may be 
 also determined by multiplying the sizes of the vectors 
 and adding their slopes. Thus : — 
 
 Size of C = 5 X 447 = 22.35 (as above) 
 Slope of C = 63° 26' -1- 36° 52' = 100° 18'. 
 
 Involution — Involution is multiple multiplication. 
 To obtain the power of a plane vector, find the power of 
 the polar value and multiply the angle by the power to 
 which the vector is to be raised. Thus, — vector A =^ 
 5 /36° 5^ : and (5 /jd" s^y = f /yf 44' = 25 
 /73° 44'- 
 
 Evolution — To find the root of a polar plane vector, 
 find the root of the polar value and then divide the 
 slope by the root desired. Thus vector A = 5 /j6° 52'; 
 and V 5 /j(5o ^2' = 2.236 /j8° 2&. 
 
 SOLUTION BY CONVERGENT SERIES 
 
 The hyperbolic formula for determining the operat- 
 ing characteristics of a transmission circuit in which 
 exact account is taken of all the electric properties of 
 the circuit is frequently expressed in the following 
 form, — 
 
 E^=E, cosh VZY + h yj^ sink ^ ZV (5/) 
 
 U = h cosh VZY + Ei L ^■"''' ^/^■ (52) 
 
 Since yZY is complex, the hyperbolic functions of 
 complex quantities are required in solving these equa- 
 tions. 
 
 In above formula, expressed in hyperbolic language, 
 the three auxiliary constants A, B and C which take into 
 account the "distributed" nature of the circuit are repre- 
 sented by the quantities — 
 
 A = Cosh yZK (53) 
 
 B = yj~ sinh yjlY (54) 
 
 C = — (=r sink \ZY. (55) 
 
 Vf. . 
 
 Equations (51) and (52) above may therefore be 
 expressed in terms of the auxiliary constants. A, B and 
 C, as follows: — 
 
 E. = Er A + L B (5(5) 
 
 I,.= h A + Er C (57) 
 
 or Er = E. A — I. B (58) 
 
 I, ^ I. A — E.C (5P) 
 
 These three auxiliary constants may be calculated 
 by convergent series as follows : — 
 
 r YZ Y'^Z^ Y^Z^ Y'Z* 1 
 
 [YZ V^ Z^ Y^ Z^ Y* Z* 1 
 
 ^ T YZ Y^Z^ Y^Z» Y*Z* 1 
 
 C= y [/ +^- +-j^ +^-j^ + J^J^+ etc.\ ... (6,) 
 
 The above series are simply expressions for the 
 auxiliary constants as previously stated. These con- 
 stants are functions of the physical properties of the 
 circuit and of the frequency only, and not of the volt- 
 age or the current. After the values for the auxiliary 
 constants have been calculated for a given circuit and 
 frequency their numerical values may be applied di- 
 rectly to any numerical values of E and / for which a 
 solution is desired. From this point on, the perform- 
 ance of the circuit may be determined either by the 
 graphical method previously described or by mathe- 
 matical calculation. 
 
 Any degree of accuracy may be obtained by the use 
 of convergent series for determining the auxiliary con- 
 stants, by simply using a sufficient number of terms in 
 the series. The rapidity of convergence of these series 
 is dependent upon the value of the argument ZY and 
 thus upon the square of the length of the circuit and 
 frequency, and also, to a lesser extent upon the product 
 of total circuit conductance and total circuit resistance. 
 
CON V URGENT SERIES SOLUTION FOR LONG LINES 
 
 8i 
 
 As far as calculations based upon the more or !es« 
 uncertain values of the fundamental constants of the 
 circuit are concerned, the use of three terms in the 
 series expression yields results in a 300 mile circuit 
 which are sufficiently close to the exact values as given 
 by the use of hyperbolic functions (infinite number of 
 terms). In the case of shorter circuits two terms will 
 give a high degree of accuracy. The number of terms 
 necessary will be determined while doing the work, for 
 it is usual to figure out the terras of the series until they 
 
 become too small to be considered when added to — - 
 
 3 
 
 V z 
 or-^. 
 
 In Table N are given values for the auxiliary con- 
 stants (expressed in rectangular co-ordinates) illustrat- 
 ing the convergence of the series for a 300 mile, 60 
 cycle circuit (Problem X), the complete calculation of 
 which will follow. 
 
 Table A'' shows that even for a 60 cycle, 300 mile 
 circuit, three terms give sufficiently accurate results 
 for determining constant A, whereas two terms are 
 sufficient for determining constants B and C. This is 
 on account of the slower convergence of the hyperbolic 
 cosine series. 
 
 TABLE N— CONVERGENT SERIES TERMS FOR PROBLEM X. 
 
 No. of 
 Terms 
 
 Constant A 
 
 Constant B 
 
 Constant C 
 
 I 
 2 
 
 3 
 
 4 
 
 Infinite 
 
 i.oooooo + j 0.000000 
 + 0.805407 -I- j 0.082057 
 -I- 0.810596 + j 0.07673s 
 + 0.810558 + j 0.076832 
 + 0.810558 + j 0.076831 
 
 105 + j 249 
 + 91.3788 + j 235.7211 
 + 91.7527 + j 235.8678 
 + 91.7486 + j 235.8680 
 + 91.7486 + j 235.8680 
 
 + j 0.001563 
 
 — 0.000043 + j 0.001462 
 
 — 0.000041 + j 0.001463 
 
 — 0.000041 + j 0.001463 
 
 — 0.000041 + j 0.001463 
 
 CALCULATION FOR THE AUXILIARY CONSTANTS BY 
 CONVERGENT SERIES 
 
 The form of solution and procedure indicated in 
 Chart XI for the calculation of the auxiliary constants 
 by convergent series is suggested as being complete and 
 easy to follow. 
 
 First the physical characteristics of the circuit and 
 the frequency are stated. These are the only features 
 having any bearing upon the value of the auxiliary con- 
 stants for a given circuit. The voltage and current to 
 be transmitted do not affect the^e constants. The re- 
 sistance, reactance, conductance, and susceptance to 
 neutral per mile are ascertained from the tables for one 
 conductor of the circuit. These values are then multi- 
 plied by the length of the circuit in miles and set down 
 as total per conductor. 
 
 The values of Y and Z must now be set down for 
 the problem in the form of complex quantities. Thus 
 Z = 7? -f ;X = 705 -f ;>./p and F = G -f /B = + 
 /o.oo/5<5? since zero leakage conductance has been as- 
 sumed for this case. Conductance G represents the 
 true power loss in the form of leakage over insulators 
 and of corona loss through the air between conductors. 
 Corona loss corresponding to the assumed atmospheric 
 conditions may be estimated by applying Peek's 
 formula (See Chapter IV on Corona). Insulator 
 
 leakage may be approximated from the most suitable 
 test data available. It is general practice in the solu- 
 tion of all but the very longest high-voltage circuits to 
 ignore the effect of the losses due 10 leakage and corona 
 effect. These losses will be ignored in this case, so that 
 C becomes zero. After Z and Y have been written 
 down in the form of complex quantities the product YZ 
 should be found as previously described for the multi- 
 plication of complex quantities. The second, third and 
 fourth power of YZ may then te found, if desired. 
 Chart XI shows the fourth power, bur on ali bur the 
 longest circuits a total of four terms will be sufficient, 
 and for most problems three terms will give sufficient 
 accuracy. The range of ciccuracy has been previously 
 indicated for a 300 mile circuit on the basis of any 
 number of terms being used up to and including in- 
 finity. The values in Chart XI are carried out to six 
 decimal places whereas four places will usually give 
 sufficient accuracy ^"r calcf'-ifinfr th*! "tlue* <^i th<» i-""- 
 stants A and B. The smallness of the value of con- 
 stant C may make six places desirable when calculating 
 its value. 
 
 After the values of YZ, Y^ Z^, 1" Z* etc., have 
 been calculated they are divided by 2, 24, 720 etc., re- 
 spectively, set down and added to i. This 
 gives the value of the auxiliary constant 
 A, as -f- 0.810558 + ; 0.076831 which is 
 also referred to as a^ -j- /oj. The abso- 
 lute value of the constant A = 0.8142 is 
 simply the square root of the sum of the 
 square of a^ and o^. The polar value of 
 A is thus 0.8142 /5° 24' 53". 
 The solution for the constant B is of the same 
 general form as the solution for the constant A, except 
 that the values of YZ, Y^*Z^, and 7' Z' etc., are divided 
 by 6, 120 and 5040 respectively. After these results 
 are added to / they are multiplied by Z, the product be- 
 ing the value of the auxiliary constant B ox h^ -\- ;'&,. 
 The absolute value of B is obtained in the same manner 
 as the absolute value of A. 
 
 The solution for C is the same as for B except that 
 in place of the constant B series being multiplied by Z 
 it is multiplied by Y and the values of C or c, + /c, 
 obtained. 
 
 AUXILIARY CONSTANTS OF VARIOUS CIRCUITS 
 
 In Chart XII are tabulated exact values for the 
 auxiliary constants for the 64 problems to which fre- 
 quent reference will be made. These auxiliary con- 
 stants have been calculated by convergent series, the re- 
 sults having been checked through the medium of three 
 separate calculations made at different times. They 
 are therefore believed exact to at least five significant 
 digits. The results have been expressed in both rec- 
 tangular and polar co-ordinates. 
 
 CALCULATIONS OF PERFORMANCE 
 
 In Chart XIII is given the complete calculation of 
 the electrical performance for problem X, starting with 
 
82 
 
 CONVERGENT SERIES SOLUTION FOR LONG LINES 
 
 CHART XI— EXAMPLE ILLUSTRATING RIGOROUS SOLUTION FOR THE AUXILIARY CONSTANTS 
 
 BY CONVERGENT SERIES FOR PROBLEM X. 
 
 PHYSICAL CHARACTERISTICS 
 
 OF CIRCUIT -FREQUENCY 
 
 LENGTH^ 300 MILES. CYCLES, 60. 
 
 CONDUCTORS — <t 000 STRANDED COPPER. 
 SPACING OF CONDUCTORS 10 X 10 X 20 FEET. 
 EQUIVALENT DELTA SPACING = l2.e FEET. 
 
 LINEAR LINE CONSTANTS 
 
 FROM TABLES - PER MILE 
 
 TABLE NO. 2, r = .350 OHM AT 25° C, 
 TABLE NO. 5,X = .830 OHM (BY INTERPOLATION). 
 TABLE NO. 10 b= 5.21 i( 10"^ MHO (BY INTERPOLATION). 
 g=(IN THIS CASE TAKEN AS ZERO). 
 
 TOTAL PER CONDUCTOR 
 R=t1 = .350 X 300 = 105 OHMS TOTAL RESISTANCE. 
 X=*l = '830 X 300 = 249 OHMS TOTAL REACTANCE. 
 B=bl = 5.2l X 300 X 10"^ =,001563 MHO TOTAL SUSCEPTANCE. 
 Q=gl= X 300 = MHO TOTAL CONDUCTANCE. 
 
 MULTIPLICATION OF YZ 
 
 v 
 
 z 
 
 = 
 
 
 +, 
 105 t- J 
 
 ,001563 
 249 
 
 
 - 
 
 
 
 . 389 1 87 + J 
 
 
 
 . 1 64 II 5 
 
 YZ 
 YZ 
 
 - 
 
 .389 187 + J 
 . 389 187 + J 
 
 . 1 64 1 1 5 
 .164115 
 
 
 + 
 
 . 151466 - 
 . 026934 - 
 
 .063871 
 .063871 
 
 y2z2 
 YZ 
 
 + 
 
 . 124532 - 
 . 389187 + J 
 
 . 127742 
 , 1641 15 
 
 
 + 
 
 . 048466 + J 
 . 020964 +j 
 
 . 020437 
 . 0497 1 5 
 
 y3z3 
 YZ 
 
 _ 
 
 . 027502 + j 
 . 389187 + j 
 
 .070152 
 . 1641 15 
 
 
 + 
 
 , 1 0703 - j 
 .011513-] 
 
 . 0045 1 3 
 . 027302 
 
 v*z* = 
 
 , 0008 I ■ 
 
 .031815 
 
 NOTE 
 
 THE AUXILIARY CONSTANTS OF 
 THE CIRCUIT (A) (B)*, (C) MAY 
 BE OBTAINED GRAPHICALLY 
 FROM THE WILKINSON CHARTS 
 
 (A)= 
 
 SOLUTION FOR (A) 
 
 - 1^ 2. 24 '' 720 ' 40.320tJ 
 
 
 1 .000 000 
 
 
 "Y = - .194593 + j .082057 
 
 
 2 2 
 ^^-|- = + .005189- j .005322 
 
 
 y3,3 
 
 ■J—- = - .000038 +j .000097 
 
 
 ^i^^ = - . 000000 -j. 000001 
 
 
 (A)= + .810558 + j .076831 
 
 
 (ai+jaj) 
 
 
 = 0.8142 /5' 24' 53" 
 
 (B) 
 
 SOLUTION FOR(B) 
 
 -Zr + ^+v2z2^Y3z3 Y^Z^I 
 
 L. 6 ' 120 ' 5.040 '^362,880* 
 
 
 1 . 000 000 
 
 
 ■^ = - . 064863 +j .027352 
 
 
 ^^ = + .001038- j .001064 
 
 
 y3,3 
 
 ^qIq = - . 000005 +j .000014 
 
 
 .^l.n - - .000000 J.OOPOOO 
 
 
 (B)= Z(+ .93617 +j. 026302) 
 
 
 Z = 1 05 + j 249 
 
 + 98.2978 +j 233.1063 
 
 
 6.5492 +j 2.7617 
 
 (B)= + 91.7486 +j 235.8680 
 
 
 (b| + j b2) 
 
 
 = 253.083 /68° 44' 4 1 OHMS 
 
 (C) 
 
 SOLUTION FOR (C) 
 
 .,r YZ^y2z2,y3z3, Y-^Z^ 1 
 6 ^ 120 6 040 362,880^ 
 
 
 (C)='>'(+ 93617 +j 026302) 
 
 
 Y = +j 001563 
 
 (C)= - 000041 +j 001463 
 
 
 '^'= (Ci+jCj) 
 
 
 =.001464 \ 91° 36 18" MHO 
 
 the values for the auxiliary constants and the receiving 
 end load conditions known. The calculations are car- 
 ried out by the employment of complex numbers, the 
 complete performance being calculated for both load 
 and zero load conditions. In order to give a more clear 
 understanding of these mathematical operations the 
 reader is referred to the vector diagrams of Fig. 37. 
 
 In Chart XIII are given the formulas for determin- 
 ing the E, and /, values under load conditions. On 
 ^'&- 37 these two same formulas are given, but in the 
 form of vector diagrams, upon which vectors the 
 numerical values corresponding to problem X are stated. 
 With the numerical values of the vectors and angles 
 stated, it should be a comparatively simple manner to 
 
 follow graphically (Fig. 37) the mathematical calcula- 
 tions shown in Chart XIII. 
 
 The formulas for £s and 7s which are stated in 
 Chart XIII and in Fig. 37 contain a complex number 
 (Cos ^r ± / sin ^r) not previously stated in connection 
 with the fundamental hyperbolic formulas for long cir- 
 cuits. The formulas previously given were based upon 
 unity power-factor. The introduction of this new com- 
 plex number is made necessary in order that the effect 
 of the power-factor of the load current may be included 
 m the calculations. The function of this new complex 
 nuniii)er is to rotate the current vector through an angle 
 corresponding to the power-factor of the load current. 
 It will be referred to as the rotating triangle. If the 
 
CONVERGENT SERIES SOLUTION FOR LONG LINES 
 CHART XII— AUXILIARY CONSTANTS OF VARIOUS CIRCUITS 
 
 H 
 
 6 
 
 z 
 
 Z 
 
 UJ 
 
 i' 
 
 n 
 
 Z 3 
 
 UJO 
 -■ o: 
 
 O 
 
 CONDUCTORS 
 
 < 
 
 -J 
 
 ? 
 
 z 
 
 i 
 
 LINEAR CONSTANTS 
 
 TOTAL PER 
 CONDUCTOR * 
 
 AUXILIARY CONSTANTS OF CIRCUIT 
 
 THESE AUXILIARY CONSTANTS TAKE INTO ACCOUNT THE EFFECT OF DISTRIBUTED 
 CAPACITANCE . THEY HAVE BEEN CALCULATED RIGOROUSLY BY CONVERGENT SERIES 
 
 rl 
 
 FROM 
 TABLE 
 NO 2 
 
 xl 
 
 FROM 
 TABLE 
 NOS 
 4&5 
 
 bl 
 
 rROM 
 
 TABLE 
 NOS 
 9 A 10 
 
 %\ 
 
 CONSTANT (A) 
 
 CONSTANT (B) 
 
 CONSTANT (C) 
 
 a, aj 
 
 b| b2 
 
 C| Cj 
 
 25CYCLES 1 
 
 / 
 
 2 
 
 3 
 
 4 
 
 20 
 
 M 
 
 It 
 
 M 
 H 
 
 3 
 
 J.J4 
 
 5.3« 
 H 
 
 
 O 
 
 
 ,'?<?'?»4 7+ J.ooo /5s 
 = .?'7'7847 /O»o'32* 
 
 5.^394 + i 5.3600 
 = 7.7 on /44'3'X7' 
 
 ° +J.000057 
 = .Oooo.?7/yo''0'o" 
 
 3 
 
 3.^4 
 
 5.36 
 
 3 7.2 
 ft 
 
 o 
 o 
 
 .9?984 7+ j.ooo /5S 
 
 5.5 394 + ) 5.360O- 
 
 = 7.70R( /44*3'27- 
 
 +3,000057 
 = .oon05-7/90»0'0» 
 
 6 
 t 
 
 7 
 8 
 
 30 
 
 ft 
 
 OOOO COf>PEIZ 
 
 • 
 
 4 
 
 8.3/ 
 
 8.5 
 
 a'." 
 
 
 
 o 
 
 .999666 + J.0OO336 
 = .9996 56 /o' no" 
 
 8. 3092 + J 8.4999 
 = //.886 /4S'3'>'lie 
 
 + j,oooo 91 
 -.OOOOBI /^o'o'cr 
 
 4 
 
 8.3 < 
 
 8.-5 
 
 8/.0 
 
 o 
 o 
 
 .9996S6 + J.00O336 
 = .999 6 56 /O' I' 10' 
 
 S.3083 + J 8.4999 
 ^ 11.986 /4S'39'/^ 
 
 4- jtfOOOO 91 
 
 zz.ooooei /9o*o'a' 
 
 y 
 
 '0 
 
 II 
 
 IZ 
 
 SO 
 
 » 
 
 OOOO COPl='£f^ 
 H 
 * 
 
 » 
 
 4 
 
 ♦ 
 
 '3.85 
 
 /4.I 
 
 /35 
 
 o 
 
 
 
 .999049 -k- j. 000935 
 = .<»<»<? 04 if /o° 3' /i" 
 
 /3.84( + ] /4.0 996 
 = I9.7S7 /4S'3I'44- 
 
 + ),ooo/ 35 
 = .ooo/.T^V5'o»<3'o- 
 
 6 
 
 '3.85 
 
 15.1 
 
 '25 
 
 o 
 o 
 
 ,999056, +• j.000866 
 = .9 99 56/0*2' 58* 
 
 13.941 3+ J 15.0991 
 = 2O.4 83 3/47''29'a0" 
 
 + i.oOO'25 
 
 = .000/J.T /9(fio'<r 
 
 13 
 /-) 
 IS 
 
 Ik 
 
 100 
 
 OOOO f.QRRER 
 
 
 27.7 
 
 3 2.2 
 
 233 
 
 o 
 o 
 
 .996248* J.003224 
 = .<?'?<^P5^ /O' n'7' 
 
 27.6307 + J 32./ 894 
 = 42.4218 /4<I'3I'29" 
 
 +j,000233 
 = .0005 ,,/90»0'0- 
 
 II 
 
 2 7.7 
 
 3 3.2 
 
 226 
 
 o 
 o 
 
 .996249+ J.003IZ6 
 = .796554 /o''/0'47' 
 
 27.6308+ j33.'874 
 = 43./84 ' /5o<>/3'/r 
 
 4. i.o 00 a 2 6 
 
 = .OOO226/?<'"'0'0- 
 
 n 
 
 /3 
 
 n 
 zo 
 
 200 
 
 * 
 
 ft 
 
 ft 
 
 II 
 
 3<7.2 
 
 64.8 
 
 4 64 
 
 o 
 
 .98499'+ i. 009049 
 - .9ff5033 /O'ai'SS" 
 
 38.808 + J64.594 
 - 7^.3 56/5^9«0'/0' 
 
 — .00000/+ 3,0 00462 
 = .oon462/f0"7'27-' 
 
 n 
 
 3 9.2 
 
 6?.2 
 
 434 
 
 o 
 o 
 
 .98-5009+ jiooB464 
 = .9ff3^05-o/0<'29'J'' 
 
 39. 8084+ j 68.965- 
 = 7^/34 /60''37'vr8' 
 
 — .00000 /+J.0 00432 
 
 = .00 04 3 2 /9C'»7'j-4!" 
 
 Zl 
 
 aa 
 
 14 
 
 3oo 
 
 • 
 * 
 
 63^M PLUM, 
 
 II 
 
 44.' 
 
 * 
 
 9/.2 
 
 74 7 
 
 o 
 
 o 
 
 ,966085 + J, 0/6285 
 = .96AZ3 3/0'S7-/~ 
 
 43./033 + j 90.409 
 - /00./S7 /64'30-36' 
 
 —.000004 + j.ooo 73 9 
 
 = .000 7.^9 /90»/7'/0* 
 
 Zl 
 
 44,; 
 
 loi 
 
 672 
 
 o 
 o 
 
 .96,6319 + J.O'465o 
 = .96 6 330 /o'S3-6" 
 
 43.1070 + j 100.077 
 = I09.966 /«»4/'48' 
 
 —.000003+ J.0OO664 
 r.000 6A4 /9cfi/S'Zr 
 
 is 
 
 it 
 ^8 
 
 4O0 
 n 
 
 * 
 
 * 
 
 6 36m flLUM, 
 
 n 
 
 .5 8.8 
 
 1 30 
 
 92 9 
 
 o 
 o 
 
 .940/61 + j. 036738 
 = .94054; /'<'37'4J- 
 
 J6.4555 + J 127.927 
 - /3<j.5?^ /66'il'/6- 
 
 -.ooobos + i. 000909 
 -.onn9a4/9o'3&l'*' 
 
 21 
 
 3 8.8 
 
 /34 
 
 896 
 
 o 
 o 
 
 .94045Z+ J.035BI9 
 = .940 80/ //'3t'S0' 
 
 J6.4664+ ]/3;.842 
 = /43.42,r/<6''-»8'54- 
 
 —.0 00008 + j.ooo 878 
 = .000 878 /90»3l'/r' 
 
 30 
 3/ 
 32 
 
 5oo 
 
 6 3 i>f^ OLUM, 
 * 
 
 n 
 
 73.^ 
 
 /63 
 
 /'60 
 
 o 
 o 
 
 .906642 + j.o4'299 
 
 = .90 7.r»T /^'S^'/.*- 
 
 68.928 + 3 '58.928 
 = /7.-?.J>^ /«<»33''3* 
 
 -.oooo'6 +j.oo'/24 
 
 Zl 
 
 -ri.S 
 
 Its 
 
 * 
 
 1 IZO 
 It 
 
 o 
 
 
 
 .907/09+ 3.03988O 
 = .9079S.^/.J°3''2- 
 
 68.9507+ j /63.76 
 = I77.684 /67'/o'0' 
 
 -.oooo'S + j.oo 1 09S 
 = .00/08.7 /9o»t -'3 3' 
 
 60 CYCLES 
 
 33 
 31 
 35 
 36 
 
 20 
 
 • 
 
 OOOO COPPC 
 
 3 
 
 5.54 
 
 /2.88 
 
 137 
 
 o 
 o 
 
 .999/19+ j, 000379 
 = .9991 IV /O' I' IE" 
 
 5.536 75 + i 1-2.9769 
 = l^.ni/.7 /6i»4ro' 
 
 + 1.000 '37 
 
 = .(100137/^°"°'°' 
 
 3 
 
 .5.54 
 
 '2.88 
 
 137 
 
 o 
 
 o 
 
 ,999//8+ J.OOO379 
 = .9991 19 /O'l' IS" 
 
 5.5-36 75- + J 11.97^9 
 = I4.0I67 /«6»44'0' 
 
 + ..000 ' 37 
 
 = .000/17/^'''0'0' 
 
 37 
 38 
 39 
 
 30 
 
 OOOO CQPPCR 
 
 4 
 
 ft 
 
 8.3' 
 
 2 0.-J 
 
 195 
 
 o 
 o 
 
 • 99901 1 + j,ooo8' 
 
 -.99 9011 /o*a'4 7" 
 
 8.299 + 120. 3887 
 = 2Z.OI4 /6TSI'i- 
 
 + j.ooo I9S 
 = .nr,n,9</90*0-i>' 
 
 ■4 
 
 8.3' 
 
 20.4 
 
 IfS 
 
 o 
 o 
 
 .9990I 1 + j.ooo 8 1 
 = .9980 /' /0»a'47' 
 
 8.299 + ].?0.3 887 
 = ZZ.OI4 /67»5/'6- 
 
 + i,OOOl9S 
 = .ooni9i/9o*0-o- 
 
 4/ 
 
 ^3 
 
 •50 
 
 m 
 
 0t 
 
 OOOO COPPCR 
 
 ft 
 
 » 
 
 * 
 
 /3.85 
 
 1 
 
 34-0 
 
 324 
 
 o 
 
 
 .994416+ j. 002539 
 = .^<7^4 9R /0'7'33' 
 
 1 3.79fZ + j 33.9479 
 = 36.64J^ /«7*52'4r 
 
 *j,poq323 
 
 = .0 00.-»3^/90«0'0' 
 
 6 
 
 /3.8i 
 
 * 
 
 36.4 
 
 30' 
 
 o 
 
 
 
 .994536+ j,00208; 
 = .<?<?4.r2a/o«7'/4- 
 
 /3.7994+ j 36.3432 
 = 3».»74./«^/» 30- 
 
 + i,ooo3oo 
 
 = .00/,^00/90<>0'0" 
 
 4S 
 
 47 
 4S 
 
 /oo 
 
 m 
 H 
 
 OOOO COPPfR 
 
 
 ^7,7 
 
 77.4 
 
 If 
 
 562 
 
 o 
 o 
 
 .97832+ j,007729 
 = .9783.C /0'Z7-/a" 
 
 27.2996 + i 76.9/16 
 = 1^1.6,119 /70»2r36' 
 
 -k0 0ooo/+ j, 00055 8 
 
 = ,000,V,T*/?0»6'"- 
 
 // 
 
 2 7.7 
 'f 
 
 7'7.7 
 
 ^4 2 
 
 o 
 o 
 
 .97847+ J.007452 
 = .978498 /0«a6-/4- 
 
 2 7.3 02 + j 7 9.196,3 
 = 83.77/70»58'J<J* 
 
 -VO0OO0/+ j.ooo 53 8 
 = .ooo.C/(8/90»<-2r 
 
 4'? 
 J-/ 
 
 20Q 
 
 » 
 m 
 m 
 
 3O0 M COPPti 
 
 * 
 
 // 
 
 3 9.2 
 
 /56 
 
 1 1 Ih 
 
 o 
 o 
 
 .9/4/28 + j.o2;243 
 = .9/437,S-//''/9'J'- 
 
 36.954/+ ilSI.79/ 
 = 1.16.13^ /76'/1-Z' 
 
 -.000008* ) .00/084 
 = .00/0 84/90''2i'.?r 
 
 /7 
 
 3,.2 
 
 /6i!> 
 
 IO-44 
 
 o 
 
 o 
 
 .9 I4SZ4+ j. 0/9876 
 = .9/474o //•/4'40' 
 
 3^.964;+ j H,I.S07 
 = /6S.t9 /7T6-3I' 
 
 -»00 000 7+ J ,00 'O (4 
 = .00/0/4 /9ti'2S4r 
 
 -J3 
 
 SS 
 Si 
 
 300 
 
 » 
 • 
 
 6 3^M f^LUM. 
 
 9 
 
 • 
 
 
 220 
 
 It 
 
 1794 
 
 o 
 o 
 
 .8088/6+ 3.03700b 
 = .909663 /Z'iVo' 
 
 38.46,55 + 3 206.359 
 
 - Zt)9.9/3 /79'Z6-Zg 
 
 -.000023 + ].0OI 67 9 
 = .00/67ff/90»47'*' 
 
 2f 
 
 • 
 
 44.1 
 
 24 3 
 
 Ikl4 
 
 o 
 o 
 
 .8IO027+ i,0333O7 
 = .S/070/ /Z'ZI'M' 
 
 38,5002+ J227.9/8 
 = Z3I.I47 /8o«J4'4J- 
 
 -.0000/8 + i.OOl SIO 
 = .ao/r/o /9o'4'*- 
 
 S7 
 60 
 
 AOO 
 H 
 
 H 
 
 n 
 
 6 3&M f1LUM» 
 
 n 
 
 • 
 
 58.8 
 
 3'4 
 
 22IZ 
 
 ff 
 
 o 
 
 
 
 ,67/70/ + j .057759 
 = .674 1 7» /4»54'J4" 
 
 45.8726+ J 280.04 
 = 2 8 3.77/ 80« 4;' JO' 
 
 -.000044+ j.oo/958 
 = .oo/9.r9 /9/«/»'0' 
 
 2J 
 
 •5-8.8 
 
 322 
 
 2/52 
 
 1* 
 
 o 
 o 
 
 ,672455+ i ,056208 
 = .i:74»on /4•4«'3r' 
 
 45-,90/3+ J287./94 
 
 -.000042* j .00/9/2 
 = .oo/«A-» /9/*/S-Z/' 
 
 (.1 
 
 63 
 
 64 
 
 ^00 
 
 m 
 
 
 636M RLUM, 
 
 m 
 /' 
 
 / 7 
 
 73.5 
 
 3 90 
 
 2785 
 
 o 
 o 
 
 .S02772+ J .094-790 
 
 = ..C/l<?ff7/ /9'34';?0- 
 
 4«.96'4 + J 325,247 
 = 32».<?l2/8/°2*'2/" 
 
 -.OOOO8S + J ,003307 
 = .oo3:^o<» f?.?*«'3a' 
 
 21 
 
 7 3. J 
 
 402 
 
 2690 
 
 o 
 o 
 
 ..5 04 852+ j ,0 8' 9 69 
 
 = .,-r//4 6 3 /9' I3-'Z- 
 
 49.06/ + j 335.4/4 
 = 3 3».9» /8''40W3- 
 
 -.000079* J .002a 3o 
 
 = .002 3T2/92'''4.J-- 
 
 *rl is the resistance in ohms at 25° C (77° F), xl the reacfcince in ohms, bl the susceptance in micromhos to neutral (multi- 
 ply by 10-' to convert to mhos). The x and b values for the 636 ocm circ. mil aluminum cable were taken as those of 700000 
 circ. mil copper on the assumption that these two conductors would have approximately the same diameter, gl, the loss restilt- 
 ing from leakage over insulators and from corona has, for simplicity, been assumed as zero. 
 
84 
 
 CONVERGENT SERIES SOLUTION FOR LONG LINES 
 
 CHART XIII— RIGOROUS CALBULATION OF PERFORMANCE WHEN RECEIVING END CONDITIONS 
 
 ARE FIXED 
 
 Er=I04 000 VOLTS 3 PHASE. 
 
 KV-Ar^I 8 000. .KWr^ I 6 200. 
 
 PER PHASE TO NEUTRAL 
 
 KV-ApM=!^ = 6 000. KWrn=!^ = 6 400. Erm=^^^=60 046. 
 
 AUXILIARY CONSTANTS OF CIRCUIT 
 
 ( A) =- -810558 + J.07683I ( B)= -91.7486 * j 235.868 
 
 (ai->ja2) = (b| + j b2) 
 
 = .8142 /s^ 24' 63' = 253.083 /68'44' 41' OHMS 
 
 PFr = 90.00% LAGGING . 
 
 I„ = 
 
 6 OOP X I OOP 
 
 = 99.92 AMPERES. 
 
 SOLUTION FOR E, 
 
 LOAD CONDITIONS 
 
 (C)= - .P0004I+ j. 001463 
 (C, 4-jC2) 
 
 = .001464 \9I 36 1 8" MHO 
 
 SOLUTION FOR L 
 
 .Es=ER(a,*ja2)+lR(coseR£jSiNeR) (b|*ib2)* ls= Ir (cosor ± jSiN0R)(a,^ja2) + Er(c, + jC2)« 
 
 * i THIS. SIGN IS MINUS WHEN THE P. F. IS LAGGING AND PLUS WHEN THE P. F. IS LEADING 
 
 (ai+jag) = + .810668 +JJ576831 
 
 xE 
 
 RN 
 
 6P046 
 
 ErnC^I^J^s) = ■•■ '•Se?! +j 4613 
 j/i36 
 
 (cos Or -j SIN Br) = 
 
 .9 
 
 99.92 
 
 »|p(COS0R-jSINeR) = 4 89.93 -J43.56 
 
 y(a|+ja2)= + .8 i0558'-j .0768ni 
 
 - 72.393 *. , 6.909 
 <■ a347 - ; 35.308 
 
 Ip (COS Or- j SIN Or) = + 89.93- j 43.58< 
 
 X (b| Vj bj) = * 91. 76 tj 235. 87 
 + 8251 + j 21212 
 
 + 10274 - j 3997 
 
 |p.(coseR-jsiN0R) (b|*-jb2) = + 18525 t- j 17215 
 
 + ERN(a|*-ja2) = <- 48671 + i 4613 
 ^SN ^ 
 
 , |p(COS0R-jSlN0f,)(a|*ja2) = -76240 -j:8399 
 
 (C|+jC2)= -.00PP4I <-j.POI463 
 X Erm = 60P46 
 
 Ern(C|»JC2) = - 2.4( 
 
 *■)' 
 
 67196 * i 21828 
 
 Si+|p|(cosoR-jsiNOR)(a|-ia2) = *76.;4o -,28.399 
 
 Iq = t 73.778 *■ j 59.451 
 
 lf(8" 96)^ +(21828)^ 
 • 70 662 VOLTS TO NEUTRAL. 
 
 ■ V(73.778)^+ (69.4511' 
 
 lg = 94.75 AMPERES. 
 
 KW,, - '87.1 96 X 73.778) * (2 1.828 X 59.45 I ) = 6.258 KW PER PHASE. EFFICIENCY = ^ ^e" 255 "" ^ 86.33%. 
 
 KV-A = (70.662 X 94.76) = 6 694 KVA PER PHASE. 
 
 PF. 
 
 6 255 X 100 
 
 93-42% LEADING . 
 
 LOSSff 6255 - 5400 = 856 KW PER PHASE. 
 
 PHASE ANGLES — at full load the voltage at the sending end leads the voltage at the receiver end by the angle 
 tan „,„° = tan '.325 = i8°oo'. and the current at the sending-end leads the voltage at the receiving -end by the angle 
 
 67196 
 
 TAN-' ^^-"^ '■ = TAN- 
 73.778 
 
 ,806 = 38" 52'. HENCE THE CURRENT AT THE SENDING-END LEaJs THE VOLTAGE AT THE SENDING-END BY THE ANGLE 38"52' 
 ANGLE I8'00' = 20"52'. THE POWER-FACTOR AT THE SENDING-END IS THEREFORE COS 20° 52' = 93.42% LEADING AT LOAD SPECIFIED. 
 
 ZERO LOAD CONDITIONS 
 
 EsNO"=''867l *J46I3 
 
 -SNO" 
 
 ' yi4867l)2-»(46l3l^ 
 = 48 889 VOLTS. 
 
 SO ■ 
 
 2.462 -i 87.85 
 
 I 
 
 = y (- 2.462)^ + 87.85)^ 
 |__ = 87.89 AMPERES. 
 
 KWs 
 
 = (48. 57 I X - 2 .462) * (4.6 I 3 X 87. 85) = 285 .43 KW PER PHASE. 
 
 KV-ArnO"*^'^^^ X 87.89 =4 297 KVA PER PHASE. 
 
 DC 285.43 X 100 _ ^ nAf i ran.iof- 
 Kh SO = T-mI 6.64% LEADING. 
 
 REGULATION 
 
 ARISE IN VOLTAGE AT THE SENDING -END OCCURS OF 70 652-48 889 = 21 763 VOLTS TO NEUTRAL WHEN THE LOAD IS INCREASED 
 FROM ZERO TO 99.92 AMPERES AT 90% POWER FACTOR LAGGING AT THE RECEIVER END WITH CONSTANT VOLTAGE AT THE RECEIVING END. 
 
 PHASE ANGLES 
 
 AT ZERO LOAD THE VOLTAGE AT THE SENDING- END LEADS THE VOLTAGE AT THE RECEIVER END BY THE ANGLE TAN"' ^g\'^| °* 
 TAN"' .0947= 5"25' AND THE CURRENT AT THE SUPPLY END LEADS THE VOLTAGE AT THE RECEIVER END BY THE ANGLE TAN"' Sf^ = 
 TAN (-36.7).= 9r 36' -HENCE THE CURRENT AT THE SUPPLY END LEADS THE VOLTAGE AT THE SUPPLY END BY THE ANGLE 19 I' 36)-i5'25 I— 
 86°ll'. THE POWER FACTOR AT THE SENDING-END IS THEREFORE COS 86" 1 1' = 6.64% LEADING AT ZERO LOAD. 
 
 
 load power- factor is lOO percent, this rotating triangle 
 ■will equal / ± j o, hence it has no effect or power to 
 rotate. If the power- factor of the load is 80 percent 
 the rotating triangle would have a numerical value of 
 0.8 ± j 0.(5. 
 
 The various phase angles given in Chart XIII show 
 whether the power-factor at the supply end is leading or 
 lagging. These various phase angles are given to make 
 the discussion complete. Actually, in order to determine 
 whether the power-factor at the supply end is leading or 
 lagging, it is only necessary to note if the supply end 
 
 current vector leads or lags behind the supply end volt- 
 age vector. At the lower end of Fig. 37 combined cur- 
 rent and voltage vectors are shown for this problem, 
 corresponding to both load and zero load conditions. 
 In Chart XIV is given a complete calculation of the 
 electrical performance of problem X, starting with the 
 values for the auxiliary constants and the sending end 
 load condition known. In other words the supply end 
 conditions which were derived by calculation in Chart 
 XIII ii:i\t' in this case been assumed as fixed, and the 
 receiver end conditions calculated. The reason that 
 
CONVERGENT SERIES SOLUTION FOR LONG LINES 
 
 8S 
 
 SOLUTION FOR Es 
 
 Estc" 8° 0*6 
 
 EsN~ 
 
 y + 18 625-' 1 * 
 
 11 , ---t-— * 
 
 -SN;>. 48671 VOLTS r I 
 
 If.. 67 186 VOLTS If 
 
 SOLUTION FOR Is 
 
 ^■.. 
 
 % 
 
 I 8M3AMP . ^ jBimi^ 
 »o\ L .810568-a, 
 
 , 00 1 464= (C) ABSOLUTE, 
 «'i.OOI4e3*C2-<^ 
 
 d- eoo4«X 
 
 ls = 
 
 +76524 AMP. 
 
 -j2t^399 
 
 ls= 
 
 -Z462 AMP 
 
 - j 28J99 jj^ *j 87.85 AMPERES 
 
 * 2.482 AMPERES 
 
 COMBINED CURRENT AND VOLTAGE VECTORS 
 
 LOAD CONDITIONS 
 
 , I 1-18.625 
 
 (^'j. 25. >Er^60 046 VOLTS 
 
 >- -j 43.56 AMPERES 
 
 a.- ZERO 
 
 I LOAD CONDITIONS 
 
 ">« >» -tO 
 
 r = 48 889 VOLTS 
 t«NO__ C3— S' 25' 
 
 *E:Rto*°°*«^^s 
 
 FIG. 37 — GRAPHIC REPRESENTATION OF PROBLEM X 
 
 Illustrating rigorous calculations of performance when receiving end conditions are fixed. 
 
 there is a slight difference between the receiving end 
 conditions as calculated on Chart XIV and the known 
 receiving end conditions is that the value for the sine 
 in the rotating triangle (0.436) in chart XIII was car- 
 ried out to only three places, whereas in Chart XIV it 
 was carried out to four places. If the values for the 
 rotating triangles had been carried out to five or six 
 places in the calculations in both charts, the receiving 
 end conditions would have checked exactly. 
 
 TERMINAL VOLTAGES AT ZERO LOAD 
 
 For a given circuit and frequency, the relation of 
 the voltage at the two ends of the circuit is fixed. The 
 ratio of sending end to the receiving end voltage is ex- 
 pressed by the constant A. The ratio of receiving to 
 
 sending end voltage is expressed by —. . For example, 
 problem X, the sending end voltage under load is 70 652 
 volts. If the load is thrown off, and this sending end 
 voltage is maintained constant at 70652 volts, the re- 
 
 70652 
 
 ceiving end voltage will rise to a value of -„ = 
 
 " ^ 0.8142 
 
 86 775 volts to neutral. The rise in percent of sending 
 
 100 X 86 775 — 70 652 
 
 end voltage is therefore 
 
 22.82 percent. 
 
 70652 
 
 PERFORMANCE OF VARIOUS CIRCUii'S 
 
 In Chart XV is tabulated the complete performance 
 of the 64 problems for which the auxiliary constants 
 are tabulated in Chart XII. The auxiliary constants 
 in Chart XII were applied to the fixed load conditions 
 as stated in Chart XV for the receiving end, and both 
 load and zero load conditions at the sending end were 
 calculated and tabulated. 
 
 The object of calculating and tabulating the values 
 for the 64 problems was two fold. First to obtain data 
 on 25 and 60 cycle problems coverin? a wide '■anee 
 which would provide a basis for constructing curves, 
 illustrating the effect that distance in transmission has 
 upon the performance of circuits and upon the auxiliary 
 constants of the circuit. Second, to give the student a 
 wide range of problems from which he could choose, 
 and from which he could start with the tabulated values 
 as fixed at either end and calculate the conditions at the 
 other end. It is believed that such problems will fur- 
 nish very profitable practice for the student and will 
 also serve as a general guide when making calculations 
 on problems of similar length and fundamental or lineal 
 constants. It is not intended that the figures given for 
 longer circuits, included in these tabulations, shall coin- 
 cide with ordinary conditions encountered in practice. 
 
86 
 
 CONVERGENT SERIES SOLUTION FOR LONG LINES 
 
 CHART XIV— RIGOROUS CALCULATION OF PERFORMANCE WHEN SENDING END CONDITIONS 
 
 ARE FIXED 
 
 K Ws" 1-8 '85 , 
 
 Es= I 22 369 VOLTS 3 PHASE . PFs=93 42% LEAOINQ . 
 
 KV-As=20 082. 
 
 PER PHASE TO NEUTRAL 
 
 KV-AsM= ^2^ = 6 694. KWs^,= '^ = 6 265. EsN= ^^f^ = 70 662 . Is° " '70 65 """ ^ ^^-'^ AMPERES. 
 
 AUXILIARY CONSTANTS OF CIRCUIT 
 
 ( A) =*■ -810558 + j.076831 ( B) = "►S I-7486 + j 236.868 
 
 (a 
 
 .8142 / 5" 24' 53' 
 
 (bi * i 62) 
 
 = 253. 083/[6814414i" OHMS 
 
 (C)= -.000041 fj. 001463 
 (0, fjCj) 
 
 = .001464 \ 9I"36' 18' MHO 
 
 SOLUTION FOR Er 
 
 LOAD CONDITIONS 
 
 SOLUTION FOR 
 
 ER=Es(ai*ia2)-lsf'^°®^s*iSiNes)(b, + ,b2)* Ir=Is(°°s®s*) siNes)(3, + ja,) -EgCCi^iCj)* 
 
 * ± THIS SIGN IS MINUS WHEN THE P. F. IS LAOOINO AND PLUS WHEN THE P. f. IS LEADING 
 
 (^I*ja2)= + .810558 ♦ j.07683 
 X Ec,, = ^ '0652 
 
 Esn(^I*J^2) = *■ 57268 tj 5428 
 
 (COSOs ^jSINes) = + .9342 +1,3567 
 
 X|< 
 
 = 4-94.75 
 
 Ig (coses •^j SIN 63) = + 88.62+ j 33.8 
 X (b, + j bj) =■ + 91.75 + 1235.9 
 
 8122+ 29882 
 7<}73 + j 3101 
 
 |g(coses + jsiNes)(b|+,b2) 
 
 EsN(ai*Ja2) = 
 
 57268 
 
 l3(C0Ses+-jSINag) = + 8S.52 ,j33.8 
 
 X (a, +,82) = * .8 I0558+, .076831 
 
 + 71.75! + , 6.801 
 - 2.597 t , 27.397 
 
 |g(coses*jsiNes)(a,+ia2) 
 
 ■ j 34.198 
 
 (C|+jC2)= -.00004 1 ».j.00l46o 
 
 xEc 
 
 5428 
 
 Esn('-1*)C2) = -2.897 tj 103.36 
 
 Ig (cosBs +j siNeg)(a| +ja2) = +69.154 + 
 
 — |g(coses + jSiNes)(b|+jb2) 149 -j 23983 <- 
 
 E^RN = *■ 67119 - ) 18555 
 
 -CHANGE SIGNS AND ADD- 
 
 ► -EsN(Cl-iC2)= + 2.897 
 \ff = 72.05 1 
 
 yi57119)^ +(13565)^ 
 Erm - 60 057 VOLTS TO NEUTRAL. 
 K\A/ -(67 119 X 72.06 l) + (I 8.665 X 69. 16) = 6 399 KW PER PHASE, 
 
 PF„ - 8 TJo°° = 9a0 1% LAGGING . 
 
 = Yl72.051)^ + lf,9.16l' 
 I - 99.87 AMPERES. 
 
 KV-A = '60.057 X 99.871 = 6 
 
 "^^ "rn 
 
 KV A PER PHASE. 
 LOSSm= 6 266-6 399 = 856 KW PER PHASE. 
 
 EFFICIENCY 
 
 6 399X100 
 
 86.32%. 
 
 PHASE ANGLES *t full load the voltage at the receiver end lags behind the voltage at the sending-end bythe 
 
 ANGLETAN-I g|-^ = tan- ' .325 = 18' 0'; AND THE CURRENT AT THE RECEIVER END LAGS BEHIND THE VOLTAGe' AT THE SENDING-END BYTHE 
 ANGLE TAN-' ^^ = TAN-1.959 =43'50'. HENCE THE CURRENT AT THE RECEIVER END LAGS BEHIND THE VOLTAGE AT THE RECEIVER END BY THE 
 ANGLE 43- 60' -ANGLE 1 8"0' = 26° 50'. THE pOWER-FACTOR AT THE RECEIVER END IS THEREFORE COS 25''50' = 90% LAGGING . 
 
 ZERO LOAD CONDITIONS 
 
 Erno - gsN0(ai-ia2) ^ 48 898(.81056-1 .07683l) ^ 39635-)3757 _ ^^ ^^^ - j 5667 = 60 058 VOLTS. 
 "^ (a2+a2) (.81056)^ + 1.076831)^ ■''°^'' 
 
 iso -^s 
 
 (C| a, + C2a2)+j (C2 ai - c, 37)^ 
 
 ((-.000041 X. 8 1056 1 + 1.00 1 463 X .07683 ll) +j ((.00 1463 X. 8 1056) .0000 41 X. 076831)). 
 
 (' 
 
 I) 
 
 |„„ =48 898 '•»-0000'92 + i.001189 L/ic 898 (.0001 1 9* j. 00 1 7941 = 48 898 X. 00 I 798-87.92 AMPERES. 
 ISO .6629 
 
 REGULATION 
 
 ARISE IN VOLTAGE AT THE SENDING-END OCCURS OF 70 652-48 898 = 21 754 VOLTS TO NEUTRAL WHEN THE LOAD IS INCREASED 
 FROM ZERO TO 99.87 AMPERES AT 90.0 1% POWER FACTOR LAGGING ATI THE RECEIVER END WITH CONSTANT VOLTAGE AT THE RECEIVING END.' 
 
 PHASE ANGLES 
 
 AT ZERO LOAD THE VOLTAGE AT THE RECEIVER END LAGS BEHIND THE VOLTAGE AT THE SENDING-END BYTHE ANGLE 
 TAN"' ^^y =TAN-' .0948-5"25'; AND THE CURRENT AT THE SENOING-END LEADS THE VOLTAGE AT THE SENDING-END BY THE ANGLE 
 
CONVERGENT SERIES SOLUTION FOR LONG LINES 
 CHART XV-CALCULATED PERFORMANCE OF VARIOUS CIRCUITS 
 
 87 
 
 u 
 
 ffl 
 
 
 0: 
 
 RECEIVING-END CONDITIONS 
 FIXED 
 
 SENDING-END CONDITIONS-CALCULATED * 
 
 LOAD CONDITIONS 
 
 LOAD CONDITIONS 
 
 ZERO LOAD 
 
 KV-A 
 
 R 
 
 3 PHASE 
 
 ITO NEUTRAL 
 
 TO NEUTRAL 
 
 TO NEUTRAL 
 
 KV-A 
 
 RN 
 
 KW 
 
 RN 
 
 Ern 
 
 Ir 
 
 PFr 
 
 % 
 
 KV-A 
 
 SN 
 
 SN 
 
 E^SN 
 
 Is 
 
 PFs 
 
 % 
 
 LINC 
 
 DROP 
 
 IN 
 
 %or 
 
 UNC 
 
 LOSS 
 
 IN 
 
 % or 
 
 KV-A 
 
 SNO 
 
 KW 
 
 SNO 
 
 ^SNO 
 
 Iso 
 
 PF 
 
 SO 
 
 % 
 
 25 CYCLES 
 
 / 
 
 2 
 
 I300 
 
 10 000 
 
 « 
 
 433.3 
 
 346.6 
 433.3 
 
 ^ 77-# 
 
 *• 
 
 7.5- 
 
 * 
 
 80 LKO. 
 100 
 
 474.63 
 465.09 
 
 377.i2 
 464.2/ 
 
 6 347 
 6 202 
 
 7'4.7y 
 74.99 
 
 7*il 
 
 94.8/ 
 
 -9.92 
 - 7.41 
 
 8.92 
 7./ 3 
 
 /.96 3 
 
 
 .5 773 
 
 • 
 
 .34 
 
 • 
 
 
 3 
 
 Sooo 
 
 20 000 
 
 II 
 
 / 333.3 
 IkUi.k 
 
 // 5SO 
 
 
 
 
 100 
 
 IS2I.9 
 1 7J6.3 
 
 /44 9..S' 
 / 783.33 
 
 12 653 
 /a 372 
 
 I43.99t 
 144.3% 
 
 7 9.i6 
 49.80 
 
 -9.SS 
 -7/2 
 
 8.7/ 
 7.00 
 
 7.42 2 
 
 
 ft S'i9 
 
 • 
 
 .66 
 
 
 
 3S00 
 
 20 000 
 
 //67 
 II 
 
 93 3 
 
 1 1 (nl 
 
 II SSO 
 n 
 
 
 80 i-flG 
 100 
 
 / 278.6 
 / J53.5 
 
 IOI7.4-S 
 /2SI.22 
 
 /2 733 
 /24 (5 
 
 loo.^l 
 100.97 
 
 7y.i8 
 99.82 
 
 -/0.J9 
 - 7.49 
 
 9.0 5 
 7.2 2 
 
 /0.85 
 u 
 
 
 1 
 
 •94 
 
 • 
 
 
 7 
 8 
 
 8000 
 
 30 000 
 
 2667 
 
 2/33 
 2 667 
 
 I73ZO 
 It 
 
 /54 
 
 So LIS. 
 /oo 
 
 2 ^agj 
 2 S<«S 
 
 2 329.8 
 2 SiO.S 
 
 /9 /2i- 
 /8 i40 
 
 IS 3.11 
 / 5 3.96 
 
 7f.« 
 99.6 » 
 
 -lOAl 
 -7.6 2 
 
 9.2 3 
 7.2 6 
 
 24. 2 f 
 
 
 17313 
 
 * 
 
 
 9 
 
 /o 
 
 Soo'o 
 
 So 000 
 
 /667 
 
 / 333 
 
 / 667 
 
 n 320 
 II 
 
 96.2 
 
 80<-fl<i 
 /oo 
 
 / 8/7.3 
 / 796.4 
 
 14 59.2 
 1 794.2 
 
 19 IS4 
 IS iSS 
 
 94.73 
 96. /4 
 
 80.2? 
 99.88 
 
 -/0.76 
 -7.89 
 
 9.47 
 7.6 3 
 
 -fO.3 2 
 
 
 /7 JO-f 
 
 m 
 
 2.33 
 
 It 
 
 
 IZ 
 
 20000 
 
 * 
 
 60 000 
 
 1 
 
 6667 
 
 .S333 
 
 i 6^7 
 
 34 640 
 
 /?2.5 
 
 100 
 
 7 303.9 
 7 /y^./ 
 
 S S4 1.0 
 7 1 81.2 
 
 3 8 490 
 37 387 
 
 / 8 9.76 
 192.37 
 
 79.97 
 4».»i^ 
 
 -11.11 
 - 7.93 
 
 9.5 3 
 
 7.7/ 
 
 /44.8 
 
 ./3 
 
 
 4.33 
 
 •09 
 
 13 
 M 
 
 22 000 
 If 
 
 83 000 
 11 
 
 7333 
 II 
 
 J' 867 
 7 333 
 
 60 8/0 
 II 
 
 
 
 SOtflQ 
 100 
 
 7 742.S 
 7 9/i.4 
 
 6 4/9.6 
 
 7 9/S.2 
 
 Si 619 
 64 S20 
 
 137.1 
 144.31 
 
 8 2.70 
 
 1OO.OC 
 
 -//.4 3 
 -7.89 
 
 9.42 
 7.94 
 
 599.3 
 
 /4-» 
 
 50 620 
 If 
 
 //.84 
 
 .32 
 
 IS 
 /6 
 
 40 000 
 
 120 000 
 n 
 
 13 333 
 
 10 hlnl 
 13 333 
 
 i,9 Z<)o 
 
 m.s 
 
 80 LUC. 
 
 100 
 
 /4 /06 
 14- 366 
 
 II 648 
 14 366 
 
 77 147 
 74 642 
 
 IS2.iS 
 I97A1 
 
 82.ir 
 
 100.00 
 
 - //.3.f 
 -7.73 
 
 9./ 9 
 7.75 
 
 /ogi 
 
 m 
 
 J.39 
 
 * 
 
 69 030 
 
 '5.66 
 
 .3/ 
 
 n 
 
 IS 
 
 2 5 000 
 
 120 000 
 
 8 333 
 
 (,i,(,7 
 8 333 
 
 69 290 
 If 
 
 1Z0.3 
 If 
 
 SOLUS. 
 /oo 
 
 7 884^ 
 9 03S.4 
 
 7 ISi.l 
 
 8 9/ao 
 
 76 7.J4 
 73 40/ 
 
 I02.7S 
 / 2 2.96 
 
 90.74 
 9S.7S1. 
 
 -lo.n 
 -S.93 
 
 7.34 
 6.96 
 
 2 IBS 
 
 It 
 
 /5.29 
 
 9 
 
 68 253 
 
 ■r 
 
 32.0/ 
 
 .70 
 
 11 
 
 ZO 
 
 40 000 
 
 (40 000 
 
 /3 333 
 
 10 667 
 /3 333 
 
 "SO 830 
 
 IbS 
 
 8oi-/?(4 
 /oo 
 
 13 270 
 /4 4S9 
 
 // 6/0 
 14412 
 
 91 76/ 
 86 86 3 
 
 I44.SZ 
 litAi 
 
 87.44 
 99.48« 
 
 -I3.S3 
 
 - 7.4 i 
 
 8.84 
 8.09 
 
 2780 
 
 n 
 
 11.44 
 
 79 622 
 
 • 
 
 3.<.92 
 
 .6 3 
 
 21 
 22 
 23 
 24 
 
 2 000 
 
 IZO 000 
 
 6 667 
 
 6 333 
 6 667 
 
 6,9290 
 ff 
 
 9t.2 
 ff 
 
 SOl-flG 
 /oo 
 
 7 is 2.7 
 
 se=42.4 
 
 7 loS,4 
 
 75 6 82 
 7/ 762 
 
 75.0 « 
 70 6.64 
 
 99.29 
 9:?.8.f. 
 
 -9.21 
 -3.S7 
 
 S.SI 
 6.57 
 
 3428 
 
 • 
 
 3 9.22 
 
 66 9SO 
 
 5/.2/ 
 
 1.14 
 
 60 000 
 
 Zoo 000 
 
 20000 
 
 !(, 000 
 2000a 
 
 IIS SOO 
 
 ■' 
 
 173.7 
 
 80/-fl«, 
 100 
 
 17 S74 
 22297 
 
 17 O-JS 
 2/ 38/ 
 
 I284S0 
 120 S74 
 
 /36.«3 
 IS4,i4 
 
 9<.99 
 <?f.9W 
 
 -11.21 
 -4.39 
 
 6.55 
 6.90 
 
 8SS9 
 
 9/.0-3 
 
 tl 
 
 III Uii 
 
 n 
 
 76.69 
 
 1.0 1 
 
 25 
 26 
 27 
 2? 
 
 20 000 
 
 i-fo 000 
 
 6667 
 
 ^333 
 6 667 
 
 so 830 
 
 szj, 
 
 80i-flG 
 /OO 
 
 S 9S9.3 
 8 808.? 
 
 Si2l.l 
 7 li-i.l 
 
 «6 404 
 8/ 64 7 
 
 68.97 
 / 7.89 
 
 9-».3 3' 
 8/.34' 
 
 -6.89 
 -1.01 
 
 5.40 
 
 7.'t7 
 
 SS.8S 
 
 I09.i 
 
 76 02.f 
 
 -3.-»7 
 
 '.96 
 
 SO 000 
 
 zoo 000 
 
 li> 667 
 
 (3 333 
 /6 667 
 
 / 16 SOO 
 
 144/1 
 II 
 
 80i-/I5 
 too 
 
 /'I29S 
 20 325 
 
 14 222 
 
 15 066 
 
 /27267 
 IIS 93^ 
 
 II2.Z2 
 
 ni.oi 
 
 49.49' 
 88.90' 
 
 -10.19 
 
 -2.S9 
 
 6.6 7 
 8.4 
 
 II OlS 
 
 202.04 
 
 108 463 
 
 101.4 
 
 '.84 
 
 29 
 30 
 
 IS 000 
 
 140 000 
 
 SOOO 
 
 4000 
 ^ 000 
 
 SO 830 
 
 GI.Zb 
 
 soLfia. 
 100 
 
 6 IS3.S, 
 8SIS.7 
 
 4237 
 5479 
 
 83 04S 
 78iSS 
 
 74.46 
 
 /oa3o 
 
 4«.32' 
 64J2- 
 
 -2.74 
 *2.i9 
 
 5.92 
 9.5 8 
 
 6 665 
 
 208.S4 
 
 73 360 
 
 90.tS 
 
 3. '3 
 
 3/ 
 32 
 
 40 000 
 
 zoo 000 
 
 /3 333 
 
 /o 667 
 /3 333 
 
 IIS SOO 
 
 llS.^ 
 II 
 
 soLue 
 /oo 
 
 /3 277 
 /<? 096 
 
 // 383 
 
 /4 6 72 
 
 123401 
 IIS 762 
 
 I07.i9 
 /6i82 
 
 85.74- 
 7t.8> 
 
 -6.85 
 
 + 0.29 
 
 6.7/ 
 
 lO.OS 
 
 73 /40 
 
 J9S.9 
 
 ft 
 
 /04 J7a 
 
 '2i.3 
 
 3.0/ 
 
 60 CYCLES 
 
 33 
 34 
 
 1 300 
 
 10000 
 
 433.3 
 
 346.6 
 
 433.3 
 
 S 774 
 
 n 
 
 75 
 If 
 
 80 '-"^ 
 100 
 
 499.03 
 
 -<<9.oi 
 
 377.44 
 4 64./S 
 
 6 702 
 6 2S9 
 
 7-f.46 
 74.94 
 
 7i.A3 
 98.96 
 
 -/6.07 
 -8.40 
 
 8.90 
 7J3 
 
 4.ii8 
 
 
 5 76 9 
 
 .79 
 
 • 
 
 
 35 
 36 
 
 s 000 
 
 20 000 
 
 /667 
 
 '333 
 /667 
 
 nSSO 
 
 ft 
 
 M4.-4 
 
 so LUC 
 
 loo 
 
 / flWi 
 
 1 800.6 
 
 I44S.9S 
 1 783.3 
 
 13 333 
 /2 4 80 
 
 /4 3J3 
 /44.2S 
 
 7S.8-i 
 99.04 
 
 -/S44 
 - 8.0s 
 
 8.70 
 4.98 
 
 18.23 
 v 
 
 
 // S40 
 
 /.iS 
 
 • 
 
 
 37 
 38 
 
 3 500 
 
 n 
 
 30 000 
 
 / /67 
 
 933 
 
 / /67 
 
 II 5S0 
 
 101 
 
 SOLflC, 
 
 loo 
 
 / 34 IJ! 
 1 264.0 
 
 / 0/6.8 
 / 2i/.2 
 
 13 482 
 12 537 
 
 99..»7 
 /0O.83 
 
 7i.82 
 98.99 
 
 -liJ3 
 -8-Si 
 
 Sit 
 
 7.2 2 
 
 2S.9 3 
 
 
 II 527 
 
 2.2s 
 
 
 Z9 
 
 40 
 
 S 000 
 
 1' 
 
 30 000 
 
 2 667 
 
 ^ /33 
 2 667 
 
 17320 
 
 IS4 
 II 
 
 8oi.fl<; 
 100 
 
 3 73.« 
 
 2 8 '-'.7 
 
 2 327.9 
 2 864./ 
 
 20 268 
 /« 830 
 
 ISI.CS 
 IS3.73 
 
 7S.74 
 9 8.94 
 
 -(7.07 
 -8.72 
 
 9-/ 3 
 7.39 
 
 58.4 3 
 
 
 '7286 
 
 338 
 
 
 4' 
 42 
 
 S 000 
 f 
 
 so 000 
 
 / 667 
 If 
 
 / 333 
 /667 
 
 n 320 
 fi 
 
 9i>.Z 
 
 f 
 
 so LHi 
 
 loo 
 
 / g79.? 
 
 / 806./ 
 
 / 4 56.2 
 / 794./ 
 
 20 33; 
 /8 845 
 
 92.43 
 9S.84 
 
 77..90 
 99.3 3 
 
 -/7.3! 
 -8.80 
 
 9.24 
 7.6 2 
 
 96.29 
 
 .2 2 
 
 /722S 
 
 5.59 
 
 V 
 
 •2 2 
 
 43 
 
 44 
 
 zoooo 
 
 io 000 
 
 6 667 
 
 5333 
 6 66 7 
 
 34 (,iiO 
 
 9 
 
 801.1s. 
 loo 
 
 7.59 7.8 
 / ? 4 3.0 
 
 i 830./ 
 7 / 80.2 
 
 40 97i 
 37 773 
 
 I8S/I2 
 I9I.7S 
 
 74.73 
 99./ 3 
 
 -18.29 
 -9.0s 
 
 9.32 
 7.70 
 
 3 5 7.9 
 
 .7S 
 
 
 
 34 4 50 
 
 Jr 
 
 /0.3<» 
 
 ■21 
 
 
 
 4i 
 46 
 
 2 2 000 
 
 88 000 
 
 7 333 
 
 .5 S67 
 
 7333 
 
 SO 8IO 
 
 144.4 
 If 
 
 SOLfS. 
 
 100 
 
 7 5 7S.7 
 7 t l^.i 
 
 6 380/) 
 
 7 9/S.3 
 
 5 9 92,f 
 54 869 
 
 I2t/n 
 /44.26 
 
 g4J8 
 /oaoo 
 
 -17.94 
 -7.99 
 
 8.74 
 7.94 
 
 / 409 
 
 If 
 
 -♦9 7/0 
 
 2 8.35 
 
 m 
 
 .6/ 
 
 47 
 48 
 
 -«4 000 
 
 120 000 
 
 /3 333 
 It 
 
 lOCil 
 1333-i 
 
 &9290 
 
 If 
 
 I92.S 
 
 so Lac, 
 /oo 
 
 13 796 
 
 /-» 3fe6 
 
 II S79 
 14 3US 
 
 81 7/0 
 74 735^ 
 
 /fc^SI 
 192.22 
 
 83.93 
 
 100.00 
 
 -/7.92 
 -7.86 
 
 8-55 
 
 7.74 
 
 2 528 
 
 14^9 
 
 ff 
 
 6 7 900 
 
 37.28 
 
 .5 7 
 
 49 
 
 2.S 000 
 
 120 000 
 
 If 
 
 8 333 
 II 
 
 6 <S67 
 8 333 
 
 69290 
 
 
 So Lite. 
 100 
 
 7 OSJ.; 
 9 4 73.0 
 
 7 07i./ 
 
 8 9-<9.6 
 
 7 9 000 
 70599 
 
 S9.ti 
 
 I3 4.IB 
 
 99.g? 
 94.47- 
 
 -'4.01 
 -1.89 
 
 iJ2 
 7.40 
 
 -» 7S9 
 
 ff 
 
 75.47 
 
 m 
 
 63 357 
 
 r 
 
 7i." 
 
 • 
 
 '•59 
 
 5' 
 S2 
 
 40 000 
 
 /fO 00c 
 
 13 333 
 
 10 667 
 /3 333 
 
 80S3O 
 
 IbS 
 
 So LKC 
 
 /oo 
 
 // 827 
 /4 6 66 
 
 // 46/ 
 /4 438 
 
 96 727 
 84 862 
 
 '22.27 
 172.82 
 
 96.fO 
 9»'.44 
 
 -I9.i1 
 -4.99 
 
 7<44 
 t.29 
 
 i 060 
 
 89.78 
 
 m 
 
 73 938 
 
 8 '.96 
 
 '.4 8 
 
 S3 
 St 
 
 20000 
 
 120 oco 
 
 6 667 
 If 
 
 5 333 
 
 6 667 
 
 (,9390 
 
 9 
 
 96.2 
 
 SOiflG. 
 
 /oo 
 
 6 972? 
 9 06/.7 
 
 s6Z6ii 
 7 231it 
 
 72 747 
 6-3 8/0 
 
 9XSS 
 142.01 
 
 80.^^ 
 79.89» 
 
 -4.99 
 + 7.91 
 
 S.SO 
 8-S9 
 
 6 523 
 
 • 
 
 208t 
 
 56 101 
 
 m 
 
 /'4.27 
 
 3.20 
 
 W 
 
 6S 
 Si. 
 
 Co 000 
 
 200 000 
 II 
 
 ZOOOO 
 
 16 000 
 20000 
 
 IIS SOO 
 
 ;73.3 
 
 ZOLte. 
 /oo 
 
 IS 72S 
 2 4 796 
 
 l<f 90s 
 21 658 
 
 /26.S4/ 
 (09/89 
 
 /-I8.00 
 227.09 
 
 90 J 8. 
 87.3'C 
 
 -f.Si 
 
 ♦ 5-47 
 
 5.68 
 8:29 
 
 /6 330 
 
 -»76.4 
 
 • 
 
 9J636 
 
 •■ '■*.'< 
 
 2.92 
 
 S7 
 SS 
 
 20 000 
 
 MO COO 
 
 fi 
 
 3 333 
 6 667 
 
 80 830 
 
 82.5 
 
 /oo 
 
 /o 089 
 // OI4 
 
 S79CJI 
 7 53?.6 
 
 74 /82 
 64 377 
 
 llt.OI 
 
 1 7/.0S 
 
 5 7.4ii 
 «8.4 j 
 
 + 8.2 2 
 + 20.3 J 
 
 8.69 
 l\09 
 
 8626 
 
 5-.i.8 
 
 54 4 9-» 
 
 /.SS.3 
 
 6.33 
 
 * * 
 
 SI 
 60 
 
 SO 000 
 
 200 COO 
 
 If 
 
 It, 667 
 
 /3 333 
 /6 667 
 
 IIS SOO 
 
 144.* 
 
 8oi.«<:. 
 loo 
 
 21 139 
 2 3 944, 
 
 14 343 
 (8 7^-7 
 
 113 606 
 96987 
 
 ist.o? 
 
 24i.i4 
 
 6 7.85 
 7?. 3 J 
 
 + A64 
 + /6.0 3 
 
 7.St 
 
 I2.S4 
 
 17 217 
 
 / Oi7 
 
 77 93« 
 
 J 2 0,* 
 
 6.'4 
 
 6/ 
 62 
 
 16 000 
 
 MO 000 
 n 
 
 ^ 000 
 * - 
 
 4 000 
 .5 000 
 
 80 830 
 
 dl.Si 
 
 m 
 
 SOLm 
 100 
 
 10 233 
 9 9 /ft* 
 
 -f ffO/J! 
 6 2 30.f 
 
 59046 
 SI 327 
 
 /7 3.30 
 '»3.2.» 
 
 4t.9> 
 6 2.8/' 
 
 + 2C9i 
 + 36J0 
 
 20j3-i 
 2*.lO 
 
 7 690 
 
 998.8 
 
 .» / 2 / 3 
 
 
 '2.9< 
 
 63 
 64 
 
 40 000 
 /I 
 
 20000c 
 
 '3 333 
 
 10 U7 
 
 /3 333 
 
 f l£ SCO 
 tt 
 
 11S.S 
 
 SOLUS. 
 
 /oo 
 
 J/f-»8 
 2i 7 SO 
 
 13 24* 
 /« 05O 
 
 93 72J 
 80 /06 
 
 23*. '7 
 J7/.J2 
 
 5i.8d 
 7 3.fc6 
 
 + /».*■' 
 + 30.*t 
 
 I4.t2 
 
 iojS 
 
 IS 223 
 
 / 907 
 
 5 9 07.# 
 
 * 
 
 2i7.7 
 
 '2. J J 
 
 The above performances are based upon values for the auxiliary constant's as Riven on Chart XII 
 
CHAPTER X 
 HYPERBOLIC FUNCTIONS 
 
 In the consideration of the hyperbolic theory as applied to transmission circuits, the writer desires to 
 express his high appreciation of the excellent literature already existing. Dr. A. E. Kennelly's pioneer 
 work and advocacy of the application of hyperbolic functions to the solution of transmission circuits has 
 been too extensive and well known to warrant a complete list of his contributions. His most important 
 treatises are "Hyperbolic Functions Applied to Electrical Engineering", igi6; "Tables of Complex Hyper- 
 bolic and Circular Functions", 1914; "Chart Atlas of Hyperbolic Functions", 1914, which provides a ready 
 means of obtaining values for complex functions, thus materially shortening and simplifying calculations, 
 and "Artificial Electric Lines", 1917. 
 
 "Electrical Phenomena in Parallel Conductors" by Dr. Frederick Eugene Pernot, 1918, is an excellent 
 treatise on the subject and contains valuable tables of logarithms of real hyperbolic functions from x := 
 to X =: 2.00 in steps of o.ooi. 
 
 An article "Long-Line Phenomena and Vector Locus Diagrams" in the Electrical World of Feb. I, 
 1919, p. 212, by Prof. Edy Velander is an excellent and valuable contribution on the subject, because of its 
 simplicity in explaining complicated phenomena. 
 
 To employ hyperbolic functions successfully in *ht solution of transmission circuits it is not necessary 
 for the worker to have a thorough understanding of how they have been derived. On the other hand it 
 is quite desirable to understand the basis upon which they have been computed. A brief review of hyper- 
 bolic trigonometry is therefore given before taking up the solution of circuits. 
 
 CIRCULAR angles derive their name from the fact 
 that they are functions of the circle, whose equa- 
 tion is x" -\- y" =^ I. Tabulated values of such 
 functions are based upon a radius of unit length. The 
 geometrical construction illustrating three of the func- 
 tions, the sine, cosine and fangent of circular angles is 
 indicated in Fig. 38. The angle AOP, indicated by full 
 lines in the positive or counter-clockwise direction, has 
 been drawn to correspond to one radian. The radian 
 is an angular unit of such magnitude that the length of 
 the arc which subtends the radian is numerically equal 
 tc that of the radius of the circle. Thus, the number 
 of radians in a complete circle is 2 t. Expressed in 
 degrees the radian is equal approximately to 57° 17' 
 44.8". The segment AOP of any angle AOP of one 
 radian has an area equal to one-half the area of a unit 
 square. Therefore the angle may be expressed in 
 radians as, — 
 
 Length o f arc s X area 
 
 (radius)' 
 
 or 
 
 Circular angle = ■ 
 
 radians 
 
 radius 
 
 Circular functions are obtained as follows,- 
 S X area 
 (radius)' 
 Y_ 
 R 
 
 R 
 Tangent = -y- 
 
 Sine = "5" 
 
 Cosine $ = -5" 
 
 The variations in the circular functions, sine, 
 cosine and tangent are indicated graphically in Fig. 39 
 for a complete revolution of 360 degrees. Since for the 
 second and each succeeding revolution these graphs 
 would simply be repeated, circular functions are said to 
 have a period equal to 2 t radians. In other words, 
 adding ^ t to a circular angle expressed in radians does 
 not change the value of a circular function. 
 
 REAL HYPERBOLIC ANGLES 
 
 Real hyperbolic angles derive their name because 
 they are functions of an equilateral hyperbola. A 
 hyperbola is a plane curve, such that the difference be- 
 tween the distances from any point on the curve to 
 two fixed points called the foci is constant. In an equi- 
 lateral hyperbola. Fig. 40, the asymptotes OS and OS' 
 are straight lines at right angles to each other and make 
 equal angles with the X-axis. The hyperbola continu- 
 ally approaches the asymptotes, and meets them at 
 infinity. The equation of such a hyperbola is x" — y* 
 
 The hyperbolic angle AOP of Fig. 40, called for 
 convenience **, has been drawn so as to correspond to 
 an angle of one hyperbolic radian, or one "hyp" as it is 
 usually designated. Hyperbolic angles are determined 
 by the area of the sector they enclose. Thtis the hyper- 
 bolic angle of one hyp AOP, encloses an area AOP of 
 one-half, or the same as the area AOP of the corre- 
 sponding circular angle of Fig. 38. It should be ob- 
 served here that although one circular radian subtends 
 an angle AOP of 57° 17* 44-8", one hyperbolic radian 
 subtends a circular angle AOP of 37° 17' 33.67" 
 (0.65087 circular radian). 
 
 In the same way as for the circle the hyperbolic 
 angle may be expressed in radians as, — 
 
 Length of arc 2 X area 
 
 p "'^ (radius)' 
 
 where p = the integrated mean radius from to AP. 
 As an illustration, the length of the arc AP, Fig, 4Q 
 
 *A "hyperbolic angle", in the sense above described, is not 
 the opening between two lines intersecting in a plane, but a 
 quantity otherwise analogous to a circular angle and the argu- 
 ment x of the function sinh x, cosh x, tanh x, etc. The use of 
 the term hyperbolic angle can only be justified by its con- 
 venience of anology. 
 
HYPERBOLIC FUNCTIONS 
 
 is 1.3167 and the mean integrated radius to arc AP is variations in hyperbolic functions are indicated 
 
 A-3I&7- aphically in Fig. 41 for hyperbolic angles up to ap- 
 
 Hyperbolic functions, distinguished from circular proximately 2.0 hyps for the sine and cosine and up to 
 
 lunctions by the letter h affixed, are obtained as fol- 3.0 hyps for the tangent, 
 
 lows: — Hyperbolic functions have no true period, but add- 
 
 y 3Vf 
 
 + YAXIS 
 
 YAXIS 
 
 XAXI3 
 
 NOTE 
 ANULb AOP IS DRAWN 
 CORRE8PONDINQ TO ONE y 
 HYPERBOLIC RADIAN OR 
 -HYP" 
 
 FIG. 38 — REAL CIRCULAR ANCLES 
 X' +¥' = J 
 
 FIG. 40 — REAL HYPERBOUC ANCLES 
 
 x* — r = / 
 
 1^1 rP 
 
 2 7T RADIANS 
 4 QUADRANTS 
 
 360* 
 
 ONE CIRCULAR RADIAN -< ' 
 FIG. 39 — GRAPHS OF CIRCULAR FUNCTIONS 
 
 6.2831863072 
 = 206264.8062' 
 ■en 7' 44.8" 
 ■ Br.28678 
 
 4.0 
 3.6 
 
 2JS 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 c 
 
 f 
 
 
 
 
 62.0 
 u. 
 
 
 
 4 
 
 
 
 
 
 
 
 7/ 
 
 
 
 
 
 
 ^^ 
 
 <. 
 
 ^ 
 
 
 
 
 
 
 
 ^ 
 
 -fSSc 
 
 b?rff 
 
 mTh) 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 5 '1. 
 
 6 1 
 
 6 2 
 
 .0 ,4 
 
 6 4. 
 
 
 
 Hyperbolic angle 6 ■■ 
 
 Cosh e 
 
 Length of arc AP 
 Length of mean radius 
 X 
 
 radians. 
 
 Sink = 
 
 Tank e = 
 
 OA 
 
 y 
 
 OA 
 Y 
 
 HYPERBOLIC RADIANS I HYPSl 
 
 KIG, 41 — GRAPHS OF HYPERBOUC 
 FUNCTIONS 
 
 ing a i" » ;' to the hyperbolic angle does not change the 
 values of the functions, hence these functions have an 
 imaginary period oi 2 ^ j. 
 
 Circular functions can be used to express the phase 
 relations of current and voltage, but not the magnitude, 
 or size, whereas hyperbolic functions, continually in- 
 
90 
 
 HYPERBOLIC FUNCTIONS 
 
 creasing or decreasing, can be used to express the mag- 
 nitude of current in a long circuit. 
 
 In Fig. 42 is shown a circular angle corresponding 
 to one circular radian divided into five equal parts, each 
 of 0.2 radian. Assuming unity radius, each of the arcs 
 will have a constant length of 0.2 and a constant mean 
 radius of i.e. In Fig. 42 is shown a hyperbolic angle 
 corresponding to one hyperbolic radian divided into five 
 equal hyperbolic angles each of 0.2 hyperbolic radian. 
 In this case the length of the arcs corresponding to each 
 subdivision increases as the hyperbolic angle increases. 
 The lengths of the corresponding integrated mean radii 
 vectors also increase with the angle. By dividing the 
 length of the arc of any of the five subdivisions by the 
 length of the mean radius for that subdivision it will be 
 seen that each subdivision represents 0.2 hyps. 
 
 From the above it will be evident that in radian 
 measure, the magnitudes of circular and hyperbolic 
 
 '^>-. 
 
 1.000 
 
 CIRCULAR RADIAN 
 
 000--. VA 
 
 HYPERBOLIC RADIAN 
 
 nc. 
 
 42 — SUBDnnSIOK OF A CIRCULAR AND A HYPERBOLIC RADIAN INTO FIVE SECTORS OF O.i 
 
 RADIAN EACH 
 
 angles are similarly defined with reference to the area 
 of circular and hyperbolic sectors. 
 
 COMPLEX ANGLES AND THEIR FUNCTIONS 
 
 A complex angle is one which is associated with 
 both a hyperbolic and a circular sector. If the complex 
 angle is hyperibolic, its real part relates to a hyperbolic 
 and its imaginary to a circular sector. On the other 
 hand, if the complex angle is circular, its real part re- 
 lates to a circular and its imaginary part to a hyper- 
 bolic sector. Complex hyperbolic trigonometry and 
 complex circular trigonometry thus unite in a common 
 geometrical relationship. 
 
 In the following treatment for the solution of trans- 
 mission circuits by hyperibolic functions, only hyperbolic 
 complex angles will enter into the solution. Such a 
 conTplex angle will then consist of a combination of a 
 "real" hyperbolic sector and a so-called "imaginary" or 
 circular sector. The circular sector will occupy a plane 
 inclined at an angle to the plane of the hyperbolic sector. 
 In other words, the complex angle will be of the three- 
 dimensional order. The construction of such a complex 
 angle may be difficult to follow if viewed only from one 
 direction. In order to illustrate the form that a com- 
 
 plex angle takes, the construction for the cosine of a 
 hyperbolic complex angle is illustrated by Fig. 43. 
 
 CONSTRUCTION FOR COSH 6 
 
 The construction, Fig. 43, assumes that the real 
 part, that is the hyperbolic sector subtends an angle of 
 one hyperbolic radian and the imaginary part, that is the 
 circular sector, subtends an angle of one circular radian. 
 This hyperbolic complex angle has therefore a numerical 
 value oi I -\- j I hyperbolic radian. These numerical 
 values embrace sectors sufficiently large for the pur- 
 pose of clear illustration. The actual construction for 
 obtaining the complex function cosh (^i -f / 6^) = cosh 
 (i -{• f I hyperbolic radians) may be carried out a.1 fol- 
 lows: — 
 
 On a piece of stiff card board lay out to a suitable 
 scale the hyperbolic sector d^ = EOC, equal to one hyp 
 as shown in the upper left hand corner of Fig. 43. This 
 may readily be plotted by the aid 
 of a table of real hyperbolic func- 
 tions for say each one tenth of a 
 hyp up to and including one hyp. 
 These are then plotted on the 
 cardboard and joined with a 
 curved line thus forming the arc 
 EC of Fig. 43. The ends of the 
 arc are then joined with O by 
 straight lines. The real part of 
 this hyperbolic complex angle is 
 then cut out of the cardboard. 
 
 The circular part ; 6^ of this 
 complex angle is traced upon the 
 cardlxjard as follows : — • With 
 radius equal to cosh 0^ (to the 
 same scale as used when trac- 
 ing the hyperbolic sector J draw the arc DOF of a 
 length such that the angle DOF is 57° 17' 44.8" (on: 
 circular radian). Join the ends of the arc to witi 
 straight lines. The circular part j 6 ^ oi this complex 
 angle is now cut out of the piece of cardboard. This 
 gives models of the two parts of the complex angle 
 which may be arranged to form the complex angle 
 I -{- j I hyps. These two models are shown at the top 
 of Fig. 43. 
 
 The two parts of the complex angle are arranged 
 as follows : — -Upon a drawing board or any flat surface 
 occupying a horizontal plane, place the hyperbolic sec- 
 tor S, in a vertical position. The plane of this 
 hyper])olic sector will then be at right angles to the 
 plane of the drawing board. The circular sector ; B^ 
 is now placed in a vertical position just back of the 
 hyperbolic sector. The toes of each sector will then 
 coincide, as well as the line OD of the circular sector 
 with the line OC of the hyperbolic sector. The top of 
 the circular sector is now turned back so that the plane 
 of the circular sector lies at an angle with the vertical 
 plane occupied by the hyperbolic sector. This displace- 
 ment angle between the planes of the two sectors is 
 
HYPERBOLIC FUNCTIONS 
 
 QI 
 
 k COSHO,- 1.643081 V 
 
 HVPERBOLIO 8E0T0R 
 
 RADIUS-CCSHe, -1.643081 -■ 
 CIRCULAR SECTOR 
 
 ^ 
 
 QUDERMANNIAN COMPLEMENT 
 OF THE COMPLEX HYPERBOLIC 
 ANGLE OF THE CIRCUIT -Bq 
 ^08 80-55^ -SECHe, 
 
 FRONT ELEVATION 
 
 LEFT END ELEVATION 
 
 _COSH^j^008_92__J+x axis) 
 
 + 6.8337~ VECTOR OF REFERENCE 
 
 TOP PLAN OF BOARD (MODEL REMOVED) 
 AHOWINO 0O8H (0, ♦{ Og ) TRACED UPON BOARD 
 
 REAR ELEVATION 
 
 MATHEMATICAL SOLUTION 
 COSH (e, + j 8] ) - (COSH 8, COS 82 + jSINH 8, SIN Sg) 
 
 LOa COSH 8, -0.1 88,389 
 
 LOG COS 82- T-732.639 
 
 7.821.026 
 
 LOG SINH 8,-0.070.1 1 2 
 LOG SIN 82 - T .926,039 
 
 1.996,161. 
 COSH (81 + j 82) -0.8337 +j 0.9889 
 -1, 297 /49* 62' 06' 
 43 — GRAPHICAL CONSTRUCTION FOE THE HYPERBOLIC COSINE OF THE COMPLEX ANGLE 
 
 «! + /*i = 1+ ii Hyperbolic Radians. 
 
 circular sector of this complex 
 angle is moved in the forward 
 direction through an angle of 
 49° 36' 18" so that the plane of 
 the circular sector assumes an 
 angle of 90° oc/ 00" — 49° 36^ 
 18" = 40° 23' 42" with the hor- 
 izontal plane of the drawing 
 board. From the end of the cir- 
 cular sector (point F) thus in- 
 clined, a plummet may tie sus- 
 pended until it meets the hori- 
 zontal plane of the drawing board 
 at the point / of the illustration. 
 In other words, the point F is pro- 
 jected orthogonally onto the hori- 
 zontal plane of the drawing board. 
 A top view of the drawing 
 board, with the model removed, 
 is illustrated in the lower left 
 hand corner of Fig. 43. The line 
 OF {1.297 /49° 52' 05'') traced 
 upon the horizontal drawing 
 board, is a vector representing 
 the complex cosine of the com- 
 plex angle 0^ -{- j 6^ — i -\- j i 
 hyperbolic radians. This com- 
 plex cosine has rectangular co- 
 ordinates of -f 0.8337 and -1- 
 ;■ 0.9889. 
 
 At the bottom of Fig. 43 is 
 given the mathematical expres- 
 sion for the exact solution for 
 the cosine of a complex hyper- 
 bolic angle following the con- 
 struction illustrated. There are 
 numerous other mathematical 
 equations with their equivalent 
 geometrical constructions which 
 will produce the same values for 
 the cosine, but the above is prob- 
 ably as easy to follow as any, and 
 will therefore be used exclusively 
 hereafter. 
 
 known as the "gudermannian complement" of the hy- 
 perbolic angle d. It will be referred to as 6o- The 
 front elevation of Fig. 43 illustrates how these two sec- 
 tors would appear when viewed from the front. To 
 the right of this illustration is shown how these two 
 sectors would appear when viewed from the left hand 
 end of tlie model. The displacement angle Ba has a 
 value for this particular complex angle of 49° 36' 18". 
 This numerical value is determined by virtue of the 
 fact that this displacement angle has a cosine of 
 
 •r T 
 
 = 0.64805 or cosine of 6^ = sech 6^ 
 
 cosh 611 1.543081 
 
 := 0.64805. It has a sine of tanh 6, = 0.76159. 
 
 The angle whose cosine is 0.64805 and whose sine 
 is 0.76159 is 49° 36' 18". Thus the top part of the 
 
 CONSTRUCTION FOR SINH 6 
 
 The construction for the sine of the complex hy- 
 perbolic angle / + ; / is indicated in Fig. 44. In this 
 case the same construction may be used for obtaining 
 the sinh as for determining the cosh of the complex 
 angle with the following two exceptions. 
 
 The circular sector is made one quadrant (90°) 
 larger. In other words the angle DOF' is 90° 4- 57° 
 17' 44.8" or 147° 17' 44.8" as indicated by Fig. 44. It 
 occupies the same plane as when determining the cosh 
 of the angle but is simply extended in the forward di- 
 rection through one quadrant, as indicated by the dotted 
 lines of Fig. 44. The plummet is again suspended, this 
 time from point F' upon the horizontal board, which it 
 
HYPERBOLIC FUNCTIONS 
 
 meets at point /'. The other difference is that the sine 
 OF' is read off from the Y axis as the vector of refer- 
 ence in place of the X axis as in the case of the cosine. 
 Thus the circular sector has been carried forward 
 through an angle of 90 degrees in the circular angle 
 plane and the vector of reference has been advanced 90 
 degrees in the horizontal plane of reference. The sine 
 of this angle is 1.446 /63° 56' 3/^ and has rectangular 
 components of 0.6349 -f- ji.sgSs. The mathematical 
 
 
 «c^^ 
 
 \ 
 
 
 ■■*r 
 
 o^ 
 
 FRONT ' ELEVATION 
 
 LOG SINH 01=0.070 112 
 
 LOO COS 02 - 7.732 638 
 
 7,802748 
 
 SINH (91+ j 02)- 0.6349 +1 1.2985 
 - I 446 /e3' 66' 37* 
 
 NOTE -THE CONSTRUOTJON FOR THE COSINE OF THE COMPLEX 
 HYPERBOLIC ANGLE. MAY ALSO BE USED FOR DETERMINING THE 
 SINE OF THE ANGLE WITH THE FOLLOWING CHANGES: - THE CIR- 
 
 ~XA-XiS _t:^+x aJ(IS CULAR SECTOR MUST BE EXTENDED THRU ONE QUADRANT AND 
 
 THE SINE MEASURED FROM THE Y AXIS AS THE VECTOR OF REFER- 
 TOP PLAN OF BOARD (MODEL REMOVED) ENCE IN PLACE OFT HE X AXIS AS IN THE CASE OF THE COSINE. 
 
 SHOWING 81NH "(©i + jej) TRACED UPON BOARD C 
 
 FIG. 44 — GRAPHICAL CONSTRUCTION FOR THE HYPERBOLIC SINE OF THE COMPLEX ANGLE 
 
 6i + jffi =^ J -^ ji hyperbolic radians. 
 
 expression for exact solution for the sine of a complex 
 angle likewise accompanies the illustrated geometrical 
 construction. 
 
 MODEL FOR ILLUSTRATING THE FUNCTIONS OF A 
 COMPLEX ANGLE 
 
 Dr. Kennelly has recently constructed a model* for 
 illustrating complex angles and for obtaining approxi- 
 mate values for the functions of such angles. Drawings 
 made from photographs of this model are shown in Figs. 
 45, 46 and 47. The construction of a complex angle as 
 above described is that employed by Dr. Kennelly in 
 building his model. Since the model is applicable to 
 tracing out numerous complex angles, it may seem a little 
 difficult at the start. It was therefore thought desirable 
 to precede the description of the model which is appli- 
 cable to the solution of so many angles with a similar 
 solution of a single definite complex angle. With the 
 procedure for the solution, as given above, for cosh and 
 sinh of I -j- y ^ hyperbolic radians in mind, it is believed 
 
 *This model was described in a paper read by him at a 
 meeting of the American Academy of Arts and Sciences in 
 April 1910. 
 
 that Dr. Kennelly's description of the model and its appli- 
 cation in determining the cosh and sinh of complex angles 
 may be followed as given in the following paragraphs. 
 
 DESCRIPTION OF MODEL 
 In this model, the cosine or sine of a complex angle, either 
 hyperbolic or circular, can be produced, by two successive 
 orthogonal projections onto the XY plane, one projection being 
 made from a rectangular hyperbola, and the other projection 
 being then made from a particular circle definitely selected 
 from among a theoretically infinite number of such circles, all 
 concentric at the origin O, which circles, however, are not 
 P coplanar. The selection of 
 
 the particular circle is deter- 
 mined by the foot of the pro- 
 jection from the hyperbola. 
 This effects a geometrical pro- 
 cess which is easily appre- 
 hended and visualized ; so that 
 once it has been realized by 
 the student, the three-dimen- 
 sional artifice is rendered su- 
 perfluous, and he can roughly 
 trace out a complex sine or 
 cosine on an imaginary draw- 
 ing board, with his eyes closed. 
 The model, however, possesses 
 certain interesting geometrical 
 properties as a three-dimen- 
 sional structure. 
 
 A drawing made from a 
 photograph of the model is 
 shown in Fig. 45. On an ordi- 
 nary horizontal drawing board 
 53-5 by 31.8 cm., is a horizon- 
 tal rod AB, which merely 
 serves to support the various 
 brass-wire semicircles, and a 
 semihyperbola, in their proper 
 positions. The axis of AB in 
 the XY plane, on the upper 
 surface of the board, is a line 
 of symmetry for the structure, 
 which, if completed, would be 
 formed by full circles and a 
 complete hyperbola. For con- 
 venience, however, only the 
 half of the structure above the 
 XY plane is presented, the 
 omission of the lower half 
 being readily compensated for 
 in the imagination. 
 The eight wire semicircles are formea with the following 
 
 2.0, 1.020. ... 1.081..., 1.185.. 
 
 MATHEMATICAL SOLUTION . 
 SINH (8| + i 92)=lSINHe,'COS02+jCOSHe.81N Oj) 
 
 LOG OOSH0| — 0.188 389 
 
 LOO SIN 02 - 7.825 039 
 
 0.113428 
 
 respective radii, in decimeters: 
 
 1.337. ••. I-S43---. l.Slo..., ana 2.150..., wmcn are me re- 
 spective cosines of o, 0.2, 0.4, 0.6, 0.8, l.o, 1.2, and 1.4 hyperbolic 
 radians, according to ordinary tables of real hyperbolic func- 
 tions. These successive semi-circles therefore have radii equal 
 to the cosines of successively increasing real hyperbolic angles 
 01, by steps of 0.2, from o to 1.4 hyperbolic radians, inclusive. 
 All of these semicircles have their common center at the origin 
 O, in the plane X Y, of the drawing board. The planes of 
 the semicircles are, however, displaced. The smallest circle of 
 unit radius (l decimeter), occupies the vertical plane X O Z, 
 
 XAXIS 
 
 FIG. 45 — DRAWING FROM A PHOTOGRAPH OF A GEOMETRICAL MODEL 
 
 For the orthogonal projection of the sines and cosines of 
 complex angles. This model was developed by A. E. Kennelly. 
 
HYPERBOLIC FUNCTIONS 
 
 93 
 
 in which also lies the rectangular semi-hyperbda X O H. 
 
 Angular distances corresponding to 0.2, 0.4 14 hyperbolic 
 
 radians, are marked off along this hyperbola at successive 
 corresponding intervals of 0.2. The cosines of these angles, 
 as obtainable projectively on the O X axis are marked off 
 between C and B along the brass supporting bar, and at each 
 mark, a semicircle rises from the X Y plane, at a certain angle 
 e„ with the vertical X Z plane. This displacement angle i« 
 determined by the relation, — 
 X 
 
 cus $a ■ 
 
 ■ ^ seek 01 
 
 cosh 81 
 
 Where ffi is the partictilar hyperbolic angle selected. This 
 means, as is well known, that the displacement angle 6a between 
 the plane of any semicircle a,nd the vertical plane Z O X 'n 
 equal to the gudermannian of the hyperbolic angle ft. 
 
 The model is, of course, only a skeleton structure of eight 
 stages. If it could be completely developed, the number of 
 semicircles would become infinite, and they would form a 
 smooth continuous surface in three dimensions. Along the 
 midplane Z O Y, all 01 these circles would have the same level, 
 raised one decimeter above the horizontal drawing board plane 
 of reference X Y. The circles would increase in radius 
 without limit, and would cover the entire X O Y plane to 
 infinity, the hyperbola extending likewise to infinity toward? 
 its asymptote O S, in the X O Z plane. The actual model is 
 thus the skeleton of the upper central sheet of the entire 
 theoretical surface, near the origin. 
 
 The semicircles are also marked off in uniform steps jf 
 circular angle. Each step is taken, for convenience, as nine 
 degrees, or one tenth of a quadrant. Corresponding angiilar 
 steps on all of the eight semicircles are connected by thin wires, 
 as shown in the illustrations. 
 
 A front elevation of the model, taken from a point on the 
 O Y axis — 15 units from O, is given in Fig. 46. It will be seen 
 that any tie wire, connecting corresponding circular angular 
 
 KG. 46 — FRONT ELEVATION OF MODEL 
 
 From a point on the O 7 axis, — 15 imits from O. 
 
 points on the semicircles, is level, and lies at a constant height 
 sin ft decimeters above the drawing board. That is, the tie 
 wire that connects all points of circular angle ft, measured from 
 O X positively towards OY , lies at the uniform height sin ft 
 decimeters above the drawing board. 
 
 A plan view of the model, taken from a point on the Z 
 axis, + 15 units above 0, is given in Fig. 47. It will be seen 
 that each semicircle forms an ellipse, when projected on the 
 Isase plane X O Y. The semi-major axis of this ellipse has 
 length cosh ft, where ft is the hyperbolic angle corresponding 
 to that semicircle. The semi-minor axis is, — 
 
 cosh ft s'm Oa = cosh ft tank ft = sink ft 
 from the well known relation that exists between a hyperbolic 
 angle and its gudermannian circular angle; namely, — 
 
 sin ftj = tanh ft 
 All of these ellipses have the same center of reference 
 Any such system, having semi-major axes cosh ft, and semi- 
 minor axes sinh ft, are well known to be confocal, and the foci 
 must lie at the points +1 and — i in the X O K plane, or the 
 points in which the innermost circle cuts that plane. 
 
 PROCEDURE FOR PROJECTING COSH ( ± ft ± /ft) 
 
 Thus premised, the process of finding the cosine of a com- 
 plex hyperbolic angle ft + /ft; that is, the process of finding 
 cosh (ft + /ft) is as follows : 
 
 Find the arc C E, Fig. 45, from C z= -\-i along the rect- 
 angular hyperbola C E H, which subtends ft radians. The 
 hyperbolic sector comprised between the radius, O V, the hyper- 
 
 bolic arc, and the radius vector E, on this arc from the 
 origin O, will then include-^ sq. dm. of area. Drop a vertical 
 perpendicular from E onto O X. It will mark off a horizontal 
 distance O D equal to cosh ft. Proceed along the circle which 
 rises at D, in a positive or counterclockwise direction, through 
 ft circular radians, thus reaching on that circle a point G wheise 
 elevation above the drawing board is sin ft decimeters. The 
 area cnclo'^ed by a radius vector from the oriein O on the circle, 
 followed between the axis O C and the circular curve, will be 
 
 — cosh 'ft sq. dms. 
 
 From G, drop a vertical plummet, as in Fig. 46, on to the 
 drawing board. In other words, project G orthogonally on the 
 plane X O Y. Let g be the point on the drawing board at 
 which the plummet from G touches the surface. Then it is 
 easily seen that Og on the drawing board is the required mag- 
 nitude and direction of cosh (ft + /ft), in decimeters, with 
 reference to O X as the initial line in the plane X O Y. It may 
 be read off either in rectangular coordinates along axes O X 
 and O y on a tracing cloth surface as shown in Fig. 47, or 'n 
 polar coordinates printed on a sheet seen through the tracing 
 cloth. 
 
 If the circular angle ft, i. e., the imaginary hyperbolic angle 
 /ft, lies between ir and 2t radians, (in quadrants 3 and 4), 
 the point G will lie on the under side of the plane X O Y, and 
 the projection onto g in that plane must be made upwards, 
 instead of downwards. 
 
 If the hyperbolic angle whose cosine is required has a nega- 
 tive imaginary component, according to the expression cosh 
 (ft — /ft), then starting from the projected point Z), we must 
 trace out the circular angle in the negative or clockwise direc- 
 tion, as viewed from the front of the model. 
 
 |l||l|||||||l|l||iy|||||iili|||||li 
 
 
 1 mtti BMi 
 
 
 
 ;;;;i-:|.:'j;!;|4:|;;i:i;i:;;i|^' 
 
 Y 
 
 ■■'i"i-ii'i''8i'' J'S'STi' 
 
 m 
 
 
 =£=» ■ 
 
 [444-;2!o|44-ii5|44 iid-j -o'54i4o444*-ci.q4>ji d-frnisl-Lp-^ (H-(+h 
 
 ""TXt 'iJ- It X A- --- --U- --iJ4--- 
 
 ' 1 . II 
 
 AH- - - + -1- -Hi-t+ -t~ c iD-t- . . 
 
 mil" 
 
 FIG. 47 — PLAN VIEW OF MODEL 
 
 From a point on the O 2 axis, 15 units from O. 
 
 If the real part of the hyperbolic angle is negative, accord- 
 ing to the expression cosh (—ft ± /ft) ; then since cosh — {Bt 
 =!= /ft) = cosh (ft =F /ft), we proceed as in the case of a posi- 
 tive real component, but with a change in the sign of the 
 imaginary component. 
 
 The operation of tracing cosh ( ± *t ± /ft) on the X Y 
 plane, thus calls for two successive orthogonal projections onto 
 that plane; namely (i) the projection corresponding to cosh 
 (± ft) as though /ft did not exist, and then (2), the projection 
 corresponding to cosh /ft = cos ft independently of ft, except 
 that the radius of the circle, and its plane, are both conditioned 
 by the magnitude of ft. 
 
 If we trace the locus of cosh (ft ± /ft), where ft is held 
 constant, it is evident from Fig. 47 that we shall remain on one 
 circle, which projects into the same corresponding ellipse on 
 the X Y plane. That is, the locus of cosh (ft i /ft) with ft 
 held constant, is an ellipse, whose semi major and minor 
 diameters are cosh ft and sinh ft respectively. If, on the other 
 hand, we trace cosh ( ± ft -f- /ft) with ft held constant, we 
 shall run over a certain tie wire bridging all the circles In the 
 model, which tie wire is sin ft dm. above the board, and it"s 
 projection on the board. In the plane X y of projection. Is part 
 of a hyperbola. 
 
 PROCEDURE FOR SINH (ft -f- /ft) 
 
 It would be readily possible to produce a modification of 
 this model here described, which would enable the sine of a 
 complex angle to be projected on the X K plane following con- 
 structions already referred to. The transition to a new model 
 for sines is, however, unnecessary. It suffices to use the cosine 
 
94 
 
 HYPERBOLIC FUNCTIONS 
 
 model here described in a slightly different way. One has only 
 to recall that 
 
 sink 
 
 e = —i cosh\9 +7-7) 
 
 or 
 
 sink (9. + M) = — y cosh [ei+j(6'i + -7)] 
 
 Consequently, in order to find the sine of a complex hyperbolic 
 angle, we proceed on the model as though we sought the cosine 
 
 TT 
 
 of the same angle, increased by -7- radians or one quadrant, in 
 
 the imaginary or circular component. We then operate with 
 — } on the plane vector so obtained; i. e., we rotate it through 
 one quadrant in the X Y plane and in the clockwise direction. 
 An equivalent step is, however, to rotate the X and Y axes of 
 reference in that plane through one quadrant in the reverse or 
 
 positive direction."*That is, we may omit the — j operation. It, 
 in dealing with sine projections, we treat O y as an O X axis, 
 and — O X as an O F axis, or read off the projections on the 
 X Y plane to the — Y O Y axis as initial line. 
 
 The only difference, therefore, between projecting the 
 cosine and the sine of a complex hyperbolic angle in the model, 
 is that in the latter case the circular component is increased by 
 one quadrant and the projected plane vector is read off to the 
 O y reference axis as initial line. The model thus gives the 
 projection of either cosh ( rt Sii ]'$•) or sinh ( ^0i '- j'S-.) 
 within the limits of +1-4 ^nd — 1.4 for 0i, and for fij between 
 the limits + <^ and — a . For accurate numerical work, refer- 
 ence would, of course, be made to the charts and tables of such 
 functions already published, and which enable such functions 
 to be obtained either directly or by interpolation, for all or- 
 dinary values of Si and 62. 
 
CHAPTER XI 
 PERFORMANCE OF LONG TRANSMISSION LINES 
 
 (RIGOROUS SOLUTION BY HYPERBOLIC FUNCTIONS) 
 
 AS STATED in the discussion of the convergent 
 series solution, the performance of an electric 
 circuit is completely determined by its physical 
 characteristics; — resistance, reactance, conductance and 
 capacitance and the impressed frequency. These five 
 quantities are accurately and fully accounted for in the 
 two complex quantities. 
 
 Impedance Z =:: R -^ jX 
 Admittance Y = G + jB 
 
 Having determined the numerical values for these 
 two complex quantities, no further consideration need 
 be given to the physical quantities of the circuit or to 
 the frequency. 
 
 In the hyperbolic theory the circuit is said to sub- 
 tend a certain complex angle, 6 = VzK. This quantity 
 represents in a sense the electrical length of the circuit. 
 The numerical value of this angle 6 is expressed in 
 hyperbolic radians. If the circuit is very long electric- 
 ally the numerical value of the angle will be compara- 
 tively large. Conversely, if the circuit is electrically 
 short, it will be comparatively small. The numeiical 
 value of the angle 6 is, therefore, a measure of the elec- 
 trical length of the circuit and an indication of how 
 much distortion in the distribution of voltage and cur- 
 rent is to be expected as an effect of the capacitance and 
 leakance of the circuit. 
 
 In order to give an idea of the extent of the varia- 
 tion in the complex 6 and its functions cosh 6 and sinh 6 
 for power transmission circuits o£ various lengths cor- 
 responding to 25 and 60 cycle frequencies approximate 
 values have been calculated, as shown in Table O. 
 
 This tabulation indicates that for circuits of f -om 
 100 to 500 miles in length, operated at frequencies of 25 
 and 60 cycles, the complex hyperbolic angle of the cir- 
 cuit (which is a plane-vector quantity) has a maximum 
 modulus, or size of 0.41 for 25 cycles and of 1.05 for 60 
 cycles. It has an argument, or slope, lying between 70 
 and 78 degrees for 25 cycles and between 80 and 85 
 degrees for 60 cycles. 
 
 In the convergent series solution, the three so-cr.Iled 
 auxiliary constants A, B and C determine the perfonn- 
 ance of the circuit. These three auxiliary constants are 
 simply expressions for certain hyperbolic functions of 
 the complex hyperbolic angle B of the circuit. 
 
 Thus 
 
 A T= cosh 
 
 r sinh 6 
 
 B = sinh 
 
 >s 
 
 7.i 
 
 = Z' 
 
 C = Sinn e — r^ — ' z — 
 
 Z * 
 
 ADDITIONAL SYMBOLS 
 
 In addition to the symbols previously listed, the 
 following will be employed in the hyperbolic treatment. 
 
 a = Linear hyperbolic angle expressed in hyps 
 per mile. It is a complex quantity consist- 
 ing of a real component at and an imaginary 
 component oc i. It is also known as the at- 
 tenuation constant or the propagation con- 
 stant of the circuit. 
 
 Oi = The real component of the linear hyperbolic 
 angle a, expressed in hyps. It is a measure 
 of the shrinkage or loss in amplitude of the 
 traveling wave, per unit length of line 
 traversed. 
 «i = The imaginary component of the linear 
 hyperbolic angle a, expressed in circular 
 radians. It is a measure of the loss in phase 
 angle of the traveling wave, per unit length 
 of line traversed. 
 
 • = The complex hyperbolic angle subtended by 
 the entire circuit, expressed in hyps. It 
 differs from a in that it embraces the entire 
 circuit, whereas a embraces unit length of 
 circuit (in this case one mile), $ =: a XL, 
 where L is the length of the circuit expressed 
 in miles. 
 
 ft = The real component of the complex hyper- 
 bolic angle of the circuit expressed in hyps, 
 and defines the shrinlcage or loss in ampli- 
 tude or size of a traveling wave, in travers- 
 ing the whole length of the line. 
 
 e, =: The imaginary component of the complex 
 hyperbolic angle of the circuit expressed ia 
 circular radians, expressing the loss in phase 
 angle or slope of the traveling wave, in 
 traversing the whole length of line, 
 e = 2.7182818 which is the base of the Napierian 
 system of logarithms. Logw = 0.4342945. 
 
 6, = Position angle at sending end. 
 
 6, = Position angle at receiving end. 
 
 0p = Position angle at point P on a circuit 
 S z= Impedance load to ground or zero potential 
 at receiving end line, in ohms at an angle. 
 
 ». =y'-^= Surge impedance of a conductor in ohms at 
 
 y 
 
 y. = — = Surge admittance of a conductor in mhos at 
 
 an angle. 
 Surge adi 
 an angle. 
 
 TABLE 0~GENERAL EFFECT OF DISTANCE AND 
 FREQUENCY UPON THE COMPLEX HYPER- 
 BOLIC ANGLE AND ITS FUNCTIONS 
 
 LENQTM Of 
 OlROUrT 
 (MILES) 
 
 z 
 
 r 
 
 
 «-/Iv 
 
 COSHt 
 
 SMH» 
 
 25 CYCLES 
 
 /oo 
 
 -f,^3, . fP 
 
 0.000 230 Lfi?' 
 
 o.cc«»4\ii«; 
 
 n. inijir 
 
 0.991012' 
 
 <jL<fl/7a» 
 
 zoo 
 
 f O.A , , st 
 
 0.000 ^30\SS' 
 
 O-OJ^tiVilfi" 
 
 0.1* lit 
 
 o.9tLa±.r 
 
 0,190X1 
 
 300 
 
 f09> 1 S 
 
 0.000 67012c 
 
 0.07303\^^£ 
 
 o.nliS 
 
 o.v*^,p.y 
 
 n.iylffl 
 
 40O 
 
 '-♦ 3 , , jfc" 
 
 0.000 900 i2Cf 
 
 0. /2 »70 VjEA." 
 
 0.3* i2i? 
 
 0.9^ if.s* 
 
 0.3523? 
 
 ■500 
 
 'SAL 2.* 
 
 0.00/ /oo ]S^ 
 
 0. ' 7/60 \/J7' 
 
 o.-ti at 
 
 o.9,li.i^ 
 
 0.3* /r9; 
 
 
 
 6 C 
 
 Y C L E 
 
 s 
 
 
 
 3.00 
 
 ■ftsLiL' 
 
 0.00/ osow^ 
 
 0./73»ia 2' 
 
 O.tiLL 
 
 l:"i£i- 
 
 ' ^'^?yf^' 
 
 30O 
 
 Z-^4179* 
 
 a. 001 iootSS! 
 
 0.3?O.»0Ui i' 
 
 0.13 11: 
 
 O.tf A. J* 
 
 O.i0 LMSl 
 
 400 
 
 ^^t.lliL'. 
 
 0.002 ISOISS.' 
 
 0.70090\^JS^ 
 
 O.ULt 
 
 0. A 7 X.f 
 
 0,7* IS^ 
 
 ■500 
 
 'i-OTULO- 
 
 0.002 700122" 
 
 
 ).osU£ 
 
 0.5/ lii." 
 
 0.97 LK27 
 
 These values are but roughly approximate to illustrate the 
 general effect for certain circuits. 
 
96 
 
 HYPERBOLIC SOLUTION FOR LONG LINES 
 
 DETERMINATION OF THE AUXILIARY CONSTANTS 
 
 It was shown in Chart XI how values for the 
 auxihary constants A, B and C may be determined 
 mathematically by convergent series form of solution, 
 u?ing problem X as an example. Chart XVI gives in- 
 formation as to how these same auxiliary constants may 
 be determined by the use of real hyperbolic functions. 
 
 The solution for the auxiliary constants by real hy- 
 p-crbolic functions is given completely for problem X 
 in Chart XVI. Vector diagrams are given to assist in 
 following the solution. In the solution for the auxiliary 
 constants by convergent series, the operations were 
 carried out by aid of rectangular co-ordinates of the 
 complex, or vector quantities. In Chart XVI, the op- 
 erations are to a large extent carried out by the aid of 
 polar co-ordinates. In the case of convergent series, 
 most of the operations consist of adding the various 
 terms of the series together. As addition and subtrac- 
 
 tion of complex quantities can be most readily carried 
 out when expressed in rectangular co-ordinates, this 
 form of expression is used for the convergent-series 
 solution. On the other hand, powers and roots of com- 
 plex quantities are most readily obtained by polar co- 
 ordinate expression. In the solution by real hyperbolic 
 functions Chart XVI, operations for powers and roots 
 predominate, and for this reason polar expressions have 
 been quite generally employed. The solution by real 
 hyperbolic functions is briefly this : — 
 
 The impedance Z and the admittance Y are first 
 set down in complex form and their product obtained. 
 
 square root of this product gives the complex angle 
 6 ==V^zyof the circuit. This angle is then exprested 
 in rectangular co-ordintes as 6^ -)- / 6^ for the purpose 
 of determining the numerical value of its real part 6^ 
 (expressed in hyps) and its imaginary or circular part 
 ^2 expressable in circular radian,^. This circulnr par' 0^ 
 
 CHART XVI— RIGOROUS SOLUTION FOR AUXILIARY CONSTANTS OF PROBLEM X 
 
 BY REAL HYPERBOLIC FUNCTIONS 
 
 CHARACTERISTICS OF CIRCUIT 
 
 LENGTH 300 MILES. CYCLES 60. 
 
 CONDUCTORS — 3 » ODD STRANDED COPPER. 
 SPACING OF CONDUCTORS 10X10X20 FEET. 
 EQUIVALENT DELTA SPACING = l2.e FT. 
 
 LINEAR CONSTANTS OF CIRCUIT 
 
 TOTAL PER CONDUCTOR 
 R - 0.360 X 300 = 106 OHMS TOTAL RESISTANCE AT 26° C. 
 X= 0.830X300 = 249 OHMS TOTAL REACTANCE. 
 Q = 6.21 X 300 X 10"^ = .001563 MHO TOTAL SUSCEPTANOE. 
 G = X 300 = MHO TOTAL CONDUCTANCE. 
 g = (IN THIS CASE TAKEN AS ZERO). 
 
 -ERN><ai * 
 
 VOLTAGE AT RECEIVING-END — 
 
 VOLTAGE DIAGRAM 
 
 SOLUTION FOR = 
 
 = N2Y 
 
 
 
 ez = TAN-'^ = TAN-l2^ = 67* 8' 8' 
 2 = R + jX = IOS + j249 
 
 
 7 i^ 
 
 4i 
 
 = 11062 + 24g2 ^ 270.233 
 - 270.233 /67" 8' 8" 
 
 eY-TAN-lB-TAN-l°-°°^S" 90- 
 Y = G + jB = + ; 0.001663 
 
 = V 0^ + 0.0015632= 0.001563 
 = 0.001563 1 90" 
 
 Y 
 
 
 
 
 ®ZY = 9z + ®Y = 67° 8' 8' + 90° = 1 57° 8' 8' 
 ZY = 270.233 X 0.001 563 = 0.4223746 
 = 0.4223746 Vl6''° 8' 8' 
 
 \ 
 
 e^. 
 
 e=VZY'= 0.6499035 As" 34' 4' HYP. 
 = O.I288l7+j 0.637009 HYP. 
 
 - "-(a.+jag) 
 
 // 
 
 
 i^^" 
 
 - 
 
 /|/Z /i;270.233 /e7-8'8' 
 IfY 1/ 0.001663 |_90° 
 
 = 7l72893\22"6l' 6^" 
 
 = 4l5.806\ll°26' 68" 
 
 WAVE LENGTH 
 
 92= +j 0.637009 HYP. 
 0^=0^^9 = 0.002,2333 
 
 WAVE LENGTH =^^=§^||ii||5I2 = 2969 MILES 
 
 SOLUTION FOR (A ) 
 
 (A)= COSH YZY=(C0SH e, COS Oj + j SINHO, SlNOj) 
 
 61 = 0.128817 HYP 0, 
 
 .360° 
 
 X 0.637009 = 36° 29' 52' 
 
 LOG COSH e|= 0.003594 
 
 LOG COS B2 =T. 905194 
 
 LOG a, =T. 908788 
 
 a,= 0.81058 
 
 LOGSINH 0| = I .111172 
 
 LOG SIN eo =T.774359 
 
 LOG 32=7.885531 
 
 r 
 
 = 0.07683 
 
 0.81066 +j 0.07683 
 (A) -0.8142 /5° 24' 62' 
 
 SOLUTION FOR(B ) 
 
 (B) •= yf SINH fZY =Y|^(SINH e, COS 92+ j COSH 0, SIN a^) 
 LOG SINH e, =7.111172 LOG COSH 0, =0.003694 
 
 LOG COS 82= j .906194 
 I .016366 
 
 LOG SIN On = 1.774359 
 7.777963 
 SINH 0| COS 02=0.10383 COSH 0, SIN 02="O.68873 
 
 TAN-I °-69973 = / oq. . „, .„, 
 0. 1 0383 ^-52_L2_*2_ 
 
 SINH e, COS 02+ j COSH 0| SIN 02 = 
 
 = O.I0383+;0.69973 
 
 = 0.60866 /80° 10' 40' 
 
 ff= 
 
 '|- = 4I5.B05 \ir 25'56' 
 
 = 415.805 \l l°25' 56* X 0.60866 /80°I0'40' 
 (B) = 253.08 /68°44'44' 
 
 SOLUTION FOR (C) 
 
 (C)= 73 SINH l/ZY 
 
 — ,,,„„ ^ , X 0.60865 /80° 10' 40' 
 416.805 \ll° 26' 58' ' 
 
 - 0.002405 /l 1° 26' 56 ' X 0.60865 /80° 1 0' 40' 
 (C)= 0.00 1 464 \ 91° 36' 36 ' 
 
 As a check against possible serious errors in the calculations, the calculated values may be compared with values read front 
 the Wilkinson Charts. The above results check exactly with those obtained by convergent series. (See Chart XI). 
 
HYPERBOLIC SOLUTION FOR LONG LINES 
 
 97 
 
 is converted to degrees by multiplying by 57° .29578. 
 The hyperbolic cosine and sine of this complex angle are 
 next obtained by the aid of logarithms of the functions 
 of the component parts of the hyperbolic complex angle 
 0. The equation for cosh and sinh 6 is given just 
 above the solution. With a vievtr of eliminating the 
 necessity of calculation for each complex angle, cosh 6 
 and sinh 0, Dr. Kennelly has prepared tables and charts 
 from which these two functions (and others) may be 
 obtained directly, thus very materially shortening the 
 solution by hyperbolic functions. Since complex angles 
 have two variable components (O^ + ; 6^) tables of 
 functions of such angles would have to be quite exten- 
 sive in order that the steps for which values for the 
 functions are given be not excessive. Although tables 
 of functions of complex angles are not as complete as 
 is desired they are a great help in the solution of ordi- 
 nary power circuits. Functions corresponding to angles 
 lying between the values for angles in these tables may 
 readily be approximated by simple proportion, giving 
 values sufficiently accurate for ordinary power trans- 
 mission circuits. They have been calculated in Chart 
 XVI for the purpose of illustrating such procedure and 
 also as a high degree of accuracy was here desired for 
 the purpose of illustrating the agreement of the results 
 as obtained by different rigorous methods. Ordinarily 
 these values would be taken from tables. 
 
 SOLUTION BY NOMINAL tt METHOD 
 
 By this method, in place of considering the admit- 
 tance of the circuit as being distributed (as it is in the 
 actual circuit) it is based upon the assumption that the 
 total conductor admittance may be lumped at two points, 
 one half being placed at each end of the circuit. Such 
 an artificial circuit is known as a "nominal -ir" circuit 
 since the nominal values of impedance and admittance 
 are ascribed to this circuit. On the above assumption, 
 the current per conductor is the vector sum of the 
 receiving end load and the receiving end condenser cur- 
 rents. The sending end current is the vector sum of 
 the conductor and the sending end condenser currents. 
 The performance of such a circuit may be determined 
 either graphically or mathematically. 
 
 If the circuit is not of great electrical length, (say 
 not over 100 miles at 60 cycles or 200 miles at 25 
 cycles) the performance of the corresponding nominal 
 w circuit will not be materially different from that of 
 the actual circuit having distributed constants which it 
 imitates. If, however, the circuit is of great electrical 
 length the performance of the nominal ir circuit no 
 longer closely imitates the performance of the actual 
 circuit which it represents, owing to an error due to the 
 lumpiness of the artificial circuit. Dr. Kennelly has 
 shown that by making certain modifications in the linear 
 or fundamental constants for the impedance and admit- 
 tance of the nominal v circuit, the lumpiness error will 
 vanish, so that the artificial circuit will then truly re- 
 present at the terminals the behavior under steady state 
 
 operation, taking distributed admittance into account. 
 Such a corrected artificial circuit is known as the 
 "equivalent" w circuit, because it then becomes externally 
 equivalent to the actual circuit, having distributed con- 
 stants, in every respect. 
 
 The complex numbers which must be applied to 
 the impedance, Z and the admittances, — and 
 
 of the nominal ir circuit in order to correct these nom- 
 inal values into the equivalent circuit are called the cor- 
 recting factors of the nominal ir circuit. The nominal 
 values of the impedance Z and the admittances — 
 
 of the circuit must be multiplied by these vector correct- 
 ing factors in order to convert them into the "equivalent" 
 
 values; thus: — 
 
 s\nh e 
 
 2 ~ 2 
 
 e 
 
 tank e/ 2 
 e/2 
 
 Where = yJZY is the hyperbolic complex angle sub- 
 tended by the circuit. 
 
 Complete tables of hyperbolic functions are not al- 
 ways available; then again, many engineers have a nat- 
 ural aversion to the use of such functions. In order 
 to avoid these objections as well as to simplify calcula- 
 tions. Dr. Kennelly has charted these "correcting 
 factors" for hyperbolic complex angles up to 
 = i.o radian in steps of o.oi in size and i degree in 
 slope. The writer is particularly indebted to Dr. 
 Kennelly for these charts, which are reproduced here- 
 with for the first time, as Charts XVIII, XIX, XX and 
 XXI. It is believed that the use of these charts will 
 greatly simplify the calculation of the performance of 
 electric power transmission circuits by hyperbolic func- 
 tions. They enable the vector values of these ratios to 
 be read to at least three decimal places in sizes and to 
 two decimal places in slope, and their availability makes 
 the use of tables of hyperbolic functions unnecessary. 
 The corrected conductor impedance Z' is the same as 
 the familiar auxiliary constant B. 
 
 EQUIVALENT ir SOLUTION FOR PROBLEM X 
 
 The solution for problem X by the equivalent r 
 method is given in Chart XVII. At the top of the sheet 
 are two diagrams, one a diagram for one conductor of 
 the circuit of problem X and the other a corresponding 
 vector diagram of the currents and the voltages at both 
 ends. The numerical values of the angles and the quanti- 
 ties pertaining to problem X are placed upon the two 
 diagrams for the purpose of assisting in following the 
 mathematical solution. 
 
 The physical properties of the circuit are first set 
 down, its linear constants obtained from the tables of 
 constants and multiplied by the length of the circuit to 
 obtain the total values per conductor. The next pro- 
 cedure is to calculate the hyperbolic angle of the 
 circuit. To do this the impedance and the admittance 
 of the circuit are set down as complex quantities in the 
 form of polar co-ordinates and multiplied together by 
 multiplying their slopes and adding their angles. The 
 square root of the resulting vector is obtained by tak- 
 
HYPERBOLIC SOLUTION FOR LONG LINES 
 
 CHART XVIII 
 KENNELLY CHART FOR IMPEDANCE CORRECTING FACTOR 
 
 (FOR ANGLES HAVING POLAR VALUES BETWEEN AND 0.40) 
 
 READ POLAR VALUE "SIZE'OF CORRECTING FACTOR HERE 
 
 COPYRIGHT 1920 A. E. KENNELLY 
 
 To find the vector "correcting factor" corresponding to any complex line angle 0, of a 
 circuit, the angle 8 is expressed in polar form with the slope in fractional degrees. The 
 correctmg factor as read from the chart will be in polar form with its slope in fractional 
 degrees. Consult Table P for rapid conversion to minutes and seconds. For example; — 
 
 e — 0.3 768° . correcting factor = a9893 /o° .Co = 0.9893 /o°36'oo" 
 
 = a2l5 /8o° .5, correcting factor = o.fKjs; /o° .14 9 — a9927 /o°o8'56" 
 
HYPERBOLIC SOLUTION I'OR LONG LINES 
 
 CHART XIX 
 KENNELLY CHART FOR IMPEDANCE CORRECTING FACTOR 
 
 V9 
 
 (FOR ANGLES HAVING POLAR VALUES BETWEEN 40 AND 1 .0) 
 
 00 
 
 CO 
 
 READ POLAR VALUE SIZE" OF CORRECTING FACTOR HERE 
 O 
 
 a> 
 6 
 
 oo 
 
 00 
 00 
 
 00 
 
 — w 
 
 ■CO 
 
 a> 
 
 03 
 
 o> 
 
 a> o 
 o> o 
 
 d — 
 
 COPYRIGHT 1920 A E KENNELLY 
 
 To find the vector "correcting factor" corresponding to any complex 
 line angle 0, of a circuit, the angle 6 is expressed in polar form with the 
 slope in fractional degrees. The correcting factor as read from the chart 
 will be in polar form with its slope in fractional degrees. Consult Table 
 P for rapid conversion to minutes and seconds. For example:— 
 
 = 0.8 /62° . correcting factor = 0.943 /.s" .TO = 0.943 /■;°ii'24'' 
 e = 0.6499 /78" .57 , correcting factor = 0.9365 /i° .6t = a9365 
 /i°.^6'.^6" 
 
leo 
 
 HYPERBOLIC SOLUTION FOR LONG LINES 
 
 CHART XX KENNELLY CHART 
 FOR ADMITTANCE CORRECTING FACTOR 
 
 (FOR ANGLES HAVING POLAR VALUES BETWEEN AND 0.20) 
 
 90°- 
 
 0.05 
 0.06 
 
 READ POLAR VALUE SIZE' 
 
 OF CORRECTING FACTOR HERE 
 
 COPYRIGHT 1920 A. E. KENNEULY 
 
HYPERBOLIC SOLUTION FOR LONG LINES 
 
 lOI 
 
 CHART XXI KENNELLY CHART 
 FOR ADMITTANCE CORRECTING FACTOR 
 
 (FOR ANGLES HAVING POLAR VALUES BETWEE;N 0.20 AND 0.50) 
 
 r80°T0° 
 
 -2.r 
 
 
 u 
 
 
 
 c 
 
 
 
 111 
 
 
 
 I 
 
 
 
 (E 
 
 
 
 
 
 
 
 t- 
 
 
 
 
 
 
 
 < 
 
 
 
 k. 
 
 
 
 
 
 
 
 Z 
 
 
 
 
 
 
 1- 
 
 
 
 u 
 
 
 
 u 
 
 
 
 (E 
 
 K u >> 
 
 V 
 
 oc 
 
 VJiXl 
 
 J3 
 
 
 
 
 
 
 fi-S-S 
 
 v^ 
 
 ll 
 
 o-c'o 
 
 
 
 
 S S-? 
 
 A 
 
 111 
 n 
 
 ^^'^ 
 
 ^ 
 
 3 
 0) 
 
 to a 
 r fo 
 then 
 
 (A 
 
 
 
 
 u 
 
 ding 
 pol 
 » is 
 
 « 
 
 < 
 
 Ul 
 
 
 K 
 
 c.E u 
 
 u 
 
 
 c 
 
 
 
 s-s- 
 
 a 
 
 
 orre 
 ress 
 the 
 
 'e 
 
 
 
 •0 
 
 a 
 
 
 ° ,., 
 
 V 
 
 
 0.2- 
 
 ^ 
 
 
 4: « c 
 
 
 
 ^-" 
 
 
 
 c ho u 
 
 
 recti 
 
 e an 
 
 Th 
 
 
 
 
 cor 
 , th 
 ees. 
 
 b 
 
 
 c« 
 
 
 
 
 
 the voctor 
 
 of a circui 
 
 ctional dcg 
 
 u 
 
 CI 
 
 C 
 
 
 
 
 •0*2 
 
 t« 
 
 
 rt 
 
 
 ^i;"** 
 
 /•^•S^ 
 
 
 id = 
 
 
 
 C-- «■ 1 ^ 
 
 
 H « 4> 
 
 
 
 u 0. 
 
 — 
 
 
 c 
 
 
 V 
 
 a 3.1 
 
 O 19 
 
 •o j; ^ 
 
 •o 
 
 •- a c 
 0.9 o 
 
 y <J »< 
 « ca « 
 
 to c 
 
 c c « 
 
 O t) «i 
 
 o « n 
 c « 
 
 be € 
 
 u^ I/l 
 
 READ POLAR VALUE "siZE" OF CORRECTING FACTOR HERE 
 
 COPYRIOHT 1 920 A. E. KENNELLY 
 
102 
 
 HYPERBOLIC SOLUTION FOR LONG LINES 
 
 ing the square root of the slope and halving the angle. 
 The result is the hyperbolic angle 6 of the circuit ex- 
 pressed in hyps. 
 
 The ratio charts XIX and XXI are next consulted 
 
 , ^, ^. , sink . tank 6/2 
 and the correcting values — ^ and corre- 
 
 sponding to ^e 'hyperbolic angle of the circuit read 
 off. Having thus obtained the correcting factors corre- 
 sponding to this circuit, the linear impedance Z and 
 linear admittance Y per conductor are multiplied re- 
 spectively by the sinh and the tanh correcting factors. 
 
 If the circuit under consideration is electrically 
 short the effect of these correcting factors upon the 
 linear constants will be small and possibly negligible 
 but, as the circuit becomes longer, their effect be- 
 comes increasingly greater. The effect of the cor- 
 recting factors for problem X is to change the 
 linear impedance Z from 270.233 /67° 08' 08" to 
 Z' = 253.083 /68° 44' 41" and to change the 
 linear admittance Y from 0.001563 /90° to Y' = 
 0.001615512 /89° 10' 45". In other words this circuit 
 will behave in the steady state at 60 cycles as though 
 its conductor resistance were reduced from 105 to 
 91.7486 ohms and its inductive reactance reduced from 
 249 to 235.866 ohms. Similarly it will behave as 
 though a non-inductive leak of 11.571 micromhos, has 
 been applied to each condenser in shunt. 
 
 In order to illustrate the exact agreement in the re- 
 sults as obtained by the equivalent tt method wath those 
 obtained by either the convergent series or pure hyper- 
 bolic solution, the ratio values used for this problem 
 were calculated and not obtained graphically. The ac- 
 curacy in the performance resulting from the use of 
 ratio values taken from the charts is well within the 
 requirements of practical power circuits. The mathe- 
 matical solution for these factors is given in Fig. 48. 
 
 Having determined the corrected values for the 
 impedance Z' and the admittance Y' which will produce 
 exact results, the remainder of the solution may be 
 carried out graphically as indicated by the vector dia- 
 gram in the upper right hand part of Chart XVII or 
 mathematically as indicated under this vector diagram. 
 
 EQUIVALENT T SOLUTION 
 
 Dr. Kennelly has shown that the correcting factors 
 which convert the nominal w into the equivalent tt of 
 the conjugate smooth line, are the same as those which 
 convert the nominal T into the equivalent T, but in in- 
 verse order; — ^that is the correcting factors for the 
 nominal T line are 
 
 tank 61 2 
 
 Z' = z- 
 
 Y' = Y 
 
 eh 
 
 sinh 
 
 Either the equivalent tt or the equivalent T solution 
 may be used by applying the two correcting factors 
 properly. Usually less arithematical work will be re- 
 quired for the equivalent tt solution. 
 
 ELECTRICAL CONDITIONS AT INTERMEDIATE POINTS 
 
 In the foregoing, the behavior of circuits at their 
 terminals has been considered. In some cases it may 
 
 be desirable to predetermine the voltage and the current 
 at points along the circuit between the terminals. This 
 may be particularly desirable in case of circuits of great 
 electrical length and consequently having a pronounced 
 bend or hump in the voltage graphs representing the 
 voltage at points along the 'ciTx;uit. In Fig. 21 voltage 
 and current graphs were shown for the circuit of prob- 
 lem X corresponding to zero load ; also load conditions. 
 Accompanying this stated was the step-by-step method 
 by which the current and voltage at these intermediate 
 points had been determined. In a corresponding man- 
 ner the intermediate electrical conditions may be de- 
 termined by the employment of hyperbolic functions. 
 It is usual, however, when employing hyperbolic func- 
 tions for determining the voltage or the current at points 
 along a smooth circuit, in the steady state, to take ad- 
 vantage of the following facts relative to the variation 
 in current and potential from point to point in such a 
 circuit. 
 
 The potentials of any and all points of such a cir- 
 cuit are as the sines and the currents as the cosines of 
 the corresponding position angles. This means that if 
 the position angles corresponding to two points of a 
 smooth circuit in the steady state are known, and the 
 voltage or the current at one of these points is also 
 known, then the voltage or current at any other point 
 will be directly proportional to the sine or the cosine 
 respectively of the corresponding position angles. In a 
 similar manner, the impedance follows the tangents, the 
 admittance the contagents and the volt-amperes the sines 
 of twice the angles. Herein lies the beauty of the 
 application of hyperbolic functions of complex angles 
 for determining the electrical performance of electric 
 circuits. The relationship expressed above (taken from 
 Dr. Kennelly 's "Artificial Electric Lines") are given in 
 equation form below for ready reference : — 
 P.„ sinh e„ 
 
 I . _ cosh 9p 
 Z _ 
 
 "z7 ~ 
 
 cosh 0. "^""^"<: ^ 
 tanh ».. 
 
 coth 0p 
 
 coth 9c 
 
 Kv-<iv\ I sinh 2 $p 
 
 ■ numeric l_ 
 
 I Kz'-OcI \sinh 2 Be 
 
 Where p and c are points along the circuit, c being 
 some point where the electrical conditions are known, 
 and p the point for which they are to be computed. The 
 vertical lines enclosing the two parts of the last equa- 
 tion are for the purpose of indicating that the "size" of 
 these complex quantities are referred to in this equation. 
 
 POSITION ANGLES 
 
 Reference has been made to the line as subtending 
 a certain complex hyperbolic angle 6. Since the circuit 
 through the load also encounters resistance and react- 
 ance, the load may be said to subtend also a certain 
 complex hyperbolic angle, so that the receiving end of 
 the circuit occupies an angular position df The total 
 
HYPERBOLIC SOLUTION FOR LONG LINES 
 
 WO 
 
 CHART XVII— RIGOROUS EQUIVALENT -t SOLUTION OF PROBLEM X 
 
 ►»*• 
 
 CHARACTERISTICS OF CIRCUIT 
 
 LENGTH. 300 MILtS. CYCLES, 60. 
 
 CONDUCTORS- 3 * 000 STRANDED COPPER. 
 SPACING OF CONDUCTORS 10 X 10 X 20 FEET. 
 EQUAVALENT DELTA SPACING = l2.e FT. 
 
 LINEAR CONSTANTS OF CIRCUIT 
 
 FROM TABLES PER MILE 
 TABLE NO. 2. r= 0-360 OHM AT 26' O. 
 TABLE NO. 6.X=0.830 OHM (BY INTERPOLATION). 
 TABLE NO. 10. b= 6.21 X 10"^ MHO (BY INTERPOLATION) 
 g=(IN THIS CASE TAKEN AS ZERO). 
 TOTAL PER CONDUCTOR 
 R - .360 X 300 = 105 OHMS TOTAL RESISTANCE. 
 X = 0.830 X 300 = 249 OHMS TOTAL REACTANCE. 
 B = 6.21 X 300 X I0"^ = .00I663 MHO TOTAL SUSCEPTANOE. 
 G = X 300 = MHO TOTAL CONDUCTANCE. 
 
 SOLUTION FOR HYPERBOLIC ANGLE 8=^^17 
 
 2=l06+j249 Y = + j°-°OI563 
 
 , = 270.233 /67°e'8' — 0.001563 I 90° 
 
 . © -=Y 270.233 /67° 8' 8' X 0.00 1 663 190° 
 -yo.4223745 /I 57' 8' 8' 
 = 0.6499036 /7B° 34' 4 ' HYP. 
 
 - 0.6499036 /78'.6e78 HYP. 
 
 — O.I288I68+J0.6370092 HYP. 
 
 FROM DR. KEN NELLY'S CHARTS 
 
 CHART XIX S!M5 _ 0.9i65366 . /it 6094 - 0.9366366 /1° 36' 33' 
 
 ruiPT yvi TANH ll , , 
 
 CHART XXI ^ = 1.033698 \0°.8208 = 1.033698 \0''49'I5- 
 
 * THESE VALUES WERE CALCUUTED IN ORDER TO OBTAIN A HIGH 
 DEGREE OF ACCURACY FOR THE PURPOSE OF DEMONSTRATING THE 
 FUNDAMENTAL ACCURACY OF THIS METHOD. 
 
 CORRECTION OF LINEAR CONSTANTS 
 
 Z' - 270.233 / 67" 8' 8* X 0.9366366 /l'36'33' 
 
 - 263.083 /68''44'4I' (WHICH IS AUXILIARY OONSTANTCB)) 
 
 - 9 1 .7486 + j 236.886 OHMS 
 
 Y'- 0.001563 [90; X 1 .033698 \0''49' 16' 
 
 - 0.001616612 /89° 10' 46' MHO 
 
 Y' 
 
 ■^ - 0.000807766 /89° 1 0' 46' 
 
 " 0.0000 I 1 67 1 + j 0.00080767 
 
 - 1238 \89°J0'45' OHMS REACTANCE. 
 
 CALCULATION OF PERFORMANCE • 
 
 PER PHASE TO NEUTRAL 
 
 2-6,000. KWrn-!^50_6,4oO. 
 
 KV-ApN-i 
 
 -RN" 
 
 p. 104.000 - 
 
 t 8,000XI000 „ 
 
 R 60,046 
 
 pPj, - 80% UGGINQ. 
 
 RECEIVING-END CONDITIONS 
 
 Iqr— 60,046X0.000807766 /b9° 1 0' 46' — 46.6026 / 89° 1 0' 45' 
 
 — 0.6848 + j 48.4973 AMP. 
 
 Ir — 99.92 (0.90 -j 0.436) + 0.6848+ j 48.4873 
 
 - 80.623+ J4.8322 AMPS. 
 = 80.76 /3° C' 55' AMPS. 
 
 PFr- COS 3° 6' 65*- 98.85% LEADING. 
 KVVqr- 60.046 (0.6948 +j 48.4973) 
 
 = 41.72 + J29I2.069 
 KWrn= 6000 (0.B0-j0.436)+41.72 + j29l2.07 
 
 = 644 1. 72 + j 296.07 
 
 12'" 90.76 /3° 6' 65' X 263.083 / 68° 44' 4 1 ' 
 « 22970 /7l°5r36' VOLTS 
 = 716l+j2l,828,VOLTS 
 
 SENDING-END CONDITIONS 
 
 EsN=e0,046 + 7l51 + j2l828 
 
 ■= 67,197+j 21,828 VOLTS 
 
 = 70,652 /l7°59'45' VOLTS 
 I - 70,662X0.000807766 /89°ID'45' 
 
 ■ 0.8 1 76+ j 57.0636 (TO SUPPLY END VOLTAGE) 
 =■ 67.0696 \lori0'30' TO VECTOR OF REFERENCE 
 = - 1 6.8604 + j 64.6089 
 
 Is- (90.023 +j4.8322) + {- 16.8604+ j 64.6088) 
 - 73.77 + 69.44 AMPS. 
 = 94.74 /38°6I'38' a'mPS. 
 
 PFs= COS 38° 61' 38'- ir 59' 46' — 93.44* LEADING 
 K WsN- 70.662 X 94.74 X .9344 =. 6266 KW PER PHASE 
 LOSS= 6255- 6400 = 865 KW PER PHASE 
 EFF" 6400X^^3533^ 
 
 ZERO LOAD CONDITIONS 
 
 REOEIVINGrENO VOLTAGE = 60,046 VOLTS 
 IP' = (0.6948 + j 48.4973) X 9 1.7486 - 64 + j 4449 VOLTS 
 |)(' ^ (-48.4973 + j0.6848)X236.866— -11438+ jl64 
 IR' + IX' - -11374+4613 
 
 E<,Nn= (60,046-ll373)+4613-48890 VOLTS /6°24'61' 
 
 ♦The above results check with those obtained by convergent series. (See Chart XIII). 
 
104 
 
 HYPERBOLIC SOLUTION FOR LONG LINES 
 
 angle of the circuit (line and load) will be 6, + ^ 
 = 0,. By similar reasoning all points lying between 
 the receiving and sending ends of a line will occupy or 
 assume an angular position 6 p. If that part of the 
 linear angle 6 of the line between the receiving end and 
 the point p be designated as 6pr, then the angular posi- 
 tion of the point p will be ^p = ^r + dp,. Thus, at 
 a point in the middle of the line, the position angle will 
 be ^p = ^, + ^p, = 0, + 6/2. 
 y If the line is grounded or short-circuited at the re- 
 
 ceiving end, there will be no load containing resistance 
 and reactance, and consequently no load angle. In such 
 case 6 1 = and the distribution of position angles along 
 the line will be purely a linear function of the total line 
 angle 6. In such a case 6^ = 0. 
 
 Load Conditions — In Fig. 49 the procedure is 
 shown which may be followed for determining by com- 
 plex functions of position angles the current and the 
 voltage vectors at points 25 miles apart along problem 
 X circuit, under load conditions. 
 
 The procedure is first to determine the complex 
 angle 6,, at the receiving end resulting from the load. 
 The mathematical determination of this load angle is 
 tedious. Such determination is given for problem X 
 circuit under stated load in Fig. 49. This complex 
 angle $, of the load (that is the position angle at the 
 receiving end) is such that its complex tangent equals 
 the impedance load S to ground, or zero potential, at 
 the receiving end of line (ohms z. ) divided by the 
 surge impedance Zg of a conductor (ohms Z- )• 
 That is, — 
 
 S 
 
 tank $T = 
 
 Zo 
 
 Since we are here interested only in the ratio be- 
 tween the load impedance and the surge impedance, the 
 values may be taken either per unit length or total per 
 conductor. Although tanh 6^ is readily calculated, as 
 may be seen by consulting Fig. 49, the subsequent cal- 
 culation for the corresponding angle ^r is tedious. After 
 having calculated the tanh 6,, the corresponding angle ti, 
 may be obtained with sufficient accuracy from a table 
 of tangents of complex angles or, more readily still, 
 from a chart of such functions.* After having de- 
 termined the angle ^r by consulting a chart of tangents 
 of complex angles, or by mathematical calculation, as 
 in Fig. 49, the position angles at points along the cir- 
 cuit may easily and readily be determined as follows : 
 
 The change in the position angle from point to 
 point along the circuit, due to the line impedance and 
 the Hne admittance is purely a linear function of the 
 line angle 0. This is the case whether the line is 
 grounded, loaded or free at the receiving end. 
 
 Referring to Fig. 49, the angular position of the 
 receiving end, due to the load conditions assumed, was 
 calculated to be 0.48047 -f j 1.06354. It is therefore 
 necessary to add this angle to each of the linear line 
 angles of the various points along the line in order to 
 obtain the position angles of the points in question. 
 
 Thus the linear line angle of the middle point of the 
 Circuit is 0.0644084 -|- j 0.3185046 and adding to this 
 the load angle 0.48047 + J 1-06354 gives 0.544874 + 
 J 1.3820446, which corresponds with the entry in the 
 tabulation of Fig. 51 for the position angle at the middle 
 of the circuit. In a similar manner position angles for 
 the load assumed are readily determined for points 25 
 miles apart. Having determined the position angles for 
 the various points along the circuit, the sines and the 
 cosines corresponding to these position angles may be 
 approximated closely from tables or charts of such com- 
 plex functions, or may be calculated accurately by fol- 
 lowing the equations at the lower left hand corner of 
 Fig. 51. Since the receiving end voltage and current 
 are known to be 60 046 volts and 99.92 amperes respec- 
 tively, the voltage and currents at all other points of this 
 circuit will be as the sines and cosines of the corre- 
 sponding position angles. From the vector quantities 
 that have been assigned to the voltage and current at 
 the points along the circuit, the power-factors at these 
 points are readily determined. 
 
 The current and voltage graphs at the bottom of 
 Fig. 51 were plotted from values as determined by the 
 use of functions of position angles. These check ex- 
 actly with similar graphs as determined by the Wilkin- 
 son charts and step-by-tep process (See Fig. 21). 
 
 Zero Load Condition — The procedure which may 
 be followed for determining the position angles under 
 zero load, their functions and the corresponding current 
 and voltage distribution is the same as given above for 
 load conditions and is shown in Fig. 50. In this case, 
 however, there is no load and consequently no real 
 part to the load angle. On the other hand the imped- 
 ance of the load is infinite, that is 8 = oc so that 0, = 
 
 lanh'^-y- =• j—r. The effect of this supersurge impedance 
 load at the receiving end at zero load is to cause a phase 
 rotation of 90 degrees or one quadrant, / — = 1.57080 
 
 circular radians. Thus, at zero load, ^ro =(0 + /— ) = 
 " + / 1 .57080 and this angle must be added to each of 
 the linear position angles of the points along the line. 
 With the position angles corresponding to zero load thus 
 obtained, and assigned to the points along the circuit, 
 the voltage will be found to follow the sines, and the 
 current the cosines, etc. of these position angles. 
 
 POLAR DIAGRAM OF CURRENT VOLTAGE 
 
 In Fig. 52 are shown the polar graphs of the volt- 
 age and the current for problem X, corresponding to 
 load, and also to zero load conditions. These polar 
 graphs were plotted from the vector values for current 
 and voltage as tabulated in Figs. 49 and 50 for each 25 
 miles of circuit. 
 
 ♦Such as that worked out by Dr. Kennelly and published by 
 the Harvard University Press. The chart atlas referred to 
 contains graphs of complex tangents of complex angles, and by 
 following the chart in the reverse from the usual direction the 
 complex angle corresponding to any complex tangent may be 
 read off directly. 
 
HYPERBOLIC SOLUTION FOR LONG LINES 
 
 MS 
 
 CURRENT & VOLTAGE DISTRIBUTION 
 
 (LOAD CONDITIONS) 
 
 I 
 
 I 
 
 (300 MILES) - 
 
 -X(MILES)- 
 
 - (D-X) MILES -•• 
 
 0pR = a(D-X) « 
 
 i^ 
 
 r 
 t 
 
 i- 
 
 NEUTRAL 
 
 , ep-ep„+0R 
 
 EpN 
 
 I 
 
 8ENDINQ 
 END 
 
 (D-X) 
 
 MILES 
 
 X 
 
 MILES 
 
 POSITION ANGLE 
 ep=0pR+0R 
 
 SINH 0p 
 
 aHE VOLTAGE FOLLOWS 
 THIS COMPLEX FUNCTION) 
 
 ^PN 
 VOLTS Z_ 
 
 COSH 0p 
 
 (THE CURRENT FOLLOWS 
 THIS COMPLEX FUNCTION) 
 
 ■p 
 
 AMPERES L. 
 
 PFp 
 
 % 
 
 O 
 
 300 
 
 6^= (.O'Si'/l" 
 
 0. Z4Z4 9+J0.9769 3 
 = l.00(>57 l7h°03'3S' 
 
 60 O-fb 
 /O'O'O" 
 
 a 54294 +^0.436 3 2 
 = 0.h9i,S4/38°47'lO' 
 
 99.92 
 \2S'S0'3I' 
 
 -90.00 
 
 25 
 
 2.7S 
 
 0.49/ZO +-;/.// 642 
 
 O.ZZ4Z(,+jl.009Z 
 = 1.0339 /77°2S'li' 
 
 (>l h70 
 1 l°Z4'4l" 
 
 0.4937Z+J0.4S937 
 = 0.67364/42°59'38' 
 
 96.64 
 \2/'>38'03' 
 
 -93.33 
 
 SO 
 
 ZSO 
 
 O.SOIf\+al.lh97l 
 01= hi" 01' 10' 
 
 0.ZO43O+ri 1.0391 
 = l.0S?O l7S.''5Z'3h' 
 
 {,3 /73 
 /Z°49'OI' 
 
 0.44 64+30.48/76 
 = 0.65288/-»7''33'0»' 
 
 93.6 6 
 V7''0-#'33" 
 
 -94.04 
 
 7S 
 
 ZZS 
 
 0.5IZb7^ + ^I.ZZZ7<f 
 ©2= 700 03' 3r 
 
 O.IZZS9 +jl.0b(,3 
 = I.0SI9 /80'>li,'S9' 
 
 H° I3'24' 
 
 • 0.3S692+J0.50333 
 = 0.63480/52° 27'25' 
 
 91. 0& 
 \IZ'I0'Z0" 
 
 -9S.94 
 
 100 
 
 ZOO 
 
 O.SZ^^l +3/.Z7Sa7 
 
 %= 73' o(>' or 
 
 O.IS9I7 ■\-jl.0909 
 = /.IOZ5 /Si'^l'SS' 
 
 ,65 770 
 /5°38'20' 
 
 0.33139 +J0.S2399 
 = 0.61999 IST'4I'Z2'' 
 
 98.94 
 \6°56'/9' 
 
 -97.60 
 
 
 . IZS 
 
 17 S 
 
 O.S34I4 +j/.3Z%9S 
 e^ 7(,°09'3%' 
 
 = 1.1207 /83° 0/4-3' 
 
 66 854 
 / 7''04'0^ 
 
 0.27447+JO-5436/ 
 = 0.(,08/S /63»/2'3<?' 
 
 87.2 4 
 \/<>25'02' 
 
 -99.90 
 
 /so 
 
 ISO 
 
 0.S4-4-S8 + J/.39Z04 
 Bf" 79° II' 07" 
 
 O.I073S+;jl.l3n 
 = /./36 8 /S^OS^-'SZ" 
 
 (>78IS 
 / 8' 31' 17' 
 
 0.2/6/8 +0O.S(>I97 
 = 0.loOZII IhS'sYsT 
 
 86.37 
 h'19'Sl' 
 
 -99.73 
 
 ns 
 
 IZS 
 
 O.SSS6I +^I.4-3SIZ 
 02= S2°/3'36' 
 
 O.07908+JI.I477 
 = I.IS04 /Sh'OZ'SO" 
 
 &8&Z6 
 /9'S9'S5" 
 
 O.IS(,i,7 +aO.S7927 
 = 0.i>OOZO /74°5/'5/' 
 
 , 86.34 
 1 10° 14' lb' 
 
 +99.99 
 
 xoo 
 
 100 
 
 O.S663S +n'/-4S8ZI_ 
 6^= SS'/i'OS 
 
 0.04-9 2i,+,jl.li>07 
 = l.li,l8 IST'34'II'' 
 
 69306 
 /ir30'3(,'' 
 
 0.096 08+JO.S9S0 8 
 = 0.(,0279 /80''49'42" 
 
 86.47 
 / lb' 12' 01' 
 
 +99- tt, 
 
 z%s 
 
 7S 
 
 O.57708 +Hl.Sf-IZ9 
 0^= ■S'S'irs^ 
 
 0.0I79S-^J I.I707 
 = /./7OS/89°07'/3'' 
 
 69S4Z^ 
 
 //a-os'atr 
 
 O.03455+JO.6O93? 
 = 0.(,O9i2 /8i°4S'/S' 
 
 87.'tS 
 122' 07'3r 
 
 +?S.7tf 
 
 ZSQ 
 
 SO 
 
 O.SS78Z+3I.S943S 
 0;j= 9I''ZI'04'' 
 
 -0.014-7 1 +J/./775 
 = I.I 77S l90''4-7:S7" 
 
 70243 
 
 / I4'39'22' 
 
 -0.02784 ■*"J0-622O7 
 = 0.62270 /92°33'44" 
 
 89.33 
 /STSb'oy 
 
 +97.32 
 
 
 Z7S 
 
 zs 
 
 0.59SSS-^iJl.i,474h 
 Bj= 94''Z3'34' 
 
 -0.O4863+j'/./8// 
 
 -I.IBZ/ /qs'zi'zs" 
 
 705/7 
 fli,°l7'S3" 
 
 -0. 09073 +-J0.63 306 
 = O.63f53/98"'09'22' 
 
 , 91.74 
 /34' ll'4l'' 
 
 ■\-9S.I7 
 
 300 
 
 
 
 0.boqz9 +JI.700SS 
 62= 97''2i-'03' 
 
 -O.OS3^I +J/.ISJ4 
 = 1.1344 /94°03'28' 
 
 70652 
 /I7°S9'63" 
 
 -0.IS4I6 +J 0.6422 6 
 ■= 0. 6 6 5 // 03*2 9'4.S' 
 
 , 94.75 
 /38'52'04» 
 
 + 99.4i 
 
 
 
 100 
 
 GRAPHS OF CURRENT AND VOLTAGE {LOAD CONDITIONS) 
 
 100 
 
 90 
 
 iso 
 
 1 70>« 
 
 80 a, 
 ui 
 
 80 S 
 
 0. 
 S 
 70 < 
 O 
 
 60 3 
 
 3 60 
 < 60 
 
 50 
 
 40 
 
 40: 
 
 g30 
 ^20 
 
 30 
 
 10 
 
 SENDING 
 END 
 
 26 
 
 60 
 
 75 
 
 100 125 160 175 200 
 
 DISTANCE FROM SENOING-END (MILES) 
 
 225 
 
 260 
 
 siNH(e|-fje2)=(SiNHe| cos ej+j cosh e, sin 02). 
 co«H(e,+je2)=(cosHe| cosOj+jSiNHe, sinoj). 
 
 20 ^ 
 
 10 
 
 275 300 
 
 RECEIVINa 
 END 
 
 ANGLE AT RECEIVING END 0r«=O. 48047+ 1.06364 
 ANGLE OF LINE =0 I2882-I- 0.63701 
 0=0+0 "0.60929+ l.7006f 
 ONE QUADRANT=I.57079632 CIRCULAR RADIANS. S " 
 
 ONE CIRCULAR RADlAN=20e264.8062'=67* 1 7' 44.8: ^ OC -0.00042939+j 0.002 1 2336 
 
 FIG. 51 — CURRENT AND VOLTAGE DISTRIBUTION 
 
 For problem X by position angles (load conditions). 
 
io6 
 
 HYPERBOLIC SOLUTION FOR LONG LINES 
 
 k-- 
 
 f— 
 
 EsNO 
 
 CURRENT AND VOLTAGE DISTRIBUTION 
 
 (ZERO LOAD CONDITION) 
 
 0(300 MILES) 
 
 (MILES)- - 
 
 ■^ 
 
 (D-X)i(MILES) >i 
 
 0pRQ=OC (D-X) *i 
 
 -i 
 
 Erno 
 
 I 
 
 NEUTRAL 
 
 6po~0pRc'''0Ro 
 
 •PNO 
 
 SENDING 
 END 
 
 RECEIVING 
 END 
 
 (D-XJ 
 MILES 
 
 X 
 
 MILES 
 
 POSITION ANGLE 
 
 6pO~0PRO'^0RO 
 
 SINH 0PO 
 
 (THE VOLTAGE FOLLOWS 
 THIS COMPLEX FUNCTION) 
 
 ^PNO 
 VOLTS Z_ 
 
 COSH0PO 
 
 (THE CURRENT FOLLOWS 
 THIS COMPLEX FUNCTION) 
 
 • po 
 
 AMPERES A. 
 
 PF-po 
 
 % 
 
 O 
 
 300 
 
 O +J/.57080 
 
 0j- 9o'oo'oo' 
 
 +y/.ooooo 
 = 1.00000 1 <7o° 
 
 60 04L> 
 
 
 |90° 
 
 /II'ZS'S^" 
 
 
 ZS 
 
 Z75 
 
 0.0/073 +JI.6238S 
 
 <9?= fs'oz'zr 
 
 0.OOO£7+J0.99Si,S 
 = Q.99Si>5 /srSS'o-^ 
 
 ,59 9<oS 
 /o'0l'S7" 
 
 0. 5 307 +J 0.0/ 070 
 = O.054\4 W°23'S8" 
 
 , 7.82 
 \9o°ol'S8'' 
 
 
 
 SO 
 
 ZSO 
 
 O.OZI-i-i, +d/.i7696 
 
 ^2= ^i'O-f'SS* 
 
 0\0O227+j0.994i>0 
 = 0.99460 /B9°S2'09'' 
 
 £9 803 
 /O" 07' £1" 
 
 O./o&98 + ;J0.OZI33 
 = 0./08U /ll'>Z2'48" 
 
 /£.&Z 
 \9o''03'08" 
 
 + 00. /2 
 
 7S 
 
 225 
 
 0.03S ZO +ai.73O04 
 
 O.OOS/I +J0.9878S 
 ~0.<7878^ /89°42'l3" 
 
 £9 317 
 /0°n'47'' 
 
 O.ISB(oi> +J0.O3I79 
 = 0.lkl8l ///Vr-f8" 
 
 23.37 
 
 \9o''o(>'oa" 
 
 +00.32. 
 
 lOO 
 
 ZOO 
 
 0.04294 +y/.7»3/3 
 
 6>2= loz'orsf 
 
 0.O0 9O6+J0.97S44 
 = 0.97847 /89°ZS'lsf' 
 
 58753 
 /0'3\'48" 
 
 0. 2,IO<70+jo.04/<?? 
 = 0.2)503 [jl" IS'3S" 
 
 , 31. OS 
 \9o° lo'zi" 
 
 +00.6/ 
 
 IZ5 
 
 17 S 
 
 O.OS3i>7+j l.'SnbZI 
 02= /Oi'lZ'Zi" 
 
 0.0 1409 +j 0.'J6638 
 -0.<fi64'S /89°09'J0' 
 
 .58033 
 /o° so' 10" 
 
 o.zi,Zi,f+jO.0£lS4Z 
 -O.Zi,77(o / WO')' so" 
 
 3 8.6 6 
 \90Vi,'0i," 
 
 +^0-9? 
 
 ISO 
 
 ISO 
 
 0.064^/ +J/.8S'730 
 02= IO'B°l4'£i," 
 
 O.OZOI S +J0.9S/6S 
 = 0. 9SIS8/8d''47'07" 
 
 £7/£ h 
 / 1° 12' £3" 
 
 0.3 I 3 SO +J0.0& /ZO 
 =0.31^70 //I'O-X'/O" 
 
 4i>.l7 
 \9o°23'4i>'' 
 
 + /.4Z 
 
 175 
 
 IZS 
 
 0.Q7SI4 +c//.9^238 
 
 02= iifiYzs' 
 
 0.0273/ +JO.93436 
 
 = o.93^7& /ae''i9'33" 
 
 £& 129 
 / 1" 40' 27" 
 
 0.3 64 17 +J0.070Oi> 
 = 0.370BSl/0''£3-'Zi' 
 
 £3.SS 
 \90°3Z'33" 
 
 +1.98 
 
 ZOO 
 
 lOO 
 
 0.0 8SS8+J/.99S-t(, 
 0^= II4°I9'£4" 
 
 0.O3J43+J0.9/4SZ 
 =■ 0.9 1 S ZZ 1 '&7°4h' £-y 
 
 £4 9£S 
 / 2° 13' 07" 
 
 0.41 3S4+J0.0783S 
 = 0.42090 /I0''43'4 i" 
 
 .60.77 
 \90°42'/5'' 
 
 ¥2.iS 
 
 Z25 
 
 75 
 
 O.O'jhkl +J2.048S4 
 e^- II7'2Z'24" 
 
 0.04449+ j 0.89218 
 =0.89328 / 37° 08'4Z" 
 
 £3 i>38 
 /Z" SI' 17" 
 
 0.4 10194- +J0.0SS93 
 ^0.4h9 8i> /lO'SZ'K," 
 
 &7-8S 
 \90''£3'40" 
 
 4-3.40 
 
 Z50 
 
 SO 
 
 0.I073S +ds.lOI(>4 
 ©2= l20°24S-i' 
 
 O.0S44S+j0.Si,73S 
 = 0. S6 90S /8&°24'28" 
 
 £2/ 83 
 /3°36'32' 
 
 0.SO9/7+JO.O9Z7S 
 
 = 0.S/7£S /icn'zi," 
 
 , 7-5'.73 
 \9/''0&'30" 
 
 f 4-33 
 
 27 S 
 
 ZS 
 
 O./ZSO? i-j 2JS473 
 02~ /23»27'22'' 
 
 o.0i,SZS + J0.840 14 
 = O.Z4Z&7 /SS'SS' 3J" 
 
 £0 599 
 /4° 2 6' 2 7" 
 
 0.SSSI4+J0.09874 
 = 0.S6385 /lO'OS'OT' 
 
 , S/.^2 
 \9l°20'49" -^^■^' 
 
 300 
 
 o 
 
 0./2 SSZ +j a.2079l 
 02= IZi?29'Si' 
 
 O.076&3 + JO.8/OS6 
 = 0.8/4Z0 /84''36'0S" 
 
 ■48 889 
 / £'24' ST 
 
 0.£9973-i-jO. 103^4 
 = O.6 8i,S/9''4 9'Z2' 
 
 S7.89 , , , 
 \9l''3&'34" +b.h4 
 
 
 GRAPHS OF CURRENT AND VOLTAGE (ZERO LOAD CONDITIONS) 
 
 <a 90 
 
 UJ 
 (£ 
 
 S 80 
 
 S 
 ■* 70 
 
 
 
 
 
 
 
 
 
 
 
 
 
 - lUO 
 
 ^--^, 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 -90 2 
 a. 
 
 on ^ 
 
 
 ' 
 
 
 
 
 
 
 
 
 
 
 
 70 "* 
 
 O '" 
 
 3 
 
 < 60 
 
 
 
 
 ""^^ 
 
 . 
 
 
 
 
 
 VOLTAGE 
 
 
 
 -60 < 
 
 < 
 f 50 
 
 " 
 
 ^ 
 
 
 
 
 ^^^ 
 
 ks2£iyr 
 
 
 
 
 
 
 -50 2 
 
 3 
 
 Z *" 
 
 Pin 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 20 
 
 
 
 
 
 
 
 
 
 ^'^^ 
 
 
 
 
 > 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 SENDING 
 END 
 
 50 
 
 100 125 150 175 200 
 
 DISTANCE FROM SENDING-END (MILES) 
 
 225 
 
 250 
 
 276 300 
 
 RECEIVING 
 END 
 
 SINH(9|+je2)=(SINHe| COS 92+ j COSH e, SIN 82). 
 
 COSH (e|+je2)=(C0SH e, cos Oj+j sinh e, sin e^). 
 
 ONE QUADRANT= 1 .67079632 CIRCULAR RADIANS 
 ONE CIRCULAR RADIAN=206264,B062"=57' I T 44.8" 
 
 OC =0.00042939+j 0.002 1 2336 
 
 ANGLEAT RECEIVING END 0Ro= +j 1.57080 
 
 ANGLE OF LINE =0. 1 2B82 + J 0.63701 
 
 eso=0+0Ro^°'^^^^''j^'^°^®' 
 
 FIG. 50 — CURRENT AND VOLTAGE DISTRIBUTION 
 
 For pri)I:/cni X by position angles (zero load conditions). 
 
HYPERBOLIC SOLUTION FOR LONG LINES 
 
 107 
 
 PnoBLEM "X" 
 
 H = IOS OHMS. X=24<?0HMS. 
 &=0«M0. Y= 0.00/563 MHO. 
 
 Z = 2 70.2 33 /^7'Og'og* 
 Y= 0.00 /S 6 3 190* 
 6 = -YzT =^0.422 374 ^ //^T'Qg'Og' 
 = 0.64"? 903 J /7^'3-t'04' 
 = 0./2S8/6S-fj0.6 3 7 0092 
 
 CALCULATION FOR *"*" 9 
 
 e 
 
 S//VH e=0-l03 S393+J O.S9973S 
 = 0.60g^.yg3 /g0V0'3g" * 
 
 SiNHd _ O.COS iS,S« 3 /80°/0'37' 
 0.(3^9 903 S /7S' 34'04' 
 
 = 0.93i53(, S //''3i'33" 
 = It^PEZinNCE CORRECTINC FACTOR. 
 
 CALCULATION rOK TAWHCB/a) 
 
 -2=0.324 95/ g /Zg'S-fO-^" 
 = 0.O644084 + jO.3( g 504 6 
 
 ■S/NH-j-=o.O*/2// 2 2 -i- J 0-3 13 796 3 
 = 0.3/f 7/0 7 /7g'37V3' ' 
 
 COSH-^=0.9S/ iVS'H-jO.OZO /83 2 
 
 fi^ SINH(0/2) o^3,g TIP 7/79*5743' 
 
 >^**^ Z CQSH(e/2) 0-9S/ 8S94 /I' /2>S4* 
 
 ■ -= Q.-33S 869 6 i72l±:il±±' 
 TfiNH(6/z)_ 0.33S i(>9(>/77°44'49' 
 
 e/Z 0.3X4 'fS/ 8 /78° 3^'0-r 
 
 = 1.033 S9 8 \0''*'I'/S' 
 
 9 6 
 
 CHECK, SlNH6 = k SINH-^COZHJ = 2X.03I97/07 17 S'sY^^' X Q.9S I SS9t- / 1° I2's^ 
 
 - 0.608 (iS8-^ / 80°/o'38' bNnicH cM£CKi with *J. 
 PIG. 48 — MATHEMATICAL DETERMINATION OF CORRECTING FACTORS FOR EQUIVALENT IT SOLUTION 
 
 PROBLEM "X" 
 
 Z=/OS + ^'Z-4f=Z70-Z'33 /67°08'0B' 0HMS. 
 Y= O +d0.O0ISi>3==O.O0ISi>3\90° MHO. 
 
 6= fzY = o./zs8U?+jo.<>37oo9Z Hyp. 
 
 KV-flf,;^ -6000 00 oV25° 50'3 /' WMTTS. 
 
 = S 400 00 0~j 2 IdI S 340. 
 Ef^l^= ho 044.4 VOLTS TO NEUTRHL. 
 lf^=99.9z60S \2S° 50'3I" 
 
 SOLUTION FOB TANH 0. 
 
 8 = ^ =&00.888 /2S'S0'3l' OHMS. 
 
 ^-0 = ^^=4 I 5. ■80S \ll'Z5'S(i," OHMS. 
 -r«MM f) - ^00-SZ8lx£° -SO'^'" 
 
 = I.44S/Z J37' Ko'27' 
 :^l.l499S+3 O.S7S2Q9 * 
 
 (Bii-dez) 
 
 SOLUTION FOR ANGLE 0- 
 
 TflNH-^{d,±j0,) = iLOGH\UL±%±^ 
 
 {i-0,f+el 
 
 + J 
 
 Tl-TRt^ 
 
 :-l/^l+l 
 
 iM 
 
 + THN' 
 
 01-1 
 ±02 
 
 f^ -2 1/ (/- 1.14 99S)Z +o.375Z09^ ^ 2 
 
 --^ LOGH 
 
 4:&2229+0.7&599 . 80''-67'' So'SS'4-9''43'zO') 
 
 <?.02 2-f 8i5+0.76«?y ^ a 
 
 = X('-OGH Z.i,/4/5) + ji,0''S6'//'' 
 =j{o.96o^39)+j/.063S397 
 
 0^= 0.4-8 04i>9S+jl.06 3S3 97 
 = O. I Z9 8/ 68-^dO-i>37009Z 
 fis= O.i>0 9 ZSk^4-jl.700S4 89 
 
 CHECK 
 
 SINH 0i^=/.OO6S7Z /7(>'03'34J' 
 
 COSH e^ =0.i>9i,S3 /38''^7'0?' 
 
 TMNH 6 = ^'^"^'^ = l-OOi>S7Z /7(,'OZ'3(, ' 
 
 COSH 9, 
 
 0.(>9(>S3 / 3 S'>'^7' 9" 
 
 — 1.14 9 9S 4-;} O.S7'SZ09 (w«/c« CHECKS vvit»*). 
 
 FIG. 49— POSITION ANGLE Ob AT RECEIVING-END 
 
 Mathematical determination at load conditions. 
 
io8 
 
 HYPERBOLIC SOLUTION FOR LONG LINES 
 
 CHOICE OF VARIOUS METHODS 
 
 Two graphical and two mathematical forms of so- 
 lution for circuits of long electrical length have been 
 described thus far. These four methods have been 
 given for the purpose of providing a choice of proce- 
 dure for the beginner. Graphical solutions are more 
 simple and more readily performed than mathematical 
 solutions and, if used correctly and made to a large 
 scale, will yield results well within the limits of per- 
 missible error for power transmission circuits. There 
 is always a possibility of error with any method, even 
 though the solution is carefully checked. For this rea- 
 son it is desirable that errors be guarded against by the 
 use of two different forms of solution. For instance 
 
 f SENDING 
 
 wave, then it will be necessary to take their effect into 
 account, if high accuracy is essential. In such a case 
 there is an independent solution required of potential 
 and current ^pr pc^ch single frequency in turn, as though 
 the others did not exist, and then the r.m.s. value at 
 any point on the line is the perpendicular sum of the 
 separate frequency values. 
 
 A detail discussion of the manner of including the 
 effect of harmonic components in the current and volt- 
 age waves is quoted below from Dr. Kennelly's "Artifi- 
 cial Electric Lines." 
 
 "The ordinary complex harmonic impressed e.m.f. contains 
 a fundamental frequency associated with multiple frequency 
 harmonics. The nth multiple of the frequency is called the nth 
 harmonic. The fundamental may thus be included as the first 
 harmonic. 
 
 "In order to deal with the plural-frequency case quantita- 
 tively, it is necessary to analyze the impressed potential wave 
 iiito its harmonic components. As is well known, the complete 
 Fourier analysis of a complex wave may be written 
 
 SENDINQ-ENO 
 
 60 MILES 
 
 -END 
 
 GRAPH OF CURRENT 
 (LOAD CONDITIONS) 
 
 • RECEIVING-END 
 FIG. 52 — POLAR DIAGRAM OF CURRF.NT AND VOLTAGE DISTRIBUTION FOR PROBLEM X 
 
 the first solution could be made by making use of the 
 Wilkinson charts followed by its accompanying graphi- 
 cal solution. The second solution could then be made 
 by means of Dr. Kennelly's ratio charts XVIII to XXI, 
 followed by its accompanying graphical solution. These 
 two methods would then yield results obtained by two 
 entirely different routes and methods of procedure. 
 The use of two such methods would constitute check 
 against errors being made in either solution. 
 
 EFFECT OF HARMONIC CURRENTS AND VOLTAGES 
 
 The foregoing discussion is based upon the assump- 
 tion that the fundamental wave is of sine shape and 
 consequently free from harmonics. If harmonics of 
 considerable magnitude are present in the fundamental 
 
 F, + V'i sin ut + V'i sin 2«/ + V, sin 3"/ + V'4 sin 4w< -h 
 -I- F", COS w/ -f V'\ cos zw/ -I- V", cos zot -i- 
 
 F"* cos 4w< + v°'*^ ('^ 
 
 where Fo is a continuous potential, such as might be developed 
 by a storage battery, ordinarily absent in an a. c. generator 
 wave, F'l, F"i, F'2, V'\, etc., maximum cyclic amplitudes of 
 the various sine and cosine components. The even harmonics- 
 are ordinarily negligible in an a. c. generator wave; so that 
 F's, V"2, V\, V",, etc., are ordinarily all zeros. If we count 
 time from some moment when the fundamental component 
 passes through zero in the positive direction, V"i = o and the 
 series becomes 
 
 F'l sin at -f- V'l sin 3<.)i + V\ sin 5w< + 
 
 F", cos 3w< + V'\cos. s<^t + .volts (2) 
 
 Compounding sine and cosine harmonic components into result- 
 ant harmonics of displaced phase, this may be expressed as 
 Fr, sin at + Vn sin (aw* -|- /9.°) + Fr. sin (S«' + P» ) + 
 volts (3) 
 
 volts (4) 
 
 where 
 
 and 
 
 V., = v"V\' + FV 
 
 tan ^n = yl 
 
 numeric (5) 
 
' HYPERBOLIC SOLUTION FOR LONG LINES 
 
 Formulas (l) and (2) give the wave analysis in sine and 
 cosine harmonics, while (3) gives it in resultant sine harmonics. 
 
 "When considering a plural -frequency alternating-current 
 line, we require to know the harmonic analysis of the impressed 
 potential, either in sine and cosine harmonics, or in resultant 
 harmonics, the latter analysis is preferable, as being shorter 
 and containing fewer terms. A decision must be made as to 
 the number of frequencies or upper harmonics which must be 
 taken into account. 
 
 "Ordinarily, the sizes of the harmonics diminish as their 
 order increases; but there are numerous exceptions to this rule, 
 as when some particular tooth frequency in the alternatipg- 
 currciit generator establishes a prominent size for that har- 
 monic. Care must therefore be exercised not to exclude any 
 important harmonics. On the other hand, the fewer the har- 
 monics to be dealt with, the better, because the labor involved 
 in correctly solving the problem increases in nearly the same 
 ratio as the number of harmonics retained. 
 
 "The rule is to work out the position angle, r.m.s. potential, 
 and r.m.s. current distributions, over the artificial or conjugate 
 smooth line, for each harmonic component in turn, as though 
 it existed alone, and then to combine them, at each position, in 
 the well-known way for root mean squares. 
 
 "Combination of Components of Different Frequencies into 
 a R.m.s. Resultant. — Let the r.m.s. value of each alternating- 
 current harmonic component be obtained by dividing its ampli- 
 tude withi/^ in the usual way, and let 
 
 y. =— ir = ^ r " +'^ ° r.m.s. voJts (6) 
 
 be the r.m.s. value of the «th harmonic. Then the r.m.s. value 
 of all the harmonics together, over any considerable number of 
 cycles, will be 
 
 V — V TvTTTTTT + r.ni.s. volts (7) 
 
 or, as is well known, the joint r.m.s. value of a plurality of 
 r.m.s. values of different frequency, is the square root of the 
 sum of their squares. If a continuous potential Fo be present, 
 this may be regarded as a r.m.s. harmonic of zero frequency, 
 and be included thus : 
 
 V — V Vo + Vi + V' -f F,' + . . .r.m.s. volts (8) 
 Moreover, from (4), it is evident that the squares of the r.m •>. 
 values of the sine and cosine terms of any harmonic may be 
 
 FIG. 53 — GEOMETRICAL REPRESENTATION OF A JOINT R.M.S. VALUE 
 
 OF PLURAL-FREQUENCY COMPONENTS DY PERPENDICULAR 
 
 SUMMATION OR "CRAB ADDITION" 
 
 substituted for the square of their resultant; or that, in this 
 respect, the sine and cosine terms may be treated as though 
 they were components of different frequencies. 
 
 "The same procedure applies to plural-frequency currents. 
 Find the r.m.s. resultant harmonics. The r.m.s. value of all 
 together will be the square root of the sum of their squares. A 
 continuous current, if present, may be included, as the r.m.s. 
 value of an alternating current of zero frequency. 
 
 "Graphical Representation of R.m.s. Plural-frequency Com- 
 bination. — The process represented algebraically in (7) or (8) 
 may be represented eraphirally by the process of successive pf >■ 
 pendicular summation, or "crab addition." An example will 
 suffice to make this clear. A fundamental alternating current 
 of 100 amp. r.m.s., is associated with a continuous current of 
 50 amp., and with two other alternating currents of other fre- 
 quencies of 20 and 10 amp. r.m.s., respectively. What will be 
 the joint r.m.s. current? Here by (8), », 
 
 / =\/ 100' -f - 50' + 20' -f 10' = V 10 000 -f 2500 -f 400 + 100 
 = 1/ 13000 ^ 114.0175 amp. r.m.s. 
 "In Fig. 53, OA represents the fundamental r.m.s. current. 
 AB, added perpendicularly to OA represents the continuous 
 current, or current of 50 r.m.s. amp. at zero frequency. The 
 perpendicular sum of OA and AB is OB = 1 11.8034 amp. 
 Adding similarly the other frequency components BC and CD, 
 
 109 
 
 the total perpendicular sum is OD = 114.0175 amp. The order 
 in which the components are added manifestly does not affect 
 the final result, and it is a matter of insignificance whether the 
 various frequencies coacting are "harmonic," t. e., are integral 
 multiples of a fundamental, or not, so long as they are different. 
 "The complete solution of an alternating-current line with 
 complex harmonic potentials and currents thus requires an in- 
 dependent solution of potential and current for each single 
 frequency in turn, as though the others were non-existent, and 
 then the r.^n.s. value at any point on the line is the perpendicular 
 sum of the separate frequency values. The powers and energies 
 of the different frequencies are independent of each other, and 
 the total transmitted energy is the sum of the energies trans- 
 mitted at the separate component frequencies. 
 
 BIBLIOGRAPHY 
 
 In order to give due prominence to some of the valuable 
 contributions on the subject of performance of electrical circuits 
 and as an acknowledgment to their authors of the assistance 
 received from a study of them, the following publications arc 
 suggested as representing a very helpful and valuable addition 
 to the library of the transmission engineer. They are given in 
 the approximate order of their publication: — 
 
 Calculation of the High Tension Line and Output and 
 Regulation in Long Distance Lines by Percy H. Thomas. 
 (Published in A. L of E. E. Trans. Vol. XXVIII, Part, I, 
 1909). The former paper introduces a so-called "wave for- 
 mula" for determining the performance of long lines having 
 considerable capacity which embodies the use of algebra ofoXy. 
 The second paper suggests the use of split conductors in order 
 to adjust the ratio of the capacity and inductance of the line 
 so that the leading and lagging components more nearly 
 neutralize each other. 
 
 Formulae, Constants and Hyperbolic Constants by W. E. 
 Miller. (Published in G. E. Review, supplement dated May 
 1910). This is a treatise; upon the subject wherein hyperbolig 
 functions of complex angles are tabulated for sinh and cosh 
 (x + jy) up to X := I, y=: I in steps of 0.02. 
 
 Transmission Line formulas by H. B. Dwight (Published 
 by John Wiley & Sons, Inc.). This book introduces what are 
 known as "Dwight's 'K' formulas," which permit the solution 
 of transmission problems without the use of mathematics 
 higher than arithmetic. It also contains working formulas 
 based upon convergent series and the solution of many problems 
 both by the K formulas and by convergent series. 
 
 Tables of Complex Hyperbolic and Circular Functions by 
 Dr. A. E. Kennelly. (Published by the Harvard University 
 Press). This book gives functions of complex angles for polar 
 values up to 3.0 by steps of o.i and for angles from 45° to 90" 
 by steps of one degree; also functions in terms of reactangidar 
 coordinates x -f- jy to x = 10 by steps of 0.05 and of y 
 virtually to infinity by steps of 0.05. 
 
 Chart Atlas of Complex Hyperbolic and Circular Functions 
 by Dr. A. E. Kennelly. (Published by Harvard University 
 Press in large charts, 48 by 48 cm.) Presenting curves for M 
 the tables published in above referred to "Tables of Complex 
 Hyperbolic and Circular Functions" for rapid graphical inter- 
 polation. J 
 
 Constant Voltage Transmission by H. B. Dwight. (Pub- 
 lished by John Wiley & Son, Inc.). Embraces a very complete 
 study of the use of over-excited synchronous motors for con- 
 trolling the voltage of transmission. 
 
 The Application of Hyperbolic Functions to Electrical 
 Engineering Problems by Dr. A. E. Kennelly. (Published by 
 the McGraw-Hill Book Company). Every student should have 
 a copy of this book because of its simplicity and completeness 
 in explaining the application of hyperbolic functions to trans- 
 mission circuit problems. It also contains a very complete 
 bibliography of publications upon this general subject. 
 
 Artificial Electric Lines by Dr. A. E. Kennelly. (Published 
 by McGraw-Hill Book Co.). This is a valuable treatise in 
 which the subject is treated in accordance with the hyperbolic 
 theory. , 
 
 Electrical Phenomena in Parallel Conductors by Dr. F. E. 
 Pernot. (Published by John Wiley & Son, Inc.). Being a 
 very recent treatise, this book contains much practical and 
 many readily understandable explanations for both the beginner 
 and those further advanced in the study of this subject. It 
 contains a six-place table of logarithms of real hyperbolic 
 functions for values of .r from 0.000 to 2.000 for intervals of 
 o.ooi in the argument. This is the most complete table of real 
 hyperbolic functions which the author has seen. 
 
tio 
 
 HYPERBOLIC SOLUTION FOR LONG LINES 
 
 FABLE P— SUBDIVISIONS OF A DEGREE 
 
 SECONDS 
 
 TO 
 DEGRE-:S 
 
 MINUTES 
 
 TO 
 DEGREES 
 
 DEGREES 
 
 TO 
 
 MINUTES AND SECONDS 
 
 1!= o 
 
 != o 
 
 °= f ff 
 
 °= f If 
 
 01 
 02 
 03 
 
 0.OOO3 
 O.OOOh 
 
 o.oooe 
 
 Ol 
 
 oz 
 
 03 
 
 O.OIbJ 
 0.0333 
 0. 500 
 
 0.00/ 
 0.002 
 0.00 3 
 
 00 03J, 
 00 07.2 
 
 00 /o.e 
 
 0.006 
 0.007 
 0.008 
 
 00 2/.6 
 00 25.2 
 2S.8 
 
 04 
 06 
 
 O.OOI 1 
 0.OO/4 
 O.OOI7 
 
 04 
 03 
 
 06 
 
 O.Okh7 
 0.O833 
 O.IOOO 
 
 0.004 
 
 o.oos 
 
 00 14.4 
 00 18.0 
 
 0.00 9 
 0.0 10 
 
 00 32.4 
 00 3t.O 
 
 07 
 OS 
 09 
 
 0.OOI9 
 
 o. oozz 
 o.oozs 
 
 07 
 08 
 09 
 
 O.llhl 
 O.I 333 
 O.I SOO 
 
 
 
 
 
 /O 
 // 
 
 O.0O28 
 0.OO3I 
 0.0033 
 
 10 
 II 
 12 
 
 0.li>i,7 
 0./833 
 O.2000 
 
 0.01 
 o.oz 
 
 00 36 
 0/ /2 
 
 O.SI 
 
 o.sz 
 
 30 36 
 
 3/ IZ 
 
 /3 
 IS 
 
 O.OOSi 
 O.0039 
 0.00-f2 
 
 13 
 14 
 IS 
 
 0.2/67 
 0.2333 
 O.2500 
 
 0.03 
 0.04 
 
 o.os 
 
 01 48 
 OZ 24 
 03 00 
 
 0.33 
 
 O.S4 
 O.SS 
 
 3/ 48 
 3i 24 
 33 00 
 
 lb 
 n 
 
 IS 
 
 0.00-+4 
 0.0047 
 0.00.50 
 
 Ih 
 17 
 18 
 
 0.2667 
 0.2833 
 0.3000 
 
 O.Oh 
 0.07 
 0.08 
 
 03 36 
 
 04 IZ 
 04 48 
 
 O.Sh 
 0.57 
 O.SS 
 
 33 36 
 
 34 IZ 
 34 48 
 
 /<? 
 
 ZO 
 Zl 
 
 0.0053- 
 0.0 055 
 0.O0.S8 
 
 19 
 ZO 
 Zl 
 
 0.3/67 
 0,3333 
 0.3SOO 
 
 0.09 
 0.10 
 0. II 
 
 05 Z4 
 
 06 00 
 06 36 
 
 O.S9 
 O.hO 
 O.hl 
 
 35 24 
 
 36 00 
 36 36 
 
 22 
 23 
 
 24 
 
 O.OOi/ 
 
 o:ooi4- 
 
 0-00,7 
 
 22 
 23 
 
 24 
 
 0.3667 
 0.3833 
 0.4000 
 
 O.IZ 
 O.I3 
 0.14 
 
 07 IZ 
 
 07 48 
 
 08 Z4 
 
 0.62 
 0.63 
 0.64 
 
 37 IZ 
 
 37 48 
 
 38 24 
 
 ■ ZS 
 
 ze, 
 
 27 
 
 O.OOh9 
 0.007Z 
 0.0073 
 
 ZS 
 
 Zh 
 27 
 
 0.4/67 
 0.4 333 
 O.4S0O 
 
 O.IS 
 
 0.1 h 
 
 0.1 7 
 
 09 00 
 09 36 
 /o /z 
 
 0.6S 
 0.66 
 0.67 
 
 39 00 
 
 39 3h 
 
 40 IZ 
 
 28 
 
 2<? 
 30 
 
 ■ 0.0078 
 
 o.ooai 
 
 O.OO 83 
 
 is 
 
 29 
 30 
 
 a4667 
 0.4833 
 0.3-000 
 
 0.18 
 0.1 9 
 0.20 
 
 10 48 
 
 11 Z4 
 IZ 00 
 
 0.68 
 0.69 
 O.70 
 
 40 48 
 ■4 1 24 
 4Z 00 
 
 3/ 
 32 
 33 
 
 0.0086 
 0.0089 
 O.O09Z 
 
 31 
 32 
 33 
 
 O.Sib7 
 0.333'3 
 0.3S00 
 
 0.2/ 
 0.22 
 0.23 
 
 IZ 36 
 /3 /Z 
 /3 48 
 
 0.7 1 
 
 0.7 z 
 
 0.73 
 
 42 36 
 
 43 /2 
 43 48 
 
 34 
 35 
 3t 
 
 0.0O9^ 
 0.0097 
 
 o.oioo 
 
 34 
 3S 
 34 
 
 0.3 667 
 0.3833 
 0. 6000 
 
 0.24 
 0.23 
 b. 2 6 
 
 14 Z4 
 
 15 00 
 /S 3t> 
 
 0.74- 
 0.7.5 
 0.76 
 
 44 Z4 
 
 45 00 
 4S 36 
 
 37 
 39 
 
 0.0I03 
 0.01 of, 
 
 o.oioe 
 
 37 
 38 
 39 
 
 0.6/67 
 0.6333 
 0.6300 
 
 0.27 
 0.2s 
 0.29 
 
 If, IZ 
 /6 48 
 17 Z4 
 
 0.77 
 
 0.7 S 
 
 0.79 
 
 46 IZ 
 4l> 48 
 
 47 24 
 
 -*< 
 ■42 
 
 O.O II 1 
 0.0//4 
 O.O/ 17 
 
 4-0 
 .41 
 ■4Z 
 
 0.6667 
 0.6 833 
 0.7000 
 
 0.3O 
 0.3/ 
 0.32 
 
 18 00 
 18 36 
 /9 /2 
 
 O.80 
 0.8I 
 0.82 
 
 48 00 
 
 48 36 
 
 49 IZ 
 
 43 
 44 
 ■4S 
 
 0.0119 
 O.OIZZ 
 
 o.oizs 
 
 43 
 
 44 
 45 
 
 0.7/67 
 Q.7333 
 0. 7.J0O 
 
 0.33 
 0.34 
 0.35 
 
 19 48 
 ZO 24 
 Zl 00 
 
 0. 83 
 0.84 
 O.SS 
 
 49 48 
 
 50 24 
 
 51 00 
 
 46 
 47 
 48 
 
 O.O 128 
 0.OI30 
 0.0/33 
 
 46 
 47 
 48 
 
 0.7hh7 
 0.7833 
 0.8000 
 
 0.36. 
 0.37 
 0.38 
 
 Zl 36 
 22 /2 
 22 48 
 
 0.86 
 0.87 
 O.SS 
 
 SI 36 
 32 IZ 
 SZ 48 
 
 .4? 
 30 
 J/ 
 
 O.O 13b 
 
 o.to;3? 
 O.O 14 1 
 
 49 
 SO. 
 SI 
 
 0.8/67 
 0.8333 
 0. &30O 
 
 0.3 9 
 
 0.4:0 
 
 ■ 0.41 
 
 ■ 23 24 
 24 00 
 24 36 
 
 0.89 
 0.90 
 0.9/ 
 
 53 24 
 34 00 
 
 54 36 
 
 .>y2 
 
 .53 
 54 
 
 O.OJ44 
 
 •0.0)47 
 
 O.OISO 
 
 SZ 
 33 
 S4 
 
 .0,86 6' 7 
 0.8833 
 0.9000 
 
 0.4Z 
 0.43 
 0.44 
 
 ZS /2 
 2S 48 
 2 6 24 
 
 0.9Z 
 0.93 
 0.94 
 
 SS IZ 
 
 55 48 
 
 56 34 
 
 ^5 
 Si 
 ,S7 
 
 O.OIS3 
 O.0IS6 
 O.OIS9 
 
 SS 
 Si, 
 37 
 
 0.9 167 
 0.9333 
 0.9SOO 
 
 0.4S 
 0.46 
 0.47 
 
 27 00 
 Z7 36 
 
 28 /2 
 
 0.9 S 
 
 0.96 
 .97 
 
 £7 00 
 
 57 36 
 
 58 IZ 
 
 .5? 
 60 
 
 O.OIhZ 
 o.oH,4 
 
 o.oihr^ 
 
 38 
 S9 
 60 
 
 0.9667 
 0.9 833 
 
 1 .0000 
 
 0.48 
 0.49 
 O.SO 
 
 2S 48 
 
 29 S4 
 
 30 00 
 
 0.93 
 0.99 
 
 /•oo 
 
 58 48 
 
 59 S4 
 
 60 00 
 
 0.^1 = 
 o\oos = 
 
 CfZ4'3h'^ 
 
 o'oo' la* 
 
 (f4i'od'- 
 
 Cf0<f4tl' - 
 
 0'.i>933. 
 O'.OIZB. 
 
CHAPTER XII 
 COMPARISON OF VARIOUS METHODS 
 
 The "localized capacitance" or "localized admittance" methods are discussed below lor the t\vo foUowmg 
 reasons. A discussion of them is of academic i^terc^t and a tabulation of the magnitude of the errors 
 in the results as obtained by these approximate methods when applied to circuits of different lengths and 
 frequencies should be helpful. These methods may be carried out either graphically or mathematically, but 
 since thoy are only approximate the simpler graphical solution should suffice. Their principle virtue is 
 the fact that they simplify the determination of performance, but this is obtained at the expense of accuracy. 
 The more accurate of these methods is somewhat tedious to carry out. The graphical solution previously 
 described in connection with the Wilkinson charts will be generally more accurate and shorter than these 
 localized capacitance methods. 
 
 T 
 
 HE LOCALIZED CAPACITANCE methods 
 are: — the single end condenser method; the 
 middle condenser or T method; the split con- 
 denser or nominal ir method and Dr. Steinmetz three 
 condenser method. These four lumped capacitance 
 methods assume the total capacitance of the circuit as 
 being divided up and "lumped" in the form of conden- 
 sers shunted across the circuit at one or more points. 
 
 PROBLEM "X" 
 
 KV-Aln~ '•°°° "^v* Ern" 80-0** W\.T^ 
 
 KWln" ''*00 XW pP|^- 90% UQQINO 
 
 ll_ - 99.91 AMPERES F - SO CYCLES 
 RECEIVING 
 END 
 
 98.92 AMPS, 
 
 methods, usually an approximation to the true value may 
 be obtained. 
 
 The middle condenser or T method assumes that 
 the total capacitance may be shunted across the circuit 
 at the middle point. On this assumption the total 
 charging current will flow over one half the length of 
 the circuit. This method is therefore more nearly ac- 
 curate than the single-condenser method. 
 
 The split condenser or ir method assumes one half 
 
 LINEAR CONSTANTS 
 R - 106 OHMS 
 X - 249 OHMS 
 B -0.001663 MHO 
 Q - (80 THAT Y-B) 
 
 •g>3'l8' 71' PF^9».9a7*LEAOINO 
 
 NEUTRAL 
 
 
 RESULTS CALCULATED BY 
 
 
 RIGOROUS SOLUTION 
 
 SINGLE END 
 
 CONDENSER 
 
 METHOD 
 
 % ERROR 
 
 EsN- 70,662 VOLTS 
 
 63,329 VOLTS 
 
 - 10.37% 
 
 Ig- 94.75 AMPS. 
 
 103.038 AMPS 
 
 + 8.8% 
 
 PFg- ♦ 93.42% 
 
 + 99.927% 
 
 + 7,0% 
 
 LOSSn-865KW 
 
 1120 KW 
 
 + 30.9% 
 
 fR-lOJI»VOLT») 
 
 93.862 AMPERES 
 
 FIG. 54 — SINGLE END CONDENSER METHOD 
 
 Problem X. 
 
 The single condenser method assumes the total ca- 
 pacitance as being lumped or shunted across the circuit 
 at the receiving-end. On this assumption the total 
 charging current for the circuit would flow over the 
 entire circuit. Actually the charging current is dis- 
 tributed along the circuit so that the entire charging 
 current does not flow over the entire circuit. Obviously 
 the assumption of the total capacitance being lumped 
 at the receiving-end will therefore give over compensa- 
 tion for the effect of the charging current upon the 
 voltage regulation of the circuit. This method of solu- 
 tion yields a voltage too low at the sending end by 
 nearly the same amount that the straight impedance 
 method gives it too high. By averaging the values, as 
 obtained by the impedance and single end condenser 
 
 the capacitance being shunted across the circuit at each 
 end. In this case one-half of the charging current .flows 
 over the entire circuit. This assumed distribution of the 
 charging current also more nearly represents the actual 
 distribution than the single-condenser method. 
 
 Dr. Steinmetz has proposed a method assuming 
 three condensers shunted across the circuit. One in the 
 middle, of two-thirds, and one at each end, each of one 
 sixth the total capacitance of the circuit. This method 
 is equivalent to assuming that the electrical quantities 
 are distributed along the circuit in a way representing 
 ap arc of a parabola. This method assumes one-sixth 
 the charging current flowing over one half the entire 
 circuit and five sixth the charging current flowing over 
 the other half of the circuit. This method gives quite 
 
112 
 
 COMPARISON OF VARIOUS METHODS 
 
 accurate results unless the circuit is very long and the 
 fiequency high. 
 
 Figs. 54-57 show leaky condensers placed at dif- 
 ferent points of the circuits, that is they indicate that 
 there is a leak G, as well as a susceptance B. For sim- 
 plicity pure condensers have been assumed in the ac- 
 companying calculations ; that is we have assumed G=o. 
 This is the usual assumption in such cases, for the 
 reason that G is usually very small, and localized ca- 
 pacitance methods are approximations at best. In the 
 equivalent w solution previously given, we have in- 
 dicated the treatment when the condensers have a leak. 
 In such case, however, the equivalent ir method pro- 
 duces exact results, and the nature of such solution may 
 demand a condenser having a material leak. 
 
 AUXILIARY CONSTANTS 
 
 Mr. T. A. Wilkinson and Dr. Kennelly have 
 worked out the algebraic expressions for the auxiliary 
 
 PROBLEM "X' 
 
 receiving-end. In such case the entire charging current 
 would flow over the total length of the circuit. 
 
 Solution by Impedance Method — The diagrams of 
 connections and corresponding graphical vector solution 
 for problem X by the single end condenser method is in- 
 dicated by Fig. 54. The current DN consumed by the 
 condenser (zero leakage assumed) leads the receiving- 
 end voltage OR by 90 degrees and is, — 
 
 /c = 0.CX)I563 X 60.046 = 93.852 amperes. 
 
 The load current of 99.92 amperes, lagging 
 25° 50' 30" (90% power-factor) has a component OA 
 of pp.p^ X o.po^= 89.928 amperes in phase with the re- 
 ceiving-end voltage and a component AD of 99.92 X 
 ^■4359 = 43-555 amperes in lagging quadrature with the 
 receiving-end voltage. This lagging component is ' 
 therefore in opposite direction to the charging current, 
 the effect of which is to neutralize an equivalent amount 
 of charging current. The remaining current AN in 
 leading quadrature with the receiving-end voltage is 
 
 KV-Aln- 8.000 kva 
 KWln- »-*ookw 
 
 ^- 89,02 AMPERES 
 
 Erm- 
 
 F- 
 
 60.048 VOLTS 
 80% LAQOINQ 
 
 R 
 
 ee.siTAMPS. 
 
 V 
 
 X 
 
 ^00000- 
 
 46.826 AMPar 
 
 fl.SeiAMPS. 
 
 LINEAR CONSTANTS 
 
 R — 106 OHMS 
 X - 249 OHMS 
 B- 0.001663 MHO 
 Q-0 (80THATY-B) 
 
 NEUTRAL 
 
 
 RESULTS CALCULATED BY 
 
 
 
 NOMINAL SPLIT 
 
 
 RIGOROUS SOLUTION 
 
 CONDENSER 
 METHOD 
 
 % ERROR 
 
 EgN- 70,662 VOLTS 
 
 72,318 VOLTS 
 
 *2.3t% 
 
 Ig- 84.76 AMPS. 
 
 81.862 AMPS. 
 
 -2.84% 
 
 PFg- + 83.42% 
 
 -•-83.878% 
 
 -►0.68* 
 
 LOSSn' b^skw 
 
 861 KW 
 
 -047% 
 
 IR- 8.449 VOLTS 
 
 60.046 VOLTS 
 
 46.926 AMPERES 
 
 FIG. SS — NOMINAL V OR SPLIT CONDENSER METHOD 
 
 Problem X. 
 
 constants corresponding to these four circuits of local- 
 ized capacitance. These are given in Table Q. It may 
 be interesting to observe to what extent each of the four 
 localized capacitance methods takes account of the three 
 linear line constants R, X and B. The rigorous or exact 
 expression for the auxiliary constants is given under 
 Table Q for comparison with the values corresponding 
 to the localized condenser methods. The numerals un- 
 der the algebraic expressions correspond to problem X ; 
 that is, to a certain 60 cycle circuit, 300 miles long. 
 They are given to illustrate for a long circuit, the ac- 
 count taken of the fundamental constants for each of 
 the five methods listed. These numerals may be com- 
 pared with the rigorous or exact values as given under 
 the rigorous expressions at the bottom of the table. 
 
 SINGLE END CONDENSER METHOD 
 
 This method assumes that the total capacitance of 
 the circuit may be concentrated across the circuit at the 
 
 93.852 — 43-555 = 50.297 amperes. The current ON 
 
 in the conductor is therefore: — 
 
 /, = V (89.928 ' -f (50.297)' 
 = 103.038 amperes. 
 
 The current at the sending-end leads the voltage; at 
 the receiving-end by the angle ^r whose tangent is, — 
 
 50-297 „ , ,„ 
 
 8^:5^ = ^ '3' 06" 
 
 The voltage consumed by the resistance, and the re- 
 actance of each conductor is, — 
 
 IR = 103.038 X 105 = 10 819 Volts (resistance drop) 
 IX = 103.038 X 249 = 25656 Volts (reactance drop) 
 
 The receiving-end conditions are thus, — 
 
 /b = 103.038 amperes 
 
 Or == 29° 13' 06" 
 Cos en — 0.8772 
 Sin Ob = 0.4881 
 
 and from (40) *•- 
 
COMPARISON or VARIOUS METHODS 
 
 "3 
 
 £« = 
 
 V.(60 046 X 0.8727 + 10 819)' + (60 046 X 0.4881 — 25 656)' 
 = 63 329 \3° 18' 27" volts to vector ON 
 = 63 329 /25° 54' 39" voiti to vector of reference. 
 PF, = Cos /3° 18' 27" = 99927 percent leading. 
 KV-A,» =: 103.038 X 63.329 = 6525 kv-a. 
 KU',« = 6S25 X 0.99927 = 6520 kiv. 
 Loss« = 6520 — 5400 =1120 kw. 
 
 Solution by Complex Quantities — From Table Q 
 the auxiliary constants corresponding to the single end 
 condenser method are found as follows: — 
 
 fli = I — XB = 0.610813 
 
 (h = RB = 0.164115 
 
 6j =: /? = 105 ohms. 
 
 bi = X =^ 249 ohms. 
 
 c, = 
 
 c, = B = 0.001563 mho. 
 
 The voltage at the sending end is determined as 
 follows : — 
 
 Jv {Cos *L — / Sin Ol) = 89.928 — ; 43.555 
 
 X (b, + ;■ b,) —20286 + 1 17819 
 
 +£.. (o« + / Os) = 36677 4- / 9854 
 
 £.. = 56963 -f- / 27673 
 
 J 
 
 end is completely determined by the load current at the 
 receiving-end and the vector addition thereto of the cur- 
 rent supplied at that end to the condenser under receiv- 
 ing-end voltage. For determining the sending-end volt- 
 age A'y = I -\- YZ and B'y = Z; but for determining 
 the sending-end current A'l = / and C'l = Y. If the 
 
 condenser were applied 
 would be identical. 
 
 symmetrically A'y and A'l 
 
 = 63 329 725° 54' 39" volts. 
 
 SPLIT CONDENSER OR NOMINAL w SOLUTION 
 
 This method assumes that the total capacitance of 
 the circuit may be concentrated at the two ends, one- 
 half being placed across the circuit at either end. In 
 this case one-half the charging current flows over the 
 entire circuit. The total resistance and the total react- 
 ance of one conductor is placed between the two ter- 
 minal condensers. 
 
 With this assumption the current consimied by the 
 condenser across the receiving-end of the circtiit is 
 added vectorially to the load current and the power-fac- 
 tor of the combined currents calculated. With these 
 new load conditions determined the conditions at the 
 
 TABLE 0— AUXILIARY CONSTANTS 
 CORRESPONDING TO CIRCUITS OF LOCALIZED CAPACITANCE 
 
 Oa 
 
 EQUIVAL£NT OONVEROEHT SCRIES 
 FOAM OF EXPRESgKJW 
 
 n 
 
 -106 
 
 A'-i 
 
 B'-i 
 
 C'-o 
 
 8INQLE END OONDENSEft 
 
 € 
 
 1 - XB 
 
 -o.aioeia 
 
 RB 
 • ♦j0.te4M6 
 
 R 
 ■I0> 
 
 -♦jM» 
 
 B 
 ♦ J0.00I683 
 
 A' - I ♦ »i B' 
 
 C- 
 
 DOUBLE END OONOCN8EK 
 
 (55 
 
 XB 
 9 
 
 3 
 •io.osio» 
 
 R 
 •lOB 
 
 4 
 - -0.000 0*41 
 
 ■-•10.001411 
 
 A'-(.»?) B'-z C'-»(i*J?) 
 
 MIDDLE OONDEN8CA 
 
 (~) NOMINAL "T Bb 
 
 J5 
 
 -Wj 0.083069 
 
 x-|(xa-R2) 
 • *i33e.06( 
 
 B 
 >jO.OOI6a9 
 
 A'-(i+^) B'-z('»^) 
 
 C" 
 
 THRCe OONOENBCn 
 
 ( gift \^ ~lfj g 
 
 '-¥-i(x'-R») 
 
 RB RXB^ 
 
 2 " 18 
 '4 JO.O7S5O0I 
 
 x-|Ua-R2) 
 
 - ►1236.721 
 
 -*jH(x'-R») 
 
 ♦jO.OOUTM 
 
 
 * THE eXACT OR RIOOROUB tXPRCBSlONS FOR THE AUXIUARV 00NSTANT8 ARE OlVtN BE LOW THE NUMERICAL FIOURE8 OORRES»ONO TO PROBLEM "X" 
 
 A — Vi+ir* 34 + 730 ' 40 330* ' B -*i'* 8 *T3r*ro40*5iO»* ' C — ''('*T^+-i30"+6345*3e5l85* ) 
 
 -•OOeH8-0.8l068*j007883 — Z.8INHe-8> 74S8*)336880 — T.BINHO- -0.000041 1«| 0.00I4884 
 
 which checks exactly with the results as obtained pre- 
 viously by the impedance method. 
 
 The current at the sending end may be determined 
 as follows: — 
 
 It. (Cos 0,. — /■ 5"i»i 01.) = 89928 — ; 43-555 
 -»- £,. (fi -f- ; f.) = o + ; 93852 
 
 /• = 89.928 -f- ;' 50.297 
 
 = 103.038 /29° 13' 06" amperes. 
 
 which also checks exactly with the result as previously 
 determined by the impedance method. 
 
 It should be noted here that in determining the 
 sending-end current, the auxiliary constant (a -f- ; a, 
 did not enter into the calculation as it does in the rigor- 
 ous solution ; this is owing to the inherent dissymmetry 
 of the single-end condenser. This is the only case in 
 which the capacitance is applied dissymmetrically, con- 
 sequently the current entering the line at the sending- 
 
 *i« 
 
 sending-end are calculated by the impedance method. 
 I'his is the only calculation required when employing 
 the nominal w method for determining the sending-end 
 voltage. The voltage at the sending-end is therefore 
 more readily calculated by this method than by the T 
 method which requires the calculation of the two sepa- 
 rate halves of the circuit. If, however, the current, 
 power-factor and kw input are required, a second calcu- 
 lation must be made to determine them. In such cases 
 the current consumed by the condenser at the sending- 
 end must be added vectorially to that of the line conduc- 
 tors. 
 
 Solution by Impedance Method — The diagrams of 
 connections and corresponding graphical vector solu- 
 tions for problem X by the nominal w method is in- 
 dicated in Fig. 55. The charging current consumsd by 
 the condenser (zero leakage assumed) at the receiving- 
 
114 
 
 COMPARISON OF VARIOUS METHODS 
 
 end of the circuit leads the receiving-end voltage by 90 
 degrees and is, — 
 
 0.001563 
 
 /cr 
 
 X 60046 = 46.926 amperes. 
 
 The current /, in each conductor is the vector sum 
 cf the load and condenser currents and may be deter- 
 mined as follows: — 
 
 /r = 1/ (9QC)2 X 0.90)' + (/cr + 9992 X -0.4359)' 
 = 89.991 /2° 08' a&" amperes. 
 PFr = Cos 2° 08' 48" = 99.33 percent leading. 
 
 The voltage consumed by the resistance, and the 
 
 reactance of each conductor is, — 
 
 IR = 89.991 X 105 = 9449 volts {resistance drop) 
 IX = 89.991 X 249 = 22408 volts (reactance drop) 
 
 and from (40), — 
 
 V'(60046 X 0.9933 -f 9449)" -f (60046 X 0.037458— 22408)' 
 
 = 72 JI9 /i6° u' 08" volts to current Vector OP. 
 
 = 72319 /i8° 19' .=i6" volts to vector of reference OR. 
 
 The charging current consumed by the condenser 
 at the sending-end (zero leakage assumed) leads the 
 voltage at the sending-end by 90° and is, — 
 
 0.001563 
 h, = X 72319 = 56.517 amperes. 
 
 The current at the sending-end is the vector sum 
 of the current in the conductor and the current con- 
 sumed by the condenser at the sending-end. It may be 
 calculated as follows: — 
 
 OT = 89.991 (.Cos 16° II' 08") = 86.424 amperes. 
 TP = 89.991 (Sin 16° 11' 08") = 25.085 amperes. 
 TN = 56.517 — 25.085 = 31-432 amperes. 
 therefore, — 
 
 /. = 1/86.424' + 31-432-' 
 
 = 91.962 /i9° so' 07" amperes to vector OS. 
 = 91.962 738° 19' 03" to vector of reference OR. 
 rr, — Cos 19° 59' 07" = 93 979 percent leading. 
 KV-A.. = 91-962 X 72.319 = 6651 kv-a. 
 KW.U = 6651 X 0.93979 = 6251 kw. 
 
 Loss, ::= 6251 — 5400 = 851 kw. 
 
 5400 X 100 
 
 Eg = 
 
 6251 
 
 = 86.37 percent. 
 
 Solution by Complex Quantities — From Table Q 
 the auxiliary constants corresponding to the nominal ir 
 method of solution are found as follows : — 
 
 XB 
 at = 1 :^— 0.8054065. 
 
 a, = —^ = 0.0820575. 
 
 61 = R = 105 ohms. 
 bt ■= X =: -\-j 249 ohms. 
 
 B'R 
 Ci ^ — — : — = — 0.0000641 mho. 
 4 
 
 Ci = B — -— — = 0.001411 mho. 
 
 The voltage at the sending-end is determined as 
 follows : — 
 
 /i (Cos Oi. — j sin Ov) = 89.928 — /43-S55. 
 
 X (fti 4- jbi) = 20286 -f yi7 8i9 volts. 
 -t- £ro (Oi + ia,) = 48361 4- i 4Q27 volts. 
 
 £„ = 68647 -t-y22 746. 
 
 = 72319 /i8° 19' 56' volts. 
 
 The current at the sending-end may be determined 
 as follows: — 
 
 /l (Cos 0L — J sin 6l) = 89.928 — ;43 555- 
 
 X (0, + jOt) = +76.003 — 727.700 amperes. 
 + £.. (C, + /CO = — 3849 + ;84.7i8 amperes. 
 
 I, = 72.154 -t- ;57-oi8. 
 
 =: 91.962 738° 19' 03" amperes. 
 
 The above results check exactly with those pre- 
 viously obtained by impedance calculations. This 
 agreement indicates that the nominal tt solution may, if 
 desired, be used with complex quantities, assuming 
 values for the auxiliary constants as indicated in Table 
 
 Q- 
 
 Convergent Series Expression — Table Q indicates 
 that the nominal tt solution is equivalent to using the 
 following values for the auxiliary constants in the con- 
 vergent series form of solution, — 
 
 .^-=(. + t?). ^' = z, ^..-O + tJ) 
 
 We will now show that the above expressions yield 
 the same values for the auxiliary constants as given in 
 Table Q. From chart XI the following values corre- 
 sponding to problem X are taken. 
 ZY = —0.389187 -f /0.164115 
 therefore, 
 
 A' = i.ooooooo 
 
 — O.I94S935 + / 0.0820575 
 
 A' = 0.8054065 -1- y 0.0820575 
 
 B' = 105 -t- y249 
 C = i.oooooo 
 
 —0.0972967 + y 0.0410287 
 
 = Y (0.9027033 -f- y 0.0410287) 
 
 C = — 0.0000641 -|- yo.001411 
 Thus the values for the auxiliary constants as de- 
 termined by the above incomplete convergent series ex- 
 pression check with those as determined above from the 
 equations in Table Q. 
 
COMPARISON OF VARIOUS METHODS 
 
 115 
 
 MIDDLE CONDENSER OR NOMINAL T METHOD 
 
 THIS METHOD assumes that the total capaci- 
 tance of the circuit may be concentrated at its 
 middle point. In such a case the entire charging 
 current would flow over half of the circuit. The re- 
 sistance and the reactance on each side of the capaci- 
 tance or condenser is equal respectively to half the total 
 conductor resistance and conductor reactance. 
 
 From an inspection of the diagram of such a cir- 
 cuit, Fig. 36, it is evident that two calculations will be 
 required. Starting with the known receiving-end con- 
 ditions, the conditions at the middle of ihe circuit are 
 first calculated by the simple impedance method. To 
 these calculated results the current consumed by the 
 condenser shunted across the middle of the circuit must 
 be vectorially added. This will give the load condition 
 at the middle of the circuit from which the sending-end 
 conditions may be calculated. 
 
 Solution by Impedance Method — The diagram of 
 ctjnnections and the corresponding graphical vector 
 solution for problem X by the nominal T method is 
 indicated by Fig. 56. The electrical conditions at the 
 middle of the circuit may be determined as follows: — 
 
 /r— = 99.92 X 52.5 = 5246 volts (resistance drop) 
 
 X 
 
 99.92 X 124.5 = 12440 volts (reactance drop) 
 
 Emu — 1/ (60046 X 0.9 + 5246)- + (60046 X 0.4J59 + 12440)' 
 
 = 70 753 X33° 04' 36" to current vector OD 
 — 70 753 /7° 14' 05" to vector of reference OR 
 
 The current consumed by the condenser (zero 
 leakage assumed) leads the voltage OM at the middle 
 of the circuit by 90 degrees and is : — 
 
 /c = 0.001563 X 70753 = 110.587 amperes 
 
 The voltage consumed by the condenser current 
 flowing back to the sending-end is : — 
 
 /c — = 110.587 X 52.5 = 5806 volts (resistance drop) 
 
 = FC 
 
 /c — = 110.587 X 124.5 
 
 percent. The voltage at the sending-end will therefore 
 be:— 
 
 £,„ = i/(57j8o X 0.77831 + 5246)' + (57280 X 0.62788 + 12440)' 
 
 = 69 467 /44° 10' 14" volts to vector OD 
 
 = 69467 y/iS" 19' 43 " volts to vector of reference OR 
 
 If desired, the receiving-end current and the con- 
 denser current may be combined and the corresponding 
 impedance triangle for the sending-end half of the cir- 
 cuit constructed on the end of vector OM as indicated 
 by the dotted lines. 
 
 The current at the sending-end may be determined 
 
 as follows: — 
 
 OB = 99.92 cos 33° 04' 36" =^ 83.727 amperes. 
 BD = 99.92 sin 33° 04' 36" = 54532 amperes. 
 BN — 110.587 — 54.532 — 56.055 amper es. 
 
 I. = ON = V (83.727)' + (56.055)' 
 
 = 100.76 /33° 48' 06" amperes to vector OB. 
 ■= 100.76 /'41'' 02' 11" amperes to vector of 
 reference OR. 
 
 The current at the sending-end leads the voltage at 
 the sending-end by the angle 41° 02' 11" — 18° 19' 43"' 
 = 22° 42' 28", which corresponds to a power-factor at 
 . the sending-end of 92.25 percent leading. . 
 
 The power at the sending-end is: — 
 Kv-a„i = 100.76 X 69467 = 7000 kv-a. 
 Kw,u = 7000 X 0.9225 = 6457 kw. 
 Lossn =6.^57 — 5400 = 1057 kw. 
 Solution by Complex Quantities — From table Q 
 the auxiliary constants corresponding to the nominal T 
 method of solution are found as follows: 
 
 13768 volts (reactance drop) 
 = FM 
 
 The voltage vector OC upon which the impedance 
 triangle corresponding to the receiving-end load cur- 
 rent /r = /l flowing over the sending-end half of the 
 circuit is constructed, may be found as follows: — 
 OC = V' (70753- i3 768)="T-l8o6= 
 
 = 57 280 /s° 49' 03" volts to vector OM 
 
 = 57 280 /iz° 03' 08" volts to vector of reference OR 
 
 The voltage OC leads the receiving-end current 
 OD by the angle 33° 04' 36" + 5° 49' 03" = 38° 53' 39" 
 which angle corresponds to a power-factor of 77-831 
 
 O, = I — 
 RB 
 
 b, = X 
 
 XB_ 
 2 
 
 RXB 
 
 2 
 V 
 
 = 0.805 406 5 
 = 0.082 057 5 
 = 84.5677 
 
 — (X' — R') = 229.081 
 
 c. = 
 
 ci ■= B = 0.001 563 
 
 The voltage at the sending-end is obtained as fol- 
 lows : — 
 
 Ik (cos Sr — / sin Or) = 89.928 — / 43-554 
 
 X (fci -f- ih) = 17582 + yi69i8 
 +Etu (flj + j a,) = 48361 -f- ;4927 
 
 £.a = 65943 + /21845 
 = 69467 /i8° 19' 43 " 
 
 The current at the sending-end may be calculated 
 as follows: — 
 
 /u (cos Sn — j sin »r) == 89.928 — / 43-554 
 
 X (Oi -f /oj) = 76.0026 — / 27.6994 
 + Etu (ci + jc) = o + y 93-8519 
 
 /, = 76.0026 4- y 66.1525 
 
 = 100.76 /4i° 02' 11" amperes 
 
ii6 
 
 COMPARISON OF VARIOUS METHODS 
 
 The above results check with those previously ob- 
 tained by impedance calculations. This agreement in- 
 dicates that the nominal T solution may, if desired, be 
 made by complex quantities, assuming values for the 
 auxiliary constants as indicated in Table Q. 
 
 Convergent Series Expression — Table Q indicates 
 that the nominal T solution is equivalent to using the 
 following values for the auxiliary constants in the con- 
 vergent series form of solution: — 
 
 =^('-^) 
 
 B 
 
 C = Y 
 
 Comparing the above expressions for the auxiliary 
 constants with the complete expression yielding rigor- 
 ous values the following difference may be noted. 
 
 For auxiliary constant A' the first two terms in the 
 complete series for the hyperbolic cosine are used and 
 
 expression!! check exactly with those as determined 
 above from the equations in Table Q. 
 
 THREE CONDENSER METHOD 
 
 This method (proposed by Dr. Chas. P. Steinmetz) 
 assumes that the admittance of the circuit may be 
 lumped or concentrated across the circuit at three 
 points, one-sixth being localized at each end and 
 two-thirds at the middle of the circuit. This is equiva- 
 lent to assuming that the electrical quantities are dis- 
 tributed along the circuit in a manner represented by 
 the arc of a parabola. It is evident that this method 
 more nearly approaches the actual distribution of the 
 impedance and the admittance of the circuit than any 
 of the three previously described localized admittance 
 methods, and therefore yields more accurate results. 
 
 From an inspection of the diagram of such a cir- 
 cuit, Fig. 57, it will be evident that it is necessary to 
 calculate the performance of the two halves of the cir- 
 
 PROBUEM 'X" 
 
 KV-Arn- 6,000 KVA Ern •= 80'<'*8 volts 
 
 * KWrn - 6,400 KW PFr - 90* LAQQINQ 
 
 \ Ip -89.92 AMPERES F - 60 CYCLES 
 
 SENDING R X R X,''E°E'^"*° 
 
 ^ ~ ~ 2 END 
 
 LINEAR CONSTANTS 
 R - 106 OHMS 
 X - 249 OHMS 
 B - 0.001663 MHO 
 Q - (SO THAT Y-B) 
 
 NEUTRAL 
 RESULTS CALCULATED BY 
 
 RIGOROUS SOLUTION 
 
 MIDDLE 
 
 CONDENSER 
 
 METHOD 
 
 % ERROR 
 
 EsN- 70.662 VOLTS 
 
 69,467 VOLTS 
 
 - 1.68% 
 
 Is- 94.76 AMPS. 
 
 100.76 AMPS. 
 
 + 6.34% 
 
 PFs- + .93.42% 
 
 * 92.26% 
 
 - 1.26% 
 
 LOSS N- 866 KW 
 
 1067 KW 
 
 «■ 23.63% 
 
 Ir 9 " 12,440 VOLTS 
 
 > 6246 VOLTS 
 
 12,440 VOLTS 
 
 F0-|c5 - 6806 VOLTS 
 - -■ 13.768 VOLTS 
 
 FIG. S6 — NOMINAL T OR MIDDLE CONDENSER METHOD 
 
 all terms beyond omitted. For auxiliary constant B' 
 the first two terms of the complete series are also used 
 except that the coefficient of the second term is given 
 as J4. whereas in the complete series it is i/6. Auxi- 
 liary constant C is equivalent to the first term only of 
 the complete expression. 
 
 We will now show that the above expressions yield 
 the same values for the auxiliary constants as given in 
 Table Q. From Chart XI the following values corre- 
 sponding to problem X are taken : — 
 
 Z = 105 + /240 
 Z y = — 0.389187 -f- ;o.i64ii5 
 Therefore /4 ' = i.oooooo 
 
 — O.I 945 935 + ;" 00 820 S7 5 
 A' = + 0.8054065 -I- ;■ 0.0 820 575 
 B' = I.oooooo 
 
 — 0.09 729 675 -I- ;" 004 102 875 
 
 = z (0.90270325 + y 0.04 102875) 
 
 B' = 84.5677 4- / 229.081 
 C = o + /o.ooi 563 
 Thus the values for the auxiliary constants as de- 
 termined by the above incomplete convergent series 
 
 cuit in order to arrive at the sending-end voltage and 
 an additional calculation will be required to determine 
 the sending-end current, power and power-factor. 
 
 Solution by Impedance Method — The diagram of 
 connections and corresponding graphical vector solu- 
 tion for problem X by the three condenser method is 
 mdicated by Fig. 57. The charging current consumed 
 by the condenser (zero leakage assumed) at the re- 
 ceiving-end leads the receiving-end voltage by 90 de- 
 grees and is: — 
 O.OOI 563 
 
 /cr = 
 
 X 60046 = 15.642 amperes. 
 
 The current per conductor for the receiving-end 
 half of the circuit is : — 
 
 /r = 1/ (99.92 X 0.9)" -I- (9992 X 0.4359 — 15-642)' 
 = 94.16 \i7° 14' 38" amperes 
 PF. = Cos \i7° 14' 38" — 95.505 lagging 
 
 The voltage consumed by the resistance and the 
 reactance per conductor between the receiving-end and 
 the middle of the circuit is: — 
 
COMPARISON OF VARIOUS METHODS 
 
 U7 
 
 /r— = 9416 X 525 = 4943-4 Volts (resistance drop) 
 
 , X 
 
 /r~= 94-i6 X 124.5 = 11723 Volts (reactance drop) 
 
 The voltage at the middle of the circuit is from 
 (30):- 
 
 £mi. = y (60046X 0.95505 + 4943-4)'+ (60046X0.29644+ 11723)" 
 
 = 68933 /25° 21' 33" volts to current vector OP 
 = 68933 y8° 06' 55" v olts to vector of reference OR 
 
 The charging current consumed by the condenser 
 (zero leakage assumed) at the middle of the circuit 
 leads the voltage at the middle of the circuit by 90 de- 
 grees and is: — 
 o.ooi 563 
 
 The current at the sending-end of the circuit may 
 be determined as follows: — 
 
 OS = Cos 10° 19' 07" X 90.73 = 89.2624 
 VS = Sin 10° 19' 07" X 90-73 = 16.2516 
 A^5" = 16. 2516 + 18.3777 = 346293 amperes. 
 I.=\/ 89.2624' + 34-6293' 
 
 = 95-744 /2i° 12' 13" to voltage vector OS. 
 
 = 95-744 /^39° 18' 56" to vector of reference OR. 
 
 Kv-a,n = 95-744 X 70-548 = 6755 kv-a 
 PF. = Cos (39° 18' 56" — 18° 06' 43") 
 
 = Cos 21° 12' 13" = 93.23 percent leading 
 Kw,n = 6755 X 0.9323 = 6298 kw 
 LosSu = 6298 — 54CX3 =: 898 kw 
 5400 X 100 
 
 Eff. 
 
 6298 
 
 85-75 percent. 
 
 /.„ 
 
 -X68933 = 71.828 amperes. 
 
 Solution by Complex Quantities — From Table Q 
 the auxiliary constants corresponding to the three con- 
 The current per conductor for the sending-end half denser method of solution are found to be:— 
 of the circuit may be determined as follows: — XB B' 
 
 OT = Cos 25° 21' 33" X 94-16 = 85.0867 amperes. 
 
 PROBLEM 'X' 
 
 O, 
 
 -~ + 
 
 36 
 
 (X' — i?') = 0.808866 
 
 
 KV-Aln" ''""^ '*^* Ern" 60,046 VOLTS 
 II - 99.92 AMPERES F= 80 CYCLES 
 
 LINEAR CONSTANTS 
 R - 106 OHMS 
 X - 248 OHMS 
 B - 0.001683 MHO 
 O - (80 THAT Y"B) 
 
 RIGOROUS SOLUTION 
 
 THREE 
 
 CONDENSER 
 
 METHOD 
 
 % ERROR 
 
 EsN- 70,662 VOLTS 
 
 70,648 VOLTS 
 
 -0.16% 
 
 Is- 94.75 AMPS. 
 
 96.744 AMPS. 
 
 •H.06% 
 
 PFs" ♦ 93.42% 
 
 •1- 93.23% 
 
 -0.21% 
 
 LOSSn- ''''<* 
 
 898 KWf 
 
 ♦ 6.03% 
 
 l6.e42AMP8.^ 
 
 riG. 57 — DR. CHAS. P. STEINMETZ S THREE CONDENSER METHOD 
 
 TP = Sin 25° 21' 33" X 94-i6 = 40.3278 amperes. 
 TV = 71.828 — 40.3 2 78 ^ 3 1.5002 amperes. 
 
 /m = 1 85.0867-' -I- 31.5002= 
 
 = 90.73 /20° 18' 55" amperes to voltage vector OM al 
 
 tniddle. 
 = 90.73 /28° 25' 50" to vector of reference OR 
 
 The voltage consumed by the resistance and the re- 
 actance per conductor between the middle and sending- 
 end of the circuit is: — 
 
 /m— X 90.73 X 52.5 = 4763-3 volts (resistance drop) 
 
 , X 
 
 /m —' X 90.73 X 124.5 = II 296 volts (reactance drop) 
 
 The voltage at the sending-end from (40) is: — 
 
 £«n = V (68933 X 0.93779 + 4763-3)' + (68933 X 0.34719 — 11 296)' 
 
 = 70548 \io° 19' 07" volts to current vector OV 
 
 = 70548 /l8° c6' 43" volts to vector of reference OR 
 
 The charging current consumed by the condenser 
 (zero leakage assumed) at the sending-end of the cir- 
 cuit leads the voltage at the sending-end by 90 degrees 
 and is: — 
 
 0.001563 
 6 
 
 b, = R 
 b, 
 c» = — 
 
 c, — B 
 
 RB _ RXB'- 
 2 '18 
 RXB 
 
 0.0785091 
 = 91.3785 
 X - -| (X' - R') = 235.7208 
 
 5 RB' RXB 
 + 
 
 0.0000347 
 
 36 ^ 108 " 
 
 5 XB' B' 
 
 ^6~ + 116 <^' - '^'> = + ooo'4794 
 
 These values for the auxiliary constants are in 
 close agreement with the rigorous values. 
 /l (Cos 0l — ;" Sin Sl) X (b, + jb,) 
 
 — 18 484 -I- y 17 218 
 
 Em (a, + JO,) = 48569 -I- y 4714 
 
 £„ = 67 053 + j 21 932 
 
 = 70 548 /iS" 06' 4 3" volts 
 
 The current at the sending-end is : — 
 
 /l (Cos Al — y Sin Sl) X (a, + i Oj) 
 = 76.159 — y 28.170 
 
 £r. (c, -1- y c) = — 2.084 -I- y 88.832 
 
 u. = 
 
 X 70548 = 18.3777 amperes. 
 
 I, = 74075 + y 60.662 
 
 = 95-744 /i9° 18' 56" amperes 
 By comparing these results with those obtained 
 
118 
 
 COMPARISON OF VARIOUS METHODS 
 CHART XXII— COMPARISON OF RESULTS BY VARIOUS METHODS 
 
 a 
 
 OD 
 
 r 
 m 
 
 s 
 
 Z 
 p 
 
 _ 
 
 Z 
 
 o 
 
 •n 
 
 O 
 
 3 
 o 
 
 c 
 
 H 
 1 
 
 S 
 
 0) 
 
 8 
 
 z 
 o 
 c 
 
 e 
 
 (0 
 
 I 
 
 z 
 o 
 
 z 
 
 ■n 
 m 
 n 
 
 D 
 
 m 
 
 5 
 
 
 LINEAR 
 
 TS 
 
 
 LOAD AT 
 
 
 
 % error IN RECEIVING-END VOLTAGE 
 AS DETERMINED BY 
 
 CON STAN 
 
 RECEIVING-END 
 
 3D 
 
 
 
 IS 
 
 3o 
 
 z 
 
 PI 
 R CI 
 
 Si 
 2 
 
 8 
 
 ? 
 
 82 
 ? 
 
 » 
 
 r 
 
 Z 
 P 
 
 X V) 
 
 3 
 
 Z 
 
 R§ 
 
 oo 
 01 
 z> 
 
 of" 
 
 m j> 
 
 nD 
 
 3JZ 
 
 3^ 
 
 !3^ 
 zm 
 
 
 
 if 
 
 S| 
 
 in J 
 
 Is 
 
 z 
 
 OD 
 
 ii 
 
 oz 
 
 §5 
 
 8 
 
 % 
 
 
 
 3 
 ■0 
 m 
 
 > 
 z 
 
 n 
 
 3 
 
 71 
 H 
 I 
 
 (3 
 
 R 
 
 X 
 
 B 
 
 X 
 
 I0"< 
 
 G 
 
 KV-A 
 
 • 
 
 ^R 
 
 ^RN 
 
 P 
 
 F 
 % 
 
 'r 
 
 ^SN 
 
 % J % 
 
 ERROf^ERROR 
 
 % 
 ERROR 
 
 ERROWERROR 
 
 % io 
 
 ERROR ERROR 
 
 ERROR 
 
 
 
 
 25 CYCLES 
 
 
 / 
 
 2 
 
 20 
 
 U 
 
 •oooo 
 COPPEK 
 
 3 
 
 &s-> 
 
 S.3i 
 
 a 
 
 ■^2 
 
 o 
 
 
 
 /j300 
 
 10,000 
 
 5,774 
 
 If 
 
 80 
 
 100 
 
 7S 
 
 6,3-^7 
 
 6,zoz 
 
 
 
 
 +.0S 
 +.03 
 
 
 
 
 
 
 
 -.0/ 
 -.0/ 
 
 
 
 
 3 
 4 
 
 30 
 
 ttoooo 
 
 COPPER 
 
 3 
 
 It 
 
 S.S4 
 
 S.3t 
 
 J7.2 
 
 o 
 
 
 
 S,ooo 
 
 z.0,000 
 
 lljSSO 
 
 80 
 100 
 
 Ml.-t 
 
 I2,&S3 
 12,372 
 
 -■04 
 -■OS 
 
 +.04 
 +.02 
 
 -■01 
 -■01 
 
 
 
 
 -.02 
 
 -.02 
 
 +.02 
 +.0Z 
 
 S 
 
 30 
 
 *oooo 
 COPPER 
 
 4< 
 
 S.3I 
 
 • 
 
 8.S 
 
 • 
 
 »/ 
 
 
 
 o 
 
 3,S0O 
 
 Z0,000 
 
 IIjSSO 
 
 80 
 100 
 
 10 1 
 
 12,733 
 I2,4IS 
 
 +■02 
 +.08 
 
 +.06 
 +.04 
 
 -■02 
 -.02 
 
 
 
 
 -.03 
 
 -.03 
 
 +.04 
 + .04 
 
 I 
 
 30 
 
 OOOOO 
 COPPER 
 
 4 
 
 " 
 
 &3/ 
 
 8.i 
 tf 
 
 ?/ 
 
 a 
 o 
 
 8,ooo 
 
 30,000 
 
 I7,3Z0 
 
 SO 
 100 
 
 IS^t 
 
 I9,IZS 
 I8,i40 
 
 i-os 
 +■08 
 
 
 
 + ■05 
 
 - .0/ 
 +.0/ 
 
 
 
 
 - .0 3 
 -.02 
 
 +.0 3 
 + .03 
 
 10 
 
 SO 
 
 •oooo 
 
 COPPER 
 
 4- 
 ti 
 
 I3SS 
 
 ft 
 
 14.1 
 ft 
 
 /3.5 
 
 o 
 
 
 
 S,ooo 
 
 3O,0OO 
 
 / 7,32 
 
 80 
 
 100 
 
 9i.z 
 
 19,184 
 IS,i8S 
 
 
 -■OS 
 
 -■0 7 
 -.03 
 
 +•03 
 +.02 
 
 
 
 
 -.08 
 -.OS 
 
 + 0.1 
 + 0.1 
 
 II 
 
 So 
 
 *oooo 
 COPPEP 
 
 k 
 
 i3.as 
 
 /S.I 
 
 /2,r 
 
 n 
 
 O 
 
 a 
 
 ZOjOOO 
 tf 
 
 60, 000 
 
 34ji40 
 
 80 
 ZOO 
 
 I92.S 
 
 3 8,490 
 3 7^387 
 
 -•03 
 -.02 
 
 -.O-f 
 -.03 
 
 +•0/ 
 +.0/ 
 
 
 
 
 
 
 -.08 
 
 -.07 
 
 + 0.1 
 
 ra.i 
 
 13 
 14 
 
 100 
 
 »oooo 
 COPPER 
 
 9 
 
 27.7 
 
 32.2 
 
 233 
 
 
 
 o 
 
 Z.3,000 
 If 
 
 SB, 000 
 
 50,810 
 
 so 
 
 100 
 
 tf 
 
 Si,il9 
 54,820 
 
 -.07 
 -.02 
 
 -.03 
 -.0.* 
 
 
 
 
 
 -.01 
 -.0/ 
 
 +.01 
 + .01 
 
 -.31 
 -.30 
 
 + 0,3 
 + a3 
 
 IS 
 U 
 
 loo 
 
 *oooo 
 COPPER 
 
 
 27.7 
 
 33.2 
 
 22^ 
 
 o 
 
 a 
 
 40,ooo 
 
 1X0,000 
 
 i?j2?0 
 
 80 
 too 
 
 l?Sfi 
 
 77,147 
 74,i>fZ 
 
 -.02 
 -.02 
 
 -■06 
 
 -.03 
 
 
 
 
 
 -.01 
 -.0/ 
 
 +.02 
 t.02 
 
 -.30 
 -.3/ 
 
 + 0.4 
 
 *<14 
 
 n 
 
 IS 
 
 ZOO 
 
 iOOtOOOCM 
 COPPER 
 
 
 3 T.J 
 
 * 
 
 ■»<-f 
 
 o 
 a 
 
 ZS,000 
 
 ft 
 
 120,000 
 
 i%Z90 
 
 80 
 
 100 
 
 no.-i 
 
 7i,7S4 
 73,401 
 
 +.oi 
 ■¥■05 
 
 — 01 
 -06 
 
 
 
 
 
 -.O.f 
 -.04 
 
 +.0i 
 +.06 
 
 -/.24 
 -/.24 
 
 '1.4 
 
 ■>■ 1.4 
 
 11 
 zo 
 
 loo 
 
 300,000 CM 
 COPPER 
 
 /7 
 
 313 
 
 A«2 
 
 -f3.f 
 
 o 
 o 
 
 40,000 
 
 I'f0,000 
 It 
 
 80,830 
 
 80 
 100 
 
 US 
 
 9l,7i' 
 8i,8i3 
 
 -.02 
 
 -.02 
 
 t.os 
 
 -■01 
 
 -.08 
 +.0/ 
 
 
 -.OJ 
 -.04 
 
 +.0* 
 +.05 
 
 -l.ll 
 -l.lt 
 
 *■ t.+ 
 
 f /.4 
 
 Zl 
 
 300 
 
 iSi,0O0 CM. 
 
 n 
 
 44.1 
 
 ?/.2 
 
 7-*7 
 
 o 
 o 
 
 Z0,O00 
 
 izo,ooo 
 
 if,Z9o 
 If 
 
 80 
 100 
 
 9i.l 
 
 7S,i8Z 
 
 7lt7iZ 
 
 
 +.08 
 
 +.02 
 -.0 2 
 
 +.0J 
 f.02 
 
 
 -.0? 
 -.08 
 
 + ./5 
 + ./3 
 
 -2.8 3 
 -2.99 
 
 + 12 
 » 3.2 
 + 3.4 
 <-3.3 
 
 S3 
 
 34 
 
 300 
 
 i3t,O0OCM. 
 
 RLUMINUM 
 
 21 
 
 ft 
 
 *4.l 
 
 1 
 
 10/ 
 
 4 72 
 
 o 
 
 
 t>OjO0O 
 
 zo 0,000 
 
 IISjSOO 
 
 80 
 
 100 
 
 /73,S 
 
 IZ8,450 
 IZO,S74 
 
 -.0 + 
 
 
 +.0+ 
 
 -.oi 
 
 +.03 
 +■01 
 
 
 -. ( / 
 -. M 
 
 + .17 
 +./5 
 
 -2.74 
 -2.82 
 
 as 
 
 400 
 
 Ut.OOO CM- 
 
 n 
 
 If 
 
 59g 
 
 130 
 
 ft 
 
 f2? 
 
 
 o 
 
 x.0,000 
 
 140,000 
 tf 
 
 80,830 
 It 
 
 80 
 100 
 
 S7.,S 
 
 8i,40^ 
 81, (-47 
 
 -■OS 
 
 -.06 
 
 
 
 +.11 
 +■05 
 
 
 
 
 -.19 
 -.18 
 
 +.2t 
 +.23 
 
 -5.08 
 -5-30 
 
 1-5.7 
 *S.7 
 
 11 
 
 ■too 
 
 63<,000CAI. 
 
 Zl 
 
 ft 
 
 SM 
 
 13-f 
 
 » 
 
 ffW 
 
 
 
 o 
 
 50,000 
 
 Z00,0OO 
 
 115,500 
 ft 
 
 80 
 100 
 
 /■*•*.'* 
 
 IZ7,Zh7 
 118,83-3, 
 
 -■OS 
 
 -.03 
 
 
 
 +■09 
 
 
 
 
 
 
 -.22 
 
 -.21 
 
 + .32 
 +.29 
 
 -4.80 
 -4.89 
 
 '5.6 
 + S.i 
 
 Z9 
 30 
 
 Soo 
 
 Oa^ooocm: 
 
 flLUM/ZVOM 
 
 n 
 ft 
 
 73^ 
 
 * 
 
 /<3 
 
 ft 
 
 //io 
 
 ft 
 
 o 
 o 
 
 15,000 
 
 140,000 
 
 90,830 
 
 80 
 100 
 
 i/.Si 
 
 83,04S 
 78,i5S 
 
 -.06 
 
 + .OS 
 
 
 
 +.06 
 + .0/ 
 
 -o-o.f 
 -o.o.« 
 
 -.27 
 -.23 
 
 + .3i 
 +.3/ 
 
 -8,22 
 -8.32 
 
 + 9-2 
 + y.3 
 
 31 
 3» 
 
 500 
 
 fii.OOOCM. 
 fiLUMINU/^ 
 
 Zl 
 
 73^ 
 
 /if 
 
 Ilio 
 
 V 
 
 o 
 
 o 
 
 40,000 
 
 Z00,000 
 
 II 5, SOO 
 
 80 
 
 100 
 
 I'S.S 
 
 123,401 
 
 ns,iiz 
 
 +-.0S 
 
 +.08 
 
 
 
 +.02 
 
 -.OJ 
 
 -O.0-* 
 -O.O.* 
 
 -.40 
 -.37 
 
 +.50 
 
 +•44 
 
 -7.65 
 -5.99 
 
 + 9.0 
 + 9.0 
 
 60 CYCLES 
 
 33 
 
 34 
 
 20 
 
 KOOOO 
 COPPER 
 
 3 
 
 5.S1- 
 
 * 
 
 /3Jlt 
 
 137 
 
 • 
 
 
 
 
 1,300 
 
 10,000 
 
 5,774 
 
 80 
 
 100 
 
 7S 
 
 6,702 
 (>,2S9 
 
 
 +.06 
 
 -■IS 
 +J3 2 
 
 
 
 +.02 
 
 
 
 
 -.07 
 -.06 
 
 + .07 
 1- 0./ 
 
 35 
 3« 
 
 20 
 
 i*oooo 
 
 COPPER 
 
 3 
 
 S.S4 
 
 /AW 
 
 >37 
 
 • 
 
 
 
 
 5,000 
 
 Z0,000 
 
 ll,SSO 
 
 80 
 100 
 
 I4t.-t 
 
 1 3, 333 
 12,480 
 
 +.02 
 +.02 
 
 -. 10 
 ■^■04 
 
 
 
 
 
 
 
 -.07 
 -.07 
 
 + 0.' 
 + 0.1 
 
 37 
 38 
 
 30 
 
 *O000 
 COPPER 
 
 4 
 
 8.3/ 
 
 X0.4 
 
 • 
 
 
 o 
 o 
 
 3jS0O 
 
 ft 
 
 Xo,ooo 
 
 llfSSO 
 
 80 
 
 100 
 
 101 
 
 13,482 
 12,537 
 
 +.07 
 
 +.OS 
 
 +.06 
 +.04 
 
 
 
 
 
 
 
 
 -.IS 
 -■16 
 
 + 0:2 
 + 0.2 
 
 39 
 
 40 
 
 30 
 
 IfOOOO 
 COPPER 
 
 4 
 
 g.3l 
 
 ao.-f 
 
 lis 
 
 o 
 o 
 
 2,000 
 
 30,000 
 
 n,3ZO 
 
 80 
 
 100 
 
 IS4 
 
 ZO,Zi8 
 18,830 
 
 +-.0 3 
 -.03 
 
 +.06 
 ^■■OS 
 
 + .01 
 +.02 
 
 
 
 
 
 
 
 
 
 
 -.IS 
 
 -. 16 
 
 ta2 
 
 + 0.2 
 
 41 
 4i 
 
 SO 
 
 ♦»oooo 
 
 COPPER 
 
 4 
 
 '3,Si 
 
 3 + 
 
 31-t 
 
 o 
 o 
 
 5,000 
 
 30,000 
 
 n,3zo 
 
 80 
 100 
 
 9i.Z 
 
 20,33/ 
 / «,84S 
 
 
 
 +-08 
 
 +■04 
 — 03 
 
 +.03 
 -.02 
 
 
 -.02 
 -.02 
 
 +.04 
 +.03 
 
 -.42 
 -.44 
 
 +0.5' 
 +a.S 
 
 43 
 44 
 
 so 
 
 *OO0o 
 COPPER 
 
 6 
 
 I3JSS 
 
 3i.* 
 
 30/ 
 
 o 
 o 
 
 S.0,000 
 
 i 0,000 
 
 34,i40 
 
 80 
 
 100 
 
 llZf 
 
 40,976 
 37,773 
 
 +.03 
 ■\-.04 
 
 -.0/ 
 — 01 
 
 +.02 
 +.03 
 
 
 -.02 
 -.02 
 
 t.03 
 +.04 
 
 -.41 
 -■4Z 
 
 + 0.5 
 + 0.5 
 
 4S 
 4i 
 
 loo 
 
 i*oooo 
 
 CapPEK 
 
 9 
 
 m 
 
 27.7 
 
 77.-f 
 
 J<2 
 
 o 
 o 
 
 2^,000 
 
 8 8,000 
 
 S0,8I0 
 
 80 
 
 100 
 
 /-f-f.-f 
 
 S9,92i 
 S4,Si9 
 
 +.03 
 
 
 -■09 
 -■13 
 
 +.0 8 
 +.05 
 
 
 
 
 -.08 
 -.07 
 
 +./3 
 +./3 
 
 -l.il 
 -1.69 
 
 + 1.91 
 + i.9i 
 
 ■♦7 
 
 100 
 
 noooo 
 
 COPPER 
 
 II 
 
 1X7 
 
 * 
 
 7».7 
 
 f-f2 
 
 o 
 
 
 
 ■40,000 
 
 /zo,ooa 
 
 6?jZ?o 
 
 80 
 100 
 
 l9Zi 
 
 f 
 
 81,710 
 74.73 s 
 
 -.08 
 -.05 
 
 -.04 
 -./ / 
 
 -.02 
 +.0' 
 
 
 
 
 
 -.OS 
 
 -.07 
 
 +■14 
 
 +.12 
 
 -i.to 
 
 -1.68 
 
 -^1.84 
 +1.92 
 
 SO 
 
 zoo 
 
 300,000 CM 
 COPPER 
 
 )i 
 
 3f.2 
 
 /Si 
 
 mi 
 
 * 
 
 
 
 
 2 S,ooo 
 
 ft 
 
 /ZO,000 
 
 (.?,Z?o 
 
 80 
 100 
 
 IZ0.3 
 
 79jO00 
 70,599 
 
 +.03 
 +.08 
 
 +./-f 
 +.07 
 
 +■18 
 +■■14 
 
 -0.04 
 -0.04 
 
 -■46 
 -.39 
 
 +.6/ 
 +.47 
 
 -6.53 
 -6.99 
 
 + 7.8 
 + 8.1 
 
 SI 
 5i 
 
 zoo 
 
 ft 
 
 30O,a0OCM 
 
 COPPER 
 
 n 
 
 ft 
 
 3f.2 
 
 /ii 
 
 ft 
 
 % 
 
 ■40,000 
 tf 
 
 M 0,000 
 
 80, 830 
 
 80 
 100 
 
 US 
 
 96,727 
 84,862 
 
 -.07 
 +.07 
 
 ■t-14 
 +.02 
 
 +.22 
 +.25 
 
 -0.04 
 -0.04 
 
 -.51 
 -.45 
 
 +.73 
 +.62 
 
 -5.96 
 -6.37 
 
 + 7.5 
 + 7-8 
 
 S3 
 
 54 
 
 300 
 
 * 
 
 iSb,000C.TA 
 RLUMINUM 
 
 II 
 
 -»-f.' 
 
 2X0 
 
 ft 
 
 I7f^ 
 
 
 
 o 
 
 ZO,000 
 
 iz 0,000 
 
 i%Z90 
 
 80 
 
 100 
 
 9i.Z 
 
 72,747 
 6 3,810 
 
 -.04 
 -.02 
 
 +.05 
 +.2/ 
 
 +.46 
 -.2.S- 
 
 -0.ZI 
 
 -O^ZI 
 
 -I.SI 
 -1.34 
 
 + /.34 
 + /.0I 
 
 -/5.68 
 -/6.43 
 
 + /9 
 + 20 
 
 ss 
 
 Si 
 
 300 
 
 i3i,OO0CiM. 
 
 RLUMINUM 
 
 21 
 
 ->*/ 
 
 It 2 
 tr 
 
 Ulf 
 
 o 
 
 
 i0,000 
 
 If 
 
 zoo, 000 
 
 115,500 
 ft 
 
 80 
 
 100 
 
 /73.2 
 
 126,541 
 109,18 9 
 
 -.05 
 -■07 
 
 +.25 
 
 +.15 
 
 +.-f4 
 +.J/ 
 
 -O.ZI 
 
 -0.2/ 
 
 -I.60 
 -1.44 
 
 + ;.4>o 
 
 + 1.29 
 
 -/4.50 
 -/5./2 
 
 + /8 
 + /9 
 
 57 
 St 
 
 +00 
 
 i36,oaoc.M. 
 
 RLUMtSUM 
 
 '; 
 
 .J 58 
 
 3'4 
 
 ZZIZ 
 
 o 
 o 
 
 zo.ooo 
 
 140,000 
 
 80^830 
 
 80 
 
 100 
 
 82.^^ 
 If 
 
 74,1 82 
 64.377 
 
 ■i-OI 
 +.03 
 
 
 
 -.08 
 .0 
 
 -0.7/ 
 -0.7/ 
 
 -39Z 
 -3.65 
 
 +2.15 
 + /.74 
 
 -29.34 
 -27./6 
 
 + 37 
 + 39 
 
 S9 
 io 
 
 400 
 
 i36,000C.M. 
 RLUMINUf^ 
 
 21 
 
 58.8 
 
 322 
 
 ZIS2 
 
 o 
 o 
 
 SOjOOO 
 
 200,000 
 
 115,500 
 
 so 
 
 too 
 
 /-tv 
 
 113,606 
 96,987 
 
 -.OS 
 
 +.0/ 
 
 — 
 
 +/.04 
 
 -0.7S 
 -0.8/ 
 
 -4.11 
 -3.92 
 
 +2.9? 
 +2.84 
 
 -2.£82 
 -2?.38 
 
 + 35 
 + 36 
 
 (,1 
 
 500 
 
 If 
 
 63i,oooc.ri 
 
 RLUMlNUtA 
 
 n 
 
 73.i 
 ft 
 
 3fo 
 
 Z7Si 
 
 I 
 
 15,000 
 
 140,000 
 
 80,830 
 
 80 
 100 
 
 il.Si 
 
 5 9,04t 
 
 S 1,337 
 
 ^-06 
 +.06 
 
 — 
 
 -/.47 
 + .2.f 
 
 -/.97 
 -/•S? 
 
 -9.I8 
 -9.64 
 
 +2.28 
 +2.54 
 
 -40.82 
 -20.62 
 
 + 70 
 + 73 
 
 is 
 6i 
 
 soo 
 
 ft 
 
 i3i,OO0C.H 
 
 Zl 
 
 73.S 
 
 40Z 
 
 2690 
 
 o 
 o 
 
 4-0,000 
 
 • 
 
 zoo, 000 
 tl 
 
 116,500 
 
 so 
 
 too 
 
 IIS.S 
 
 93,725 
 80,106 
 
 +.06 
 +.06 
 
 — 
 
 -.28 
 +3.8 
 
 -/.84 
 -I-7Z 
 
 -9.32 
 -8.84 
 
 + 4.44 
 + 5-+3 
 
 -3S.S4 
 -/3.53 
 
 + 64 
 + 65 
 
 *It would be commercially impractical to transmit such small amounts of power some of the extreme dis- 
 tances indicated by the tabulation. The problems are stated simply for the purpose of illustrating in an approxi- 
 mate manner the effect distance of transmission has upon the voltage drop as calculated by various methods. 
 
COMPARISON OF VARIOUS METHODS 
 
 119 
 
 by the impedance method of procedure, it will be seen 
 that they are in exact agreement. 
 
 Convergent Series Expression — Dr. F. E. Pemot 
 in "Electrical Phenomena in Parallel Conductors," Vol. 
 I, shows that the above described three condenser solu- 
 tion is equivalent to using the following values for the 
 auxiliary constants in the convergent series form of 
 solution : — 
 
 A' = (^1 + -^^ + 
 
 B" 
 
 2 
 
 36 
 
 C 
 
 
 36 ^ 216 / 
 Comparing the above expressions for the auxiliary 
 constants with the complete expressions yielding rigor- 
 ous values, the following differences may be not^d. 
 For constant A' the first two terms are the same as in 
 the complete series, but the third term is less than in 
 the complete series, and all terms beyond the third are 
 omitted. For constant B' the first two terms are the 
 same as in the complete series, but all terms beyond the 
 second are omitted. For constant C both the ZY and 
 the Z- Y= terms are smaller than in the complete series 
 and all terms beyond the third are omitted. 
 
 The above expressions yield the same values for 
 the auxiliary constants as given in Table Q. Thus 
 from chart XI, the following values corresponding to 
 problem X are taken: — 
 
 ZY = — 0.389187 + j 0.164115 
 Z^Y" = + 0.124532 — } 0.127742 
 Therefore 
 
 A' = i.oooooo 
 
 — 0.194593 + J 0.0820575 
 0.003459 — y 0.0035484 
 
 A' = 0.808866 -I- y 0.0785091 
 B' = I.oooooo 
 
 — 0.0648645 + j 0.0273525 
 
 z (0.9351355 + y 0.0273525) 
 B' = 913785 + y 235.7208 
 
 C = I.oooooo 
 
 — 0.0540538 + j 0.0227938 
 + 0.0005765 — y 0.0005914 
 
 y (0.9465227 -f- / 0.0222024) 
 c = — 0.0000347 -1- y 0.0014794 
 
 It will be seen that the above convergent series ex- 
 pression for the auxiliary constants check exactly with 
 those as determined by the equations in Table Q. 
 
 COMPARATIVE ACCURACY OF VARIOUS METHODS 
 
 In order to determine the inherent error in various 
 methods of solution, when applied to circuits of in- 
 creasing length; also for frequencies of both 25 and 
 60 cycles, 64 problems were solved. These problems 
 embrace thirty-two 25 cycle circuits, varying in length 
 from 20 to 500 miles and in voltage from 10 000 to 
 200000 volts. Fixed receiving-end load conditions 
 were assumed for unity, and also for 80 percent power- 
 factor lagging. These same problems were also solved 
 for a frequency of 60 cycles. 
 
 These 64 problems with corresponding linear con- 
 stants and assumed load conditions are stated on Chart 
 
 XXII. This is followed by columns in which have 
 been tabulated the error in voltage at the sending-end 
 of these circuits as determined by nine different 
 methods. The errors are expressed in percent of re- 
 ceiving-end voltage. Obviously the inherent error 
 corresponding to various methods will vary widely for 
 conductors of various resistances and to some extent 
 for different receiving-end loads. The tabulated values 
 should therefore be looked upon as comparative rather 
 than absolute for all conditions. 
 
 Rigorous Solution — The column headed "Rigorous 
 Solution" contains values for the sending-end voltage 
 which are believed to be exact. These values were ob- 
 tained by calculating values for the auxiliary constants 
 by means of convergent series and then calculating the 
 performance mathematically. The calculations were 
 carried out to include the sixth place and terms in con- 
 vergent series were used out to the point where they 
 did not influence the results. 
 
 The first values calculated were checked by a 
 second set of values calculated independently at an- 
 other time and where differences were found the cor- 
 rect values were determined and substituted. This 
 corrected list of values was again checked by a third 
 independent calculation. It is therefore believed that 
 the values contained in this column are exact, repre- 
 senting 100 percent. 
 
 Semi-Graphical Solution— The next column con- 
 tains the error in the results as derived by the combi- 
 nation of an exact mathematical solution for the 
 auxiliary constants and a graphical solution from there 
 on. This combination gave results in which the maxi- 
 mum error does not exceed eight one hundredths of one 
 percent of receiving-end voltage for either frequency. 
 In other words, since the values for the auxiliary con- 
 stants used in this method were exact, the maximum 
 error of eight one hundredths of one percent occurs in 
 the construction and reading of the graphical construc- 
 tions. 
 
 Complete Graphical Solution — This solution em- 
 ploys Wilkinson's charts for obtaining graphically the 
 auxiliary constants, the remainder of the solution being 
 also made graphically as previously described. It will 
 be seen that the maximum error as obtained by this 
 complete graphical solution is seven hundredths of one 
 percent for the 25 cycle and twenty-five hundredths 
 of one percent for the 60 cycle circuits. These errors 
 represent the combined result of various errors. First 
 there is a slight fundamental error in the basis upon 
 v.'hich the Wilkinson Charts are constructed when 
 used for circuits employing conductors of various sizes 
 ?nd spacings, the introduction of this error making 
 possible the simplification attained. Then there is the in- 
 herent limitation of precision obtainable in the construc- 
 ton and reading of the charts and vector diagrams. 
 
 These results show that the inherent accuracy of 
 this simplified, all graphical solution is sufficiently ac- 
 curate for all practical power circuits up to 300 miles 
 long. 
 
120 
 
 COMPARISON OP VARIOUS METHODS 
 
 Dwight's "K" Formulas — The high degree of ac- 
 curacy resulting by the use of H. B. Dwight's "K" 
 formulas should be noted. This error is a maximum 
 of eleven hundredths of one percent for these 32 
 twenty-five cycle problems. The statement is therefore 
 justified that these "K" formulas are sufficiently accur- 
 ate for all 25 cycle power circuits. 
 
 For the 60 cycle problems the maximum error by 
 the "K" formulas for problems up to and including 200 
 miles is one-fourth of one percent of receiving-end volt- 
 age. For 300 mile circuits this error is one-half of one 
 percent and increases rapidly as the circuit exceeds 300 
 miles in length. The accuracy of the "K" formulas for 
 60 cycle circuits is therefore well within that of the 
 assumed values of the linear constants for circuits up 
 to approximately 300 miles in length. 
 
 The "K" formulas are based upon the hyperbolic 
 formula expressed in the form of convergent series. 
 In the development of these formulas, use was made of 
 the fact that the capacitance multiplied by the reactance 
 of non-magnetic transmission conductors is a constant 
 quantity to a fairly close approximation. This assump- 
 tion has enabled the "K" formulas to be expressed in 
 comparatively simple algebraic form without the use 
 of complex numbers. To those not familiar or not in 
 position to make themselves familiar with the operation 
 of complex numbers, such as is used in the convergent 
 
 scries or hyperbolic treatments, the availability of the 
 Dwight "K" formulas will be apparent.* 
 
 Localised Capacitance Methods — The next four 
 columns contain values indicating the error in results 
 as determined by the four different localized capaci- 
 tance methods previously described in detail. It is in- 
 teresting to note the high degree of accuracy inherent 
 in Dr. Steinmetz's three condenser method. It is also 
 interesting to note that three of these methods over 
 compensate (that is, give receiving-end voltages too 
 low) and one (the split condenser method) gives under 
 compensation. 
 
 Impedance Method— Th^ values of the sending-end 
 voltage as obtained by the impedance method (which 
 takes no account of capacitance) are always too high 
 when applied to circuits containing capacitance. The 
 results by this method are included here simply to serve 
 as an indication of how great is the error for this 
 method when applied to circuits of various lengths and 
 frequencies of 25 and 60 cycles. Some engineers pre- 
 fer to use this method for circuits of fair length and 
 allow for the error. These tabulations will give an 
 approximation of the necessary allowance to be made. 
 
 *Thesc have been included with much other valuable ma- 
 terial in "Transmission Line Formulas" by H. B. Dwight, pub- 
 lished by D. Van Nostrand Co. of New York City. 
 
CHAPTER XIII 
 
 CABLE CHARACTERISTICS 
 
 Heating Limits for Cables 
 
 THE MAXIMUM safe-limiting temperatures in 
 degrees C at the surface of conductors in cables 
 is given in the Standardization Rules of the 
 A. I. E. E. (1918) as follows: — 
 
 For impregnated paper insulation (85 — E) 
 For varnished cambric (75 — E) 
 For rubber insulation (60—0.25 E) 
 
 Where E represents the effective operating e.m.f. 
 in kilovolts between conductors and the numerals 
 represent temperature in degrees C. Thus, at a work- 
 ing pressure of 5 kv, the maximum safe limiting 
 temperature at the surface of the conductors in a 
 cable would be: — 
 
 For impregnated paper insulation (80 degrees C) 
 For varnished cambric insulation (70 degrees C) 
 For rubber compound insulation (58.75 degrees C) 
 
 The acttial maximum safe continuous current load 
 for any given cable is determined primarily by the tem- 
 perature of the surrounding medium and the rate of 
 radiation. This current value is greater with direct 
 than with alternating current and decreases with in- 
 creasing frequency, being less for a 60 cycles than 
 for 25 cycles. The carrymg capacity of cables will 
 therefore be less in hot climates than in cooler climates 
 and will be considerably increased during the winter. 
 
 Cables immersed in water, carry at least 50 per- 
 cent more than when installed in a four-duct line, and 
 when buried in the earth 15 to 30 percent more than 
 in a duct line, depending upon the character of soil 
 moisture, etc. Circulating air or water through 
 conduits containing lead covered cables will increase 
 their capacity. From the above it is evident that no 
 general rule relative to carrying capacity can be formu- 
 lated to apply in all cases, and it is necessary, there- 
 fore, to consider carefully the surroundings when de- 
 termining the size of cables to be used. 
 
 The practicability of tables which specify car- 
 rying capacity for cables installed in ducts will gen- 
 erally be questioned, for the reason that operating con- 
 ditions are frequently more severe than those upon 
 which table values are based. A duct line may op- 
 erate at a safe temperature throughout its entire length, 
 except at one isolated point adjacent to a steam pipe 
 or excessive local temperatures due to some other 
 cause. If larger cables are not employed at this point, 
 burnouts may occur here when the remainder of the 
 cable line is operating well within the limits of safe 
 operating temperature. The danger in using table 
 values for carrying capacity without carefully consid- 
 ering the condition of earth temperatures throughout 
 the entire duct length is thus evident. 
 
 HEATING OF CABLES TABLE XXIV 
 
 The basis upon which the data in Table XXIV 
 has been calculated is covered by foot notes below the 
 table. The kv-a values are determined from the cur- 
 rent in amperes and are based upon 30 degree C rise 
 and a maximum of 3CXX) volts.* Expressing the car- 
 rying capacity of cables in terms of kv-a (corrected 
 for the varying thickness of insulation required for 
 various voltages) may be found more convenient 
 than the usual manner of expressing it in amperes. 
 It will be noted that the kv-a values of the table are 
 on the basis of a four-duct line and that for more 
 than four ducts in the line the table kv-a values will 
 be reduced to the following : — 
 
 For a 4 duct line — 100 percent. 
 For a 6 duct line — 88 percent. 
 For an 8 duct line — 79 percent. 
 For a ID duct line — 71 percent. 
 For a 12 duct line — 63 percent. 
 For a 16 duct line — 60 percent. 
 
 When applied to all sizes of cables, the above 
 values are only approximate. The reduction of car- 
 rying capacity caused by the presence of many cables 
 is more for large cables than for small ones. Also, 
 where load factors are small, the reduction due to the 
 presence of many cables is less than the value assigned, 
 although the carrying capacity of a small number of 
 cables is only slightly affected. 
 
 REACTANCE OF THREE-CONDUCTOR CABLES 
 
 Tables XXV and XXVI contain values for the 
 inductance, reactance and impedance of round three- 
 conductor cables of various sizes and for the thick- 
 nesses of insulation indicated.All values in the tables 
 are on the basis of one conductor of the cable one mile 
 long. 
 
 The table values were calculated from the funda- 
 mental equation (4), 
 
 D 
 
 L = 0.08047 -f- 0.741 logu—S- 
 
 where L = the inductance in millihenries per 
 mile of each conductor, R the actual radius of the 
 conductor and D the distance between conductor cen- 
 ters expressed in the same units as R. As indicated in 
 Section I, under Inductance,** this formula has been 
 derived on the basis of solid conductors. In the case 
 of cables, the effective radius is actually slightly less 
 than that of the stranded conductor. The values for 
 
 ♦These current values are taken from General Electric 
 Bulletin No. .49302 dated March 1917. They are in general 
 slightly higher than those published by the Standard Undei^ 
 ground Cable Company in their Hand Book dated 1906. 
 
 ♦♦Chapter I. 
 
122 
 
 CABLE CHARACTERISTICS 
 
 TABLE XXIV— CARRYING CAPACITY OF INSULATED COPPER CONDUCTORS 
 
 The following values for carrying capacity must not be assumed unless it is positively known that the conditions upon 
 which they are based will not be exceeded in service. 
 
 
 
 
 
 
 THREE 
 
 CON DUCTO R 
 
 
 CABLES 
 
 
 
 
 
 
 
 
 
 
 
 B & S NO. 
 
 AREA 
 
 IN 
 
 CIRCULAR 
 
 MILS 
 
 XX 
 
 0*RH»iNG 
 
 CftPAC'TV IN 
 -AMPERES 
 
 Direct - 
 
 CURRENT 
 BASED UPON 
 30« C « ' S E 
 
 AND A 
 
 MAXIMUM OF 
 
 3000 rfOLT& 
 
 PAPER IN. 
 
 8ULATION 
 
 K.V A W 
 LEAD OOVERE 
 THAT ALL DUC 
 
 FOR A 6 
 TO 79 PER CE 
 (4 WIDE AND t 
 
 HIGH MAY BE TRANSMITTED AT THREE PHASE AND THE FOLLOWING VOLTAGES OVER PAPER INSULATED 
 D CABLES INSTALLED IN A FOUR DUCT LINE WITH 30° C RISE IN TEMPERATURE BASED UPON THE ASSUMPTION 
 DTS CARRY LOADED CABLES AND UPON A NORMAL EARTH TEMPERATURE OF 20° C 
 DUCT LINE THESE K.V.A VALUES WOULD BE REDUCED TO APPROXIMATELY 88 PER CENT FOR AN 8 DUCT LINE 
 
 NT FOR A 10 DUCT LINE TO 71 PER CENT FOR A 12 DUCT LINE TO 63 PER CENT AND FOR A I6 DUCT LINE 
 HIGH) TO 60 PER CENT OF THE TABLE VALUES X X X X . 
 
 220 
 VOLTS 
 
 440 
 VOLTS 
 
 550 
 VOLTS 
 
 1100 
 
 VOLTS 
 
 2200 
 VOLTS 
 
 3300 
 VOLTS 
 
 4000 
 VOLTS 
 
 6000 
 VOLTS 
 
 6600 
 VOLTS 
 
 10000 
 VOLTS 
 
 II 000 
 VOLTS 
 
 12000 
 VOLTS 
 
 13200 
 VOLTS 
 
 15000 
 VOLTS 
 
 20000 
 
 VOLTS 
 
 22000 
 vQlTS 
 
 25000 
 VOLTS 
 
 /o 
 
 is 
 
 30 
 
 1 
 
 It 
 
 17 
 
 Z3 
 
 / 7 
 Zl 
 
 28 
 
 1^7 
 
 4,f 
 
 114 
 
 /OS 
 IZS 
 
 17 1 
 
 IZ4 
 3oi 
 
 /8-^ 
 
 22.5- 
 J07 
 
 202 
 
 300 
 347 
 jroo 
 
 328 
 
 S47 
 
 3Sk 
 43S 
 S9S 
 
 390 
 
 Vsl 
 
 438 
 S3 6 
 73 
 
 il°s 
 
 ISO 
 
 .430 
 
 7X7 
 /OZS 
 
 693 
 
 847 
 
 I'SS 
 
 ■1 
 
 70 
 
 fS 
 
 30 
 
 11 
 
 38 
 
 7i 
 i04 
 133 
 
 l5Z 
 ZOl 
 Z&i 
 
 ZZ8 
 314 
 4O0 
 
 37S 
 378 
 
 481 
 
 4IO 
 S&2 
 71 S 
 
 Vil 
 
 TVS 
 
 Li7 
 iVtI 
 
 730 
 /OOO 
 I37S 
 
 793 
 
 10 fo 
 
 8*7 
 1190 
 /S30 
 
 9 7S 
 /34O 
 IT 10 
 
 iZiS 
 
 1740 
 331s 
 
 138a 
 /89S 
 3410 
 
 IS4 
 2/20 
 2 700 
 
 1 
 o 
 
 ISO 
 
 II 
 
 7i 
 
 t1 
 
 il's 
 
 133 
 
 I90 
 ZO<) 
 
 2-4 7 
 
 3&I 
 418 
 4fS 
 
 S4S 
 7-tO 
 
 LSS 
 
 970 
 //ZS 
 /330 
 
 /OLS 
 /230 
 l4tO 
 
 IS8S 
 I840 
 2/70 
 
 1730 
 
 zoso 
 
 ZS70 
 
 / 8 90 
 ZI80 
 3S80 
 
 ZOio 
 3390 
 Z830 
 
 33IO 
 Zb80 
 3I70 
 
 3000 
 3480 
 41 30 
 
 3370 
 3780 
 *470 
 
 34.ro 
 
 -»2+0 
 sooo 
 
 oo 
 
 OOO 
 
 oooo 
 
 /so 
 I70 
 
 i.oo 
 
 7i 
 
 1 ,* 
 ISO 
 IS2 
 
 143 
 liZ 
 I90 
 
 ZtS 
 323 
 3»o 
 
 S70 
 i47 
 7iO 
 
 ess 
 
 t70 
 II4C 
 
 /03O 
 II70 
 I3 7S 
 
 /S3S 
 I740 
 Z040 
 
 1 7/0 
 /940 
 ZZ40 
 
 2 5^00 
 2»f0 
 3340 
 
 Z740 
 3IOO 
 3iSO 
 
 3970 
 3370 
 39*° 
 
 33iO 
 3690 
 4340 
 
 36£o 
 4ISO 
 49TO 
 
 47SO 
 S37S 
 li37S 
 
 SI7 
 S8SO 
 6900 
 
 S770 
 6 SSO 
 7700 
 
 Z so OOO 
 300 OOO 
 3 so OOO 
 
 Z2S 
 
 iso 
 a to 
 
 I's 
 
 lot 
 
 I7Z 
 /10 
 ZIS 
 
 2/4 
 238 
 24<. 
 
 -►28 
 
 8S7 
 aso 
 lOiS 
 
 I39S 
 I430 
 lioo 
 
 /SSO 
 I7ZO 
 /93S 
 
 2 300 
 2S60 
 38io 
 
 3 SZO 
 2 8-00 
 3I40 
 
 37SO 
 4170 
 4i70 
 
 4100 
 4 SSO 
 SI 00 
 
 ■44SO 
 ■49SO 
 SSSO 
 
 4880 
 S43S 
 i070 
 
 S470 
 ilOO 
 
 Aexs 
 
 7IZS 
 7TOO 
 8 SSO 
 
 77SO 
 8600 
 96SO 
 
 86SO 
 
 963S 
 
 /otoo 
 
 ■too OOO 
 
 4So OOO 
 ^ooooo 
 
 3/0 
 3-fO 
 3tO 
 
 lit 
 
 IZi 
 137 
 
 £St 
 
 zsa 
 
 Z74 
 
 Z^S 
 32 3 
 
 34Z 
 
 i%°7 
 
 6es 
 
 use 
 IZfS 
 /37<r 
 
 I770 
 /94 
 20S0 
 
 2/30 
 2340 
 Z480 
 
 3l 70 
 348o 
 3fc80 
 
 3480 
 3810 
 4O40 
 
 SI70 
 St70 
 iooo 
 
 Si SO 
 6ZOO 
 6 SSO 
 
 il SO 
 i7SO 
 7/30 
 
 t730 
 7370 
 7800 
 
 7SSO 
 8300 
 8 7 70 
 
 9SOO 
 /0 7S0 
 // 400 
 
 /0700 
 // 700 
 /Z4^0 
 
 //9SO 
 /3 100 
 /3SSO 
 
 too OOO 
 700 OOO 
 
 570 
 
 iSi. 
 
 /7» 
 
 sn 
 
 3»o 
 
 447 
 
 7i0 
 89S 
 
 /SZO 
 
 nfo 
 
 2280 
 3 no 
 
 33 io 
 3Z40 
 
 -♦ loo 
 49CO 
 
 4SOO 
 S370 
 
 A6SO 
 78ZO 
 
 7300 
 
 as7o 
 
 79ZO 
 930 
 
 8 tea 
 /ozoo 
 
 9TSO 
 //-wo 
 
 /Z 6SO 
 /4 8SO 
 
 /3 800 
 /6ZOO 
 
 'S40O 
 /t/OO 
 
 SINGLE CONDUCTOR CABLES 
 
 b^SNo 
 
 AREA 
 
 IN 
 
 CIRCULAR 
 
 MILS 
 
 CARRYING CAP) 
 AMPERES DIRECT 
 
 \CITY IN 
 
 K V A WHICH MAY BE TRANSMITTED AT THREE PHASE AND THE FOLLOWING VOLTAGES OVER THREE 
 
 CURRENT 
 
 PAPER INSULATED LEAD COVERED CABLES INSTALLED IN A FOUR DUCT LINE with 30" C RISE IN TEMPERATURE 
 
 N E CODE 
 
 ■.INTEmOR OON0UCT0R8 
 
 XX> 
 
 c 
 
 ■ON 
 *A 
 
 OP 
 T8 
 
 BASED UPON THE ASSUMPTION THAT ALL DUCTS CARRY LOADED CABLES AND UPON A NORMAL EARTH 
 TEMPERATURE OF 20° C 
 
 FOR A 6 DUCT LINE THESE K.V.A. VALUES WOULD BE REDUCED TO APPROXIMATELY 88 PER CENT FOR 
 AN 8 DUCT LINE TO 79 PER CENT FOR A 10 DUCT LINE TO 71 PER CENT: FOR A 12 DUCT LINE TO 63 PER CENT 
 
 AND FOR A 16 DUCT LINE (4 WIDE AND 4 HIGH) TO 60 PER CENT OF THE TABLE VALUES x X x x 
 
 X 
 
 TABLt » 
 
 RUBBER 
 
 INSULATION 
 
 TABLe a 
 
 OTHER 
 INSULATION 
 
 BASED Uf 
 30-CRISt 
 MAXIMUM 
 30O0 VOL 
 
 INSULATION 
 
 220 
 
 VOLTS 
 
 440 
 VOLTS 
 
 550 
 
 VOLTS 
 
 1100 
 
 VOLTS 
 
 2200 
 
 VOLTS 
 
 3300 
 VOLTS 
 
 4000 
 VOLTS 
 
 6000 
 VOLTS 
 
 6600 
 
 VOLTS 
 
 10000 
 
 VOLTS 
 
 II 000 
 
 VOLTS 
 
 12000 
 
 VOLTS 
 
 13200 
 
 VOlTS 
 
 15000 
 VOLTS 
 
 20000 
 
 VOLTS 
 
 22000 25000 
 
 VOLTS VOITS 
 
 ',1 
 
 lO 
 
 ii 
 
 IS 
 
 JO 
 
 Z-* 
 
 JO 
 
 -fo 
 
 /'^ 
 
 18 
 23 
 30 
 
 23 
 28 
 38 
 
 ■ffc 
 
 93 
 
 114 
 ISX 
 
 /37 
 17/ 
 33S 
 
 US 
 ZOh 
 Z7S 
 
 34S 
 3oi 
 408 
 
 270 
 357 
 
 .♦J-o 
 
 400 
 ^00 
 ii7 
 
 +37 
 
 S47 
 730 
 
 47 S 
 
 ■520 
 
 ^so 
 868 
 
 sas 
 
 73 3 
 97 S 
 
 7.r8 
 
 948 
 I36S 
 
 824 
 
 /O3o 
 /37» 
 
 IS4 
 
 * 
 
 3S 
 
 so 
 
 so 
 70 
 
 ss 
 
 7S 
 
 2 / 
 X8 
 
 4Z 
 S7 
 
 SZ 
 7/ 
 
 /04 
 
 I4Z 
 
 Z09 
 394 
 
 314 
 438 
 
 378 
 SiS 
 
 S60 
 7t,S 
 
 i_/7 
 8+2 
 
 9_/7 
 
 izso 
 
 /OIO 
 
 1270 
 
 1 040 
 
 I48S 
 
 1 I4S 
 /i30 
 
 I3A0 
 1830 
 
 IT40 
 2370 
 
 I8_9S 
 ZS8C 
 
 ZIZO 
 2890 
 
 3 
 
 ss 
 Vc 
 
 to 
 »o 
 
 /OO 
 
 tf 
 
 3t, 
 
 72 
 
 to 
 
 181 
 
 3iZ 
 
 cT^O 
 
 is3 
 
 970 
 
 1 obS 
 
 isas 
 
 1 730 
 
 1880 
 
 ZObO 
 
 Z320 
 
 JOOO 
 
 32T0 
 
 3660 
 
 Z 
 
 t 
 
 o 
 
 to 
 loo 
 
 IXS 
 
 Its 
 ISO 
 
 ZOO 
 
 /as 
 /so 
 
 /70 
 
 47 
 
 Vs 
 
 9S 
 114 
 /30 
 
 119 
 /+3 
 liZ 
 
 237 
 2 8.r 
 J23 
 
 47S 
 
 i'/7 
 
 713 
 8SS 
 970 
 
 860 
 I030 
 1 170 
 
 IZ7S 
 /S30 
 1740 
 
 140s 
 
 /68S 
 /910 
 
 zogj 
 
 Z500 
 2 830 
 
 Z280 
 Z7IS- 
 31 OO 
 
 2480 
 2970 
 3370 
 
 27/0 
 3ZiO 
 369° 
 
 30SO 
 30,0 
 4ISO 
 
 39 SO 
 
 4300 
 
 SI7C 
 S8SC 
 
 4820 
 
 S780 
 6 SSO 
 
 oo 
 
 OOO 
 
 oooo 
 
 ISO 
 ITS 
 ziS 
 
 zzs 
 %Vs 
 
 200 
 230 
 Z70 
 
 il 
 
 ISZ 
 I7S 
 iOS 
 
 19° 
 ZI9 
 3 Si 
 
 Jfo 
 
 437 
 SI3 
 
 /02S 
 
 //4o 
 /3IO 
 /S40 
 
 13 BO 
 /SSO 
 /860 
 
 Z040 
 Z3SO 
 37to 
 
 2280 
 242C 
 3030 
 
 3330 
 
 3 840 
 4-SOO 
 
 3iSO 
 -4200 
 -f920 
 
 39iO 
 4SSO 
 S3SO 
 
 43SO 
 S^OOO 
 SSSO 
 
 4880 
 S(,oo 
 6 SSO 
 
 *320 
 7270 
 8S30 
 
 6 86c 
 
 79 30 
 9300 
 
 7700 
 88(0 
 10400 
 
 a. so OOO 
 
 3 00 OOO 
 
 3 so OOO 
 
 Z7S 
 
 400 
 
 •300 
 3-»o 
 390 
 
 114 
 /Zl 
 /••■» 
 
 ZXS 
 
 zs? 
 
 Zgf 
 
 3SS 
 3Z4 
 342 
 
 S70 
 i-»7 
 723 
 
 1141 
 /3fS 
 /44S 
 
 I7/0 
 /f4c 
 3l70 
 
 3.0i0 
 Z340 
 ZhZO 
 
 3070 
 3470 
 J8SO 
 
 3370 
 3830 
 4Z70 
 
 sooo 
 Si70 
 
 A330 
 
 ,r-»7o 
 ^200 
 <h93o 
 
 S9SO 
 i730 
 7S3 
 
 6SOO 
 73 so 
 8ZSO 
 
 73Z0 
 B3O0 
 9360 
 
 9 SOO 
 / 07 so 
 /300O 
 
 I03S0 
 Il 700 
 I3IO0 
 
 II SSO 
 13 200 
 I41.S0 
 
 ^OO OOO 
 ASO OOO 
 SOO OOO 
 
 3ZS 
 
 SOO 
 
 too 
 
 ■*IO 
 tSO 
 
 ■1-so 
 
 /St 
 
 3IZ 
 J-t3 
 365" 
 
 390 
 438 
 4 St 
 
 7g'0 
 
 ess 
 
 9/3 
 
 /S&O 
 /7/ O 
 
 /ess 
 
 3340 
 3S&0 
 3740 
 
 ZS30 
 3IOO 
 ^300 
 
 4180 
 4-600 
 4-900 
 
 4i,oo 
 
 SOSO 
 
 .5-foo 
 
 6eso 
 
 7 SOO 
 
 sooo 
 
 74-80 
 8 ZOO 
 8 7S0 
 
 8/ZO 
 8900 
 fSOO 
 
 8900 
 9770 
 10400 
 
 10000 
 10900 
 11700 
 
 /Z9S0 
 
 /4 300 
 /SZOO 
 
 14 ISO 
 ISSOO 
 16 SSO 
 
 JSSao 
 
 l73Sa 
 ISSOO 
 
 Aooooo 
 
 TOO OOO 
 900 OOO 
 
 4 so 
 JOO 
 
 ^ sso 
 
 <sro 
 
 7«0 
 »40 
 
 sso 
 6IO 
 <470 
 
 zss 
 
 4IS 
 4^4 
 Sio 
 
 S3 3 
 S80 
 
 /04S 
 // LO 
 /Z7S 
 
 3090 
 2320 
 3 SSO 
 
 Jli-O 
 J480 
 ■3 SZO 
 
 3790 
 42O0 
 4ioO 
 
 SiZO 
 &Z30 
 t>8SO 
 
 ^I80 
 
 teso 
 
 7 SZO 
 
 9170 
 
 10 ISO 
 
 11 ISO 
 
 10000 
 /1 100 
 IZ ZOO 
 
 10900 
 IZIOO 
 /3ZSO 
 
 II9S0 
 I3ZS0 
 I4SSO 
 
 13400 
 I48S0 
 I63S0 
 
 17400 
 19300 
 
 ZIZOO 
 
 l89St! 
 Ziooo 
 23/00 
 
 31x00 
 
 23 SOO 
 
 zsaoo 
 
 900 000 
 
 f OOO OOO 
 
 i foo OOO 
 
 too 
 
 6 so 
 <90 
 
 9ZO 
 /OOO 
 
 lo^o 
 
 73 o 
 7£o 
 
 Z74 
 297 
 
 S48 
 S94 
 
 d,es 
 74 Z 
 
 /370 
 I4SO 
 
 3740 
 39to 
 
 4I0O 
 44S0 
 
 49 SO 
 •5-3 70 
 
 7 3 SO 
 79SO 
 
 eioo 
 
 87SO 
 
 IZOOO 
 
 13000 
 
 /3/SO 
 I4ZOO 
 
 I43SO 
 IS4SO 
 
 /SiSO 
 /tiso 
 
 I7SS0 
 I9000 
 
 ZZ70C 
 246S0 
 
 24800 
 2^80C 
 
 Z770C 
 30IOC 
 
 i XOOOOQ 
 i 2SOOOO 
 /300 OOO 
 
 730 
 770 
 
 nso 
 
 IZZO 
 
 ^oo 
 
 34Z 
 
 6is- 
 
 8SS 
 
 /7/0 
 
 34ZO 
 
 SI3C 
 
 *200 
 
 9X00 
 
 loioo 
 
 IS 000 
 
 /i4O0 
 
 17 800 
 
 I9SSC 
 
 zitao 
 
 Z8400 
 
 31 000 
 
 34700 
 
 /'fOO OOO 
 / ^OO OOO 
 / 600 000 
 
 810 
 BSO 
 f»o 
 
 izto 
 l3to 
 l-*30 
 
 <030 
 
 39Z 
 
 F£S 
 
 9eo 
 
 /TlO 
 
 3920 
 
 S870 
 
 7IOO 
 
 losoo 
 
 //SSO 
 
 173 00 
 
 18800 
 
 ZO400 
 
 2Z400 
 
 zsioo 
 
 3zioO 
 
 3SS00 
 
 stFoo 
 
 / 7SOOOO 
 1 900 OOO 
 ZOOO OOO 
 
 /OIO 
 lOSO 
 
 /no 
 
 /*7o 
 
 / I30 
 
 izto 
 
 430 
 
 ■iTo 
 
 S60 
 
 /07S 
 IXOO 
 
 ZiSO 
 Z400 
 
 ■4 3 00 
 4-SOO 
 
 &4S0 
 72.00 
 
 7770 
 
 sTto 
 
 /I SSO 
 
 /zaso 
 
 /Z7O0 
 
 i-iTso 
 
 /8 800 
 
 xTooo 
 
 aoioo 
 
 33 000 
 
 Z34O0 
 3SOOO 
 
 Z4SOO 
 27300 
 
 Z7SO0 
 307O0 
 
 3S700 
 3980C 
 
 3900a 
 433ai 
 
 -f3«0<l 
 
 4e6ot 
 
 X For purposes of comparison these values are given for interior conductors. 
 
 XX For four conductor cables these ampere values would be reduced by 12.5 percent. 
 
 XXX For solid conductors these ampere ratings would be reduced by seven percent. For two conductor cables made up 
 cither round or flat, they would be reduced by 15 percent. For two conductor concentric cables they would be reduced by 25 
 percent. They will also be reduced in the case of the larger conductors when used on alternating-current circuits on account 
 of skin effect, unless special cables having non-conducting cores are used. These special cables should be used for 700000 
 circ. mils and larger for 60 cycle and i 000 000 circ. mils and larger for 25 cycle service. 
 
 XX XX For the higher voltage cables the kv-a values of the table have been reduced by one percent for each 2000 
 volts that the working pressure exceeds 3000 volts, that is by 11 percent for a 25000 volt cable. For insulated aluminum 
 conductors the safe carrying capacity (based upon 61 percent conductivity) is 79.3 percent of the above table values with the 
 same kind of insulation. These kv-a values are based upon the current in columns headed by XX and XXX. 
 
CABLE CHARACTERISTICS 
 
 123 
 
 TABLE XXV— INDUCTANCE, REACTANCE AND IMPEDANCE, AT 25 CYCLES, PER MILE OF 
 SINGLE CONDUCTOR FOR THREE CONDUCTOR CABLES 
 
 AREA 
 
 IN 
 
 CIRCULAR 
 
 MILS 
 
 B & S NO. 
 
 a: CO 
 
 UJ UJ 
 
 1- I 
 
 UJ 
 
 S z 
 
 <;~ 
 OS 
 
 RESISTANCE 
 
 PER MILE 
 IN OHMS )f 
 
 INSULATION THICKNESS IN 64THS OF AN INCH * • 
 
 ^ BY ^ 
 
 4 nv 4 
 67 BY 6^ 
 
 A BY ^ 
 
 ^ BY ^ 
 
 IND. 
 M.H. 
 
 REAC. 
 OHMS 
 
 IMP. 
 OHMS 
 
 IND. 
 M.H. 
 
 REAC. 
 OHMS 
 
 IMP. 
 OHMS 
 
 IND. 
 M.H. 
 
 REAC. 
 OHMS 
 
 IMP. 
 OHMS 
 
 IND. 
 M.H. 
 
 REAC. 
 OHMS 
 
 IMP. 
 OHMS 
 
 •Soo ooo 
 -4S0 ooo 
 •^oo ooo 
 
 .772 
 • 7XS 
 
 .1/6 
 ./3.<f 
 ./4S 
 
 .33S 
 .340 
 .343 
 
 .O.5'30 
 
 .0534 
 .0537 
 
 .128 
 .140 
 .ISS 
 
 .349 
 .3S 1 
 .3S4 
 
 .0.547 
 .ossz 
 
 .OSS4 
 
 .129 
 .140 
 .iss 
 
 .360 
 .362 
 .367 
 
 .0S6S 
 • 0S68 
 ,0576 
 
 ./27 
 .14 1 
 • ISS 
 
 .370 
 .3 73 
 .377 
 
 .0.5 80 
 .0J85 
 .0.572 
 
 .130 
 .142 
 .IS7 
 
 3S0 ooo 
 3oo ooo 
 ZSo ooo 
 
 .hei 
 .630 
 •S75 
 
 .ihi, 
 
 .233 
 
 .3-J6 
 .34'! 
 .3S3 
 
 .0S43. 
 .0S47 
 .OSS4 
 
 .ns 
 .204 
 
 .240 
 
 .3S7 
 .361 
 .366 
 
 .0.560 
 
 .056 7 
 
 .0.575 
 
 .176 
 .Z04 
 
 .240 
 
 .370 
 .3 74 
 .38/ 
 
 • OSSI 
 .0587 
 .0577 
 
 • 176 
 
 •20s 
 .240 
 
 .3SO 
 ,386 
 .394 
 
 .OS76 
 ,060S 
 .OAI9 
 
 • 177 
 .207 
 .242 
 
 oooo 
 ooo 
 oo 
 
 ..528 
 .■^70 
 .418 
 
 .27-5' 
 
 .357 
 .362 
 .369 
 
 ,osio 
 
 .OSi7 
 
 .OS 79 
 
 .28/ 
 .3.52 
 
 .44/ 
 
 .372. 
 .379 
 .388 
 
 .058.5 
 .OS9S 
 .0607 
 
 .Z8I 
 .3.52 
 
 .442 
 
 .3 8 7 
 .377 
 .406 
 
 .0607 
 .0623 
 .o637 
 
 .282 
 .352 
 .44 2 
 
 .403 
 .41 1 
 .42 3 
 
 .0633 
 .0645 
 .0665 
 
 .282 
 • 353 
 .442 
 
 o 
 / 
 
 .373 
 
 .332 
 .292 
 
 .sso 
 
 .693 
 .87? 
 
 .377 
 .384 
 .393 
 
 ,os9Z 
 .0603 
 .06/ 7 
 
 .SS2 
 .697 
 .882 
 
 .398 
 .40s 
 
 .417 
 
 .QtXS 
 ,063S 
 .0655 
 
 .SS4 
 .698 
 .882 
 
 .4/7 
 .42? 
 •441 
 
 ,o653 
 .0673 
 .0691 
 
 .55-4 
 .678 
 .882 
 
 .432 
 .447 
 .463 
 
 .06-'7 
 • 0700 
 .0727 
 
 •SS4 
 • 699 
 .882 
 
 3 
 
 .a.to 
 
 .232 
 
 ./8-f 
 
 li^ 
 
 .403 
 .413 
 .437 
 
 .0633 
 .06 + 8 
 .068-S- 
 
 l.ll 
 1.40 
 2.21 
 
 .43/ 
 .442 
 .470 
 
 .06 7.5 
 .069S 
 .0737 
 
 l.ll 
 J.40 
 2.2/ 
 
 .454 
 .4 67 
 ..50 / 
 
 .07/2 
 .0736 
 ,07SS 
 
 /.ll 
 1.4 
 2.2/ 
 
 .476 
 .494 
 .S29 
 
 .0746 
 .077S 
 .0830 
 
 I.I 1 
 1.40 
 2,21 
 
 
 
 T aw 1 
 64 BY 64 
 
 ^ BY^ 
 
 ^ BY ^ 
 
 ^byI| 
 
 IND. 
 M.H. 
 
 REAC. 
 OHMS 
 
 IMP. 
 OHMS 
 
 IND. 
 M.H. 
 
 REAC. 
 OHMS 
 
 IMP. 
 OHMS 
 
 IND. 
 M.H. 
 
 REAC. 
 OHMS 
 
 IMP. 
 
 OHMS 
 
 IND. 
 M.H. 
 
 REAC. 
 OHMS 
 
 IMP. 
 OHMS 
 
 ■SOO ooo 
 ■4SO ooo 
 -400 ooo 
 
 .91-* 
 .772 
 .728 
 
 .116 
 .129 
 .I4S 
 
 .379 
 
 .354 
 .389 
 
 .OS9S 
 .0602 
 ,o6lo 
 
 .130 
 .143 
 .ISS 
 
 • 389 
 
 • 393 
 
 • 396 
 
 .O&IO 
 
 .06/7 
 ,0622 
 
 . 131 
 .143 
 .IS8 
 
 .378 
 .403 
 .409 
 
 .otss 
 .0634 
 .0642 
 
 ./32 
 
 .144 
 .IS9 
 
 .407 
 .4/' 
 .-4'7 
 
 .0640 
 .0645 
 
 .0 655 
 
 •/33 
 ./45 
 ./60 
 
 3SO ooo 
 300 ooo 
 3 so ooo 
 
 .630 
 .S7S 
 
 .166 
 .l'>4 
 .233 
 
 .39s 
 ,399 
 
 .409 
 
 ,06X0 
 ,Oi>2i, 
 ,0642 
 
 .177 
 .24i 
 
 • 402 
 .409 
 
 .419 
 
 .06 30 
 
 .0642 
 
 .0 6 5S 
 
 .178 
 .2oS 
 .242 
 
 .4IS 
 .42 1 
 .430 
 
 .0652 
 .0660 
 .067S 
 
 .178 
 .2 OS 
 .242 
 
 .423 
 .442 
 
 .0664 
 .0675- 
 .0673 
 
 • 179, 
 .206 
 .24.3 
 
 oooo 
 
 ooo 
 
 oo 
 
 •470 
 
 .27.5 
 .346 
 .437 
 
 .4 IS 
 .429 
 .439 
 
 .06S2 
 .0673 
 .0690 
 
 .283 
 .3.5 3 
 .44 3 
 
 .427 
 
 • 440 
 
 • 4SS 
 
 .0673 
 .0670 
 .0714 
 
 .2 8 3 
 .3 5^3 
 .443 
 
 .44 1 
 .4SS 
 .469 
 
 .0690 
 .0714 
 .073S 
 
 .2 84 
 .354 
 
 .443 
 
 ,452 
 .466 
 .483 
 
 .0708 
 .0730 
 .0758 
 
 .28S 
 .3SS 
 
 .444 
 
 o 
 
 i 
 
 .373 
 .332 
 .292 
 
 .SSO 
 .69S 
 .e79 
 
 .4S3 
 .46^ 
 .483 
 
 .0712 
 .0732 
 .07SS 
 
 .SS4 
 
 .466 
 .483 
 
 .SO 2 
 
 .0731 
 .07S7 
 .0787 
 
 .SS4 
 .677 
 .882 
 
 .4SS 
 .SOI 
 .S2I 
 
 .O760 
 .078S 
 ,OS/6 
 
 .sss 
 
 ■699 
 .833 
 
 .478 
 ..5/6 
 .S37 
 
 .0780 
 .08/0 
 .0843 
 
 .ssi 
 
 .700 
 .883 
 
 1 
 
 .260 
 .232 
 
 ./ 84 
 
 1,1 1 
 /,40 
 2.ZI 
 
 .499 
 .SIS 
 .J -5 7 
 
 .orez 
 ,oe/3 
 
 .0873 
 
 l.ll 
 
 /,4o 
 
 3,2 1 
 
 .SI9 
 .S3e 
 •S80 
 
 .O814 
 
 .oe4S 
 
 .0910 
 
 I.I 1 
 1.40 
 2.ZI 
 
 .S3 8 
 .SS8 
 
 .601 
 
 .0 84 5 
 .087.5 
 .0743 
 
 l.ll 
 1.40 
 2.2/ 
 
 .SS8 
 
 .0 8 74- 
 .090s 
 .077.5 
 
 l.ll 
 1.40 
 2.2/ 
 
 
 
 1 nv II 
 64 BY 64 
 
 ^BY^ 
 
 ^BYi| 
 
 ^BYli 
 
 IND. 
 M.H. 
 
 REAC. 
 OHMS 
 
 IMP, 
 
 OHMS 
 
 IND. 
 M.H. 
 
 REAC. 
 OHMS 
 
 IMP. 
 OHMS 
 
 IND. 
 M.H. 
 
 REAC. 
 OHMS 
 
 IMP. 
 OHMS 
 
 IND. 
 M.H. 
 
 REAC. 
 OHMS 
 
 IMP. 
 OHMS 
 
 ■Soo ooo 
 ^So ooo 
 ^oo ooo 
 
 .SI4 
 .772 
 .72S 
 
 .116 
 .IZ9 
 .MS 
 
 .417 
 
 .423 
 .429 
 
 .O&SS 
 
 .066S 
 .0673 
 
 ./33 
 ./4S 
 .160 
 
 .427 
 .43/ 
 .436 
 
 .0670 
 
 .067S 
 .06S3 
 
 ./33 
 • I4S 
 .I60 
 
 .434 
 .437 
 .446 
 
 .068/ 
 ,0670 
 .0700 
 
 ./34 
 ./46 
 .161 
 
 .44/ 
 .447 
 .457 
 
 .067/ 
 
 .0705 
 
 .07/7 
 
 .I3S 
 .147 
 .162 
 
 3^o ooo 
 300 000 
 ZSO 000 
 
 .6S/ 
 .630 
 .S7S 
 
 .166 
 • 194 
 .233 
 
 .436 
 .444 
 .4S4 
 
 .o6es 
 
 .0697 
 
 .0712 
 
 .ISO 
 .206 
 .244 
 
 .446 
 .4.S6 
 .4 6S 
 
 .0700 
 .07/5 
 .0730 
 
 • ISO 
 
 .206 
 
 .244 
 
 .453 
 
 .46/ 
 .47J- 
 
 •07I0 
 .0722 
 .074S 
 
 .ISO 
 .207 
 
 .24.r 
 
 .464 
 .473 
 ,486 
 
 .0727 
 .0742 
 
 ,0762 
 
 ■181 
 .208 
 .34S 
 
 0000 
 000 
 
 00 
 
 .SZ8 
 
 .470 
 ^/■8 
 
 •2 7.5- 
 .346 
 .437 
 
 .46S 
 .491 
 .478 
 
 ,0730 
 
 .QTSS 
 .0780 
 
 .28.i- 
 .3SS 
 .44S 
 
 .476 
 .473 
 • SIO 
 
 .074S 
 .077S 
 
 .oeoe 
 
 <2 8J 
 .3SS 
 •44S 
 
 .486 
 • 503 
 
 ,S2 1 
 
 .0760 
 .0790 
 .08 16 
 
 .2 86 
 .3SS 
 .44S 
 
 .498 
 .S16 
 ,S3S 
 
 .0782 
 
 .08/0 
 
 .O840 
 
 .287 
 .356 
 .446 
 
 
 
 k 
 
 .373 
 .331 
 .292 
 
 .sso 
 
 .69S 
 .879 
 
 .SI4 
 .S3I 
 .SS4 
 
 .0 90S 
 .0830 
 .0970 
 
 .SS6 
 .700 
 .882 
 
 .S28 
 • S4 6 
 ,S70 
 
 .0838 
 
 .OBSS 
 .OS9S 
 
 .SS6 
 • 70 
 .882 
 
 .S39 
 .SS9 
 .S83 
 
 .084S 
 .0877 
 .091S 
 
 .SS6 
 .700 
 .883 
 
 .SS4 
 .573 
 .S98 
 
 .08 70 
 .0700 
 
 .0738 
 
 .557 
 ■78%% 
 
 3 
 
 1 
 
 .Zio 
 .232 
 
 ./84 
 
 /.40 
 X.ZI 
 
 .S74 
 .S96 
 .643 
 
 .0900 
 .093S 
 ,1010 
 
 l.ll 
 1.40 
 2.2/ 
 
 .S9I 
 .6/3 
 .til 
 
 .0927 
 
 •0962 
 
 ./037 
 
 /.ll 
 /.40 
 2.2/ 
 
 .606 
 .627 
 .678 
 
 .09 SO 
 .0783 
 ./<J63 
 
 I.I 1 
 /.40 
 .2.2 1 
 
 .6/8 
 
 .0970 
 .1 010 
 .1090 
 
 2.21 
 
 1 
 
 
 ^BYil 
 
 -Ifi- RV 18 
 64 BY 63 
 
 20 BY 20 
 64 ^^ 64 
 
 21 'aw 22 
 f4 BY 64 
 
 IND. 
 M.H. 
 
 REAC. 
 OHMS 
 
 IMP. 
 
 OHMS 
 
 IND. 
 M.H. 
 
 REAC. 
 OHMS 
 
 IMP. 
 OHMS 
 
 IND. 
 M.H. 
 
 REAC. 
 OHMS 
 
 IMP. 
 OHMS 
 
 IND. 
 M.H. 
 
 REAC. 
 OHMS 
 
 IMP. 
 OHMS 
 
 •Soo 000 
 '4SO 000 
 -400 000 
 
 .914 
 .773. 
 .728 
 
 .Ilk 
 ./Z9 
 . I4S 
 
 .4S7 
 .462 
 .47' 
 
 .07/7 
 .07^S 
 .0738 
 
 ./36 
 ./48 
 ./63 
 
 .4V4 
 .48/ 
 .487 
 
 •0744 
 .07 S4 
 
 • 0764 
 
 .138 
 .ISO 
 .1 64 
 
 .487" 
 
 .476 
 
 .SOS 
 
 .0764 
 .0778 
 
 .0772 
 
 • /40 
 .IS' 
 .I6S 
 
 .so 1 
 .S09 
 .S19 
 
 .0785 
 
 .0800 
 ,08/5 
 
 .141 
 .ISZ 
 .166 
 
 3 so 000 
 
 300 000 
 ZSO 000 
 
 .(,81 
 .630 
 .S7S 
 
 .166 
 .194 
 .233 
 
 .4 SO 
 '491 
 .SOS 
 
 .07S3 
 .0770 
 .079a. 
 
 .182 
 .XOS 
 .24 6 
 
 -4 7 6 
 .SI 1 
 .S24 
 
 • 0778 
 .Ot02 
 ,09%2 
 
 .1 83 
 
 .SI 3 
 .S26 
 
 .S4I 
 
 .08O5 
 .OS25 
 .0 848 
 
 .I8S 
 
 .Zl 1 
 .248 
 
 .527 
 .S4I 
 .SS7 
 
 .0830 
 .0 848 
 
 .0875 
 
 • IS6 
 .2/2 
 .247 
 
 0000 
 000 
 
 00 
 
 .SZ8 
 .470 
 
 .418 
 
 .27.5- 
 .3->6 
 .437 
 
 .sn^ 
 ,J36 
 ..162 
 
 *OBIO 
 .O840 
 .OSiS 
 
 .287 
 .3 SI 
 .446 
 
 •S3 6 
 .SSi 
 •S7S 
 
 .0840 
 .08 70 
 .0707 
 
 .288 
 
 .3.57 
 .446 
 
 .ssi 
 
 .S7S 
 .S99 
 
 .0870 
 .09OS 
 .0940 
 
 .287 
 .3S8 
 
 .447 
 
 .573 
 .572 
 .618 
 
 -0700 
 .0730 
 .0770 
 
 .270 
 .360 
 .448 
 
 
 / 
 z 
 
 .373 
 .332 
 .292 
 
 .SSO 
 
 .69S 
 .■879 
 
 .S7S 
 .S98 
 .623 
 
 .0902 
 .093S 
 .097S 
 
 .sss 
 .700 
 
 .8 84 
 
 .60 / 
 .623 
 
 .647 
 
 .o74i 
 
 .O980 
 
 • ion 
 
 .558 
 .700 
 .8 84 
 
 .62/ 
 .64S 
 .674 
 
 .0772 
 .1010 
 .1060 
 
 .SSS 
 .70/ 
 
 • kes 
 
 ■.itL 
 
 .1 OOS 
 .I04S 
 ,I08S 
 
 .55-7 
 .702 
 .886 
 
 3 
 
 1 
 
 .260 
 .232 
 
 ./S4 
 
 I.I 1 
 J.40 
 2.2/ 
 
 .&49 
 .673 
 
 .10 19 
 .loss 
 
 .II3S 
 
 1.41 
 2,2 2 
 
 .674 
 .701 
 . 7.S4 
 
 .1060 
 .1100 
 
 .1 1 90 
 
 I.I 1 
 
 r.4i 
 
 2.22 
 
 .678 
 .72 5- 
 .780 
 
 .I09S 
 .1 138 
 
 ,I2XS 
 
 1.1 1 
 1.41 
 
 2.2 Z 
 
 '73. 1 
 
 .1130 
 
 • 1 no 
 ,1270 
 
 1.12 
 1.41 
 2.2 2 
 
 ♦Resistance based upon loo percent conductivity at 25 degrees C (^77 degrees F), including two percent allowance for 
 spiral of strands and two percent allowance for spiral of conductors. For a temperature of 65 degrees C (149 degrees F) 
 these resistance values would be increased 15 percent. 
 
 **The inductance is in millihenries; the reactance and the impedance are in ohms. 
 
 The table values were derived from the equation L = 0.08047 + 0.741 Logi„ -^ where R is the radius of conductor, D 
 
 the distance between centers of conductors expressed in the same terms as /?. and Z. the inductance in millihenries per mila 
 of each conductor. All values in the table are single-phase and based upon a single conductor one mile long. 
 
124 
 
 CABLE CHARACTERISTICS 
 
 TABLE XXVI— INDUCTANCE, REACTANCE AND IMPEDANCE, AT 60 CYCLES, PER MILE OF 
 SINGLE CONDUCTOR FOR THREE CONDUCTOR CABLES 
 
 
 AREA 
 
 IN 
 
 CIRCULAR 
 
 MILS 
 B & S NO. 
 
 oc CO 
 liJ UJ 
 
 f-x 
 
 UJO 
 
 oz 
 
 RESISTANCE 
 
 PER MILE 
 IN OHMS ^ 
 
 INSULATION THICKNESS IN 64THS OF AN INCH •• 1 
 
 
 ^ BY-^ 
 
 4 Qw 4 
 
 64 BY 67 
 
 ^ BY ^ 
 
 ^BY^ 1 
 
 
 IND. 
 M.H. 
 
 REAC. 
 OHMS 
 
 IMP. 
 OHMS 
 
 IND. 
 M.H. 
 
 REAC. 
 OHMS 
 
 IMP. 
 OHMS 
 
 IND. 
 M.H. 
 
 REAC. 
 
 OHMS 
 
 IMP. 
 OHMS 
 
 IND. 
 M.H. 
 
 REAC. 
 OHMS 
 
 IMP. 
 OHMS 
 
 
 Soo ooo 
 •^So ooo 
 -f oo ooo 
 
 .8)4 
 .772 
 .728 
 
 • iz-r 
 
 .145 
 
 .338 
 .340 
 .3-»3 
 
 .127 
 .128 
 .129 
 
 .172 
 .181 
 .I9S 
 
 .349 
 
 .3SI 
 ,354 
 
 . 13 1 
 .132 
 .134 
 
 .ns 
 ./84 
 .197 
 
 .340 
 .342 
 .347 
 
 ,/34 
 .137 
 .138 
 
 .178 
 .199 
 .201 
 
 .370 
 ,373 
 .377 
 
 ./40 
 .141 
 .142 
 
 ./82 
 •J2V4 
 
 
 3S0 ooo 
 300 ooo 
 2.SO ooo 
 
 .68/ 
 .i30 
 .S7J 
 
 ./94 
 .233 
 
 ,3-,^ 
 .349 
 .3^3 
 
 .130 
 .132 
 .13 3 
 
 .21 1 
 .235 
 .2 48 
 
 .357 
 .3 4/ 
 .344 
 
 .I3S 
 ./3i 
 .138 
 
 .214 
 .2 37 
 .27/ 
 
 .370 
 .3 74 
 .3 8/ 
 
 ,/40 
 .141 
 .144 
 
 .2/7 
 .240 
 .274 
 
 .380 
 -384 
 .3 94 
 
 .143 
 .14 S 
 .149 
 
 .22 
 .244 
 ,277 
 
 
 OOOO 
 
 ooo 
 oo 
 
 .S28 
 
 .470 
 
 .418 
 
 .275^ 
 .34i, 
 .437 
 
 .3.57 
 .342 
 .3k9 
 
 .I3S 
 .13b 
 .139 
 
 .308 
 
 .373 
 .440 
 
 .3 72 
 .379 
 ,3 88 
 
 ,140 
 .143 
 .146 
 
 .30f 
 .375- 
 .44/ 
 
 ,387 
 ,397 
 ,404 
 
 .146 
 ./SO 
 ./53 
 
 .3 13 
 .378 
 .4 44 
 
 .403 
 .4 1 1 
 .423 
 
 .152 
 .ISS 
 .HO 
 
 .3/6 
 .381 
 .466 
 
 
 o 
 
 / 
 z 
 
 .373 
 .331 
 
 .a?2 
 
 .SSO 
 .69S 
 
 .879 
 
 .377 
 .384 
 .393 
 
 .142 
 .I4S 
 .148 
 
 .S4>9 
 .711 
 .893 
 
 .398 
 .405" 
 .4/7 
 
 .ISO 
 ■.'1V7 
 
 ,57/ 
 
 • 4/7 
 ,429 
 ,44/ 
 
 ./57 
 ./42 
 .166 
 
 .572 
 .7/5 
 .896 
 
 ,432 
 ,447 
 ,4 43 
 
 .163 
 ./48 
 ./74 
 
 .573 
 .7/4 
 .896 
 
 
 3 
 
 .2«o 
 .232 
 ./8-f 
 
 i.2i 
 
 .403 
 .4/3 
 .437 
 
 ./S2 
 .IS6 
 .Its 
 
 1.12 
 1.41 
 2.2 2 
 
 .43/ 
 .442 
 .470 
 
 ./42 
 .167 
 .177 
 
 I./3. 
 /.4I 
 2.22 
 
 ,454 
 .44^ 
 .SOI 
 
 .171 
 .177 
 .189 
 
 1.12 
 /.41 
 2.2Z 
 
 .474 
 .4 94 
 ,52 9 
 
 .18a 
 .186 
 .200 
 
 J.I2 
 /■t9l 
 2,2 2 
 
 
 
 
 
 67 BY -^ 
 
 ^ BY ^ 
 
 A BY ^ 
 
 ^BYi^, 
 
 
 IND. 
 M.H. 
 
 REAC. 
 OHMS 
 
 IMP. 
 OHMS 
 
 IND. 
 M.H. 
 
 REAC. 
 OHMS 
 
 IMP. 
 OHMS 
 
 IND. 
 M.H. 
 
 REAC. 
 OHMS 
 
 IMP. 
 OHMS 
 
 IND. 
 M.H. 
 
 REAC. 
 OHMS 
 
 IMP 
 OHMS 
 
 
 •SOO ooo 
 ASo ooo 
 •400 ooo 
 
 .ZI4 
 .772 
 .738 
 
 .Hi, 
 
 .129 
 
 .I4S 
 
 •379 
 .384 
 .3 89 
 
 ,/43 
 
 .I4S 
 .147 
 
 ./84 
 .194 
 .20h 
 
 .38-9 
 ,393 
 .394 
 
 ,/4<i 
 .148 
 .149 
 
 .186 
 .I9S 
 ,208 
 
 .398 
 .403 
 
 .409 
 
 ./50 
 ./52 
 ,/54 
 
 •190 
 .200 
 .2/2 
 
 ,407 
 .411 
 .4/7 
 
 ,/53 
 .ISS 
 .IS7 
 
 .192 
 .202 
 ,2 30 
 
 
 3^o ooo 
 300 ooo 
 2S0 ooo 
 
 .tei 
 
 .i,30 
 .S7S 
 
 ./a 
 
 .194 
 .2*33 
 
 .395 
 .399 
 .409 
 
 .149 
 
 .ISO 
 
 .IS4 
 
 .222 
 .24S 
 .279 
 
 .402 
 .409 
 .4/9 
 
 .ISI 
 
 •.'/st 
 
 ,224 
 .24 b 
 .282. 
 
 .4/5 
 ,42/ 
 ,430 
 
 .1S7 
 ,/S8 
 .162 
 
 .229 
 
 f2i's 
 
 .423 
 
 •43/ 
 .442 
 
 ./60 
 .162 
 .166 
 
 ,23/ 
 ,2 54 
 .2 84 
 
 
 OOOO 
 
 ooo 
 oo 
 
 .470 
 
 .4 1 8 
 
 .37S 
 .34(, 
 .437 
 
 .41S 
 •421 
 .439 
 
 .IS7 
 .li,2 
 .Hi 
 
 .3/8 
 .383 
 
 .447 
 
 .427 
 .440 
 .4SS 
 
 .171 
 
 .320 
 ,385 
 .4 7/ 
 
 .44 1 
 .4 55 
 .449 
 
 .166 
 .172 
 .177 
 
 .32 3 
 .388 
 .473 
 
 .452 
 ,444 
 .483 
 
 .170 
 .176 
 .182 
 
 .32 3 
 ■319 
 
 .4 74 
 
 
 O 
 
 I 
 
 s. 
 
 .373 
 .332 
 .3J2 
 
 .SSO 
 .69S 
 .879 
 
 .453 
 .446 
 .4 83 
 
 .171 
 .I7i 
 .182 
 
 .578 
 .7/8 
 .900 
 
 .444 
 .4 83 
 .S02 
 
 .nt 
 
 .182 
 .189 
 
 ,578 
 .497 
 .90a 
 
 .485 
 .SOI 
 .521 
 
 ./83 
 ./89 
 .19 6 
 
 .580 
 .720 
 .902 
 
 .499 
 .516 
 .537 
 
 .18 9 
 .195 
 .202 
 
 .582 
 
 .72/ 
 ■ 902 
 
 
 3 
 
 .ZbO 
 .232 
 .1 84 
 
 l.ll 
 1.40 
 S.2I 
 
 .499 
 .SIS 
 .IS 7 
 
 ./8» 
 .I9S 
 .210 
 
 1.13 
 /.4I 
 2.2 2 
 
 .SI 9 
 .538 
 .580 
 
 .I9S 
 .203 
 .2/9 
 
 /.1 3 
 1.4 1 
 2.22 
 
 .538 
 
 .SS8 
 
 .&0 1 
 
 .203 
 .2 10 
 .224 
 
 1.13 
 J. 42 
 2.2 2 
 
 .558 
 .577 
 .422 
 
 .2/ / 
 .218 
 .234 
 
 /./3 
 /.42 
 2.2 2 
 
 
 
 
 
 e^BYi^ 
 
 ^BYi| 
 
 ^BYil 
 
 14 nv 14 
 64 BY 24 
 
 
 IND. 
 M.H. 
 
 REAC. 
 OHMS 
 
 IMP. 
 OHMS 
 
 IND. 
 M.H.I 
 
 REAC. 
 OHMS 
 
 IMP. . 
 OHMS 
 
 IND. 
 M.H. 
 
 REAC. 
 OHMS 
 
 IMP. 
 OHMS 
 
 IND. 
 M.H. 
 
 REAC. 
 OHMS 
 
 IMP 
 OHMS 
 
 
 Soo OOO 
 
 ^£o ooo 
 400 ooo 
 
 .214 
 .772 
 .728 
 
 .11^ 
 .129 
 .145 
 
 .4/7 
 .423 
 .429 
 
 .IS7 
 .HO 
 
 .HI 
 
 ./95 
 .204 
 .2/4 
 
 .4 2 7 
 .43/ 
 .434 
 
 .HI 
 ,/42 
 .H4 
 
 .198 
 .208 
 .220 
 
 .434 
 
 ■.V4I 
 
 .164 
 .I6S 
 .168 
 
 .202 
 .21 1 
 .222 
 
 .44/ 
 
 .449 
 .437 
 
 ./44 
 ./70 
 ./72 
 
 .202 
 ■211 
 
 .224 
 
 
 3£0 ooo 
 300 ooo 
 
 2SO ooo 
 
 .S7S 
 
 .Hi 
 .194 
 .233 
 
 .434 
 .444 
 ,4S4 
 
 ./44 
 ,/47 
 .171 
 
 .235 
 .254 
 .2 89 
 
 ,444 
 ,454 
 ,445 
 
 .H8 
 ■ 172 
 .I7S 
 
 .2 3 7 
 .260 
 .292 
 
 .453 
 .44/ 
 .475 
 
 .17 1 
 .174 
 .179 
 
 .240 
 
 ,2 4 2 
 .29s 
 
 .444 
 .473 
 .4»4 
 
 .I7S 
 .178 
 .183 
 
 ■Z'^o 
 .2h'\ 
 .294 
 
 
 OOOO 
 
 ooo 
 
 oo 
 
 .J28 
 
 .470 
 
 .418 
 
 •437 
 
 .4iS 
 .4 8 1 
 .4 9 8 
 
 .I7S 
 ,181 
 .188 
 
 .32 8 
 .392 
 .4 7 4 
 
 .474 
 .493 
 .SIO 
 
 .I80 
 ./S4 
 ./92 
 
 .330 
 .395 
 .479 
 
 .4 84 
 .503 
 .52/ 
 
 .183 
 .190 
 .196 
 
 .332 
 .394 
 .480 
 
 .498 
 .5/4 
 .535 
 
 .l?S 
 .194 
 .202 
 
 .334 
 ■398 
 .482 
 
 
 o 
 / 
 
 2 
 
 .373 
 
 .332 
 .2 92 
 
 .SiO 
 .i,9S 
 .879 
 
 .SI4 
 .S3I 
 .SS4 
 
 .194 
 .200 
 .209 
 
 .5S4 
 
 .724 
 .90s 
 
 .528 
 ,544 
 .J70 
 
 .199 
 .2oi 
 
 .2/J" 
 
 .584 
 .7ZS 
 .904 
 
 .539 
 .SS9 
 .5 8 3 
 
 .20 3 
 .21 1 
 .220 
 
 .589 
 .724 
 .908 
 
 .554 
 .573 
 .S98 
 
 .209 
 .216 
 .2 2 5 
 
 .590 
 .728 
 .9/0 
 
 
 1 
 
 .3i,o 
 .232 
 ./ 84 
 
 'r.^'o 
 
 2.2/ 
 
 .374 
 .S9i 
 ,443 
 
 .2li 
 .224 
 .243 
 
 1.13 
 /.42 
 2.2 2 
 
 .59/ 
 .4/3 
 .44/ 
 
 .222 
 .23/ 
 .24f 
 
 /./ 3 
 /.4 2 
 2.2 2 
 
 .4 4 
 ,427 
 ,478 
 
 .2 2» 
 .234 
 .254 
 
 1.13 
 /'W2 
 2.2 2 
 
 .618 
 .443 
 .494 
 
 .2 3 3 
 .242 
 .242 
 
 1.14 
 1.42 
 2'.2 3 
 
 
 
 
 
 
 
 
 
 
 ^BV^ 
 
 ^byI| 
 
 20 BY 20 
 64 °' 64 
 
 00 00 
 
 JA RY ±A 
 64 °^ 64 
 
 
 IND. 
 M.H. 
 
 REAC. 
 OHMS 
 
 IMP. 
 OHMS 
 
 IND. 
 M.H. 
 
 REAC. 
 OHMS 
 
 IMP. 
 OHMS 
 
 IND. 
 M.H. 
 
 REAC. 
 OHMS 
 
 IMP. 
 OHMS 
 
 IND. 
 M.H. 
 
 REAC. 
 OHMS 
 
 IMP. 
 OHMS 
 
 
 ^oo ooo 
 
 A so ooo 
 •^oo ooo 
 
 .814 
 .772 
 .728 
 
 .Ilk 
 .129 
 .I4S 
 
 .4.S7 
 
 .442 
 .4 7 1 
 
 ,/7 2 
 ./74 
 ./V8 
 
 .208 
 
 .2(8 
 
 .230 
 
 *474 
 .411 
 .487 
 
 .179 
 .ISI 
 .IS3 
 
 .2/2 
 .2 24 
 .235 
 
 ,487 
 
 ,494 
 
 ,sas 
 
 •/8 3 
 
 ./87 
 
 ./90 
 
 .2/7 
 
 ,2 2 8 
 .240 
 
 .50 1 
 .50 9 
 .519 
 
 .l«9 
 .192 
 .196 
 
 .2 2 2 
 .232 
 .2^4 
 
 
 3.SO ooo 
 3oo ooo 
 
 2^a ooo 
 
 .h8l 
 .630 
 .S7S 
 
 .Ikk 
 .194 
 .233 
 
 .480 
 .491 
 .SOS 
 
 .181 
 
 ■!,Vo 
 
 .2 4 4 
 .27 
 .302 
 
 ,494 
 ,5/ / 
 .524 
 
 ,/8 7 
 .192 
 .197 
 
 .252 
 .2 74 
 .3o€ 
 
 .5/3 
 ,524 
 ,54/ 
 
 .19 3 
 .19 8 
 .204 
 
 .2S4 
 .279 
 
 .3// 
 
 .529 
 
 .54/ 
 .357 
 
 .200 
 .204 
 .21 
 
 .240 
 ,282 
 
 .314 
 
 
 OOOO 
 
 ooo 
 
 oo 
 
 .SZ8 
 .•470 
 .418 
 
 ,27^ 
 
 .34 h 
 .437 
 
 .5/7 
 ,334 
 ,552 
 
 .I9S 
 
 .202 
 
 ,20s 
 
 .3 3 8 
 
 .403 
 .4 84 
 
 .534 
 ,554 
 .57e 
 
 ,2 2 
 ,209 
 ,2/8 
 
 .3 4 2 
 .40 4 
 ,490 
 
 ,554 
 ►575 
 .399 
 
 .2/0 
 .2/7 
 ,2 2 4 
 
 ,34? 
 ,4/ 
 .494 
 
 .573 
 ,59 2 
 .618 
 
 .2/4 
 .223 
 ,2 3 3 
 
 ,35 2 
 .4/5 
 .494 
 
 
 o 
 1 
 2 
 
 .373 
 .332 
 .292 
 
 .S^O 
 ■.(,9S 
 .879 
 
 .573- 
 .59 S- 
 .423 
 
 .2 1 7 
 .2 2 5 
 .23s 
 
 .592 
 .732 
 .912 
 
 .40/ 
 .423 
 .i19 
 
 ,224 
 ,23 5 
 ,245 
 
 ,594 
 .734 
 .9/4 
 
 .42/^ 
 .445 
 .4 74 
 
 .2 3 4 
 .243 
 .2S4 
 
 ,599 
 
 .737 
 ,9/7 
 
 .641 
 .666 
 .693 
 
 ,242 
 ,2 3/ 
 .2 4/ 
 
 .402 
 .740 
 .920 
 
 
 3 
 
 6 
 
 .sua 
 
 .232 
 
 ./ S4 
 
 I.I 1 
 1.40 
 3.21 
 
 .449 
 .473 
 .72J- 
 
 .245 
 
 ,2 54 
 .273 
 
 /.I4 
 1.42 
 2,2 2 
 
 .474 
 .70/ 
 .754 
 
 .254 
 ,2 44 
 .2ff4 
 
 1.14 
 1.4 3 
 2.2 3 
 
 .498 
 .725 
 
 .7yo 
 
 .2 6-3. 
 •2 7 3 
 .294 
 
 ;:41 
 
 2,2 3 
 
 • 72/ 
 
 • 744 
 ,S09 
 
 ,2 72 
 ■281 
 ,305 
 
 1.14 
 1.4 3 
 2.23 
 
 ♦Resistance based upon 100 percent conductivity at 25 degrees C (77 degrees F), including two percent allowance ^r 
 spiral of strands and two percent allowance for spiral of conductors. For a temperature of 65 degrees C (149 degrees F) 
 these resistance values would be increased 15 percent. 
 
 **The inductance is in millihenries; the reactance and the impedance are in ohms. 
 
 The table values were derived from the equation L = 0.08047 + 0.741 Logw-JJ where R is the radius of conductor, D 
 the distance between centers of conductors expressed in the same terms as R, and L the inductance in. millihenries per mile 
 of each conductor. All values in the table are single-phase and based upon a single conductor one mile long. 
 
 inductance, as determined by the fundamental formu- 
 la, would thus tend to give values several percent 
 less than the actual when applied to three-conductor 
 cable calculations. On the other hand spiraling the 
 conductors of three conductor cables tends to increase 
 their reactance bv several percent. It may, therefore, be 
 
 assumed that the use of the fundamental formula in 
 the case of three-conductor cables give results ap- 
 proximately correct. Skin effect on the larger cables 
 will, however, tend to decrease the reactance slightly, 
 particularly at 60 cycles. 
 
C.IIU.E CHARACTERISTICS 
 
 125 
 
 I ' CAPACITANCE OF 3 CONDUCTOR CABLES 
 
 Formulas for determining the approximate capaci- 
 tance of three-conductor cables are cumbersome. They 
 give reasonably accurate results only in the case of 
 a homogeneous dielectric and in cases where the con- 
 ductors are small compared to the radius of the sheath. 
 They give inaccurate results in cases of large conduc- 
 tors closely spaced. Fig. 58* illustrates the various 
 
 MODEL 
 
 RECIPROCAL MODEL 
 
 FIG. 58 — REPRESENTATION OF CAPACITANCES OF A SYMMETRICAL 
 THREE-PHASE CABLE 
 
 capacitances of a three-conductor cable. Formulas 
 taken from Russel's "Alternating Currents" have been 
 combined and converted to common logarithms and 
 are given below. They were derived by the method 
 of images and on the assumption that the conductors 
 are round and symmetrically spaced with respect to 
 the axis of the sheath. 
 
 I 
 
 li' — d' + 
 
 C. 
 
 13.82 logi, g ^1 ^ . 
 
 ^ , {I73d_ ^ R'-d- \ 
 
 &.91 log,, \^ ^ X (^R' ^ K' d' + d'YA ) 
 
 C\..= —^ 
 
 R' — d' 
 13.82 log,, .R'd'r 
 
 X 0.179 X K. (70) 
 
 1.73'^ ., 
 
 R' 
 
 (R' + R' (P + d 
 
 "W 
 
 X 0.179 X K (71) 
 
 13-82 log,o ( — — 
 
 Where — 
 
 R = inside radius of sheath in centimeters (Fig. 59). 
 
 r = radius of conductor in centimeters. 
 
 d = distance between axis of conductor and axis of 
 sheath in centimeters. 
 
 K =: the dielectric constant. For impregnated paper in- 
 sulation it varies between 3 and 4; for varnished 
 cambric insulation it varies between 4 and 6; for 
 rubber insulation it varies between 4 and 9. 
 
 Ci =: capacitance in microfarads per mile between one 
 conductor and the other two conductors plus the 
 sheath. 
 
 Ci-, = mutual capacitance in microfarads per mile between 
 any two conductors. The capacitance to neutral is 
 twice this value. 
 
 C]2 is used in determining the capacitance for various 
 combinations or arrangements as explained below. 
 
 CAPACITANCE AND SUSCEPTANCE — TABLE XXVII 
 
 Table XXVII contains values for capacitance and 
 susceptance of three conductor paper insulated cable 
 for the various sizes of conductors and thicknesses of 
 insulation indicated. All values are based upon a value 
 for K of J.5 and, as indicated, a thickness of insulation 
 for the jacket the same as that surrounding each con- 
 
 *Reproduccd from Alexander Russel's "Alternating Cur- 
 rents." 
 
 ductor. The values were calculated by equations (70) 
 and (71). 
 
 The susceptance values given for 25 and 60 cycles 
 are to neutral. In calculating the voltage regulation 
 of circuits, it is general practice to calculate the regu- 
 lation on the basis of one conductor to neutral. The 
 susceptance between two of the conductors would be 
 half the table values to neutral. The values for suscep- 
 tance were calculated from the equation, — 
 
 Susceptance to neutral in micromhos = 2 ir f C 
 Thus No. o three-conductor cable with 7/64 and 
 7/64 insulation has a capacitance between conductors of 
 0.195 microfarads (0.39 microfarads to neutral). The 
 susceptance to neutral at 60 cycles therefore is, — 
 2 Tr 60 X 0-39 = 147 microfarads, as indicated by the 
 table. 
 
 INTER-RELATION OF CAPACITANCE OF THREE- 
 CONDUCTOR CABLES 
 
 The following equations for determining the effec- 
 tive capacitance for various arrangements of the three 
 conductors and the sheath are given in Russell's "Alter- 
 nating Currents." 
 
 Capacitance between I and .? = J/^ (G — Co) (72) 
 
 Capacitance between I and 2, 3=. % (Ci — Cu) .... (73) 
 Capacitance between I and S {3 and 3 insulated) = 
 (C, — U) (C, + zC») 
 
 (74) 
 
 Capacitance between i and S, 2 (3 insulated) ■= 
 (C\ — C«) (C. 4- C„) 
 
 G 
 
 Capacitance between I and S, 2, 3 ^ C, 
 
 Capacitance between S and i, 2, (3 insulated) = 
 2 (Ci — C«) (C. 4- ^ C.0 
 
 G 
 
 (75) 
 (76) 
 
 iyi) 
 
 Capacitance between I, S and 2, 3 ::= 2 (Ci + Ca) (78) 
 
 Capacitance between S and i, 2, 3 = 3 (C, + 2C1,) . . . (79) 
 
 It Q J 
 
 FIG. 59 — DIMENSIO-VS OF A SYMMETRICAL THREE-PHASE CABLE 
 
 Cj (76) may be measured in the ordinary way, 
 by reading the throw of a mirror galvanometer and 
 comparing with the throw given by a standard con- 
 denser. A further measurement of (78) or (79) will 
 give a simple equation to find C,2. For instance, if 
 measurements were taken of (78) and (79) and were 
 found to be: — 
 
126 
 
 CABLE CHARACTERISTICS 
 
 TABLE XXVII— CAPACITANCE AND SUSCEPTANCE PER MILE OF THREE CONDUCTOR 
 
 PAPER INSULATED CABLES 
 
 AREA 
 
 IN 
 
 CIRCULAR 
 
 MILS 
 
 B & S NO. 
 
 INSULATION THICKNESS IN e4THS OF AN INCH 
 
 3 pv 3 
 64 B^ 64 
 
 64 BY ^ 
 
 ^ BY ^ 
 
 ^ BY ^ 
 
 CAPACITANCE 
 
 SUSCEPTANCE 
 TO NEUTRAL 
 
 CAPACITANCE 
 
 SUSCEPTANCE 
 TO NEUTRAL 
 
 CAPACITANCE 
 
 SUSCEPTANCE 
 TO NEUTRAL 
 
 CAPACITANCE 
 
 SUSCEPTANCE 
 TO NEUTRAL 
 
 C 
 1 
 
 C 
 12 
 
 C 
 I&2 
 
 25 
 
 CYCLES 
 
 60 
 
 CYCLES 
 
 C 
 
 1 
 
 c 
 
 12 
 
 c 
 
 I&2 
 
 25 
 
 CYCLES 
 
 60 
 
 CYCLES 
 
 C 
 
 1 
 
 
 
 12 
 
 C 
 I&2 
 
 25 
 
 CYCLES 
 
 60 
 
 CYCLES 
 
 C 
 
 1 
 
 C 
 
 12 
 
 C 
 I&2 
 
 25 
 
 CYCLES 
 
 60 
 
 CYCLES 
 
 ■Soo ooo 
 ^So ooo 
 -^oo ooo 
 
 .680 
 .hi>7 
 .iSl 
 
 -.2/7 
 -.197 
 -.114 
 
 .448 
 .432 
 .42.S 
 
 /4/ 
 I3(, 
 133 
 
 JZO 
 
 .4/3 
 .S90 
 .S70 
 
 -.I7S 
 -.It9 
 -.159 
 
 ■394 
 .379 
 .344 
 
 IZ4 
 119 
 11-4 
 
 Z97 
 Z-Bi 
 274 
 
 ..S.S.S- 
 ..538 
 
 .517 
 
 -./S4 
 -.149 
 -.142 
 
 .354 
 .343 
 .329 
 
 III 
 /O8 
 /03 
 
 267 
 259 
 248 
 
 .SOS 
 .488 
 .4 75 
 
 -./3 7 
 -.130 
 -.IZS 
 
 .32/ 
 
 .309 
 .300 
 
 101 
 
 97 
 94 
 
 24i 
 233 
 224 
 
 sso ooo 
 300 ooo 
 zso ooo 
 
 .&10 
 ,6o6 
 .SfO 
 
 -.I7(. 
 -.171 
 
 .4/4 
 .391 
 .3 80 
 
 130 
 IZ3 
 /ZO 
 
 313 
 Z94 
 ZS6 
 
 .SbO 
 .S4S 
 
 .sie 
 
 -.IS8 
 -.153 
 -./42 
 
 .359 
 .34? 
 
 .330 
 
 113 
 
 no 
 
 104 
 
 270 
 Z49 
 
 .50 h 
 
 .490 
 .468 
 
 -.138 
 
 -.13 1 
 -./ZS 
 
 .322 
 .310 
 .294 
 
 / 1 
 97 
 93 
 
 242 
 234 
 223 
 
 .440 
 .446 
 .427 
 
 -.119 
 -.116 
 -•I09 
 
 .289 
 .28/ 
 .268 
 
 91 
 88 
 ?4 
 
 2/8 
 2/2 
 202 
 
 oooo 
 ooo 
 oo 
 
 .S70 
 .S3S 
 .513 
 
 -.IkO 
 -./47 
 -./40 
 
 .34.5 
 
 .34/ 
 .327 
 
 / II 
 107 
 103 
 
 ZkS 
 ZS7 
 Z4t 
 
 .soo 
 
 .A7S 
 .447 
 
 -./34 
 -./25 
 -.lit 
 
 .317 
 ,300 
 .ZSI 
 
 1 00 
 1% 
 
 239 
 224 
 ZIZ 
 
 .44c 
 .420 
 .398 
 
 -.IIS 
 -.107 
 -.101 
 
 .280 
 .262 
 
 .249 
 
 88 
 fl 
 
 211 
 I9S 
 IS! 
 
 .407 
 .384 
 .3 64 
 
 -.103 
 -.09S 
 -.088 
 
 .2 55 
 .239 
 .226 
 
 •go 
 75 
 7/ 
 
 /92 
 /SO 
 /70 
 
 z 
 
 .■>20 
 
 -./23 
 -.114 
 '.107 
 
 .308 
 .290 
 .Z(,3 
 
 97 
 %'3 
 
 232 
 ZI9 
 1 98 
 
 .422 
 .398 
 .3 73 
 
 -.107 
 -.099 
 -.091 
 
 .Zit 
 .Z1S 
 .232 
 
 83 
 78 
 
 199 
 /87 
 / 7,r 
 
 .3 74 
 .3.5 4 
 .3 3 2 
 
 -.090 
 -.o84 
 -.077 
 
 .2 3 2 
 .22/ 
 .203 
 
 7.3 
 69 
 64 
 
 I7S 
 1 67 
 /53 
 
 -342 
 .323 
 .30s 
 
 -.08/ 
 -.0 74 
 -.070 
 
 .Zll 
 .198 
 .187 
 
 66 
 42 
 59 
 
 141 
 
 3 
 
 6 
 
 .402 
 .378 
 .3-t2 
 
 -.101 
 -.100. 
 -.08/ 
 
 .ZSI 
 .23 9 
 .2/ / 
 
 Vs 
 44 
 
 IS9 
 /SO 
 /S9 
 
 .362 
 .330 
 .30/ 
 
 -.084 
 -.077 
 -.o43 
 
 .2/8 
 
 .203 
 
 ./S2 
 
 ■57 
 
 /hS 
 /S3 
 137 
 
 .0/4 
 .Z9S 
 .244 
 
 -.072 
 -.064 
 - OJ-4 
 
 .193 
 .ISO 
 ./40 
 
 6/ 
 57 
 SO 
 
 MS 
 136 
 /Zl 
 
 .284 
 .270 
 
 .239 
 
 -.06Z 
 -.059 
 -.OSO 
 
 .173 
 .164 
 .144 
 
 54 
 52 
 45 
 
 131 
 124 
 /OS 
 
 
 
 -^ BY -^ 
 64 °^ 64 
 
 ^ BY ^ 
 
 ^BY^ 
 
 JO. RV 10 
 64 BY g^ 
 
 CAPACITANCE 
 
 SUSCEPTANCE 
 TO NEUTRAL 
 
 CAPAblTANCE 
 
 SUSCEPTANCE 
 TO NEUTRAL 
 
 CAPACITANCE 
 
 SUSCEPTANCE 
 TO NEUTRAL 
 
 CAPACITANCE 
 
 SUSCEPTANCE 
 TO NEUTRAL 
 
 C 
 
 1 
 
 °I2 
 
 c 
 
 I&2 
 
 25 
 
 CYCLES 
 
 60 
 
 CYCLES 
 
 C 
 
 1 
 
 c 
 
 12 
 
 c 
 
 I&2 
 
 25 
 
 CYCLES 
 
 60 
 
 CYCLES 
 
 
 
 1 
 
 C 
 
 12 
 
 C 
 I&2 
 
 25 
 CYCLES 
 
 60 
 
 CYCLES 
 
 C 
 1 
 
 C 
 
 12 
 
 c 
 
 I&2 
 
 25 
 
 CYCLES 
 
 60 
 
 CYCLCS 
 
 Soo ooo 
 -4--50 ooo 
 ^oo ooo 
 
 .468 
 .454 
 .442 
 
 =:'/f| 
 
 -.Hi 
 
 .Z9i 
 .284 
 .27f 
 
 93 
 II 
 
 2 24 
 ZU 
 ZIO 
 
 .43.5- 
 
 .427 
 .■4IS 
 
 -.IIS 
 -.107 
 -,I0S 
 
 .27.^- 
 .247 
 .2 40 
 
 84 
 84 
 82 
 
 207 
 
 20/ 
 /96 
 
 -4/0 
 .405 
 .392 
 
 -./04 
 -./03 
 -.099 
 
 .257 
 .2 54 
 .245 
 
 8/ 
 79 
 77 
 
 /93 
 191 
 /84 
 
 .39 i 
 
 .3SO 
 .368 
 
 -.097 
 -.093 
 -.090 
 
 .244 
 .236 
 .229 
 
 77 
 74 
 72 
 
 /84 
 
 ','7% 
 
 3SO ooo 
 
 300 ooo 
 
 zso ooo 
 
 .42^ 
 .4/.5 
 .400 
 
 -.108 
 -.los 
 -.101 
 
 .247 
 .2 60 
 .ZSO 
 
 79 
 
 ZOI 
 I9t 
 
 /ss- 
 
 .348 
 .390 
 .370 
 
 -.099 
 -.o9t 
 ~.OS9 
 
 .248 
 .243 
 .22'? 
 
 78 
 74 
 72 
 
 IB7 
 J83 
 173 
 
 .380 
 .36S 
 .352 
 
 -.093 
 -.089 
 -.087 
 
 .2 3 4 
 .2 27 
 .2 19 
 
 74 
 
 'l'7'l 
 I6S 
 
 .358 
 .34? 
 .332 
 
 -.osi 
 
 -.OS 1 
 
 -.078 
 
 .222 
 .2/5 
 .20.S 
 
 70 
 68 
 <S5 
 
 '/tl 
 
 /ss 
 
 Oooo 
 ooo 
 oo 
 
 .380 
 .358 
 .33e 
 
 -.044 
 -.o84 
 -.080 
 
 .237 
 .222 
 .ZOS 
 
 7S 
 70 
 
 I7S 
 /48 
 /57 
 
 .3S4 
 .332 
 .3/3 
 
 -.OSS 
 -.079 
 -.07I 
 
 .2 2 
 .ZOS 
 .192 
 
 4-? 
 ^4 
 60 
 
 /66 
 ISS 
 MS 
 
 .334 
 .3/^ 
 .Z9S 
 
 -oih 
 
 -.0 7 3 
 -.067 
 
 ^OS 
 .194 
 ./SI 
 
 64 
 61 
 S7 
 
 ISS 
 146 
 136 
 
 .3/4 
 .294 
 .27S- 
 
 -073 
 -.066 
 
 -.061 
 
 ./94 
 .181 
 .169 
 
 6/ 
 57 
 53 
 
 146 
 136 
 127 
 
 1 
 z 
 
 .3 17 
 .Zft 
 .Z7-} 
 
 -.073 
 
 -.048 
 -.04 2 
 
 ./9S 
 ./S3 
 .170 
 
 61 
 SB 
 .54 
 
 /47 
 /38 
 /28 
 
 .Z43 
 .280 
 .264 
 
 -.OkS 
 
 -.oti 
 
 -.OS6 
 
 .179 
 
 .no 
 
 .IbO 
 
 Sb 
 
 St 
 
 I3S 
 IZS 
 IZI 
 
 .279 
 .ZhI 
 .247 
 
 -.0 4/ 
 -.0.54 
 -.0.52 
 
 .no 
 
 ,/S8 
 ./SO 
 
 S4 
 SO 
 
 47 
 
 1/3 
 
 .243 
 .24 7 
 .2 3 3 
 
 -.OS6 
 -.OSS 
 -.048 
 
 ./S9 
 ./S / 
 ./40 
 
 SO 
 'J7 
 44 
 
 120 
 114, 
 106 
 
 i, 
 
 .2 44 
 .2.50 
 .2 2/ 
 
 -.06 4 
 -.O.S3 
 -.0 4S 
 
 .l(,0 
 .151 
 ./33 
 
 .50 
 
 47 
 
 /2/ 
 11-4 
 too 
 
 .248 
 .233 
 
 .209 
 
 -.osz 
 -.048 
 -.04 / 
 
 ./SO 
 
 .MO 
 .IZS 
 
 4 7 
 44 
 39 
 
 113 
 loi 
 94 
 
 .232 
 .22 / 
 ./42 
 
 -.048 
 -.04S 
 -.037 
 
 .140 
 .133 
 .117 
 
 44 
 42 
 37 
 
 106 
 loo 
 88 
 
 .2 2 2 
 .2/0 
 .ISS 
 
 -.044 
 -.04 1 
 -.034 
 
 ./33 
 ./ZS 
 .// Z 
 
 42 
 
 39 
 3S 
 
 100 
 
 Vs 
 
 
 
 II RY II 
 
 64 ^^ 64 
 
 J2. BY -L?- 
 64 "^ 64 
 
 13 Rv 13 
 64 BY 64 
 
 14 RY 14 
 64 BY 54 
 
 CAPACITANCE 
 
 SUSCEPTANCE 
 TO NEUTRAL 
 
 CAPACITANCE 
 
 SUSCEPTANCE 
 TO NEUTRAL 
 
 CAPACITANCE 
 
 SUSCEPTANCE 
 TO NEUTRAL 
 
 CAPACITANCE 
 
 SUSCEPTANCE 
 TO NEUTRAL 
 
 
 
 1 
 
 c 
 
 12 
 
 C 
 I&2 
 
 25 
 
 CYCLES 
 
 60 
 
 CYCLES 
 
 C 
 
 1 
 
 c 
 
 12 
 
 c 
 
 I&2 
 
 25 
 
 CYCLES 
 
 60 
 
 CYCLES 
 
 C 
 
 1 
 
 °,2 
 
 C 
 i&2 
 
 26 
 
 CYCLES 
 
 60 
 
 CYCLES 
 
 C 
 
 1 
 
 C 
 
 12 
 
 
 I&2 
 
 25 
 
 CYCLES 
 
 60 
 
 CYCLES 
 
 •Soo ooo 
 -4 so ooo 
 ■^oo ooo 
 
 .37/ 
 
 .344 
 .3Si, 
 
 -.08^ 
 -.o«7 
 -.OSS 
 
 .230 
 .22.? 
 .220 
 
 72 
 1'9 
 
 /73 
 /44 
 
 ■.in 
 
 .33S 
 
 -.o9S 
 -.OSS 
 -.080 
 
 .220 
 
 .2/8 
 .209 
 
 i,4 
 
 48 
 44 
 
 / 64 
 /44 
 157 
 
 .343 
 .332 
 .324 
 
 -.OS2 
 -.0 7 8 
 -074 
 
 .2/2 
 .2 05 
 •ZOI 
 
 67 
 64 
 i63 
 
 /6O 
 
 /ss 
 
 /SZ 
 
 .329 
 .32 / 
 .3/0 
 
 -.078 
 -.075 
 -.07/ 
 
 .203 
 .198 
 ./90 
 
 64 
 62 
 <SO 
 
 ISS 
 
 149 
 143 
 
 3So ooo 
 
 300 ooo 
 
 2 so ooo 
 
 .340 
 .311 
 .3/4 
 
 -.080 
 -:o7 8 
 
 -;o72 
 
 .2/0 
 .Z0 3 
 .199 
 
 44 
 44 
 43 
 
 /S8 
 /S3 
 /So 
 
 .328- 
 .3/3 
 . 98 
 
 -.077 
 -.071 
 -.068 
 
 .202 
 • 192 
 ./S3 
 
 63 
 £0 
 
 152 
 14 S 
 /3 8 
 
 .3/7 
 .303 
 .288' 
 
 -.073 
 -04f 
 -.044 
 
 •I9S 
 .186 
 .176 
 
 61 
 S9 
 
 /47 
 
 /40 
 /33 
 
 .300 
 .290 
 .276 
 
 -.068 
 -.06S 
 -.06/ 
 
 ./84 
 .177 
 ./68 
 
 SS 
 S6 
 S3 
 
 139 
 /33 
 /2 7 
 
 oooo 
 ooo 
 oo 
 
 .302 
 .232 
 .247 
 
 -.Ob4 
 -,o&i 
 -,058 
 
 ./es 
 
 .171 
 .ItZ 
 
 .58- 
 
 .54 
 ~SI 
 
 MO 
 129 
 IZZ 
 
 .t3S 
 .27/ 
 .2SS 
 
 -obj 
 -.040 
 -.Oi4 
 
 ./74^ 
 .IS-t 
 
 SS 
 SZ 
 48 
 
 133 
 124 
 /Ik 
 
 .2 7* 
 .24/ 
 .24 7 
 
 -.04/ 
 — OSL 
 -.OS2 
 
 ■ 169 
 ./£S 
 
 ./SO 
 
 S3 
 SO 
 
 47 
 
 IZ7 
 119 
 / 13 
 
 .244 
 .2SI 
 .237 
 
 -.OS 6 
 -.05 3 
 -.04 8 
 
 .160 
 .IS? 
 .142 
 
 SO 
 ■48 
 45 
 
 /Zl 
 //S 
 /07 
 
 o 
 / 
 2 
 
 .250 
 .2 3 7 
 .2 2 2 
 
 -.0-5-3 
 
 -oso 
 -;o4.r 
 
 ./SI 
 .It 3 
 .133 
 
 48 
 45 
 42 
 
 "t 
 108 
 /OO 
 
 .228 
 .2/4 
 
 -,O.SO 
 
 -.047 
 -.044 
 
 • MS 
 .137 
 ./30 
 
 44. 
 '43 
 
 ■41 
 
 109 
 103 
 9S 
 
 .233 
 .220 
 .2oS 
 
 -.048 
 -.044 
 -.041 
 
 .MO 
 .132 
 .IZ4 
 
 ■44 
 42 
 39 
 
 /Oh 
 
 •222 
 
 .2/2 
 .199 
 
 -t044 
 -.042 
 -.039 
 
 .133 
 .127 
 .118 
 
 42 
 AO 
 37 
 
 / 00 
 
 ■4- 
 
 .2/2 
 .20/ 
 
 -.042 
 -034 
 -.033 
 
 ,rz7 
 
 .IZO 
 .107 
 
 ■40 
 3^ 
 
 96 
 9/ 
 
 .204 
 .192. 
 ./74 
 
 -.039 
 -.037 
 -03 1 
 
 • /Zl 
 ./M 
 ./oz 
 
 3S 
 3h 
 32 
 
 9/ 
 
 77 
 
 ./TS 
 ./86 
 ■ /he 
 
 -.037 
 -034 
 —.030 
 
 .lit 
 .099 
 
 3i 
 3S 
 31 
 
 88 
 83 
 
 75- 
 
 ■/90 
 ./SO 
 ./63 
 
 -.0 34 
 -1033 
 -.029 
 
 • 113 
 .106 
 .096 
 
 36 
 33 
 30 
 
 8S 
 
 BO 
 
 73 
 
 
 
 ^Bvi| 
 
 ^Bvil 
 
 20 RV 20 
 64 BY fi4 
 
 HbyII 
 
 CAPACITANCE 
 
 SUSCEPTANCE 
 TO NEUTRAL 
 
 CAPACITANCE 
 
 SUSCEPTANCE 
 TO NEUTRAL 
 
 CAPACITANCE 
 
 SUSCEPTANCE 
 TO NEUTRAL 
 
 CAPACITANCE 
 
 SUSCEPTANCE 
 TO NEUTRAL 
 
 C 
 
 1 
 
 c 
 
 12 
 
 c 
 
 I&2 
 
 25 
 
 CYCLES 
 
 60 
 
 CYCLES 
 
 C 
 
 1 
 
 c 
 
 12 
 
 c 
 
 I&2 
 
 25 
 
 CYCLES 
 
 60 
 
 CYCLES 
 
 C 
 
 1 
 
 °I2 
 
 C 
 I&2 
 
 25 
 
 CYCLES 
 
 60 
 
 CYCLES 
 
 C 
 
 1 
 
 C 
 12 
 
 C 
 I&2 
 
 25 
 
 CYCLES 
 
 60 
 
 CYCLES 
 
 soo ooo 
 ■4-So ooo 
 ^oo ooo 
 
 .30S 
 .302 
 .2?2 
 
 -07/ 
 -.0&9 
 -.oti 
 
 .ie9 
 ,iss 
 
 .179 
 
 S9 
 SB 
 v54. 
 
 /43 
 MO 
 /3S 
 
 .2S8 
 .282 
 .274 
 
 -.06S 
 -.042 
 -.oio 
 
 ./74 
 • 172 
 ./48 
 
 S4 
 S3 
 
 /33 
 I30 
 /27 
 
 .27^ 
 .249 
 
 .2i,0 
 
 -.04/ 
 
 -.058 
 -OSS 
 
 .ihi 
 
 .163 
 .157 
 
 53 
 
 5/ 
 50 
 
 1 27 
 IZ3 
 1/8 
 
 .263 
 .255 
 .249 
 
 -.05 7 
 -.OS4 
 -.053 
 
 ./ho 
 ./54 
 ./SI 
 
 SO 
 48 
 47 
 
 /Z 1 
 1/6 
 113 
 
 3SO ooo 
 
 3oo ooo 
 zso ooo 
 
 .282 
 .27/ 
 .240 
 
 -.obi 
 -.OS 4 
 -.OSS 
 
 .17/ 
 .liS 
 .IS7 
 
 S4 
 SZ 
 
 ■49 
 
 /Z9 
 I2S 
 
 lie 
 
 • 244 
 .254 
 .244 
 
 -OS 7 
 -.O.S4 
 
 -OS 1 
 
 ./tl 
 ,ISS 
 • M7 
 
 44 
 
 IZI 
 
 in 
 
 III 
 
 .252 
 • 24 2 
 .23 / 
 
 -OS 3 
 -oso 
 
 -.04S 
 
 ./SZ 
 ./46 
 .139 
 
 4I8 
 4^ 
 
 / /S 
 
 no 
 
 /OS 
 
 .240 
 .232 
 .2 2 2 
 
 -.04^ 
 -.047 
 -.044 
 
 .144 
 .139 
 ./33 
 
 4S 
 
 44 
 42 
 
 /OS 
 /OS 
 /oo 
 
 oooo 
 ooo 
 oo 
 
 .24 8 
 .234 
 .222 
 
 -osz 
 
 -.o4'S 
 -.044 
 
 ■ ISO 
 .141 
 .133 
 
 44 
 ■42 
 
 113 
 
 1 ok 
 J 00 
 
 .233 
 .22/ 
 .Z09 
 
 -.049 
 
 -.0 + 4 
 -.04/ 
 
 .Ml 
 .132 
 .IZS 
 
 44 
 
 106 
 100 
 
 94 
 
 .2 2 2 
 .2 11 
 .199 
 
 -,044 
 -.041 
 -.038 
 
 .133 
 ./26 
 .112 
 
 -f2 
 37 
 
 /oo 
 9S 
 
 .21 1 
 .zoo 
 
 .191 
 
 -.04/ 
 -.038 
 -.034 
 
 ./26 
 ./I9 
 .//3 
 
 40 
 38 
 36 
 
 96 
 90 
 
 s-s 
 
 I 
 
 .7.10 
 
 -.041 
 -.038 
 -.034 
 
 ./ZS 
 I'l'lt 
 
 3? 
 37 
 J4 
 
 1! 
 
 .199 
 ./S8 
 ./79 
 
 -.038 
 -.03 .5 
 -.032 
 
 . IIS 
 .III 
 .lOS 
 
 37 
 ¥3 
 
 89 
 
 .I8S 
 .lei 
 .170 
 
 -.034 
 -.0 34 
 -.030 
 
 .112 
 ./07 
 ./OO 
 
 3S 
 33 
 32 
 
 es 
 
 .ISO 
 .172 
 .163 
 
 -.030 
 -.030 
 -.029 
 
 .lOS 
 
 .10 1 
 .096 
 
 30 
 
 79 
 76 
 73 
 
 
 .179 
 .no 
 ./S-f 
 
 -.034 
 -.030 
 -.OZS 
 
 • /07 
 ./OO 
 
 .08? 
 
 34 
 32. 
 28 
 
 11 
 
 67 
 
 .171 
 ./42 
 ./47 
 
 -.03/ 
 -.027 
 -.024 
 
 .101 
 
 .09-4 
 
 32 
 30 
 27 
 
 74 
 1'4 
 
 ./42 
 ./.54 
 
 .141 
 
 -,02 8 
 -0 26 
 -.023 
 
 .095 
 .090 
 .082 
 
 30 
 ZS 
 26 
 
 1% 
 
 62 
 
 ./S5 
 .148 
 .136 
 
 —.024 
 -.025 
 -.02 1 
 
 .090 
 .084 
 .07? 
 
 Z8 
 
 rs 
 
 ^8 
 6S- 
 S9 
 
 Capacitance— The values in table for capacitance were derived by formulas in Alexander Russel's "Alternating Currents." 
 These values are as follows : — Cx values are the capacitance in microfarads per mile between one conductor and the other 
 tv70 conductors plus sheath. C1-2 values are the mutual capacitance in microfarads per mile between any two conductors. 
 The capacitance to neutral is twice these values. C12 values per mile are used in the application of Russel's formulas for 
 determining the capacitance corresponding to various arrangements of the three conductors and the sheath. 
 
 The Charging Current in amperes per mile for each conductor to neutral = susceptance in micromhos to neutral (taken 
 from Table) X volts to neutral X 10 ''. 
 
 Dielectric Constant — All of the above table values are based upon a value for the dielectric constant K of 3.5. For all 
 other values of K the table values will change in direct proportion. Values for K will usually be found between the follow- 
 ing limits; for impregnated paper 3.0 to 4.0; for varnished cambric 4.0 to 6.0 and for rubber 4.0 to 9.0. 
 
CABLE CHARACTERISTICS 
 
 127 
 
 TABLE XXVIII— THREE-PHASE CHARGING KV-A PER MILE OF THREE-PHASE CIRCUIT Of 
 
 THREE CONDUCTOR PAPER INSULATED CABLES 
 
 
 
 
 
 25 
 
 C Y C L 
 
 E S 
 
 
 
 
 
 AREA 
 
 IN 
 
 CIRCULAR 
 
 MILS 
 
 B & S NO. 
 
 CHARGING KVA PER MILE (EXPRESSED IN KV-A 3 PHASE) FOR PAPER INSULATED THREE 
 CONDUCTOR CABLES BASED UPON A VALUE FOR'K'OF 3.5 AND UPON A THICKNESS OF 
 INSULATION SURROUNDING THE CONDUCTORS AND OF THE JACKET INDICATED. 
 
 220 
 VOLTS 
 
 ■4 
 64 
 
 440 
 
 VOLTS 
 
 4 
 
 64 
 
 560 
 
 VOLTS 
 
 4 
 
 64 
 
 1100 
 
 VOLTS 
 -5. 
 64 
 
 2200 
 
 VOLTS 
 
 6 
 
 64 
 
 4400 
 VOLTS 
 
 6000 
 VOLTS 
 
 6600 
 VOLTS 
 
 6900 
 
 VOLTS 
 
 10 
 
 63 
 
 ^ 
 
 14 
 
 64 
 
 10 
 64 
 
 14 
 64 
 
 Soo ooo 
 ■^50 ooo 
 4_oo ooo 
 
 •00 600 
 ,00^75" 
 
 ,oosso 
 
 .03^0 
 .0230 
 .0220 
 
 .037i 
 .0360 
 ,034i 
 
 .134 
 .131 
 .I2S 
 
 .438 
 .4t9 
 .4SS 
 
 1.66 
 1.62 
 /.SS 
 
 2.76 
 .2.66 
 2.58 
 
 2.2i 
 2.22 
 2.IS 
 
 3.3S 
 3.2 2 
 3.13 
 
 2.7-? 
 2.70 
 2.tl 
 
 J.66 
 
 3.52 
 3.42 
 
 J40 ooo 
 
 •300 ooo 
 
 zso ooo 
 
 ,00537 
 ,O0S02 
 
 .021S 
 .0213 
 .0201 
 
 .0342 
 .0333 
 .OJIS 
 
 .122 
 
 .11 7 
 .113 
 
 .440 
 .42s 
 .40h 
 
 /~SI 
 1.4 7 
 1.39 
 
 2.5/ 
 2.44 
 2.33 
 
 2.08 
 
 Z.OI 
 1.90 
 
 3.04 
 2.91, 
 J.S3 
 
 2.S2 
 2.44 
 2.3/ 
 
 3.33 
 J.2 3 
 
 3.0 9 
 
 aooo 
 ooo 
 oo 
 
 
 .01 f 3 
 .018 2 
 .0170 
 
 .0303 
 .02-SS 
 .02i,(, 
 
 .lot 
 
 .Oil 
 M14 
 
 .387 
 .363 
 .343 
 
 1.3 3 
 1.24 
 I.I 6 
 
 2.11 
 2.04 
 1.10 
 
 1.79 
 
 /.72 
 /.6I 
 
 2.&S 
 2W8 
 
 2.3/ 
 
 Z.I8 
 2.09 
 1.16. 
 
 Z.90 
 2.71 
 2.SZ 
 
 o 
 
 k 
 
 ,00400 
 ,0027^ 
 ,O03S2 
 
 .OlkO 
 .OISI 
 .OMI 
 
 .02S0 
 .0236 
 ,0221 
 
 .0 883 
 .oS3i 
 .0 77S 
 
 .311 
 .300 
 .286 
 
 /.OS 
 IM4 
 .96S 
 
 1.79 
 /.63 
 /,S8 
 
 /.SI 
 
 2.18 
 2.0s 
 
 1.12 
 
 1.83 
 1.74 
 1.61 
 
 2.3 7 
 A2 3 
 2.09 
 
 3 
 
 1 
 
 .00333 
 .0030^ 
 .OOS.7^ 
 
 ,0133 
 .012^ 
 .0110 
 
 .0201 
 .0114 
 .0173 
 
 .0740 
 .0610 
 
 .otos 
 
 .Zil 
 •2S2 
 .218 
 
 .908 
 .8S0 
 .7SS 
 
 LSI 
 1.40 
 1.26 
 
 1.08 
 
 1.83 
 1.70 
 I.SZ 
 
 /.57 
 /.44 
 /.3/ 
 
 2,00 
 /.8S 
 /.66 
 
 
 2 5 CYCLES 
 
 10.000 
 
 VOLTS 
 
 i2. 
 64 
 
 IIOOO 
 VOLTS 
 
 13.200 
 VOLTS 
 
 16.500 
 VOLTS 
 
 14 
 
 64 
 
 20.000 
 VOLTS 
 
 22.000 
 VOLTS 
 
 18 
 
 64 
 
 26000 
 
 VOLTS 
 
 ^ 
 
 14 
 64 
 
 ^ 
 
 ^ 
 
 ^ 
 
 IS 
 64 
 
 18 
 64 
 
 F4 
 
 soo ooo 
 4 so ooo 
 
 4ooooo 
 
 
 ■g.3S 
 8.2 3 
 
 7.18 
 
 7.7 7 
 7..S0 
 7.26 
 
 1 2.00 
 11.80 
 II. So 
 
 I0.2S 
 
 /O.IO 
 
 9.7 S 
 
 I7.3S 
 16.80 
 /6.2S 
 
 2 3.6 
 2 3.3 
 22.4 
 
 22.0 
 Zl.h 
 21.2 
 
 2 6.6, 
 26.1 
 2S.6 
 
 34.£ 
 33.8 
 33.2 
 
 3 3.3 
 32.0 
 
 3 1.4 
 
 3S0 ooo 
 .300 ooo 
 
 2SOO0O 
 
 6.33 
 6.02 
 ^.82 
 
 7.6 2 
 
 7.2 7 
 7.oi 
 
 7.02 
 0.7S- 
 6.4 Z 
 
 10.1s 
 
 /0.4S 
 / o.ia 
 
 9.40 
 9.0s 
 8.52 
 
 IS.7S 
 
 IS. 20 
 14.40 
 
 21. b 
 20.8 
 
 19.6 
 
 20.4 
 ll.b 
 /8.4 
 
 24.6 
 2 3.7 
 2 2.2 
 
 J2.0 
 
 30.7 
 28.8 
 
 30.1 
 23.8 
 27.6 
 
 OOOO 
 OOO 
 OO 
 
 ^•2 3 
 -J.?2 
 
 6.3a 
 S.^0 
 
 6.0s 
 AS2 
 .5.4s 
 
 1.5 h 
 9.0S 
 8.3 i, 
 
 S.I 7 
 7.6S 
 7.30 
 
 I3.SS 
 I3.0S 
 I2.2S 
 
 18.8 
 17.6 
 76.8 
 
 /7.6 
 /6.8 
 /S.6 
 
 21.3 
 20.3 
 /S.S" 
 
 27.6 
 26.3 
 24.4 
 
 26.4 
 2S.I 
 2 3.2 
 
 o 
 
 t 
 
 2 
 
 -f.i2 
 -f.3 2 
 
 •S.S7 
 .5.21 
 .4.17 
 
 4.4 S 
 
 8.00 
 7.49 
 7.13 
 
 6.78- 
 6.43 
 6.26 
 
 11.40 
 /0.8S 
 
 /o.oS 
 
 'lt% 
 14.4 
 
 /4.8 
 14.0 
 13.2 
 
 /7.S 
 16.9 
 /J. 9 
 
 -^3.2 
 21.1 
 20.7 
 
 Z2.0 
 20.7 
 20.1 
 
 \ 
 
 3.S2 
 3.62 
 3.2 2 
 
 't.kO 
 
 •4.3<. 
 J.S7 
 
 4.3i> 
 4.00 
 3.6 3 
 
 6.hO 
 6.2 7 
 S.i7 
 
 S.92 
 S.S7 
 4.8 7 
 
 9.7 S 
 8.9S 
 
 S.IS 
 
 13.6, 
 12.8 
 II. 2 
 
 12.8 
 
 13.0 
 
 /0.8 
 
 /S.S 
 I4.S 
 /3.I 
 
 20.1 
 IB-S 
 16.9 
 
 18.8 
 /7.6 
 /i.3 
 
 60 CYCLES 
 
 
 220 
 
 VOLTS 
 
 A 
 
 64 
 
 440 
 
 VOLTS 
 
 4 
 
 64 
 
 550 
 
 VOLTS 
 
 4 
 
 64 
 
 1100 
 
 VOLTS 
 
 2200 
 VOLTS 
 
 4400 
 VOLTS 
 
 6000 
 VOLTS 
 
 6600 
 VOLTS 
 
 6900 
 
 VOLTS 
 
 JO. 
 
 64 
 
 a 
 
 ^ 
 
 10 
 64 
 
 J4. 
 64 
 
 soo ooo 
 
 ASo ooo 
 ■4oo ooo 
 
 .Oi43 
 .0/38 
 .0132 
 
 .OS7f 
 .OSS4 
 ,OS30 
 
 ,o8SS 
 .O830 
 
 .323 
 .313 
 .301 
 
 1.17 
 
 /.I3 
 1.01 
 
 4.00 
 
 3.8 S 
 
 3.79 
 
 6.sa 
 6.40 
 6.20 
 
 S.I 3 
 
 8.00 
 
 7.75 
 7.53 
 
 6.65- 
 6.4 8 
 6.2 2 
 
 1:41 
 
 8.2 2 
 
 3SO ooo 
 3 00 000 
 ZSOOOQ 
 
 ,013 1 
 .0127 
 ,0120 
 
 .OS23 
 .0510 
 .0483 
 
 .OS18 
 .0718 
 .076S 
 
 •292 
 .283 
 
 .270 
 
 1.0 S 
 
 1.02 
 .97S 
 
 3.S4 
 J.3S 
 
 S.98 
 S.80 
 s.ss 
 
 4.98 
 4.7 8 
 4.S6 
 
 7.2.r 
 
 7.0 s 
 
 6.75 
 
 6.07 
 5.8 
 5.5 2 
 
 7.9 3 
 7.70 
 
 7.37 
 
 OOOO 
 OOO 
 
 oo 
 
 ,OJ IS 
 .0/0^ 
 .OIOZ 
 
 .0463 
 .0437 
 .0410 
 
 .072 S 
 
 .oi93 
 .01:43 
 
 .2SS 
 .240 
 .226 
 
 .9ZS 
 .870 
 .820 
 
 -3.2 2 
 3.00 
 2.80 
 
 ^.24 
 -f.e8 
 
 4.SS 
 
 4.33 
 f.'8% 
 
 6.3S 
 S.92 
 S.3Z 
 
 5.2 6 
 S.OO 
 4.6S 
 
 6.93 
 
 ifol 
 
 o 
 
 i 
 
 ■ OOft 
 
 .0090 
 
 .DOS') 
 
 .038S 
 ,03k2 
 .033Cf 
 
 .Ot02 
 .OS66 
 .OS30 
 
 .212 
 .202 
 ./ SS 
 
 .768 
 .720 
 .680 
 
 2.61 
 3.48 
 2.3 -»■ 
 
 4.30 
 4.08 
 3. so 
 
 3.S9 
 3.4 4 
 3.19 
 
 5.2 2 
 4.1s 
 ^.6 
 
 4.3S 
 
 4.17 
 3.87 
 
 S.70 
 
 i^ol 
 
 
 ,0080 
 .007-? 
 ,OOh6 
 
 .O320 
 • Qift 
 .026S 
 
 .OSOO 
 .OliiS 
 .04 IS 
 
 .176 
 .US 
 
 .147 
 
 .6 3 2 
 .600 
 ..S22 
 
 2.18 
 2.0s 
 1.82 
 
 3.SI 
 
 3.37 
 3.0s 
 
 3.0 s 
 2.87 
 .?.6 2 
 
 4.35 
 4.0 8 
 3.70 
 
 3.7 
 3.48 
 3.17 
 
 t4i 
 4.0 s 
 
 
 60 CYCLES 
 
 10,000 
 
 VOLTS 
 12 
 64 
 
 11.000 
 
 VOLTS 
 
 13.200 
 VOLTS 
 
 16.500 
 VOLTS 
 
 14 
 
 64 
 
 20.000 
 VOLTS 
 
 22.000 
 VOLTS 
 
 IB. 
 
 64 
 
 25000 
 VOLTS 
 
 12 
 64 
 
 '^ 
 
 ^ 
 
 a 
 
 il 
 
 18 
 64 
 
 a 
 
 15 
 
 tSoo ooo 
 
 4SO ooo 
 400 ooo 
 
 /&.7 
 /t.S 
 /S.« 
 
 20.1 
 
 11.8 
 11.0 
 
 I9.S 
 18.1 
 17.3 
 
 2 8.1 
 2g.S 
 27.2 
 
 2 4.9 
 24.4 
 23.S 
 
 4I.S 
 40.3 
 38.8 
 
 57.3 
 
 S6.0 
 S4.0 
 
 5 3.3 
 52.0 
 S0.9 
 
 6.f.2 
 62.7 
 6/. 3 
 
 83.3 
 8/.5- 
 7y.5 
 
 71.S 
 77.0 
 74.0 
 
 3£0 ooo 
 
 3oo ooo 
 
 Z30 ooo 
 
 IS.3 
 
 /3.1 
 
 18.4 
 
 n.i, 
 
 lt.7 
 
 lt.8 
 lh.1 
 15.4 
 
 2t^ 
 2S.2 
 Z4.0 
 
 2 2.4 
 2 1.8 
 20.S 
 
 3 7.6 
 3 6.0 
 34.S 
 
 i'i:°3 
 
 48.S 
 4 6.8 
 44.3 
 
 S6.S 
 5-3.6 
 
 75.8 
 73.3 
 
 a.9.s 
 
 72.2 
 69.0 
 65.7 
 
 OOOO 
 
 ooo 
 
 OO 
 
 /3.1 
 I2.S 
 1 1.7 
 
 It.l 
 is.o 
 
 14.1 
 
 14.7 
 I3.f 
 13.0 
 
 2 3.2 
 Zl.h 
 20.2 
 
 I9.i, 
 18.4 
 17.4 
 
 32.e 
 31.2 
 29.0 
 
 4S.3 
 4Z.S 
 .40.0 
 
 42.5 
 40.0 
 37.6 
 
 5-/.2 
 48.3 
 'fS.4 
 
 6 6.S 
 62.7 
 S9.0 
 
 6 2.7 
 S9.S 
 SS.7 
 
 O 
 
 i 
 
 II. 
 10.4 
 
 a.es 
 
 13.2 
 12. S 
 11.^ 
 
 12.1 
 
 n.i, 
 
 10.8 
 
 18.1 
 17.1 
 
 n.o 
 
 16.3 
 /S.S 
 14.9 
 
 27.2 
 Z6.0 
 
 24.1 
 
 37.1 
 
 3S.6 
 
 34.S 
 
 35.7 
 33.^ 
 
 3/.7 
 
 43.0 
 40~S 
 3 8.2 
 
 J 6.8 
 S2.S 
 41.S 
 
 S3.3 
 S0.7 
 47.6. 
 
 i 
 
 ".IS 
 
 8.6 S 
 
 7.7 i^ 
 
 II.O 
 
 1 0.3 
 9.7 
 8.8 
 
 IS.8 
 IS.O 
 
 /3.-( 
 
 14.1 
 /3.Z 
 ll.-J 
 
 23.0 
 
 2 1.7 
 11.8 
 
 32.4 
 30.4 
 2 6.1 
 
 30.4 
 28 A 
 2S.6 
 
 36.7 
 34.3 
 30.9 
 
 47.6 
 44.S 
 40.2 
 
 4ia 
 
 42.6 
 3 8.f 
 
 The values in Table XXVIII are based upon a value for the dieleclric constant K of 3.5. For all other values of K the 
 table values will change in direct proportion. Values for K will usually be found between the following limits; for impreg- 
 nated paper 3.0 to 4.0; for varnished cambric 40 to 6.0 and for rubber 4.0 to 9.0. 
 
 2 Ci-j- 2 Ca = 0.410 mf. per mile (78) 
 
 And 5 C, + 6 C12 = 0.450 mf. per mile (79) 
 
 Therefore d = o.z6 mf. per mile 
 C,2 =: — 0.055 mf per mile 
 
 Numerical Examples — From Table XXVII for a 
 250000 circ. mil., three-conductor cable having a band 
 of insulation surrounding each conductor of 16/64 of 
 an inch and an insulation jacket surorunding all three 
 conductors of the same thickness, the following values 
 are obtained : — 
 
 Cx = 0.260 mf. per mile. 
 
 Cii = — 0.055 '"/. /"''' "I'l^^- 
 Then, in the order in which the capacitance increases, — 
 
 Capacitance between I and 2 = 0.157 »'/■ per mile (72) 
 
 Capacitance between i and 2, 3 := 0.210 mf. per mile. . (73) 
 Capacitance betv.een I and S (2 and 3 insulated) = 
 
 0.230 mf. per mile (74) 
 
 Capacitance between I and S, 2 (3 insulated) = 0.248 
 
 mf. per mile (75) 
 
 Capacitance between I and S, 2, 3 ^ 0.260 mf. per 
 
 mile (76) 
 
 Capacitance between S and I, 2 (3 insluated) = 0.363 
 
 mf. per mite (77) 
 
128 
 
 CABLE CHARACTERISTICS 
 
 Capacitance between I, S and 3, 3 = 0.410 mf. per 
 
 mile (78) 
 
 Capacitance between S and i, 2, 3 =: 0.450 mf. per 
 
 mile (79) 
 
 COMPARISON OF CALCULATED CAPACITANCE WITH 
 TEST RESULTS 
 
 The difference between measured results of capaci- 
 tance and the results calculated by the above formulas 
 are given in Fig. 60. It will be seen that in all cases 
 these calculated results are less than the corresponding 
 test results, the discrepancy being greater as the con- 
 ductor becomes larger and the separation less. The dif- 
 ferences vary from zero to as much as eleven percent 
 for the largest cable, at the minimum spacing shown. 
 The discrepancy is greatest with the minimum thickness 
 of insulation. Since such cables would be used only 
 for low-voltage service, the charging current would be 
 small and consequently this error would probably be of 
 little importance. For 6600 volt cables the results by 
 the formula would seem to be approximately five per- 
 cent too low. 
 
 4 6 
 
 6 7 8 9 10 I r 12 13 14 15 16 17 18 19 20 
 THICKNESS OF INSULATION SURROUNDING CONDUCTOR AND OF 
 JACKET IN 64THS OF AN INCH 
 
 FIG. 60 — COMPARISON OF CALCULATED AND MEASURED CAPACITANCES 
 
 Tests made on three conductor paper insulated cables, K = 3.5. 
 
 The cause of the discrepancy between the formula 
 and test results is as follows: — In order to obtain a 
 mathematical solution, Russell found it necessary to 
 make certain approximations to the true physical condi- 
 tions. Thus the resulting mathematical formula can- 
 not give exact results. The approximation made by 
 Russell is very close to the actual physical fact where 
 the conductors are small compared with the insulation 
 thickness, but it is not very close where the conductors 
 are large compared with the insulation. 
 
 CHARGING KV-A TABLE XXVIII 
 
 Table XXVIII contains values for charging cur- 
 rent (expressed in kv-a, three-phase) for three-conduc- 
 tor paper insulated cables, both 25 and 60 cycles, based 
 upon a value for K of 3.5. For other values of K, the 
 table values would vary in proportion. For other 
 thicknesses of insulation, the kv-a values would vary as 
 the susceptance values corresponding to the thickness of 
 insulation (See Table XXVII). In some cases, such 
 for instance, as grounded neutral systems, the thickness 
 
 of insulation of the jacket may be less than that sur- 
 rounding the conductors. In such cases it might be de- 
 sirable to calculate the susceptance and charging cur- 
 rent, if accurate results were desired. The values for 
 charging current corresponding to two thicknesses of 
 insulation are included for some of the commonly em- 
 ployed transmission voltages. 
 
 These kv-a values were calculated by using the 
 values for susceptance in Table XXVII which, in turn, 
 were derived from the capacitance in the same table ob- 
 tained by formulae (70) and (71). Thus a 350000 
 circ. mil cable with 10/64 and 10/64 paper insulation 
 has a 60 cycle susceptance to neutral of 167 micromhos 
 per mile. Since the charging current in amperes to 
 neutral equals the susceptance to neutral X volts to 
 neutral X lo"* and assuming 6600 volts, three-phase 
 between conductors, we have: — 
 
 6600 
 
 167 X -^-rr^X 10-* = 0.637 amperes to neutral. 
 
 Charging kv-a — 0.637 X 3815 X 3 = 7-25 kv-a, 
 as indicated in Table XXVIII. 
 
 VALUES FOR K 
 
 The capacitance of any cable depends upon the di- 
 electric constant of the insulating material and a dimen- 
 sion term or form factor. The dielectric constant 
 should be determined from actual cables and not from 
 samples of material. The usual range in value for K 
 is given below. 
 
 Value of K 
 
 Impregnated Paper 3.0 to 4.0 
 
 Varnish Cambric 4.0 to 6.0 
 
 Rubber 4.0 to 9.0 
 
 All values in Tables XXVII and XXVIII are based 
 upon a value of K of 3.5. For all other values of K 
 all table values will vary in the same proportion as their 
 K values. The actual value of permittivity of most 
 paper insulation runs about ten percent less than the 
 value 3.5 which has been used in calculating the accom- 
 panying table values. The true alternating-current ca- 
 pacitance is always considerably lower than the capaci- 
 tance measured with ballistic galvanometer. 
 
 REFERENCES 
 
 "Electric Power Conductors," by W. S. Del Mar. 
 "Electric Cables," by Coylc and Howe, London, England. 
 "The Heating of Cables with Current," by Melsoni & Booth. 
 Journal I. E. E., Vol. 47 — 191 1. 
 
 "Current and Rating of Electric Cables," by Atkinson S.- 
 Fisher, Trans. A. I. E. E., 1913, p. 325. 
 
 "The Heating of Cables Carrying Current," bv Dushman. 
 Trans. A. I. E. E., 1913, p. 333. 
 
 "Effect of Moisture in the Earth on Temperature of Under- 
 ground Cables," by Imlay, Trans. A. I. E. E., 1915, Part I, p. 
 2^3- 
 
 "Temperature Rise of Insulated Lead Covered Cables," by 
 Richard A. Powell, Trans. A. I. E. E., 1916, Part II, p. 1017. 
 
 "The Restoration of Service After a Necessary Interrup- 
 tion," by Rickets, Trans. A. I. E. E., 1916, Part II, p. 635. 
 
 "The Dielectric Field in an Electric Power Cable," by 
 Atkinson, Trans. A. I. E. E., June 1919. 
 
 "The Current Carrying Capacity of Lead Covered Cables," 
 by Atkinson, Journal A. I. E. E., Sept. 1920. 
 
CHAPTER XIV 
 
 SYNCHRONOUS MOTORS AND CONDENSERS FOR 
 
 POWER-FACTOR IMPROVEMENT 
 
 BEFORE discussing the employment of syn- 
 chronous machinery for improving the power- 
 factor of circuits, it may be desirable to review 
 how a change in power-factor affects the generators 
 supplying the current. 
 
 Fig. 6i shows the effect of in-phase, lagging and 
 leading components of armature current upon the field 
 strength of generators*. A single-coil armature is il- 
 lustrated as revolving between the north and south 
 poles of a bipolar alternator. The coil is shown in 
 four positions 90 degrees apart, corresponding to one 
 complete revolution of the armature coil. The direc- 
 tion of the field flux is assumed to be constant as in- 
 dicated by the arrows on the field poles of each illus- 
 tration. In addition to this field flux, when current 
 flows through the armature coil another magnetic flux 
 is set up, magnetizing the iron in the armature in a di- 
 rection at right angles to the plane of the armature coil. 
 This will be referred to as armature flux. 
 
 This armature flux varies with the armature cur- 
 rent, being zero in a single-phase generator when no 
 armature current flows, and reaching a maximum 
 when full armature current flows. It changes in direc- 
 tion relative to the field flux as the phase angle of the 
 armature current changes. 
 
 The revolving armature coil generates an alternat- 
 ing voltage the graph of which follows closely a sine 
 wave, as shown in Fig. 61. When it occupies a verti- 
 cal plane marked start no voltage is generated, for the 
 reason that the instantaneous travel of the coil, is 
 parallel with the field flux.** As the coil moves for- 
 ward in a clockwise direction, the field enclosed by 
 the armature coil decreases; at first slowly but then 
 more rapidly until the rate of change of flux through 
 the coil becomes a maximum when the coil has turned 
 90 degrees, at which instant the voltage generated be- 
 comes a maximum. As the horizontal position is passed 
 the voltage decreases until it again reaches zero when 
 the coil has traveled 180 degrees or occupies again a 
 vertical plane. As the travel continues the voltage 
 again starts to increase but since the motion of the coil 
 
 ♦For a more detailed discussion of this subject the reader 
 is referred to excellent articles by F. D. Newbury in the 
 Electric Journal of April 1918, "Armature Reaction of Poly- 
 phase Alternators" ; and of July 1918. "Variation of Alternator 
 Excitation with Load". 
 
 **For the sake of simplicity this and the following state- 
 ments are based upon the assumption that armature reaction 
 does not shift the position of the field flux. Actually, under 
 load, the armature reaction causes the position of the field flux 
 to be shifted toward one of the pole tips, so that the position 
 of the armature coil is not quite vertical at the instant of zero 
 voltage in the coil. 
 
 relative to the fixed magnetic field is reversed the volt- 
 age in the coil builds up in the reverse direction dur- 
 ing the second half of the revolution. When the coil 
 has reached the two 270 degree position the voltage 
 has again become maximum but in the opposite direc- 
 tion to that when the coil occupied the position of 
 90 degrees. When the coil returns to its original posi- 
 tion at the start the voltage has again dropped to zero, 
 thus completing one cycle. » 
 
 If the current flowing through this armature coil 
 is in phase with the voltage, it will produce cross mag- 
 netization in the armature core, in a vertical direction, 
 as indicated by the arrows at the 90 and 270 degree 
 positions. The cross magnetization neither opposes 
 nor adds to the field flux at low loads and therefore has 
 comparatively little influence on the field flux. At 
 heavy loads, however, this cross magnetization has con- 
 siderable demagnetizing effect, due to the shift in ro- 
 tor position resulting from the shifting of the field flux 
 at heavy loads. 
 
 If the armature is carrying lagging current, this 
 current will tend to magnetize the armature core in 
 such a direction as to oppose the field flux. This ac- 
 tion is shown by the middle row of illustrations of 
 Fig. 61. Under these illustrations is shown a current 
 wave lagging 90 degrees representing the component 
 of current required to magnetize transformers, induc- 
 tion motors, etc. When the lagging component of cur- 
 rent reaches its maximum value the armature coil will 
 occupy a vertical position (position marked start, 180 
 degrees and 360 degrees) and in this position the arma- 
 ture flux will directly oppose the field flux, as indicated 
 by the arrows. The result is to reduce the flux thread- 
 ing the armature coil and thus cause a lowering of the 
 voltage. This lagging current encounters resistance 
 and a relatively much greater reactance, each of which 
 consumes a component of the induced voltage, as 
 shown in Fig. 62. When the armature current is lag- 
 ging, the voltage induced by armature inductance is in 
 such a direction as to subtract from the induced volt- 
 age, and thus the voltage is still further lowered, as a 
 result of the armature self induction. In order to 
 bring the voltage back to its normal value it will be 
 necessary to increase the field flux by increasing the 
 field current. Generators are now usually designed of 
 sufficient field capacity to compensate for lagging 
 loads of 80 per cent power-factor. 
 
 If the armature is carrying a leading current this 
 leading component will tend to magnetize the armature 
 core in such a direction as to add to the field flux. 
 
130 
 
 SYNCHRONOUS MOTORS AND CONDENSERS FOR POWER-FACTOR CORRECTION 
 
 This action is shown by the bottom row of illustrations 
 of Fig. 6i. Under these illustrations is shown a current 
 wave leading the voltage wave by 90 degrees. When 
 the leading component of current reaches its maximum 
 values, the armature coil will again occupy vertical 
 positions, but the armature flux will add to that of the 
 field flux, as indicated by the arrow. The resulting 
 flux threading the armature coil is thus increased caus- 
 ing a rise in voltage. This leading current flowing 
 through the generator armature encounters resistance 
 and a relatively much greater reactance, each of which 
 consumes a component of the induced voltage, as 
 shown in Fig. 62. When the armature current is lead- 
 
 OROSS MAQNETIZINQ EFFECT OF IN-PHASE ARMATURE CURRENT 
 
 laUCH U OONSUMCO ST INOANOESCCNT LAMPt' 
 
 KM 5iB 53 53 EiS 
 
 SUSTRACTIVE EFFECT OF A LAGGING ARMATURE CURRENT 
 
 '■UCM M THE MAONrtlliNa CUBOCMT Of TDAMftTOOMIOS AND INDUCTtOH HOTOMi 
 
 Eia g'S K:3 5 3 E:S 
 
 ADDITIVE EFFECT OF A LEADING ARMATURE CURRENT 
 
 itUCH U THI CHUtOlNO OunnCNT OF TRANSMISSION UNU (J* OVM I>OITEX> STN MOTDBS 
 
 I I I 
 
 KiS 5S E:B BS E:S 
 
 FIG. 61- 
 
 -EFFECT OF ARMATURE CURRENT UPON FIELD EXCITATION OF 
 ALTERNATING-CURRENT GENERATORS 
 
 ing, the voltage induced by armature inductance is in 
 such a direction as to add to the induced voltage und 
 thus the voltage at the alternator terminals is .still 
 further increased as the result of armature self-induc- 
 tion. In order to reduce the voltage to its normal 
 value it is necessary to decrease the field flux by de- 
 creasing the field current. 
 
 With alternators of high reaction the magnetizing 
 or de-magnetizing effect of leading or lagging current 
 will be greater than in cases where the armature reac- 
 tion is low. For instance if the alternator is so de- 
 signed that the ampere turns of the armature at full 
 armature current are small compared to its field am- 
 pere turns, the voltage of such a machine would be less 
 disturbed with a change in power-factor of the arma- 
 
 ture current than in an alternator having armature 
 ampere turns large compared with its field ampere 
 turns. 
 
 Modern alternators are of such design that when 
 carrying rated lagging current at zero power-factor 
 they require approximately 200 to 250 percent of their 
 no-load field-current and when carrying rated leading 
 current at zero power-factor they require approximate- 
 ly — 15 to -|-I5 percent of their no-load field current. 
 Thus with lagging armature current the iron will be 
 worked at a considerable higher point on the satura- 
 tion curve and the heating of the field coils will in- 
 crease because of the greater field current required. 
 
 The voltage diagrams of Fig. 62 are intended to 
 show only the effect of armature resistance and arma- 
 ture reactance upon voltage variation. Voltage regu- 
 
 FIG. 62 — VECTORS ILLUSTRATING THE EFFECT OF ARMATURE REACT- 
 ANCE AND RESISTANCE UPON THE TERMINAL VOLTAGE FOR IN-PHASE, 
 LEADING AND LAGGING CURRENTS 
 
 lation is the combined effect of armature impedance 
 and armature reaction. Turbogenerators have, for 
 instance, very low armature reactance but their arma- 
 ture reaction is higher, so that the resulting voltage 
 regulation may not be materially different from that of 
 a machine with double the armature reactance. 
 Under normal operation armature reaction is a more 
 potent factor in determining the characteristics of a 
 generator than armature reactance. In the case of a 
 generator with a short circuit ratio of unity, this total 
 reactive effect may be due, 15 percent to armature re- 
 actance and 85 percent to armature reaction. 
 
 For the case illustrated by V\g. 62 the field flux 
 corresponds to the induced voltage indicated, but the 
 field current does not. The field current corresponds 
 to a value obtained by substituting the full synchron- 
 ous impedance drop for that indicated. 
 
SyNCHKONOUS MOTORS AND CONDENSEKS I'OK POWER-FACTOR CORRECTION 
 
 I3« 
 
 SYNCHRONOUS CONDENSERS AND I'HASE MODIFIERS 
 
 The term "synchronous condenser" applies to a 
 synchronous machine for raising the power- factor of 
 circuits. It is simply floated on the circuit with its 
 fields over excited so as to introduce into the circuit a 
 leading current. Such machines are usually not 
 intended to carry a mechanical load. When this dou- 
 ble duty is required they are referred to as synchron- 
 ous motors for operation at leading power-factor. 
 On long transmission circuits, where synchronous con- 
 densers are used in parallel with the load for varying 
 the power-factor, thereby controlling the transmission 
 voltage, it is sometimes necessary to operate them with 
 under excited fields at periods of lightloads. They are 
 then no longer synchronous condensers but strictly 
 speaking become synchronous reactors. 
 
 Whether synchronous motors for operation at 
 leading power-factor, synchronous condensers or syn- 
 chronous reactors be used they virtually do the same 
 thing, that is; their function is to change the power- 
 factor of the load by changing the phase angle between 
 the armature current and the terminal voltage. They 
 
 TABLE R— SYNCHRONOUS CONDENSER LOSSES 
 
 Kv-a 
 
 Loss (Kw) 
 
 Kv-a 
 
 Loss (Kw) 
 
 100 
 
 12 
 
 3500 
 
 180 
 
 200 
 
 18 
 
 5000 
 
 220 
 
 300 
 
 22 
 
 7500 
 
 320 
 
 500 
 
 3-' 
 
 lOOOO 
 
 420 
 
 750 
 
 47 
 
 15000 
 
 620 
 
 1000 
 
 55 
 
 20000 
 
 820 
 
 1500 
 
 70 
 
 25000 
 
 1000 
 
 2000 
 
 IJO 
 
 35000 
 
 1400 
 
 2500 
 
 130 
 
 50000 
 
 2000 
 
 are, therefore, sometimes referred to as "phase modi- 
 fiers." This latter name seems more appropriate when 
 the machine is to be operated both leading and lagging, 
 as when used for voltage control of long transmission 
 lines. 
 
 Rating — Sjmchronous condensers as regularly 
 built may be operated at from 30 to 40 percent of their 
 rating lagging, depending upon the individual design. 
 Larger lagging loads result in unstable operation on 
 account of the weakened field. Phase modifiers can 
 be designed to operate at full rating, both leading and 
 lagging, but they are larger, require larger exciters, 
 have a greater loss and cost 15 to 20 percent more 
 than standard condensers. 
 
 Starting — Condensers are furnished with squir- 
 rel-cage damper windings, to prevent hunting, which 
 also provides a starting torque of approximately 30 
 percent of normal running torque. They have a pull- 
 in torque of around 15 percent of running torque. 
 The line current at starting varies from 50 to 100 per- 
 cent of normal. The larger units are sometimes 
 equipped for forced oil lubrication, which raises the 
 rotor sufficiently to permit of oil entering the bearing, 
 thus reducing the starting current. 
 
 Mechanical Load — Synchronous condensers are 
 generally built for high speeds and equipped with 
 shafts of small diameter. If they are to be used to 
 transmit some mechanical power it may be necessary 
 to equip them with larger shafts and bearings, particu- 
 larly if belted rather than direct connected. If a 
 phase modifier is to furnish mechanical energy and at 
 the same time to operate lagging at times of light load 
 for the purpose of holding down the voltage on an un- 
 loaded transmission line there may be danger of the 
 machine falling out of step, if a heavy mechanical load 
 occurs when the machine is operating with a weak 
 field. 
 
 Losses — At rated full load leading power-factor 
 the total losses, including those of the exciter, will vary 
 from approximately 12 percent for the smallest capaci- 
 ty to approximately four percent for the larger capaci- 
 ty 60 cycle synchronous condensers. The approximate 
 
 76 1 00 1 26 
 
 FIELD AMPERES 
 
 FIG. 63— V-CUKVES OF A PHASE MODIFIER 
 
 values given in Table R may be of service for prelimin- 
 ary purposes. 
 
 "V" Curves — The familiar V curves shown in 
 Fig. 63 serve to give some idea of the variation in field 
 current for a certain phase modifier when operating 
 between full load lagging and full load leading kv-a.* 
 For this particular machine the excitation must be in- 
 creased from 112 amperes at no load minimum input 
 or unity power-factor to 155 amperes at full kv-a out- 
 put leading or a range of 1.4 to i in. field excitation. 
 For operation between full lagging and full leading, 
 with no mechanical work done, the range of excitation 
 is from 67 to 155 or 2.3 to i. 
 
 Generators as Condensers — Ordinary alternators 
 may be employed as synchronous condensers or syn- 
 chronous motors by making proper changes in their 
 field poles and windings to render them self-starting 
 
 ♦These curves have been reproduced from H. B. Dwight's 
 book "Constant Voltage Transmission". 
 
13^ 
 
 SYNCHRONOUS MOTORS AND CONDENSERS FOR POWRR-fACTOR CORRECTION 
 
 and safely insulated against voltages induced in the 
 field when starting. 
 
 Where transmission lines feed into a city net work 
 and a steam turbine generator station is available these 
 generating units can serve as synchronous condensers 
 by supplying just enough steam to supply their losses 
 and keep the turbine cool. When operated in this way 
 they make a reliable standby to take the important load 
 quickly in case of trouble on a transmiss>on line. 
 
 Location for Condensers — The nearer the center 
 of load that the improvement in power-factor is made 
 the better, as thereby the greatest gain in regulation,, 
 greatest saving in conductors and apparatus are made 
 since distribution lines, transformers, transmission 
 lines and generators will all be benefited. 
 
 How High to Raise the Power-Factor — Theoreti- 
 cally for most efficient results the system power 
 factor should approach unity. The cost of synchron- 
 ous apparatus having sufficient leading current capaci- 
 ty to raise the power-factor to unity increases so 
 rapidly as unity is approached, as to make it unecono- 
 mical to carry the power-factor correction too high. 
 Not only the cost but also the power loss chargeable to 
 power-factor improvement mounts rapidly as higher 
 power-factors are reached. This is for the reason that 
 the reactive kv-a in the load corresponding to each per- 
 cent change in power-factor is a maximum for power- 
 factors near unity. It usually works out that it 
 doesn't pay to raise the power factor above 90 to 95 
 percent, except in cases where the condenser is used 
 for voltage control, rather than power-factor improve- 
 ment. 
 
 DETERMINING THE CAPACITY OF SYNCHRONOUS MOTORS 
 AND CONDENSERS FOR POWER-FACTOR IMPROVEMENT 
 
 A very simple and practical method for determining 
 the capacity of synchronous condensers to improve the 
 power-factor is by aid of cross section paper. A very 
 desirable paper is ruled in inch squares, sub- ruled into 
 10 equal divisions. With such paper, no other equip- 
 ment is required. 
 
 With a vector diagram it is .astonishing how easy 
 it is to demonstrate on cross section paper, the effect 
 of any change in the circuit. A few typical cases are 
 indicated in Fig. 64. These diagrams are all based up- 
 on an original circuit of 3000 kv-a at 70 percent power- 
 factor lagging, shown by (i). It is laid off on the 
 cross section paper as follows. The power of the cir- 
 cuit is 70 percent of 3000 or 2100 kw, which is laid off 
 on line AB, by counting 21 sub-divisions, making 
 each sub-division represent 100 kw or 100 kv-a. Now 
 lay a strip of blank paper over the cross section paper 
 and make two marks on one edge spaced 30 sub-divi- 
 sions apart. This will then be the length of the line AC. 
 This blank sheet is now laid over the cross section 
 paper with one of the marks at the edge held at the 
 point A. ■'The other end of the paper is moved down- 
 ward until the second mark falls directly below the 
 point B thus locating point C. The length of the 
 
 line BC represents the lagging reactive kv-a in the cir- 
 cuit, in this case 2140 kv-a. 
 
 Diagram (2) shows the effect of adding a 1500 
 kv-a synchronous condenser to the original circuit. 
 The full load loss of this condenser is assumed as 70 
 kw. The resulting kv-a and power- factor are de- 
 termined as follows: Starting from the pomt C trace 
 to the right a line 0.7 of a division long. This is 
 parallel to the line AB for the reason that it is true 
 power, so that there is now 2170 kw true energy. The 
 black triangle represents the condenser, the line CD, 
 15 divisions long, representing the rating of the con- 
 denser. In this case, however, the vertical line is 
 traced upward in place of downward, because the con- 
 denser kv-a is leading. This condenser results in de- 
 creasing the load from 3000 kv-a at 70 percent power- 
 factor to 2275 kv-a at 95.4 percent power-factor. The 
 line AD represents in magnitude and direction, the re- 
 sulting kv-a in this circuit. The power-factor of the 
 resulting circuit is the ratio of the true energy in kw to 
 the kv-a or 95.4 percent, in this case. Since the line 
 AD lays below the line AB, that is in the lagging direc- 
 tion, the power- factor is lagging. 
 
 Diagram (3) is the same as (2) except that the 
 condenser is larger, being just large enough to neu- 
 tralize all of the lagging component of the load, result- 
 ing in a final load of 2215 kw at 100 percent power-fac- 
 tor. Diagram (4) is similar to (3) except that a 
 still larger condenser is shown. This condenser not 
 only neutralizes all of the lagging kv-a of the load but 
 in addition introduces sufficient leading kv-a into the 
 circuit to give a leading resultant power-factor of 91 
 percent with an increase in kv-a of the resulting cir- 
 cuit from 2215 of (3) to 2400 kv-a of (4). 
 
 Diagram (5) illustrates the addition to the original 
 chxuit of a 100 percent power-factor synchronous 
 motor of 600 hp. rating As this motor has no leading 
 or lagging component, there is no vertical projection. 
 The power- factor of the circuit is raised from 70 to 
 y7 percent as the result of the addition of 500 kw true 
 power (load plus loss in motor) to the circuit. A re- 
 sistance load would have this same effect. 
 
 Diagram (6) shows a 450 kw (600 hp.) syn- 
 chronous motor of 625 kv-a input at 80 percent lead-j 
 ing power-factor added to the original circuit. The 
 input to this motor (including losses) is assumed to be 
 500 kw. The resulting load for the circuit is 3150 
 kv-a at 82.5 percent lagging power-factor. 
 
 The Diagram (7) shows an 850 kw, (1140 hp.) 
 synchronous motor generator of 1666 kv-a input at 60 
 percent power-factor leading added to the original cir- 
 cuit. This gives a resulting load of 3200 kv-a at 96.9 
 percent lagging power- factor. 
 
 Diagram (8) shows the addition to the original 
 circuit of the following loads, including losses. 
 
 A 550 kw synchronous converter at 100 percent power- 
 factor. 
 
 A 650 kw inljction motor at 70 percent lagging 
 power-factor. 
 
 .\ 500 kw '\ iirhronoiis mi>tor. 
 
SYNCHRONOUS MOTORS AND CONDENSERS FOR POWER-FACTOR CORRECTION 
 
 133 
 
 The resultant load of this circuit is 3800 kw, and 
 if a power-factor of 95 percent lagging is desired the 
 total kv-a will be 4000. The line AD may be located 
 by a piece of marked paper and the capacity of the 
 necessary synchronous motor scaled oflf. This is 
 found to be 1650 kv-a at 30.3 percent power-factor. 
 
 The Circle Diagram — The circle diagram in Fig. 65 
 shows the fundamental relations between true kw, reac- 
 tive kv-a and apparent kv-a corresponding to different 
 power-factors, the values upon the chart being read to 
 any desired scale to suit the numerical values of the 
 problem under consideration. This diagram is suffi- 
 
 
 
 
 
 
 
 - 
 
 - 
 
 
 " 
 
 
 
 r 
 
 - 
 
 
 
 
 ' 
 
 
 "1 1 1 1 1 1 1 1 M 1 ITT 
 
 
 -I 
 
 -l 
 
 r-I 
 
 -1 
 
 
 
 
 -T 
 
 
 
 
 -1 
 
 
 -| 
 
 -| 
 
 
 
 
 -[ 
 
 
 "T 
 
 
 
 -1 
 
 
 
 T-| 
 
 -T-| 
 
 
 ~n 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 1 1 1 1 SOALC 1 1 1 1 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 « 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 fMTH 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ~ 
 
 
 
 
 
 
 
 
 ■fi 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 > 
 
 ttk 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 V> 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 9SO0 
 
 in 
 
 
 - 
 
 r 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 * 
 
 
 
 
 
 
 
 
 
 
 (W» 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ' rf 
 
 » 
 
 
 
 ,, 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 A 
 
 S 
 
 
 
 
 
 
 
 ,[ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 " 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ■ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 s 
 
 
 
 
 
 
 /I 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 s 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 s 
 
 
 
 
 
 % 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 s 
 
 
 
 
 1 
 
 f 
 
 
 
 
 
 
 
 
 
 
 
 _i 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 s 
 
 s 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 '~ 
 
 
 
 
 
 
 
 
 
 s 
 
 
 
 '' 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 s 
 
 s 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 '~ 
 
 
 
 
 
 
 
 
 
 
 s 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 s 
 
 
 s 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 s 
 
 
 
 
 
 
 
 
 
 
 
 
 M 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 s 
 
 ^« 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ~ 
 
 
 
 
 
 
 
 
 
 M 
 
 
 
 
 
 
 
 
 
 
 
 £ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 VV' 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 1 
 
 
 
 
 
 
 
 
 
 
 
 fis 
 
 
 
 
 
 
 
 
 
 
 5 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 *x 
 
 ■ 
 
 •7 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ">• 
 
 
 
 
 
 
 
 
 
 
 g 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 *»M 
 
 A 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 p" 
 
 ^ 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^\ 
 
 
 S*r 
 
 
 
 
 
 
 
 
 
 
 
 
 '~ ~ 
 
 
 ~ " 
 
 
 
 
 
 
 
 
 
 
 
 
 ■<> 
 
 "s 
 
 
 
 
 
 
 
 
 3 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 f 
 
 
 Is' 
 
 ^- 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 '0 
 
 
 
 
 
 
 
 
 * 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^M 
 
 
 >^ 
 
 h 
 
 
 
 
 
 
 
 
 
 
 
 
 
 [_ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 « 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 "A 
 
 
 
 ? 
 
 ■n 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 L 
 
 " 
 
 
 
 
 's 
 
 
 
 
 
 t* 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 f\ 
 
 
 
 s 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 OROINAl 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 r 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 WK 
 
 Aim 
 
 
 
 
 
 s 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 S 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 <s 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 _. 
 
 
 
 
 
 
 
 s, 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ujurni 
 
 
 
 
 
 
 \ 
 
 
 
 S 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 s 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 CUITTO 
 
 
 aTT«LAQQINa 
 
 
 
 
 
 
 s 
 
 
 
 S 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 - 
 
 TS 
 
 
 
 
 
 
 \ 
 
 
 
 
 ^ 
 
 
 
 " 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 m 
 
 D 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 " 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 - 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ' 
 
 
 
 
 
 
 
 
 - 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 e 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 r 
 
 A 
 
 k 
 
 ^ 
 
 
 
 - 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 f^ 
 
 .^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 "s 
 
 N, 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ■ 
 
 
 
 
 
 
 '"( 
 
 -A 
 
 ,» 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 "s 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 . 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 '-* 
 
 4 
 
 r-^ 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 s 
 
 
 
 
 « 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ; 
 
 
 
 
 
 
 
 
 s! 
 
 
 
 
 '-> 
 
 »i 
 
 V 
 
 
 s 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 s 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ■ 
 
 
 
 
 
 — 
 
 _ 
 
 
 r 
 
 -^ 
 
 l_J 
 
 — 
 
 _ 
 
 \- 
 
 
 
 "155^= 
 
 ■-r- 
 
 
 - 
 
 
 _ 
 
 
 - 
 
 
 
 
 
 
 
 - 
 
 
 _ 
 
 
 
 _ 
 
 - 
 
 - 
 
 
 
 s 
 
 r 
 
 _ 
 
 
 
 
 
 - 
 
 
 
 
 _ 
 
 
 
 
 
 _ 
 
 _ 
 
 
 
 -_ 
 
 1 
 
 
 
 - 
 
 
 — 
 
 ~ 
 
 
 
 
 "1 
 
 < 
 
 - 
 
 — 
 
 - 
 
 
 - 
 
 
 - 
 
 
 
 
 — 
 
 -- 
 
 
 - 
 
 - 
 
 - 
 
 
 - 
 
 
 
 
 
 
 
 
 
 - 
 
 
 
 n 
 
 - 
 
 - 
 
 
 
 r 
 
 
 •s 
 
 
 fn- 
 
 T 
 
 
 
 
 
 
 - 
 
 r 
 
 -- 
 
 
 
 - 
 
 - 
 
 
 -- 
 
 - -1 
 
 
 
 
 
 
 
 
 
 
 
 s 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 < 
 
 't 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 _ 
 
 _ 
 
 
 
 
 
 _ 
 
 L) 
 
 % 
 
 __ 
 
 
 ^ 
 
 
 _ 
 
 _ 
 
 
 _ 
 
 __ 
 
 
 -_ 
 
 - 
 
 _ 
 
 _ 
 
 
 _ 
 
 
 
 
 
 
 _ 
 
 
 
 _ 
 
 
 _ 
 
 _ 
 
 _ 
 
 _ 
 
 
 
 
 -, 
 
 
 
 ^ 
 
 
 
 _ 
 
 
 ^ 
 
 &' 
 
 _ 
 
 J 
 
 
 
 
 _ 
 
 
 
 __ 
 
 - L. 
 
 
 
 
 
 
 
 - 
 
 
 
 
 - 
 
 - 
 
 
 ~1 
 
 
 '\ 
 
 -I 
 
 
 ISOO 
 IcOMM 
 
 KV-A 
 
 t 
 
 
 - 
 
 - 
 
 ~ 
 
 -^ 
 
 - 
 
 
 
 
 
 
 
 
 - 
 
 - 
 
 
 r 
 
 - 
 
 ^ 
 
 - 
 
 
 
 L- 
 
 - 
 
 
 
 " 
 
 J 
 
 >s 
 
 - 
 
 h 
 
 
 
 Sj 
 
 5?*v 
 
 (-- 
 
 
 
 - 
 
 
 
 -- 
 
 -^ 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 '\ 
 
 E»«ERJ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Xi 
 
 
 
 N"! 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ■ f Aj \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 •*\ 
 
 
 
 
 
 
 *s 
 
 
 
 
 
 
 
 ■ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ■?\ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^> 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 laoo K 
 
 V-A 
 
 .^3NO£NSER 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 s 
 
 
 
 
 
 
 
 s 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 RAieeS THE P.F.OF THE 
 
 
 s 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 IS> 
 
 
 
 
 
 
 s 
 
 
 
 
 
 
 
 
 . 
 
 
 1 
 
 
 
 
 
 
 
 
 C>RC-TTo"^0 = ««i. 
 
 
 
 s, 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 s 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ia:ic 
 
 2276 
 
 
 
 
 
 s 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 SYNCHRONOUS MOTOR IftOO KW 
 INPUT ASSUMED) RAISES THE RF 
 
 
 
 
 ^ 
 
 
 
 A 
 
 y 
 
 
 
 
 
 
 
 
 
 
 
 UPHi 
 
 
 
 
 
 i 1 1 
 
 
 
 
 N 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 s 
 
 
 -v^ 
 
 
 
 8 3 KV-A 
 STNCHRONOUS 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 . 
 
 
 
 
 
 
 - 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / - 
 
 -- 
 
 
 
 
 
 
 
 
 '~ 
 
 
 
 
 
 
 
 . 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 , 
 
 
 
 
 
 1 IN CONDENSErI* 
 
 rfo 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 _l I-l 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 L2 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 8 218 KV-A 
 
 " a a 1 B KW 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 r3 
 
 M 
 
 KW' 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 *i 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 [• 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 •■ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 A 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 B 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ™ 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 s 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 ^ 
 
 
 
 '1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 s 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 rf 
 
 
 
 - 
 
 JL 
 
 Kl 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 s 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 s, 
 
 
 
 
 
 
 
 
 
 
 F 
 
 
 & 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ■v 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 v 
 
 
 
 
 
 
 
 
 
 
 
 
 
 r" 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 s 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 I_L- 
 
 
 
 
 
 
 
 s 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 s 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 V 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ■■ ■^ 
 
 
 
 
 1 ' 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 f^ 
 
 
 1 
 
 
 
 »s 
 
 
 
 
 
 
 
 
 
 
 
 ■S 
 
 
 
 
 
 - — j 
 
 
 
 ' 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 % 
 
 V 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 j 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 5> 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 <2 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 - 
 
 
 
 
 
 
 
 
 
 
 s 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 "-N 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 ■[_ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 r^ 
 
 > 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 "' 
 
 ■ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 " 
 
 s 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 "J 
 
 
 
 
 
 
 
 
 
 
 
 
 iijf' 
 
 
 
 
 J 
 
 'X" 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ♦A 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 V 
 
 
 
 
 
 ■' 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 s 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 (71 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 *i 
 
 ; 
 
 
 
 
 
 
 
 _ 
 
 _ 
 
 _ 
 
 - 
 
 
 _ 
 
 _ 
 
 1.1) 
 
 444- 
 
 
 ps 
 
 
 __ 
 
 ._ 
 
 
 — 
 
 _ 
 
 _ 
 
 
 _ 
 
 
 
 
 
 
 
 
 
 _ 
 
 
 _ 
 
 esoKWiiaee 
 
 KVJk 
 
 AT0O»l!F,'8TI' 
 
 
 
 s 
 
 _, 
 
 
 _ 
 
 
 
 ^% 
 
 'i: 
 
 
 _. 
 
 
 
 
 
 - 
 
 --] 
 
 
 - 2140 KV 
 
 A CO 
 
 NDCNSER 
 
 
 \ 
 
 s 
 
 ■ 
 
 
 „ 
 
 ~ 
 
 ~ 
 
 
 - 
 
 
 
 
 
 
 
 
 
 - 
 
 
 - 
 
 IIOOO KW INPUT ASSUMED) RAISE 
 
 B' 
 
 
 
 S 
 
 
 - 
 
 
 -v 
 
 - 
 
 
 --- 
 
 =:: 
 
 laesKv-A 1 
 
 STNCHRQ.'JOUa 1 
 
 motor-gcnerator[ 
 
 
 - 
 
 - 
 
 
 
 RAI8E8 Tht r.r. «r . ni 
 
 
 
 \ 
 
 J 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 THE P.F. OF THE CIRCUIT TO ■s:^::^ " 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ''J 
 
 
 __ 
 
 \- 
 
 _ 
 
 
 i_ 
 
 
 
 
 
 
 
 
 
 _ 
 
 
 _ 
 
 
 
 p. 
 
 **>'W^ 1 1 1 1 T 1 J- 
 
 
 
 _ 
 
 
 
 ^ 
 
 
 _ 
 
 
 
 __ 
 
 _ 
 
 
 1 [ 
 
 
 
 - 
 
 4- 
 
 
 I— 
 
 H 
 
 - 
 
 Mil; 
 
 
 " < 
 
 
 - 
 
 
 - 
 
 
 h 
 
 
 
 
 
 
 
 
 
 - 
 
 
 - 
 
 ~ 
 
 - 
 
 
 
 
 
 - 
 
 
 
 
 - 
 
 
 - 
 
 - 
 
 
 
 
 - 
 
 - 
 
 - 
 
 - 
 
 
 ~ 
 
 1000 • 
 
 ■w- 
 
 - 
 
 
 -- 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 " '' 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ■ 
 
 
 
 
 
 
 
 
 
 
 
 " 
 
 
 *- 
 
 
 
 
 
 
 
 
 
 
 
 
 ■* 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 IMc 
 
 
 .• ' 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 i 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ,Vtt 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ' 
 
 -- 
 
 ^ 
 
 
 -1 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 > 
 
 *J 
 
 '> 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 <*!■ 
 
 n' 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 t 
 
 ►1 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 s 
 
 
 
 
 
 
 jyw 
 
 a\ ' 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 '■ 
 
 
 
 
 ■*'^ 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 s. 
 
 
 
 
 
 
 
 
 r' 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 *- 
 
 '■ ' 
 
 'X 
 
 
 J 
 
 
 JllL!^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 fi 
 
 u 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 X-^' ' ' 
 
 
 ■■ 
 
 
 
 " ' 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 <!& 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 I I 1 
 
 
 
 
 
 
 
 r 
 
 
 
 B 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 s 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 " 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 M 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ■«. 
 
 
 
 
 
 
 
 
 
 
 "s 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 5 
 
 !» 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 , 
 
 
 
 
 
 
 
 
 V 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 y 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 . 
 
 ■> ^ 
 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ~ ~ 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 Up^. 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^h 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ** 
 
 .^ 
 
 
 
 
 
 
 
 K 
 
 
 
 
 
 
 
 
 
 
 DENSER ! 
 
 - Z * 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 V 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 v 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 li :: 
 
 
 
 
 
 
 
 
 
 
 
 
 A 
 
 
 
 
 
 
 2 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^4 
 
 
 
 
 
 
 
 
 
 
 
 
 
 qf 
 
 
 
 
 
 
 
 
 
 • 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 tSftOK>AA AT 3a.3«P.F.- 
 SYNCHROMOUO MOTOR 
 
 ,- 
 
 
 
 
 
 
 
 
 
 
 s 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 > 
 
 
 
 f 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 I 
 
 
 
 
 
 
 
 
 
 
 
 1, 1 
 
 
 
 
 
 i_,7ll 1 1 
 
 
 
 1 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ADD TU IHt UMiMI"-!. \^r«,u. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 1 
 
 
 
 
 
 
 f 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 IWv» ri 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 ,-? 
 
 >, 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 -600 KW SYNCHRONOUS MOTOR 
 
 DETERMINE RATING OF SrNCMRONOUS MOTOR 
 -WHICH WILL OrVE A RESULTANT RF. FOR THE CtR. 
 
 - 
 
 - 
 
 _ 
 
 
 1 
 
 
 
 ■ I ■» 
 
 
 •/ 
 
 
 
 
 
 1, 
 
 1 
 
 
 
 
 
 
 
 
 n1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 '<"f°""':'-|,-'"°,""- .' 
 
 
 
 
 
 
 INOCNKR 
 
 
 
 
 s 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 SrNCHRONO 
 
 
 / 
 
 
 
 . RAISES THE RF. OF THE 
 
 
 
 
 
 \ 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ua ^ 
 
 V 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 a BOO I 
 
 
 
 
 
 cc 
 
 •NVERTCn ■ j •■ 
 
 Jn' 
 
 
 r 
 
 
 
 OiHCUITTO^^ 
 
 ■liAWNQ -pp 
 
 
 
 p 
 
 p 
 
 
 ^ 
 
 - 
 
 
 - 
 
 
 ^ 
 
 
 
 - 
 
 - 
 
 - 
 
 
 - 
 
 
 
 - 
 
 - 
 
 
 
 
 
 - 
 
 V ■ ■ »4000 KV-A AT Sftft KF. 
 
 - 
 
 
 - 
 
 - 
 
 - 
 
 - 
 
 - 
 
 
 
 « 
 
 nr 
 
 4- +^7^ 
 
 \-/ 
 
 - 
 
 
 
 - 
 
 - 
 
 
 
 
 - 
 
 - 
 
 - 
 
 -- 
 
 
 i»»kWiJ«| 
 
 
 r 
 
 
 - 
 
 - 
 
 - 
 
 - 
 
 
 
 - 
 
 - 
 
 - 
 
 - 
 
 
 
 - 
 
 
 ' 
 
 h 
 
 - 
 
 - 
 
 ' 
 
 - 
 
 - 
 
 - 
 
 - 
 
 
 
 - 
 
 - 
 
 - 
 
 - 
 
 - 
 
 
 - 
 
 - 
 
 - 
 
 - 
 
 -- 
 
 - 
 
 - 
 
 ' 
 
 -- 
 
 MOTORS AT - »rHCMRONOU» 
 
 
 ^ 
 
 
 
 
 1 
 
 
 
 
 
 ~ 
 
 — 
 
 
 
 — 
 
 
 -T^-HT" 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 r 
 
 "■n 1 1 
 
 FIG. 64— EXAMPLES IN POWER-FACTOR IMPROVEMENT 
 
134 
 
 SYNCHRONOUS MOTORS AND CONDENSERS FOR POIVER-EACTOR CORRECTION 
 
 ciently accurate for ordinary power-factor problems. 
 In place of drawing out the vector diagrams as just 
 explained they are traced out with a pin point on the 
 circle diagram. 
 
 Assume again a load of 2100 kw at 70 percent 
 power-factor lagging, and that the power-factor is to 
 be raised to 95.4 percent as in (2) of Fig. 64, and that 
 the loss in the condenser necessary to accomplish this 
 is again taken as 70 kw. The capacity of the syn- 
 chronous condenser may be traced on the circle dia- 
 gram as follows: From the true power load of 2100 
 kw (top horizontal line) follow vertically downward 
 
 of the condenser would be the hypotenuse rather than 
 the vertical projection. The error in assuming the 
 vertical projection as the rating of the condenser is 
 negligible unless the condenser furnishes mechanical 
 power, in which case the hypotenuse should be marked 
 on a separate strip of paper and its length determined 
 from the kv-a scale. 
 
 ADVANTAGE OF HIGH POWER-FACTOR 
 
 Less Capacity Installed — Low power-factors de- 
 mand larger generators, exciters, transformers, switch- 
 ing equipment and conductors. Loads of 70 percent 
 
 1000 
 
 2000 
 
 3000 
 
 POWER KW 
 4000 5000 
 
 6000 
 
 7000 
 
 8000 
 
 9000 
 
 10.000 
 
 % POWER-FACTOR 
 FIG. 65 — RELATION BETWEEN ENERGY LOAD, APPARENT LOAD AND REACTIVE KV-A FOR DIFFERENT POWER FACTORS 
 
 until the diagonal line representing 70 percent power- 
 factor is reached. This is opposite 2140 kv-a reactive 
 component. From the point thus obtained, go hori- 
 zontally to the right a distance representing 70 kw 
 power. From this point go vertically upward until 
 the diagonal line representing 95.4 percent power-fac- 
 tor is reached. Then read the amount of reactive kv-a 
 (640) corresponding to this last point. The original 
 lagging component of 2140 — 640=^1500 kv-a which is 
 approximately the capacity of the condenser necessary 
 to accomplish the above results. Actually the rating 
 
 power-factor demand equipment of 28 percent greater 
 capacity than would be required if the power-factor 
 were 90 percent. The cost of apparatus for opera- 
 tion at 70 percent power-factor would be approxi- 
 mately 15 percent greater than the cost of similar ap- 
 paratus for 90 percent power-factor operation, since 
 the capacity of apparatus to supply a certain amount 
 of energy is inversely proportional to the power-factor. 
 Higher Efficiency — Assume that the power-factor 
 of a 1000 kv-a (700 kw at 70 percent power- factor) 
 transmission circuit is raised to 90 percent. As the cop- 
 
SYNCHRONOUS MOTORS AND CONDENSERS FOR POWER-FACTOR CORRECTION 
 
 '35 
 
 per loss varies as the square of the current, raising the 
 power- factor reduces the copper loss approximately 
 40 percent. If we assume an efficiency for the genera- 
 tor of 93 percent (one percent copper loss) ; for com- 
 bined raising and lowering transformers 94 percent 
 (three percent copper loss) and for the transmission 
 line 92 percent, the saving in copper loss correspond- 
 ing to 90 percent power-factor operation would be as 
 follows : 
 
 Generators 0.4 percent 
 
 Transformers 1.2 percent 
 
 Transmission line .... 3.2 percent 
 
 Total 4.8 percent or approximately 33 kw. 
 
 To raise the power-factor to 90 percent would re- 
 quire a synchronous condenser of 375 kv-a capacity. 
 This size condenser would have a total loss of about 
 30 kw, resulting in a net gain in loss reduction of 
 three kw. Against this gain would be chargeable, the 
 interest and depreciation of the condenser cost with its 
 accessories, also any cost of attendance which there 
 might be in connection with its operation. It is evi- 
 dent that in this case it would not pay to install a con- 
 denser if increased efficiency were the only motive. 
 
 TABLE S— COST OF POWER-FACTOR CORRECTION 
 WITH SYNCHRONOUS MOTORS 
 
 Syn. Motor 
 Kv-a 
 
 Motor Will Furnish 
 
 Chargeable to 
 Power-Factor Correction 
 
 Mech. 
 Kw 
 
 Leading 
 Kv-a 
 
 Loss 
 Kw 
 
 Difference 
 in Price 
 
 140 
 280 
 420 
 700 
 1050 
 1400 
 
 100 
 200 
 300 
 500 
 750 
 
 1000 
 
 100 
 200 
 300 
 500 
 750 
 1000 
 
 1.6 
 
 2.5 
 50 
 8.0 
 9.0 
 14.0 
 
 $500.00 
 
 500.00 
 
 500.00 
 
 800.00 
 
 1000.00 
 
 1200.00 
 
 The improvement in power-factor can be more 
 cheaply and efficiently obtained by the installation of 
 one or more synchronous motors designed for opera- 
 tion at leading power-factor. Sufficient capacity of 
 these will give, in addition to mechanical load, suffi- 
 cient leading current to raise the power-factor to 90 
 percent. The extra expense and increased loss of 
 synchronous motors enough larger to furnish the nec- 
 essary leading component for power-factor correction 
 is very small. Table S gives in a very approximate 
 way, some idea of the amount of loss and proportional 
 cost of synchronous motors chargeable to power-factor 
 improvement when delivering both mechanical power 
 and leading current. 
 
 Thus if a synchronous condenser is used on the 
 above circuit there is a loss of 30 kw, chargeable to 
 power-factor improvement, whereas if a synchronous 
 motor of sufficient capacity (530 kv-a) to give 375 kw 
 mechanical work and at the same time the necessary 
 375 kv-a leading current for power-factor improve- 
 ment, the extra loss chargenble to power-factor im- 
 provement would be something like six kw. The in- 
 creased cost of a synchronous motor to furnish 375 
 kv-a leading current in addition to 375 kw power 
 would be about $600 vi'hereas the cost of a 375 kv-a 
 
 condenser would be in the neighborhood of $4000. 
 Varying costs and designs make cost and loss values 
 unreliable. They are given here only to illustrate the 
 points which should be considered when considering 
 synchronous motors vs synchronous condensers. 
 
 Improved Voltage Regulation — The voltage drop 
 under load for generators, transformers and trans- 
 mission lines rapidly increases as the power-factor goes 
 down. Table T gives an idea of the variation in volt- 
 age drop corresponding to various power-factors at 60 
 cycles. 
 
 Automatic voltage regulation may be used to hold 
 the voltage constant at the generators or at some other 
 point, but it cannot prevent voltage changes at all 
 points of the system. 
 
 Increased Plant Capacity — The earlier alternators 
 were designed for operation at 100 percent power-fac- 
 tor with prime movers, boilers, etc installed on the 
 same basis. Increasing induction motor loads have 
 resulted in power-factors of 70 and 80 percent. As a 
 result, some of the older generating stations are being 
 operated with prime movers, boilers etc. underloaded 
 because the 100 percent power-factor generators which 
 
 TABLE T— EFFECT OF POWER-FACTOR ON VOLTAGE 
 DROP 
 
 Percent Power-Factor. 
 
 100 
 
 90 
 
 80 
 
 70 
 
 Generators *(oldcr design) 
 Transformers 
 Transmission line 
 
 S,o 
 
 1.2 
 
 7-9 
 
 4-1 
 130 
 
 1 
 25.0 
 
 4-9 
 14.2 
 
 1 - 
 5-5 
 
 15-2 
 
 ihey drive limit the amount of power that can be gen- 
 erated without endangering the generator windings. 
 This condition some times makes it necessary to oper- 
 ate three units, where two might be sufficient to carry 
 the load at unity power-factor. The shutting down of 
 r unit would result in a considerable saving in steam 
 consumption. A recent case came up of a transmis- 
 sion line 30 miles long, fed at each end by a small gen- 
 erating station. On account of heavy line drop it was 
 necessary to operate both stations to furnish the com- 
 paratively light night load. Investigation developed 
 that by installing a synchronous condenser at one of 
 these tenninal stations for reducing the voltage drop 
 in the line, one generating station could be shut down 
 during the night, thereby resulting in a very large 
 annual saving in coal and labor bills. 
 
 A station may have some generating units de- 
 signed for 100 percent power-factor and other units 
 designed for 80 percent power- factor; or again, where 
 two generating stations feed into the same transmission 
 system, one may have 100 percent power-factor gen- 
 erating units and the other 80 percent power-factor 
 
 *The present-day design of maximum rated generators with 
 a short-circuit ratio of about unity will barely circulate full- 
 load current with normal no-load excitation. Under such con- 
 ditions the terminal voltage would be practically zero regardless 
 of the oower-factor. 
 
136 
 
 SVXCHRONOCS MOTORS AND CONDENSERS fOR POWER-FACTOR CORRECTION 
 
 generating units. In such cases, the field strength of 
 the generators may be so adjusted as to cause the 80 
 percent power-factor units to take all the lagging cur- 
 rent, thus permitting the 100 percent power-factor 
 units to be loaded to their full kw rating. 
 
 BEHAVIOR OF A. C. GENERATORS WHEN CHARGING A 
 TRANSMISSION LINE* 
 
 It has been shown above how leading armature 
 current, by increasing the field strength, causes an in- 
 crease in the voltage induced in the armature of an 
 alternator and consequently an increase in its terminal 
 voltage. It was also shown that the terminal voltage 
 is further increased as result of the voltage due to 
 self induction adding vectorially to the voltage induced 
 in the armature. 
 
 If an alternator with its fields open is switched 
 onto a dead transmission line having certain electrical 
 characteristics, it will become self exciting, provided 
 there is sufficient residual magnetism present to start 
 the phenomenon. In such case, the residual magnet- 
 ism in the fields of the generator will cause a low volt- 
 age to be generated which will cause a leading line 
 charging current to flow through the armature. This 
 leading current will increase the field flux which in 
 turn will increase the voltage, causing still more charg- 
 ing current to flow, which in turn will still further in- 
 crease the line voltage. This building up will continue 
 until stopped by saturation of the generator fields. 
 This is the point of stable operation. Whether or not 
 a particular generator becomes self exciting when 
 placed upon a dead transmission line depends upon the 
 relative slope of the generator and line characteristics. 
 
 In Fig. 66 are shown two curves for a single 
 45 000 kv-a, 1 1 000 volt generator, the charging current 
 of the transmission line being plotted against genera- 
 tor terminal voltage. One curve corresponds to zero 
 excitation, the other curve to 26.6 percent of normal 
 excitation. A similar pair of curves correspond to 
 two duplicate generators in parallel**. The straight 
 line representing the volt-ampere characteristics of the 
 transmission line fed by these generators corresponds 
 to a 220 kv, 60 cycle, three-phase transmission cir- 
 cuit, 225 miles long, requiring 69 000 kv-a to charge it 
 with the line open at the receiving end. 
 
 The volt-ampere charging characteristic of a 
 transmission line is a straight line, that is, the charg- 
 ing current is directly proportional to the line voltage. 
 On the other hand the exciting volt-ampere character- 
 istic for the armature has the general slope of an 
 ordinary saturation curve. 
 
 *For a more detailed discussion of this subject see the fol- 
 lowing articles : — "Characteristics of Alternators when Excited 
 by Armature Currents" by F. T. Hague, in the Journal for Aug. 
 1915 ; "The Behavior of Alternators with Zero Power-Factor 
 Leading Current" by F. D. Newbury, in the Journal for Sept. 
 1918; "The Behavior of A. C. Generators when Charging a 
 Transmission Line" by W. O. Morris, in the General Electric 
 Revinv for Feb. 1920. 
 
 **It is assumed that with the assumed field current such 
 generators can be synchronized p.n<\ held together during the 
 process of charging the line. 
 
 If the alternator characteristic lie above the line 
 characteristic at a point corresponding to a certain 
 charging current the leading charging current will 
 cause a higher armature terminal voltage than is re- 
 quired to produce that current on the line. As a re- 
 sult the current and voltage will continue to rise until, 
 on account of saturation, the alternator characteristic 
 falls until it crosses the line characteristic. At this 
 point the voltage of the generator and that of the line 
 are the same for the corresponding current. If on the 
 other hand the alternator characteristic falls below the 
 line characteristic the alternator will not build up with- 
 out permanent excitation. 
 
 As stated previously, whether or not a generator 
 becomes self-exciting when connected to a dead trans- 
 mission line depends upon the relative slopes of gener- 
 ator and transmission line characteristics. The rela- 
 tive slopes of these curves depend upon: — 
 
 a — The magnitude of the line charging current. 
 
 h — The rating of the generators compared to the full voltage 
 charging kv-a of the line. 
 
 c — The armature reaction. High armature reaction, (that is 
 low short-circuit ratio) favors self-excitation of the gener- 
 ators. 
 
 d — The armature reactance. High armature reactance also 
 favors self-excitation of the generators. 
 
 Methods of Exciting Transmission Lines — If the 
 
 relative characteristics of an alternator and line are 
 
 such as to cause the alternator to he self-exciting, this 
 
 condition may be overcome by employing two or more 
 
 1000 2000 3000 4000 500C 
 
 CHARGING CURRENT OF LINE 
 
 (AMPERES PER GENERATOR TERMINAL) 
 
 FIG. 66 — VOLT AMPERE CHARACTERISTICS OF ONE 45 OOO KV-A, 1 1 COO 
 
 VOLT generator; TWO DUPLICATE 45 COO KV-A GENERATORS; AND A 
 
 THREE-PHASE, SINGLE-CIRCUIT, 220 KV TRANSMISSION LINE 
 
 alternators (provided they are available for this pur- 
 pose) to charge the transmission line. The combined 
 characteristics of two or more alternators may be such 
 as to fall under the line characteristic, in which case 
 the alternator will not be self-exciting. In such case, 
 the alternators could be brought up to normal speed, 
 and given sufficient field charge to enable them to be 
 
SYNCHRONOUS MOTORS AND CONDENSERS FOR POWER-FACTOR CORRECTION 
 
 137 
 
 TABLE U— INSTALLATIONS OF LARGE PHASE MODIFIERS (1921) 
 By American Manufactnrers 
 
 synchronized and held in step, after which they could Fig. 66, and there were sufficient residual magnetism 
 be connected to the dead transmission line and their to start the phenomenon, the generator voltage would 
 voltage raised to normal. rise to approximately double normal value before the 
 
 Generators as normally designed will carry ap- point of staple operation is reached. If, however, 
 proximately 40 percent of their rated current at zero two generators having 26.6 percent of normal excita- 
 leading power-factor. If more than this current is tion were paralleled and connected to this circuit, a 
 demanded of them they are likely to become unstable point of staple operation would be reached at a 
 in operation. By modifying the design of normal terminal voltage of approximately 15 500 volts, 
 alternators so as to give low armature reaction, they Actually stable operation would be reached at a some- 
 may be made to carry a greater percentage of leading what less terminal voltage for the reason that the line 
 current. If the special design is such that with zero would probably not be open at the receiving end, but 
 
 would probably have the 
 lowering transformers con- 
 nected to it. In such case 
 the magnetizing current 
 required for lowering trans- 
 formers would lower the 
 receiving end voltage, re- 
 sulting in less line charging 
 current. 
 
 In either case the 
 curves of Fig. 66 show that 
 either more than two gen- 
 erators will be required to 
 charge the line when un- 
 loaded, or some other meth- 
 od of charging must be re- 
 sorted to. Reactance coils 
 could be used at the receiv- 
 ing end t ) furnish lagging 
 current for neutralizing some 
 of the line charging cur- 
 voltage field excitation when carrying half the line rent, but there might be difficulty in removing these 
 charging kv-a, the armature voltage will not exceed 70 from the circuit when the line is fully charged At 
 percent of normal, this reduced voltage will result in the present time it is expected that the problem of 
 a line charging kv-a of half of normal value. Spe- charging long transmission lines may usually be solved 
 cially designed alternators usually result in larger and by starting one or more generators with sufficient field 
 more costly machines and the gain resulting in the spe- strength to permit them to be synchronized and held in 
 cial design is usually not sufficient to warrant the extra step. One or more phase modifiers with under-excited 
 ^°^^- , fields may then be connected to the line at the receiving 
 
 If a single generator with its field circuit open end and brought up to normal speed with the genera- 
 were connected to a dead transmission circuit such as tors. Such a method of solving this problem has been 
 the one whose volt-ampere characteristics are shown in employed by the Southern California Edison Company. 
 
 Kv-a 
 
 a.p.M. 
 
 Volts 
 
 Cycles 
 
 No. of 
 Units 
 
 Pate of 
 Order 
 
 NAMF. AND LOCATION 
 
 80 000 
 
 600 
 
 6600 
 
 50 
 
 1 
 
 1919 
 
 So. Csl. Ed. Co., Los Angeles, Cal. 
 
 
 20 000 
 
 600 
 
 11 000 
 
 60 
 
 2 
 
 1921 
 
 Pacific Gas & Elec. 
 
 
 15 000 
 
 875 
 
 6600 
 
 50 
 
 
 1912 
 
 Southern Oal. Ed. Co., Los Ang., Cal. 
 
 
 15 000 
 
 375 
 
 6600 
 
 50 
 
 1 
 
 1912 
 
 Pacific Lt. & Pr. Co. 
 
 
 112 500 
 
 500 
 
 22 000 
 
 50 
 
 2 
 
 1918 
 
 Andhra Valley, India 
 
 
 7500 
 
 400 
 
 6600 
 
 60 
 
 2 
 
 1913 
 
 Utah Pr. & Lt. Co., Salt Lake, Utah 
 
 
 7500 
 
 400 
 
 6600 
 
 60 
 
 2 
 
 1916 
 
 Canton El. Co., Canton, Ohio 
 
 
 7500 
 
 600 
 
 13 800 
 
 60 
 
 1 
 
 1917 
 
 Blackstone Valley Gas & Elec. Co., Pawtucket, R. 
 
 I. 
 
 7500 
 
 600 
 
 13 800 
 
 60 
 
 1 
 
 1917 
 
 New England Pr. Co., Worcester, Mass. 
 
 
 7500 
 
 720 
 
 13 800 
 
 60 
 
 1 
 
 1918 
 
 New England Pr. Co., Fitchburg, Mass. 
 
 
 7500 
 
 800 
 
 11 500 
 
 40 
 
 1 
 
 1918 
 
 Adirondack El. Pwr. Corp., Watervliet, New York 
 
 
 7500 
 
 750 
 
 U 000 
 
 50 
 
 1 
 
 1919 
 
 Energia Electrica de Cataluna, Barcelona, Spain 
 
 
 7500 
 
 600 
 
 11 000 
 
 60 
 
 1 
 
 1920 
 
 Duquesne Light Co. 
 
 
 7500 
 
 600 
 
 1200 
 
 60 
 
 2 
 
 1918 
 
 J. G. White, Engineers 
 
 
 7500 
 
 600 
 
 11 000 
 
 60 
 
 1 
 
 1918 
 
 Puquesne Light Co. 
 
 
 7500 
 
 600 
 
 11 000 
 
 60 
 
 1 
 
 1916 
 
 Duquesne Light Co. 
 
 
 7500 
 
 600 
 
 11 000 
 
 60 
 
 2 
 
 1917 
 
 Puquesne Light Co. 
 
 
 6500 
 
 750 
 
 2200 
 
 50 
 
 1 
 
 1917 
 
 Shanghai Municipal Council, Shanghai, China 
 
 
 6000 
 
 500 
 
 16 500 
 
 50 
 
 1 
 
 1914 
 
 So. Cal. Ed. Co., Los Angeles, Cal. 
 
 
 5000 
 
 600 
 
 7200 
 
 60 
 
 1 
 
 1916 
 
 Pac. Pwr. & Lt., Kennewick, Wash. 
 
 
 5000 
 
 500 
 
 6600 
 
 50 
 
 2 
 
 1915 
 
 Tata Hydro El. Pr. & S. Co., India 
 
 
 6000 
 
 750 
 
 6600 
 
 50 
 
 3 
 
 1917 
 
 Ebro Irrigation & Pr. Co., Barcelona, Spain 
 
 
 5000 
 
 750 
 
 11 500 
 
 50 
 
 1 
 
 1919 
 
 Societa Lombarda Distribuziona Energia Elettrica 
 
 Italy 
 
 6000 
 
 600 
 
 2300 
 
 60 
 
 1 
 
 1918 
 
 TurnbuU Steel Co., Warren, Ohio 
 
 5000 
 
 720 
 
 2300/ 
 4000 
 
 60 
 
 
 1921 
 
 Public Service of N. 111. 
 
 
 5000 
 
 720 
 
 11000 
 
 60 
 
 1 
 
 1921 
 
 Takata & Co., Japan. 
 
 
 5000 
 
 600 
 
 13 200 
 
 60 
 
 1 
 
 1919 
 
 Conn. Lt. & Pr. Co. 
 
 
CHAPTER XV 
 PHASE MODIFIERS FOR VOLTAGE CONTROL 
 
 WITH alternating-current transmission there is 
 a voltage drop resulting from the resistance 
 of the conductors, which is in phase with the 
 current. In addition there is a reactance voltage drop ; 
 that is a voltage of self-induction generated within the 
 conductors which varies with and is proportional to 
 the current, and may add to or decrease the line volt- 
 age. If the line is long, the frequency high or the 
 amount of power transmitted large, this induced volt- 
 age will be large, influencing greatly the line drop. 
 By employment of phase modifiers the phase or direc- 
 tion of this induced voltage may be controlled so that 
 it will be exerted in a direction that will result in the 
 desired sending end voltage. 
 
 A certain amount of self-induction in a transmission 
 circuit is. an advantage, allowing the voltage at the re- 
 ceiving end to be held constant under changes in load 
 by means of phase modifiers. It may even be made to 
 reduce the line voltage drop to zero, so that the voltage 
 at the two ends of the line is the same for all loads. 
 Self-induction also reduces the amount of current 
 which can flow in case of short-circuits, thus tending to 
 reduce mechanical strains on the generator and trans- 
 former windings, and making it easier for circuit 
 breaking devices to function successfully. On the 
 ether hand, high self-induction reduces the amount of 
 power which may be transmitted over a line and may, 
 in case of lines of extreme length, make it necessary 
 to adopt a lower frequency. It also increases the ca- 
 pacity of phase modifiers necessary for voltage con- 
 trol. High reactance also increases the surge over- 
 voltage that a given disturbance will set up in the sys- 
 tem. 
 
 On the long lines, the effect of the distributed 
 leading charging current flowing back through the line 
 inductance is to cause, at light loads, a rise in voltage 
 from generating to receiving end. At heavy loads, the 
 lagging component in the load is usually sufficient to 
 reverse the low-load condition; so that a drop in volt- 
 age occurs from generating to receiving end. The 
 charging current of the line is, to a considerable extent, 
 an advantage; for it partially neutralizes the lagging 
 component in the load, thus raising the power-factor 
 of the system and reducing the capacity of synchron- 
 ous condensers necessary for voltage control. 
 
 The voltage at the receiving end of the line should 
 be held constant under all loads. To partially meet 
 this condition, the voltage of the generators could be 
 varied to a small extent. On the longer lines, how- 
 ever, the voltage range required of the generators 
 would be too great to permit regulation in this 
 
 manner. In such cases, phase modifiers operating in 
 parallel with the load are employed. The function of 
 phase modifiers is to rotate the phase of the current at 
 the receiving end of the line so that the self-induced 
 voltage of the line (always displaced 90 degrees from 
 the current) swings around in the direction which will 
 result in the desired line drop. In some cases a phase 
 modifier is employed which has sufficient capacity not 
 only to neutralize the lagging component at full load, 
 but, in addition, to draw sufficient leading current from 
 the circuit to compensate entirely for the ohmic and re- 
 actance voltage drops of the circuit. In this case, the 
 voltage at the two ends of the line may be held the 
 same for all loads. This is usually accomplished by 
 employing an automatic voltage regulator which oper- 
 ates on the exciter fields of the phase modifier. The 
 voltage regulator may, if desired, be arranged to com- 
 pound the substation bus voltage with increasing load. 
 
 CHECKING THE WORK 
 
 A most desirable method of determining line per- 
 formance is by means of a drawing board and an en- 
 gineer's scale. A vector diagram of the circuit under 
 investigation, with all quantities drawn to scale, greatly 
 simplifies the problem. Each quantity is thus repre- 
 sented in its true relative proportion, so that the re- 
 sult of a change in magnitude of any of the quanti- 
 ties may readily be visualized. Graphical solutions 
 are more readily performed, and with less likelihood 
 of serious error than are mathematical solutions. The 
 accuracy attainable when vector diagrams are drawn 
 20 to 25 inches long and accurate triangles, T squares, 
 straight edges and protractors are employed is well 
 within practical requirements. Even the so-termed 
 "complete solution" may be performed, graphically 
 with ease and accuracy. A very desirable virtue of 
 the graphical solution which follows is that it exactly 
 parallels the fundamental, mathematical solution. For 
 this reason this graphical solution is most helpful even 
 when the fundamental mathematical solution is used, 
 for it furnishes a simple check against serious errors. 
 The result may be checked graphically after each in- 
 dividual mathematical operation by drawing a vector 
 in the diagram paralleling the mathematical operation. 
 Thus, any serious error in the mathematical solution 
 may be detected as soon as made.* 
 
 *A method of cfiecking arithmetical operations which 
 requires little time and is an almost sure preventative of errors 
 is that known as "casting out the nines." This method is given 
 in most older arithmetics but has been dropped from many of 
 the modern ones. A complete discussion is given in Robinson's 
 "New Practical Arithmetic" published by The American Book 
 Company. 
 
PHASE MODIFIERS I'OR VOLTAGE CONTROL 
 
 139 
 
 When converting a complex quantity mathematic- 
 ally from polar to rectangular co-ordinates, or vice 
 versa, the results may readily be checked by tracing 
 the complex quantity on cross-section paper and 
 measuring the ordinates and polar angle, or for ap- 
 proximate work the conversion may be made graphic- 
 ally to a large scale. For instance, in using hyperbolic 
 functions, polar values will be required for obtaining 
 powers and roots of the complex quantity. For 
 approximate work much time will be saved by ob- 
 taining the polar values graphically. 
 
 In the graphical solution of line performance it 
 will usually be desirable to check the line loss by a 
 mathematical solution in cases which require exact loss 
 values. Since the line loss may be five percent or less 
 of the energy transmitted, a small error in the overall 
 results might correspond to a large error in the value 
 of the line loss. 
 
 EFFECT OF TRANSFORMERS IN THE CIRCUIT 
 
 Usually long transmission circuits have trans- 
 formers installed at both ends of the circuit and one 
 or more phase modifiers in parallel with the load. 
 Such a transmission circuit must transmit the power 
 loss of the phase modifiers and of the receiver trans- 
 formers. In addition to this power loss, a lagging re- 
 active current is required to magnetize the transformer 
 iron. A complete solution of such a composite cir- 
 cuit (generator to load) requires that the losses of the 
 phase modifiers and transformers be added vectorially 
 to the load at the point where they occur so that their 
 complete effect may be included in the calculation of 
 the performance of the circuit. A complete solution 
 also requires that three separate solutions be made for 
 such a circuit.* First with the known or assumed con- 
 ditions at the load side of the lowering transformers 
 the corresponding electrical conditions at the high volt- 
 age side of the transformers is determined by the usual 
 short line impedance methods. With the electrical 
 conditions at the receiving end of the high-tension line 
 thus determined, the electrical conditions at the send- 
 ing end of the line are determined by one of the vari- 
 ous methods which take into account the distributed 
 quantities of the circuit. With the electrical condition 
 at the sending end thus determined the electrical con- 
 ditions at the generating side of the raising transform- 
 ers are determined. The above complete method of 
 procedure, is tedious if carried out mathematically, 
 but if carried out graphically is comparatively simple. 
 
 It is the general practice to neglect the effect of 
 condenser and lowering transformer loss in traveling 
 over the line, but to add this loss to the loss in the 
 high-tension line after the performance has been calcu- 
 lated. If the loss in condensers and lowering trans- 
 formers is five percent of the power transmitted the 
 
 *A method for calculating a transmission line with trans- 
 formers at each end in one solution is given in the articles by 
 Messers. Evans and Sels in the Journal for July, August, Sept- 
 ember, ct scq. 1921. 
 
 error in the calculated results would probably be less 
 than 0.5 percent, a rather small amount. 
 
 In order to simplify calculations, it is the general 
 practice to consider the lumped transformer impedance 
 as though it were distributed line impedance by add- 
 ing it to the linear constants of the line and then pro- 
 ceeding with the calculations as though there were no 
 transformers in the circuit. This simplifies the solu- 
 tion but at the expense of accuracy, particularly if the 
 line is very long, the frequency high or the ratio trans- 
 former to line impedance high. This simplified solu- 
 tion introduces maximum errors of less than two per- 
 cent in the results for a 225 mile, 60-cycle line. 
 
 It has been quite general practice to disregard the 
 effect of the magnetizing current consumed by trans- 
 formers. The magnetizing current required to excite 
 transformers containing the older transformer iron was 
 about two percent and therefore its effect could 
 generally be ignored. Later designs of transformers 
 employ silicon steel, and their exciting current varies 
 from about 20 percent for the smaller of distribu- 
 tion type transformers, to about 12 percent on trans- 
 formers of 100 kv-a capacity and about five percent for 
 the very largest capacity transformers. The average 
 magnetizing current for power transformers is between 
 six and eight percent. This magnetizing current is im- 
 portant for the reason that it is practically in opposition 
 to the current of over-excited phase modifiers used to 
 vary the power-factor. If in a line having 100 000 
 kv-a transformer capacity at the receiving end, the 
 magnetizing current is five percent, there will be a 
 5000 kv-a lagging component. If the capacity of 
 phase modifiers required to maintain the proper volt- 
 age drop under this load is 50000 kv-a the lagging 
 magnetizing component of 5000 kv-a will subtract this 
 amount from the effective rating of the phase modi- 
 fiers, with a resulting error of ten percent in the ca- 
 pacity of the phase modifiers required. 
 
 In the diagrams and calculations which follow, the 
 transformer leakage, consisting of an in-phase com- 
 ponent of current (iron loss) and a reactive lagging 
 component of current (magnetizing current), is con- 
 sidered as taking place at the low-tension side of the 
 transformers. A more nearly correct location would 
 be to consider the leak as at the middle of the trans- 
 former, that is, to place half the transformer imped- 
 ance on each side of the leak. To solve such a solu- 
 tion it would be necessary to solve two complete im- 
 pedance diagrams for the transformers at each end of 
 the circuit. The gain in accuracy of results would 
 not, for power transmission lines, warrant the in- 
 creased arithmetical work and complication necessary. 
 
 In the case of lowering transformers, it would 
 seem that the magnetizing current would be supplied 
 principally from synchronous machines connected to 
 the load. If phase modifiers are located near the 
 lowering transformers, the transformers would prob- 
 ably draw most of their magnetizing current from 
 
140 
 
 PHASE MODIFIERS FOR VOLTAGE CONTROL 
 
 them rather than from the generators at the distant 
 end of the Hne. Partly for this reason, but more 
 particularly for simplicity, the leak of the lowering 
 transformers will be considered as taking place at the 
 lead side of the transformers. On this basis we first 
 
 current also from the low side; that iS from the gen- 
 erators. Both the complete and the approximate 
 methods of solving long line problems which follow, 
 include the effect of not only the magnetizing current 
 consumed by the transformers, but also the losses in 
 
 TABLE V— COMPARISON OF RESULTS AS OBTAINED BY FIVE DIFFERENT METHODS OF CALCULATIONS 
 
 
 
 
 
 
 76,000 KW (88,236 KVA AT 85% PF) 3 PHASE 60 CYCLES RECEIVER VOLTAGE HELD CONSTANT AT 220 KV, 60.000 KV-A CONDENSER AT RECEIVING END 
 LENGTH OF TRANSMISSION 226 MILES ALL TABULATED VALUES REFERRED TO NEUTRAL 
 
 
 
 
 AREA 
 
 IN 
 
 CIRCULAR 
 
 MILS 
 
 a 
 
 RECEIVING END TO NEUTRAL 
 
 8E1IDINQ END TO NEUTRAL 
 
 LOSSES IN KW TO NEUTRAL 1 
 
 LOW TENSION SIDE OF 
 TRANSFORMERS. 
 
 HIGH TENSION SIDE OF 
 TRANSFORMERS 
 
 HIGH TENStON SIDE OF 
 TRANSFORMERS 
 
 LOW TENSION SIDE OF 
 TRANSFORMERS 
 
 LOWERING 
 TRANSFORMERS 
 
 CONDENSER 
 
 HIGH 1 RAISING 
 TENSION LINEJTRANSFORMERS 
 
 TOTAL LOSS 
 
 VOLTS 
 
 4MP8 
 
 PF. 
 
 LEAD 
 
 VOLTS 
 
 AMPS 
 
 Ir 
 
 PF„ 
 
 LAO 
 
 VOLTAGE 
 
 CURRENT 
 
 PFs 
 
 LEAD 
 
 VOLTAGE 
 
 CURRENT 
 
 P*bEN 
 
 LEAD 
 
 IRON 
 
 COPPER 
 
 KW, 
 
 LOSS 
 
 %or 
 
 IRON 
 
 COPPER 
 
 KW, 
 
 LOSS 
 
 »or 
 
 EsN 
 
 % 
 
 Is 
 
 % 
 
 ^GEN-N 
 
 % 
 
 'gen 
 
 % 
 
 ioS ooo 
 
 e 
 
 12703.0 
 
 4oi.3 
 
 ":.'° 
 
 /27 SSi, 
 
 20-f.9 
 
 Itts 
 
 129 OfO 
 /2-t 247 
 
 ;00 
 
 963 
 
 22 7 8 
 736S 
 
 lOO 
 
 103-9 
 
 9377 
 
 73 3A 
 
 I3(, 970 
 /2.f t£7 
 
 100 
 
 ~98.y 
 
 22< / 
 232 3 
 
 /OO 
 /~03% 
 
 97 *9 
 9S 1'* 
 
 2 35 
 
 /30 
 
 lt(. 
 
 /t3-* 
 
 /J a 3 
 
 6 53 
 
 
 172 
 
 29 73 
 
 3079 
 302 1 
 
 /2 3' 
 /208 
 
 II 
 
 
 
 " 
 
 - 
 
 
 
 
 136 783 
 
 <??.-f 
 
 ?Jg7 
 
 lOO^ 
 
 9-f 32 
 
 
 Tools 
 
 
 993 
 
 9J-8 7 
 
 " 
 
 - 
 
 
 <SS3 
 
 tSlO 
 
 6 71 
 
 6.0^ 
 
 - 
 
 Iht. 
 /60 
 
 393s 
 3936 
 
 // 7-f 
 
 l77-53r 
 
 7 7^-6 
 
 ■71 S SOO 
 
 a 
 
 c 
 
 
 :; 
 
 II 
 
 /a 7 ssi 
 
 so-ii 
 
 fit! 
 
 /27?// 
 '?3 0-*/ 
 
 lOO 
 9«2 
 
 2 38 7 
 2373 
 
 /OO 
 
 93 09 
 
 126 668 
 
 /23 'J09 
 
 100 
 99 7 
 
 336 i 
 233 / 
 
 /OO 
 /02.8 
 
 9 7^6 
 9'?.93 
 
 
 : 
 
 ■: 
 
 I3SO 
 MOV 
 '338 
 
 S38 
 
 S63 
 S 3S 
 
 : 
 
 Ih6 
 /■SO 
 /'3 
 
 77S7 
 38S4 
 
 27/7 
 
 // 0' 
 
 ■■ 
 
 D 
 
 C 
 
 
 
 ■- 
 
 
 
 
 '2S £76 
 
 ?g 3 
 
 22?7 
 
 'S£j- 
 
 ft-oi 
 
 I7t 2?2 
 
 /OO s 
 
 2 25.-J 
 
 9?^ 
 
 9i47 
 
 •■ 
 
 
 '■■ 
 
 :ir3 
 
 £ 40 
 £.o9 
 
 - 
 
 /68 
 /fc2 
 
 3 783 
 270 ' 
 
 //./3 
 /O.JO 
 
 79S ooo 
 
 a 
 c 
 
 
 ;; 
 
 ;; 
 
 137 iSt 
 
 20't-9 
 
 99 f3 
 
 177 196 
 1X1 3/3 
 
 
 3-39-i 
 23BO 
 
 /OO 
 
 /OS 8 
 
 9333 
 97 94 
 
 /y-f 9o9 
 137 648 
 
 /OO 
 
 9S2 
 
 227-f 
 233,5 
 
 /OO 
 /02.7 
 
 97 2-? 
 9-f.?0 
 
 
 
 
 // 77 
 
 4 77 
 .50+ 
 4.71 
 
 ; 
 
 /67 
 / 8' 
 
 1 74 
 
 7625 
 2 70/ 
 26/7 
 
 /OSO 
 
 /o tz 
 
 /t} 47 
 
 
 f 
 
 
 - 
 
 
 
 
 
 /2-f S-frf 
 
 98 1 
 
 Z30j 
 
 /004 
 
 93.9-? 
 
 
 /OOS 
 
 
 W3 
 
 
 ■• 
 
 ■ 
 
 
 //?? 
 
 //2fi 
 
 4 79 
 4S0 
 
 
 'Al 
 
 2^33 
 
 255-* 
 
 /O 53 
 
 /o 22 
 
 /7£ £Z7 
 
 2 24.ff 
 
 9^.5£ 
 
 tS'i ooo 
 
 a 
 
 c 
 
 
 
 
 117 SSi 
 
 2o*y 
 
 99t3 
 
 /^i 1311 
 
 131 2/2 
 
 too 
 96 1 
 
 2304 
 2 3? -f 
 
 /ao 
 
 /03 9 
 
 9293 
 9SSS 
 
 123 74 O 
 m 488 
 
 /OO 
 9S2 
 
 22 8.4 
 235/ 
 
 /OO 
 /02 9 
 
 96 99 
 94 S/ 
 
 :; 
 
 
 
 976 
 /Q£9 
 
 /030 
 
 390 
 4 73 
 4 08 
 
 
 /»9 
 /83 
 
 250ff 
 2^ 6 3 
 
 964 
 /0.03 
 
 
 £ 
 
 ■■ 
 
 " 
 
 " 
 
 
 
 
 /33 737 
 
 981 
 
 23/ S 
 
 /OO £ 
 
 93 St 
 
 /2-f 36? 
 
 /OO £ 
 
 ?"2 7 3 
 
 99'S 
 
 9S3I 
 
 ■■ 
 
 
 
 984 
 
 4 OS 
 3 73 
 
 
 /6S 
 
 V-)jo 
 34 'S 
 
 II4 
 
 *A — Transformer impedances treated as lumped at the ends of the line. This is the most nearly accurate of the five methods. 
 It is referred to in the text as the complete solution. 
 
 B — This assumes the impedance of the lowering transformers as line impedance. It takes no account of the leakage of the 
 lowering transformers. 
 
 C — This assumes the impedance of both lowering and raising transformers as line impedance — It takes no account of the leak- 
 age of the lowering and raising transformers. 
 
 D — This is the same as B except that the leakage of the lowering transformers has been added to the load — It is referred to 
 in the text as the approximate solution. 
 
 E — This is the same as C except that the leakage of the lowering transformers has been added to the load. 
 
 have a load current expressed in rectangular co- 
 ordinates with the load voltage as a temporary vector 
 of reference. To this we add algebraically a phase 
 modifier current (loss -|- j or leading) and to this we 
 add the transformer leakage (loss — j or lagging). In 
 other words, these three components of current at the 
 leceiving end of the line add up algebraically upon a 
 
 transformers and phase modifiers flowing over the line. 
 For the purpose of determining the magnitude of 
 errors in the calculated results corresponding to sim- 
 plified methods of calculation where transformers are 
 required at both ends of the line, the calculations shown 
 in Table V were made. Five methods of calculations 
 were made for each of four sizes of cable. A con- 
 
 TABLE W-PERCENTAGE ERRORS IN RESULTS, AS DETERMINED BY VARIOUS METHODS OF CALCULATION. 
 These methods do not take complete account of the effects of the transformers in the circuit 
 
 Method 
 
 At Generator 
 Percent Error 
 
 At Sending^ End 
 Percent Error 
 
 Line Loss 
 
 Percent 
 
 Error 
 
 Transformer Account 
 
 
 Egen 
 
 Igen 
 
 PPe... 
 
 E, ! I, ' pFk 
 
 A 
 
 
 
 
 
 
 
 0,0 
 
 1 
 
 „ ; Complete method — Assumed for comparison as resulting in 100 
 percent values. 
 
 B 
 
 
 
 
 -3.7 
 
 +3.9-0.42 
 
 , „ „- Leak of lowering transformers ignored. Impedance of lowering 
 "•""•"^ transformers assumed as line impedance. 
 
 C 
 
 -1.8 
 
 +2.8 
 
 -2.35 
 
 ... 
 
 
 
 ,f. ._ Leaks of raising and lowering transformers ignored. Impedance 
 "■" of raising and lowering transformer assumed as line impedance. 
 
 D 
 
 
 
 
 -1.6 
 
 +0.4 
 
 +0.55 
 
 , r, „. ! Same as B except that the transformer leak has been added to 
 +0-°"^ the load. 
 
 E 
 
 +0.5 
 
 -0.7 
 
 -1.62 
 
 1 
 
 Same as C except that the transformer leak has been added 
 -^•^^ to the load. 
 
 common vector of reference, thus making it very easy 
 to obtain the resulting load at the receiving end of the 
 line. 
 
 The transformers at the sending end of the line 
 
 stant load, load voltage and condenser capacity were as- 
 sumed for all calculations and the results of these calcu- 
 lations are tabulated in Table W. Thus method D which 
 does not take anv account of the lowering transformer 
 
 have been considered as receiving their magnetizing magnetizing current and assumes the transformer im- 
 
PIlASli MODfl-IKRS FOR VOLTAGE CONTROL 
 
 MI 
 
 pedance as line impedance, gives the sending ena volt- 
 age too low by 3.7 percent and the current too high by 
 3.9 percent. 
 
 Table X contains approximate data upon trans- 
 formers of various capacities 25 and 60 cycles. Since 
 such data will vary greatly for different voltages it 
 must be considered as very approximate but may be 
 found useful in the absence of specific data for the 
 problem at hand. 
 
 Fig. 67 shows complete current and voltage dia- 
 grams for both short and long lines. The diagram 
 illustrating short lines is based upon the current hav- 
 ing the same value and direction at all points of the 
 circuit. On this basis the IR drops of the line and of 
 the raising and lowering transformers will be in the 
 same direction. Likewise their individual IX drops 
 will also be in the same direction. It is evident, 
 therefore, that, for short lines where the capacitance 
 
 voltage circuit in order to combine properly with the 
 linear constants of the line. Although all calculations 
 are made in terms of the high-voltage circuit the re- 
 sults may, if desired, be converted to terms of the low 
 voltage circuit, by applying the ratio of transformation. 
 The transformer impedance to neutral is one- 
 third the equivalent single-phase value. The reason 
 for this is that the I^R and I^X for one phase is 
 identical whether to neutral or between phases. Since 
 the current between phases is equal to the current to 
 neutral divided by VS. the square of the phase current 
 would be one-third the square of the current to neu- 
 tral ; therefore, R and X to neutral will be one-third the 
 phase values. Another way of looking at this is that 
 the resistance and reactance ohms vary with the square 
 of the voltage, and since the phase voltage is V^ times 
 the voltage to neutral, the phase resistance and phase 
 reactance would be three times that to neutral. In 
 
 TABLE X— APPROXIMATION OF RESIS'^ANCE AND REACTANCE VOLTS, OF IRON AND COPPER LOSSES 
 AND OF MAGNETIZING CURRENT FOR TRANSFORMERS OF VARIOUS CAPACITIES 
 
 Capacity 
 
 of 
 
 Transformer 
 
 KV-A 
 
 
 60 CYCLES PER SEC.O.VD 
 
 
 
 25 CYCLES PER SECO.ND 
 
 
 Percen : 
 'Resistance 
 
 Percent 
 ■Reactance 
 
 Percent Loss 
 
 Percen t 
 
 Magnetizing 
 
 Current 
 
 Percent 
 Resbtance 
 
 Percent 
 Re ictance 
 
 Percent Loss 
 
 Pirienl 
 
 Muunclizing 
 
 Current 
 
 Iron Copper 
 
 Iron j Copper 
 
 200 
 30J 
 500 
 
 1.5 
 
 1.3 
 1.2 
 
 5.5 1.4 
 
 5.6 1.3 
 6.0 1.2 
 
 1 
 
 l.S 
 1.3 
 1.2 
 
 10 
 9 
 
 8 
 
 2.6 
 2.15 
 1.85 
 
 4.0 1.1 2.6 
 4.0 1.0 2.1s 
 i.l 1.0 , 1.85 
 
 i 
 
 10 
 10 
 9 
 
 780 
 1000 
 1500 
 
 1.1 
 1.1 
 0.9 
 
 6.3 
 6.5 
 7.0 
 
 1.0 
 0.9 
 0.8 
 
 1.1 
 1.1 
 0.9 
 
 8 
 7 
 6 
 
 l.«S 
 1.55 
 1.4 
 
 i2 0.9 1.65 
 t.O 0.8 1.55 
 6.2 0.8 1.4 
 
 9 
 8 
 8 
 
 2000 
 3000 
 MlflO 
 
 0.8 
 
 0.75 
 
 0.65 
 
 7.0 
 7.0 
 7.0 
 
 0.7 
 0.7 
 0.6 
 
 0.8 
 
 0.75 
 
 0.65 
 
 6 
 6 
 6 
 
 1.3 
 1J2 
 1.1 
 
 6.4 
 6.8 
 7.2 
 
 0.7 1.3 
 0.6 ISt 
 0.5 1.1 
 
 8 
 7 
 7 
 
 7500 
 10000 
 15000 
 
 0.6 
 0.6 
 0.55 
 
 3.0 
 
 8.0 
 8.5 
 
 0.6 , 0.6 
 0.5 0.6 
 0.5 0.55 
 
 5 
 5 
 5 
 
 1.0 
 1.0 
 0.96 
 
 7.8 
 8.0 
 8.0 
 
 0.5 1.0 
 0.5 1.0 
 0.6 0.95 
 
 7 
 6 
 6 
 
 25000 
 3.5000 
 60000 
 
 0.5 
 0.5 
 0.5 
 
 9.0 
 9.5 
 10.0 
 
 0.6 1 0.5 
 0.6 i 0.5 
 0.6 0.5 
 
 S 
 5 
 5 
 
 0.9 
 0.9 
 0.9 
 
 8.0 1 O.t 0.9 
 9.0 0.6 0.9 
 9.0 0.6 0.9 
 
 t 
 6 
 6 
 
 •The actual ohms resistance and ohms reactance will vary as the square of the voltage. The values in above table- must be considered as only roughly 
 approximate. They will vary materially with transformers wound for different voltages 
 
 is neglible, the transformer impedance may be added 
 directly to the line impedance, provided the electrical 
 characteristics on the high-tension side of the trans- 
 formers are not required. 
 
 As the line becomes longer, the current changes in 
 both amount and direction from point to point, as a re- 
 sult of the superimposed distributed charging current 
 of the line. The result of this is that the impedance 
 triangles of the line and of lowering and raising trans- 
 formers change in both size and relative position; so 
 that their individual impedances can no longer be 
 added together and considered as all line impedance, 
 without accepting an error in the results thus obtained. 
 The complete diagram for long lines shown by Fig. 67 
 will be considered later. 
 
 TRANSFORMER IMPEDANCE TO NEUTRAL* 
 
 Transformer constants are referred to the high 
 
 *The writer desires to expre.ss his appreciation of helpful 
 assistance and useful data on transformer characteristics receiv- 
 ed from Mr. J. F. Peters. 
 
 calculating the impedance to neutral, the results will be 
 the same whether star or delta connection is used. 
 
 Even if the transformers at both ends of the 
 transmission line are duplicates their impedance will 
 not be the same if operated on different taps of the 
 windings to accommodate different voltages. In such 
 cases, their impedances will vary as the square of the 
 voltages. For instance, if they are operated at 220 
 and 230 kv at the receiving and sending end respec- 
 tively, then their impedances will have the relation of 
 
 220^ ' 
 
 5^= o.oit;. In other words, if the resistance and 
 
 230^^ 
 
 reactance of the receiving end transformers is 3.185 
 and 39.82 ohms respectively, the sending end trans- 
 formers will have resistances and reactances of 3.481 
 and 43.52 ohms respectively; provided transformer 
 taps corresponding to this higher voltage are used. <« 
 The impedance in ohms of an 18 000 kv-a, three- 
 phase, or of three 6000 kv-a single-phase transform- 
 ers, connected in a bank, may be determined as fol- 
 
J42 
 
 PHASE MODIFIERS FOR VOLTAGE CONTROL 
 
 lows. Assume that they are operated at 104000 volts 
 between conductors (60 046 to neutral) and that the 
 resistance voltage is 1.04 percent and reactance voltage 
 is 4.80 percent. 
 
 The single-phase values are: — 
 6 000 000 
 
 >?.■ 
 
 A-,= 
 
 10^ 000 
 10 i 000 X 0.0104 
 
 104000 X 0.04S 
 
 = sy.j amperes 
 
 57-7 
 
 = 1S.7S ohms resistance 
 
 = S6.52 ohms reactanee 
 
 The values to neutral are, as stated above, one- 
 third of the above ; but, for the sake of uniformity in 
 determining values to neutral, should preferably be de- 
 termined as follows: — 
 
 6 000 000 
 ^ . = 99.92 amperes to neutral 
 
 ^tn = 
 
 ^Vtn 
 
 60 046 X o.oro4 
 
 99.92 
 60046 X 0.0480 
 
 99-92 
 
 = 6.25 ohms resistance to neutral 
 
 2S.S4 ohms reactance to neutral 
 
 If two or more banks operate in parallel, the re- 
 sulting impedance Z,. can be obtained by taking the re- 
 
 to the same kv-a base. For instance, if a 6000 kv-a 
 and a 3000 kv-a transformer each have a resistance of 
 1.04 percent and a reactance of 4.8 percent, their im- 
 pedance is 4.91 percent. Before combining the imped- 
 ances, that of the 3000 kv-a unit should be put in terms 
 of the 6000 kv-a, and the resultant would be: — 
 4.91 X 9.S2 
 
 Zr = 
 
 ■■ J .27 percent at 6000 kva. 
 
 4.91 -f 9.S2 ■ 
 = 0.159 percent resistance volts at 6000 kva. 
 — 3 .'9 percent reactance volts at 6000 kva. 
 
 If the impedance triangles of the two banks to be 
 paralleled are considerably different (that is their ratio 
 of resistance to reactance) it will be necessary to ex- 
 press the impedances in complex form. We have as- 
 sumed above that the triangles are proportional, other- 
 wise they would not divide the load evenly at all 
 power- factors. Solving the preceding problem for 
 the resultant impedance by complex notation, we get: 
 
 {1.04^ J4^) X {2.o8-\-J9.6) 
 ^' - {1.04 -h J4.8) -I- {2.0S + J9.6) 
 ^ -43-9'7 -^ j^9.968 
 3.12 + j/4.4 
 
 COMPLETE DIAGRAM FOR 
 SHORT LINES 
 
 COMPLETC DIAGRAM POB 
 LOWG LINES 
 
 IMPEDANCe OF 
 
 RAISING 
 
 TRANSFORMERS 
 
 LEAKAGE OF 
 
 RAISING 
 
 TRANSFORMERS 
 
 IMPEDANCE OF 
 
 RAISING 
 
 TRANSFORMERS 
 
 IMPEDANCE OF 
 
 LOWERING 
 TRANSFORMERS 
 
 THE BROKEN LINES REPRESENT THE VOLTAGE 
 DIAGRAM FOR THE APPROXIMATE SOLUTION 
 USUALLY employed; THAT IS, WHEN THE IMPEO- 
 ANCE OF THE LOWERING TRANSFORMERS IS ADD- 
 ED TO AND CONSIDERED AS DISTRIBUTED LINE 
 IMPEDANCE-A SIMILAR DIAGRAM MAV BE DRAWN 
 TO INCLUDE THE IMPEDANCE OF BOTH THE LOW- 
 ERING AND THE RAISING TRANSFORMERS- IN THE 
 CASE ILLUSTRATED ' A 220 K« 225 MILE 6Q CYCLE 
 CIRCUIT! THE APPROXIMATE SOLUTION GIVES * 
 VOLTAGE AT THE SENDING END TOO LOW BY t 93% 
 (THE DISTANCE BETWEEN S'- S). 
 
 FIG. 67 — VECTOR DIAGRAMS FOR SHORT AND LONG LINES 
 
 ciprocal of the sum of the reciprocals of the individual 
 impedance. Thus: — 
 
 Z\ Zi 
 
 Zr = 
 
 Z, -t-Za 
 
 In the above example Zt = T I.O^ -\- 4.8^ = 
 4.91 percent . 
 
 To parallel two banks containing transformers 
 duplicates of the above, we get, by the above rule, the 
 following resultant impedance: — 
 
 Z,= 
 
 4.91 X 4.91 
 
 = 2.4^ percent 
 
 4-9' -H 4-9' ' 
 
 Which is just half the impedance of a single bank, 
 as is evident without applying the rule. 
 
 Where two or more banks are to be operated in 
 parallel consisting of transformers not duplicates, then 
 the above rule must be applied to determine the re- 
 sultant impedance. If the impedances are expressed 
 in percent, as is usual, then they must be both referred 
 
 ^ 48. 25 \iS5''32'5S" 
 
 '4.734 /77''46'29" 
 " 3.27 /77''l6'29" ohms 
 = 0.69 + j 3.19 ohms 
 
 Which checks with the results determined above 
 on the percentage basis. 
 
 THE AUXILIARY CONSTANTS 
 
 The graphical construction for short lines repre- 
 sented typically by the Mershon Chart is so generally 
 known and understood that a similar construction 
 modified to take into accurate account the distribution 
 effect of long lines will readily be followed. Both the 
 short and the long line diagrams are reproduced in Fig. 
 68. From these diagrams it will be seen how the 
 three auxiliary constants correct or modify the short 
 line diagram adapting it to long line problems. The 
 two mathematical and three graphical methods of ob- 
 taining the auxiliary constants are Indicated a* the 
 
PHASE MODIFIERS FOR VOLTAGE CONTROL 
 
 143 
 
 bottom of this figure. Since the auxiliary constants 
 are functions of the physical properties of the circuit 
 and of the frequency only, they are entirely independ- 
 ent of the voltage or the current. Having determined 
 
 VECTORS BASED UPON ONe VOLT AND ONE AMPERe AT B5» POWER f ACTOR BEINO 
 OeUVEREO AT THE RECEIVING END-THE DIAGRAMS CORRESPOND TO A LONG LINE 
 
 E8-ER(a,*ia2)*|R(t>,*it'2) 
 ls-lR(3fja2)*ER(c,*jC2) 
 
 QIAORAM rOR SHORT LINES 
 
 REACTANCE VOLTS 
 
 OF H T LINE 
 
 IN QUADRATURE 
 
 V^iTH CURRENT 
 
 RESISTANCE rfOLTS 
 
 OFH.T LINE 
 
 IN PHASE 
 
 WITH CURRENT 
 
 DtAORAM FOR LONG LINES 
 I ZERO LOAD) 
 - CURRENT AT SENDlNG-END 
 
 ■ CONSTANT (CI 
 
 ^^^m^^msm^^M^m^^ 
 
 (VECTOR OF REFERENCE) 
 
 CHARGING CURRENT 
 RESISTANCE VOLTS 
 OF H,T LINE- R'-F 
 
 CHARGING CURRENT 
 REACTANCE VOLTS 
 OF H T, LINE - F - R 
 
 ■ VOLTAGE AT REOEIVtNQ-ENO - 1 VOLT- 
 
 DIAQRAM FOR LONG LINES 
 (LOAD CONDITION) 
 
 THE DOTTED LINES 
 
 CORRESPOND TO A CIRCUIT 
 
 DEVOID OF CHARGING CURRENT. 
 
 THAT 18-A SHORT LINE 
 
 HOW THE AUXILIARY CONSTANTS MAY BE OBTAINED 
 
 = COSHB (BY REAL HYPERBOLIC FUNCTIONS-SEE CHART XVI) 
 = y^^ - gjg <GRAPHICAL-SEE KENNELLY CORRECTING FACTOR CHARTS XVIIIXIX-XX-XXI) 
 ," COSH e (GRAPHICAL-SEE KENNELLY S CHART ATLAS. HARVARD PRESS) 
 - COSH e (ALL GRAPHICAL FROM WILKINSON 8 CHART "A '-SEE CHART V) 
 
 (B)-(^l*i''2)= z['* -6-+ ^725- ♦g^,*352:geo*ETC.l(BY CONVERGENT SERIES-SEE CHARTXI> 
 
 - Iy SINH e (BY REAL HYPERBOLIC FUNCTIONS-SEE CHART XVI) 
 
 = Z 51!^ (GRAPHICAL-SEE KENNELLY S CORRECTING FACTOR CHARTS XVIII ■ XIX I 
 
 - |y SINH e (GRAPHICAL-SEE KENNELLYS CHART ATLAS, HARVARD PRESS) 
 
 - 1^ SINH e (ALL GRAPHICAL FROM WILKINSON S CHART B' -SEE CHART VI) 
 
 / \ / \ r YZ Y^Z' Y^Z^ Y*Z* T 
 
 (CJ-(*=l'i*^)- ^ L" 6'*T20' * 6:3So*362ll8(f ^"'■^J'^^ CONVERGENT SERIES-SEE CHARTXI) 
 
 SINH 9 ;BY REAL HYPERBOLIC FUNCTIONS-SEE CHART XVI) 
 
 51Mi (GRAPHICAL -SEE KENNELLY S CORRECTING FACTOR CHARTS XVUI ■ XIX ) 
 SINH e (GRAPHICAL -SEE KENNELLY'3 CHART ATLAS. HARVARD PRESS) 
 
 SINH S (ALL GRAPHICAL FROM WILKINSONS CHART ' C"-8EE CHART Vli ) 
 
 - Y 
 
 WHERE 9 - Vz7 1^ 
 
 FIG. 68 — HOW THE AUXILIARY CONSTANTS MODIFY SHORT LINE 
 DIAGRAMS ADAPTING THEM TO LONG LINE PROBLEMS 
 
 by any of the five methods referred to, the value for 
 the auxiliary constants corresponding to a given cir- 
 cuit, the remainder of the solution for any receiving 
 end current or voltage is readily performed graphically. 
 
 Constants Oj and a^ — If the line is short electric- 
 ally the charging current, and consequently its effect 
 upon the voltage regulation is small. In such a case 
 constant a^ would be unity and constant Oj would be 
 zero, and the line impedance triangle would be attached 
 to the end of the vector ER representing the receiving 
 end voltage, since this vector also represents the send- 
 ing end voltage at zero load. 
 
 If, however, the circuit contains appreciable ca- 
 pacitance, the e.m.f. of self-induction resulting from 
 the charging current will result in a lower voltage at 
 zero load at the sending end than at the receiving end 
 of the line. Obviously, the load impedance triangle 
 must be attached to the end of the vector representing 
 the voltage at the sending end of the circuit at zero 
 load. This is the vector ER' of the long line diagrams 
 of Fig. 68. In such a circuit the effect of the charg- 
 ing current is sufificiently great to cause the shifting of 
 the point R for a short line to the position R' for the 
 long line. The constants Oj and a^ therefore, deter- 
 mine the length and position of the vector representing 
 the sending end voltage at zero load. Actually the 
 constant Oj represents the volts resistance drop due to 
 the charging current for each volt at the receiving end 
 of the circuit. That is, the line FR' equals approxi- 
 mately one-half the charging current times the resist- 
 ance R, taking into account, of course, the distributed 
 nature of the circuit. For a short line, it would be 
 sufficiently accurate to assume that the total charging 
 current flows through one-half the resistance of the 
 circuit. To make this clear, it will be shown later 
 that, for a 220 kv problem, the resistance per conduc- 
 tor is R = 34.65 ohms and the auxiliary constant C^ = 
 O.001211 mho. Thus, this line will take 0.001211 am- 
 pere charging current, at zero load, for each volt main- 
 tained at the receiving end, and since FR' ;= approxi- 
 R 
 
 mately I^e X — we have FR' or 
 
 0.001211 X 
 
 (A)-(ac*jaj)» [i • y*-54- ♦755"* SOm* "°n'°''°°''^^''°E'''''8E"'E8-8EE0t«RTXI) 34-05 
 
 = 0.020980. The exact value of a^ as calculated 
 
 by hyperbolic functions, taking into account the dis- 
 tributed nature of the circuit is 0.020234. Since the 
 charging current is in leading quadrature with the 
 voltage ER, the resistance drop FR' due to the charg- 
 ing current is also at right angles to ER. 
 
 The length of the line FR or (one-Oj), represents 
 the voltage consmed by the charging current flowing 
 through the inductance of the circuit. This may also 
 be expressed with small error if the circuit is not of 
 
 X 
 great electrical length as I^c X — • The reactance per 
 
 conductor for the 220 kv problem is 178.2 ohms. 
 
 Therefore, FR = 0.001211 X 
 
 178.2 
 
 = 0.107900 and 
 
 a, = I — 0.107900 = 0.892100. The exact value 
 of a, as calculated rigorously, is 0.893955. 
 
 Constants fc, and b^ — These constants represent 
 respectively the resistance and the reactance in ohms. 
 
144 
 
 PHASE MODIFIERS FOR VOLTAGE CONTROL 
 
 as modified by the distributed nature of the circuit. 
 The values for these constants, multiplied by the cur- 
 rent in amperes at the receiving end of the circuit, give 
 the IR and IX volts drop consumed respectively by the 
 resistance and the reactance of the circuit. To illus- 
 trate this, the values of R and X for the 220 kv prob- 
 lem are 34.65 ohms and 178.2 ohms per conductor. 
 The distributed effect of the circuit modifies these 
 linear values of R and X so that their effective values 
 are b-^ = j^./pS" and tj = 172.094 ohms. The line 
 impedance triangle, as modified to take into exact ac- 
 count the distributed nature of the circuit, is therefore 
 smaller than it would be if the circuit were without 
 capacitance. 
 
 Constants c^ and c^ — These constants represent 
 respectively the conductance and susceptance in mhos 
 as modified by the distributed nature of the circuit. 
 The values for these constants, multiplied by the volts 
 at the receiving end of the circuit, give the current con- 
 sumed respectively by the conductance and the suscep- 
 tance of the circuit. To illustrate, the linear value of c^ 
 for the 220 kv problem is 0.001211 mho. The distribu- 
 tion effect of the circuit modifies this linear value so 
 that its effective value Cj = 0.001168. The value of c^ 
 is so small that its effect is negligible for all except for 
 long circuits. An exception to this statement would 
 be that if the line loss is very small compared to the 
 amount of power transmitted the percent error in the 
 value of line loss may be considerably increased if the 
 effect of Cj is not included in the solution. If c^ is 
 ignored, Cj will represent the charging current at zero 
 load per volt at the receiving end. Thus Cj multiplied 
 by the receiving end voltage, gives the charging current 
 at zero load for the circuit. For the 220 kv problem 
 Cj = 0.001168 and this multiplied by 127020, the re- 
 
 ceiving end voltage to neutral, gives 148.36 amperes 
 charging current per conductor. 
 
 Referring to the formulas at the top of Fig. 68, 
 •'^r (Oi -|- / 02) is that part of E^ which would have to 
 be impressed at the sending end if /r = 0, or the line 
 was freed at the receiving end with E^ steadily main- 
 tained there. It may be called "free" component of 
 £s*. Again /, {b^ -\- j b^) is that other part of £, 
 which would have to be impressed at the sending end, 
 if £r ^ 0, or the line was short-circuited at the receiv- 
 ing end, with I^ steadily maintained there. It may be 
 called the "short" component of E^. 
 
 Similarly, the term /, (Oi + / a^) is the compon- 
 ent of /s necessary to maintain /, at the receiving end 
 without any voltage there (£, = 0) ; while E^ (q + 
 y Cj) is the component of I^ necessary to maintain E^ 
 at the receiving end without any current there (/, = 
 0). The reason that c^ is likely to be negative in ordi- 
 nary power lines is because the complex hyperbolic 
 angle of any good power transmission line has a large 
 slope, being usually near 88 degrees. The sinh of such 
 an angle, within the range of line lengths and sizes of 6 
 ordinarily present, is also near 90 degrees in slope. 
 
 The surge impedance Zo = -^ — of such a line is not 
 
 far from being reactanceless ; but it usually develops a 
 
 small negative or condensive slope. This means that the 
 
 I 
 surge admittance Fo ^ ^ usually develops a small 
 
 positive slope. Consequently, C or the product E^ 
 (Tj -}- / '^2) usually slightly exceeds 90 degrees in 
 slope ; or c^ becomes a small negative rectilinear com- 
 ponent. 
 
 *See paper by Houston and Kennelly on "Resonance in A. 
 C. Lines" in Trans. A. I. E. E. April. 1895 
 
CHAPTER XVI 
 A TYPICAL 220 KV PROBLEM 
 
 TO illustrate the method of determining the per- 
 formance of long lines requiring phase modifiers 
 for voltage control, the following 220 kv prob- 
 lem will be considered, which is typical of many like- 
 ly to be considered in the near future. A line necessi- 
 tating such large expediture would warrant 
 a thorough investigation before determining the final 
 design. The conclusions are given only for the pur- 
 pose of illustrating the procedure. 
 
 The Problem — It is assumed that 300000 kw at 
 85 percent lagging power-factor is to be delivered a 
 distance of 225 miles, at 220 kv, three-phase, 60 cycles. 
 Two lines will be required, so that in case one is under 
 repair, the other will transmit the entire 300000 kw 
 load. Since the self -induced voltage would be exces- 
 sive if the 300000 kw were transmitted in emergency 
 over a single-circuit tower line, we will as- 
 sume that each tower line will support two 
 three-phase circuits. The cost of two three- 
 phase circuits per tower line will not be 
 greatly in excess of a single circuit tower 
 line employing conductors of double the 
 cross-section. On this basis each of the four 
 three-phase circuits will normally transmit 
 75 000 kw and, under emergency condition, 
 each of the two circuits on one tower line 
 
 In Table Y is shown a comparison of values of cap- 
 italized losses vs. first costs of conductors for four 
 sizes of aluminum-steel cables considered in connection 
 with this 220 kw problem**. The cost of power losses 
 is based upon rates of 0.3, 0.4 and 0.5 cents per kw 
 hour, an average load corresponding to 80 percent of. 
 the full load loss and a capitalization of these losses 
 at 15 percent. The cost of the cables is based upon 
 29 cents per pound for the complete cable (aluminum 
 plus the steel). All tabulated data is based upon four 
 three-phase circuits. The losses include those in the 
 high voltage line only. If the capacity of transformers 
 or phase modifiers varies materially for different con- 
 ductors, the difference in their losses should be included. 
 If the base load power generated in such a large 
 amount by water power costs 0.3 of a cent per kw-hr., 
 
 will transmit 150000 kw. Such a transmis- 
 sion is illustrated by Fig. 69* 
 
 Economic Size of Conductors — For a 
 fixed transmission voltage and material of 
 conductors, the most economic size of conductor will be 
 found by applying Kelvin's law extended to include, in 
 addition to the cost of conductors, that part of the cost 
 of towers, insulators, line construction, phase modi- 
 fiers, etc. which increases directly with the cost of con- 
 ductors. Kelvin's law is as follows: — 
 
 "The most economical section of a conductor is 
 that which makes the annual cost of the PR losses equal 
 to the annual interest on the capital cost of the conduct- 
 ing material plus the necessary annual allowance for 
 depreciation". Stated another way, "The annual cost 
 of the energy wasted, added to the annual allowance 
 for depreciation and interest on first cost shall be a min- 
 imum". 
 
 *Tht calculations and the illustrations in this article were 
 made in such a way as to tx? equally suited for the series of 
 articles on "Electrical Characteristics of Transmission Circuits" 
 and the Superpower Survey. Figs. 69, 70, 72 and 75 and Charts 
 XXIII. XXV and XXVII appear also in the report of the latter, 
 which is printed as Pnifcsmmal I'afcr 123 by the United States 
 Geological Survey. Similarly, Charts XXIV. XXVI and XXVIII 
 appear in the Paper bv L. E. Imlav in the Jnitrihil of the A. I. 
 /■:. /•;. for June. 1921. (Ed.) 
 
 FIG. 69 — THE TR.ANSFORMEE AND CONDENSER ARRANGEMENT UPON WHICH 
 THE CALCULATIONS FOR THE 220KV PROBLEM HAVE BEEN BASED. 
 
 It is not intended that this arran(cment would, upon a complete study 
 of the problem, be found to be the most desirable. If single-phase 
 transformers were selected, possibly three banks for each double circuit 
 would be found more desirable than four banks, as indicated above. 
 
 the values in Table Y show that the smallest size cable, 
 605 000 circ. mil. will be the cheapest to install. At 0.3 
 cents per kw-hr. the power loss for this cable, capitalized 
 at 15 percent, represents the equivalent of an investment 
 of $2 593 000 for the four three-phase circuits, where- 
 as the cost of the conductors is $3 224000. If the cost 
 of power loss is taken as 0.4 cents per kw-hr., the next 
 larger cable will be the most economical size to use, 
 provided that there is no increased cost of towers, in- 
 sulators, etc. If the losses in transformers or conden- 
 sers vary for the different sizes of cables compared 
 such losres should be included with the conductor losses. 
 There is always a question as to what price should 
 be charged in Kelvin's equation in estimating the cost 
 of power loss. If all power saved could be promptly 
 sold, the cost to allow might be considered the cost at 
 the consumers meter. If, on the contrary', none of the 
 power saved can be sold under any circumstances, 
 
 **An interesting graphic presentation of Kelvin's Law is 
 .civen in the article by Mr. L. J. Moore in the Electrical World 
 for Sept. 24, 1921, p. 612. 
 
146 
 
 A TYPICAL 220 KV PROBLEM 
 
 then the cost to allow is the cost at the generating (Charts XVIII, XIX, XX and XXI). When using 
 switchboard. Intermediate cases may occur. charts it is desirable to read the results from them, at 
 
 The conductor losses of Table Y were taken from two different times as a check against errors in reading, 
 
 the calculated values by the complete method A listed 
 in Table V*. It is usually sufficient to calculate the 
 
 TABLE Y— APPLICATION OF KELVIN'S LAW 
 
 Conductors 
 Circ. Mill. 
 
 Total Loss 
 
 in 12 
 
 Conductors 
 
 Kw 
 
 Cost of Power lost 
 In 12 Conductors, Capitalized at isJS 
 
 Cost of 12 
 Conductors 
 
 at 2QC 
 
 per Lb. 
 
 At 0.3c 
 per Kw-hr. 
 
 At 0.4c 
 per Kw-hr. 
 
 At 0.5c 
 per Kw-hr. 
 
 ♦605 000 
 715500 
 
 18504 
 15S40 
 
 $2 593 coo 
 
 $■2 220000 
 
 |3 458 000 
 $2 960000 
 
 $\ 322 000 
 $3700000 
 
 $3224000 
 $3 837 000 
 
 795000 
 954000 
 
 14304 
 II 712 
 
 |2 040000 
 $1 641 COO 
 
 $2 673 000 
 $2 188 000 
 
 $i 341 000 
 $2 736 000 
 
 $4 244000 
 $50:1 000 
 
 *Thi8 is the smallest conductor which is, in this case, permissible 
 on account of corona limitations. These tabulations are total for four 
 three-phase circuits. It will usually be sufficiently accurate to calcu- 
 late the conductor I'^R loss for one size of conductor and assume 
 that the loss for other sizes will be proportional to their resistances. 
 This assumes that the distribution of current throughout the length 
 of circuit will be approximately the same for the different sizes of 
 conductors compared. The above data is based upon 75 000 kw at 
 85 percent power-factor, three-phase, 60 cycles, delivered over each of 
 the four circuits a distance of 225 miles at 220 kv with a 50 000 kva 
 condenser in parallel with the load on each of the four circuits and an 
 average load equivalent to 80 per cent of full load. It should be noted 
 that the third, fourth and fifth columns do not give the actual cost 
 of the power lost, but give instead the values at which these losses 
 are capitalized. 
 
 loss in the conductors for one size of cable and to esti- 
 mate it for other sizes of cable, assuming that this loss 
 varies as the resistance of the conductors, that is, for 
 a given line, frequency, load, delivery voltage and con- 
 denser capacity the current distribution in the line is 
 approximately the same for various sizes of conductors 
 likely to be considered. Since the conductor loss varies 
 as the square of the current and directly as 
 the resistance, it will be sufficient to estimate 
 
 or the constants may be read from both the Wilkinson 
 
 and Kennelly charts and the results compared. From 
 
 Table V we find r = 0.154 ohms, so that R =0.154 'X 
 
 225 = 34.65 ohms and x = 0.792 so that X = 0.792 'X 
 
 225 = 178.2 ohms. From Table X we obtain b = 5.38 
 
 X10-* so that B =: 5.38 X 225 X io-^= 0.001211 mho. 
 
 G is assumed here as zero. 
 
 From Wilkinson Charts — 
 
 fli = 0.892 
 and since rb = 0.828 
 
 32 = 0.020 
 
 b\ = 32.2 ohms 
 62 = 173-5 ohms 
 and since r6^ = 4-457 
 
 Ci = (too small to read) 
 
 C2 = O.OOII75 
 
 From Dr. Kennelly's Charts — We must first obtain 
 the hyperbolic complex angle of the circuit as fol- 
 lows : — 
 
 Z= 34-65 +7178.1 
 = 181-54 (78 59'46'' 
 
 Y = o +y 0.00121 1 
 = 0.00121 1 190° 
 
 ZY = 0.21984 \i6£s9U§Z 
 
 e = i/2r = 0.4689 /84'29^53'^ 
 Shift e 
 
 From Chart XIX, — 7— = 0.964 /o-4° 
 
 From Chart XXI, 
 
 
 
 Tank 6 
 6 
 
 0.964 /d'lAr'oa'' 
 
 = 1.0785 \o.88" 
 = 1.0785 \o"52'48" 
 
 TABLE Z— CABLE AND CIRCUIT CONSTANTS CORRESPONDING 
 TO A THREE-PHASE, 60 CYCLE CIRCUIT, 225 MILES LONG 
 the loss for other conductors as being in- CONSISTING OF FOUR SIZES OF ALUMINUM CABLES OF AN 
 
 versely proportional to their resistance. ARRANGEMENT EQUIVALENT TO 21 FEET DELTA 
 
 The various constants corresponding to 
 the four sizes of conductors considered are 
 listed in Table Z. It may be interesting to 
 note the variation in these constants corre- 
 sponding to the different sizes of cable for 
 the high-tension line alone, and also when the 
 transformer impedances are included with the 
 line impedance. 
 
 SOLUTION OF THE 220 KV PROBLEM 
 
 Assuming that 605 000 circ. mil. alumi- 
 num-steel cables work out as the most econo- 
 mical size, the next step is the determination 
 of the auxiliary constants A, B, and C for 
 this size of conductor, spacing and 60 cycles. 
 (These constants would have previously been 
 determined when determining the most eco- 
 nomical size). Mathematically these constants 
 may be calculated by real hyperbolic func- 
 tions (Chart XVI) or by convergent 
 series (Chart XI). Graphically, they may be ob- 
 tained from Wilkinson's charts (Charts V, VI and 
 VII) or through the medium of Dr. Kennelly's charts 
 
 AREA Of CONDUCTORS (C M p 
 
 DIA 
 OF 
 
 ALUM 
 COND. 
 
 STRANDS 
 
 LINEAR CONSTANTS OF LINE 
 TO NEUTRAL 
 
 IMPEDANCE TO NEUTRAL 1 
 OF SO 000 KVA BANK OF | 
 
 ALUM 
 
 BTttL 
 
 TOTAL 
 
 COPPER 
 
 EQUIV 
 
 AL 
 
 ST 
 
 r 
 
 X 
 
 g 
 
 .\. 
 
 R 
 
 X 
 
 G 
 
 B„ 
 
 ■ 10-8 
 
 R,» 
 
 Xr. 
 
 Gtn 
 
 ^i;-. 
 
 (toS.coc 
 
 7B.00O 
 
 68 3,iC0 
 
 i-fj^j,^ 
 
 0.9 SI 
 
 S* 
 
 ' 
 
 0./34 
 
 0.7 •}! 
 
 
 
 6.38 
 
 J-S.*! 
 
 / '*..- 
 
 
 
 IZ<' 
 
 4.37 
 
 'f.it 
 
 o 
 
 
 
 y'i.sQo 
 
 <ii.foo 
 
 eoa.'9oo 
 
 AiO.OOO 
 
 ,..,. 
 
 S-* 
 
 7 
 
 0./3' 
 
 0.792 
 
 
 
 S.fS 
 
 : 9.-9 8 
 
 I7S.9 
 
 
 
 '234, 
 
 «.3' 
 
 7 9.t4 
 
 
 
 o 
 
 793,000 
 
 103. lOQ 
 
 BtS.ioo 
 
 eg^.^ 
 
 I.09Z 
 
 J-* 
 
 7 
 
 o.'<; 
 
 0,TS 
 
 
 
 S.-9 9 
 
 ii,33 
 
 '7t.9- 
 
 
 
 /93S 
 
 i.-i'' 
 
 >9.6 9 
 
 
 
 o 
 
 96-^.000 
 
 'i3.7O0 
 
 ',07 7. 700 
 
 ioo.ooo 
 
 i.i9(> 
 
 S4 
 
 7 
 
 0.097g 
 
 O.7.. 
 
 
 
 S.58- 
 
 2 3.00 
 
 '7 '.9 
 
 
 
 /2S6 
 
 i.3r 
 
 .■«.*■# 
 
 o 
 
 c 
 
 
 UNEAH CONST AhfTS 
 
 HYPERBOLIC QUANTITIES 
 
 AUXILIARY CONSTANTS OF CiRCUiT 
 
 R 1 X [GLS. 
 
 e-Yz^ 1 zo-yi 
 
 (A) 1 (B) 1 (C) 
 
 ♦ MI0HTlM(OI(l.lN[,tOHCUT«*ii | 
 
 i^QS.OCQ 
 
 ^■*^<, 
 
 '7g3 
 
 " 
 
 >3,> 
 
 
 J8 
 
 
 .^93->SS*3.C3o:>3', 
 
 .•-:;;j,-:^' 
 
 ~-'°cV.!'^°*\9o-T*''2*-f' 
 
 7.2\5'J0'C7- 
 
 7/ 5. SCO 
 
 if-iS 
 
 ItSt 
 
 
 
 IZ34, 
 
 .02g,9^J.-,t.99 
 = .th7b/9S''-> =11 
 
 3a 
 
 
 
 = in.oilBa'so'"" 
 
 -.cao DflJ^j.OO-.Si 
 
 .'t W*-)5'i5' 
 
 79S.OOC 
 
 Xi33 
 
 17^4 
 
 c 
 
 >Z:>S 
 
 -.^it7/S^^93-Zf 
 
 37 
 
 
 = ,ffS,4//-0'/t- 
 
 .',ycV. I'sV^v^y 
 
 ::T.-VA9i=f8-s:' 
 
 
 9 6 -f. 000 
 
 Z300 
 
 mf 
 
 " 
 
 /SS6 
 
 ■ 039£»-f*j-'*tS'6 
 ^.Aht.) isJ 3r,z' 
 
 J7 
 
 
 ■ f939'*t 1 j.O/ai3« 
 
 
 .'°oo.°°!'(2^^' 
 
 
 
 
 (.OS.OOO 
 
 J?83 
 
 3itia 
 
 
 
 IZ:l 
 
 .0444. 7*3.5. S?f? 
 
 ■♦3 
 
 
 
 3*.Sii3^jJ09..J 
 ^3,'.(.7jBC-3i-0i- 
 
 = .O0.i5«V.'0-Jiil- 
 
 7.-t7\-)'53'y«* 
 
 TS.SOO 
 
 3S 6i 
 
 3,S?3 
 
 
 
 ilti 
 
 
 
 
 =..87079 II' 'i"3f,' 
 
 31.837 fjZOt.SfS 
 
 = ZOI.>.9/i''9t.--1- 
 
 
 
 
 (f/*" 
 
 -.^^..,X\9o-iVi*' 
 
 
 79S.COO 
 
 i9si 
 
 3M31 
 
 
 
 laiS 
 
 .03S-i*7rj.S'SSf 
 
 4'S.-»-#\.}*JiVJ" 
 
 
 Ji..*j-fi-»j_jo;.ojfl* 
 
 -,.e>o,.ai\9o-3-'IS' 
 
 9S^.0C0 
 
 3S 19 
 
 3..U 
 
 ° 
 
 l3Si 
 
 
 4/J 
 
 
 
 S3:.1fl->-j3oa.SBB 
 
 -ao-i.99lsV3rso' 
 
 - ,oo "01 \90'mi* 
 
 o/\j-a3-?s' 
 
 
 bOS.OOC 
 
 ',102 
 
 IS7 » 
 
 
 
 ,3n 
 
 
 ■*«. 
 
 
 .f^''f=*J.033Sit- 
 :.89S' fii 31- 
 
 3i.»l-*U-'f.93^ 
 -.Z47.l.7/8rZi-3r 
 
 -.noii99\90-2ro3' 
 
 .3W3.-..' 
 
 7'S.SOO 
 
 3S'9S 
 
 tsss 
 
 
 
 /13(, 
 
 .O39'Jt3.St/03 
 s:..S42-f /8fOO^ 
 
 ■*s 
 
 
 .8f 73i-*-j.oaoff-»4 
 
 ^.a-Ji /''39'3-il 
 
 = 3^4.8 ihu£i.t' 
 
 ~J.OO"l.-i\9'o''s'i-'*3- 
 
 
 79S.0O0 
 
 32 7C 
 
 3SA 
 
 
 
 1X3^ 
 
 = .St3t/g*-'9-SS' 
 
 ^i5.1^3--fO'<J-*' 
 
 
 , 241.9 jsroyzs' 
 
 -.oo,.7/l^o-?J-i»" 
 
 9S^.0DC 
 
 ...T 
 
 ZSI s 
 
 
 
 ""■ 
 
 -:°l'ifVi*'-vV'iJ'' 
 
 -f-*8.9Fl3*/3'oa 
 
 ^.8tfi/rc8-3r 
 
 ^^.,0/S3'J-p4 3' 
 
 ~.000 C07t ].00'l9/ 
 
 ..oo"f'\:io:3c:53i 
 
 ♦Since two 50000 kv-a banks of transformers will be required at 
 each end the corresponding values tor impedance will be half these 
 amounts. 
 
 Sinhdie 0.964 h^M'^o" 
 
 ♦In the Journal for Dec. 1921, p. 544. 
 
 Tankdie 1.0785 \o''52'48" 
 = ^0.8939 A°i6^48^" 
 a, = 0.8937 
 
 fl2 ■= 0.01096 
 
A TVPICAl. 220 KV PROBLEM 
 
 147 
 
 B - Z^^^ - 181.54 /78°59'46" X 0.964 /d'u'oo" 
 
 
 — 175.0 l-jifi3'^6" ohms 
 
 b\ — 31.2 ohms 
 dj » 172 ohms 
 
 C- K = 0.001211 [go^ X 0.964 ^.^i^/ca" 
 
 = 0.001167 \ 90°24''oo^^ mho 
 O = — 0.000008 mho 
 f2 ■■ o. 001 167 mho 
 
 The auxiliary constants as obtained graphically 
 and by exact mathematical solution, are given in Table 
 ZZ. It is thus seen that the Kennelly charts, although 
 primarily intended for correcting the linear impedance 
 and the linear admittance of circuits for the equivalent 
 IT solution, are highly adaptable to determining the 
 values of the auxiliary constants to a very close degree 
 of accuracy. The use of these charts for obtaining 
 auxiliary constants requires more arithmetical work 
 than the use of the Wilkinson charts. For instance 
 the hybolic angle, Q = yzT of the circuit must first 
 be calculated before the charts can be employed. The re- 
 sults, read from charts, must then be multiplied by the 
 impedance and the admittance of the circuit for obtain- 
 ing auxiliary constants B and C. Auxiliary constant 
 A cannot be taken directly from a single Kennelly chart. 
 To obtain this auxiliary constant from these charts it 
 is necessary to divide the values read from two of these 
 
 sinh 6/e 
 
 charts since A= - — . „ , . . Chart tanh 6/6 is con- 
 tanh 6/6 
 
 structed for angles up to and including 0.50 polar 
 values. This makes it adapted to angles up to i.O polar 
 value when used for determining correcting factors for 
 the equivalent tt solution. This is for the reason that 
 for obtaining such correcting factors we enter this chart 
 with 6/^. However for obtaining auxiliary constant 
 A by means of values read from these charts we must 
 enter this chart with 6 in place of 6/2. This limits the 
 use of the Kennelly charts for obtaining auxiliary con- 
 stant A to circuit angles not exceeding 0.5 polar values. 
 In case the circuit angle has a polar value greater than 
 0.5, Wilkinson chart A may be used provided the line 
 is not over 300 miles long. If the circuit is over 300 
 miles long the auxiliary constants should be determined 
 by mathematical calculation. 
 
 In the following discussion the calculated values 
 for the auxiliary constants will be used, since exact re- 
 sults are required for the purpose of comparing the re- 
 sults with those obtained by the approximate method, 
 a description of which follows the complete solution. 
 
 NORMAL LOAD — COMPLETE SOLUTION 
 
 The complete solution for normal load is given by 
 Chart XXIII. At the top is illustrated the circuit dia- 
 gramatically. Underneath this is stated the load con- 
 ditions, linear and the auxiliary constants for this cir- 
 cuit. The transformer data and method of determining 
 the amperes iron loss, magnetizing current and im- 
 pedance to the neutral of the lowering transformer is 
 
 also shown. Actually the impedance of raising and 
 lowering transformers, even when duplicates, is slight- 
 ly different when the connections are not made to simi- 
 lar taps. This difference is so slight (and so far as the 
 raising transformer is concerned so unimportant) that 
 for simplicity, we are assuming that both raising and 
 lowering transformers have the same impedance. This 
 comprises all the data required for a complete mathe- 
 matical or graphical solution of this circuit. 
 
 Following the data is a complete graphical vector 
 solution of this circuit with symbols placed on all vec- 
 tors indicating the manner of obtaining their values. 
 At the lower left hand corner is placed a complete 
 mathematical solution of the problem, which parallels 
 the graphical solution (one method of solution check- 
 ing the other). In the calculations of the high-voltage 
 circuit the curretjt, in order to include the power-fac- 
 tor, must always be expressed in complex form referred 
 to the vector of reference, as indicated by a dot under 
 the symbol I. 
 
 At the lower right hand comer a method is indi- 
 cated of determining the transmission loss from the cal- 
 culated quantities. The loss in the high-tension line 
 
 TABLE ZZ— AUXILIARY CONSTANTS FOR 220 KV 
 PROBLEM APPROXIMATE SOLUTION 
 
 
 Calculated 
 
 From 
 Wilkinson Chart 
 
 I-"rom 
 Kennelly Chart 
 
 ai 
 
 Ci 
 
 c= 
 
 0.893955 = 100% 
 0.020234 — 100% 
 32.198=100% 
 172.094^ 100% 
 -0.000008=100% 
 0.001168=100% 
 
 0.892 = 9978% 
 0.020 = 98.85 % 
 32.2= 100% 
 1735= 100.82% 
 
 can't read 
 0.001175 = 100.6% 
 
 0.8937 = 9997% 
 0.01996 = 98.65 % 
 32.2=100% 
 172 = 99.95% 
 -0.000008=100% 
 
 o.ooi 167 = 99.91 % 
 
 can be determined graphically by scaling off the voltage 
 and the current at each end of the high-tension line and 
 measuring the angle between the vectors representing 
 the current and the voltage. The current times the 
 voltage times the cosine of this angle will give the power 
 at the point considered and the difference between the 
 power as so determined at the two ends of the high- 
 tension line is the line loss. The losses in transformers 
 and condensers are known and stated at the top of the 
 chart. 
 
 The complete vector diagram is constructed as fol- 
 lows : First draw the horizontal line representing Ei.s, 
 the voltage at the load to neutral. This should be 
 drawn to as large a scale as possible. All other voltage 
 vectors will of course be drawn to the same scale. The 
 vector /l representing the load current is now drawn 
 to as large a scale as can be used without mixing the 
 current vectors with the voltage vectors. This is drawn 
 at an angle of 31° 47' from £ln in the lagging direction, 
 corresponding to a lagging load of 85 percent power- 
 factor. It usually works out that for normal load the 
 power-factor at the receiving end should be slightly 
 lagging and at the sending end slightly leading so that 
 the average power-factor of the line will be close to 
 unity. This will necessitate a phase modifier in paral- 
 lel with the load, having approximately the capacity of 
 the lagging kv-a in the load. 
 
148 
 
 A TYPICAL 220 KV PROBLEM 
 
 The lagging kv-a in the load is equal to the kv-a of 
 the load times the sine of the angle of the load. In this 
 case it is 88235 X sin 31° 47' = 46500 kv-a. The 
 vector diagram is constructed on the basis of a 45 000 
 kv-a condenser in parallel with the load. This con- 
 denser has a power loss of 4.72 amperes to neutral and 
 since this is in phase with the load voltage, we trace from 
 the end of the load current vector horizontally to the 
 right a distance representing 4.72 amperes by the cur- 
 rent scale. The current per terminal for the condenser is 
 118.09 amperes so that the leading component of the cur- 
 rent input of the condenser is 118.00 amperes. Since this 
 is leading it is drawn vertically upward from the last 
 point determined. Actually we will not need to deter- 
 mine the 1 18 amperes leading component, but will com- 
 plete the solid black condenser triangle, since the length 
 of the input line is 1 18.09 amperes. To the vector sum 
 of load and condenser currents thus determined we now 
 add the leakage current of the lowering transformers, 
 the lagging component of which materially effects the 
 capacity of the phase modifiers required because of 
 its nearly direct opposition to it under load. We have 
 assumed that the leakage current required by the low- 
 ering transformers will be supplied by the phase modi- 
 fier on account of its close electrical proximity to the 
 lowering transformers. On this assumption the trian- 
 gle representing this transformer leakage will be located 
 as indicated. There is a loss current of 1.85 amperes in 
 phase with the load voltage and a magnetizing current 
 of 13.9 amperes in lagging quadrature with the load 
 voltage. We thus find that the current /r at the receiv- 
 ing end of the line is 204.17 amperes, lagging 5° i' 16" 
 behind the load voltage. In this case the magnetizing 
 current of the lowering transformer reduces the effec- 
 tive capacity of the phase modifier by an amount of 
 13.9 amperes ; that is by 5.3 percent of the total capa- 
 city of the lowering transformers. 
 
 We next determine the voltage at the high-voltage 
 side of the lowering transformers; that is the voltage 
 jETrn at the receiving end of the transmission line. 
 Knowing the resistance and reactance of the lowering 
 transformer banks to neutral and the curent /r, the 
 transformer resistance voltage drop is plotted in phase 
 with the current /r and the reactance voltage drop in 
 quadrature with the resistance drop as indicated. The 
 voltage at the sending end £sn of the transmission line 
 is next determined by applying auxiliary constants A 
 and B to the voltage and current respectively of the re- 
 ceiving end. 
 
 The base of the impedance triangle for the high- 
 tension line In X ^1 represents the resistance drop of 
 the high-tension line in phase with the receiving end 
 current. In quadrature to this is the reactance volts 
 drop of the line /r X ^2- The voltage at the sending 
 end is thus determined to be 131 858 volts which cor- 
 responds to slightly less than 230 000 volts between con- 
 ductors. An arc of a circle corresponding to the volt- 
 age to be maintained at the sending end will serve as 
 
 a guide in determining the proper capacity condenser 
 necessary to maintain this sending end voltage. An in- 
 crease in condenser capacity rotates the vector /r in 
 a counter-clockwise direction, swinging the line im- 
 pedance triangle also in a counter-clockwise direction 
 thus decreasing the voltage £sn and reducing the line 
 drop. A decrease in condenser capacity rotates the 
 vector /r in a clockwise direction, swinging the line 
 impedance triangle also in a clockwise direction, thus 
 increasing the voltage £sn and increasing the line drop. 
 Thus the effect upon line voltage drop may be readily 
 determined for condensers of various capacities. 
 
 The next step is to determine the current at the 
 sending end. This is done by applying auxiliary con- 
 tants A and C to the current and voltage respectively 
 of the receiving end. It will be noted that the charg- 
 ing current is drawn as leading by 90 degrees the high- 
 tension voltage at the receiving end, which voltage is 
 taken as the vector of reference as in previous discus- 
 sions. The current at the sending end is thus deter- 
 mined to be 220.34 amperes leading the vector of ref- 
 erence by 35° 12'. The impedance triangle for the rais- 
 ing transformers may now be drawn in, the resistance 
 drop of same being drawn parallel with /g. This then 
 gives the voltage at the generators. The current at the 
 generators is determined by adding vectorally to /g the 
 leakage of the raising transformers. It is assumed that 
 the raising transformers will receive their excitation 
 from the generators, in which case the leakage trian- 
 gle will occupy the position shown, resulting in a cur- 
 rent at the generators of 218.88 amperes. 
 
 NORMAL LOAD — APPROXIMATE SOLUTION 
 
 The approximate solution for normal load is given 
 in Chart XXIV. It differs from the complete solu- 
 tion in that the impedance of the lowering transform- 
 ers is added to and considered as a part of the line im- 
 pedance so that there are no transformer impedance 
 triangles to construct. It differs also in that, in the 
 case illustrated, the conditions at the sending end only 
 are obtained, whereas in the complete diagram the con- 
 ditions at both sending end and generators were deter- 
 mined. If the condition at the generators in place of 
 at the sending end is required, the impedance of the 
 raising transformers would also be added to that of 
 the line, the general construction of the diagram remain- 
 ing the same as for the complete solution. 
 
 If it is not necessary to know conditions at both 
 sides of the raising and lowering transformer banks, 
 then it will be seen from a comparison of the two dia- 
 grams that the approximate solution will be simpler, al- 
 though the results will be somewhat incorrect. For in- 
 stance, for the 220 kv problem illustrated, the errors 
 in the results will, according to tabulations in the lower 
 right hand corner, vary from 0.88 to 2.38 percent. If 
 the losses in condensers and transformers were not 
 added to the load (as they are in both these complete 
 and approximate methods) and the transformer mag- 
 
A TYPICAL 220 KV PROBLEM 
 
 i» 
 
 CHART XXIII— 220 KV PROBLEM— NORMAL LOAD 
 
 (COMPLETE SOLUTION) 
 
 ■<LOW TENSION VALUES REFERRED TO THE HIQM TENSION OlROUITl 
 
 RAISING TRANSFORMERS 
 Ztn~ 3' 186^3 39. 62 OHMS 
 
 HIGH TENSION UNE 
 
 "Z ■ 34.66 ♦) I 78.2 OHMS 
 Y- »i .001211 MHO 
 
 LOWERINO TRANSFORMERS 
 ."Zjft - 3 1 86 *! 39.82 OHMS 
 
 /<L* '0_ 
 
 rr ^f-] [ T T T I 1 I 1 1 I T T y T T iT-"^1 
 
 SENDING END 
 
 NORMAL LOAD 
 
 PER 3 PHASE 
 
 PER PHASE 
 
 ciRbuiT 
 
 TO NEUTRAL 
 
 KV-AL-e6 238 
 
 KV-Aln-"^i» 
 
 KW| • 76.000 
 
 Kvy/LN-"""" 
 
 PF,_- 86% LAG. 
 
 PFln-'""^°- 
 
 E|_" 220.000 
 
 Eln-i""" 
 
 1|_-23l 66 
 
 IlN-231 66 
 
 eo OVOLES 
 
 CONDENSER 
 
 (ONE REQUIRED) 
 
 3 PHASE 
 
 TO NEUTRAL 
 
 NEUTRAL 
 
 LINEAR CONSTANTS 
 
 Z - 34.86 ^j I 78.2 OHMS 
 Y " +j .001211 MHO 
 
 AUXILIARY CONSTANTS 
 
 (a) ■ (3| » j 82) - COSH e - 893.965 *j 020.234 - e94io /rir4r 
 
 (B).(b|«,Bj).zS!!|l«. fIsiNHe.32 198 4^172.064 OHMS 
 (C)-(O|<-»02)-lfSi^-=l=8INH8--.000.00»»j.00l.ie8MHO 
 WHERE e-V^ I* 
 
 RCOEnmacNO 
 
 TRANSFORMERS 
 
 rrVW) BANKS IN PARALLEL AT EACH END OF THE LINE) 
 
 I RESISTANCE VOLTS - 668 * 
 
 ' REACTANCE VOLTS - 8 226 » 
 
 1 MAGNETIZING CURRENT - 6.3O0 % 
 [iron LOSS -0.706% 
 
 VALUES TO NELITRAL. 
 KV-Ato"33.333. Etn"'""" lT„-SeS.4 
 
 ON BASIS OF 
 
 100.000 KV-A RATING 
 
 FOR TWO BANKS 
 
 -TN 
 
 'TN- 
 
 Xtn' 
 
 .00868 » 127.020 
 
 282.4 
 .08226 it 1 27.020 
 
 282.4 
 
 - 3. 1 86 OHMS RESISTANCE 
 
 - 39.82 OHMS REACTANCE 
 
 MAGNETIZING CURRENT - ""lay";"^'" - 1 3.9 AMPS AT 1 27.020 VOLTS 
 
 KV-Ag- 46.000 K V-Agu = 1 6.000 
 
 Eq- 220.000 Eon" 1 27.020 
 
 lQ-na.09 Iqn-iis.os 
 
 KWc-.n«-l600 KWo-Loaa-N-600 
 In 
 
 IRON LOSS 
 IRON LOSS 
 
 .,-4.72 
 
 •O-CO88-N 
 ■C-LOSS"*" 'o-L03a-N"<" 
 
 NOTE - THE CONDENSER INDICATED BY BROKEN 
 UNE CIRCLE SERVES AS A SPARE DURING NORMAL 
 OPERATION BUT IS REQUIRED FOR THE EMERGENCY 
 CONDITION. 
 
 LINE CHARACTERISTICS 
 
 LENGTH OF TRANSMISSION 226 MILES 
 
 CONDUCTORS- 3-606.000 CM ST. REINFORCED ALUMINUM 
 
 DIAMETER OF CONDUCTORS 963- 
 
 MAXIMUM ELEVATION 600 FEET 
 
 SPACING- EQUIVALENT 10 21' DELTA SPACING' 
 
 IMPCOANCC Of 
 
 RAISING 
 riUNSFOAMCMa 
 
 -m 1.B6 AMPS AT 1 27.020 VOLTS 
 
 , .00706 "33. 333.333 
 127.020 
 . - 236 KW TO NEUTRAL 
 NOTE-FROM THE VECTOR DIAGRAM IT MAY BE SEEN THAT THE RAISINO 
 TRANSFORMERS WILL BE EXCITED BY A VOLTAGE EQUIVALENT TO 129.918- 
 THE LEAKAGE CURRENT REQUIRED TO EXCITE THE RAISING TRANSF0RMCR8 
 VfILL THEREFORE BE 
 \ MAGNETIZING CURRENT -13.8 AMPS. AT 1 28.8 1 6 VOLTS 
 
 IRON LOSS - 1 .6 1 AMPS. AT 1 29.9 1 8 VOLTS 
 
 CALCULATION 
 FOR RECEIVING-END 
 CURRENT AND VOLTAGE 
 
 I|_-l96.82-j 121.97 AMPS. TO VECTOR £, 
 
 •o' 
 
 Ij- 1.88 
 lo- 203.39 
 
 4.72«jllS.OO 
 1.88-1 13.90 
 
 COS 6*01 I6'«.996I8 
 
 SIN 6* or 18' -.087523 
 
 COS 8'3e' l9'-.98e74 
 
 SIN 8*38' 19*- 14983 
 
 -204.17 \6' 1 ■ 1 8* AMPS. TO VECTOR E.^^, 
 
 - 204.1 7 \8* 36 1 8' AMPS. TO VECTOR OF REFERENCE 
 
 "i 
 
 •201.87 -i 30.66 
 
 E(,N " Yd 27.020 « 998I8*204.I7»3.I86)'*<I27.020« 087623*204.17 « 39.82)* 
 - 128.630 /8'38' 1 9* VOLTS TO VECTOR If, 
 
 CALCULATION FOR HIGH TENSION CIRCUIT 
 
 ■ Il4.999»j 2.803 
 I.767»j33.767 
 
 !r(B)- 
 
 EsN*'3S'7*>*i38.38ff 
 
 - 1 3 1.86S /|8*00'24* VOLTS 
 
 Jr(A)-I8I.08-j 23.23 
 
 Epf|(C)- -l.03*i 160.24 
 
 l3-IB0.06*jl27.0l 
 
 - 220.34 /36*I2 00* AMPft 
 
 DETERMINATION OF LOSSES 
 
 KWln - 26.000 
 
 I^WrN - 1 28.830 « 304. 1 7 1003 8' 98' I r) - 26.988 
 
 KWg^ - 13l.86a«220.34(OO319* 1 1' 38-) - 27.439 
 
 KW(jEN-N" I29.9I6«2I8.88 (00811*62' 28*) - 27.827 
 
 LOSSES TO NEUTRAL 
 
 LOWERING TRANSFORMERS AND CONDENSER 
 
 26.988-28.000 - 888 
 
 HIGH TENSION LINE 27.430 -26.088 - 1471 
 
 RAISINO TRANSFORMERS 27.827 - 27.439 . 888 
 
 TOTAL LOSS TO NEUTRAL - 2827 
 
 AS A PARTIAL CHECK 
 
 LOWERING TRANSFORMERS 
 
 (IRON LOSS 1. 86 XI 27.02 ^- 9SB 
 
 (COPPER LOSS 204.1 7' x 3. 1 e«) - 132 
 
 SYNCHRONOUS CONDENSER ■ SOO 
 
 RAISINO TRANSFORMERS 
 
 (IRON LOSS 1.8 I « 129.918)' ■ 19S 
 
 (COPPER LOSS 220.342x3 186) - 184 
 
 HIGH TENSION LINE 
 
 (VALUE ABOVE ASSUMED AS CORRECT - 1471 
 2827 
 
 1 8.00 AMPS 
 
 EFFICIENCY 
 
 CALCULATION FOR GENERATOR VOLTAGE AND CURRENT 
 
 EFFICIENCY. (HIGH TENSION UNE) 
 EFFICIENCY. (GENERATORS TO LOAD) ' 
 
 g» 966 ^ ».,.- 
 27.439 •*"» 
 26.000 
 27.827 
 
 ' 89 84* 
 
 Eqen^-V (I3I.868X 94442 + 220.34x3. I86)2*(I3I 868 x 32878-220.34x39 82)2 
 - 1 29.0 1 6 /l 6* 26 03* VOLTS TO VECTOR Ig 
 • 128.918 /l9*466r VOLTS 
 
 K V-Ag " 1 3 1 . 868 X 220 34 X 3 - 87. 1 83 PER 3 PHASE CIRCUIT 
 
 I^V-Agj^- I 29.9 18x21 8 88 x 3 - 86.308 PER 3 PHASE CIROOIT 
 
 l^y ./\ QQ-I3I868X|60.24«3- 69.43 1 PER 3 PHASE CIRCUIT 
 
 220.34 (.98394 * j .288 1 3) 
 -21 2.39 ♦i 58.84 
 . 1.81 -j 13 80 (LIAIOUie OF RAISINO TRANSFORMERS) 
 
 '0EN-N-"*"*i"°* 
 
 - 2 1 6.88 /ir62' 28* AMPERES TO VECTOR Eqe^ 
 • 318.88 /3r 38' 26* AMPERES 
 
150 
 
 A TYPICAL 220 KV PROBLEM 
 
 CHART XXIV— 220 KV PROBLEM— NORMAL LOAD 
 
 (APPROXIMATE SOLUTION) 
 
 THIS APPnOXIMATC SOLUTION ASSUMES THAT THE IMPEDANCE OF THE LOWERING TRANsroflMERS MAY BC ADDED TO THE LINE IMPEDANCE AND TREATED AS THOUGH IT WERE DtSTRIBUTCO LINE IM< 
 P£DANCE--THIS ASSUMPTION SIMPLIFIES THE SOLUTION AT THE EXPENSE OF ACCURACY (SEE LOWER RIGHT HAND CORNER OF PAGE; ALSO TEXTl.-THE SOLUTION BELOW IS BASED UPON THE VOLTAGE BEING 
 HELD CONSTANT AT THE LOAD SIDE OF THE LOWERING TRANSFORMERS AND AT THE MICH TENSION SIDE OF THE RAISING TRANSFORMCRS.-'IF THE VOLTAGE IS TO BE HELD CONSTANT AT THE GENERATOR BUS, 
 THE IMPEDANCE OF THE RAISING TRANSFORMERS MUST ALSO BE ADDED TO THAT OF THE UNE-ALL LOW TENSION VALUES ARE REFERRED TO THE HIGH TENSION CIRCUIT, 
 
 RAISING TRANSFORMERS 
 Z-,. = 3, 1 SB + i 39, 82 OHMS 
 
 HIGH TENSION LINE 
 
 Z = 
 
 34,B6«i I 78.2 OHMS 
 -fj .001211 MHO 
 
 rr^^^^^ I »»iii)M I '^^^''^TTraiiCOTp^'^^ i »» »M» p^^^ I »»i)Mi) |''^^^ iiWMiiT ^^'^^ I CMMa I '■"■'VApnfjim^w^ 
 
 'T T T 
 
 U. 
 
 LOWERING TRANSFORMERS 
 Ztn'^'SS^J'^ 820HMS 
 
 zikjo 
 
 I I I T I I I 
 
 SENDING END 
 
 NORMAL LOAD 
 
 PER 3 PHASE PER PHA 
 
 RECEIVING END 
 
 CIRCUIT 
 
 TO NEUTRAL 
 
 KV-Al- 88.236 K V-Aln = 29.4 1 2 
 
 K Wl= 76.000 K Wln ■ 26.000 
 
 PF|_=86%LAQ. PFln=B6%LAG. 
 
 El= 220.000 Eln = 1 27.020 
 
 1^=231.66 Iln=23I.66 
 
 60 CYCLES 
 
 CONDENSER 
 
 (ONE REQUIRED! 
 3 PHASE TO NEUTRAL 
 
 KV-A(3N=I6.000 
 ~ 127.020 
 
 LINEAR CONSTANTS 
 
 Z " 37.836 + j 2 1 8.02 OHMS * 
 Y = +j .001 21 1 MHO 
 •k THIS INCLUDES IMPEDANCE OF LOWERING TRANSFORMERS 
 
 AUXIL IARY CONSTANTS 
 
 (a) " (3| + j 82) " COSH a = .870783 -» j ,02 1 9 1 I 
 
 (B) = (b| t j bj) . z ^"^"° = ^ 8INH - 34,6663 ♦ j 208.83 OHMS 
 
 (C) - (C| + j C2) = Y 5!^ - ^ SINH 9 — .000.009 tj .00 1 1 68 MHO 
 WHERE e- yZY l? 
 
 TRANSFORMERS 
 
 (TWO BANKS IN PARALLEL AT EACH END OF THE LINE) 
 
 ON BASIS OF 
 
 1 00.000 KVA RATING 
 
 FOR TWO BANKS 
 
 RESISTANCE VOLTS . -0.668% 
 
 REACTANCE VOLTS =8,225% 
 
 MAGNETIZING CURRENT =6,300% 
 
 IRONLOSS =0,706% 
 
 KV-A 
 R 
 X 
 
 VALUES TO NELITRAL 
 ,^^-33.333. E.^jj= 127.020 Itn=262.4 
 
 00668X127,020 
 
 KV-Ao'^ooo 
 Eq= 220.000 
 
 'TN 262.4 
 
 .08226 X I 27.020 
 'TN 262,4 
 
 ,0630X33,333,333 
 27,020 
 
 IRONLOSS . 00706X33,333,333 , 
 
 IRON LOSS • 836 KW TO NEUTRAL 
 
 MAGNETIZING CURRENT = - 
 
 3,186 OHMS RESISTANCE 
 
 39,82 OHMS REACTANCE 
 
 13.9 AMPS AT 1 27,020 VOLTS 
 : 1 ,86 AMPS AT 1 27.020 VOLTS 
 
 -ON 
 
 KWo-LO38-l800 KWo_LOSS-N-«<'0 
 ln-ins.<!='>-72 lr'_ins.o-N"*-72 
 
 NOTE - THE CONDENSER INDICATED BY BROKEN 
 LINE CIRCLE SERVES AS A SPARE DURING NORMAL 
 OPERATION BUT IS REQUIRED FOR THE EMERGENCY 
 CONDITION. 
 
 LINE CHARACTERISTICS 
 
 LENGTH OF TRANSMISSION 226 MILES 
 
 CONDUCTORS- 3-606,000 CM ST, REINFORCED ALUMINUM 
 DIAMETER OF (XINDUCTORS 963- rfS- 
 
 MAXIMUM ELEVATION 600 FEET , !# " 
 
 SPACING- EQUIVALENT TO 2 r DELTA SPADING 
 
 CALCULATION FOR 
 RECEIVING-END CURRENT 
 
 , -■l,9e.82-j 1 2 1 .97 AMPS. TO VECTOR Ern 
 |(j- 4.72 tj 1 18.00 
 
 •t- 
 
 lp-203.39-j 17.87 
 
 = 204.17 \6'or 16' AMPERES 
 
 CALCULATION FOR HIGH TENSION CIRCUIT 
 
 E (A)-ll0.607-Fj 2.783 Id(A) = I 77,60 -( 11,11 
 
 Id(B)= I 0.762 -f 1 41,666 
 • " 
 
 Es((=l2l,3e9+j44,e39 
 
 -129.318 /20'ir36' VOLTS 
 
 „(C)° -ll4Ti 147,09 
 = 176,36*1 136,98 
 
 = 222.69 /3r Se'Of AMPERES 
 
 13,9 AMPS. 
 
 DETERMINATION OF LOSSES 
 
 KWln" 26,000 
 
 KWrn" 127020X304,17(003 e'OI'IO'l =26,836 
 
 KWgN" 129,316 X 222,69 (003 1 7' 2e' 26-) - 27,474 
 
 LOSSES TO NEUTRAL 
 
 CONDENSER ■• " 600 
 
 LOWEBINO TRANSFORMERS 
 
 (IRONLOSS) • "S'' 
 
 (COPPER LOSS 204, 1 7^x3, 1 86) ». ■ 133 
 
 RAISING TRANSFORMERS 
 
 (IRONLOSS) . " 236 
 
 (COPPER LOSS) 222.892x3,185 - IBS 
 
 HIGH TENSION LINE 27.474 - (25.835 + 1 33) ■ 1 506 
 
 TOTAL LOSS TO NEUTRAL - 2887 
 
 EFFICIENCY 
 
 25.966 
 2 7,474 
 
 EFFICIENCY. (GENERATORS TO LOAD) = jf^ff 
 
 = 94.52 
 = 89 7r. 
 
 KV-As- 
 KV-Aoo" 
 
 I 29.318 x222.69> 
 1 29.3 I 8 X 1 47.09 ' 
 
 ' 86.393 PER 3 PHASE CIRCUIT 
 ■■ 67.064 PER 3 PHASE CIRCUIT 
 
 4.72 AMPS,- 
 1 16.00 AMPS' 
 
 ERROR IN RESULTS 
 
 VOLTAGE AT SENDING END =-1,93% 
 
 CURRENT AT SENDING END =+1.06% 
 
 POWER FACTOR AT SENDING END = *- 1,02% 
 KV-A AT SENDING END =-088% 
 
 LOSS IN HIGH TENSION LINE -•2.38% 
 
 / NOTE- THESE PERCENTAGES OF ERROR ARE BASED UPON 
 THTASSUMPTION THAT THE COMPLETE METHOD PRO- 
 DUCES 100% VALUES. -THE MINUS SIGNS SIGNIFY RE- 
 SULTS TOO LOW: THE PLUS SIGNS RESULTS TOO HIGH. 
 
A TYPICAL 220 KV PROBLEM 
 
 iji 
 
 netizing current were not taken into account, (as it also 
 
 is in both these methods) the error resulting from the 
 use of the approximate method would be considerably 
 greater than the above values. 
 
 The simplified graphical approximate solution illus- 
 trated by Chart XXIV will yield results sufficiently ac- 
 curate for preliminary work, although for final results 
 it should be supplemented by a mathematical solution 
 and, in cases of very long lines, a complete mathema- 
 tical solution might be desirable. A complete solution 
 as given by Chart XXIII may be followed as a guide in 
 such cases. 
 
 The method of obtaining the auxiliary constants 
 corresponding to the approximate solution is given be- 
 low. The linear constants of the circuit including 
 transformer impedance are determined as follows: — 
 
 A = 
 
 Sinh 01$ 0.957 /o°27'oo" 
 
 Tank did 1.098 \i°oo'oo" 
 = 0.8716 /x^^foo" 
 
 a\ = 0.8713 
 
 as = 0.02206 
 Sinh 
 
 B = Z = 221.28 feo<'o9'23" X 0.957 fa°vi'oo'' 
 
 g 
 
 = 211.76 /io'sG'zf' ohms 
 bi = 34.561 ohms 
 bi = 208.92 ohms 
 
 C= Y =0.001211 I90 X0.957 MtYoo" 
 
 Line 
 
 Transformers 
 
 Resistance 
 (Ohms) 
 
 ...34.650 
 
 ... 3-i8S 
 
 Reactance 
 (Ohms) 
 
 178.20 
 39.82 
 
 Total 37-835 218.02 
 
 Dividing these total values by 225 we obtain the 
 
 following as the impedance per mile of the combined 
 
 circuit. 
 
 r = o.i68i ohms 
 X = 0.969 ohms 
 
 TABLE ZZZ— AUXILIARY CONSTANTS FOR 220 KV 
 PROBLEM, APPROXIMATE SOLUTION 
 
 Calculaled 
 
 From 
 Wilkinson Chart 
 
 From 
 Keniielly Chart 
 
 ai = 0.870783 = 100 % 
 
 aj =^0.02191 1 = 100% 
 bi = 34-5653 = 100% 
 bs = 2o8.83 = ioo% 
 
 Ci=r - 0.000000 ^ 100 % 
 Ci = 0.001158= 100% 
 
 0.8921=102.44% 
 
 0.868 = 99.68% (corrected) 
 0.0221 = 100.86% 
 34-3 = 99-23% 
 211.2^ 101.14% 
 
 -O.OOOOI ^ III. II % 
 
 o.ooi 163 = 100.43 % 
 
 0.8713 = 100.05 % 
 
 0.02206= 100.68% 
 
 34.561=99-99% 
 208.92= 100.04% 
 
 -0.000009 =: 100 % 
 
 O.OOI 159 = 100.09% 
 
 The admittance per mile is assumed the same as 
 before namely : — 
 
 
 b = 
 
 = 5- 
 
 .38 X 10- 
 
 -'' mh< 
 
 
 £r = 
 
 = 
 
 
 
 'ilkinson 
 
 's Charts 
 
 
 
 
 «i 
 
 = 0.892 
 
 
 and 
 
 since 
 
 rb 
 ai 
 
 = 0.904 
 
 = 0.121 
 
 
 
 
 bi 
 
 = 34.3 ohms 
 
 
 
 bi 
 
 = 211. 2 
 
 ohms 
 
 and 
 
 since 
 
 rb-' 
 
 = 4.865 
 
 
 
 
 C\ 
 
 = — 0.000010 
 
 
 
 1*2 
 
 = O.OOI 
 
 63 
 
 From Dr. Kennelly's Charts 
 
 z = 37.835 +y 218.02 
 
 = 221.28 /8o°09^23''^ 
 
 Y •= o +y 0.001 211 
 
 = 0.001 211 190° 
 
 ZY = 0.26797 \i7o°09^23" 
 
 9 = VTY = 0.5177 |85''04^4l'''' 
 
 Sinh e 
 from Chart XIX — ■_ — = 0.957 /fo.45° 
 
 d 
 
 = 0.957 ^lYoo" 
 
 TanhO 
 
 from Chart XXI = 1.098 \i°oo'oo"* 
 
 = 0.0011589 l90°27^oo^' 
 Ci = — 0.000 009 
 Ct => 0.001 159 
 
 The auxiliary constants as obtained graphically and 
 by exact mathematical results are given in Table ZZZ. 
 The same remarks in regard to use of the Kennelly 
 charts for obtaining the auxiliary constants as given 
 under the complete solution also apply when the ap- 
 proximate solution is used. Wilkinson chart A, if used 
 when transformer impedance is added to the line im- 
 pedance, as in the approximate method, requires a cor- 
 rection to constant a^. Constant a^ as read from this 
 chart will be correct but constant a.^ as read from the 
 chart will be too high for the following rea- 
 son. Constant c^ accounts for the rise in 
 voltage along the line at zero load due to the 
 charging current flowing through the line 
 inductance adding directly to the sending 
 end voltage. The section of Wilkinson chart 
 A applying to constant a^ is based upon dis 
 tance and frequency only, so that values read 
 from this section would be the same for a giv- 
 en distance and frequency regardless of whether or not 
 transformer impedance is included with the line con- 
 stants. This section of chart A therefore takes acount 
 only of the voltage lowering effect of the charging cur- 
 rent flowing through the line inductance. In addition 
 to this, it flows also through the transformer inductance, 
 which further lowers the value of a^. The value of a, 
 read from the chart must therefore be reduced. From 
 the chart, Oj = 0.892 volt corresponding to a voltage 
 rise of 0.108 volt which results from a linear conduct- 
 ance reactance of 178.02 ohms. Actually the reactance 
 of the circuit including lowering transformers is 218.02 
 ohms or 22.5 percent greater. Increasing 0.108 volt 
 by 22.5 percent we get 0.132 volt rise, so that Oj becomes 
 i.ooo — 0.132 = 0.868, which is 99.68 percent of the 
 calculated results. 
 
 In the following solutions calculated values for the 
 auxiliary constants are used since exact results are re- 
 quired for the purpose of comparing the results with 
 those previously obtained by the complete solution. 
 
 ♦This was interpolated since this angle lies beyond the range 
 of this chart. 
 
152 
 
 A TYPICAL 220 KV PROBLEM 
 
 EMERGENCY LOAD COMPLETE SOLUTION 
 
 The complete solution for emergency load condi- 
 tions shown by Chart XXV follows the same construc- 
 tion as covered by Chart XXIII for normal load. The 
 difference being that the load is doubled and the con- 
 denser capacity for a circuit increased nearly four 
 times. Thus to force double the amount of power 
 through the line and transformer impedance, with the 
 same voltage drop, it is necessary in this case, nearly 
 to quadruple the condenser capacity per circuit. Thus 
 to meet the emergency condition nearly double the total 
 condenser capacity will be required. This large in- 
 crease in condenser capacity necessitated drawing the 
 current vectors to one half the scale used for current 
 vectors in the normal load diagram. 
 
 EMERGENCY LOAD APPROXIMATE SOLUTION 
 
 The approximate solution for emergency load 
 shown by Chart XXVI follows the same construction as 
 in Chart XXIV for normal 
 load with the exception of 5 
 increased load and condenser 
 capacity. 
 
 ZERO LOAD — COMPLETE 
 SOLUTION 
 
 The complete solution 
 for zero load is shown by 
 Chart XXVII. In this case y 
 the load is made up of a 
 lagging phase modifier load 
 and the leakage of the low- 
 ering transformers. The same 
 constructions are used as for 
 the other complete solutions. 
 
 ZERO LOAD APPROXIAIATE 
 
 SOLUTION 
 
 The approximate solu- 
 tion for zero load is shown by Chart XXVIII. It may 
 be seen from the tabulated errors that this approximate 
 method produces at zero load larger errors than the cor- 
 responding errors for loaded conditions. This is usual- 
 ly of little importance, however, as the light load con- 
 ditions are generally not important. 
 
 PHASE MODIFIER CURVES 
 
 Frequently the normal and maximum amount of 
 power to be transmitted is known ; that is the transmis- 
 sion line, condensers and transformers are designed for 
 a certain maximum load and it is of little importance 
 what condenser capacity would be required for other 
 loads or for various sending end voltages. At other 
 times, especially in preliminary surveys, such data may 
 be very necessary. 
 
 In Fig. yo are plotted curves* showing the phase 
 modifier capacity required to produce certain voltages 
 at the sending end corresponding to various receiving- 
 end loads at 85 percent power-factor and 220 kv. At 
 85 percent power- factor and 220 kv 200000 kw is ap- 
 proximately the maximum amount of power which may 
 be transmitted through the lowering transformers and 
 over this line of three 605 000 circ. mil. cables if the 
 sending end voltage is not permitted to exceed 230 lc\'. 
 This is indicated by the fact that the curve correspond- 
 ing to this load becomes flat when it reaches the 230 
 kv horizontal line. To deliver this maximum load at 
 220 kv through the impedance of this line will require 
 a total condenser capacity of about 300000 kv-a. The 
 economic capacity of the line is reached at loads very 
 much below the maximum theoretical limit of 200 000 
 kw. 
 
 The sending end voltages corresponding to various 
 
 
 
 z 
 
 
 ~ 
 
 
 
 
 
 
 
 
 r 
 
 
 ... 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 z 
 
 i- 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 \ 
 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 no Z 
 
 
 
 
 
 
 s 
 
 
 
 ^ 
 
 
 
 V 
 
 
 
 
 X 
 
 
 
 
 ^ 
 
 N 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 if, z 
 
 M 
 
 
 ^ 
 
 
 s 
 
 \ 
 
 
 \ 
 
 
 
 \ 
 
 
 
 ^ 
 
 \ 
 
 
 
 \ 
 
 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 a 
 
 fe"" 
 
 \ 
 
 
 \ 
 
 
 
 \ 
 
 
 \ 
 
 
 
 ^v 
 
 
 
 '\ 
 
 k 
 
 
 
 \ 
 
 ^'( 
 
 
 
 ^. 
 
 *«« 
 
 
 
 
 
 
 
 
 
 
 HO "* 
 
 j^... 
 
 
 \ 
 
 
 \ 
 
 
 
 K 
 
 
 N 
 
 ^'t 
 
 
 
 ^ 
 
 
 
 
 f^. 
 
 
 
 V. 
 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 \ 
 
 
 ^ 
 
 
 
 \ 
 
 
 
 \ 
 
 
 
 \ 
 
 \\ 
 
 
 
 ^ 
 
 ^ 
 
 
 
 
 ^;^ 
 
 
 
 ' "~ 
 
 
 "-^ 
 
 
 
 
 
 
 3 
 
 Q 3*0 
 
 z 
 
 O«o 
 
 \ 
 
 
 \ 
 
 
 \ 
 
 "''l 
 
 
 % 
 
 
 
 -«^ 
 
 <C'. 
 
 
 
 Kl 
 
 
 
 
 ^ 
 
 N. 
 
 
 
 
 ^ 
 
 
 
 
 
 '~~' 
 
 ~- 
 
 -^ 
 
 
 
 Z 
 
 
 
 
 ) 
 
 \ 
 
 
 ^ 
 
 K 
 
 
 H 
 
 1** 
 
 
 
 fS 
 
 \, 
 
 
 \ 
 
 N 
 
 
 
 -N 
 
 
 
 
 "-^ 
 
 \ 
 
 
 
 
 
 
 
 
 ™2 
 
 u *" 
 
 
 
 
 % 
 
 
 ^ 
 
 < 
 
 
 % 
 
 
 'ss 
 
 f^ 
 
 
 ^ 
 
 k. 
 
 
 
 
 =^ 
 
 
 
 
 s.,,^^ 
 
 >.^ 
 
 
 
 
 
 ;;;:; 
 
 ■--, 
 
 ^ 
 
 
 
 
 Hi 
 
 > 
 
 f<" 
 
 
 
 
 
 % 
 
 '•■ 
 
 H 
 
 f*t 
 
 
 \ 
 
 
 
 Xl 
 
 
 N 
 
 
 
 
 ■~- 
 
 
 
 
 
 
 
 
 
 
 
 L 
 
 
 ~ 
 
 a '"* 
 
 
 
 
 
 
 
 
 N 
 
 
 
 
 N 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 x 
 
 M 
 
 o55~ 
 
 30 
 
 Doa 
 
 ' 5 ».ooo ■ t 
 
 
 lAOl* 
 
 
 
 
 PHASE MODIFIER CAPACITY REQUIRED-KV-A 
 FIG. 70 — PHASE MODIFIER CAPACITY REQUIRED TO MAINTAIN CONSTANT RECEIVER VOLTAGE. 
 
 These curves indicate for a constant load power-factor of 85 percent lagging and con- 
 stant load voltage of 220 kv, the amount of energy which may be delivered to the load 
 over one 225 mile, 60 c}'cle, three-phase circuit consisting of three 605 000 circ. mil alumi- 
 num-steel conductors corresponding to various voltages between conductors at the high-ten- 
 sion side of the raising transformers. The values by which these curves were drawn were 
 determined graphically. For 230 kv at the sending end the maximum amount of power which 
 can be transmitted is approximately 200 000 kw and to force this amount of power through 
 the line impedance will require approximately 300000 kv-a capacity in phase modifiers. 
 
 capacities of phase modifiers in parallel with different 
 receiving end loads for drawing curves such as shown 
 by Fig. 70 are most readily obtained by the following 
 graphical procedure. After auxiliary constants A and 
 B for the circuit under investigation have been deter- 
 mined (preferably through the medium of both the 
 Wilkinson and Kennelly charts) a tabulation of the cur- 
 rent to neutral corresponding to each load for which 
 curves are desired is made. A further tabulation of 
 current to neutral for condensers of various capacities 
 is made. The current to neutral which represents the loss 
 in the various condensers, is also tabulated. The resist- 
 
 *Such curves were suggested by Mr. F. W. Peek, Jr. in 
 an article on "Practical Calculations of Long Distance Trans- 
 mission Line Characterictics" in the General Electrical Review 
 for June, 1913, p. 430. 
 
A TYPICAL 220 KV PROBLEM 
 
 155 
 
 CHART XXV-220 KV PROBLEM— EMERGENCY LOAD 
 
 RAISING TRANSFORMEBS 
 Zy^'l 6S26«il9 tlOHMS 
 
 (COMPLETE SOLUTION) 
 
 (THIS IS DOUBLE LOAD- 1 50.000 KW AT B5% P F LAQOING) 
 
 H10H TtNSION UNE 
 
 2 " M •» • j I '8.2 OHMS 
 Y • «j 0012 1 1 MHO 
 
 U3WERIN0 TRANSfORMERS 
 Z-]^"' M36^jl0.9IOHM8 
 
 HENDINQ END 
 
 EMERGENCY LOAD 
 
 PER 3 PHASE PER PHASE 
 
 Tl [ T y y 1 T T T T 1 i T T If- 
 
 TO NEUTRAL 
 
 NEUTRAL 
 
 TRANSFORMERS 
 
 (FOUR BANKS IN PARALLEL AT EACH END OF THE UNE) 
 
 KV-Al-I'8.470 
 KWi^- 1 60.000 
 PF|_-85*LAO. 
 El' 220,000 
 ||_-463.l 
 
 KV-Aln= 68.824 
 KWln"60 000 
 PFln"86%LAG. 
 Eln-12'020 
 
 Iln»483.I 
 00 CYCLES 
 
 CONDENSERS 
 
 R- 1^ = 1 .6926 OHMS RESISTANCE 
 
 19.91 OHMS REACTANCE 
 
 MAGNETIZING CURRENT - 27 8 AMPS ATI 27 020 VOLTS 
 
 IRON LOSS 3.70 AMPS. AT I 27.020 VOLTS 
 
 IRON LOSS • 470 KW TO NEUTRAL 
 
 MAGNETIZING CURRENT 
 IRON LOSS 
 
 RECEIVING END 
 
 LINEAR CONSTANTS 
 
 Z ■ 34 86 « i I 78.2 OHMS 
 V *j .001211 MHO 
 
 '" ™ AUXILIARY CONSTANTS 
 
 (a) - (a, ♦ j Sj) . .893,956 * j .020.234 
 (B) - C"! 'j ''2) " 82.198 ♦} 172.0*4 OHMS 
 
 \(C)-(C|*J52)-- ooo.ooetj.ooi.i»«i»« 
 
 FOUR 45.000 KV-A SYNCHRONOUS 
 CONDENSFRS PER 3 PHASE CIRCUIT WHEN 
 OPERATING AT FULL LOAD PROVIDE MORE 
 COMPENSATION THAN REQUIRED FOR MEETING 
 THE EMERGENCY CONDITIONS-SINCE BOTH 
 3 PHASE CIRCUITS ON THE SAME TOWERS 
 WILL BE OPERATED IN PARALLEL 7 CONDENSERS 
 MAY BE USED FOR THE TWO CIRCUITS. THE 8TH 
 CONDENSER BEING AVAILABLE AS A SPARE. UPON 
 THIS BASIS THE CONDENSER DATA PER CIRCUIT IS 
 AS FOLLOWS: 
 
 KW, 
 
 I, 
 
 CALCULATION 
 FO R RECEIVING-END' 
 CURRENT AND VOLTAGE 
 
 l|_» 393 64 -j 243.94 AMPS. TO VECTOR E| 
 
 • IS.63tJ4t3.0l 
 T- 3.70 -i 27.80 
 
 o-4l3.87 + i 141 27 
 
 COS 1 8" 60' 49' = ,94638 
 SIN 1 8- 60' 49' ".32304 
 
 COS 1 4' 68' r 
 
 SIN I4'68'0I'>. 26828 
 
 - 437 32 / 1 8' 60' 49' AMPS. TO VECTOR E, 
 
 - 437.32 /I4'68"0I AMPS. TO VECTOR OF REFERENCE 
 
 - 422.49 tj 112.94 
 
 Ern " 7 ( I 27.020 X .94638 + 437.32 x 1.5926)^ + { I 27.020 x .32304 - 437.32 x I 9.9 1)2 
 - 126,152 /l4'680r VOLTS TO VECTOR | 
 
 CALCULATION FOR HIGH TENSION CIRCUIT 
 
 632 jp(A)- 376,42 ♦j I 09.61 
 
 Lg^ -1 06,047 ^j 78,876 
 
 -I32.l64 /3e'38'3r VOLTS 
 
 Is»37442tj266 69 
 
 '463,40 / 34' 1 9' 44' AMPERES 
 
 DETERMINATION OF LOSSES 
 
 KWln . 60.000 
 
 '^Wrn - 1 26. 1 62 X 437.32 (COS 14' 68' IT - 82.874 
 
 KWgN - 132.184X463.40(008 2" I8•47^ - 69.874 
 
 '^Wqei^N- 133.662X460.47(003 9'2400T - 80.871 
 
 LOSSES TO NEUTRAL 
 
 LOWERING TRANSFORMERS AND CONDENSERS 
 
 62.874-60.000 - 3874 
 
 HIGH TENSION LINE 69.874 -62.874 ■ 7000 
 
 RAISING TRANSFORMERS 80.97 1 - 69.874 -__7»7 
 
 TOTAL LOSS TO NEUTRAL -10.671 
 
 AS A PARTIAL CHECK 
 
 LOWERING TRANSFORMERS 
 
 (IRON LOSS 3-7 X 127.02 ." 470 
 
 (COPPER LOS3437 32^x1,(936) . J04 
 
 SYNCHRONOUS CONDENSERS ._ _- ..^ • 3100 
 
 RAISING TRANSFORMERS 
 
 (IRON LOSS 3,52 X 133,662) - 470 
 
 (COPPER LOSS 463 4' X 1 6926) - 327 
 
 VIIQH TENSION LINE 
 
 (VALUE ABOVE ASSUMED AS CORRECT - 7000 
 413.01 AMPS io«77 
 
 i8 53AMPa EFFICIENCY 
 
 EfflOIENOY (HIGH TENSKW LINE) - |f||J • 88.3I« 
 tFFlOlENCY, (GENERATORS TO LOAD)- |g g°° - 62.41* 
 
 CALCULATION FOR GENERATOR VOLTAGE AND CURRENT 
 
 ^OEN-N " Vl 1 32. 1 84 x .999 1 8 ♦ 463.4 x 1 .6926)'' + ( I 32. 1 84 x .040359 t 463.4 x 1 9.9 1 )' 
 =■ 1 33.652 Iff 10' 23" VOLTS TO VECTOR Ig 
 - 1 33.662 Ao' 30' OT VOLTS 
 
 KV-As" I 32. 1 84 X 463,40 x 3 - I 70.789 PER 3 PHASE CIRCUIT 
 KV-AgEN" I 33.662 X 460 47 x 3 - | 64.490 PER 3 PHASE CIRCUIT 
 ■^V-A cc * ' 32' I 84 X 1 48.1 8 X 3 - 67.969 PER 3 PHASE CIRCUIT 
 
 4634 (.9942 -j. 1 0763) 
 
 = 460.77 -j 48 76 
 
 i 28 46 (LEAKAGE OF RAISING TRANSFORMERS) 
 
 '0EN-N"«4.=9-j76 2l 
 
 ■ 480.47 \9' 24' 00' AMPERES TO VECTOR Eq£h 
 -4S0>«7 /31'06 07' AMPtRe3 
 
I.S4 
 
 A TYPICAL 220 KV PROBLEM 
 
 CHART XXVI— 220 KV PROBLEM— EMERGENCY LOAD 
 
 (APPROXIMATE SOLUTION) 
 
 THIS APPROXIMATE SOLUTION ASSUMES THAT THE IMPEDANCE OF THE LOWERING TRANSFORMERS MAY BE ADDED TO THE LINE IMPEDANCE AND TREATED AS THOUGH IT WERE DISTRIBUTED LINE IM- 
 PEDANCE--THIS ASSUMPTION SIMPLIFIES THE SOLUTION AT THE EXPENSE OF ACCURACY (SEE LOWER RIGHT HAND CORNER OF PAGE; ALSO TEXTl.— THE SOLUTION BELOW IS BASED UPON THE VOLTAGE BEINO 
 HELD CONSTANT AT THE LOAD SIDE OF THE LOWERING TRANSFORMERS AND AT THE HIGH TENSION SIDE OF THE RAISING TRANSFORMERS.--IF THE VOLTAGE IS TO BE HELD CONSTANT AT THE GENERATOR BUS. 
 THE IMPEDANCE OF THE RAISING TRANSFORMERS MUST ALSO BE ADDED TO THAT OF THE LINE-ALL LOW TENSION VALUES ARE REFERRED TO THE HIGH TENSION CIRCUIT. 
 
 RAISING TRANSFORMERS 
 Z.^"l 6926 + j I9.ai OHMS 
 
 \l (000000 I 
 
 HIGH TENSION LINE 
 
 34.66 +j 178.2 OHMS 
 +j .001211 MHO 
 
 LOWERING TRANSFORMERS 
 
 SENDING END 
 
 EMERGENCY LOAD 
 
 PER 3 PHASE PER PHASE 
 
 T I T T y T 1 1 y I I ' 
 
 KV-Al= I 76.470 
 
 KWl" 1 60,000 
 
 PFl=86%LAG. 
 
 Il=463. I 
 
 TO NEUTRAL 
 
 KWln- 60.000 
 PF|N = e6%LAG. 
 = 127.020 
 = 463.1 
 
 NEUTRAL 
 
 LINEAR CONSTANTS 
 
 Z "37.S36+J2I8.02OHMS* 
 Y= +j. 001 21 1 MHO 
 * THIS INCLUDES IMPEDANCE OF LOWERING TRANSFORMERS 
 
 AUXILIARY CONSTANTS 
 
 (a) = (3| + j ^2) = COSH e " -870783 + j .02 1 9 1 
 
 RECEIVING END 
 
 TRANSFORMERS 
 
 (FOUR BANKS IN PARALLEL AT EACH END OF THE LINE) 
 
 H-jtf" 1 .6926 OHMS RESISTANCE 
 
 Xt^|= I 9.9 1 OHMS REACTANCE 
 
 MAGNETIZING CURRENT - 27.8 AMPS. AT 1 27,020 VOLTS 
 
 IRON LOSS = 3.70 AMPS. AT I 27.020 VOLTS 
 
 IRON LOSS ..... - 470 KW TO NEUTRAL 
 
 60 CYCLES 
 
 CONDENSERS 
 
 FOUR 45,000 KVA SYNCHRONOUS 
 CONDENSERS PER 3 PHASE CIRCUIT WHEN 
 OPERATING AT FULL LOAD PROVIDE MORE 
 COMPENSATION THAN REQUIRED FOR MEETING 
 THE EMERGENCY CONDITIONS-SINCE BOTH 
 3 PHASE CIRCUITS ON THE SAME TOWERS 
 WILL BE OPERATED IN PARALLEL 7 CONDENSERS 
 MAY BE USED FOR THE TWO CIRCUITS. THE 8TH 
 CONDENSER BEING AVAILABLE AS A SPARE, UPON 
 THIS BASIS THE CONDENSER DATA PER CIRCUIT 
 IS AS FOLLOWS 
 
 3 PHASE TO NEUTRAL 
 
 470 
 
 aos 
 
 CONDENSERS 
 
 LOWERING TRANSFORMERS 
 
 (IRON LOSS! ~ 
 
 (COPPER LOSS 437,32' « 1. 6926) ... 
 RAISING TRANSFORMERS 
 
 (IRON LOSS! .„-■ - 470 
 
 (COPPER LOSS) 452.532 X 1. 5925 " 33^ 
 
 HIGH TENSION LINE 60.248 - (52.570 ♦ 306) ■ 7373 
 
 TOTAL LOSS TO NEUTRAL - 1 1 .044 
 
 EFFICIENCY 
 
 EFFICIENCY. (HIGH TENSION LINE) , 
 EFFIOIENOV. (CiENERATORS TO LOAD) 
 
 62,876 
 60.248 
 60.000 , 
 61.044 
 
 87.76* 
 6I.«I« 
 
 % ERROR IN RESULTS 
 
 ■ 1 34.004 /44' 36' 07" VOLTS 
 KV- As = I 34.004 X 462,63 X 3 = I 8 1 .922 PER 3 PHASE CIRCUIT 
 KV-Aco- 134.004X147.09X3= 69. 1 32 PER 3 PHASE CIRCUIT 
 
 VOLTAGE AT SENDING END "+1.39% 
 
 CURRENT AT SENDING END "-0.20% 
 
 POWER FACTOR AT SENDING END - - 0,66% 
 KV.A AT SENDING END -+1.18% 
 
 LOSS IN HIGH TENSION LINE = + 6,33% 
 
 NOTE - THESE PERCENTAGES OF ERROR ARE BASED UPON 
 THE ASSUMPTION THAT THE COMPLETE METHOD PRO- 
 ^— DUCES 100% VALUES, -THE MINUS SIGNS SIGNIFY HE- 
 \ SULTS TOO LOWi THE PLUS SIGNS RESULTS TOO HIGH. 
 ^ie.53AMP3 
 
 Note : — Linear constant Z, as used in this chart, incorrectly includes impedance of two banks, whereas it should have included 
 four banks of transformers. This error will not, however, materially affect the result. 
 
A TYPICAL 220 KV PROBLEM 
 
 155 
 
 ance, reactance, iron loss and magnetizing currents of 
 the transformer banks to neutral should also be deter- 
 mined for all capacity transformer banks required. 
 With the above data tabulated any draughtsman can be 
 instructed how to draw vector diagrams of the circuit 
 to determine the sending end voltages corresponding to 
 
 ficient to locate the curve, although more points were 
 calculated for drawing the curves of Fig. 70. This 
 method of obtaining condenser capacities correspond- 
 ing to sending end voltages is a cut and try method. 
 Il has one important advantage in its favor. That is, 
 the results check each other, so that an error in one 
 
 CHART XXVM— 220 KV PROBLEM—ZERO LOAD 
 
 (COMPLETE SOLUTION; 
 (THIS OORRESPONOS TO NORMAL LOW OONNECTJONSI 
 
 *T ZCRO 1.0*0. WITH 230.000 VOITS MAINTAINED BETWCCN CONDUCTORS (132.T8T VOLTS TO NtUTRALl. AT THE MIGM TENSION SIDE OF THE RAISING TRANSfOHMERS THE VOtTAGC AT THE HIGH TCI*. 
 SION SIDE Of THE LOWERING THANSEORMERS [NEGLECTING THE EFFECT OF THE LAGGING MAGNETIZING CURRENT OF THE LOWERING TRANSFORMERSi WILL RISE TO 230.000 DIVIDED ST Ai- 230,000 DIVIDED ST 
 .89416 ■ 25T.219 VOLTS BETWEEN CONDUCTORS HAB.SIO VOLTS TO NEUTRAL). ACTUALLY THE GREATLY INCREASED LAGGING MAGNETIZING CURRENT OF THE LOWERING TRANSFORMERS WHEN EXCITED BT AB- 
 NORMALLY HIGH VOLTAGE WILL NOT PERMIT OF THE RECEIVING END VOLTAGE REACHING SUCH A HIGH VOLTAGE UNLESS THE GENERATOR VOLTAGE RAISES MOMENTARILY TO A VALUE GREATLY IN EXCESS Of 
 230,000 VOLTS. IF HOWEVER THE LOWERING TRANSFORMERS ARE DISCONNECTED FROM THE CIRCUIT, THE INCREASED LEADING CHARGING CURRENT OF THE LINE, REACTING UPON THE GENERATOR FIELDS, 
 CCVleiNED WITH A MOMENTARY OVER SPEED OF THE GENERATORS MAY CAUSE THE RECEIVING END VOLTAGE TO GREATLY EXCEED THE ABOVE VALUE. 
 
 IN ORDER TO HOLD THE VOLTAGE AT THE RECEIVING END CONSTANT AT 220,000 VOLTS BETWEEN CONDUCTORS 1127.020 VOLTS TO NEUTRALI AT ZERO LOAD IT WILL BE NECESSARY TO PLACE AN ARTL 
 FICIAL LAGGING LOAD AT THE LOAD END OF THE LINE--THIS IS ACCOMPLISHED BY OPERATING ONE OF THE SYNCHRONOUS CONDENSERS WITH ITS FIELDS UNDER EXCITED— BY CONSTRUCTING SEVERAL VECTOR 
 DIAGRAMS FOR THIS CIRCUIT EACH BASED UPON DIFFERENT VALUES OF REACTOR LOAD. A CURVE MAY BE DRAWN BY PLOTTING THE REACTOR LOADS AGAINST THE CORRESPONDING BCNOING END VOLTAOCS.— 
 FRON THIS CURVE THE REACTOR CAPACITY CORRESPONDING TO 23O.0O0 VOLTS BETWEEN CONDUCTORS AT THE SENDING END WILL BE SEEN TO BE APPROXIMATELY 30.000 RV-A 
 
 LEAKAGE OF 
 
 RAISING 
 TRANSFORMERS 
 
 CONDENSER 
 
 (ONE RE(3UIRED) 
 3 PHASE TO NEUTRAL 
 
 KV-A(;= 30.000 KV-AoN = io.ooo 
 
 Ec=220.000 EcM=U7.O20 
 
 0=78,73 
 
 KWc 
 
 KW, 
 
 'ON- 
 
 0-LOSS-N ' 508 
 'o-LOSS-N'=*00 
 
 LINEAR CONSTANTS 
 
 Z = 34,66 tj 178,2 OHMS 
 
 Y= +j .001211 MHO gig 2g 
 
 AUXILIARY CONSTANTS ?>f, M 
 
 (a) - (8l +j 82) -OOSH 8 = . 893.966 +j. 020.234" .BS4I8 /l'(7'47' " j _,^ 
 (B).(b|*jbj) = 25Mil=yisiNH6.32.l98tjl72.094OHM8 |S |S 
 
 ,= -l.04+jl6269 AMPERES 
 'q£N-N =57.1 I /si'Se'CA" AMPERES 
 
 PFqen"'*°*i-"°"*° 
 
 (C)-(c,*jOj).y§^. 
 
 WHERE 6" fZY 
 
 = siNH8- -.ooo.ooe+j.ooi,ie8 MHO 5s 
 
 0SO" 
 
 84- 44' 3 I • PFso - 9 ' ** LEADING 
 
 /^gen-no" "°-'°° / 0' 30' 1 6" VOLTS 
 
 ^-(vfclOHOmt^ERENOE) VE,^- 1 27.020 VOLTa "^EgNo " ' 
 
 CALCULATION FOR 
 RECEIVING-END CURRENT AND VOLTAGE 
 
 Ico" *00-i'8.e3 COS, 86' 22' 67- -.063096 COS. Be" 21' 27- - .08363 
 |.^P= 1.86-j 13,90 SIN. 86' 22' 67" =9980 I SIN. 86" 21' 27' - .99798 
 |p|-l= 6 86- j 92.63 AMPERES 
 
 = 92.71 \86' 22' 67" AMPERES TO VECTOR E^no 
 
 i.»7« /0'ir36 ' VOLTS 
 
 LEAKAGE OF 
 
 LOWERING 
 
 TRANSFORMERS 
 
 Ion - 92.7 I \86'2|-27- AMPERES TO RECTOR OF REFERENCE 
 - 6.89 -j 92.62 AMPERES 
 
 Erno" 7 " ''•°^° " °"°^5 + 92,7 1 k 3. 1 86)^ + ( I 27.020 « .9980 1 + 92.7 1 « 39.62)^ 
 • 1 30.724 LSS11112i" VOLTS TO VECTOR Ifio 
 
 CALCULATION FOR HIGH TENSION CIRCUIT 
 
 DETERMINATION OF LOSSES 
 KWlno -0 
 
 I^Wrno - 130.724 k 92.7 1 ((XJ8 6«" 8 r 27n • 7T0 
 ^'^ma >l32.974K703a((X»W'44'3n - MT 
 KWqe,^.,^- 1 30 206 K67.il (008 8 r 26' 4**) - MOt 
 
 LOSSES TO NEUTRAL 
 
 tOWERINO TRANSFORMERS AND CONDENSER - 770 
 
 tW3H TENSION LINE 667-770 , " 87 
 
 RAiaiNQ TRANSFORMERS I 1 08-867 - 861 
 
 'TOTAL LOSSjrO NEUTRAL - 1 108 
 
 AS A PARTIAL CHECK 
 
 -RN0(A)= I 16.861 +j2e46 
 
 1ro(B) = ie.iii-ii966 
 
 EsNO' 1 32.972 tj 680 
 
 Iro(A)= 7.13 - j 8269 
 
 ^RN0(C)= - 1.04 *i 162 69 
 
 |gn= 8.09 + j 70.10 
 
 I C(}S. -.091642 
 
 - 132.974 / 0' I 7' 36 ' vni T.-i - 70.36 / 86' 02' 06" AMPERES 
 
 CALCULATION FOR GENERATOR VOLTAGE AND CURRENT 
 
 ^QEN-NO- I (I 32,974 X 09 I 642 4. 70,36 K 3, 1 SSl'' * ( I 32,974 x .99679 - 70.36 x 39.82)' 
 
 LOWERINO TRANSFORMERS 
 
 (IRON LOSS 1,86X127.02 ... - 
 
 93S 
 
 (COPPER LOSS 82,71 '»1.H6) - 
 
 SYNCHRONOUS CONDENSER 
 
 RAISINO TRANSFORMERS 
 
 (IRON LOSS 1,80 X 130,308) . «. - 
 
 (COPPER LOSS 70 3«'x3.||6) - 
 
 HIGH TENSION LINE 
 
 (VALUE ABOVE ASSUMED AS CORRECT - _ 
 
 IT 
 •0( 
 
 iis 
 
 16 
 
 87 
 
 1108 
 
 EFFICIENCY 
 
 
 EFFICIENCY, (HIOH TENSION LINE) "517" 
 
 ef>.8(» 
 
 1 30.206 /84'3r 60' VOLTS TO VECTOR 
 1 30.206 / 0*30' 16' VOLTS 
 
 70.38 (.0963 1 6 -fj. 99646) 
 = 6,71 +j 70,04 
 
 - 1,80 -j 13,67 (LEAKAGE OF RAISINO TRANSFORMERS) 
 'qEN-N- 8 61 *j 66,47 
 
 -67,11 /8r26'48' AMPERES TO VECTOR Eqe^-no 
 
 - 67, 1 I /8I'66'04' AMPERES 
 
 . (COS. = .096316 
 I SIN. -.99646 
 
 C(3S.SI'26'48'- 14902 
 
 KV-Aso° ' 32 974 X 70.36 x 3 - 28.068 PER 3 PHASE CIRCUIT 
 
 KV-Aqen-o" ' 30.206 X 67. 1 I X 3 - 22.308 PER 3 PHASE CIRCUIT 
 
 KV-Aco" 132.974(001 168 xU8,6IO)x 3 = 69.197 PER 3 PHASE CIRCUIT * 
 ABASED UPON H.T. LINE BEING OPEN AT RECEIVING END. 
 
 the various receiving end loads and different phase mod- 
 ifier capacities. 
 
 The graphical method used in determining the 
 values to plot the curves of Fig. 70, is illustrated by Fig. 
 71. Three solutions are illustrated, two with condens- 
 ers of different size and one without condensers. 
 Three such solutions for each load will usually be suf- 
 
 of the graphical constructions corresponding to a given 
 load will be detected, since the point will not lay in the 
 curve and an error in a curve corresponding to a given 
 load will be detected by the curves of Fig. 72. 
 
 CAPACITY OF PHASE MODIFIERS 
 
 The curves of Fig. 70 show that, for a constant 
 delivered load, power-factor and voltage, the leading 
 
156 
 
 A TYPICAL 220 KV PROBLEM 
 
 capacity of phase modifiers required goes down as the 
 Hne drop increases. For instance 75 000 kw at 85 per- 
 cent power-factor and 220 kv can be delivered over this 
 line with 230 kv sending end voltage, if 43 000 kv-a 
 condenser capacity is placed in parallel with the load. 
 If, however, a line drop of 20 kv is selected in place 
 of 10 kv, the sending end voltage will be 240 kv and 
 the corresponding condenser load will be reduced to 
 approximately 30000 kv-a. On the other hand this 
 increased line drop will necessitate a greater capacity 
 
 The dotted line in Fig. 70 is simply the zero load line 
 thrown over to the leading load side to facilitate studv 
 in phase modifier capacity. For instance, projection 
 from the points where the dotted line intersects a load 
 curve will give the minimum capacity of phase modifier 
 on the bottom scale and the corresponding sending end 
 voltage on the vertical scale to the left. Thus with a 
 load of 75 000 kw, intersection of the dotted line with 
 this load curve indicates that 33 000 kv-a phase modi- 
 fier capacity will be required both at this load and at zero 
 
 CHART XXVMI— 220 KV PROBLEM— ZERO LOAD 
 
 (APPROXIMATE SOLUTION) 
 
 (THIS CORRESPONDS TO THE NORMAL LOAD CONNECTIONS) 
 
 THIS APPROXIMATE SOLUTION ASSUMES THAT THE IMPEDANCE OF THE LOWERING TRANSFORMERS MAY 8E ADDED TO THE LINE IMPEDANCE AND TREATED AS THOUGH IT WERE DISTRIBUTED UNE IM. 
 PCDANCE-THIS ASSUMPTION SIMPLIFIES THE SOLUTION AT THE EXPENSE OF ACCURACT (SEE LOWER RIGHT HAND CORNER OF PAGE: ALSO TEXTI.-THE SOLUTION BELOW IS MSEO UPON THE VOLTAGE BEINC 
 MELD CONSTANT AT THE LOAD SIDE OF THE LOWERING TRANSFORMERS AND AT THE HIGH TENSION SIDE OF THE RAISING THANSFORMERS.-IF THE VOLTAGE IS TO BE HELD CONSTANT AT THE GENERATOR BUS 
 THE IMPEDANCE OF THE RAISING TRANSFORMERS MUST ALSO BE ADDED TO THAT OF THE LINE-ALL LOW TENSION VALUES ARE REFERRED TO THE HIGH TENSION CIRCUIT ■ Ht glhehator BUS. 
 
 AT ZERO LOAD. WITH ?3O.000 VOLTS MAINTAINED BETWEEN CONDUCTORS (I32.T87 VOLTS TO NEUTRAL). AT THE HIGH TENSION SIDE OF THE RAISING TRANSFORMERS THE VOLTAGE AT THE HIGH TEN- 
 8.0N SIDE OF THE LOWERING TRANSFORMERS .NEGLECTING THE EFFECT OF THE LAGGING MAGNETIZING CURRENT OF THE LOWERING TRANSFORMERS? vlrLL RISE TO 230 OOOOIvVoEDByTa"/^^^ DmSED BY 
 
 ■ 7I0S -ZSA.OAS VOLTS BETWEEN CONDUCTORS 152 »5tVOLTS TO NtUTRALi. ACTUALLY THE GREATLY INCREASED LAGGING MAGNETIZING CURRENT OF THE LOWErTnG TRANSFORMERS WHEN EXcfTED BY AB- 
 NORMALLY HIGH VOLTAGE WILL NOT PERMIT OF THE RECEIVING END VOLTAGE REACHING SUCH A HIGH VOLTAGE UNLESS THE GENERATOR VOLTAGE RAISES mSmentARIlV TO A VaIuE GREATLY IN EXCESS OF 
 i^^^.Z:'^'-!^^". 'i?,'lV'«S".J?'„l;?S'=="^;?Jj;''#i?".!t=„"Aj;S.°J^£?~~,'^"^'' = '>/''°" ■'"■^ CRCUIT.THE increased leading charging CURRENT OF THE LIiSe. SeACtIngui^NtSe GENERATOR F|"dS, 
 COMBINED WITH A MOMENTARY OVER SPEED OF THE GENERATORS MAY CAUSE THE RECEIVING END VOLTAGE TO GREATLY EXCEED THE ABOVE VALUE 
 
 IN ORDER TO HOLD THE VOLTAGE AT THE RECEIVING END CONSTANT AT 220.000 VOLTS BETWEEN CONDUCTORS (127,020 VOLTS TO NEUTRAL! AT ZERO LOAD IT WILL BE NECESSARY TO PLACE AN ARTI. 
 riCIAL LAGGING LOAD AT THE LOAD END OF THE LINE-THIS IS ACCOMPLISHED BY OPERATING ONE OF THE SYNCHRONOUS CONDENSERS WITH ITS FIELDS UNDER EXCITeS .-BY CONSTRlJCTING SEVERAL VECTOR 
 DIAGRAMS FOR THIS CIRCUIT EACH BASED UPON DIFFERENT VALUES OF REACTOR LOAD, A CURVE MAY BE DRAWN BY >Co-fTING THE REACTOR LOADS AcjArNST THE COR^ END "oLTAMB ." 
 
 FHON THIS CURVE THE REACTOR CAPACITY CORRESPONDING TO 230.000 VOLTS BETWEEN CONDUCTORS AT THE SENDING END WiLl BE SEEN TO BE APPROXIMATELY 30 OOO RV-A ^ * "* VOLTAWB... 
 
 CONDENSER 
 
 (ONE REQUIRED) 
 
 'c-LOSS'*<"' 
 
 to neutral 
 KV-A(;n=i<"""' 
 
 KW, 
 
 ON'"" 
 
 c-loss-n"'"' 
 'c-Loss-Nr*"" 
 
 LINEAR CONSTANTS 
 
 Z ' 37.S36 *j7\ 8.03 OHMS -k 
 Y - +i.OOI2ll MHO 
 1l THIS INCLUDES IMPEDANCE OF LOWERINO TRANSFORMERS 
 
 AUXILIARY CONSTANTS 
 
 (A) = (ai*iaa)=' cosh e = . 670783 -tj. 021 91 1» 87108 /r ae' 7%- 
 
 SINH6 
 
 
 /•C-SNO 
 
 (B).(bcjbj).z 
 {C)-(0i*iCs).y5!Mie 
 
 WHERE e - ^ZY 
 
 130.134 • /0"!2II7' VOLTS 
 
 ■ft 
 
 SINH e - 34.8883 + j 208.83 OHMS 
 8INH a " - .000.008 «j .00 1 1 88 MHO. 
 
 CALCULATION FOR 
 
 MVECTOR OF REFERENCE) 
 
 DETERMINATION OF LOSSES 
 KW,«o-o 
 
 RECEIVING-END CURRENT 
 
 RNO' 
 
 I27.030XS2.7I (COS Be' 22' 67-|-743 
 
 .0«° pfno 
 
 
 'CO 
 
 'to 
 'ro 
 
 = 4.00 -j 78.83 
 • 186-113.80 
 
 6.88 -j 92.63 AMPERES 
 92.7 1 \88'2287- AMPERES 
 
 CALCULATION FOR HIGH TENSION CIRCUIT 
 
 LEAKAGE OF 
 
 LOWERING 
 TRANSFORMERS 
 
 ErNO (A)= I 1 0,607 +j 2783 
 
 lRf%(B)° I9S26-J 1977 
 
 ESNO' I30.l32*j 806 
 
 = 130.134 /O' 21' 17' VOLTS 
 
 !ro(A)- 
 
 Erno^C)= 
 
 'so" 
 
 ■ ),l4+j 147,09 
 6.98 +j 86.66 
 88.92 / 84- 62 23' AMPERES 
 
 KV-A so - I 30. 1 34 X 68.92 X 3 - 26. 1 26 PER 3 PHASE CIRCUIT 
 
 KV-A CO = I 30. 1 34 (.00 II 68 X I 62,46 I ) « 3 = 69,5 I 6 PER 3 PHASE CIRCUIT •*■ 
 if BASED upon H.T. LINE BEINQ OPEN AT RECEIVING END. 
 
 KW, 
 
 KWjNo" 1 30. 1 34 « 66 82 (008 84"3r08")-B3l 
 
 LOSSES TO NEUTRAL 
 
 CONDENSER - 808 
 
 LOWERINQ TRANSFORMERS 
 
 (IR0NL(^8> -» 836 
 
 (COPPER LOSS 92.7l'»3.ie6) - ■ «7 
 
 RAISING TRANSFORMERS 
 
 (IR0NL0S3) > ,..- 218 
 
 (COPPER LOSS 66 82^x3.186) .» 14 
 
 HIGH TENSION LINE 83 1 -(743 + 27) " 6^ 
 
 TOTAL LOSS TO NEUTRAL - 1080 
 
 EFFICIENCY 
 
 EFFICIENCY. (HIOH TENSION LINE) .- ^-92,88% 
 
 % ERROR IN RESULTS 
 
 VOLTAGE AT SENDING END " - 2.13% 
 
 CURRENT AT SENDING END "- 4.86% 
 
 POWER FACTOR AT SENDING END - ♦ 4,24% 
 
 KV-A AT SENDING END --- 6,93* 
 
 LOSS IN HIGH TENSION LINE ..--2088% 
 
 NOTE - THESE PERCENTAGES OF ERROR ARE BASED UPON 
 THE ASSUMPTION THAT THE COMPLETE METHOD PRO- 
 DUCES 100% VALUES.-THE MINUS SIGNS SIGNIFY RE- 
 SULTS TOO LOV*( THE PLUS SIGNS RESULTS TOO HIGH. 
 
 at zero load in order to maintain 240 kw constant at 
 the sending end. Thus with 230 kv at the sending end, 
 about 30 000 kv-a reactor load will be required at zero 
 load, whereas with 240 kv at the sending end, about 
 40 000 kv-a reactor load will be required at zero load. 
 
 Obviously the smallest phase modifier capacity 
 possible to maintain regulation is one in which full ca- 
 pacity leading will be required under maximum load 
 and full capacity lagging under zero load. At half load 
 such a phase modifier would operate at near zero kv-a. 
 
 load and that the corresponding sending end voltage will 
 be approximately 236 kv. At 100 000 kw load, nearly 
 50000 kv-a phase modifier capacity will be required, 
 and the corresponding sending end voltage would be 
 250 kv. 
 
 As previously stated, phase modifiers which may be 
 operated at rated load both lagging and leading are spe- 
 cial, and cost more than standard phase modifiers. On 
 account of unstable operation due to weakened field, 
 standard condensers usually cannot be operated at lag- 
 
A TYPICAL 220 KV PROBLEM 
 
 1S7 
 
 ging loads above approximately 70 percent of their full 
 load leading rating. To deliver 75 000 kv-a at 85 percent 
 power-factor requires approximately 42000 kv-a in 
 phase modifier capacity with 230 kv at the sending end. 
 To maintain the sending end voltage of 230 kv at zero 
 
 liMM 
 
 
 
 
 
 
 
 
 
 
 
 .^^ 
 
 
 — ' 
 
 
 
 
 
 
 
 
 
 
 A0 
 
 
 
 >:::; 
 
 '-^^ 
 
 
 
 
 
 *;-. 
 
 
 
 
 
 
 
 '^ 
 
 
 
 
 
 
 
 1 
 
 
 
 
 ^^ 
 
 
 
 
 
 
 
 
 
 
 
 
 !< 
 
 
 
 
 ^ 
 
 ^_,«^ 
 
 
 
 
 
 
 
 
 
 
 
 
 y 
 
 ^;:>^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 " 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 M 
 
 BM U 
 
 
 
 
 
 
 
 
 
 
 
 which determines the total capacity of phase modifiers, 
 for the 220 kv problem. For instance at normal load, 
 43 000 kv-a in capacity is required, whereas for the 
 double or emergency load 157000 kv-a capacity 
 (nearly four times) is required This large increase 
 is due to the fact that the line charging cur- 
 rent (which tends to reduce phase modifier 
 capacity under load) has not changed, and 
 that the line impedance volts has become 
 twice as much, making it necessarj' to turn 
 the line impedance triangle through a large 
 angle in the counter-clockwise direction in 
 order that the sending end voltage be not in- 
 
 creased. 
 
 PHASE MODIFIER CAPACITTf REOUrRED-KV-A -LEADING 
 
 FIG. 72 — PHASE MODIFIER CAPACITY REOUIRICD FOR THE VARIOUS LOADS 
 
 These curves are plotted from values read from the curves of Fig. 
 70 and are on the basis of a constant load voltage of 220 kv. 
 
 load requires approximately 30000 
 kv-a lagging. This is 70 percent of 
 the capacity leading, thus permitting 
 of employing a standard 43 000 kv-a 
 condenser. To provide margin a 
 45 000 kv-a standard condenser might 
 be selected for this normal load con- 
 dition. 
 
 Under emergency conditions 
 (that is, double or 150000 kw load 
 at 85 percent power-factor) 157000 
 kv-a phase modifier capacity will be 
 required if 230 kv is not to be ex- 
 ceeded at the sending end. If the 
 generator can be operated during the 
 emergency condition at increased 
 voltage of, for instance, 240 kv, the 
 phase modifier capacity could be re- 
 duced to approximately 140000 
 kv-a. However, too much liberty in 
 variation of generator operating 
 voltage should not be taken. If the 
 voltage is held constant at the high- 
 voltage side of the raising transform- 
 ers, the generator operating voltage 
 will have to be varied to compensate 
 for the regulation of the sending end 
 transformers, and to provide a still 
 greater range in generator operating 
 voltage might impose a hardship on 
 the generator designers. The volt- 
 age drop through the transformers 
 is small under load conditions, since 
 the power-factor will be near unity, 
 but under zero load condition the 
 drop will be considerable, due to the 
 low power-factor, especially if a 
 large phase modifier load is re- 
 quired at zero load. It will be seen 
 that it is the emergency condition 
 
 CONSTANTS 
 
 E^- VOLTUC AT LOAD HElA OOMSTAnT at 3,0 oOO VOLTS 
 KWt"''00OHW*Ti«PFOON»T*NT 80TMAT 1^ • III \]|-4r*M^ 
 
 (A) - ••**; 0I03.(S|.j«j) 
 
 (B) -3J J'ilH0MM8.(b, .jD,) 
 
 (C) - l»*OT ItEQUmED MEHEt 
 W'TM M OCC KV » COWDENaEI C*y*OITV 
 
 TRANaroAMER nESlSTANCt - 3 < ■ OHMS TO MFUTWAL 
 TTUNHFOAMER REACTANCE • 3* •] OHMS TQ NEUTRAl 
 TRANSrOAMER LEAKAGE • I » - j 1 3 » AMPS TO NEUTRM. 
 
 FIG. 71 — GRAPHIC METHOD FOR DETERMINING THE 
 VOLTAGE AT THE SENDING END. 
 
 Corresponding to different condenser 
 loads in parallel with a constant power load of 
 75 000 kw at 85 percent power- factor and 220 
 kv. The results as plotted in Fig. 70 were 
 obtained by similar constructions. 
 
 LINE 1I2.S MILES LONG 
 (A)-.T..,«.i 
 
 LINE 22S MILES LONG 
 (A). •••.,«•» 
 (B)-...-, 
 
 LINE 337.6 MILES LONG 
 
 E-" 
 
 FIG. Ti — VECTOR DIAGRAMS SHOW- 
 ING THE EFFECT OF THE LENGTH 
 OF THE LINE ON THE PHASE MODI- 
 FIER CAPACITY REQUIRED 
 
 The diagrams represent a 
 three-phase, 60 cycle circuit, con- 
 sisting of three 605 000 circ mil 
 aluminum steel reinforced con- 
 ductors, when delivering 75 000 
 kw at 85 percent lagging power- 
 factor at a load voltage of 220 kv 
 with a sending end voltage of 230 
 kv. 
 
158 
 
 A TYPICAL 220 KV PROBLEM 
 
 The zero load curve on Fig. 70 is drawn for the 
 normal load connection; that is, for two 50000 kv-a 
 transformer banks in parallel. For the emergency load 
 four transformer banks in parallel will be required. 
 The result of the increased magnetizing current con- 
 sumed by four in place of two transformer banks will 
 be to reduce the capacity of phase modifiers required 
 under zero load. A second zero load line could be 
 added, covering four transformer banks. Such a line 
 would lie directly above the one for two transformer 
 banks but would not materially affect the results. For 
 load conditions of 100 000 kw at 85 percent power-fac- 
 
 LINC 112.6 MILES LON& 
 
 PHASE MOOinER CAPACITY RESUIREO-KVA 
 FIG. 74— CURVE SHOWING THE RELATION BET\VEEN PHASE MODIFIER 
 CAPACITY AND SENDING END VOLTAGE 
 
 For various receiving end loads of 85 percent lagging 
 power-factor and a constant load voltage of 220 ky. These 
 carves apply to a three phase, 60 cycle circuit consisting of 
 three 605 000 circ. mil aluminum steel conductors. The vector 
 construction of these four lines is shown in Fig. 73. 
 
 tor and above, the points for the curves were determined 
 on the basis of four transformer banks. 
 
 In the above it was assumed that the power-factoi 
 of the load would be 85 percent lagging. A long line 
 such as this would probably feed into an extended dis- 
 tribution net work, having numerous load centers. At 
 these load centers synchronous condensers would pro- 
 bably be located for the purpose of holding the voltage 
 constant. This would necessitate operating the con- 
 denser leading at heavy loads thus raising the power- 
 factor of the entire system under load, and in effect 
 reducing the capacity of phase modifiers required for 
 voltage control at the receiving end of the line. This 
 point should be investigated where a long line such as 
 
 this feeds a net work on which condensers are required 
 for voltage control. 
 
 It may be desired to investigate the effect of line 
 charging current on phase modifier capacity for lines of 
 different lengths. For this purpose the vector dia- 
 grams Fig. 73, and the phase modifier curves, Fig. 74, 
 were prepared. These vector diagrams and curves are 
 based upon a constant load of 75 000 kw at 85 percent 
 power-factor delivered at 220 kv and a line drop of 10 
 kv. In other words the only variable for the four 
 different lines is the length and this varies in equal in- 
 crements. 
 
 The vector diagrams of Fig. 73 show the influence 
 of line charging current upon condenser capacity. As 
 the length of the line increases, the influence of the in- 
 
 
 
 
 — 1 
 
 I 
 
 
 — ' 
 
 — 
 
 
 
 
 ^, 
 
 
 
 
 
 
 
 
 
 sv 
 
 " 
 
 
 
 
 
 
 
 
 ^ 
 
 F, 
 
 
 
 
 
 
 
 ■ 
 
 
 
 
 
 
 
 
 
 ?'""" 
 
 
 ^ 
 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 
 
 
 iSM.OW 
 
 
 
 V 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 §344 060 
 
 5 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 |3«0 000 
 
 
 
 
 
 
 
 
 
 
 
 
 
 •s, 
 
 
 
 
 lauooo 
 
 
 
 
 
 ^^\ 
 
 
 
 
 
 
 
 ^ 
 
 ^-j 
 
 
 
 E»tooo 
 
 
 
 
 
 
 
 
 
 
 
 
 
 X ^ 
 
 
 
 
 
 
 
 
 'v. 
 
 
 
 
 
 
 
 
 
 
 * 
 
 Vl.POO 
 
 
 
 
 
 
 
 
 s. 
 
 
 
 
 
 
 
 
 s. 
 
 
 
 
 
 
 
 
 S ' 
 
 
 
 
 
 
 
 
 s 
 
 ato.oeo 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 flM.OM 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 FIG. 75 — CURVES SHOWING THE VOLTAGE ON EACIt 
 SIDE OF THE RAISING TRANSFORMERS 
 
 Corresponding to condenser loads of vari- 
 ous capacities in parallel with a constant load 
 of 75 000 kw at 85 percent power factor lagg- 
 ing and 220 kv. The vertical distance between 
 the two voltage lines is the voltage drop or 
 voltage rise through the raising transformers. 
 For condenser loads up to 15000 kv-a there is 
 a drop in voltage through the raising transfor- 
 mers. For condenser loads above 15000 ky-a 
 there is a rise in voltage through the raising 
 transformers. 
 
 creased line charging current is toward a reduction in 
 condenser capacity; that is the line itself furnishes a 
 large part of the leading current necessary to maintain 
 the proper line voltage drop. If this line were longer 
 than 450 miles, the line charging current at a certain 
 length would be sufficient in itself to maintain the de- 
 sired voltages at the two ends of the line without the 
 aid of condensers. In such a case, however, a large 
 reactor capacity would be required at zero and low 
 loads to hold the receiving end voltage at a constant 
 value. 
 
 The reason that a short line may necessitate more 
 condenser capacities for voltage control than a long line 
 is simple. For the 112.5 mile line the charging current 
 will be about one half as much as for a 225 mile line. 
 Since the line is only half as long this smaller charging 
 curent will flow through only half the inductance so that 
 the net result of half the line charging current and half 
 the inductance will be about one fourth the voltage 
 
A TYPICAL 220 KV PROBLEM 
 
 159 
 
 boosting effect due to line charging current. On the 
 other hand the line impedance will be only half as grear, 
 but the net result will be more condenser capacity for 
 the short line. A large part of the condenser capacity 
 is required for neutralizing the lagging reactor com- 
 ponent of the load. 
 
 Auxiliary constant A, as previously explained, ac- 
 counts for the effect of the line charging curent flowing 
 through the impedance of the circuit; that is, the volt- 
 age boosting effect of the charging current. Thus for 
 the 1 12.5 mile line (Fig. 73) a^ which accounts for the 
 line charging current flowing through the inductance 
 of the circuit is near unity and a„ near zero, but for the 
 450 mile line a^ drops to 0.594 and a, increases to 
 0.07508. As the length of line increases, constant A 
 moves the line impedance triangle to the left and raises 
 its toe somewhat. The increased line impedance and 
 
 slightly increased current at the receiving end increases 
 the size of the line impedance triangle. 
 
 The curves of Fig. 74 show the relation between 
 phase modifier capacity and sending end voltage for dif- 
 ferent receiving end loads of 85 percent lagging power- 
 factor and a constant load voltage of 220 kv. It is inter- 
 esting to note the effect of distance for fixed size con- 
 ductors upon the maximum amount of power which can 
 be transmitted over a circuit, as evidenced by the load 
 curves bending upward as the line length increase. It is 
 also interesting to note the decrease in phase modifiers 
 leading capacity and increase in phase modifier lagging 
 capacity as the line becomes larger, as evidenced of the 
 load curves shifting to the right. The curves. Fig. 75, 
 show the voltage at each side of the raising transformer, 
 corresponding to various condenser capacities in paral- 
 lel with a constant load of 75 000 kw at 85 percent lag- 
 ging power- factor and 220 kw. 
 
 H. B. DWIGHT'S METHOD. 
 
 In the various methods for determining the per- 
 formance of transmission lines which are described 
 above, current and voltage vectors or corresponding 
 vector quantities have been employed throughout. It 
 was believed that solutions embodying the use of current 
 and voltage vectors would be the more easily followed by 
 the young engineer, for the assistance of whom this book 
 has been primarily written. 
 
 H. B. Dwight worked out and published in book 
 form formulas for determining the complete performance 
 of circuits by the employment of quantities not gener- 
 ally employed in the methods described above. These 
 quantities require a new set of symibols applicable to 
 his method. Partly to prevent confusion in symbols but 
 principally because his method has been so completely 
 and clearly set forth and illustrated with numerous ex- 
 amples worked out in the two books referred to his 
 method has not been detailed in this book. To include 
 it here would simply be a duplication of what is al- 
 ready available in very complete form. 
 
 THE CIRCLE DIAGRAM 
 
 Various forms of circle diagrams as an aid in 
 determining the performance of short transmission lines 
 have been frequently described by writers, notably by 
 R. A. Philip 'thru the medium of the A. I. E. E. trans- 
 actions of February 1911. Following this H. B. Dwight 
 worked out a solution and construction for a circle 
 diagram which accurately takes into account the effect 
 of capacitance in transmission lines that is, a circle 
 diagram for long high voltage lines. This circle dia- 
 gram consists of curves which indicate the phase modi- 
 
 fier capacity (leading or lagging) required to maintain 
 a certain reciving end voltage corresponding to all values 
 of delivered load up to the maximum capacity of the 
 line. In other words it gives data such as is given 
 by the curves of Fig. No. 70. 
 
 The next step in the development of the circle dia- 
 gram was to so alter the constants upon which it is con- 
 structed that it will take accurately into account the 
 localized impedance and loss in raising or lowering 
 transformers or in both. Of course^ the transformer 
 impedance may be added to the line impedance as is 
 frequently done and considered as distributed line im- 
 pedance; Such procedure, will, however, in the case of 
 the circle diagram for the line alone result in objection- 
 able errors in the results. In order to correctly apply 
 the circle diagram to long lines so as to accurately in- 
 clude the effect of transformers in the circuit it is neces- 
 sary to develop new formulas for obtaining values for 
 the constants by which the circle diagram is constructed. 
 See articles on transmission line constants by R. D. 
 Evans and H. K. Sels in the Electric Journal, page 306 
 July 1921, page 356 August 1921 and page 530 Decem- 
 ber 1921. 
 
 To the exi>ert who spends much time investigating 
 transmission problems the general use of the circle dia- 
 gram should be of great assistance. It indicates per- 
 formance at all loads, which with other methods would 
 hav^e to be obtained by a separate calculation or vector 
 diasfram construction for each load. 
 
 ♦Transmission Line Formulas, 1913. D. Van Nostrand Co- 
 New York City and Constant-voltage Transmission, 1915, John 
 Wiley & Sons Inc., New York City. 
 
INDEX 
 
 Pages. 
 
 Accuracy of Various Methods — Comparative 119 & 140 
 
 Admittance Correcting Factor for Kennelly Equivalent 
 
 V Solution— Charts XX and XXI 100-101 
 
 Advantages of High Power Factor 134 
 
 Angle — Hyperbolic 88 
 
 Application of Tables to the Solution of Short Lines — 
 
 Chart II 52 
 
 Long Lines— Chart VIII 70 
 
 Armature Current — Effect Upon Field Excitation of A. 
 
 C. Generators 129-130 
 
 Armature Impedance — Effect Upon Voltage of A. C. 
 
 Generators 130 
 
 Auxiliary Constants — Corresponding to Localized Capa- 
 citance methods 1 12-1 13 
 
 Comparison of Results When Taken from Wilkinson 
 
 and Kennelly Charts 146-lSl 
 
 Definition of 64-142 
 
 Determination by Convergent Series 80 & 82 
 
 Determination by Hyperbolic Functions 96 
 
 Five Different Methods of Determining 143 
 
 How They Modify Short Line Diagrams 143 
 
 Tabulation for 64 Different Circuits 83 
 
 Wilkinson Chart for Obtaining A — Chart V 67 
 
 Wilkinson Chart for Obtaining B— Chart VI.../. 68 
 
 Wilkinson Chart for Obtaining C— Chart VII 69 
 
 Behavior of A. C. Generators when Charging a Trans- 
 mission Line 136 
 
 Bibliography on Solution of Circuits 109 
 
 On Cable 128 
 
 Cables — Capacitance of 3-Conductor 125-126 
 
 Capacitance and Susceptance of 3-Conductor — Table 
 
 XXVir 126 
 
 Charging Kv-a for 3-Conductor— Table XXVIII 127 
 
 Effect of Stranding and Spiraling Upon Inductance 9 
 
 Heating Limits 121-122 
 
 Inductance, Reactance, Impedance of 3-Conductor 
 
 Cable at 25 Cycles— Table XXV 123 
 
 Inductance, Reactance, Impedance of 3-Conductor 
 
 Cable at 60 Cycles— Table XXVI : 124 
 
 Reactance of 3-Conductor Cable 121 
 
 Capacitance — Charging Current, Inductance — Chapter II 10 
 
 Definition of H 
 
 Formula 11 & 20 
 
 Relation to Inductance 21 
 
 Susceptance Bare Conductors at 25 Cycles — Table IX 17 
 
 Susceptance Bare Conductors at 60 Cycles — Table X 18 
 
 Three Conductor Cable 125-126 
 
 To Neutral per 1000 Feet of Single Bare Conductor 
 
 —Table VIII 16 
 
 Capacity of Synchronous Motor and Condensers for 
 
 Power Factor Improvement 132 
 
 of Phase Modifiers for Voltage Control Chapter XV 138 
 of Bare Conductors in air (heating limit) Table 
 
 XXIII 43 
 
 Susceptance to Neutral Per Mile of Single Bare Con- 
 ductor at 25 Cycles— Table IX 17 
 
 Susceptance to Neutral Per Mile of Single Bare Con- 
 ductor at 60 cycles — Table X 18 
 
 Charging Current — At Zero Load 20 
 
 Capacitance, reactance — Chapter II 10 
 
 Effect upon Conductor Loss 34 
 
 Of Short Lines 62 
 
 Relation in Single and Three Phase Circuits 20 
 
 Charging Kv-a — In three Phase Circuits Per Mile of 
 
 3 Bare Conductors — Table XI 19 
 
 of 3-Conductor Cables 127 
 
 Charging Transmission Lines — Behavior of A. C. Gen- 
 erators when 136 
 
 Chart I Inductance 6 
 
 11 Application of Tables to Short Transmission 
 
 Lines 52 
 
 III Mershon Chart for Determining Line Drop in 
 Short Lines 54 
 
 IV Dwight Chart for Determining Line Drop in 
 Short Lines 56 
 
 Pages. 
 
 V Wilkinson Chart A for Determining Auxiliary 
 
 Constants a.^ and aj 67 
 
 VI Wilkinson Chart B for Determining Auxiliary 
 
 Constants K and h, 68 
 
 VII Wilkinson Chart C for Determining Auxiliary 
 Constants Ci and Cj 69 
 
 VIII Applicatiort of Tables to Long Transmission 
 Lines 70 
 
 IX Peters Efficiency Chart for Transformers 74 
 
 X Peters Regulation Chart for Transformers 75 
 
 XI Determination of Auxiliary Constants by Con- 
 vergent Series 82 
 
 XII Auxiliary Constants of 64 Different Circuits 83 
 
 XIII Calculation of Performance (Receiver End 
 Conditions Fixed) 84 
 
 XIV Calculation of Performance (Sending End 
 Conditions Fixed) 86 
 
 XV Calculated Performance of 64 Different Circuits 87 
 XVI Determination of Auxiliary Constants by Hy- 
 perbolic Functions 96 
 
 XVII Equivalent 'V' Solution of Problem X 103 
 
 XVIII Kennelly Chart for Impedence Correcting Fac- 
 tor (Angles to .40) 98 
 
 XIX Kennelly Chart for Impedance Correcting Fac- 
 tor (Angles .40 to 1.0) 99 
 
 XX Kennelly Chart for Admittance Correcting Fac- 
 tor (Angles to .20) 100 
 
 XXI Kennelly Chart for Admittance Correcting Fac- 
 tor (Angles .20 to .50) 101 
 
 XXII Comparison of Results by Various Methods 118 
 
 XXIII 220 Kv. Problem— Normal Load (Complete 
 
 Solution) 149 
 
 XXIV 220 Kv. Problem— Normal Load (Approxi- 
 mate Solution) 150 
 
 XXV 220 Kv. Problem— Emergency Load (Com- 
 plete Solution) 153 
 
 XXVI 220 Kv. Problem — Emergency Load (Approxi- 
 mate Solution) 154 
 
 XXVII 220 Kv. Problem— Zero Load (Complete So- 
 lution) 155 
 
 XXVIII 220 Kv. Problem— Zero Load (Approximate 
 Solution) 156 
 
 Checking the Work 138 
 
 Choice of Various Methods 108 
 
 Circuits — Paralleling 41 
 
 Electric, Dielectric and Magnetic 1-2 
 
 Circular Functions 88 
 
 Sines, Cosines, Tangents — Table K 57 
 
 Common Transmission Voltages — Table H 48 
 
 Comparison — Accuracy of 9 Different Methods 118-119 
 
 Accuracy of 5 Methods of Including Transformers.... 140 
 of Calculated Capacitance of 3-Conductor Cables with 
 
 test results 128 
 
 of 9 Different Methods— Chapter XII Ill 
 
 Complex Angle — Definition 90 
 
 Complex Hyperbolic Functions — Definition 90-94 
 
 Dr. Kcnnelly's Model 92 
 
 Complex Quantities — Definition 78 
 
 Condensers and Phase Modifiers 131 
 
 Condensers and Synchronous Motors — For Power Factor 
 
 Improvement — Chapter XIV 129 
 
 Determination of Capacity 132-134 
 
 Condenser.? — Generators as 131 
 
 Installations of Large Capacity — Table U 137 
 
 Location 132 
 
 Methods of applying — Chapter XII Ill 
 
 Mechanical Load Carried 131 
 
 Ratings, Starting, "\"' Curves, Losses, Etc 131 
 
 Conductors — Capacitance (See Capacitance) 
 
 Economic Size 45 & 145 
 
 Effect of Unsymmetrical Spacing 10 
 
 Effect of Stranding and Spiraling upon Inductance 9 
 
 Flux Distribution Around 3 
 
 Heating of Bare Conductors in Air 42-43 
 
 Heating of 3-Conductor Cables 121-122 
 
 Inductance (See Inductance) 
 Impedance (See Impedance) 
 
INDEX 
 
 161 
 
 Pages. 
 
 Reactance (See Reactance) 
 Resistance (See Resistance) 
 Skin Effect (See Skin Effect) 
 
 Weight of Bare— Table E-1 46 
 
 Constants — Auxiliary (See Auxiliary Constants) 
 
 Methods of Determining the Linear Constants 51 
 
 Transformer Constants Taken Into Account 7i & 139 
 
 Convergent Series for Determining the Auxiliary Constants 80-82 
 
 Copper Loss of Transformers — Table X 141 
 
 Copper Conductors (See Conductors) 
 
 Corona — Effect of — Chapter IV 35 
 
 Formulas , 36-37 
 
 Voltage Limitation— Table XXII 38 
 
 Correcting Factors — Charts for Impedance for Equivalent 
 
 T Solution Charts XVIII and XIX 98-99 
 
 Charts for Admittance for Equivalent ■^ Solution^ 
 
 Chart XX and XXI 100-101 
 
 Mathematical Determination for Equivalent tt Solu- 
 tion 107 
 
 Cosines, Sines and Tangents of Angles — Table K 57 
 
 Cost — Relative Cost of High Tension Apparatus 46 
 
 Power Factor Improvement by Synchronous Motors 135 
 
 Current and Voltage Determination Along Circuit — By 
 
 Auxiliary Constants 62-64 & 76 
 
 by Hyperbolic Position Angles 102-106 
 
 Degree — Subdivisions of, Table P 110 
 
 Determination of — Capacity of Synchronous Motors and 
 
 Condensers 132-134 
 
 Correcting Factor for Equivalent tt Solution Mathe- 
 matically 107 
 
 Frequency and Voltage 45 
 
 Dielectric Circuit 2 
 
 Distribution of Current and Voltage Along the Circuit — 
 
 By Auxiliary Constants 62-64 & 76 
 
 by Hyperbolic Position Angles 102-106 
 
 by a Polar Diagram 108 
 
 Dwight Chart for Short Lines 55-56 
 
 Economies Size of Conductors 45 & 145 
 
 Effect of Armature Current Upon Field Excitation of 
 
 A. C. Generators 129-130 
 
 Armature Impedance upon Terminal Voltage of 
 
 A. C. Generators 130 
 
 Corona — Chapter IV 35 
 
 Charging Current on Conductor Loss 34 
 
 Harmonies in Current and Voltage 108 
 
 Power Factor Upon Voltage Drop 135 
 
 Spiraling and Stranding of Conductors Upon Induc- 
 tance 9 
 
 Transformers in the Line 7i & 139 
 
 Corona — Chapter IV 35 
 
 Effective Spacing of Conductors 10 
 
 Efficiency Chart for Transformer — Peters Chart IX 74 
 
 Electric Circuit — The 2 
 
 Electric Propogation — Speed of 40 
 
 Electric Wave— Length of 40 
 
 Equivalent tt Method — Charts for Impedance Correcting 
 
 Factor— Charts XVIII & XIX 98-99 
 
 Charts for Admittance Correcting Factor — Charts 
 
 XX & XXI 100-101 
 
 General 97 
 
 Example of Calculation— Chart XVII 103 
 
 Mathematical Determination of the Correcting Fac- 
 tors 107 
 
 Equivalent "T" Solution— General 102 
 
 Estimating Tables — Quick — Chapter III 23 
 
 Exciting Transmission Lines — Methods of 136 
 
 Field Excitation — Effect of Armature Current Upon 129-130 
 
 Flux — Effect of Armature Flux Upon Field Excitation 
 
 of A. C. Generator 129-130 
 
 Distribution Around Conductor 3-6 
 
 Formulas 7 & 9 
 
 Formulas — Capacitance 11 & 20 
 
 Carrying Capacity of Bare Conductors in Air 44 
 
 Convergent Series 80 
 
 Corona 36-37 
 
 Inductance 7 & 9 
 
 Transmission Line in Terms of the Auxiliary Con- 
 stants 80 
 
 Transmission Lines of Short Length 59 
 
 Transmission Line in Terms of Hyperbolic Functions 80 
 
 Frequency Determination 45 
 
 Pages. 
 
 FuiKtions— Complex Functions of Hyperbolic Angles 90-94 
 
 Circular 88 
 
 Hyperbolic— Real 88 
 
 Hyperbolic — Complex 91 
 
 Sines, Cosines and Tangents of Circular Angles 57 
 
 Generators — As Synchronous Condensers 131 
 
 Behavior When Charging Transmission Lines 136 
 
 Effect of Field Excitation Upon A. C. Generators....l29-130 
 
 Graphical Solution (See Solutions) 
 
 Harmonic Currents and Voltages — Effect of 108 
 
 in Quarter Wave Resonance 41 
 
 Heating — Bare Conductors in Air 42-43 
 
 Limits for Cables, General 121-122 
 
 Tabulated Values for Cables— Table XXIV 122 
 
 High Power Factor — Advantages 134 
 
 High Tension Apparatus — Relative Cost — Table F 46 
 
 How High to Raise the Power Factor 132 
 
 Hyperbolic Angles — Real 88 
 
 Complex 9(^-94 
 
 Hyperbolic Functions — Applied to the Solution of Line 
 
 Performance — Chapter XI 95 
 
 Chapter X 88 
 
 Complex Angles 91 
 
 Formulas for Long Lines 80 
 
 for Determining the Auxiliary Constants 96 
 
 Symbols Pertaining to 95 
 
 Inductance — Capacitance — Charging Current — Chapter II 10 
 Effect of Spiraling and Stranding of Conductors 
 
 Upon 9 
 
 Formulas 7 & 9 
 
 (General 3 
 
 Graphic Solution — Chart 1 6 
 
 Per 1000 Feet of Single Conductor— Table III 8 
 
 Reactance and Impedance of 3-Conductor Cables at 
 
 25 Cycles— Table XXV 123 
 
 Reactance and Impedance of 3-Conductor Cables at 
 
 60 Cycles— Table XXVI 124 
 
 Relation to Capacitance 21 
 
 Skin Effect and Resistance — Chapter 1 1 
 
 Variation from the Fundamental Formula 7 
 
 Installations of Large Phase Modifiers — Table U 137 
 
 Impedance — Correcting Factor for Equivalent tt Solu- 
 tion—Charts XVIII and XIX 98 & 99 
 
 Effect of Armature Impedance Upon Voltage of A. 
 
 C. Generators „.... 130 
 
 Effect of Transformer Impedance in the Circuit 73 & 139 
 
 Inductance and Reactance of 3-Conductor Cables 
 
 at 25 Cycles— Table XXV 123 
 
 Inductance and Reactance of 3-Conductor Cables 
 
 at 60 Cycles— Table XXVI 124 
 
 Transformer Impedance to Neutral 141 
 
 Iron Loss in Transformers — Table X 141 
 
 Kennelly— Charts for Impedance Correcting Factors for 
 
 Equivalent ,r Solution— Charts XVIII and XIX 98-99 
 
 Charts for Admittance Correcting Factors for Equiva- 
 lent T Solution— Charts XX and XXI lOO-lOl 
 
 Equivalent ,r Solution — General 97 
 
 Equivalent ,r Solution— Example of Solution— Chart 
 
 XVII 103 
 
 Model for Explaining Functions of Complex Angles 92 
 
 Light Speed — Relation to Inductance and Capacitance 21 
 
 Location for Synchronous Condensers 132 
 
 Localized Capacitance Methods — Chapter XII Ill 
 
 Auxiliary Constants Corresponding to 113 
 
 Losses in Transformers — Table X „ 141 
 
 Synchronous Condensers 131 
 
 Magnetic Circuit — The 2 
 
 Magnetizing Current of Transformers 141 
 
 Mechanical Load Carried by Synchronous Condensers.... 131 
 
 Mershon Chart — General 53 
 
 Mershon Chart — Chart III 54 
 
 Where it Falls in Error When Applied to Long 
 
 Lines 72 & 143 
 
 Methods of Exciting Transmission Lines 136 
 
 Solution (See Solution) 
 
 Middle Condenser or Nominal T Solution 115 
 
 Model for Explaining Complex Functions of Complex 
 
 Hyperbolic Angles 92-94 
 
162 
 
 INDEX 
 
 Pages. 
 
 Nominal 'V" or Split Condenser Solution 97 & 113 
 
 "T" or Middle Condenser Solution 11 S 
 
 Paralleling Transmission Lines 41 
 
 Peeks Corona Formulas 36 
 
 Performance of Short Lines — Chapter VII 49 
 
 Composite Lines 50 
 
 Formulas for 59 
 
 Graphical Methods 52 
 
 Mathematical Methods 57 
 
 Procedure in Determining — Chart II 52 
 
 Performance of Long Lines — By Hyperbolic Functions — 
 
 Chapter XI 95 
 
 by Convergent Series — Chapter IX 77 
 
 by Graphical Method — Chapter VIII 61 
 
 by Localized Capacitance Methods— Chapter XII Ill 
 
 Equivalent jr Method 97 
 
 Procedure in Determining — Chart VIII 70 
 
 Tabulated Performance of 64 Circuits — Chart XV.... 87 
 
 Typical 220 Kv Problem— Chapter XVI 145 
 
 Performance of Long Lines Including Transformers 145 
 
 Peters Efficiency Chart for Transformers — Chart IX.... 74 
 
 Regulation Chart for Transformers — Chart X 75 
 
 Phase Modifiers for Voltage Control — Chapter XV 138 
 
 and Synchronous Condensers 131 
 
 Installations of 137 
 
 Curves of 152 
 
 Capacity of 155 
 
 Polar Diagram of Voltage and Current Distribution for 
 
 Problem X 108 
 
 Position Angles — Explanation of 102-107 
 
 Mathematical Determination 107 
 
 Solution for Voltage and Current Along the Circuit 
 
 by Hyperbolic 102-106 
 
 Power Factor — Advantages in High Power Factor 134 
 
 Cost of Improvement 135 
 
 Examples of Determination of Improvement 133 
 
 Effect on Voltage Drop 135 
 
 How High to Raise 132 
 
 Improvement by Synchronous Motors and Condensers 
 
 —Chapter XIV 129 
 
 Propogation — Speed of Electric 40 
 
 Quick Estimating 23-33 
 
 Quarter Wave Resonance 40 
 
 Quantities — Complex 78 
 
 Ratings of Synchronous Condensers 131 
 
 Ratio of Reactance to Resistance at 25°C. and 25 Cycles 
 
 —Table VI 14 
 
 60 Cycles— Table VII 15 
 
 Reactance and Resistance of Copper and Aluminum Con- 
 ductors at 25 Cycles, Table IV 12 
 
 60 Cycles— Table V 13 
 
 A High Reactance Problem 60 
 
 Reactance — Capacitance and Charging Current — Chap- 
 ter II 10 
 
 Inductance and Impedance of 3-Conductor Cables at 
 
 25 Cycles— Table XXV 123 
 
 Itiductance and Impedance of 3-Conductor Cables at 
 
 60 Cycles, Table XXVI 124 
 
 Ratio of Reactance to Resistance at 25° C. and 25 
 
 Cycles— Table VI 14 
 
 Ratio of Reactance to Resistance at 25° C. and 60 
 
 Cycles— Table VII 15 
 
 Three-Conductor Cables 121 
 
 Transformers 141 
 
 Regulation Chart — Dwights — Chart IV 56 
 
 Mershons — Chart HI 54 
 
 Peters Transformer— Chart X 75 
 
 Relation of Inductance to Capacitance 21 
 
 Inductance and Capacitance to Speed of Light 21 
 
 Charging Current in Single and Three Phase System 20 
 
 Relative Cost of High Tension Apparatus — Table F 46 
 
 Resistance — Copper Conductors — Genera! 2 
 
 Copper Conductors — Per 1,000 Feet — Table 1 4 
 
 Copper Conductors — Per Mile — Table II 5 
 
 and Reactance of Copper and Aluminum Conductors 
 
 —25 Cycles— Table IV 12 
 
 and Reactance of Copper and Aluminum Conductors 
 
 60 Cycles— Table V 13 
 
 Ratio Reactance at 25 Cycles to— Table VI 14 
 
 Ratio Reactance at 60 Cycles to— Table VII 15 
 
 Transformers 141 
 
 Resonance — Guarter Wave 40 
 
 Paces. 
 
 Self Induction — Effect Upon Voltage Regulation 62 
 
 Short Lines — Formulas 59 
 
 Performance 49 
 
 Symbols 50 
 
 Single End Condenser Method 112 
 
 Sines, Cosines and Tangents of Circular Angles 57 
 
 Skin Effect — In Conductors 2 
 
 Resistance and Inductance — Chapter 1 1 
 
 Tabulation of Increase in Resistance Due to — Table B 3 
 
 Solution— A Typical 220 Kv. Problem 145 
 
 Choice of 108 
 
 Comparison of Various Methods — Chapter XII Ill 
 
 Comparative Accuracy of Various Methods 118-119 
 
 Comparison of Short and Long Line Diagrams 72 & 142 
 
 Complete for Long Lines Including Transformers.... 145 
 
 Dwight Chart 55-56 
 
 Equivalent ,r — General 97 
 
 Equivalent t, — Exainple — Chart XVII 103 
 
 Equivalent T 102 
 
 Graphical vs. Mathematical 51 
 
 Graphical for Short Lines 52 
 
 Graphical for Problem "X" 70 
 
 Graphical for Long Lines Including Transformers.... 145 
 
 Hyperbolic Functions — Chapter XI 95 
 
 Localized Capacity — Chapter XII Ill 
 
 Mershon Chart 53-54 
 
 Middle Condenser or Nominal T Method 115 
 
 Nominal tt or Split Condenser Method 97 & 113 
 
 Position Angles 102-106 
 
 Single End Condenser Method 112 
 
 Split Condenser or Nominal ^ Method 113 
 
 Three Condenser or Dr. Steinmetz's Alethod 116 
 
 Typical 220 Kv Problem— Chapter XVI • 145 
 
 Spacing of Conductors — Equivalent 10 
 
 Split Condenser or Nominal ,r Solution 113 
 
 Speed of Electric Propogation 40 
 
 Speed of Light — Relation of Inductance and Capacitance 
 
 to 21 
 
 Steinmetz's Three Condenser Method 116 
 
 Sub-division of a Degree — Table P 110 
 
 Susceptance — Three Conductor Cables — Table XXVII.... 126 
 
 Overhead Conductors at 25 Cycles^Table IX 17 
 
 Overhead Conductors at 60 Cycles — Table X 18 
 
 Symbols — Corona 36 
 
 Line 50 
 
 Hyperbolic 95 
 
 Synchronous Condensers — (See Condensers) 
 
 Table I Resistance of Copper Conductors at Various 
 
 Temperatures per 1000 feet 4 
 
 II Resistance of Copper Conductors at Various 
 
 Temperatures Per Mile 5 
 
 III Inductance of Single Conductors per 1000 feet 8 
 
 IV Resistance and 25 Cycle Reactance per Mile.... 12 
 V Resistance and 60 Cycle Reactance Per Mile 13 
 
 VI Ratio of 25 Cycle Reactance to Resistance at 
 
 25° C 14 
 
 VII Ratio of 60 Cycle Reactance to Resistance at 
 
 25° C. 15 
 
 VIII Capacitance of Single Conductor per 1000 feet 16 
 IX 25 Cycle Capacity Susceptance of Single Bare 
 
 Conductors Per Mile 17 
 
 X 60 Cycle Capacity Susceptance of Single Bare 
 
 Conductors Per Mile 18 
 
 XI Charging Kv-a, 3 Phase Circuits, Bare Conduc- 
 tors Per Mile 19 
 
 XII Quick Estimating Table for 220 and 440 Volts 24 
 
 XIII Quick Estimating Table for 550 and 1100 Volts 25 
 XIV Quick Estimating Table for 2200, 4000 and 
 
 4400 Volts 26 
 
 XV Quick Estimating Table for 6000, 6600, 10000 
 
 and 11000 Volts 27 
 
 XVI Quick Estimating Table for 12,000, 13,200, 15,000 
 
 and 16,500 Volts 28 
 
 XVII Quick Estimating Table for 20,000, 22,000, 30,000 
 
 and 33.000 Volts 29 
 
 XVIII Quick Estimating Table for 40,000, 44,000, 50.000, 
 
 and 60,000 Volts 30 
 
 XIX Quick Estimating Table for 66,000, 70,000, 
 
 80,000 and 88,000 Volts 31 
 
 XX Quick Estimating Table for 100,000, 110,000, 
 
 120,000, 132,000 and 140,000 Volts 32 
 
 XXI Quick Estimating Table for 154,000, 187,000 
 
 and 220,000 Volts 33 
 
 XXII Approximate Voltage Limitations from Corona 38 
 
INDEX 
 
 163 
 
 Pages. 
 
 XXIir Heating Capacity for 40° C. Rise Bare Con- 
 ductors 43 
 
 XXIV Carrying Capacity of Insulated Copper Con- 
 ductors 122 
 
 XXV Inductance, Reactance, Impedance at 25 
 
 Cycles, 3-Conductor Cables 123 
 
 XXVI Inductance, Reactance, Impedance at 60 Cycles 
 
 3-Conductor Cables 124 
 
 XXVII Capacitance and Susceptance Per Mile of 3- 
 Conductor Paper Insulated Cables 126 
 
 XXVIII Charging Kv-a of 3-Conductor Cables Per 
 
 Mile 127 
 
 Tangents, Sines and Cosines of Circular Angles (Table 
 
 K) 57 
 
 Three Condenser or Dr. Steinmetz Method 116 
 
 Transformers — Constants 141 
 
 Effect in Circuit 73 & 139 
 
 Iron Loss, Impedance, Magnetizing Current 141 
 
 Peters Efficiency Chart 74 
 
 Peters Regulation Chart 75 
 
 Reactance, Resistance 141 
 
 Transmission Lines — Behavior of A. C. Generators When 
 
 Charging 136 
 
 Common Voltages 48 
 
 Paces. 
 
 Excitation of 136 
 
 Paralleling 41 
 
 Performance (See Performance and Solutions) 
 
 Determination of Frequency and Voltage 45 
 
 Typical 220 Kv Problem 145 
 
 Symmetrical Spacing of Conductors 10 
 
 Various Methods — Comparison of Ill 
 
 Variation of Current and Voltage Along the Circuit 62-64 & 76 
 
 Vector Operations — General 78-80 
 
 As Applied to Problem "X" 85 
 
 "V" Curve of Synchronous Condenser 131 
 
 Voltage — Control by Means of Phase Modifiers — Chap- 
 ter XV 138 
 
 Common Transmission 48 
 
 Distribution Along the Circuit 62-64 & 76 
 
 Effect of Armature Impedance Upon Generator Volt- 
 age 130 
 
 Limitations Due to Corona Effect — Table XXII 38 
 
 Determination of 45 
 
 Wave Length 40 
 
 Wilkinson Charts— Charts A, B and C for the Auxiliary 
 
 Constants 67-69 
 
 Weight of Bare Copper Conductors 46 
 
 PRINTED IN 
 U. S. A. 
 
THIS BOOK IS DUE ON THE LAST DATE 
 STAMPED BELOW 
 
 BOOKS REQUESTED BY ANOTHER BORROWER 
 ARE SUBJECT TO RECALL AFTER ONE WEEK. 
 RENEWED BOOKS ARE SUBJECT TO 
 IMMEDIATE RECALL 
 
 LIBRARY, UNIVERSITY OF CALIFORNIA, DAVIS 
 
 Book Slip-8eries 458 
 
Vestinghousi elect. & 
 
 I T.anufact. c< )_. 
 
 qTK3nQl 
 
 West i>^Atfic5e 
 
 w4 
 
 ±Q?>ZAX^