FROM -THE- LI BRARY- OF 
 WILLIAM -A HILLEBRAND 
 
 PHYSICS OEPT. 
 
ELECTRIC TRANSIENTS 
 
c MsQra&)'3/ill Book & 1m 
 
 PUBLISHERS OF BOOKS F O R^ 
 
 Electrical World ^ Engineering News-Record 
 Power v Engineering and Mining Journal-Press 
 Chemical and Metallurgical Engineering 
 Electric Railway Journal v Coal Age 
 American Machinist ^ Ingenieria Internacional 
 Electrical Merchandising v BusTransportation 
 Journal of Electricity and Western Industry 
 Industrial Engineer 
 
ELECTRIC TRANSIENTS 
 
 BY 
 
 CARL EDWARD MAGNUSSON 
 
 AUTHOR OF "ALTERNATING CURRENTS," PROFESSOR OF ELECTRICAL ENGINEERING, 
 DEAN OF THE COLLEGE OF ENGINEERING, DIRECTOR OF THE ENGINEERING 
 
 EXPERIMENT STATION, UNIVERSITY OF WASHINGTON 
 
 A. KALIN 
 
 INSTRUCTOR IN ELECTRICAL ENGINEERING, UNIVERSITY OF WASHINGTON 
 
 J. R, TOLMIE 
 
 INSTRUCTOR IN ELECTRICAL ENGINEERING, UNIVERSITY OF WASHINGTON 
 
 FIRST EDITION 
 
 McGRAW-HILL BOOK COMPANY, INC. 
 NEW YORK: 370 SEVENTH AVENUE 
 
 LONDON: 6& 8 BOUVERIE ST., E. C. 4 
 
 1922 
 
COPYRIGHT, 1922, BY THE 
 MCGRAW-HILL BOOK COMPANY, INC. 
 
 THE MAPLE PRESS - YORK PA 
 
PREFACE 
 
 Transient electric phenomena generally increase in 
 commercial importance with the size and complexity of 
 electric systems, and a knowledge of the fundamental 
 principles of electric transients and their application to the 
 solution of quantitative problems is as essential to the 
 successful operation of large power and communication 
 systems as a mastery of the basic laws of direct and alter- 
 nating currents. 
 
 This work is an outline of an introductory lecture and 
 laboratory course given during the past twelve years to 
 electrical engineering students in the University of Wash- 
 ington. The purpose of the book is to aid the student in 
 gaining clear concepts of the fundamental principles of 
 electric transient phenomena and their application to 
 quantitative problems. The course as outlined is pro- 
 fessedly of an elementary character with emphasis placed 
 on the physical properties of electric transients. The text 
 is illustrated and supplemented by a large number of 
 oscillograms of transients that occur in the various types of 
 machines and electric circuits in common use in electrical 
 engineering laboratories. The problems are based on 
 quantitative data obtained from laboratory experiments 
 under circuit conditions that may easily be reproduced by 
 the student. 
 
 Quantitative laboratory work is essential in order to 
 readily gain insight into the physical nature of transient 
 electric phenomena. It is advisable to require the student 
 to devote at least two-thirds of the time allotted to a course 
 in electric transients to the taking of oscillograms. Adjust- 
 ing an oscillograph so as to obtain sharply defined, well 
 proportioned oscillograms of electric transients is an 
 effective method for acquiring due appreciation of quanti- 
 
 995863 
 
vi PREFACE 
 
 tative values, both absolute and relative, of the factors 
 involved. The quality of the photographic record depends 
 as much on painstaking care in handling the films and in 
 developing and printing the oscillograms as on skilful 
 operation of the oscillograph. Many pitfalls in the photo- 
 graphic part of the work may be avoided by carefully 
 following the directions given in the Appendix. 
 
 No attempt is made to give references to original investi- 
 gations or to papers and books dealing with the various 
 phases of electric transient phenomena, as the principles 
 discussed are well established and the material is arranged 
 in text book form. A distinctive feature of the book lies in 
 the illustrations. All of the oscillograms were taken by 
 A. Kalin and J. R. Tolmie or by students in the course 
 under their direction in the electrical engineering 
 laboratories of the University of Washington. 
 
 C. EDWARD MAGNUSSON. 
 
 SEATTLE, WASH., 
 March, 1922. 
 
CONTENTS 
 
 PAGE 
 
 PREFACE v 
 
 CHAPTER I. INTRODUCTION 1 
 
 Magnetic circuit Dielectric circuit Electric circuit. 
 
 CHAPTER II. OSCILLOGRAPHS 10 
 
 Three element oscillographs Timing wave from oscillator genera- 
 tor Oscillograms Problems and experiments. 
 
 CHAPTER III. SINGLE ENERGY TRANSIENTS. DIRECT CURRENTS . 23 
 Single energy circuits The exponential law The time constant 
 Dissipation or attenuation constants The exponential curve 
 Initial transient values Current, voltage and ^magnetic flux 
 transients Problems and experiments. ^'4 
 
 CHAPTER IV. SINGLE ENERGY TRANSIENTS. ALTERNATING 
 
 CURRENTS 40 
 
 Single phase, single energy load circuit transients Three phase, 
 single energy load circuit transients Starting transient of a poly- 
 phase rotating magnetic field Polyphase short circuits. Alter- 
 nator armature and field transients -Single phase short circuits. 
 Alternator armature and field transients Single phase short cir- 
 cuits on polyphase alternators Problems and experiments. 
 
 CHAPTER V. DOUBLE ENERGY TRANSIENTS 75 
 
 Double energy circuits Surge or natural impedance and admit- 
 tance Frequency of oscillations in double energy circuits 
 Dissipation constant and damping factor in simple double energy 
 circuits Equations for current and voltage transients Problems 
 and experiments. 
 
 CHAPTER VI. ELECTRIC LINE OSCILLATIONS. SURGES AND TRAVEL- 
 ING WAVES 101 
 
 Artificial transmission lines Time, space and phase angles 
 Natural period of oscillation Length of line Velocity unit 
 of length Surge impedance Voltage and current oscillations and 
 power surges General transmission line equations Traveling 
 waves Compound circuits Problems and experiments. 
 
 CHAPTER VII. VARIABLE CIRCUIT CONSTANTS 130 
 
 Variable resistance Variable inductance Variable conductance 
 Variable condensance Problems and experiments. 
 
 vii 
 
viii CONTENTS 
 
 PAGE 
 
 CHAPTER VIII. RESONANCE 145 
 
 Voltage resonance Current resonance Coupled circuits Direct 
 coupling Inductive coupling Condensive coupling Coupling 
 coefficient Multiplex resonance Resonance growth and decay 
 Problems and experiments. 
 
 CHAPTER IX. OSCILLOGRAMS 168 
 
 Starting transients of a D.C. lifting magnet Opening of D.C. and 
 A.C. circuit breakers due to overload T.A. regulator operating 
 transients Short circuits on series generators "Bucking 
 broncho" transients Current transformer transients Single 
 phase short circuit on a two phase alternator Undamped oscillo- 
 graph vibrator oscillations Starting transients on a three phase 
 induction motor Starting transients on a repulsion-induction 
 motor and a split phase motor Single phase operation of a three 
 phase induction motor Transients in three phase induction motor 
 due to short circuit on stator terminals Short circuits on a rotary 
 converter Synchronizing a rotary converter from 85 per cent 
 synchronous speed Synchronous motor falling out of step due to 
 overload The magnetic flux distribution of a synchronous motor 
 when slipping a pole Problem and experiments. 
 
 APPENDIX 189 
 
 Instructions' for developing and printing oscillograms. 
 INDEX. . 193 
 
ELECTRIC TRANSIENTS 
 
 CHAPTER I 
 INTRODUCTION 
 
 The laws for direct currents, as usually -expressed, state 
 the relations of the several factors involved under continu- 
 ous or permanent conditions, and cannot be correicyy 
 applied while the current or voltage is increasing or decreas- 
 ing. Similarly, alternating currents are expressed as 
 continuous phenomena by means of effective values and 
 complex quantities, on the basis that the successive cycles 
 are of the same magnitude and wave shape. Observations 
 and test data for both the direct-current and alternating- 
 current systems are ordinarily taken only during steady 
 or permanent conditions. The equations derived, and the 
 data obtained from tests, apply only to permanent or 
 constant conditions and cannot be correctly applied during 
 transition periods when the conditions vary. Transient 
 electric phenomena, as the term implies, are usually of 
 short duration and relate to what occurs in an electric 
 circuit between periods of stable conditions. This defini- 
 tion is, however, not rigidly adhered to in electrical discus- 
 sions. Frequently other disturbances that militate against 
 successful operation of electric systems, such as unstable 
 electric equilibrium, permanent instability, resonance and 
 cumulative oscillations are included with the true transients 
 under the caption of transient electric phenomena. 
 
 It is important that the student should realize that 
 electric transients are of very frequent occurrence in all 
 commercial electric systems. Any change, such as the 
 starting or stopping of a motor, the turning on of a lamp, 
 or any change in the operating conditions necessitates a 
 
2 ELECTRIC TRANSIENTS 
 
 re-adjustment of the energy content in the whole system 
 and produces electric transients just as truly as a stroke 
 of lightning or a short circuit. In the operation of street 
 car systems the changes in load, and hence the transients 
 on the system, are so frequent that they overlap and occupy 
 by far the greater part of the time; hence, for street railway 
 systems, it might appear simpler to define the permanent or 
 steady conditions as short periods occurring between succes- 
 sive series of overlapping transients. 
 
 Electrical ^engineering deals with the transmission and 
 transformation of electric energy. During permanent 
 conditions- the flow of energy is uniform and continuous; 
 any change in the power indicates a transient condition. 
 Changes in the current and voltage factors imply a cor- 
 responding change in the energy content of the electric 
 field, since a magnetic field surrounds all electric currents, 
 and an increase or decrease in the current necessitates a 
 corresponding change in the stored magnetic energy. 
 Similarly, any change in voltage between conductors must 
 be accompanied by a corresponding re-adjustment in the 
 energy stored in the dielectric field of the system. 
 
 Magnetic Circuit. In the study of transient phenomena, 
 as well as of all phases of the electric field, Faraday's 
 concept of magnetic and dielectric lines of force is of funda- 
 mental importance. All magnetic lines are continuous and 
 closed on themselves. Ohm's law applies to the magnetic 
 circuits in the same way as to the electric circuit. The 
 magnetic flux produced is equal to the magneto-motive 
 force divided by the reluctance. 
 
 -,, ,. a magneto-motive force 
 
 Magnetic flux = - : 
 
 reluctance 
 
 cy 
 
 $ = - or ff = (R$ (1) 
 
 The magnetic field is produced by, and is proportional to, 
 the electric current. 
 
 $ = Li (2) 
 
INTRODUCTION 
 
 The proportionality factor L is called the inductance of 
 the circuit. 
 
 The reluctance varies directly as the length and inversely 
 as the cross section of the magnetic circuit. The specific 
 reluctance per cm. 3 is the reciprocal of the permeability ju. 
 If the magneto-motive force is expressed in ampere turns, 
 the resultant field intensity is given by the equation. 
 
 H ---- 4irnl lO^per cm. (3) 
 
 This magnetizing force produces a magnetic flux density 
 of B lines per cm. 2 in materials having /* permeability. 
 
 B = [J.H lines per cm. 2 (4) 
 
 The permeability is the reciprocal of the specific reluct- 
 ance in the magnetic circuits and corresponds to the specific 
 
 FIG. 1. Magnetic field of single conductor. 
 
 Magnetic field of circuit. 
 
 conductivity of the conductor in the electric circuits. In 
 empty space ^ = 1 and for all non-magnetic materials it is 
 very nearly equal to unity. For magnetic materials the 
 permeability is greatly increased and may reach several 
 thousand for soft iron and steel. The factor 4?r comes from 
 the definition of a unit magnetic pole as having one line 
 per cm. 2 on the surface of a sphere of unit radius. The 
 10" 1 factor results from the definition of the ampere. 
 
 In building up a magnetic field, lines of force cut the 
 conductor and thus produce a counter e.m.f., or inductance 
 
4 ELECTRIC TRANSIENTS 
 
 voltage, L e, which is equal to the time rate of change of the 
 interlinked magnetic flux. 
 
 d$ T di 
 
 * - dt = L dt < 5 > 
 
 Necessarily an equal opposite voltage must be impressed 
 to force the current through the electric circuit. The prod- 
 uct of the voltage and the current represents the power 
 required to generate the field. Hence, the energy stored in 
 a magnetic field by a current, 7, in a circuit having an induc- 
 tance, L, is given by equations (6) and (7). 
 
 C w C 1 C 1 . 
 
 I dw = I L eidt = L I idi (6) 
 
 Jo Jo Jo 
 
 w-% 
 
 The energy is stored magnetically in the electric field 
 surrounding the conductor and is proportional to the square 
 of the current. When the current decreases the energy is 
 returned to the circuit, for if i and therefore <f> decrease, 
 di/dt and hence L e are negative, which means that the energy 
 is returned to the electric circuit. 
 
 The practical unit of inductance, L, is the henry. In 
 any consistent system of units a circuit possesses one unit 
 of inductance, if a unit rate of change of current in the 
 circuit generates or consumes one unit of voltage. If the 
 current changes at the rate of one ampere per second, and 
 the voltage generated or consumed is one volt, then the 
 inductance is one henry. 
 
 Dielectric Circuit. For the dielectric field similar rela- 
 tions exist. All dielectric lines of force are continuous and 
 end on conductors. Ohm's Law may be applied to the 
 dielectric circuit in the same manner as to the magnetic 
 and electric circuits. 
 
 Dielectric flux = , - ^ -; 
 elastance 
 
 * = = Ce (8) 
 
INTRODUCTION 
 
 The dielectric flux is directly proportional to the voltage 
 between the conductors and inversely proportional to the 
 elastance of the dielectric circuit. The elastance, S, is the 
 reciprocal of the condensance, C, and varies directly as 
 the length, x, and inversely as the cross section, A, of 
 the dielectric circuit. It corresponds to resistance of the 
 electric circuit and to reluctance of the magnetic circuit. 
 
 FIG. 3. Dielectric field of single 
 conductor. 
 
 FIG. 4. Dielectric field of circuit. 
 
 S = . ; C = - '- in c.g.s. electrostatic units 
 K.A 4:irX 
 
 (9) 
 
 S = - ', C = -. -in electromagnetic units (10) 
 
 ' 
 
 i , 
 
 - 1 -a; darafs 
 
 , A 1 O 9 * A 
 
 C = - = 88.42 *~10- 15 farads 
 
 X 
 
 C = 88.42 " 10- 9 microfarads 
 
 JU 
 
 (12) 
 (13) 
 
 The permittivity K is unity for empty space and very 
 nearly equal to unity for air and many other materials. 
 In Table I is given the permittivity constants for the more 
 common dielectrics used in electric apparatus. The con- 
 stant v = 3-10 10 cm/sec., the velocity of the propagation 
 of an electric field in space (equivalent to the velocity of 
 light) , is the ratio of the units used in the electromagnetic 
 
6 
 
 ELECTRIC TRANSIENTS 
 
 and electrostatic systems. The factor 4w comes from the 
 definition of a unit line of dielectric force. 
 
 TABLE I 
 
 Material 
 
 Permit- 
 tivity 
 
 Material 
 
 Permit- 
 tivity 
 
 Air and other gases .... 
 
 1.0 
 
 Olive oil 
 
 3 . to 3 2 
 
 Alcohol, amyl 
 Alcohol, ethyl 
 Alcohol, methyl 
 Asphalt 
 Bakelite 
 Benzine 
 Benzol 
 
 15.0 
 24.3 to 27.4 
 32.7 
 4.1 
 6.6 to 16.0 
 1.9 
 2 2 to 24 
 
 Paper with turpentine 
 Paper or jute impreg- 
 nated 
 Paraffin 
 Paraffin oil 
 Petroleum 
 Porcelain 
 
 2.4 
 
 4.3 
 2.3 
 1.9 
 2.0 
 5 3 
 
 Condensite 
 Glass (easily fusible) . . 
 Glass (difficult to fuse) 
 Gutta-percha 
 Ice 
 Marble 
 
 6.6 to 16.0 
 2 . to 5.0 
 5.0 to 10.0 
 3 . to 5.0 
 3.0 
 6.0 
 
 Rubber 
 Rubber vulcanized. . . 
 Shellac 
 Silk 
 Sulphur 
 Turpentine 
 
 2.4 
 
 2 . 5 to 3 . 5 
 2.7 to 4.1 
 1.6 
 4.0 
 2 2 
 
 Mica 
 
 5.0to 7.0 
 
 Varnish 
 
 2.0 to 4.1 
 
 Micarta 
 
 4.1 
 
 
 
 The charging current, c i, storing energy in the dielectric 
 circuit is equal to the time rate of change in the dielectric 
 flux. 
 
 ^ r<^ e n/n 
 
 = dT L dt 
 
 Hence the energy stored in the dielectric field by a voltage, 
 E, in a circuit having a condensance, C, is given by equa- 
 tions (15) and (16): 
 
 JW (*E / 
 
 dw = I c iedt = C I 
 
 E 
 
 ede 
 
 CE* 
 2 
 
 (15) 
 (16) 
 
 The energy stored dielectrically in the electric field sur- 
 rounding a conductor is proportional to the square of the 
 voltage. When the voltage decreases the energy is 
 returned to the electric circuit, for if e and therefore ^ 
 
INTRODUCTION 7 
 
 decreases, then de/dt and hence c i are negative, which 
 means that the energy is returned to the electric circuit. 
 The unit of condensance (capacitance), C, is the farad. 
 In any consistent system of units a circuit possesses one 
 unit of condensance if a unit rate of change of voltage 
 produces (or consumes) one unit of current. If the voltage 
 changes at the rate of one volt per second and the current 
 produced (or consumed) is one ampere, the condensance 
 
 FIG. 5. Electric field of conductor. 
 
 FIG. C. Electric field of circuit. 
 
 of the circuit is one farad. The farad is too large a unit for 
 practical purposes and hence in commercial problems the 
 condensance is usually measured in microfarads. 
 
 1 farad = 10 6 microfarads (17) 
 
 Electric Circuit. The electric circuit relates specifically 
 to the conductor carrying the electric current although the 
 term is frequently made to include the dielectric and mag- 
 netic fields, since the electric, dielectric and magnetic 
 circuits are interlinked. Under steady or permanent 
 conditions in a direct current system the electric circuit 
 transmits the energy without causing any change in the 
 energy stored magnetically and dielectrically in the space 
 surrounding the electric circuit. In starting the system a 
 transient condition exists until the magnetic and dielectric 
 fields have been supplied with the required amount of 
 energy as determined by the magnitude of the current and 
 voltage and the circuit constants. 
 
8 ELECTRIC TRANSIENTS 
 
 If the electric circuit be considered as something separate 
 and apart from the surrounding magnetic and dielectric 
 fields, no storage of energy would be involved and hence 
 no transients could exist, since all the changes would be 
 instantaneous. But the electric circuit is interlinked with 
 the dielectric and magnetic circuits. Changes in the cur- 
 rent and voltage in the electric circuit are accompanied by 
 changes in the energy stored in the dielectric and magnetic 
 fields, thus necessitating a readjustment of the energy con- 
 tent in the whole electric system. The transfer of energy 
 requires time and thus the transient period is of definite, 
 although often of extremely short, duration. 
 
 The close analogy existing between electric, dielectric 
 and magnetic circuits may be shown to advantage by 
 arranging the corresponding quantities in tabular form as 
 in Table II. 
 
 For convenience in solving problems the energy equations 
 are expressed in the units used in commercial work: 
 Energy in a Magnetic Field : 
 
 = W (joules) = ^ en ^)il(am_peres) (lg) 
 
 2i 
 Energy in a Dielectric Field : 
 
 TTT/. i N CYmicr of arads) e 2 (volts) /irvv 
 
 w oouies) = -~ 
 
 Energy in a Moving Body: 
 
 = TF(ergs) = M (g rams ) ^ (meters per sec.) 
 
 2i 
 Energy in a Moving Body : 
 
 = TF(joules) = M(k S') v 2 (meters per sec.) 
 
 
 Energy in a Moving Body : 
 
 = TFfft Ib } = -' v " ( ft -_Per_sec.) 
 
 2 X 32.2 
 1 joule = 1 watt-sec. = 10 7 ergs = 0.7376 ft.-lb. 
 
 - 0.2389 g.-cal. ='0.102kg.-m. = 0.0009480 B.t.u. (21) 
 1 ft.-lb. = 1.356 joules = 0.3239 g. = 0.1383 kg.-m. 
 
 = 0.001285 B.t.u. = 0.0003766 watt-hour (22) 
 1 B.t.u. - 1,055 joules = 778.1 ft.-lb. = 252 g.-cal. 
 
 - 0.2930 watt-hour (23) 
 
INTRODUCTION 
 
 TABLE II 
 
 Electric circuit 
 
 Dielectric circuit 
 
 Magnetic circuit 
 
 Electric current: 
 
 Dielectric flux (dielectric 
 
 Magnetic flux (magnetic cur- 
 
 
 current): 
 
 rent): 
 
 i = Ge = electric current. 
 
 
 
 R 
 
 * = Ce = * lines of di- 
 
 </> = Li 10 8 lines of mag- 
 
 
 o 
 
 netic force. 
 
 
 electric force. 
 
 
 Electromotive force, voltage: 
 
 Electromotive force: 
 
 Magnetomotive force: 
 
 e = volts. 
 
 e = volts. 
 
 5 = 4-irni ampere-turns. 
 
 
 
 gilberts. 
 
 Conductance: 
 
 Condensance, capacitance, 
 
 Inductance: 
 
 
 permittance or capacity 
 
 n<j> _ <b 
 
 G = mhos. 
 e 
 
 C = farads. 
 
 5 
 
 
 e 
 
 henrys 
 
 Resistance: 
 
 Elastance: 
 
 Reluctance: 
 
 R = ohms. 
 
 S = Q = ^ darafs. 
 
 R = , oersteds. 
 
 Electric power: 
 
 Dielectric energy: 
 
 Magnetic energy: 
 
 'P = R?' 2 G^ = ic watts. 
 
 Ce 2 tye 
 
 T i 2 (hi 
 
 
 w = == joules. 
 
 w = - = n 10~ 8 joules. 
 
 i 
 
 Electric-current density: 
 
 Dielectric-flux density: 
 
 Magnetic-flux density: 
 
 J = = yG amp. per cm. 2 
 
 D = = KK lines per cm. - 
 A. 
 
 B = . - = pH lines per cm. 2 
 
 Electric gradient: 
 
 Dielectric gradient 
 
 Magnetic gradient: 
 
 G' = volts per cm. 
 
 G' = volts per cm. 
 
 / = -j amp. -turns per cm. 
 
 Conductivity: 
 
 Condensivity, permittivity 
 
 Permeability: 
 
 j 
 
 or specific capacity: 
 
 
 7 = mho. -cm. 3 
 
 D 
 
 B 
 
 
 ~ K 
 
 M = H 
 
 Resistivity: 
 
 Elastivity: 
 
 Reluctivity: 
 
 1 G , 
 P = = ohm-cm. 3 
 
 1 K 
 k D 
 
 r = B 
 
 Specific electric power: 
 p = p /2 = 7 G 2 = GI watts 
 
 Specific dielectric energy: 
 kG'* G'D. 
 
 Specific magnetic energy- 
 
 0.47TU/ 2 fB. 
 
 per cm. 3 
 
 cm. 3 
 
 Condensance, permittance, 
 capacitance current: 
 
 d = ,, = C - - amperes. 
 at at 
 
 Dielectric-field intensity: 
 K = - 2 lines of dielectric 
 force per cm. 2 
 
 w = ^ = -10- 
 
 joules per cm. 3 
 Inductance voltage: 
 
 Magnetic-field intensity. 
 // = 47T/10- 1 lines of mag- 
 netic force per cm. 2 
 
CHAPTER II 
 OSCILLOGRAPHS 
 
 The oscillograph is the most important apparatus for 
 obtaining quantitative data on electric transient phenom- 
 ena. To gain clear concepts of the relative magnitude of 
 the physical quantities involved it is highly desirable for 
 
 FIG. 7. Magnetic field and vibrating elements. 
 
 the student to take oscillograms of a number of typical 
 transients. For this purpose an oscillograph with a photo- 
 graphic attachment is necessary. 
 
 While several types of oscillographs are in commercial 
 use all operate on the same basic principle. The essential 
 
 10 
 
OSCILLOGRAPHS 
 
 11 
 
 element of the oscillograph is the galvanometer, an insu- 
 lated loop of wire, placed in a magnetic field, through which 
 the electric current flows. The direction and magnitude 
 of the currents cause a proportional turning movement of 
 a small mirror attached to both sides of the loop. The 
 deflection of a beam of light thrown on the mirror indicates 
 the angular position of the mirror and hence the magnitude 
 and direction of the current flowing through the loop. 
 
 Three Element Oscillographs. The three element, port- 
 able type oscillograph manufactured by the General Electric 
 Co. is shown in Figs. 7 to 13. The arrangement of the 
 
 Mirror- 
 
 FIG. 8. Vibrating element. 
 
 FIG. 9. Cross section of vibrating element. 
 
 electromagnetic field and the three vibrating elements is 
 shown in Fig. 7. One of the vibrating elements removed 
 from its magnetic field is shown in Fig. 8 and its vertical 
 cross-section in Fig. 9. The three vibrators are indepen- 
 
12 
 
 ELECTRIC TRANSIENTS 
 
 dent units and insulated so as to carry three separate elec- 
 tric currents. The vibrating strips and mirrors are of 
 silver. The vibrating element can be turned around a 
 vertical axis, passing through the center of the mirror, by 
 the screw Q. The containing cell for the whole vibrating 
 element is also movable around a horizontal axis, passing 
 through the center of the mirror, by means of the screw S. 
 Hence the beam of light reflected from the vibrating mirror 
 may be directed to any desired spot and so adjusted as to 
 pass through the cylindrical lens to the slit in front of the 
 rotating photographic film. 
 
 In Figs. 7 and 8, the letters TT' mark the terminals of 
 the vibrating strips marked ST in Fig. 9. The mirror 
 with the vibrating portion of the loop lies between the 
 supports BB 1 '. The size of the mirror is about 20 by 10 mils 
 and the vibrating element has a natural period of approxi- 
 
 XA 
 
 FIG. 10. Optical train horizontal projection. 
 
 FIG. 11. Optical train vertical projection. 
 
 mately one five-thousandth of a second (0.0002 sec.). By 
 immersing the vibrating element in oil the instrument is 
 made dead-beat. 
 
OSCILLOGRAPHS 
 
 13 
 
 The horizontal projection of the optical train for photo- 
 graphic work is shown in Fig. 10 and a vertical projection 
 in Fig. 11. The arc lamp is at A and the arrows indicate 
 the directions of the beams of light. PI, P 2 , P* are right- 
 angled prism mirrors; Si, $ 2 , $ 3 , adjustable slits; h, 1%, h 
 condensing lenses; VMi, VM^, VM Z the vibrating mirrors ; 
 CL a cylindrical lens for bringing the light beams to a sharp 
 focus on the photographic film on the surface of the revolv- 
 ing cylinder in the film holder. 
 
 FIG. 12. Oscillograph on operating stand. (Gen. Elec. Co.) 
 
 In Fig. 12 the oscillograph is shown mounted on a con- 
 veniently arranged operating stand. The positions of the 
 arc lamp, film motor, film holder, controlling rheostats, 
 time wave oscillator and other accessory appliances for 
 recording electric transients are clearly indicated. The 
 corresponding wiring diagram, with quantitative circuit 
 data, is shown in Fig. 13. 
 
14 
 
 ELECTRIC TRANSIENTS 
 
 The three-element, portable oscillograph of compact 
 design, manufactured by the Westinghouse Elec. & Mfg. 
 Co., is shown in Figs. 14 to 16. The photographic film 
 drum and driving pulley with rheostats, switches, etc., are 
 shown on the right side of Fig. 14 a, while on the left are the 
 
 FIG. 13. Wiring diagram three element oscillograph. (Gen. Elec. Co.) 
 
 three sets of dial resistances, one for each vibrating element, 
 with switches, binding posts and fuses. In Fig. 146 is 
 shown the driving motor with control apparatus for opera- 
 ting the film holder at several speeds. Light for making 
 the photographic record is obtained from a low voltage 
 incandescent lamp of special design. For high speed records 
 an arc lamp is used, in place of the incandescent lamp, to 
 gain the greatest possible light intensity. 
 
 The galvanometer, with one of the three elements re- 
 
OSCILLOGRAPHS 
 
 15 
 
 FIG. 14a. Front and resistance panel side of portable oscillograph. (Westing- 
 house Elec. & Mfg. Co.} 
 
 FIG. 14&. Front view of portable oscillograph coupled to motor. (Westinghouse 
 
 Elec. & Mfg. Co.) 
 
16 ELECTRIC TRANSIENTS 
 
 moved, is shown in Fig. 15. The moving element consists 
 of a single turn or oblong loop of wire forming two parallel 
 conductors. A tiny mirror is attached to both conductors 
 and placed in a strong magnetic field. Hence when a 
 current passes down one conductor and up the other, one 
 tends to move forward and the other backward. The 
 mirror bridging these conductors is given an angular deflec- 
 tion proportional to the current. 
 
 FIG. 15.- -Three element galvanometer. (Westinghouse Elec. & Mfg. Co.) 
 
 The design of the electromagnetic field circuit is unique. 
 A direct current passing through a single coil sets up a mag- 
 netic flux which passes through the three vibrating elements 
 in series. To insulate the elements from each other and 
 from the main magnetic core and yokes four insulating 
 gaps are used, thus placing seven air gaps in series in the 
 path of the magnetic flux. The three gaps in the galvano- 
 meter elements are J^ 2 in. long, giving sufficient space for 
 the vibrators and producing uniform distribution of the 
 magnetic flux. The four insulating gaps are KG in. long 
 but of large cross-sectional area so as to give comparatively 
 low reluctance in the magnetic circuit. The field excitation 
 requires 6 volts, direct current. 
 
 A view of the trip magnet and shutter release mechanism 
 
OSCILLOGRAPHS 
 
 17 
 
 is shown in Fig. 16, in which the trip magnet holds the long 
 shutter finger so that the short finger does not quite touch 
 the shutter tripping arm. The shutter is a tube with two 
 opposite longitudinal slots. The tube rotates and when 
 the slots are in a horizontal plane the beams of light, re- 
 flected from the tiny mirrors of the galvanometer vibrators, 
 pass through the cylindrical condensing lens and are 
 focused on the revolving photographic film. This occurs 
 between the time the short finger falls from the shutter 
 
 FIG. 16. Trip magnet and shutter release mechanism. (Westinghouse Elec. & 
 
 Mfg. Co.) 
 
 tripping arm and the time the variable finger falls from the 
 arm one revolution later. The shutter is actuated by the 
 spiral spring seen just beyond the finger hub. A pin on 
 the shutter shaft strikes an arm on the lamp extinguishing 
 switch. On the hub are attached laminated copper strips 
 which complete the lamp circuit when the shutter is set and 
 which break the circuit when the shutter snaps closed. 
 The tripping device can be adjusted so as to start exposures 
 at any desired part of the film. 
 
 Timing-waves from the Oscillator Alternator. The 
 time factor is of special importance in electric transient 
 phenomena and some means for recording the time elapsed 
 
18 
 
 ELECTRIC TRANSIENTS 
 
 is necessary. In taking oscillograms in which the transient 
 current or voltage recorded does not give directly an indica- 
 tion of the time consumed it is customary to impress an 
 alternating current timing wave of known frequency on 
 one of the vibrators. 
 
 Current for the timing wave may be takefh directly 
 from any available power circuit, but the frequencies of 
 commercial systems are to some extent variable and the 
 indicating frequency meters may not be sufficiently accu- 
 
 : /VWK&&& 
 
 1ZOV.D.C 
 
 FIG. 17. Oscillator alternator circuit diagram. 
 
 rate for this purpose. A convenient source of supply for 
 timing wave current of constant frequency is found in the 
 oscillator generator. The circuit diagram of a simple porta- 
 ble form used in the electrical engineering laboratories of 
 the University of Washington is shown in Fig. 17. The 
 alternator consists of an audion tube connected to condens- 
 ance, resistance and inductance, as shown in the circuit 
 diagram, of the following quantitative values: 
 
 LI = 0.756 henry s R\ = 99 ohms 
 
 L 2 = 0.756 henrys E 2 = 99 ohms 
 
 L(total) == 2.38 henrys R 3 = 96 ohms 
 
 C = 1.06 microfarads R = 50 ohms 
 
 The impressed d.c. voltage was 110 volts but other values 
 may be used by adjusting the resistance, R s . The ampli- 
 tude of the timing wave may be varied by means of the 
 
OSCILLOGRAPHS 
 
 19 
 
20 ELECTRIC TRANSIENTS 
 
 resistance, R*. The alternating current produced by the 
 oscillator is of simple sine wave form and has a constant 
 frequency of 100 cycles per second. 
 
 Oscillograms. Great care must be taken in making 
 the adjustments on the oscillograph in order to produce 
 good oscillograms. The speed of the revolving drum carry- 
 ing the sensitized film and the amplitude of the galvano- 
 
 FIG. 19. Starting transient of induction motor with secondary resistance in 
 circuit. See Fig. 167, Chap. IX. 
 
 meter mirror vibrations must be adjusted to meet the 
 conditions imposed by the transient under investigation. 
 Thus the relative drum speed for the oscillograms shown 
 in Figs. 18, 19, and 20 was as 4 : 1 : 29, and the amplitude 
 adjusted in each case so as to use the film area to good 
 advantage. 
 
 The time lag of tripping devices and shutter operating 
 mechanism must be determined so as to expose the film at 
 the instant the transient occurs. The optical train must 
 be adjusted so as to give a spot of light sharply focused on 
 the sensitized film. 
 
 Instructions for developing the films and for printing the 
 oscillograms are given in the Appendix. A circuit diagram 
 
OSCILLOGRAPHS 
 
 21 
 
22 
 
 ELECTRIC TRANSIENTS 
 
 with quantitative data should be attached to each film. 
 It is important to show the circuit position of each vibrator 
 
 FIG. 21.- 
 
 -Transfer of oscillating energy in inductively coupled circuits. 
 Chap. VIII. 
 
 See 
 
 FIG. 22. Same circuit as in Fig. 21. Primary opened at the instant all the oscil- 
 lating energy was in the secondary circuit. 
 
 so that the record will indicate precisely where the transient 
 appearing on the film was taken. 
 
OSCILLOGRAPHS 
 
 23 
 
 Problems and Experiments 
 
 1. Examine the oscillograph with care; trace all the circuits; operate the 
 arc lamp; adjust the optical train until the mirror on each vibrator throws 
 a spot of light through the slit and this is sharply focused on the ground 
 glass screen. Arrange a circuit with variable inductance and condensance 
 as indicated in Fig. 23. Connect vibrator V\ by means of a shunt, S, so 
 
 (VWWVW 
 
 FIG. 23. 
 
 as to indicate the current wave and vibrator Vz with a resistance, R\, in 
 series to show the voltage wave. By means of the small synchronous motor 
 operate the large oscillating mirror throwing the beams of light on the 
 mica screen on top of the oscillograph. By varying the resistance, induct- 
 ance and condensance in the circuit the time phase relations of the voltage 
 and current may be changed from lag to lead. 
 
 2. Reproduce, as nearly as available equipment will permit, the oscillo- 
 gram in Fig. 18 
 
CHAPTER III 
 SINGLE ENERGY TRANSIENTS. DIRECT CURRENTS 
 
 Transient electric phenomena are produced by changes 
 in the magnitude, distribution and form of the energy 
 stored in electric systems. The simplest types of electric 
 transients are found in electric circuits having only one 
 kind of energy storage that is, either the magnetic or the 
 dielectric field, but not both. A condenser discharging 
 through a non-inductive resistance, as illustrated by the 
 oscillogram in Fig. 24, gives electric transients of the 
 simplest type. Since the resistance in the circuit is con- 
 stant the current is at all instants directly proportional 
 to the voltage across the terminals of the condenser. The 
 curve on the oscillogram can therefore be used as represent- 
 ing either the current-time or the voltage-time relation as 
 indicated by the two scales in the figure. 
 
 The Exponential Law. The energy stored in the con- 
 denser is at any instant equal to Ce 2 /2. The rate of 
 discharge is ei, which must be equal to the Ri 2 rate of energy 
 dissipation into heat in the resistance. The rate of energy 
 discharge is therefore at any instant proportional to the 
 energy stored in the condenser and the rate of change in the 
 current is at any instant directly proportional to the magnitude 
 of the current. 
 
 Let i and i' represent the currents at any two points on 
 the current-time curve of the oscillogram, in Fig. 24. Then : 
 
 di di' . ., 
 
 dt : dT ::i:i (24) 
 
 Let the line OP be drawn from starting point perpendi- 
 cular to the X axis. Let the line OQ be drawn tangent to 
 the curve at and intersecting the X axis at Q. The time 
 represented by the distance PQ is called the time constant, 
 
 24 
 
DIRECT CURRENTS 
 
 25 
 
 .s 
 
26 ELECTRIC TRANSIENTS 
 
 T, of the circuit. Since i' may be any point on the curve, 
 let it be taken at the starting point, ; then 
 
 t'-7 and ^ = -^ (25) 
 
 From (24) and (25) 
 
 di . IQ . . - . T 
 
 df ~T % ' Io (26) 
 
 Separating the variables and taking the limits of integra- 
 tion from the starting point, 0, to any point (i, t) oil the 
 curve : 
 
 r*. 
 
 JJ 
 
 (27) 
 (28) 
 
 t = Joe-* (29) 
 
 Similarly, for the corresponding voltage-time curve: 
 
 - F f -'f (30) 
 
 e J^Q 
 
 Equations (29) and (30) show that the fundamental 
 relations in simple electric transients are expressed by the 
 exponential equation. The minus sign is used as di/dt is 
 negative. The exponential curve represented by equations 
 (29) and (30) is as fundamental in the study of electric 
 transients as the sine wave in alternating currents. 
 
 If the energy stored in a magnetic field is released by 
 short circuiting through a resistance and dissipated into 
 heat, the same relations exist. Oscillograms of the current- 
 time or voltage-time curves similar to Fig. 24, may be 
 obtained by discharging a magnetic field through a resis- 
 tance, Fig. 27. Likewise, the electric transients existing 
 while a condenser is charged, or while a magnetic field is 
 established, obey the exponential law. In Fig. 25 is shown 
 the oscillogram of a current-time transient obtained while 
 establishing a magnetic field in the circuit shown in the 
 diagram. Let the line OP be drawn through the starting 
 point at right angles to the X axis. Let PS be drawn 
 
DIRECT CURRENTS 
 
 27 
 
 parallel to the X axis and be an asymptote to the current- 
 time curve. Let the line OQ be drawn tangent to the curve 
 
 FIG. 25. Forming a magnetic field. 
 E= 109 volts;/ - 0.36 amps.; R = 303ohms;L = 5.1 henrys; T = 0.017 seconds. 
 
 /v 
 
 / 
 
 ! w 
 
 FIG. 26. Showing permanent, transient and instantaneous values for oscillogram 
 
 in Fig. 25. 
 
 at and intersecting the line PS at Q. The line OP repre- 
 sents the value of the permanent current / which is equal 
 
28 ELECTRIC TRANSIENTS 
 
 to E/R. The time measured by the line PQ is the time 
 constant, T, of the circuit. The rate of storing the mag- 
 netic energy at any instant is proportional to the remaining 
 magnetic storage facilities in the circuit under the given 
 conditions. Therefore the rate of change in the current is at 
 any point on the curve proportional to / - - i, and equation 
 (31) is derived in the same manner as equations (24) and (26) 
 
 (33) 
 
 (34) 
 The transient which by definition represents the change 
 
 from one permanent condition to another is in equation 
 
 _ t 
 
 (34) represented by the factor Ie T . Before the circuit 
 was closed the value of the current was zero, while the 
 final permanent value is I. 
 
 In Fig. 26 the permanent or final value of the current is 
 represented by OP, the distance of the line PS from the X 
 axis. The transient values are given by the ordinates to 
 the broken curve WVX, while the instantaneous current 
 which must at any instant be equal to the algebraic sum of 
 the permanent and transient values is given by the ordi- 
 nates to the curve OYS, which is the curve photographed 
 on the oscillogram in Fig. 25. It is important to note that 
 the photographic record of the actual instantaneous values 
 gives at each point the resultant or the algebraic sum of 
 the corresponding permanent and transient ordinates. 
 Thus for any time, t: 
 
 RY = RU + (-RV) (35) 
 
 The Time Constant. From the starting point of the 
 current-time curve in the oscillogram, Fig. 27, which shows 
 the discharge of a magnetic field through a resistance, the 
 line OP is drawn at right angles to the X axis ; the line OQ tan- 
 
DIRECT CURRENTS 
 
 29 
 
 gent to the curve at the point and intersecting the X axis at 
 Q' } the line QN perpendicular and ON parallel to the X axis. 
 From the principle of the conservation of energy, the 
 energy stored in the magnetic field must be equal to the 
 amount dissipated as heat in the resistance of the circuit 
 when the field is discharged. 
 
 /00 / 
 
 = R I tfdt = RI Q 2 I 
 
 Jo t/o 
 
 RIJ-T 
 
 (36) 
 
 (37) 
 
 In circuits having resistance and inductance in series, as 
 in Fig. 27, the time constant is equal to the inductance 
 divided by the resistance. 
 
 FIG. 27. Discharge of a magnetic field through a constant resistance. 
 E = 60 volts; I = 0.21 amps.; R = 28.6 ohms; L= 0.89 henrys; T = 0.031 
 seconds. 
 
 In a similar manner the expression for the time constant, 
 T, in terms of the circuit constants may be found for circuits 
 having condensance and conductance, Fig. 24. The energy 
 stored in the condenser when the discharge starts must be 
 equal to the energy expended as heat in the Ri 2 losses. 
 
30 ELECTRIC TRANSIENTS 
 
 CF 2 f ra f ra 2t J?T 2 T 
 
 ^p_ = R tfdt = RI 2 e~ Tdt = ^ (38) 
 
 /o /o 
 
 Hence, 
 
 T = CR = C G (39) 
 
 In circuits having resistance and condensance in series, 
 as in Fig. 24, the time constant is equal to the condensance 
 divided by the conductance. 
 
 Equations (37) and (39) are of fundamental importance 
 in the study of transient phenomena. The exponential 
 equation for the transients in Figs. 24 and 26 may be 
 rewritten using the value of T as given in (37) and (39) and 
 the data in the circuit diagrams. 
 
 Fig. 24, equations (29), (39) 
 
 / C 1 
 
 i = I Q ~T = 7 oc -c' = 4.18 " : '"'amperes ( 40 ) 
 Fig. 24, equations (30), (39) 
 
 e = E Q e~ * = E$. = 120.6 e~ 38 - 5 Volts ( 41 ) 
 
 Fig. 25, equations (34), (37) 
 
 i = I - Ie~ T = I - 7e~*' = 0.36 - 0.36e~ 58 ' 7 amperes (42) 
 The reciprocals of the time constants appearing in the 
 exponential equations as R/L and G/C or such combinations 
 of circuit constants as the complexity of the system may 
 require, are often called the dissipation constants or the 
 attenuation constants of the circuit. 
 
 The expressions for the time constants in equations (37) 
 and (39) may be derived from the current-time and voltage- 
 time curves instead of basing the equations directly on the 
 principle of the conservation of energy. In Fig. 24 the 
 quantity of electricity (coulombs) in the condenser when 
 starting the transient must be equal to the total amount 
 expended when the condenser is discharged, as represented 
 by the area between the current-time curve and the X axis. 
 
 rco /oo 
 
 idt = 7 I e~ * dt = I Q T (43) 
 
 Jco / 
 
 idt = 7 1 
 */ o 
 
DIRECT CURRENTS 31 
 
 Hence, 
 
 T = C E T = CR = C n = 0.026 seconds (44) 
 
 -/O (JT 
 
 Similarly, in Fig. 27, the magnetic flux in the field when 
 starting the transient may be equated to the total number 
 of lines of force cutting the circuit when all the magnetic 
 energy in the field changes into heat in the resistance. 
 
 Hence, 
 
 Jf- r~ t 
 
 edt = R\ idt = RI Q I e T dt = RI T (50) 
 t/o /o 
 
 T = = 0.31 seconds (51) 
 
 H 
 
 If the initial rate of discharge in Fig. 24 be continued 
 unchanged, the current-time curve would coincide with the 
 line OQ and the condenser would be completely discharged 
 in the time represented by PQ or T. Hence the area of the 
 rectangle OPQN must be equal to the area between the 
 current-time curve and the X axis. 
 
 Similarly, in Fig. 27, if the initial rate of discharge 
 continued unchanged, the current-time curve would coin- 
 cide with the line OQ and all the energy stored in the mag- 
 netic field would appear as heat in the resistance in the time 
 represented by PQ or T. Hence the area of the rectangle 
 OPQN must be equal to the area between the current-time 
 curve and the X axis. 
 
 Expressions for the transient current and voltage as given 
 in (40), (41) and (42) are derived without using the time 
 constant term. The customary differential equations 
 giving the. basic relations, with expressions for the transient 
 currents, are given in (52) to (55). 
 
 For circuits having resistance and condensance, Fig. 24, 
 while the condenser is discharged through a constant 
 resistance : 
 
 Ri + f- == 0; hence i == / O e~*c == 1$ (52) 
 
32 ELECTRIC TRANSIENTS 
 
 For circuits having resistance and condensance, similar 
 to Fig. 24, the transient current while charging from to 
 the voltage E is the same as for discharging through the 
 resistance. 
 
 Ri + ( l ~ ---- E', hence i = 7 e"V (53) 
 
 For circuits having resistance and inductance, Fig. 19, 
 while the magnetic field is formed: 
 
 Ri +L- = E-, hence, i = f - ^"^ = / - /e~ (54) 
 at K H 
 
 For circuits having resistance and inductance, Fig. 21, 
 and a magnetic field supplied by a current 7 , while the 
 field discharges through a short circuit: 
 
 Ri + L~ = 0; hence, i = 7 e~ ^ (55) 
 
 Cvt' 
 
 The Exponential Curve. Oscillograms of simple electric 
 transients give a photographic record of the current-time 
 factors. The amplitude of the curve varies directly with 
 the magnitude of the current passing through the vibrator 
 and the strength of the magnetic field in which the vibrator 
 moves. The length of film used for any given unit of 
 time depends on the speed of the revolving drum carrying 
 the film. It is evident that both the amplitude of the 
 mirror vibrations and the speed of the film may be adjusted 
 independently of the circuit in which the transient occurs. 
 By examining exponential equations representing simple 
 electric transients it is apparent that if the value of the time 
 constant, T, be used as the unit of length on the X axis and 
 the initial value of the variable as the unit of measure for 
 the ordinates, then all exponential transients will have the 
 same shape and may be represented by the numerical 
 values of the exponential equation, y = e~ x . The same 
 space unit need not be used on both axes to represent the 
 unit values of current and time, but the scale may be 
 selected so as to secure a convenient shape for the available 
 
DIRECT CURRENTS 
 
 33 
 
 space. In Fig. 28 is shown a current-time curve in which 
 the unit representing the initial value of the transient 
 current is five cm., while the unit used on the X axis, that 
 is, for the time constant of the circuit, is one cm. 
 
 FIG. 28. The exponential curve. Current-time transient. 
 
 By using the initial value of the transient as the unit of 
 ordinates and the time constant of the circuit as the unit 
 
 TABLE III 
 
 y = e *; = 2.71828 
 
 y 
 
 0.00 1.000 1.2 
 
 0.301 
 
 0.05 
 
 0.951 
 
 1.4 
 
 0.247 
 
 0.10 
 
 0.905 
 
 1.6 
 
 0.202 
 
 0.15 
 
 0.860 
 
 1.8 
 
 0.165 
 
 0.20 
 
 0.819 
 
 2.0 
 
 0.135 
 
 0.30 
 
 0.741 
 
 2.5 
 
 0.082 
 
 0.40 0.670 
 
 3.0 
 
 0.050 
 
 0.50 
 
 0.607 
 
 3.5 
 
 0.030 
 
 0.60 0.549 
 
 4.0 
 
 0.018 
 
 0.70 
 
 0.497 
 
 4.5 
 
 0.011 
 
 0.80 
 
 0.449 
 
 5.0 
 
 0.007 
 
 0.90 
 
 0.407 
 
 6.0 
 
 0.002 
 
 1.00 
 
 0.368 
 
 7.0 
 
 0.001 
 
34 ELECTRIC TRANSIENTS 
 
 of abscissae, all exponential transients are of the same shape 
 and if plotted to the same scale would be identical with the 
 curve in Fig. 28. The numerical relations between y and 
 x in the exponential equation y = e~ x are given in Table 
 III. While the plotting of transients may be facilitated 
 by the selection of the above units, the actual initial values 
 of the transient quantity, expressed in amperes or volts, 
 may be of any magnitude as determined by the circuit 
 conditions. 
 
 Initial Transient Values. In simple electric transients 
 the initial value of the variable quantity depends on both 
 the permanent value and on the relative magnitude of the 
 circuit constants. Thus equations (33) and (42) show that 
 the time constant of a magnetic field depends on the induc- 
 tance and resistance in the circuit. If the energy stored in 
 the magnetic field be discharged by short circuiting the 
 terminals of the field, the initial value of the transient volt- 
 age will be equal in magnitude but opposite in direction to 
 the previously permanent value. But if the discharge be 
 made through an additional resistance, R^ the initial 
 voltage transient will be greater in magnitude in the ratio 
 of Ri + Rz : Ri, when Ri represents the resistance of the 
 field winding. The time constant of the circuit in which 
 the transients appear would be, 
 
 T = D when the field is short circuited, 
 HI 
 
 and TI = - D -~ '=- when the additional resistance # 2 is 
 
 Hi T- Hz 
 
 inserted in the discharging circuit. With the same amount 
 of energy stored in the magnetic field, the products of the 
 initial value and the corresponding time constants must 
 be equal. 
 
 E T = E 'T' (56) 
 
 Hence, 
 
 E Q :E Q f ::R l :R 1 + R 2 (57) 
 
 Eo > = E Q R R * (58) 
 
 Hi 
 
DIRECT CURRENTS 
 
 35 
 
 The initial induced discharge voltage is therefore greater 
 than the permanent impressed voltage in the proportion 
 of the resistances in circuit for the two cases. In the 
 voltage-time curve, Fig. 29, the initial discharge voltage, 
 Eo", is that part of the induced voltage, E Q ', due to the 
 Ri 2 drop. 
 
 EV/ 
 111 Q = 
 
 = E 
 
 R 
 
 (59) 
 
 This relation is of great importance in the design and opera- 
 tion of electrical machinery. In breaking electric circuits, 
 as induction coils, motor and generator fields, transmission 
 
 wwwwvwww 
 
 FIG. 29. Magnetic field discharging through additional resistance, Ri. 
 
 lines, etc., in which energy is stored magnetically, the air 
 or oil gap in the switch introduces a rapidly increasing 
 resistance. The faster the contact points of the switch or 
 circuit breaker separate, the more rapidly the resistance is 
 inserted and the higher the induced voltage.' 
 
 In Fig. 30 is shown the voltage-time and current-time 
 oscillogfams for breaking the field circuit of a direct- 
 current motor. In opening the switch an arc is formed 
 by which a resistance of rapidly increasing magnitude is 
 
36 ELECTRIC TRANSIENTS 
 
 introduced into the circuit. The oscillograrn shows that in 
 about >f 5 of a second the induced voltage increased to more 
 than twenty-eight times the voltage impressed on the termi- 
 nals of the field before the switch was opened. Although 
 the voltage applied to the motor field was only 31.5 volts 
 the opening of the switch in the field circuit produced a 
 transient stress of over 900 volts on the field insulation. 
 
 FIG. 30. Breaking field circuit of direct current motor. Current and voltage 
 
 transients. 
 
 Since the transient induced voltage on the motor field 
 winding is directly proportional to the rate of cutting lines 
 of force the shorter the time used in opening the switch, 
 or the faster the resistance is inserted in the circuit the 
 greater the transient voltage-stress tending to puncture the 
 field insulation. If the circuit breaker operates in steps by 
 which resistances of known value are introduced into the 
 circuit in rapid succession the transient induced voltage 
 will be proportionately lower and the destructive action 
 of the arc greatly reduced. Since the energy stored in an 
 
DIRECT CURRENTS 37 
 
 electromagnetic field depends on the current flowing in the 
 field windings, it must be converted into some other form 
 when the current is interrupted. 
 
 Current, Voltage and Magnetic Flux Transients. In 
 electromagnetic circuits having constant permeability the 
 current, voltage and flux transients have the same shape 
 and are expressed by the exponential equation. Referring 
 to Fig. 27 
 
 e = Ri (60) 
 
 The curve in the oscillogram therefore represents either 
 the current or voltage transients and the quantitative 
 values are obtained by applying the corresponding ampere 
 and volt scales. From the law of electromagnetic induc- 
 tion the induced voltage is equal to the rate of cutting 
 lines of force. 
 
 e = . 10~ 8 volts (61) 
 
 at 
 
 Hence, 
 
 edt = 
 
 = 10 s I e 
 
 - R t 
 
 = constant e L lines of flux (62) 
 
 The flux transient therefore is an exponential curve of 
 the same form as the current and voltage transients. 
 
 In Fig. 31 is shown the corresponding transients for the 
 current, voltage and flux in forming an electromagnetic 
 field in a magnetic circuit of constant permeability. The 
 transients are shown by the dotted lines, the permanent 
 values by the broken lines and the instantaneous values 
 by the full line curves. 
 
 / _ / - (63) 
 
 / _t\ _R t 
 
 = E + \-Ee T ) = E - Ee L (64) 
 
 _* 4 
 = $ _ $ e L (65) 
 
38 
 
 ELECTRIC TRANSIENTS 
 
 T 
 
 -/ 
 
 / 
 
 L 
 
 T 
 
 1 
 
 'N 
 
 T 
 
 $ 
 
 \\ 
 
 1 
 
 f N 
 
 FIG. 31. Single energy voltage, current and magnetic flux transients in forming a 
 magnetic field in a magnetic circuit of constant permeability. 
 
DIRECT CURRENTS 39 
 
 Equations (61), (62) and (63) express the instantaneous 
 values as equal to the algebraic sum of the permanent and 
 transient quantities. At any instant in time, t, as indicated 
 in Fig. 31: 
 
 PQ = the instantaneous value, i, e or < (66) 
 
 PS = the permanent or final value, /, E or < (67) 
 
 t t t 
 
 PN -- the transient value, I e~ T , E Q e~ T , 3> Q e~ T (68) 
 In discharging energy stored in a magnetic field, in a 
 magnetic circuit of constant permeability, through a con- 
 stant resistance, the transient and the instantaneous values 
 are equal as the permanent value is zero. 
 
 i = 1^^ = I^~ Lt (69) 
 
 -' - R t 
 
 e = E e T = E e L (70) 
 
 -' - R t 
 
 = $ oe T = $ oe L (71) 
 
 Problems and Experiments 
 
 1. A condenser of 115 mfds, charged to 500 volts, is discharged through a 
 constant resistance of 425 ohms. 
 
 (a) Derive the equations for the current-time curve. 
 (6) Find the time constant of the circuit. 
 
 (c) Plot the voltage across the terminals of the condenser-time curve. 
 Ordinates in volts and abscissae in seconds. 
 
 (d) Draw an ampere scale of ordinates so that the curve plotted in (c) 
 will represent the current transient. 
 
 2. The time constant of an inductance coil is found by taking an oscillo- 
 gram to be 0.04 seconds. The resistance in circuit was 15.8 ohms. 
 
 (a) Find the inductance in henrys. 
 
 (6) With 110 volts impressed on the coil plot the starting current 
 transient. 
 
 (c) Write the equation for the current-time curve in (6). 
 
 3. Take an oscillogram of the starting current transient of the field of a 
 laboratory motor or generator. 
 
 (a) From the oscillogram find the time constant of the field. 
 (6) Measure the resistance of the circuit and calculate the inductance of 
 the field. 
 
 4. With the vibrators connected as shown in the circuit diagram in Fig. 
 30 taken an oscillogram showing the current and voltage transients pro- 
 duced by breaking the field circuit of a motor or generator. 
 
 6. By means of oscillograms determine the time required for the opera- 
 tion of automatic circuit breakers. Arrange the connection for the vibrators 
 so as to show the time consumed by each step in the operation. 
 
CHAPTER IV 
 
 SINGLE ENERGY TRANSIENTS. ALTERNATING 
 CURRENTS 
 
 In direct-current systems the transient electric phenom- 
 ena described in the preceding chapter, are due to the 
 storage of energy in magnetic and dielectric fields. If a 
 constant direct-current voltage is impressed on a circuit 
 having constant resistance but neither inductance or 
 condensance the current would instantly reach its perma- 
 nent value, and any change in the voltage would at the same 
 time cause a proportional change in the current. With 
 either condensance or inductance in the circuit a short 
 period of time is required for the current to reach its perma- 
 nent value after any change in voltage, and the current- 
 time curve during the transition period is expressed by the 
 exponential equation. The value of the variable current 
 is at any instant equal to the algebraic sum of the per- 
 manent and transient values; and single energy transients 
 for direct currents in circuits having constant resistance 
 and inductance or constant resistance and condensance, 
 may be expressed by exponential equations similar in form 
 to (63), (64) and (65). 
 
 Single Phase, Single Energy Load Circuit Transients. 
 The same principle applies to single energy transients in 
 alternating-current systems. In circuits having constant 
 resistance but neither inductance nor condensance, no 
 transients appear. Any change in the voltage produces 
 instantly a proportionate change in the current. In 
 circuits having either inductance or condensance a tran- 
 sient period for the readjustment of the energy content of 
 the system follows any change in voltage. At any instant 
 the transient current or voltage is the algebraic sum of the 
 
 40 
 
ALTERNATING CURRENTS 
 
 41 
 
 . > II 
 
42 ELECTRIC TRANSIENTS 
 
 corresponding permanent and transient values. The per- 
 manent term, i' is the alternating current wave assumed 
 to be of simple sine form as expressed in equation (72) with 
 71 as the time phase angle at the starting moment. 
 
 i' = +Vsin (co - 7l ) (72) 
 
 e ' = +-# sin (co - T2 ) (73) 
 
 The transient term, i" is expressed by the exponential 
 equation of the same form as in direct currents with an 
 initial value equal in magnitude but opposite in time phase 
 to what would have been the permanent value at the start- 
 ing point if the circuit has been closed at some previous time. 
 
 i" = Vsin 7, e~r (74) 
 
 e" -- M E sin 72 ^ T (75) 
 
 The instantaneous value of the current or voltage would 
 therefore be expressed by (76) and (77) : 
 
 i = t' + i" a! + V sin (ut - 71) "I sin 71 e~* (76) 
 e = e ' + e " = M I s in (ut - 72) + "E sin Tl e~ T (77) 
 
 The oscillogram in Fig. 32 shows the current-time curve, 
 i, produced in a circuit having 20 ohms resistance and 0.575 
 henrys inductance when a 60 cycle alternating-current volt- 
 age, e, was impressed by closing a switch at the instant in 
 time presented by the OY axis. 
 
 The voltage impressed is represented by the sine wave, 
 
 e = M E sin (co - T2 ) (78) 
 
 The actual current flowing is represented by the curve 
 i, which practically coincides, after completing 6 cycles, 
 with the permanent value i, shown by the dotted sine wave. 
 The transient, i", is shown as the broken line whose initial 
 value, 
 
 ON = -OP = Vsin Tl (79) 
 
 At any instant, t, after the closing of the switch, i is equal 
 to the algebraic sum of i' and i" . 
 
ALTERNATING CURRENTS 
 
 43 
 
 I = I + I 
 = 0.835 
 
 = V sin co - 
 
 wt 2 
 
 + 0.835 sin 
 
 -4 
 
 '/ sin Tl e L (80) 
 
 -35* 
 
 (81) 
 
 It is evident that the initial value of the transient may 
 vary from - M I to + A 7, depending at what point of the 
 voltage-time curve the circuit is closed. If the switch be 
 thrown at the instant when the permanent current wave 
 would be zero, 7 = 0, no transient would appear and the 
 
 E 
 
 FIG. 33. Single phase, single energy, current transient. 
 129 volts; M E = 182.5 volts; I = o'.59 amps.; "I .835 amps.; 
 R = 20 ohms; L = 0.575 henrys; / = 60 cycles; 71 = 0. 
 
 permanent and actual current time curves would coincide 
 throughout as shown in Fig. 33. 
 
 i = i' = M I sin (0 (82) 
 
 The transient current would have a maximum initial 
 value if the circuit is closed at the instant the permanent 
 current wave is at a maximum, that is when sine 71 = 90% 
 The time constant for the current transient would be the 
 
44 ELECTRIC TRANSIENTS 
 
 same at whatever point in the cycle the circuit is closed, as 
 it depends on the resistance and inductance in the load 
 circuit. 
 
 Three-phase, Single Energy, Load Circuit Transients. 
 For three-phase circuits similar relations exist. The start- 
 ing current transients in three-phase systems in which 
 energy may be stored either magnetically or dielectrically 
 follow the same laws as discussed for single-phase circuits. 
 
 In Figs. 34 and 35 are shown oscillograms of the three 
 starting load currents in a three-phase system, star con- 
 nected and having 9.0 ohms resistance and 0.205 henrys 
 inductance in each phase to neutral. The corresponding 
 permanent current waves and transient currents were 
 traced on the oscillogram in Fig. 34. In Fig. 35 the circuits 
 were closed at the instant the current in v\ was of zero 
 value. 
 
 Phase 1 : 
 
 Impressed voltage, 
 
 61 = "Ei sin (co 72) (83) 
 
 Permanent current, 
 
 i'i = M Ii sin (ut 71) (84) 
 
 Initial value starting current transient, 
 
 OQ 1 = -OP, = "/sin 71 (85) 
 
 Transient current, 
 
 _Ri 
 
 i"i = "I i sin 71 e L I (86) 
 
 Actual current, oscillogram, 
 
 ^ = i\ + i'\ = Vi sin (ut - 71) + Vi sin yie ^ ( 87 ) 
 Time constant, 
 
 T = - = 0.023 seconds (88) 
 
 Phase 2: 
 
 Impressed voltage, 
 
 e, = E 2 sin (ut - 72 - 120) (89) 
 
 Permanent current, 
 
 i\ = / 2 sin (ut - 71 - 120) (90) 
 
ALTERNATING CURRENTS 
 
 45 
 
 IS 
 
46 
 
 ELECTRIC TRANSIENTS 
 
ALTERNATING CURRENTS 47 
 
 Initial value starting current transient, 
 
 OQz = -OP 2 = "I z sin (71 - 120) (91) 
 
 Transient current, 
 
 t" 2 = V 2 sin (71 -- 120)~5 (92) 
 
 Actual current, oscillogram, 
 
 i 2 = i' 2 + i" 2 = / 2 sin (co - 72 -- 120) 
 
 +V 2 sin (71 - 12Q)e~ (93) 
 Time constant, 
 
 T 2 = f 2 = 0.023 seconds (94) 
 
 -^2 
 
 Phase 3: 
 
 Impressed voltage, 
 
 e, = M Ez sin (* -- 72 - 240) (95) 
 
 Permanent current, 
 
 i' 8 = "/ 8 sin (* -- 71 - 240) (96) 
 
 Initial value starting current transient, 
 
 OQ 3 = -OP, = M I Z sin (71 - 240) (97) 
 
 Transient current, 
 
 t" 8 == "I* sin (71 -- 240)e *' (98) 
 
 Actual current, oscillogram, 
 
 i 3 = i' 8 + i'% = V 8 sin (co^ - 71 - 240) 
 
 + M / 3 sin (71 - 240) e~^ (99) 
 Time constant, 
 
 T z =^ = 0.023 seconds (100) 
 
 /i3 
 
 It is of interest to note that the sum of the instantaneous 
 values of the three currents, ii + i z + i$, is equal to zero 
 during the starting period as well as after the permanent 
 state has been reached. This is evidently the case since 
 under permanent conditions the sum of the currents is at 
 any instant equal to zero and hence at the instant the 
 circuit is closed, OPi + OP 2 + OP 3 = 0. Therefore, the 
 sum of the initial values of the transient currents, OQi + 
 + OQz = 0, and as the time constants of the three 
 
48 ELECTRIC TRANSIENTS 
 
 transients are equal the sum of the actual currents in the 
 three phases must at any instant be equal to zero. 
 
 Starting Transient of a Polyphase Rotating Magnetic 
 Field. In the preceding illustrations the single energy 
 transients are due to changes in the amounts of energy 
 stored in the given circuits, and the current-time curves 
 show a continuous decrease of the current as expressed by 
 the exponential equation. If the permanent condition 
 relates to interconnected circuits which permit a transfer 
 of energy from one circuit to another, although the total 
 amount of energy stored in the magnetic field is constant, 
 as in the rotating field of a polyphase induction motor, 
 pulsations will appear during the transition period, that 
 merit attention. 
 
 In Fig. 36 let a vector of constant length, ON, rotating in 
 a counter clockwise direction represent a constant rotating 
 magnetic field, as would be produced by three equal mag- 
 netizing coils, placed 120 deg. apart, and excited by three- 
 phase currents, as, for example, in a three-phase induction 
 motor. For simplicity let the rotor be removed and con- 
 sider the stator circuit and the magnetic flux during the 
 starting transition period in which the rotating field is 
 built up to its constant permanent value. Let the switch, 
 connecting the stator circuit to three-phase mains, be 
 closed at the instant the rotating magnetic flux vector, ON, 
 lies along the X axis, ON in Fig. 36 ; as would have been the 
 case if the switch had been closed at some previous 
 time. The actual value of the rotating flux at the instant 
 the circuit is closed is zero. The permanent value is 
 represented by ON Q and since the initial value of the transi- 
 ent flux must be equal in magnitude but of opposite time 
 phase, it is represented by the vector OQ . 
 
 OQ Q =-- (-(W ) (101) 
 
 From its initial values OQ the transient flux decreases 
 in magnitude, as indicated by the exponential flux-time 
 curve in Fig. 37, but continues fixed in space direction 
 
ALTERNATING CURRENTS 49 
 
 along the X axis. After the time, ti, has elapsed, repre- 
 sented by the time angle N ONi, the transient has a value 
 OQi. The actual value of the flux must be the vector sum 
 of the permanent value ONi and the transient OQi, or the 
 resultant OPi. In the time t 2 , the transient has decreased 
 to OQ 2 and the permanent flux vector reached the position 
 ON 2 . The actual flux OP 2 is the resultant of OQ 2 and ON 2 . 
 Similarly OP 3 is the resultant of OQ 3 and ON Z ] OP 4 , of 
 
 FIG. 36. Permanent, transient and instantaneous values of the magnetic flux in 
 starting a rotating magnetic field. Polar coordinates. 
 
 OQ 4 and ON^ etc. From the vector diagrams, Figs. 36 
 and 38, it is evident that the actual starting flux will oscil- 
 late having values greater and smaller than the permanent 
 value, the number of oscillations depending on the time 
 constant of the circuit. The maximum value of the flux 
 in the starting period will in any case be less than double the 
 permanent value, as the transient flux continuously 
 decreases from an initial value equal to the permanent flux 
 in magnitude and different by 180 deg. in time-phase. 
 
50 
 
 ELECTRIC TRANSIENTS 
 
 The flux-time curve in Fig. 39 gives in rectangular coordi- 
 nates the same relation as shown by the flux vector OP in 
 the polar diagram in Fig. 38. 
 
 Polyphase Short Circuits. Alternator Armature and 
 Field Transients. Consider a three-phase alternator carry- 
 ing a constant balanced load of constant power factor. 
 The three-phase currents flowing in the armature produce 
 a resultant constant armature flux or armature reaction. 
 
 FIG. 37. Starting magnetic flux transient from Fig. 36. Rectangular coordi- 
 nates. 
 
 With respect to the field the armature flux is stationary but 
 with respect to any diameter of the armature taken as a 
 reference axis, the armature currents produce a constant 
 rotating field of the same nature as the constant rotating 
 field of a three-phase induction motor. For a machine in 
 which the field is on the rotating spider while the armature 
 is stationary, the resultant flux producing armature 
 reaction rotates synchronously with the field. For an 
 alternator with the field stationary and the armature 
 rotating the resultant constant flux rotates at the same 
 
ALTERNATING CURRENTS 
 
 51 
 
 speed but in direction opposite to the rotation of the arma- 
 ture and therefore is stationary with respect to the frame 
 of the machine. In either case the armature flux or the 
 armature reaction is stationary, if referred to the alternator 
 field, but is a synchronously rotating field with respect to 
 the armature. 
 
 FIG. 38. Starting a rotating magnetic field. Polar vector diagram of magnetic 
 flux for three cycles of Fig. 36. 
 
 Due to the close proximity of the armature conductors 
 to the field poles a large part of the magnetic flux produced 
 by the armature currents passes through the field magnetic 
 circuit. This causes a reduction in the field flux and 
 therefore in the amount of energy stored magnetically by 
 the exciting current in the field. Hence, although a con- 
 stant direct-current voltage is impressed on the field circuit 
 the useful flux is greatly reduced by the armature reaction, 
 and as a consequence the generated armature voltage 
 decreases in the same proportion. To effect any change in 
 the amount of energy stored magnetically takes time and 
 therefore the interaction of the armature flux with the field 
 magnetic circuit produces electric transients. 
 
 During the transition period following the instant the 
 short circuit occurs, two distinct causes are therefore 
 superimposed in producing transient phenomena in the 
 
52 
 
 ELECTRIC TRANSIENTS 
 
 interlinked electric and magnetic circuits of polyphase 
 alternators. 
 
 (a) The armature transient which is equivalent to the 
 starting transient of a rotating magnetic field including full 
 frequency pulsations as illustrated in Figs. 38, 39. 
 
 (b) A field transient due to the reduction of the field flux 
 by the armature reaction. 
 
 FIG. 39. Same data as in Fig. 38. Rectangular coordinates. 
 
 The transients produced under (a) and (b) differ in 
 duration, the ratio being in each case determined by the 
 relative time constants of the armature and field circuits. 
 In general the time constant in the field circuit is greater 
 than in the armature circuits. Large turbo-alternators 
 have very slow field transients as compared to the duration' 
 of the armature transients. 
 
 In Fig. 40 is shown an oscillogram of transients produced 
 by a short circuit on all three phases of a 7.5 kw., 240 volt, 
 60 cycle, three-phase, star-connected alternator running 
 idle and with 40 per cent normal field excitation. A similar 
 oscillogram of short circuit transients for the same machine 
 while carrying 50 per cent of full load is shown in Fig. 41. 
 
ALTERNATING CURRENTS 53 
 
 As indicated in the circuit diagrams, Figs. 40 and 41, 
 vibrator Vi records the armature voltage across one pair of 
 slip rings, e a \ vibrator i> 2 , the current, i a , in one armature 
 circuit, and vibrator v s the field current, i f . As the short 
 circuit is directly across v\, the voltage e a instantly drops to 
 zero. The transient in the field winding is due to the com- 
 bined action of the starting transient of the rotating field 
 
 FIG. 40. Short circuit transients from no load. Three-phase alternator, star- 
 connected. 
 
 E, no load = 109 volts; 7, short circuit = 12.0 amps.; I, field = 1.25 amps.; 
 
 / = 60 cycles. 
 
 in the armature, which produces the full frequency pulsa- 
 tions, and the slower field transient resulting from the 
 reduction of the field flux by the armature reaction. In 
 breaking the short circuit the field transient alone will 
 appear in the field winding, as shown by the oscillograms 
 in Figs. 42 and 43. Necessarily the transient is reversed 
 in direction from what is represented in Figs. 40 and 41, 
 when the short circuit is made. It should be noted that 
 breaking the armature short circuit was not instantaneous 
 
54 
 
 ELECTRIC TRANSIENTS 
 
 a 
 
 o ^ 
 
 II 
 
 
 
ALTERNATING CURRENTS 
 
 55 
 
 O ^H 
 .^ CO 
 
 bC 3 
 
 a 
 
56 
 
 ELECTRIC TRANSIENTS 
 
ALTERNATING CURRENTS 
 
 57 
 
 as arcs formed at the switch and continued the circuit 
 during the time, a b, Fig. 42, approximately for % of 
 a cycle or J^oo f a second. During this period the 
 energy stored magnetically in the armature circuits was 
 dissipated. Much more time, over 10 complete cycles, 
 was required to restore full excitation in the field poles. 
 
 FIG. 44. Armature current transients. Short-circuit on three-phase, star-con- 
 nected alternator, no load. 
 
 E = 280 volts; /, short circuit = 28.7 amps.; 7, field = 3.3 amps.; / = 59 
 cycles. 
 
 The direct relation of the voltage generated in the arma- 
 ture to the variable useful field flux is shown by the voltage 
 wave, e a , and the field transient, i f , in Fig. 42. When the 
 short circuit is made the same transients occur, but reversed 
 in time, as is evident from oscillograms in Figs. 40, 41, 44 
 and 45. 
 
 In the operation of alternators the relative value of the 
 initial or momentary to the final or permanent short circuit 
 currents is of great importance. At any instant the short 
 circuit current obeys Ohm's law, that is in magnitude it 
 
58 ELECTRIC TRANSIENTS 
 
 will be directly as the voltage generated and inversely as 
 the impedance of the armature circuit. 
 
 ia = ^ --~^-^-'~ x - ( 102 ) 
 
 Since the armature resistance, R a , is small compared to 
 the armature reactance L x a , equation (102) may be written 
 as in (103). 
 
 ia =- ~ (103) 
 
 For constant speed the generated voltage, e , is directly 
 proportional to the useful flux. At the instant the short 
 circuit occurs and the alternator carries no load, as in Fig. 
 40, the useful flux depends on the direct current voltage 
 impressed on the field winding and produces an armature 
 voltage, O e a , and hence the initial or momentary value of 
 the short circuit current, 
 
 O ia = -?- a - (104) 
 
 joX a 
 
 If expressed in effective values as if the current sine wave 
 continued at the initial magnitude, 
 
 ./. = -- (105) 
 
 ifi a 
 
 During the transient period following the short circuit 
 the armature reaction reduces the field flux and as a conse- 
 quence the voltage generated in the armature decreases 
 in the same ratio. With the expiration of the field tran- 
 sient the useful flux, <j> u , is constant and hence the gener- 
 ated voltage, E a , and the armature current, I a , are constant 
 or have permanent values. 
 
 Ia == (106) 
 
 I a E a $ M _ field excitation armature reaction 
 Ia E a & u field excitation 
 
 (107) 
 
 Although the decrease in the armature current from its 
 initial to the permanent value, as shown in Figs. 44 and 45, 
 
A L TERN A TING C URREN TS 
 
 59 
 
 is due to a reduction in the useful field flux and hence in the 
 generated armature voltage it is customary to consider the 
 voltage constant and ascribe the change to a fictitious 
 increase in the reactance of the armature circuit. The 
 combined effect of the armature reaction and the true arma- 
 ture reactance is represented by the so-called synchronous 
 reactance s x a . 
 
 FIG. 45. Armature current transients. Short-circuit on three-phase, star- 
 connected alternator, 50 per cent, of full load. 
 
 The permanent short circuit current may therefore be 
 expressed by equation (108) and the ratio of the permanent 
 to the initial or momentary values by (109) 
 
 Let : I a = permanent short circuit armature current. 
 I a = initial short circuit armature current. 
 L x a = armature reactance. 
 
 s x a = synchronous reactance = armature reactance 
 + armature reaction. 
 
 T a 
 
 a ~ r 
 
 S'^ffl 
 
 J- a L %a 
 
 a* a sX a 
 
 (108) 
 (109) 
 
60 ELECTRIC TRANSIENTS 
 
 If it be assumed that the permeability of the magnetic 
 circuits remains constant for the changes in flux density, 
 the field current-time curve may also be expressed in the 
 form of an equation in terms of the circuit constants and 
 the initial value of the transients. 
 
 Let, R f = resistance of field circuit. 
 L f = inductance of field circuit. 
 R a = resistance of armature circuit. 
 L a = inductance of armature circuit. 
 t= time from the instant short circuit occurs. 
 oo = 27r/; / = frequency in cycles per second. 
 i/= instantaneous value of field current. 
 //= permanent value of field exciting current 
 
 before short circuit occurs. 
 i'/= instantaneous value of current in field circuit 
 
 due to field transients. 
 /'/ initial value of i' f . 
 i' a f= instantaneous value of current in field circuits 
 
 due to armature transient. 
 I' af= initial value of i' ' af 
 
 (110) 
 
 - R *t 
 
 i' af = r af sin (0 (111) 
 
 In Figs. 42, 43: 
 
 if=I f - i' s = If - r f e J (112) 
 
 In Figs. 40, 41: 
 
 i af = i f +i' f + i' af -.= J + rr^V/'e'^'sinarf (113) 
 
 Short circuit currents, particularly under normal field 
 excitation, produce so great changes in flux density that 
 the permeability is not constant and hence L a and L f are 
 not constant. The purpose of the equation is however, 
 merely to state in concise form the factors involved without 
 taking into consideration the complications due to 
 variations in the permeability of the magnetic circuits. 
 
ALTERNATING CURRENTS 
 
 61 
 
 While short circuits produce electric transients of greater 
 magnitude than the changes that occur during normal 
 operation of alternators, it should be kept in mind that any 
 modification in the armature currents, as, for example, an 
 increase or decrease in the load, produces transients having 
 the same characteristics as those produced by short circuits. 
 Any change in the amount of energy stored magnetically 
 in the armature or field circuits requires time and during 
 the period of readjustment electric transients are produced 
 in the interlinked electric and magnetic circuits. 
 
 FIG. 46. Short circuit transients, single phase alternator. Symmetrical. No 
 
 load. 
 
 E = 108 volts; 7, short circuit = 10.4 amps.; /, field = 2.6 amps.; / = 60 
 cycles. 
 
 Single -phase Short Circuits. Alternator Armature and 
 Field Transients. In polyphase alternators the permanent 
 armature field produced by the balanced armature currents, 
 and hence the armature reaction, is constant in value and, 
 with respect to the alternator field poles, fixed in position. 
 In single-phase alternators the magnetic field produced by 
 the armature currents, and therefore the armature reaction, 
 
62 ELECTRIC TRANSIENTS 
 
 pulsates synchronously with the armature rotation. The 
 pulsations of the armature reaction necessarily appear in 
 the field circuit. As the armature rotates 180 electrical 
 degrees for each half cycle of the armature current, the 
 pulsations of the armature reaction with respect to the 
 field poles will have double the frequency of the armature 
 currents. Therefore, the field current has a permanent 
 double frequency pulsation as shown in Figs. 46 and 47. 
 
 FIG. 47. Short circuit transients, single phase alternator. Symmetrical. Load. 
 E, load = 106 volts; /, load = 16.83 amps.; /, short circuit = 21.5 amps.; 
 7, field = 2.6 amps.; / = 60 cycles. 
 
 Since the armature reaction is pulsating and not constant, 
 as in polyphase alternators, the initial value of the starting 
 transient of the armature flux will depend on the point on 
 the current wave at which the short circuit occurs. Thus 
 in Figs. 46 and 47 the short circuiting switch closed nearly 
 at the instant the armature current was zero and hence 
 only a very small armature transient was produced. With 
 the armature transient absent the field current-time oscit- 
 lograms, as illustrated in Figs. 46 and 47, are symmetrical 
 
ALTERNATING CURRENTS 63 
 
 showing the permanent double frequency pulsations of the 
 armature reaction superimposed on the field transient. 
 If the short circuit occurs at other than the zero points 
 on the armature current wave, an armature transient of full 
 frequency is produced for the same reason as explained for 
 short circuits in polyphase alternators. The oscillograms 
 in Figs. 48 and 49 show the asymmetrical field current- 
 
 FIG. 48. Short circuit_transients, single phase alternator. Asymmetrical. No 
 
 load. 
 
 E = 57 volts; /, short circuit = 23.0 amps.; I, field = 1.3 amps.; / = 60 
 cycles. 
 
 time curves on which are superimposed the double fre- 
 quency permanent armature reaction, the field transient, 
 and the full frequency pulsation produced by the armature 
 transient. The combination of the full frequency arma- 
 ture transient pulsation with the permanent double 
 frequency armature reaction produces the asymmetry 
 in the curves. The ordinates for the odd numbers of the 
 double frequency waves add to the full frequency values, 
 while for the even number of waves the difference in the 
 ordinates produces the wave recorded by the oscillograph. 
 
64 
 
 ELECTRIC TRANSIENTS 
 
ALTERNATING CURRENTS 65 
 
 Hence, during the transition period the peaks of the odd 
 numbered waves decrease, while the even numbered peaks 
 increase, and at the expiration of the armature transient 
 all reach the permanent constant pulsation produced by 
 the pulsating armature reaction. While the field current 
 pulsates as a result of the double frequencyarmature reac- 
 tion and the full frequency armature transients, the voltage 
 across the field terminals will pulsate to a greater or less 
 degree depending on the amount of external resistance and 
 inductance in series with the field circuit. With much ex- 
 ternal resistance or impedance the voltage at the terminals 
 of the field winding may reach high values which may 
 puncture the insulation and cause a short circuit in the 
 field exciting circuit. 
 
 The field transient separated from the armature reaction 
 may be shown by taking an oscillogram of the field current 
 when the short circuit on the single phase alternator is 
 broken, as shown in Figs. 50 and 51. The armature tran- 
 sient is dissipated during the opening of the switch, indi- 
 cated by the time a b on the oscillogram, while several 
 complete cycles are required before the field flux, and as a 
 consequence the armature voltage, regains its full value. 
 In the transition period following the closing or opening of 
 the short circuiting switch the oscillograms of the field 
 currents show the effects of the energy changes taking 
 place in both the field and armature circuits. 
 
 Under the assumption that the permeability of the mag- 
 netic circuits is constant the field-current-time curve in 
 Figs. 46 to 51 may be expressed in terms of the circuit 
 constants and the initial values of the transients : 
 Let: R f = resistance of field circuit. 
 
 L f = inductance of field circuit. 
 
 R a = resistance of armature circuit. 
 
 L a = inductance of armature circuit. 
 t = time from the instant short circuit occurs. 
 co = 27r/; / = frequency in cycles per second. 
 if = instantaneous value of field current. 
 
66 
 
 ELECTRIC TRANSIENTS 
 
ALTERNATING CURRENTS 
 
 67 
 
 J3 05 
 
 a- 1 
 
 83 
 
68 ELECTRIC TRANSIENTS 
 
 I f = permanent value of exciting current before 
 
 short circuit occurs. 
 i' f = instantaneous value of current in field circuit 
 
 due to field transient. 
 /'/ = initial value of i f /. 
 i a f = instantaneous value of current in field circuit 
 
 due to armature reaction. 
 la/ = maximum value of i a /. 
 i f af = instantaneous value of current in field circuit 
 
 due to armature transient. 
 Iaf = maximum initial value of i' af . 
 
 71 = phase angle of i af . 
 
 72 = phase angle of i' /. 
 
 iaf = M I a /' sin (2 cot 71) (115) 
 
 i 
 In Fig. 50: 
 
 _R O( 
 'af = "I'af6 La sin (ut - 72) (116) 
 
 -i 
 
 i, = // - 7/e L ' (117) 
 
 In Fig. 46: 
 
 if ---- If + r f * L ' + "J a/ sin (2^ - 71) (H8) 
 In Fig. 48: 
 
 - Rf t 
 
 i f = I f + r f e L! + M I af sin (2wf -- 71) 
 
 t 
 
 -7a) (119) 
 
 - R "t 
 
 As indicated by the difference in the upper and lower 
 halves of the double frequency pulsation the permeability 
 of the magnetic circuit changed with the flux density. 
 Under full field excitation the short circuit transients 
 would produce much greater changes in the flux density 
 and hence in the permeability of the steel in the armature 
 and field poles. For this reason the equations are not 
 directly applicable to commercial problems but state the 
 
ALTERNATING CURRENTS 
 
 69 
 
 relations of the factors involved provided the permeability 
 of the iron core is constant. 
 
 Single -phase Short Circuit on Polyphase Alternators. 
 If all phases of polyphase alternators are short circuited 
 simultaneously the armature transients appear in the field 
 circuit as full frequency pulsations produced by the rotating 
 magnetic field, as illustrated for three-phase machines in 
 Figs. 40 to 43. 
 
 FIG. 52. Single phase short circuit on three phase alternator. 
 
 If one phase only is short circuited the effect on the field 
 circuit is essentially the same as illustrated for single phase 
 alternators in Figs. 46 to 48. In Fig. 52 is shown the 
 transient of the field current of a three-phase alternator 
 after short circuiting one phase. The field current-time 
 curve shows the effects produced by the field and armature 
 transients and the permanent double frequency pulsations 
 due to the armature reaction. In Fig. 53 is shown an 
 oscillogram for a single-phase short circuit on a three-phase 
 alternator which after 4 cycles is followed by a short circuit 
 
70 
 
 ELECTRIC TRANSIENTS 
 
 on all three phases. While only one phase is short circuited 
 the field current shows the double frequency pulsations 
 combined with both the armature and field transients. 
 After the three-phase short circuit occurs the field current 
 shows the full frequency pulsations of the armature tran- 
 sient combined with the slower field transient. 
 
 FIG. 53. Single phase short circuit on three phase alternator followed by a three 
 
 phase short circuit. 
 
 E = 118 volts; 7, load =10 amps.; /, short circuit = 22.5 amps.; I, field = 
 2.6 amps. 
 
 Oscillograms of transients in polyphase systems produced 
 by single-phase short circuits necessarily differ with the 
 type of machine and the way the transient magnetic fluxes 
 interlink with the electric circuit to which the oscillograph 
 vibrator is connected. Thus in Fig. 54 the open phase 
 voltage of a two-phase alternator with short circuit on one 
 phase shows a triple frequency harmonic, while the field 
 current shows the double frequency pulsation combined 
 with the field and armature transients of the same charac- 
 teristics as for single-phase alternators. 
 
ALTERNATING CURRENTS 
 
 71 
 
72 
 
 ELECTRIC TRANSIENTS 
 
 Problems and Experiments 
 
 1. Let the sine wave curve in Fig. 55 represent the 60 cycle alternating 
 current that would flow in a circuit having 3.0 ohms resistance, 0.05 henrys 
 inductance for a given voltage. 
 
 (a) Let the switch impressing the voltage on the circuit be closed at the 
 instant marked (a) in the diagram. Draw in rectangular coordinates: 
 
 1. The permanent current sine wave as in Fig. 55. 
 
 2. The starting transient. 
 
 3. The actual current flowing in the circuit during the first ^ second 
 after the switch is closed. 
 
 \ 
 
 FIG. 55. Single phase current, sine wave, 60 cycle starting transient. 
 
 (6) Similar to (a) except the voltage is impressed at the instant marked 
 
 (6). 
 
 2. In a circuit having 60 ohms resistance and 0.045 henrys inductance a 
 25 cycle current is flowing, as represented by the sine wave on the left side 
 in Fig. 56. At the instant marked (a) the impressed voltage is suddenly 
 changed so that it will produce a permanent 60 cycle current shown by the 
 dotted line sine wave in the figure. 
 
 (a) Draw in rectangular coordinates: 
 
 1. The sine current waves as in Fig. 56. 
 
 2. The starting transient. 
 
 3. The actual 60 cycle current for the first ^lo second after the voltage 
 was changed. 
 
 (6) Same as (a), except the change is made at some other point along the 
 time axis. 
 
 3. Take an oscillogram of the starting current in a circuit of known 
 resistance and inductance. Calculate the starting transient and draw it 
 on the oscillogram. Check by combining the ordinates for the actual current 
 
ALTERNATING CURRENTS 
 
 73 
 
 recorded by the oscillograph with the corresponding values of the calcu- 
 lated transient and compare the resulting curve with the permanent cur- 
 rent sine wave. 
 
 4. Let the sine waves in Fig. 57 represent the permanent value of the 
 currents flowing in a balanced three-phase system, whose time constant is 
 M>,ooo f & second. 
 
 FIG. 56. Single phase current, sine wave, 25 cycles to 60 cycles transient. 
 
 FIG. 57. Three phase current, sine wave, 60 cycle starting transients. 
 
 (a) Let the voltage be impressed at the instant marked (a). Draw in 
 rectangular coordinates : 
 
 1. The permanent current sine waves as in Fig. 57. 
 
 2. The starting transients for the three phases. 
 
74 ELECTRIC TRANSIENTS 
 
 3. The actual currents as would be recorded by an oscillograph 
 if a vibrator was connected to each of the three phases so as to 
 record the current-time curves. 
 
 (6) Same as (a), except the voltage is impressed at the instant marked (6). 
 
 6. Take an oscillogram of the starting currents in a three-phase system 
 
 connecting the vibrators as in the circuit diagram in Fig. 34. From the 
 
 oscillogram and the circuit constants plot the starting transients and check 
 
 with the permanent current waves as explained in Prob. 3. 
 
 6. Make oscillograms similar to Figs. 40 and 42 or 41 and 43. Obtain 
 the necessary data to draw the scale in amperes or volts for each vibrator. 
 A circuit diagram showing the position of each vibrator should be attached 
 to each film. 
 
 7. Make oscillograms similar to Figs. 46 and 48 or 47 and 49. Quanti- 
 tative data should be obtained for each vibrator and for the circuit constants. 
 
 8. Make an oscillogram similar to Fig. 53. 
 
CHAPTER V 
 DOUBLE ENERGY TRANSIENTS 
 
 Single energy transients occur in electric circuits or 
 other apparatus in which energy can be stored in only one 
 form. Any change in the amount of energy stored produces 
 transients and whether the stored energy is decreased or 
 increased the transient itself is a decreasing function with 
 its maximum value at the first instant. In magnetic, 
 electric and dielectric circuits in which the resistance, 
 inductance and condensance are constant during the transi- 
 tion period, single energy transients may be expressed by 
 the exponential equation as discussed in Chaps. Ill and IV. 
 
 In apparatus having two forms of energy storage as a 
 pendulum or electric circuits having both inductance and 
 condensance, a series of oscillations may take place by 
 which the energy is transferred from one form to the other, 
 while the dissipation of the stored energy into heat proceeds 
 in much the same manner as in single energy systems. 
 Thus a pendulum, freely suspended in air, will swing back 
 and forth over arcs of decreasing amplitude, with energy 
 changing from the kinetic to the potential form and back 
 to the kinetic twice for each cycle. The amplitude of each 
 swing is less than for the one preceding since part of the 
 energy has been dissipated into heat by friction during the 
 intervening time. The pendulum comes to rest when all 
 the stored energy is dissipated into heat. 
 
 In electric circuits having both dielectric and magnetic 
 storage facilities the energy stored in one form may change 
 to the other and back and forth in a series of oscillations 
 of definite frequency. This is illustrated by the oscillogram 
 in Fig. 58. The energy stored in a condenser is discharged 
 through a resistance in series with an inductance. In 
 
 75 
 
76 
 
 ELECTRIC TRANSIENTS 
 
 o < 
 
 iO ^ 
 
DOUBLE ENERGY TRANSIENTS 77 
 
 passing from the dielectric field to the magnetic field or 
 the reverse, the energy goes through the resistance and a 
 part is dissipated by the Ri 2 losses. Hence the amplitude 
 of each oscillation is less than for the one preceding. By 
 referring to the timing wave on the oscillogram, Fig. 58, 
 it is found that the frequency of oscillation was 1,070 cycles 
 per second, and that practically all the energy was dissi- 
 pated into heat by the Ri 2 losses in 50 cycles, or approxi- 
 mately Ho of a second. 
 
 Surge or Natural Impedance and Admittance. If no 
 energy is dissipated during the transfer the stored energy in 
 the dielectric field when the voltage is a maximum must be 
 equal to the quantity stored in the magnetic field when the 
 current is a maximum. Hence from (7) and (16) 
 
 C f L f (120) 
 
 Therefore, from (120) 
 
 M j = ^\~ = n z< the surge or natural impedance of 
 
 the circuit (121) 
 
 "I 1C 
 
 fY, = A/7 n y, the surge or natural admittance of 
 rj \L 
 
 the circuit (122) 
 
 The quantity, \/L/VC, is in the nature of an impedance 
 and is called the surge or natural impedance of the circuit, 
 and its reciprocal, \/C/\/L, the natural or surge admit- 
 tance of the circuit. 
 
 Frequency of Oscillation in Simple Double Energy 
 Circuits. Consider circuits "a" and "6" in Fig. 59. Let 
 the inductance, L, the condensance, C, and the resistance, 
 /2,'be constant and of the same value in the two cases. Let 
 an alternating current voltage be impressed on the ter- 
 minals and let the frequency be varied until the current is 
 in phase with the voltage at the terminals. All the energy 
 absorbed by the Ri 2 losses is supplied from the a.c. mains. 
 
78 
 
 ELECTRIC TRANSIENTS 
 
 In circuit (a) under the given conditions: 
 
 J L % jcX = 
 
 Hence, 
 
 Likewise for circuit (b) 
 
 j c b - j L b = 
 
 Hence, 
 
 (121) 
 
 (122) 
 
 (123) 
 
 
 
 (124) 
 (125) 
 
 (126) 
 
 The expressions in equations (123) and (126) are generally 
 used to determine the "resonance frequency" of the cir- 
 
 (a) (b) 
 
 FIG. 59. Simple series and parallel double energy circuits. 
 
 cuits. As shown in Chap. VIII a strict application of the 
 definition for resonance gives a different value for the true 
 resonance frequency unless the resistance is negligible. 
 If all the resistance were removed from the circuits in 
 Fig. 59 no energy would be supplied from the bus bars and 
 the stored energy would be transferred back and forth 
 between the inductance and the condensance. With no 
 losses the frequency of the natural or free oscillations 
 would be the same as the " resonance frequency" given in 
 equations (123) and (126). 
 
DOUBLE ENERGY TRANSIENTS 
 
 79 
 
 In circuit a, Fig. 60, and for the oscillograms in Figs. 
 61 to 65 the condenser is charged from a direct current 
 supply main after which the switch " S" is thrown to the 
 right so as to form an independent closed circuit with the 
 condenser, C, resistance, R, and inductance, L, in series. 
 The energy stored in the condenser is dissipated into heat 
 by the Ri 2 losses during a series of oscillations between the 
 dielectric and magnetic fields. 
 
 FIG. 60: Simple oscillatory double energy circuits. 
 
 From Kirchoff's Laws the voltage in the closed circuit, 
 Figs. 60 to 64, while the energy originally stored in the 
 condenser is dissipated into heat, is expressed by equations 
 
 (127) or (128). 
 
 = (127) 
 
 dt* + R di + (J = (128) 
 
 This is a homogeneous differential equation of the second 
 order and its general solution is given by equation (129), 
 in which A\ and A 2 are the arbitrary constants. 
 
 (129) 
 (130) 
 
 In equation (129) 
 
 B 
 
 LC 
 
 R 
 
 LC 
 
 (131) 
 
80 
 
 ELECTRIC TRANSIENTS 
 
 FIG. 61. Double energy transient. 
 
 E = 120 volts; R = 40 ohms; G = 0; L = 0.205 henrys; C 
 farads; timing wave 100 cycles. 
 
 0.813 mtero- 
 
 FIG. 62. Double energy transient. 
 
 E = 120 volts; R = 75 ohms; G = 0; L = 0.205 henrys; C = 0.873 micro- 
 farads; timing wave 100 cycles. 
 
DOUBLE ENERGY TRANSIENTS 
 
 81 
 
 FIG. 63. Double energy transient. 
 
 E = 120 volts; R = 150 ohms; G = 0; L = 0.205 henrys; C = 0.813 micro- 
 farads; timing wave 100 cycles. 
 
 FIG. 64. Double energy transient. 
 
 E = 700 volts; R = 770 ohms; G = 0; L = 0.205 henrys; C 
 farads; timing wave 100 cycles 
 6 
 
 0.813 micro- 
 
82 ELECTRIC TRANSIENTS 
 
 In order to more readily keep the dissipation or damping 
 factors separate from the parts indicating oscillations, 
 equations (130), (131) are rewritten in (132), (133): 
 
 R 
 
 Ul = ~ n r + 
 
 R . I I 
 -- 3 - 
 
 From (129), (132), (133): 
 
 _ 2" \ orr/ 2 " , _ m _ ;\i L T ~& t 
 
 i = An 2L e 3 \LC 4L'+^ ae 2L e 3 \LC 4L f (134) 
 
 But from Euler's equation for the sine and cosine: 
 
 v/JL _ *! 
 \LC 4Li = ; " = ^ + j sin co^ (135) 
 
 rf - j sin ut (136) 
 Hence from (134), (135), (136): 
 
 _Rt _Rt 
 
 i = Aie ^[cos ut +' sin ut] + A 2 e 2L 
 
 n (137) 
 
 [cos ut j sin at] 
 From (135), (136): 
 
 co == ^f=^- C ~f^ ( 138 ) 
 
 Hence, 
 
 / = 2W/!c- (139) 
 
 In circuits for which the quantity under the radical is 
 real, oscillations occur at a definite frequency as determined 
 by equation (139) and as illustrated by the oscillograms in 
 Figs. 61 to 63. 
 
 If the resistance, 
 
 . R >2 J5 (140) 
 
DOUBLE ENERGY TRANSIENTS 83 
 
 the quantity under the radical sign in (139) becomes imagin- 
 ary, and hence the circuit is non-oscillatory. All the 
 energy initially stored in the condenser is dissipated into 
 heat as the voltage and current decrease to zero. This 
 condition is illustrated by the oscillograms in Figs. 64 and 
 65. 
 
 For circuits having comparatively little resistance the 
 naturalfrejjuency of oscillations, as given in equation (139), 
 is ver$; Dearly the same as the " resonance frequency" 
 given by equations (123) or (126). Thus for the circuit 
 data in Fig. 62 the nafuFal frequency of oscillation, using 
 equation (139), is given in equation (141), while the cor- 
 responding " resonance frequency" from equation (126), 
 is given in equation (142). 
 
 1 / 1 
 
 ir\LC " 
 
 4/ 2 cycles per second 
 
 f = n /T ^ = 391 cycles per second (142) 
 
 For circuits corresponding to the conditions that would 
 exist if the condenser in Fig. 606 were leaky, similar equa- 
 tions may be obtained. The voltage equation, based on 
 KirchofPs Laws for circuits of the type shown in Fig. 606, 
 and in Figs. 66 to 70, is expressed by equations (143) and 
 (149). 
 
 L % + *$ + c = (143) 
 
 Using the notation shown in the circuit diagram, Fig. 
 606, letting c e be the voltage across the condenser terminals, 
 and applying Ohm's and KirchofPs Laws. 
 
 d = i + O i (144) 
 
 d = G L e (145) 
 
 dj 
 c e = Ri'+L (146) 
 
 Hence, 
 
 di 
 
 c i = i + GRi +GL (147) 
 
84 
 
 ELECTRIC TRANSIENTS 
 
DOUBLE ENERGY TRANSIENTS 85 
 
 From (143) and (147), 
 
 LC d jl + (RC + GL)~.+ (1 + GK)i = (148) 
 at ctt 
 
 or, 
 
 M ' '*<** '^-M = (149) 
 
 Equation (149) is a homogeneous differential equation 
 of the second order of the same form as equation (133). 
 Hence, the same general solution applies to both equations, 
 as expressed by equation (150), in which B } and B 2 are the 
 two arbitrary constants. 
 
 + B 2 e v * (150) 
 
 * - - 
 
 Rewriting (151), (152) so as to more clearly indicate the 
 damping and oscillation factors, equations (153), (154) are 
 obtained. 
 
 From (150), (153), (154), 
 
 ~\(L + c)' - y "N/Lc - KL - ?)' (155) 
 
 From Euler's equation, 
 
 _ _ 
 
 C 4- c< = e ^ = cos ut + sin 
 
 V_L _ i/j? _ o\ 
 c LC iU e^ = e wf = cos ut _j s in ^ (157) 
 
 Hence, 
 
 [cos cot + j sin cotj 
 
 ^ 2e 2^L c/ r cos ^1 j sin 
 
86 
 
 ELECTRIC TRANSIENTS 
 
 FIG. 66. Double energy transient. 
 
 ~E = 625 volts; R = 4.5 ohms; G = 1.67 10~ 4 mhos.; L = 0.205 henrys; C 
 0.813 microfarads; timing wave 100 ovnles. 
 
 FIG. 67. Double energy transient. 
 
 E = 640 volts; R = 4,.5 ohms; G =3.33 10" 4 mhos; L = 0.205 henrys; C 
 0.313 microfarads; timing wave 100 cycles. 
 
DOUBLE ENERGY TRANSIENTS 
 
 87 
 
 From (156), (157), 
 
 ' = 2 & - 
 
 Zir\LL 4\ 
 
 fi 
 
 L 
 
 G 
 
 (159) 
 
 (160) 
 
 (161) 
 
 The circuit is non-oscillatory if 
 R _G _2_ 
 L C > \LC 
 For circuits in which the quantity under the radical sign 
 is greater than zero, the energy in the condenser will be 
 dissipated into heat during a series of oscillations of definite 
 frequency as determined by equation (160) and as illus- 
 trated by the oscillograms in Figs. 66, 67 and 68. 
 
 FIG. 68. Double energy transient. 
 
 E = 625 volts; R = 4.5 ohms; G = 6.66 10~ 4 mhos; L = 0.205 henrys; 
 C = 0.813 microfarads; timing wave 100 cycles. 
 
 If the resistance and the conductance are of such values 
 relatively to the inductance and the condensance that the 
 quantity under the radical sign in (160) becomes imaginary, 
 
88 
 
 ELECTRIC TRANSIENTS 
 
 FIG. 69. Double energy transient. 
 
 E = 550 volts; R = 4.5 ohms; G = 1.67 10~ 3 mhos; 
 C = 0.813 microfarads; timing wave 100 cycles. 
 
 0.205 henrys; 
 
 FIG. 70. Double energy transient. 
 
 E = 550 volts; R = 4.5 ohms; G = 4.4 110~ 3 mhos; 
 C = 0.813 microfarads; timing wave 100 cycles. 
 
 L = 0.205 henrys; 
 
DOUBLE ENERGY TRANSIENTS 89 
 
 the circuit would be non-oscillatory. The energy initially 
 stored in the condenser is dissipated into heat while the 
 voltage and current decrease to zero, as illustrated by the 
 oscillograms in Figs. 69 to 70. 
 
 The circuits in which the resistance and conductance 
 are comparatively small the natural frequency of oscilla- 
 tion is very nearly the same as the resonance frequency or 
 the natural frequency of circuits in which R and G are 
 equal to zero. Thus for the circuit in Fig. 67 the natural 
 oscillation frequency, 
 
 f = ~ := 391 CyCl6S PCT S6COnd (162) 
 
 Considering R and G as negligible in determining the 
 frequency of oscillation, 
 
 / = - 7 = 391 cycles per second (163) 
 
 2ir\/LC 
 
 A very interesting circumstance is revealed by equation 
 (160). A circuit having resistance greater than the critical 
 value for oscillatory discharges as given in (161), may be 
 made oscillatory by increasing the conductance across the 
 terminals of the condenser without changing the resistance. 
 This is illustrated by the oscillograms in Figs. 71, 72, 73 
 and 74. 
 
 For the given circuit constants in Fig. 71 the circuit is 
 non-oscillatory. Letting R, L and C remain constant and 
 of the same value as in Fig. 71 but increasing the conduc- 
 tance, G, the circuit is made oscillatory in Fig. 72 although the 
 damping factor is greater than for the circuit in Fig. 71. 
 In Fig. 73 the oscillation was greatly reduced and by still 
 further increasing the conductance while R, L and C remain 
 constant, the circuit is again made non-oscillatory as shown 
 by the oscillogram in Fig. 74. 
 
 Dissipation Constant and Damping Factor in Simple 
 Double Energy Circuits. In the solution for the current in 
 double energy circuits, Fig. 60a and Figs. 61 to 65, as 
 given in equation (137), the damping factor and the dissi- 
 
90 
 
 ELECTRIC TRANSIENTS 
 
 FIG. 71. Double energy transients. 
 E = 700 volts; R = 150 ohms; G = mhos; L = 0.205 henry 
 microfarads; timing wave 100 cycles. 
 
 C = 36 
 
 FIG. 72. Double energy transients. 
 
 E = 700 volts; R = 150 -ohms; G = 4.35 X 10~ 3 mhos; L 
 C = 36 microfarads; timing wave 100 cycles. 
 
 0.205 henry s; 
 
DOUBLE ENERGY TRANSIENTS 
 
 91 
 
 FIG. 73. Double energy transients. 
 
 E = 400 volts; R = 150 ohms; G = 1.31 10~ 2 mhos; L = 0.205 henrys 
 C = 18 microfarads; timing wave 100 cycles. 
 
 FIG. 74. Double energy transients. 
 
 E = 700 volts; R = 150 ohms; G = 2.63 10~ 2 mhos; 
 C = 36 microfarads; timing wave 100 cycles. 
 
 0.205 henrys; 
 
92 ELECTRIC TRANSIENTS 
 
 pation constant have already been found. Similarly 
 for circuits in Fig. 606 and in Figs. 66 and 67, the factors 
 may be obtained from equation (158) 
 
 Dissipation or damping constant = J(T + f ) 
 
 Damping factor = ~2\L + c) i (1(35) 
 
 While the above expressions are obtained mathematic- 
 ally by the solution of the differential equation of the cir- 
 cuit, it is important that the student gain a clear concept of 
 the physical phenomena involved. 
 
 In Chap. Ill it was shown that for single energy tran- 
 sient in circuits having resistance and inductance in series 
 the time constant is directly proportional to the inductance 
 and inversely to the resistance. 
 
 ,T 1 -- ^ (166) 
 
 Similarly for circuits having" condensance in parallel 
 with conductance, the time constant is directly proportional 
 to the condensance and inversely to the conductance. 
 
 C T, -- g (167) 
 
 In double energy circuits the energy is alternately stored 
 in the magnetic and dielectric fields. In circuits having 
 inductance, resistance, condensance and conductance, 
 arranged as shown in the circuit diagrams in Figs. 66 to 70, 
 energy is dissipated into heat both in the resistance and in 
 the conductance. The rate of dissipation is greatest in 
 the conductance, (re 2 , when the voltage across the condenser 
 is a maximum, that is, at the instant all the energy is 
 stored in the dielectric field. Similarly the rate of dissipa- 
 tion in the resistance, Ri 2 , is a maximum, when the current 
 is a maximum, that is, when all the energy is stored in the 
 magnetic field. It is evident that since the energy is 
 oscillating it will be in the dielectric field half of the time 
 and in the magnetic field half of the time. Since the energy 
 
DOUBLE ENERGY TRANSIENTS 93 
 
 is in the dielectric form only half the actual time, the 
 rate of dissipation in the conductance will be equal to half 
 of what would be the case for the same circuit constants 
 in the corresponding single energy transient. Hence, the 
 time constant, C T^ for the dielectric half of the double energy 
 circuit would be twice the time constant, c Ti, in the corre- 
 sponding single energy transient. 
 
 9C 
 
 J\ = 2 o r, = ^T (168) 
 
 (jT 
 
 Similarly the time constant, ,7%, for the inductance- 
 resistance part of the double energy circuit would be twice 
 the time constant, L Ti, for the corresponding single energy 
 transient. 
 
 or 
 
 ,T 2 = 2 L T 1 = - (169) 
 
 K 
 
 Under the given circuit conditions with R, L, G and C 
 constant, the proportionality law applies to double energy 
 transients on the same basis as for single energy transients. 
 The transient term is therefore expressed by the exponential 
 equation and appears as a factor in the current-time and 
 voltage-time equations and represents the dissipation of 
 energy into heat by the resistance, Ri 2 , and the conduc- 
 tance, Ce 2 , in the circuit. 
 
 Let u represent the dissipation constant of double energy 
 circuits. The damping factor is therefore, 
 
 1 l - R t --t 
 
 ut = ,T> C T 2 = 2L 2G 
 
 c (17!) 
 
 \/R . G\ /ihr<n\ 
 
 -2(1 + c) (172) 
 
 This is the same value as obtained in equation (158). 
 For circuits in which G = 0, as illustrated by Figs. 61 
 to 65, the term G/C would be zero. 
 
94 ELECTRIC TRANSIENTS 
 
 Hence, 
 
 u' -g- (173) 
 
 - R t 
 e u't = e 2L Q74) 
 
 This corresponds to the value of the damping factor in 
 equation (137). 
 
 For circuits similar to Figs. 66 to 70 but in which R = 0, 
 the term R/L would be zero. 
 
 Hence, 
 
 u" = - c (175) 
 
 _G_ t 
 
 e u"t = 2c (175) 
 
 As it is not possible to completely eliminate the resis- 
 tance in circuits having inductance, the conditions for u" 
 can not be fully realized experimentally. 
 
 Equations for Current and Voltage Transients. For 
 simple double energy circuits, with R, L and C in series, 
 as in Fig. 60a, the general equation (137) for the current is, 
 
 _Rt 
 
 i = Aie 2Z/ [cos cot + j sin cofl 
 
 _ Rt 
 
 + A 2 e~ 2L [cos cot j sin co(] (177) 
 
 In equation (177) A\ and A 2 are the arbitrary constants, 
 which for any specific case are determined by the perman- 
 ent circuit conditions preceding and following the transient 
 period. Equation (177) may be written in a more compact 
 form as in (178), in which A 3 and A 4 are the arbitrary 
 constants which for any specific case may be determined 
 from the given limiting conditions under which the tran- 
 sient occurred. 
 
 _Rt 
 
 i = e 2L [A 3 cos ut + A sin co] (178) 
 
 The voltage across the terminals of the condenser, 
 
 c e = -Ri - L (179) 
 
DOUBLE ENERGY TRANSIENTS 95 
 
 From (178), (179), 
 c e = e 2L I rt~[A 3 cos co + A 4 sin co] 
 
 + wL[A 4 cosco^ A 3 sincofl (180) 
 
 For the transients in Figs. 61 to 65 the starting conditions 
 are: 
 
 t = 0; i = 0; c e == # (181) 
 
 From (178), (180), (181), 
 
 Hence, 
 
 p -* 1 
 
 L sin w< (183) 
 
 CO-L 
 
 Rt 7? 
 
 2L [cos co^ + ^ sin co] (184) 
 
 T> 
 
 For the given circuit constants, ^ j is very small and 
 hence, 
 
 _Rt 
 
 c e = Ee 2L sin ut (very nearly) (185) 
 
 To illustrate the application of equations (183) and (185) 
 
 for the solution of numerical problems, equations (186) 
 
 and (187) give the value of the current and voltage in 
 
 amperes and volts for the oscillograms in Fig. 62. 
 
 i = -0.24 ~ 182 "sin (170760*) amperes (186) 
 
 e =-. 120. e~ 182 'cos (1707600 volts (187) 
 
 For the circuit in Fig. 606 the equations are of a similar 
 nature. The general equation (158) for the current is 
 given in (188) and may be written in a more compact 
 form as in equation (189), in which B s and B 4 are constants 
 that in each case depend on the permanent circuit condi- 
 tions preceding and following the transient period. 
 
 
 [cos coZ + j sin <*t] 
 
 2L c 
 
 [cos ut - j-sin.ut] (188) 
 
96 ELECTRIC TRANSIENTS 
 
 - t 
 
 i = 2 \ L c > [B 3 cos ut + B 4 sin <*t] (189) 
 
 c e = -Ri _ L-- (190) 
 
 di 
 -- 
 
 From (188) and (190), 
 
 cos ut + ^ 4 sin 
 
 + <o[/?4 cos cot J5 3 sin ool (191) 
 
 For the transients in Figs. 66 to 70 the initial conditions 
 are: 
 
 t = 0;t = 0; c e = E (192) 
 
 From (189), (191) and (192), 
 
 3 = 0; B, = Jr (193) 
 
 Hence, 
 i=- 4 '^* + ^'sin (194) 
 
 (195) 
 
 If in equation (195), 
 
 _ , 
 
 c e = Ee 2\L + c) cos ut (197) 
 
 and 
 
 (198) 
 
 As an illustration of the application of equations (194) 
 and (197) to the solution of a specific problem, let the 
 numerical values of the circuit constants in Fig. 67 be 
 used. Equations (199) and (200) give the instantaneous 
 values of the current and voltage for the oscillogram in 
 Fig. 67. 
 
 _ 9 1 r, i 
 
 i=-l.28e' sin (170760*) amperes (199) 
 
 c e = 640 e' cos (1707600 volts (200) 
 
DOUBLE ENERGY TRANSIENTS 
 
 97 
 
 o 
 
 o 
 
 LO^ 
 
 II J 
 ^'E 
 
 fi 
 
 O c3 
 
 2 3 
 
 o 
 w> S 
 
 03 
 
98 
 
 ELECTRIC TRANSIENTS 
 
 FIG. 76. Starting current and voltage transients. 
 
 E = 125 volts; R = 5.0 ohms;(? = 0.00167 mhos;L = 0.205 henrys; C = 9.0 
 microfarads; timing wave 100 cycles. 
 
 FIG. 77.- Starting current and voltage transients. 
 
 E = 125 volts; R = 5.0 ohms; G = 0.0025 mhos; L = 0.205 henrys; C 
 microfarads; timing wave 100 cycles. 
 
 9.0 
 
DOUBLE ENERGY TRANSIENTS 
 
 99 
 
 FIG. 78. Starting current and voltage transients. 
 
 E = 125 volts; R = 5.0 ohms; G = 0.005 mhos;L = 0.205 henrys; C 
 microfarads; timing wave 100 cycles. 
 
 FIG. 79. Starting current and voltage transients. 
 
 E = 125 volts; R = 5.0 ohms; G = 0.0132 mhos; L = 0.205 henrys; C = 9.0 
 microfarads; timing wave 100 cycles. 
 
100 ELECTRIC TRANSIENTS 
 
 The oscillogram in Fig. 74 shows the current and voltage 
 transients in a circuit having a high damping factor but 
 in which the frequency of oscillation is the same as if both 
 R and G w r ere zero. The data in Fig. 74 show that the 
 circuit constants were of such values as to satisfy equation 
 (196). 
 
 For the oscillograms in Figs. 75 to 79 the circuits are of 
 the same type as in Figs. 66 to 70, but the permanent con- 
 ditions preceding and following the transition period are 
 different. The oscillograms show the starting current and 
 voltage transients at the points in the circuit indicated by 
 the positions of the vibrators in the circuit diagram and for 
 the values of R, L, G and C, as given in each figure. 
 
 Problems and Experiments 
 
 1. Given a circuit similar to Fig. 60 (a) having R, L, and C in series. Let 
 R = 20 ohms, L = 0.31 henrys, C = 1.2 microfarads and E = 120 volts, 
 the initial condenser discharge voltage. 
 
 (a) Find the natural period of oscillation of the circuit. 
 
 (6) Find the time constant, and the damping factors. 
 
 (c) Write the equation for the transient condenser discharge current. 
 
 (d} For what values of R would the circuit be non-oscillatory. 
 
 2. Derive the equations for e e, the transient voltage across the condenser 
 terminals in Fig. 62. Trace the voltage-time curve for c e on rectangular 
 coordinates, using the same time scale on the X axis as in the oscillogram. 
 
 3. Take a double energy oscillogram similar to Fig. 58. Obtain all the 
 necessary data and write the equations for the transient current. 
 
 4. Write the equations for the voltage and current curves of the oscillo- 
 gram in Fig. 61 similar to equations (199) and (200) for Fig. 67 in the text. 
 
 5. Take a series of oscillograms similar to Figs. 66 to 70. Find the values 
 of the circuit constants and place on the film ampere and volt scales for the 
 current and voltage curves. 
 
 6. For the oscillogram in Fig. 75 with the given circuit conditions: 
 (a) Write the expression for c e and i similar to equations (194), (195). 
 (6) Insert the numerical values of circuit constants and express c e and 
 
 i in volts and amperes, similar to equations (199) and (200). 
 
CHAPTER VI 
 
 ELECTRIC LINE OSCILLATIONS, SURGES AND 
 TRAVELING WAVES 
 
 Electric lines whether designed for poWer*tVaVL&fnissioTi or 
 telephone service, may be considered "as^cpnisisjifig^f 8it 
 infinite series of infinitesimal double energy 'Circuits o*f the 
 simple types discussed in Chap. V. Each infinitesimal 
 length of line may be represented by the resistance and 
 inductance in one of the series circuit elements in Fig. 80 
 and the corresponding portion of the dielectric between 
 the conductor and neutral by the conductance and con- 
 densance in the adjacent parallel circuit. The line con- 
 stants, R, L, G and (7, depend on the size and spacing of the 
 conductors and the electrical properties of the dielectric 
 and conductor materials. To readily gain clear concepts 
 of transmission line phenomena it is essential for the student 
 to conduct experiments and obtain quantitative test data. 
 Commercial transmission lines are seldom available for 
 experimental work but artificial lines having the electrical 
 characteristics of actual lines can be readily constructed 
 of convenient design for operation in the laboratory. 
 
 Artificial Electric Lines. Since the operating charac- 
 teristics of transmission lines are determined by the line 
 constants, the resistance, inductance, conductance and 
 condensance and are independent of the space and mass 
 factors, much of the experimental work can to good advan- 
 tage be performed on equivalent artificial electric lines. 1 
 The oscillograms of electric line transients used for illus- 
 trations in this chapter were obtained from an artificial 
 transmission line, 2 one section of which is shown in Fig. 
 
 1 DR. A. E. KENNELLY, "Artificial Transmission Lines." 
 
 2 Trans. A. I. E. E., vol. 31, p. 1137. 
 
 101 
 
102 
 
 ELECTRIC TRANSIENTS 
 
 81. This line is of the lumpy "T" type of design, which 
 means that each unit has resistance and inductance in 
 series combined with condensance and conductance in 
 parallel as shown in Fig. 82. If the insulation is sufficiently 
 high the conductance factor may be omitted and the section 
 circuit diagram would be as in Fig. 83, which represents 
 the circuit diagram for the "T" unit in Fig. 81. 
 
 R L 
 
 C =tr^G C=t^G C== 
 
 R L L R 
 
 R L 
 
 FIG. 80. Transmission line circuit diagram showing three elements. 
 
 In the lumpy types the line constants R, L, G and C, are 
 massed instead of uniformly distributed as in actual lines. 
 As the lumpy type only approximates a uniform distribu- 
 tion of the resistance, inductance, conductance and con- 
 densance in the line, the size of each unit must be small in 
 comparison to the total length of the line. In Fig. 81 
 is shown one of the twenty ten-mile units in the artificial 
 transmission line in the electrical engineering laboratories 
 of the University of Washington. In each unit the line con- 
 stants may be adjusted within the following limits: 
 
 Resistance, minimum value, 2.59 ohms. 
 
 Inductance, maximum value, 0.021 henry. 
 
 Condensance, 0.1 to 1.0 microfarad. 
 
 The resistance may be increased to any desired amount by 
 moving the clamp on the resistance loop or by inserting 
 resistance elements between the units; the inductance 
 may be decreased by turning the right hand coil and by 
 taps in the lower coil; and the condensance may be varied 
 in steps by using ten or a less number of condensers in 
 
OSCILLATIONS, SURGES AND TRAVELING WAVES 103 
 
 series. Adjustments can be made so as to give to the artifi- 
 cial line the electrical constants equivalent to an actual 
 transmission line of any size of wire up to No. 0000 A.W.G. 
 hard-drawn copper and for any spacing up to 120 inches. 
 
 FIG. 81. Section of artificial electric line, University of Washington. 
 
 The line may also be adjusted so as to be equivalent to 
 commercial telephone lines. 
 
 Time, Space and Phase Angles. In Chap. V the equations 
 for the current and voltage transients were derived for 
 
104 
 
 ELECTRIC TRANSIENTS 
 
 simple double energy circuits, Fig. 60, in which the circuit 
 constants, R, L, G and C are massed. Evidently the 
 energy transfer between the magnetic and dielectric fields 
 would be of essentially the same nature if the inductance 
 and resistance were intermixed with the condensance and 
 conductance or uniformly distributed as in a transmission 
 line. However, one important difference must be noted 
 which necessitates an additional factor in the expression 
 for the transient current and voltage. In circuits having 
 massed circuit constants the maximum value of the voltage 
 
 f=f /? 
 
 FIG. 82. T-circuit with leaky condenser. 
 
 FIG. 83. T-circuit. 
 
 will be impressed on all of the condensance at the same 
 instant, and all parts of the magnetic field reach a maximum 
 at the instant the current is a maximum. On the other 
 hand, with R, L, G and C distributed, as in long transmission 
 lines, the time required for the electric wave to travel along 
 the length of the line enters into the problem. If a constant 
 direct current voltage is impressed at one end of an electric 
 line a short but definite time will elapse before the voltage 
 reaches the other end of the line. If an alternating current 
 is transmitted over the line the successive waves travel over 
 the line at definite velocity in the same manner as the 
 impulse from the direct current voltage. The maximum 
 point of any wave travels at a definite velocity as deter- 
 mined by the distribution of the resistance, inductance, 
 conductance and condensance in the line. In trans- 
 mission lines with air as the dielectric and with copper or 
 aluminum conductors the speed at which a wave or impulse 
 
OSCILLATIONS, SURGES AND TRAVELING WAVES 105 
 
 travels is approximately the same as the velocity of propa- 
 gation of an electromagnetic wave in space or the velocity of 
 light. 
 
 v = 3-10 10 cm. per second (205) 
 
 In a medium having a permeability ^ and a permittivity /c, 
 
 3-10 10 
 
 v' = - -=,- cm. per second (206) 
 
 v M* 
 
 The time required for the voltage wave to travel a 
 distance x along the line having distributed line constants, 
 depends on the distance and velocity of propagation. 
 
 
 (207) 
 
 In comparing the transient voltage and current conditions 
 at any two points on an electric line, x distance apart, con- 
 sideration must be given to the time required for the 
 propagation of the electric wave over the given distance 
 and hence the factor t, must be included in the equations. 
 In double energy circuits having massed R, L, G and C, 
 as in the oscillograms in Figs. 66 to 69, and for oscillations 
 produced by the discharge of energy initially stored in the 
 condensers, the instantaneous values of the voltage and 
 current, under the stated conditions, are given in equations 
 (194), (197). Under similar conditions, as illustrated by 
 the oscillograms in Figs. 84 to 91, and by the introduction 
 of space angles, the equations may be considered as apply- 
 ing to circuits having distributed R, L, G and C, as in trans- 
 mission lines. 
 
 To simplify the notations, let 
 
 = K? - D 
 
 (208) 
 
 I = - E (209) 
 
 co-L 
 
 y = time phase angle for t = (210) 
 
 For oscillations in circuits with massed R, L, G and C, 
 under the stated assumptions: 
 
106 
 
 ELECTRIC TRANSIENTS 
 
 e = Ee~ ut cos (ut 7) 
 
 (211) 
 (212) 
 
 FIG. 84. Electric line oscillations. 
 
 E = 500 volts; R = 52.14 ohms; G = 0; L = 0.427 henrys; C = 3.66 micro- 
 farads; length = 232 miles; timing wave 100 cycles. 
 
 & V 
 
 1C) 
 
 FIG. 85. Circuit and wave diagram for Fig. 84, 
 
OSCILLATIONS, SURGES AND TRAVELING WAVES 107 
 
 For oscillations in circuit with distributed R, L, G and C, 
 under similar conditions: 
 
 FIG. 86. Electric line oscillations. 
 
 E = 500 volts; R = 26.12 ohms; # 2 = 26.02 ohms; Gi = 0; Gz = 0; Li = 0.2128 
 henrys; Li = 0.2146 henrys; Ci = 1.831 microfarads; Cz 1.834 microfarads; 
 timing wave 100 cycles. 
 
 FIG. 87. Circuit and wave diagram for Fig. 86. ~ j 
 
 i = I e -ut s i n [ w ( t _ tl ) - y] (213) 
 
 e = Ee~ ut cos [u(t - ti) -- y] (214) 
 
108 
 
 Substituting 
 
 ELECTRIC TRANSIENTS 
 
 c for coi : 
 
 = e-ut sin (o> -- 0x 7) (215) 
 
 e = Ee~ ut cos (ut - <$>x - 7) (216) 
 
 In equations (215), (216) ut is the time angle, <f)X the 
 space angle and 7 the phase angle. 
 
 FIG. 88. Electric line oscillations. 
 
 E = 500 volts; Ri = 31.28 ohms; R 2 = 15.64 ohms; Gi = 0; G-i = 0; Li = 
 0.2564 henrys; L 2 = 0.1282 henrys; C\ = 2.201 microfarads; C 2 = 1.10 micro- 
 farads; timing wave 100 cycles. 
 
 FIG. 89. Circuit and wave diagram for Fig. 88. 
 
 Natural Period of Oscillation. Since the space angle, 
 4>x, in equations (215), (216) is directly proportional to 
 
OSCILLATIONS, SURGES AND TRAVELING WAVES 109 
 
 the distance, x, from the origin, it is evident that the phase 
 of the current, i, and the voltage, e, changes progressively 
 along the line. At some distance, 1 Q , the current and volt- 
 
 FIG. 90. Electric line oscillations. 
 
 E = 500 volts; Ri = 39.10 ohms; #2 = 13.04 ohms; Gi = 0; Gz = 0; Li = 
 0.3204 henrys; Z/ 2 = 0.1070 henrys; Ci = 2.748 microfarads; (7 2 = 0.917 micro- 
 farads; timing wave 100 cycles. 
 
 VBWWtfWtf^^ 
 
 FIG. 91. Circuit and wave diagram for Fig. 90. 
 
 age are displaced 360 deg. from their starting point values. 
 The distance, 1 , is called the wave length and is the distance 
 
110 ELECTRIC TRANSIENTS 
 
 the electric field travels during the time, t Q , required for the 
 completion of one cycle or complete wave. 
 
 If /is the frequency of oscillations in cycles per second, 
 
 t Q = - seconds (217) 
 
 J 
 
 h = vto (218) 
 
 The fundamental frequency or natural period of free 
 oscillation depends on the length of the line and on the 
 imposed circuit conditions. For the oscillations recorded 
 in the oscillogram in Fig. 84, the line is open at the receiver 
 end and connected through the vibrator circuit at the 
 generator end. The diagram in Fig. 856 shows that 
 under these conditions the length of the line is one-fourth 
 wave length of the fundamental oscillations. In Fig. 85c is 
 shown the wave diagram for the ninth harmonic which 
 appears as ripples on the fundamental oscillation. 
 
 In Fig. 86 the vibrator is connected at the middle point 
 leaving both ends open. The corresponding wave diagram 
 in Fig. 876 shows that the length of the line is two quarter- 
 wave lengths or one-half wave length, and the frequency of 
 the fundamental oscillation is twice that in Fig. 84. Simi- 
 larly in Fig. 88, in which the vibrator is connected at one- 
 third the distance from one end, each of the two parts 
 becomes a vibrating element giving fundamental oscilla- 
 tions. The frequency of the oscillation of the shorter 
 part is twice as great as for the longer portion. In Fig. 90, 
 with the vibrator connected at one-fourth the distance from 
 one end of the line, the short end oscillates at three times 
 the frequency of the long end. In all cases the voltage 
 and current vary progressively along the line so that at 
 any instant the average voltage, instead of the maximum 
 value, is impressed on the condensers and the average 
 current, in place of the maximum value, flows through the 
 inductance. 
 
 The same results would be obtained in circuits with 
 massed R, L, G and C in which the maximum voltage is 
 
OSCILLATIONS, SURGES AND TRAVELING WAVES 111 
 
 impressed on all the condensance simultaneously or all of 
 the magnetic field reaches a maximum at the instant the 
 current is a maximum, by reducing the condensance and 
 inductance in the ratio of the maximum to the average 
 values. This ratio is ir/2 for sine waves. 
 
 The frequency for free oscillations in simple circuits 
 with massed R, L, G and C was derived in Chap. V, equation 
 (162). 
 
 ~ 2 cycles per second (219) 
 
 The frequency of free oscillations in circuits having 
 distributed R, L, G and C and a sine wave distribution of 
 the voltage and current may be obtained by multiplying 
 L and C in equation (219) by Tr/2, the ratio of the maximum 
 to the average value. 
 
 ~ c cycles per second (220) 
 
 In commercial electric lines the quantity 
 
 negligibly small in comparison with 1/LC. For practical 
 problems the frequency of the fundamental oscillations or 
 surges in transmission lines with uniformly distributed 
 R, L, G and C may therefore be obtained by equation (221). 
 
 / = . . n cycles per second (221) 
 
 Thus the fundamental frequency of oscillation for the 
 transmission line in Fig. 84, 
 
 ThuS = /= - 2 - CydeS Per S6COnd (222) 
 
 This may be checked by measurements on the oscillogram 
 in Fig. 84. On the original film (the cut in the text is 
 reduced in size) 10 cycles of the timing wave measured 
 14.3 cm., while 10 cycles of the transient oscillations 
 measured 7.1 cm. Hence the frequency, 
 
 / = -j 4 ^ == 200.1 cycles per second (223) 
 
112 ELECTRIC TRANSIENTS 
 
 Since L and C represent the total inductance and conden- 
 sance of the line the frequency depends on the total length 
 of the line or the length of time in which the oscillation 
 occurs, as illustrated by the oscillogram in Figs. 84, 86, 88 
 and 90. The transmission line, or other circuits of dis- 
 tributed R, L, G and C, therefore, have a fundamental 
 frequency at which the whole line oscillates, but as any 
 fractional part of the line may also oscillate independently 
 of the whole line, particularly if the oscillating section is 
 short as compared to the entire line, oscillations of any 
 frequency may occur. At high frequencies the successive 
 waves are so close together that a small variation in the 
 time constants will cause them to overlap. Since R, L, 
 G and C are not perfectly constant high frequency oscilla- 
 tions interfere with each other, and on this account reso- 
 nance phenomena occur only at low or moderate frequencies. 
 
 Length of Line. In ordinary transmission lines, with air 
 as the dielectric and conductors of copper or aluminum, an 
 electric wave or impulse travels approximately 3 10 10 cm. 
 per second, the velocity of propagation of an electromag- 
 netic field in free space, equation (205). This fact is of 
 much practical importance in transmission line calculations. 
 If the length of the line is known the frequency of the 
 fundamental oscillation and of the harmonics can readily 
 be determined. The length of the line is one quarter wave 
 length of the fundamental oscillation as illustrated by the 
 oscillogram in Fig. 84 and corresponding diagrams in Fig. 
 85. 
 
 v =- Wo (224) 
 
 Conversely, if the frequency of the oscillation is known 
 the length of the oscillating section may be determined. 
 In artificial transmission lines with the frequency of the 
 fundamental oscillation obtained from oscillograms the 
 equivalent length of the line can be calculated. Thus from 
 measurements on the oscillogram in Fig. 84, equation (223), 
 / = 200 cycles per second. Hence the length of the line, 
 
OSCILLATIONS, SURGES AND TRAVELING WAVES 113 
 
 v 3-10 10 
 /o = 4 ,- = cm. = 375 km. == 233 miles (225) 
 
 From equations (205), (221) and (224), relations are 
 obtained by which L or C may be calculated if the length 
 of the line, I in cm., and either C or L are known. 
 
 v = 3-10" - 4/7 = ~ (226) 
 
 Hence, 
 
 For cables or circuits in which the permeability, /*, and 
 the permittivity, K, are greater than unity the corresponding 
 relations are obtained from equations (206), (221) and 
 (224). 
 
 3-10 10 I 
 
 These equations are useful in the calculation of the con- 
 densance of circuits in which the inductance can be more 
 easily determined, as in complex overhead systems and in 
 calculating inductance in cables or other circuits in which 
 the condensance may readily be measured. 
 
 Velocity Unit of Length. Surge Impedance. In hand- 
 books and tables the values of R, L, G and C are given for 
 some unit of length as cm., km., 1,000 ft., mile, etc. In 
 discussions and calculations of transient phenomena the 
 velocity unit of length is sometimes used. For overhead 
 structures the unit of length, I, on this basis would be v t or 
 3-10 10 cm. Hence from equation (227), and under the 
 assumptions made in its derivation, 
 
 L, = ~ (230) 
 
 O v 
 
 The natural or surge impedance from equations (121), 
 (230) : 
 
114 ELECTRIC TRANSIENTS 
 
 = J'-=L. ==V (231) 
 
 \ U v ^v 
 
 By the use of the velocity unit of length investigations 
 on transmission systems having sections of different con- 
 stants and hence of different wave length are greatly 
 simplified. In systems having overhead lines, cables, 
 coiled windings, as in transformers, arresters, etc., the 
 wave length becomes the same in the velocity measure of 
 length. 
 
 Voltage and Current Oscillations and Power Surges. 
 It has been shown that in free or stationary oscillation 
 transmission lines or other electric circuits having uniformly 
 distributed R, L, G and C the current and voltage are essen- 
 tially in time quadrature. From equations (215), (216) : 
 
 i = It-ut s i n ( w $ __ X __ T ) (234) 
 
 e = Ee~ ut cos (coZ - 4>x - 7) (235) 
 
 Hence, the instantaneous power, p, at any point in the 
 circuit is given by equation (236) : 
 
 TjJT 
 
 p = ei = - - e~ ut sin 2(J - <f>x - 7) (236) 
 
 tU 
 
 The direction of the flow of power changes 4/ times each 
 second since the sine function becomes alternately positive 
 and negative for successive r time degrees. That is, a 
 surge of power occurs in the circuit of double the frequency 
 of the current or voltage oscillations, although the average 
 flow of power along the line is zero. 
 
 Average power, p Q = (237) 
 
 General Transmission Line Equations. In the preceding 
 paragraphs various phases of the electric transients that 
 occur during the free or natural oscillations of electric 
 circuits have been discussed. The general problem, in which 
 transient phenomena occur while continuous power is 
 supplied to the system and transmitted along the line, is 
 
OSCILLATIONS, SURGES AND TRAVELING WAVES 115 
 
 necessarily much more complex. In transmission lines 
 or other electric circuits having uniformly distributed resis- 
 tance, inductance, conductance and condensance, with 
 R, L, G and C the constants per unit length of line, the 
 voltage and current relations in time may be expressed by 
 partial differential equations as in (238), (239): 
 
 -< + ft (239) 
 
 Differentiating (238) with respect to x and (239) with 
 
 d^i 
 
 respect to / and eliminating equations (240), (241) 
 
 (7 JU(J L 
 
 may be derived: 
 
 ^2 P x2 p 3p 
 
 LC + (RC + GL) + BGe (240) 
 
 + (RC + GL) | + RGi (241) 
 
 A general solution for these equations is given in equation 
 (242) , one term of which represents the sum of the outgoing 
 and the other the sum of the incoming waves. 
 
 e = Aie at e b * sin (at + 0x + 71) 
 
 + A 2 e at e- b * sin (at + $x + T2 ) (242) 
 
 In order to determine the values of A\ t A 2 , a, 6, a, |8, 71, 
 and 72, the specific conditions under which the line operates 
 must be given. It is, however, of first importance to 
 understand the purpose or functions of each term in the 
 equation. On the basis of energy flow and dissipation in 
 a line transmitting power the following interpretation of 
 the symbols in equation (242) may be helpful. 
 
 A i, and A 2 are constants whose values are determined by 
 the limiting conditions of each specific problem. 
 
 e~ at may be called the time damping factor and a the time 
 dissipation constant for the transient oscillations. 
 
116 ELECTRIC TRANSIENTS 
 
 This factor represents the same form of energy 
 dissipation as e~ wi in Chap. V. Ordinarily the trans- 
 formation of electric energy into heat by the Ri 2 and 
 Ge 2 losses is non-reversible and therefore the sign of 
 the dissipation constant must be negative. 
 e bx may be called the distance damping factor and b the 
 distance dissipation constant. It relates both to the 
 losses along the line in the steady flow of energy, as in 
 transmission lines carrying permanent load, and to the 
 flow of transient energy in the system as with travel- 
 ing waves or in the oscillations of compound circuits. 
 at is the time angle. Under permanent or steady condi- 
 tions with a simple sine voltage, M E sin ut, impressed 
 at the generating station a = w and has only one 
 value. However, if the impressed voltage is a 
 complex wave or during transition periods between 
 two permanent conditions while transient currents 
 and voltages are flowing in the system, a would have 
 more than one value. 
 
 fix is the space or distance angle with x as the distance 
 along the line from the origin. If waves of more 
 than one frequency are passing over the line |8 
 would have more than one value. 
 71 and 72 are phase angles for t = 0. 
 Traveling Waves. Traveling waves are in many respects 
 similar to free oscillations or standing waves as the transfer 
 of energy between the dielectric and magnetic fields is the 
 basis for all double energy electric phenomena. The essen- 
 tial difference is that in traveling waves power flows along 
 the line while in free oscillations or standing waves the 
 energy oscillates between the two fields but does not travel 
 from one line element to another. Oscillograms of the cur- 
 rent and voltage factors in traveling waves are shown in 
 Figs. 92 to 97. It should be noted that the current is 
 in time phase with the voltage for the outgoing waves 
 and differs by 180 deg. for the returning waves. In both 
 cases a flow of power occurs along the line. 
 
OSCILLATIONS, SURGES AND TRAVELING WAVES 117 
 
 In Fig. 92 the receiver end of the line is short circuited. 
 The reflected voltage wave is in opposite time phase to the 
 outgoing wave while the corresponding current waves are 
 in the same direction. 
 
 In Fig. 93 the receiver end of the line is open and as a 
 consequence the reflected current wave reverses in sign 
 while the corresponding voltage wave is in the same direc- 
 tion as the outgoing wave. 
 
 FIG. 92. Traveling waves on artificial transmission line. Receiver end short 
 
 circuited. 
 
 Eo = 120 volts, d.c.; Ei = 5 volts; 7i = 19.5 amps.; R =56.1 ohms; G = 0; 
 L = 0.418 henrys; C = 3.053 microfarads; timing wave 100 cycles. 
 
 For the circuit in Fig. 94 a resistance equal to the surge 
 impedance of the circuit, VL/VC, is inserted at the 
 receiver end of the line. All the energy of the traveling 
 wave was dissipated into heat by the Ri^ losses at the 
 receiver end of the line and as a consequence there was no 
 reflected voltage or current waves or return flow of power. 
 From the timing wave and known length of line it is found 
 that the velocity of propagation of the impulse is equal to 
 
118 
 
 ELECTRIC TRANSIENTS 
 
 v or 3-10 10 cm. per second, the velocity of propagation of 
 an electromagnetic field in free space. 
 
 A traveling wave in an electric line is sometimes trans- 
 formed into a standing wave, as illustrated by the oscillo- 
 grams in Figs. 95, 96 and 97. In Fig. 95, with the 
 receiver end of the line open, both the voltage and current 
 waves show that the traveling wave passed from the genera- 
 tor to the receiver end of the line and back again four 
 times before it was changed into a standing wave. During 
 this period the voltage and current waves are in phase or 
 
 FIG. 93. Traveling waves on artificial transmission line. Receiver end open. 
 Eo = 120 volts d.c.; Ei = 5 volts; /i = 19.5 amps. R =56.1 ohms; G = 0; 
 L = 0.418 henrys; C = 3.053 microfarads; timing wave 100 cycles. 
 
 180 deg. apart, showing a flow of power along the line, but 
 when the traveling wave is changed to an oscillation the 
 current leads the voltage (note position of vibrators in 
 the circuit diagram) by 90 deg. If the current leads or 
 lags 90 deg. with respect to the voltage, the power in the 
 circuit is reactive and therefore the average flow of power 
 along the line is equal to zero. 
 
OSCILLATIONS, SURGES AND TRAVELING WAVES 119 
 
 Similarly, the oscillograms in Figs. 96 and 97 show 
 impulses which after passing over the lines several times 
 as traveling waves are transformed into standing waves or 
 oscillations. In each case the impulse starts as a traveling 
 wave with the current and voltage in phase and a flow of 
 power along the line. The oscillogram shows that the 
 traveling wave was converted into an oscillation or stand- 
 ing wave, in which the current and voltage differ by 90 deg. 
 in time phase, in less than one hundredth of a second, and 
 that the energy then oscillated between the magnetic and 
 dielectric fields without flow of power along the line. 
 
 FIG. 94. Traveling waves on artificial transmission line. 
 
 Receiver resistance =\/L /\/C; Eo = 120 volts d. c.; Ei = 5 volts; I\ = 19.5 
 amps.; R = 56.1 ohms;G= 0; L = 0.418 henrys; C = 3.053 microfarads; timing 
 wave 100 cycles. 
 
 In Fig. 97 the vibrator connections for the voltage wave, 
 t> 3 , were reversed; the voltage and current were in phase 
 instead of 180 apart as indicated by the oscillogram. 
 
 The change in frequency when the traveling wave is 
 converted into a standing wave should be noted. In the 
 
120 ELECTRIC TRANSIENTS 
 
 traveling wave the inductance and condensance of the 
 line alone determines the velocity of propagation while 
 for the oscillations or standing waves the line and trans- 
 former oscillate together as a compound circuit. 
 
 In determining the instantaneous values for the current 
 and voltage at any point on the system the power flow must 
 be taken into consideration in addition to the dissipation 
 of the transient electric energy into heat as expressed by 
 the damping factor e~ ut . It is evident that the flow of 
 power may be increasing, decreasing or unvarying in the 
 direction of propagation. 
 
 If the power flow is uniform the expressions for the cur- 
 rent and voltage are in the simplest form (244), (255), as 
 the power transfer factor does not appear in the equations. 
 
 i = Io~ ut cos (ut + </> 7) (244) 
 
 e = Eoe~ ut cos (ut + <f>x - 7) (245) 
 
 p = # /oe-<[l -- sin 2 (ut + 4>x - 7)] (246) 
 
 F 1 J 
 Average power, p = ~ e~ 2ut (247) 
 
 Uniform flow of transient power is infrequent but may 
 occur in special cases. Thus if a transformer line and load, 
 as in Fig. 100, are disconnected from the power supply and 
 left to die down together, uniform flow of power in the line 
 may be realized provided the dissipation constant of the 
 line is equal to the average dissipation constant of the 
 whole system. Consider the transformer as having stored 
 in the magnetic field a comparatively large quantity of 
 energy while its resistance and conductance are relatively 
 small compared to the corresponding value for the line. 
 Likewise assume that the load part of the circuit has very 
 little energy stored in its magnetic and dielectric fields and 
 that its dissipation constant is large as compared to that 
 of the line. Under these conditions the dissipation of 
 energy is most rapid in the load part of the circuit and 
 slowest in the transformer. Hence a 'flow of energy will 
 occur from the transformer to the load. If the rate of 
 
OSCILLATIONS, SURGES AND TRAVELING WAVES 121 - 
 
 energy dissipation of the line is midway between the corre- 
 sponding rates for the load and transformers the energy 
 dissipated in the line would be equal to the amount initi- 
 ally stored in the line while part of the energy originally 
 stored in the transformer flows through the line and is dissi- 
 pated in the load part of the circuit. The flow of power in 
 the line would be uniform as it delivers to the load part of 
 the circuit all the energy received from the transformer. 
 
 FIG. 95. Traveling waves changing to standing waves on artificial transmission 
 
 line. 
 
 R = 55.32 ohms; G = 0; L = 0.419 hemys; C = 3.05 microfarads; Length = 
 207 miles; 4/0 copper; 96 in spacing; Transformer L = 37.8 henrys; timing wave 
 100 cycles. , 
 
 The flow of power decreases along the line in the direction 
 of propagation, if energy is left in the circuit elements as 
 the traveling wave passes along the line. That is, the 
 traveling wave scatters part of its energy along its path 
 and thus decreases in intensity with the distance traveled. 
 This decrease is expressed by a power transfer constant, s, 
 comparable to the power dissipation constant u. If no 
 energy were supplied to the line by the traveling wave the 
 
122 ELECTRIC TRANSIENTS 
 
 voltage and current would decrease by the dissipation 
 factor t~ ut . With power supplied by the flow of energy 
 the decrease would be slower and would be expressed 
 by a combination of the damping and power transfer 
 factors. 
 
 For decreasing flow of power: 
 
 Damping factor = e~ ut (248) 
 
 Power transfer factor = e +st (249) 
 
 Combined damping and power transfer factor 
 
 ' (250) 
 
 Similarly if the flow of power increases along the line in 
 the direction of propagation the traveling wave receives 
 
 FIG. 96. Traveling waves changing to standing waves of artificial transmission 
 
 line. 
 
 Eo= 110 volts; 7i= 19.8 amps.; R = 52.9 ohms; G = 0; L = 0.412 henrys; 
 C = 3.03 microfarads; timing wave 60 cycles. 
 
 energy from the line elements and the actual decrease in 
 the voltage and current is greater than indicated by the 
 dissipation constant. The power transfer would in this 
 case be negative, and the combined damping and power 
 transfer factor would be expressed by equation (253). 
 
OSCILLATIONS, SURGES AND TRAVELING WAVES 123 
 
 For increasing flow of power : 
 
 Damping factor = e- ut (251) 
 
 Power transfer factor = e~ st (252) 
 
 Combined damping and power transfer factor 
 
 = e- (M + s) ' (253) 
 
 To express the instantaneous values of the current and 
 voltage at any point in the circuit a distance factor must 
 be included. For if the traveling wave either scatters or 
 gathers in energy as it travels along the line the voltage and 
 current factors decrease at a lesser or greater rate, as the 
 case may be, in the direction of propagation than if the flow 
 of power were uniform. In order to use only one power 
 transfer constant, s, in the equation, let X = the distance x 
 expressed in velocity measure (254) 
 
 For decreasing flow of power along the line : 
 
 the distance damping factor = e~' (255) 
 
 For increasing flow of power along the line : 
 
 the distance damping factor = e sX (256) 
 
 The instantaneous values of the transient current, voltage 
 and power under conditions producing a flow of power along 
 the line from the point of reference, in the direction of propa- 
 gation may be expressed by equations (257), (258), or 
 (259), (260). 
 
 i = I e~ ( ' e + ?X cos (cot + X -- T ) (257) 
 
 e = E e "c cos (cot + 0X 7) (258) 
 
 i = Le~ ut e cos (cot + 0X - 7) (259) 
 
 e = E e ~ ut e~ cos (cot + </>X -- 7) (260) 
 
 T -2ut 2s(t - X) . 
 
 p = loEoe e [I Sin 2 (cot + 0X 7)! 
 
 (261) 
 
 Average power, p = Lfj>r** - (262) 
 
 2i 
 
 The upper sign of 4>X applies to waves traveling in the 
 direction of increasing values of X and the lower sign for 
 returning waves, for which X is decreasing. For s = 
 
124 
 
 ELECTRIC TRANSIENTS 
 
 which represents a constant flow of power, equations 
 (259) and (260) become identical with equations (244) and 
 (245). Referring to Fig. 100, already used for illustrating 
 the flow of constant power, it is evident that if the dissipa- 
 tion constant for the line is less than the average dissipa- 
 tion constant for the system the flow of power from the 
 transformer will be such as to increase the power stored 
 in the line, while if the line dissipation constant is greater 
 than the average the reverse would be the case. 
 
 FIG. 97. Traveling waves changing to standing waves on artificial transmission 
 
 line. 
 Eo = 120 volts; Length = 200 miles; 4/0 copper; 120 in. spacing; timing wave 
 
 100 cycles. 
 
 Traveling waves are of very frequent occurrence in elec- 
 tric power systems. . Not merely such violent disturbances 
 as direct strokes of lightning or short circuits, but practi- 
 cally every change in load or circuit conditions produce 
 transient waves that travel over the system. Simple travel- 
 ing waves as illustrated by the oscillograms in Figs. 92 to 
 
 101 are frequently called impulses. In the first part of the 
 
OSCILLATIONS, SURGES AND TRAVELING WAVES 125 
 
 FIG. 98. Oscillation of compound circuit. Starting transient of artificial 
 
 transmission line and step-up transformer. 
 
 Length of line = 52 miles; 4/0 copper; 96 in. spacing, R = 13.84 ohms ; G = 
 0; L = 0.105 henrys; C = 0.764 microfarads; transformer L = 37.8 henrys; 60 
 cycle supply. 
 
 FIG. 99. Oscillation of compound circuit. Starting transient of (artificial 
 
 transmssion) line and transformers. 
 
 Length of line 52 miles; 4/0 copper; 96 in. spacing; R = 13.84 ohms; G = 0; 
 L = 0.105 henrys; C = 0.764 microfarads; 60 cycle supply. 
 
126 ELECTRIC TRANSIENTS 
 
 impulse as it passes along a line the wave energy increases 
 at a rate depending on the steepness of the wave front, and 
 after the maximum value is reached the wave energy 
 decreases. While the wave energy increases the combined 
 dissipation and power transfer factor is represented by 
 c~' '* as in equation (253), and during the decreasing 
 stage by e }t as in equation (250). The steepness of 
 
 the wave front which corresponds to the sharpness or 
 suddenness of a blow is often a more important factor in 
 causing damage to the electric system than the quantity 
 of energy involved. 
 
 Compound Circuits. In commercial systems the trans- 
 mission line is not an independent unit but merely a link 
 between the generator and load circuits. Step-up and 
 step-down transformers, generators and load circuits, 
 lightning arresters and regulating devices, and all the 
 apparatus necessary for the operation of the system are 
 electrically interconnected into one unit. In the several 
 parts of the system the circuit constants differ in relative 
 
 magnitude and hence the velocity of 
 propagation of an electric impulse 
 varies and no two sections may have 
 the same natural period of oscilla- 
 tion. While the whole system may 
 
 FIG. 100. circuit diagram oscillate as a unit partial oscilla- 
 
 of a compound circuit. , . ,. , ,. 
 
 tions are of much more frequent 
 occurrence. 
 
 In Figs. 98, 99, 101 and 102 are shown the oscillations of 
 compound circuits consisting of an artificial transmission line 
 and transformers. The ripples on the current wave, Vi, indi- 
 cate a wave traveling over the transmission line alone. From 
 measurements on the film, Fig. 101, the length of the line is 
 found to be 207 miles. The line and transformers oscillate 
 as a compound circuit at a frequency of 10.5 cycles per 
 second. In Fig. 102 the length of the second half wave is 
 longer than for the first half wave. This is due to a varia- 
 tion in the permeability of the iron in the transformer core. 
 
OSCILLATIONS, SURGES AND TRAVELING WAVES 127 
 
 o i 
 
 60 
 
 I 
 
 en 
 
 sl 
 
 & 
 
 O iO 
 
 B " 
 
 03 .0 
 
 II 
 
 fl ft 
 
 S o 
 
128 
 
 ELECTRIC TRANSIENTS 
 
 ^ g 
 
 " 
 
 ^3 O 
 
 fl >> 
 
 g'a 
 +3 a 
 
 <Jo 
 to 
 
 3 ,y 
 
 o o 
 *-i 03 
 
 '3 a 
 
 o o 
 fl" 
 
 a 
 
 
 ^ o 
 
 a 
 o 
 
OSCILLATIONS, SURGES AND TRAVELING WAVES 129 
 
 Problems and Experiments 
 
 1. Given a transmission line, 80 miles long, of No. 0000 copper, spaced 
 12 ft. and with the receiver end open. From handbook tables obtain the line 
 constants. Find the fundamental oscillation frequency of the line. Check 
 the results by solving for the frequency from the known velocity of propaga- 
 tion of an electric wave in space and use the given length of the line. 
 
 2. Make a series of oscillograms similar to Figs. 84, 86, 88 and 90, on an 
 artificial transmission line. From the oscillograms determine the equiva- 
 lent length of actual line. Check by determining the natural frequency of 
 oscillation from the line constants. 
 
 3. From the oscillogram in Fig. 95 or 97 determine the frequency of oscil- 
 lation of the transmission line alone and the transmission line combined 
 with the transformer. Assume the condensance of the transformer equal to 
 zero. From the data given calculate the inductance of the transformer. 
 
 4. In the oscillogram in Fig. 101 the ripples on the voltage wave indicate 
 reflections of traveling waves in the transmission line with the receiver end 
 open. Calculate the length of the line. 
 
 6. From the oscillograms and data in Fig. 101 calculate the inductance 
 in the transformer in the compound circuit. Assume the condensance of 
 the transformer equal to zero. It should be noted that the inductance is 
 essentially massed while the condensance is distributed and hence for the 
 
 combined circuit/ = / - 
 
 ^ v 2irLC 
 
 6. From the data given in Fig. 102 calculate the average inductance of 
 the transformers during the first half cycle after the current and voltage 
 wave lines cross; also during the second half cycle. 
 
 7. Make oscillograms of the oscillations of compound circuits, similar to 
 Figs. 99, 100, 101, and 102. 
 
CHAPTER VII 
 VARIABLE CIRCUIT CONSTANTS 
 
 In the preceding chapters the fundamental laws of tran- 
 sient electric phenomena are derived under the assumption 
 that in any given circuit the resistance, inductance, con- 
 ductance and condensance, the so-called circuit constants, 
 remain constant in value during the transition period under 
 discussion. The transients are due to changes in circuit 
 condition or in the impressed voltage, but during the period 
 required for the dissipation of the stored energy, or the 
 readjustment of the energy content in the system the values 
 of R, L, G and C are assumed constant. The oscillograms, 
 Chaps. Ill to VI inclusive, of electric transients were 
 obtained from circuits in which the resistance, inductance, 
 conductance and condensance remained essentially constant. 
 
 It is evident that if the circuit constants do not remain 
 constant during the period the transients occur but vary 
 rapidly over a wide range of values the nature of the result- 
 ing electric phenomena must be correspondingly more 
 complex. The laws for the variations in R, L, G and C are 
 not always known or are so complex that they can not be 
 represented in the form of equations. For example, data 
 for the quantitative ratios between the magnetomotive 
 force and the resulting magnetic flux in iron clad circuits, 
 as indicated by the hysterises loop, may readily be obtained 
 experimentally but it has not been possible to express the 
 relation in the form of a mathematical equation. The 
 empirical equations in common use are limited in their 
 application and give only approximate values. 
 
 Variable Resistance. Change in temperature is the 
 most important factor in producing variations in the resis- 
 tance of electrical conductors, the R circuit constant. For 
 
 130 
 
VARIABLE CIRCUIT CONSTANTS 
 
 131 
 
 metals the specific resistance is a linear function of the 
 temperature over a fairly wide range. 
 
 ---- po + at (270) 
 
 = specific resistance at t C. 
 specific resistance at C. 
 = temperature coefficient. 
 
 Pt 
 
 PO 
 
 For rapid changes in temperature the rate of change in 
 the resistance may be large. This is illustrated by the 
 oscillograms in Figs. 103, 104. For the tungsten incandes- 
 cent lamp, Fig. 103, a starting transient appears in the 
 current due to a rapid increase in the resistance of the fila- 
 
 FIG. 103. Starting current transient of a 60-watt, 120 volts, tungsten incan- 
 descent lamp. Resistance variable; timing wave 100 cycles. 
 
 ment as the temperature rises. When the switch is closed 
 the filament is at room temperature and the resistance low. 
 The current flowing through the lamp rapidly heats the 
 filament to incandescence with an accompanying increase in 
 the resistance and a decrease in the current. The timing 
 
132 
 
 ELECTRIC TRANSIENTS 
 
 wave shows that it required about 0.02 of a second for the 
 lamp to reach full brilliancy. During this period the resis- 
 tance of the filament increased by 400 per cent of its initial 
 value. 
 
 For carbon the resistance decreases with an increase in 
 temperature, or the temperature coefficient is negative, as 
 illustrated in Fig. 104, showing that the time required for 
 the resistance to reach a constant value was approximately 
 0.5 of a second and that the resistance of the incandescent 
 carbon filament is about 70 per cent of its value at room 
 temperature. 
 
 FIG. 104. Starting current transient of a 50 watt, 120 volts, carbon incandescent 
 lamp. Resistance variable; timing wave 100 cycles. 
 
 The temperature of the lamp filament increases until the 
 dissipation of heat by radiation from the lamp is equal to 
 the heat generated by Ri 2 losses. For a direct current sup- 
 ply with constant impressed voltage the constant tem- 
 perature condition is quickly reached. For alternating 
 currents the power supplied to the lamps pulsates with double 
 
VARIABLE CIRCUIT CONSTANTS 133 
 
 the current frequency and as the lamp emits or radiates 
 heat continuously the temperature, and therefore the resis- 
 tance of the filament, pulsates. This is illustrated by the 
 oscillogram in Fig. 105. Alternating currents are impressed 
 on two pairs of tungsten and carbon lamps, arranged as 
 shown in the circuit diagram, with the vibrator of the oscil- 
 lograph in the bridge connection. Since the resistance of 
 
 FIG. 105. Pulsating resistance of tungsten and carbon lamps, alternating 
 currents; 60 cycle supply. 
 
 the tungsten lamp increases and the carbon lamp decreases 
 with an increase in temperature, the pulsations in the Ri 2 
 losses unbalance the bridge as indicated by the pulsations 
 in the currents flowing through the vibrator. 
 
 The resistance of the electric arc depends on many factors 
 and may vary over a wide range with extreme rapidity. 
 Since the resistance of the arc decreases with the increase 
 in temperature the arc alone is unstable and hence must be 
 provided with a " ballast" to make continuous operation 
 possible. On alternating currents an inductance placed in 
 
134 
 
 ELECTRIC TRANSIENTS 
 
 series with the arc serves as the stabilizer and the variations 
 in the resistance of the arc are counterbalanced by the 
 induced voltage in the inductance. In direct current arc 
 lamps a series resistance serves the same purpose. 
 
 In commercial systems the electric arcs that affect the 
 series resistance, the R circuit constant, occur chiefly in 
 the opening of switches. In breaking the circuit under 
 load, especially when a large quantity of energy is stored 
 magnetically in the circuit, arcs form in which the resistance 
 varies rapidly from zero at start to infinity when the circuit 
 is open. 
 
 ^^^^^^^^^^^MM^^^^MMM^^^^^^MMM^M^^^M 
 
 FIG. 106. Transformer magnetizing current; no starting transient. 
 Vi = 106 volts; v-2 = primary current; 03, calibration current = 10.0 amps.; 10 
 KVA. transformers;/ = 60 cycles. 
 
 This is illustrated by the oscillogram in Fig. 110. In 
 the opening of the switch an arc forms whose resistance 
 rapidly increases, approaching infinity when the circuit 
 opens, which occurs at the point of maximum value in the 
 voltage curve. The increase in the resistance can be deter- 
 mined quantitatively from the oscillogram by combining 
 
VARIABLE CIRCUIT CONSTANTS 135 
 
 data from the rapidly increasing voltage and decreasing 
 current curves. 
 
 Variable Inductance. In iron-clad circuits as in trans- 
 formers the magnetic flux is not directly proportional to the 
 ampere turns or magnetizing force. Hence the inductance, 
 the L circuit constant is not constant but varies with the 
 permeability of the iron. Moreover, the variation in the 
 inductance is different for decreasing and increasing flux 
 
 FIG. 107. Starting transient of magnetizing current in iron-clad circuit. 
 vi, primary voltage = 236 volts; vz = primary current; % calibration current 
 = 13.0 amps.; 10 KVA.. transformers;/ = 60 cycles. 
 
 values and depends on the maximum flux density as indi- 
 cated by the form of the hysteresis loop. As no satisfac- 
 tory mathematical expression has yet been found for the 
 hysteresis cycle, solutions of practical problems are obtained 
 by a series of approximations. As a first step in obtaining 
 the shape of transients in iron-clad circuits, neglecting the 
 difference between increasing and decreasing flux values, 
 Frohlich's formula is generally used. 
 
 H B = v + *H (271) 
 
136 
 
 ELECTRIC TRANSIENTS 
 
 The formula is based on the assumption that the permea- 
 bility of the iron is proportional to its remaining magnetiza- 
 bility and states that the reluctivity of an iron-clad circuit 
 is a linear function of the field intensity. 
 
 FIG. 108. Starting transient of magnetizing current in iron-clad circuit. 
 i, primary voltage = 106 volts; 02, primary current; 03, calibration current 
 = 10.0 amps.; 10 KVA. transformer; / = 60 cycles. 
 
 The effect of variable inductance in iron-clad circuits 
 may be illustrated by the starting transients of alternating 
 current transformers. The magnitude of the starting 
 current transient depends more on conditions affecting the 
 value of the inductance in the circuit than on what point 
 on the voltage cycle the switch is closed. The direction 
 and magnitude of the residual magnetism are important 
 factors as a combination of much residual flux with an 
 additional magnetizing force in the same direction may 
 bring the flux density in the core beyond the saturation 
 point and hence greatly reduce the inductance in the 
 circuit. 
 
 For the oscillograms in Figs. 106 to 109 a constant alter- 
 nating current voltage of sine wave shape was impressed on 
 
VARIABLE CIRCUIT CONSTANTS 137 
 
 the transformer terminals. In Figs. 106, 107 and 108 the 
 residual magnetism in the iron core was, in each case, 
 removed before the oscillogram was taken. The three 
 oscillograms form a series showing the transient current due 
 to the closing of the switch at different points of the voltage 
 cycle. In Fig. 106 the switch was closed at an instant the 
 magnetizing current would have been zero (maximum point 
 on the voltage wave), if the circuit had been closed earlier, 
 
 FIG. 109. Starting transient in transformer magnetizing current. Residual 
 
 magnetism. 
 
 vi, primary voltage = 150 volts; 02, primary current; vs, calibration current 
 = 2.5 amps.; / = 60 cycles. 
 
 and hence no starting transient. In Figs. 107 and 108 the 
 switch was thrown at other than the zero point of the mag- 
 netizing current cycle. The impressed voltage was less than 
 normal and the change in the flux density is not large and 
 hence the inductance at the maximum points of the mag- 
 netizing current wave is practically constant. The starting 
 current transients under the given conditions may be expres- 
 sed by an exponential equation as explained in Chap. IV. 
 
138 
 
 ELECTRIC TRANSIENTS 
 
 The starting transient in Fig. 109 differs greatly both in 
 form and magnitude, as compared to Fig. 108, although the 
 circuits were closed in the two cases at approximately the 
 same point on the voltage wave. In Fig. 109 the impressed 
 voltage was higher than the rating of the transformer and 
 the residual magnetism in the iron core was in the same 
 direction as the flux produced by the magnetizing current 
 during the first half cycle. Above saturation of the iron 
 
 FIG. 110. Breaking generator field circuit. Field current and voltage 
 
 transients. 
 
 vi = 100 cycle timing wave; v?, impressed voltage =31.5 volts; va, field cur- 
 rent = 4.0 amps. 
 
 core the transformer inductance is relatively small and 
 hence the first half cycle shows a correspondingly large 
 current transient. A smooth curve drawn through the 
 successive maximum values of the starting transient in 
 Figs. 107 or 108 could with a fair degree of accuracy be 
 expressed by the exponential equation ; but the correspond- 
 ing curve drawn through the successive maximum values 
 of the current wave in Fig. 109 would have a much steeper 
 
VARIABLE CIRCUIT CONSTANTS 
 
 139 
 
 gradient due to the variation in the inductance, L, of the 
 transformer winding. 
 
 The same effect, due to variable inductance, may be 
 obtained in breaking the field circuit of a direct current 
 generator as illustrated by the oscillogram in Fig. 110. 
 The change in the voltage and current curves from the 
 instant the jaws of the switch separate to the peak value 
 of the voltage is largely due to a change in the arc resistance. 
 After the arc breaks, at the peak of the voltage curve, the 
 
 FIG. 111. Building up generator field. Field current and armature voltage 
 
 transients. 
 
 vi = generator terminal voltage; vz = field current; vs 100 cycle timing 
 wave. 
 
 vibrator circuit provides a path for the dissipation of the 
 energy stored in the field. As the resistance in the vibrator 
 circuit is constant the voltage curve also represents the 
 transient current. The dotted curve traced on the oscillo- 
 gram shows the exponential curve conforming with the 
 latter part of the actual voltage or current curves. The 
 relative magnitude of the peak value of the voltage to the 
 
140 
 
 ELECTRIC TRANSIENTS 
 
 corresponding initial value of the dotted curve indicates 
 the change in magnitude of the inductance in the field 
 winding. 
 
 The corresponding variation in the inductance when the 
 generator field is formed is evidenced by the starting field 
 current and armature voltage curves shown in Fig. 111. 
 
 FIG. 112. Arcing grounds on transmission line. 
 
 Ground at generator end. Impressed voltage = 90 volts;/ = 60 cycles; ^2 = 
 arc voltage; 03 = arc current. 
 
 Variable Conductance. In the calculations on power 
 transmission lines and in general for constant potential 
 systems in good condition the leakage through the insulation 
 is small, so that the conductance is negligible and the G 
 circuit constant may be taken as equal to zero. The insula- 
 tion of electric circuits deteriorate with varying rates and 
 the conductance and leakage increase and may become very 
 large, as for example, if the insulation completely breaks 
 down and a short circuit is formed. A rupture of the insula- 
 tion or any sudden change in the conductance of the electric 
 circuit will of necessity cause violent disturbances in the 
 
VARIABLE CIRCUIT CONSTANTS 
 
 141 
 
 FIG. 113. Arcing- grounds on transmission line. 
 
 Semi-continuous copper-carbon arc 114 miles from generator end of 207 
 mile artificial transmission line. 4/0 copper, 96 in. spacing. v\ = arc voltage; 
 #2 = arc current; vs line current. 
 
 FIG. 114. Arcing grounds on transmission line. 
 
 Arc at receiver end. vi = 100 cycles timing wave; vz = current receiver end: 
 = voltage receiver end. 
 
142 
 
 ELECTRIC TRANSIENTS 
 
 system. Arcing grounds or intermittent arcs, as illustrated 
 by the oscillograms in Figs. 112 to 115, are prolific sources 
 of electric transients. It is evident that momentary 
 short circuits, as would be produced by an intermittent 
 arcing ground with the conductance varying practically 
 from zero to infinity at an extremely rapid rate, would give 
 rise to oscillations of any frequency and produce waves and 
 impulses that would travel to all parts of the system. 
 
 FIG. 115. Arcing grounds on transmission line. 
 
 Arc at middle of line. v\ 100 cycle timing wave; vi = arc current; va arc 
 voltage. 
 
 Variable Condensance. Under ordinary conditions and 
 for low voltages, air is very nearly a perfect insulator. In 
 other words, the conductivity of air is practically zero, the 
 permittivity, unity and the energy loss extremely small. 
 If the voltage is increased until the limit of the insulating 
 strength of the air is reached important changes occur 
 in both the electric and dielectric circuit constants. With 
 the occurrence of visual corona in high voltage circuits the 
 conductivity of the air in the space filled by the corona is 
 
VARIABLE CIRCUIT CONSTANTS 
 
 143 
 
 increased. Thus in circuits with parallel wires as high 
 tension transmission lines a voltage gradient above 29.8 ky. 
 per cm. will produce corona in the air surrounding the 
 conductor surface and this space filled by the corona glow 
 becomes semi-conducting. This produces a change in the 
 circuit condensance as with the appearance of the corona the 
 effective size of the conductor, and hence of the condenser 
 surface, is increased. For alternating currents the visual 
 
 FIG. 116. Variable condensance. Corona. 
 
 Single phase line 135 ft. long, 10 in. spacing, No. 24 A.W.G. steel wire. Line 
 voltage = 3400 volts; line current = 0.0008 amps. 
 
 corona, is intermittent, appearing only near the peaks of 
 the successive voltage waves, when the instantaneous volt- 
 age gradient exceeds 29.8 kv. per cm., the required value 
 for producing visual corona. As a consequence the con- 
 densance of the alternating current circuit when corona 
 occurs is variable, pulsating with double the frequency of 
 the voltage. This is illustrated by the oscillograms in 
 Figs. 116, 117. If an alternating current voltage of sine 
 wave shape is impressed on a circuit having constant con- 
 
144 
 
 ELECTRIC TRANSIENTS 
 
 densance the charging current would also follow the 
 sine law. If the condensance, the C circuit constant, varies 
 during the voltage cycle, a corresponding change is produced 
 in the wave shape of the charging current. 
 
 FIG. 117. Variable condensance. Corona. 
 
 Line covered with snow and swaying in the wind. Line constants same as 
 for Fig. 116. 
 
 Problems and Experiments 
 
 1. Take oscillograms, similar to Figs. 106, 108 and 109, showing the 
 starting transients of transformers. 
 
 2. Take oscillograms showing the variable condensance of an arcing 
 ground for direct and alternating currents on a transmission line. 
 
 3. Take oscillograms similar to Figs. 116 and 117, showing the change in 
 condensance produced by corona. 
 
 4. Take an oscillogram similar to Fig. 110, showing the voltage across 
 the terminals. Compare the operating voltage with the maximum value 
 when the switch is opened. 
 
CHAPTER VIII 
 
 RESONANCE 
 
 Electric resonance phenomena have essentially perman- 
 ent or stable characteristics but are closely related to, and 
 frequently accompanied by, true electric transients. The 
 conditions required for producing resonance and expres- 
 sions for the frequency at which resonance occurs, in simple 
 electric circuits, are referred to in Chap. IV in connection 
 with the derivation of the equations for the natural fre- 
 quency of free oscillations. Resonance in an electric circuit 
 implies a forced oscillation of energy between the magnetic 
 and dielectric fields, during which the energy dissipated as 
 heat by the Ri 2 and Ge' 2 losses, is supplied from some outside 
 source. Distinction is usually made between voltage reson- 
 ance occurring in series circuits, and current resonance that 
 may be produced in parallel circuits. 
 
 Voltage Resonance. In series circuits voltage resonance 
 occurs at that frequency of the impressed voltage for which 
 the impedance of the circuit is a minimum. In series circuits, 
 as in Fig. 118, the impedance is a minimum when the con- 
 densive and inductive reactances are equal. 
 
 ,x = c x; 27T/L = (275) 
 
 /= \ (276) 
 
 2irVLC 
 
 z ---- VR 2 + ( L x - c x) 2 =R (277) 
 
 Frequently the assumption is made that a circuit is in 
 resonance when the current and the impressed voltage are 
 in phase, as illustrated by the vector diagram in Fig. 119. 
 For straight series circuits the conditions required for unity 
 power factor of the power supplied to the circuit are iden- 
 10 145 
 
146 
 
 ELECTRIC TRANSIENTS 
 
 tical with the requirements for minimum impedance, but 
 in complex circuits or for current resonance in parallel 
 circuits this is not always the case. 
 
 FIG. 1 18. Series circuit for voltage resonance. 
 
 Equation (276) gives the optimum condition for reson- 
 ance in series circuits for given values of the R, L and C, the 
 circuit constants. Resonance phenomena are, however, 
 
 Y 
 
 FIG. 119. Vector diagram for voltage resonance in series circuit Fig. 118. 
 
 not limited to the exact frequency determined by equation 
 (276), but persist over a range of frequencies, more or less 
 sharply defined, depending on the relative magnitude of 
 the resistance and the inductive or condensive reactance. 
 The voltage-frequency relation for given constant values 
 of R, L and C, is shown in Fig. 120. The feature of special 
 
RESONANCE 
 
 147 
 
 interest is the large increase in L E and C E, the voltages 
 across the inductance and the condensance under resonance 
 conditions. If the resistance is small ,E arid C E may rise 
 
 FIG. 120. Voltage resonance for series circuit as in Fig. 118 
 
 to many times the value of the impressed voltage E n . 
 Voltage resonance in power circuits is undesirable as the 
 increase in voltage above the normal operating value 
 endangers the insulation. 
 
 The effect of varying the resistance on the sharpness of 
 resonance is illustrated by Fig. 121. The smaller the resis- 
 tance the higher and sharper the voltage and current reson- 
 ance peaks. The sharpness of resonance may be defined 
 as the ratio of the inductive reactance or the condensive 
 reactance at resonance frequency to the resistance in the 
 circuit. 
 
 Sharpness of resonance = ^ = C D (278) 
 
 ti H 
 
 Reactance Curves. Curves in rectangular coordinates 
 showing graphically the changes in magnitude of the 
 
148 
 
 ELECTRIC TRANSIENTS 
 
 inductive reactance and the condensive reactance produced 
 by varying the frequency of the impressed voltage are of 
 much value for giving a clear insight into resonance phe- 
 nomena. The ordinates of the curves in Fig. 122 represent 
 respectively the inductive reactance, L x, the condensive 
 reactance, c x, and the total reactance, x, with the frequency 
 of impressed voltage as the other variable represented 
 
 / \ 
 
 FIG. 121. Resonance curves for series circuit with different resistances. 
 
 along the X axis. Since resonance occurs when the 
 impedance of the series circuit is a minimum, the resonance 
 frequency is indicated by the intersection of the total 
 reactance curve, in Fig. 122, with the X axis. 
 
 Current Resonance. Forced oscillatory transfer of 
 energy between dielectric and magnetic fields is the basis of 
 resonance phenomena in parallel circuits in much the 
 same manner as in series circuits, but the resultant voltage 
 and current values are different. In simple parallel 
 circuits, as illustrated by Figs. 123 and 127, current reson- 
 ance occurs at that frequency of the impressed voltage for 
 which the total admittance is a minimum. In discussions of 
 resonance phenomena it is frequently assumed that the 
 
RESONANCE 
 
 149 
 
 conditions for current resonance in parallel circuits are 
 met when the inductive and condensive susceptances 
 are equal, that is, when the impressed current and voltage 
 are in phase. That this assumption is not in full accord 
 with the above definition of current resonance for all 
 
 FIG. 122. Reactance curves. Series circuit. 
 
 values of R in the circuits shown in Figs. 123 and 127, may 
 readily be seen from the corresponding vector diagrams in 
 Figs. 124 and 128. For the circuit in Fig. 123 current 
 resonance occurs when ,b = ( b under the condition that 
 R = 0. From the vector diagram in Fig. 124: 
 
 R 
 
 J == E Q (g - jjb) ---- E ^ 
 
 J = j c bE Q = juCE 
 I == J + J == E [g +j( c b - L b)] 
 I = 1 
 
 (279) 
 
 (280) 
 (281) 
 (282) 
 
150 
 
 ELECTRIC TRANSIENTS 
 
 The total current, 7, will be in phase with the impressed 
 voltage, E, if 
 
 * = *><*<> =*** (284) 
 
 Hence for unity power factor supply, the frequency for 
 the circuit in Fig. 123, 
 
 ~ 
 
 (285) 
 
 
 ' 
 
 FIG. 123. Parallel circuit for current resonance. 
 
 FIG. 124. Vector diagram for circuit in Fig. 123. 
 
 For maximum current resonance the total admittance 
 of the circuit must be a minimum and hence for constant 
 impressed voltage, E Q , the total current must be a minimum. 
 Therefore, the resonance frequency may be obtained by 
 
RESONANCE 
 
 151 
 
 equating the first derivative of / to co, L, or C, as the case 
 may be, in equation (283) to zero. Taking co as the variable 
 factor with R, L, C, and E constants for the circuit in Fig. 
 123: 
 
 Letting C be the variable factor with R, L, co, and E 
 constant: 
 
 1 /I "722 
 
 / = 
 
 (287) 
 
 FIG. 125. Current resonance. Variable u>. For Fig. 123, Equation (286). 
 
 Letting L be variable with R, C, co, and E constants: 
 
 (288) 
 
 f = l - 1 - 
 
 J o \ or r< 
 
 UY 
 
 V C / 
 
 In a similar manner expressions may be obtained for 
 unity power factor frequency and maximum current resonance 
 frequency for co, C or L respectively as the variable with the 
 other factor constants for the circuit in Fig. 127. 
 
152 
 
 ELECTRIC TRANSIENTS 
 
 j = E (g ~ jjb) 
 J == E (G + job) 
 t --= J+ J ---- E Q [(g + G) + j( c b - L b)} 
 
 (289) 
 (290) 
 (291) 
 
 FIG. 126. Vector diagram. Variable C. For Fig. 123, equation (287). 
 
 The total current, /, will be in phase with the impressed 
 voltage, EQ if 
 
 E>9 | 97" 2 
 
 it -J- CO JLJ 
 
 (293) 
 
 Hence, the frequency required to give unity powerfactor 
 for the circuit in Fig. 127 is the same as for Fig. 123. 
 
 L 2 
 
 (294) 
 
 The frequency for maximum current resonance if w is 
 variable while R, L, (7, G and E'o are constant, Figs. 127, 128: 
 
 
 
RESONANCE 153 
 
 If C be the variable, while R, L, G, u and E Q are constant: 
 
 (296) 
 
 1 / 1 ~ R' 2 
 
 L' 2 
 If L be the variable, while R, C, G, co and E^ are constant: 
 
 = 27r\ 2LC + [r 4 + CL 3 4L 2 C 2 
 
 (297) 
 
 o 
 
 o 
 
 L 
 
 G O 
 
 FIG. 127.- -Parallel circuit with leaky condenser. 
 
 In tuning ratio receiver sets resonance is obtained by 
 varying C or L as expressed by equations (296) (297). 
 
 FIG. 128. Vector diagram for circuit in Fig. 127. 
 
 Changes in the inductance by varying the number of turns, 
 also changes the ohmic resistance but the conditions 
 
154 
 
 ELECTRIC TRANSIENTS 
 
 required for equation (297) may be obtained experiment- 
 ally for circuits in which the change in L may be produced 
 by varying the mutual or self-induction between parts of 
 the inductance in circuit. 
 
 The smaller the resistance in the resonating circuit the 
 greater the increase in the resonance current and voltage. 
 Resonance phenomena are of commercial importance only 
 when 'the resistance in circuit is small as compared to the 
 inductance and condensance. 
 
 FIG. 129. Susceptance curves for parallel circuit. 
 
 In most cases and particularly those of greatest impor- 
 tance, the resistance is negligibly small. If R and G are 
 taken equal to zero all the resonance frequency equations 
 (295) to (297) become identical in form. 
 
 Resonance frequency, massed circuit constants (approxi- 
 mate value) : 
 
RESONANCE 155 
 
 > = 2.VLC (298) 
 
 In commercial work equation (298) is in general use, giv- 
 ing with sufficient accuracy the resonance frequency for 
 simple circuits having massed condensance, inductance and 
 resistance. 
 
 For distributed circuit constants, as in long transmission 
 lines, the space distribution of the voltage and current 
 waves must be taken into consideration, the approximate 
 resonance frequency is given by equation (299), as explained 
 in Chap. VI on Transmission Line Oscillations. 
 
 Resonance frequency, uniformly distributed circuit con- 
 stants (approximate value) 
 
 f - 4VLC (299) 
 
 In power circuits resonance conditions must be avoided 
 or the resistance in circuit be sufficiently large to prevent 
 any marked increase due to resonance in the current and 
 voltage. 
 
 Coupled Circuits. Resonance phenomena are of funda- 
 mental importance in the operation of radio communica- 
 tion apparatus. The circuits in commercial use are more 
 complex than the forms discussed above but may be con- 
 sidered as combinations of simple circuits. In general 
 the component simple circuits have certain parts in 
 common. 
 
 The couplings or connections may be made in a number of 
 ways. For two circuit apparatus the coupling is generally 
 made in one of the following ways: 
 
 1. By direct connection across an inductance coil. 
 Direct coupling as in Fig. 130. 
 
 2. By magnetic induction. Inductive or magnetic coup- 
 ling as in Fig. 131. 
 
 3. By dielectric induction. Condensive, capacitative 
 or dielectric coupling as in Fig. 132. 
 
loG 
 
 ELEC TRIG TEA NSIEN TS 
 
 The inductive interaction of the voltages and currents 
 in tAvo resonating coupled circuits and the transfer of the 
 
 PTXRP 1 nRHT^-lf 
 
 M 
 
 FIG. 130. Direct coupling. 
 
 oscillating energy between the primary and secondary 
 circuits are illustrated by the oscillograms in Figs. 133 to 
 
 FIG. 131. Inductive or magnetic coupling. 
 
 138. The oscillations of the energy between the dielectric 
 and magnetic fields of each circuit are combined with a 
 
 FIG. 132. Condensive or dielectric coupling. 
 
 rapid to and fro transfer of the energy between the mag- 
 netically or dielectrically coupled circuits. In Fig. 133 
 the energy was initially stored in the condenser in the pri- 
 
RESONANCE 157 
 
 mary circuit. By closing the switch oscillations are set up 
 between the dielectric and magnetic fields in both the 
 primary and secondary circuits, and these are combined 
 with a rapid to and fro transfer of the energy between the 
 two circuits. The oscillogram shows that the frequency 
 of oscillation between the magnetic and dielectric fields in 
 both the primary and secondary was 790 cycles per second, 
 while the frequency of transfer between the circuits was 
 approximately 99 cycles per second. That is, the time 
 required for the transfer of the energy from the primary to 
 the secondary through the magnetic coupling and back 
 again was approximately equal to eight complete oscilla- 
 tions between the magnetic and dielectric fields of either the 
 primary or the secondary circuits. The oscillations 
 decrease in magnitude due to the Ri 2 losses and practically 
 all of the energy was dissipated into heat in ^ of a second. 
 
 For the oscillogram in Fig. 134 the primary circuit was 
 opened at the instant all the energy had been transferred 
 from the primary to the secondary circuit, thus preventing 
 its return to the primary circuit. Hence the secondary 
 continues to oscillate until all the energy has been dissi- 
 pated as heat by the Ri 2 losses. 
 
 The oscillogram in Fig. 135 shows the starting oscillatory 
 transient of two inductively coupled circuits when an 
 alternating current of resonance frequency is impressed on 
 the primary. Similar oscillograms showing the oscillatory 
 transfer of energy between the primary and secondary of 
 dielectrically coupled circuits are shown in Figs. 136, 137 
 and 138. The difference in form in the three oscillograms 
 is due to change in the degree of coupling as indicated by 
 the quantitative data in each case. 
 
 Coupling Coefficient. In coupled circuits as in Figs. 
 130 and 131, the interaction will depend on what part of 
 the total magnetic flux interlinks both circuits. The degree 
 of coupling which is often termed "loose" or " close, " 
 depending on whether a small or large fraction of the flux 
 interlinks both circuits, is quantitatively expressed [by 
 
158 
 
 ELECTRIC TRANSIENTS 
 
RESONANCE 
 
 159 
 
160 
 
 ELECTRIC TRANSIENTS 
 
 the coupling coefficient. This is defined as the ratio of 
 the mutual reactance to the square root of the product of 
 the primary and secondary circuit reactances. 
 
 FIG. 135. Transient oscillations. Inductive or magnetic coupling. Resonant 
 
 charge. 
 
 Impressed frequency = 750 cycles; R = 6.5 ohms; L = 0.205 henrys; C = 
 0.2 microfarads; coefficient of coupling = 11 percent; timing wave 100 cycles; 
 natural frequency 790 cycles when K = 0. 
 
 Inductive coupling coefficient, Fig. 131: 
 a m M 
 
 (300) 
 
 M = mutual inductance 
 
 L^ = inductance of primary with the secondary open 
 
 or removed 
 L 2 = inductance of secondary with the primary open 
 
 or removed. 
 Condensive coupling coefficient, Fig. 132: 
 
 cX, 
 
 vc 
 c, 
 
 C 1 I 
 
 = V(C. 
 
 c r 
 
 \j a\* / 
 
 (C. + C.) (C. + C 
 C m = condensance in common condenser 
 
 -, (301) 
 
RESONANCE 
 
 161 
 
 , 
 
 II 
 
 0> 
 
 I! 
 
 o3 o 
 
 11 
 
162 
 
 ELECTRIC TRANSIENTS 
 
 O 
 
 d 
 
 I 
 
 O 
 
 S b 
 
 I" 
 
 II 
 
 g 3 
 
 2 2 
 fl 
 
RESONANCE 
 
 163 
 
 s 
 -i* 
 
 "Eg 
 
 s 
 
 -^ o 
 
 i s 
 
 ^ 
 
1 64 ELECTRIC TRANSIENTS 
 
 C a = condensance in primary circuit 
 Ci = condensance in secondary circuit 
 
 C C 
 
 d = -^j ~ m r - = total condensance in primary 
 
 C o ~\~ C m 
 
 C C 
 Cz = ~/V = total condensance in secondary. 
 
 Cb ~\- (j m 
 
 Multiplex Resonance. In complex circuits or series of 
 double energy loops the conditions for resonance may be 
 satisfied for more than one frequency of the impressed 
 voltage. The degrees of freedom, or the number of fre- 
 quencies at which resonance may occur, depends on the 
 number and interconnection of the elemental double 
 energy circuits in the system. Thus, a transmission line 
 having uniformly distributed R, L, G and C, and hence to 
 be considered as consisting of an infinite series of infinitesi- 
 mal double energy circuits, would resonate for the funda- 
 mental frequency of the line as a unit and for any multiple 
 or harmonic of the fundamental frequency. As the line 
 constants are not perfectly constant and the distribution 
 of R, L, G and C not quite uniform, resonance is limited to 
 the fundamental and a few of the lower harmonics. 
 
 Resonance Growth and Decay. As stated in the begin- 
 ning of this chapter resonance in electric circuits implies a 
 forced oscillation of energy between magnetic and dielectric 
 fields, at such frequencies of the impressed voltage as to make 
 the total impedance or admittance a minimum. To supply 
 the resonating circuit with the oscillatory energy necessitates 
 a transient starting period during which the amplitude of 
 each oscillation is greater than the one preceding. For 
 systems having constant finite circuit constants in which 
 the resonance phenomena reach permanent values, the 
 growth of the transient follows the exponential law. This 
 increase in the magnitude of the oscillations during the 
 starting period is illustrated by the oscillograms in Figs. 
 139 and 140. In these oscillograms the power supply was 
 cut off when the resonance had reached the permanent 
 stage. The decay parts of the oscillograms in Figs. 139 and 
 
RESONANCE 
 
 165 
 
 B^ 
 
 03 O) 
 
 a a 
 
 a T2 
 
 c3 to 
 
 I 
 
166 
 
 ELECTRIC TRANSIENTS 
 
 FIG. 140. Resonance in high speed signaling. 
 
 R = 10 ohms; L = 89 millihenrys; C = 0.25 microfarads; timing wave 100 
 cycles; frequency = 1070 cycles; decrement = 0.052. 
 
 FIG. 141. Resonance limited by spark gap discharge. 
 
 R = 15 ohms; L = 89 millihenrys; C = 0.25 microfarads; timing wave 100 
 cycles; frequency = 1070 cycles; decrement = 0.079. 
 
RESONANCE 167 
 
 140, represent, therefore, free oscillations with a decrease in 
 amplitude as the electric energy is dissipated into heat. 
 
 In Fig. 141 the starting period is of the same form as in 
 Fig. 139 or 140, but not the decay stage. It is evident from 
 the circuit connections that the decay of the resonating 
 currents or voltages will differ in shape depending at what 
 instant in the cycle the short circuit occurs. The oscillo- 
 gram in Fig. 141, for which the short circuit was produced 
 by spark-over, occurred near the maximum point of the 
 voltage wave with practically all of the oscillating energy 
 initially stored in the dielectric field of the condenser. 
 
 Problems and Experiments 
 
 1. Take oscillograms showing the transients accompanying the growth 
 and decay of cumulative resonance in circuits similar to Figs. 139, 140 
 and 141. 
 
 2. Take oscillograms of the transient oscillations of two inductively 
 coupled circuits similar to Figs. 133, 134 and 135. 
 
 3. Take oscillograms of the transient oscillations in two dielectrically 
 coupled circuits similar to Figs. 136, 137 and 138. 
 
CHAPTER IX 
 OSCILLOGRAMS 
 
 In the preceding chapters the fundamental principles 
 of electric transient phenomena are illustrated by a number 
 of oscillograms, many of which the student should repro- 
 duce in order to gain the necessary appreciation of the 
 quantitative value of the factors involved. However, the 
 laboratory work in the course should not be restricted to 
 the reproduction of oscillograms appearing in the text for 
 which quantitative data are provided, or to the taking of 
 other oscillograms that merely illustrate the fundamental 
 principles. For while the gaining of clear concepts of the 
 basic laws of transient electric phenomena is of primary 
 importance, training in applying the principles to practical 
 engineering problems is likewise an essential part of the 
 work. Ample material for this purpose is available in all 
 electrical engineering laboratories. The oscillograms in this 
 chapter, Figs. 142 to 161, which were selected from the labo- 
 ratory reports of students in the introductory course in 
 electric transients, may be taken as typical examples. The 
 students were required to outline the problem, to select the 
 necessary apparatus and instruments, to make preliminary 
 calculations and to predict the form and shape of the 
 transients to be recorded. They made all the adjustments 
 on the oscillograph, obtained experimentally the recorded 
 quantitative data, took the oscillograms, developed the 
 films and prepared a report on the transients photographic- 
 ally recorded by the oscillograph. Each oscillogram repre- 
 sents a separate problem to be analyzed on the basis of the 
 principles discussed in the preceding chapters. 
 
 168 
 
OSCILLOGRAMS 
 
 169 
 
 . 
 
 2 a 
 
 . 
 
 i^ 
 
170 
 
 ELECTRIC TRANSIENTS 
 
 C 
 O > 
 
CSCILLCGRAMS 
 
 171 
 
172 
 
 ELECTRIC -TRANSIENTS 
 
 FIG. 145. T. A. regulator operating transients. 
 
 Fi = exciter field current; V* = alternator field current; Va = alternator 
 terminals. 
 
 FIG. 146. Undamped oscillograph vibrator oscillations. 
 
 Vi = timing wave, 100 cycles; Vz = Oscillations of undamped oscillograph 
 vibrator superimposed on tungsten lamp starting transient. Vz = starting 
 transient (vibrator damped) of tungsten lamp, imperfect contact. 
 
OSCILLOGRAMS 
 
 173 
 
174 
 
 ELECTRIC TRANSIENTS 
 
OSCILLOGRAMS 
 
 175 
 
 "C o 
 
 c o 
 
 
 ^ 
 
 1? 
 
 II 
 
 11 1 
 
 03 
 
 :! 
 
 is, 
 
 s 
 
176 
 
 ELECTRIC TRANSIENTS 
 
 FIG. 150. Current transformer transients. 
 
 Vi = secondary current; Vz = secondary voltages; Va = primary current; 
 primary / = 60 amps.; secondary / = 3.5 amps.; core undersaturated before 
 transient. 
 
 FIG. 151. Single phase short circuit on a two-phase alternator. 
 Open phase voltage = 605 volts; short circuit current = 23 amps.; E, field = 
 500 volts; I, field = 3.25 amps.; frequency = 60 cycles; Vi = open phase voltage; 
 Vz = short circuit current; V = field current; brushes sparking. 
 
OSCILLOGRAMS 
 
 477 
 
 te -j 
 ^ s 
 
 J o 
 
 ll 
 
 >> 
 
 Cj H^ 
 
 -+ 
 
 8 g 
 
 
 a 
 
 O 03 
 
 12 
 
178 
 
 ELECTRIC TRANSIENTS 
 
 ii 
 
 CD S 
 
 03 o 
 
 -as 
 
 = 
 * a 
 
 C 3 
 
 Ss 
 3 
 
 Si 
 
 o & 
 
 ^ s 
 
OSCILLOGRAMS 
 
 179 
 
 ^.-a 
 is 5 
 
 5 II 
 
 fl T3 
 O (3 
 
 an 
 C3 
 O <N 
 
 il 
 
180 
 
 ELECTRIC TRANSIENTS 
 
 It- 
 
 So 
 
OSCILLOGRAMS 
 
 181 
 
 3c 
 
182 
 
 ELECTRIC TRANSIENTS 
 
 ll 
 
 o 
 
 O (H 
 
 S g 
 
OSCILLOGRAMS 
 
 183 
 
 tif 
 
 o g 
 
 H 
 
 is a 
 
184 
 
 ELECTRIC TRANSIENTS 
 
OSCILLOGRAMS 
 
 185 
 
 N 
 
 35 
 
 O o 
 
 ga 
 
 o ft 
 
 8 a 
 a * 
 
186 
 
 ELECTRIC TRANSIENTS 
 
 
 12 
 
OSCILLOGRAMS 187 
 
 Problems and Experiments 
 
 1. Take oscillograms of a number of transients in circuits of the types 
 shown in this chapter. In each case obtain quantitative data and pre- 
 pare'a report giving an explanation of the transients appearing in the oscillo- 
 gram based on the fundamental principles of transient electric phenomena. 
 
 2. Find several electric transients in the laboratory under different 
 circuit conditions from those described in the book. For each case draw 
 diagrams of the proposed circuit connections showing the location of the 
 vibrators; make preliminary calculations as to the amount of resistance 
 required in each vibrator circuit; the most desirable speed of the film drum, 
 etc., to give a well proportioned oscillogram; take the oscillogram; record 
 the quantitative, data; develop the film and make prints. Compare the 
 predicted forms of the curves with the photographic record and check the 
 preliminary calculations with the final circuit data. Prepare a report on 
 the transients recorded on the oscillogram. 
 
APPENDIX 
 
 Developing and Printing Oscillograms. The finished 
 oscillogram, even if perfect electrically, is often disappoint- 
 ing photographically. Care and cleanliness in the manipu- 
 lation of the photographic film and printing paper will reduce 
 these failures to a negligible quantity. 
 
 Starting with the unexposed film, the photographic proc- 
 ess will be traced to the completed print, ready for the files. 
 Cleanliness is essential. During no part of the process 
 should the hands come in contact with the sensitized side 
 of the negative. In order to accomplish this, the film and 
 its black protecting paper should be placed on the drum as 
 a unit, with the black paper on the outside. After the film 
 and paper have been adjusted to the proper position, the 
 paper may be removed from the drum. In this way the 
 hands have not touched the surface of the film. 
 
 Unlike most photographic work, the permissible time of 
 exposure for oscillograms is limited, especially in high speed 
 work. Stray light of any nature is injurious. For this 
 reason it is highly desirable to load the film-holders in 
 complete darkness and to develop for the first two or three 
 minutes without even the ruby light. After a little practice 
 the student will have no trouble in working without the 
 darkroom light. 
 
 Any metol-hydrochinon film developer may be used with 
 varying degrees of success. Where only a few T negatives 
 are made at odd times, Eastman's " Special" developer is 
 satisfactory. This developer will give better results if 
 some of the used developer be added to the fresh solution. 
 In our laboratories the following stock solution is used: 
 water 64 oz., metol one drachm, hydrochinon one-half oz., 
 sodium sulphite 2 oz., sodium carbonate 3 oz., potassium 
 bromide 30 grains. This stock solution is diluted in the 
 proportion of two parts stock solution to one part water. 
 
 188 
 
APPENDIX 189 
 
 It is very important that the developer be used at a tem- 
 perature of 65 deg. F. The hydrochinon is inactive at lower 
 temperatures, resulting in slow development and a flat 
 negative which lacks density and contrast. If used at a 
 higher temperature, the negative will gain density rapidly 
 but will be lacking in contrast and show a decided tendency 
 to fog in the unexposed portions. 
 
 The exposed negative should be given maximum develop- 
 ment possible without fogging the unexposed portions. 
 The image should be allowed to develop until it appears 
 quite definite on the reverse side of the negative. A good 
 rule to follow is to develop until by comparison with the 
 back of the negative, the sensitized side appears quite 
 gray. The gray tone will disappear in the fixing bath and 
 further development is detrimental. 
 
 Care should be taken to fix and wash the negatives prop- 
 erly. The film should be left in the standard fixing bath 
 at least five minutes longer than is necessary to dissolve the 
 last visible trace of un-reduced silver salts. After careful 
 fixing, the film should be washed for at least twenty minutes 
 in running water. It is desirable to rinse off the surface 
 with a tuft of cotton before hanging up to dry. The hurry 
 which often comes in the completion of the day's work in 
 the laboratory, results in haste in the darkroom. If the 
 fixing and washing processes are slighted, the film, though 
 apparently good at the time, becomes worthless in a few 
 months on account of staining. 
 
 The same developer may be used for the printing paper, 
 except that it should be always mixed fresh just before 
 using. The best results are obtained by following the 
 printed instructions accompanying the photographic paper. 
 
 In order to get the maximum contrast in the finished 
 print, it is necessary to use the most contrasting photo- 
 graphic paper. The paper which has proven the best is 
 the Eastman "Azo," grade No. 4, glossy, although others 
 may satisfy the individual user. If this is purchased in ten 
 yard rolls, twenty inches wide and cut on a circular saw or 
 
190 APPENDIX 
 
 band-saw to four and one-half inch widths, four small rolls 
 result with a two inch roll left over for use in testing 
 exposure. 
 
 Prints should be given normal exposure so that with 
 normal or full development the background reaches good 
 density without appreciable reduction of the silver in the 
 highlights. As usual, prints should be fixed fifteen minutes 
 in a standard hypo bath and washed for at least twenty-five 
 minutes in running water. The best finish is obtained by 
 drying the prints on ferro-type plates, which imparts high 
 gloss to the surface. 
 
INDEX 
 
 Alternating current transients, 40 
 Alternator field transients, 50, 61 
 Alternators, single-phase, 61-69 
 
 three-phase, 53-60, 69 
 
 two-phase, 71, 176 
 Arcing grounds, 141 
 Armature reactance, 59 
 
 reaction, 58 
 
 transients, 50, 61 
 Artificial electric lines, 101 
 Asymmetrical field transients, 63 
 Attenuation constant, 30 
 
 B 
 
 Breaking field circuit, 138 
 "Bucking broncho," 175 
 
 Capacitance, 7, 9 
 Carbon lamps, 132 
 Circuit breakers, 170 
 
 constants, 101, 130 
 Compound circuits, 126 
 Condensance, 5, 7, 9 
 
 variable, 142 
 Condensive coupling, 156 
 Conductance, 9 
 Corona, 143 
 Coulombs, 9 
 Coupled circuits, 154 
 Coupling coefficient, 155 
 Current resonance, 148 
 
 D 
 
 Damping factor, 92, 115 
 Developing oscillograms, 188 
 Dielectric circuit, 4 
 coupling, 156 
 
 Dielectric field intensity, 9 
 
 flux, 4, 9 
 
 gradient, 9 
 Direct coupling, 156 
 
 current transients, 24 
 Dissipation constant, 30, 92, 115 
 Distance angle, 116 
 Double energy transients, 21. 75-100 
 
 Elastance, 4, 9 
 Electric circuit, 7 
 
 line oscillations, 101 
 Energy, 8, 9 
 Exponential curve, 32 
 
 law, 24 
 
 Farad, 7, 9 
 
 Faraday's lines of force, 2 
 Forming magnetic field, 27, 139 
 Frequency, distributed R, L, G 
 
 and C, 110 
 
 massed R, L, G and C, 111 
 Frohlich's formula, 135 
 
 Galvanometers, 14 
 Generator field transient, 138 
 Gilbert, 9 
 
 II 
 
 Henry, 4, 9 
 
 High frequency signalling, 166 
 
 Impedance, 9 
 Impulses, 124 
 
 191 
 
192 
 
 INDEX . 
 
 Inductance, 3, 9 
 
 variable, 135 
 
 Induction motors, 20, 48, 177, 181 
 Initial values, 33 
 Intermittent arcs, 141 
 
 Joule, 8, 9 
 
 Permittivity, 4, 6 
 Phase angle, 103 
 Polyphase short circuit, 50 
 Power surges, 114 
 
 transfer factor, 123 
 Printing oscillograms, 189 
 Pulsating condensance, 142 
 
 inductance, 135 
 
 resistance, 133 
 
 K 
 Kirchoff's Laws, 79, 83 
 
 L 
 
 Leaky condenser, 83, 104 
 Length of line, 112 
 Lifting magnet transient, 169 
 Line constants, 101 
 oscillations, 101 
 Lumpy line, 101 
 
 R 
 
 Reactance curves, 149, 154 
 Reluctance, 3, 9 
 Repulsion induction motor, 178 
 Resonance, 145 
 "Resonance" frequency, 78 
 growth and decay, 164 
 Resistance, 9, 130 
 Resistivity, 9 
 
 Rotary converter, 182-187 
 Rotating magnetic field, 48 
 
 M 
 
 Magnetic circuit, 2 
 coupling, 156 
 field intensity, 3, 9 
 flux, 2, 9 
 
 Microfarads, 7 
 
 Multiplex resonance, 164 
 
 N 
 
 Natural admittance, 77 
 impedance, 77 
 period of oscillation, 108 
 
 Oersted, 9 
 Ohm's law, 2, 4, 9 
 Oscillator alternator, 17 
 Oscillatory circuits, 79 
 Oscillograms, 20 
 Oscillographs, 10 
 
 Permeability, 3, 9 
 Permittance, 9 
 
 Series generator, 173 
 Sharpness of resonance, 147 
 Short circuits, polyphase, 50 
 
 single-phase, 61 
 Single energy transients, a.-c., 40 
 
 d.-c., 23 
 
 Space angles, 103, 116 
 Split-phase motor, 178 
 Standing waves, 118 
 Surge admittance, 77 
 
 impedance, 77, 113 
 Susceptance curves, 154 
 Synchronous reactance, 58 
 
 T. A. regulator, 172 
 "T" circuits, 104 
 Three-phase transients, 44 
 Time angles, 103, 116 
 
 constant, 28 
 Timing waves, 17 
 "T" line, 102 
 Transformers, 135, 176 
 
INDEX 
 
 Transmission line, artificial, 101 Units, 9 
 
 constants, 101 V 
 
 equations, 114 
 
 oscillations, 101 Variable circuit constants, 130 
 
 Traveling waves, 116 Velocity unit of length, 113 
 
 Tungsten lamp, 1'31 Voltage resonance, 145 
 
 U W 
 
 Undamped vibrator, 172 Watt, 8 
 
 193 
 
7 DAY USE 
 
 RETURN TO DESK FROM WHICH BORROWED 
 
 PHYSICS LIBRAKT 
 
 This publication is due on the LAST DATE 
 stamped below. 
 
 General I 
 University o 
 
 Berkele, 
 
 RB 17-60m-12,'57 
 (703slO)4188 
 
995868 
 
 THE UNIVERSITY OF CALIFORNIA LIBRARY