THE APPLICATION OF HYPERBOLIC FUNCTIONS TO ELECTRICAL ENGINEERING PROBLEMS THE OF HYPERBOLIC FUNCTIONS TO ELECTRICAL ENGINEERING PROBLEMS BEING THE SUBJECT OF A COURSE OF LECTURES DELIVERED BEFORE THE UNIVERSITY OF LONDON IN MAY AND JUNE 1911 BY A. E. KENNELLY, M.A., D.Sc. PROFESSOR OF ELECTRICAL ENGINEERING AT HARVARD UNIVERSITY lonfrm: Ittnfversft? of Xon&on preas PUBLISHED FOR THE UNIVERSITY OF LONDON PRESS, LTD. BY HODDER & STOUGHTON, WARWICK SQUARE, E.G. 1912 HODDER AND STOUGHTON PUBLISHERS TO THE UNIVERSITY OF LONDON PRESS s\ & y o $ f 1 PREFACE HYPERBOLIC functions have numerous, well recognized uses in applied science, particularly in the theory of charts (Mercator's projection), and in mechanics (strains). But it is only within recent years that their applications to electrical engineering have become evident. Wherever a line, or series of lines, of uniform linear constants is met with, an immediate field of usefulness for hyperbolic functions presents itself, particularly in high-frequency alternating-current lines. The following pages are intended to cover the scope and purport of five lectures given for the University of London, at The Institution of Electrical Engineers, Victoria Embankment, by kind permission of the Council May 29 to June 2, 1911, bearing the same title as this book. The central ideas around which those lectures, and this presentation, have been framed are (1) That the engineering quantitative theories of continuous- currents and of alternating-currents are essentially one and the same ; all continuous-current formulas for voltage, current, resistance, power and energy being applicable to alternating- current circuits, when complex numbers are substituted for real numbers. Thus there appears to be only one continuous-current formula in this book (277) which is uninterpretable vectorially in alternating-current terms ; namely, as shown in Appendix J, that which deals with the mechanical forces developed in a telegraph receiving instrument, such forces being essentially "real" and not complex quantities. (2) That there is a proper analogy between circular and hyperbolic trigonometry, which permits of the extension of the notion of an " angle " from the circular to the hyperbolic sector. The conception of the "hyperbolic angle" of a continuous- vi PREFACE current line is useful and illuminating, leading immediately, in two-dimensional arithmetic, to an easy comprehension of alternating-current lines. The subject, which is very large, very useful, and very beautiful, is only outlined in the following pages. There are many directions in which accurate and painstaking research is j needed, in the laboratory, the factory, and the field. Fortunately there are already a number of workers in this field, and good progress is, therefore, to be looked for. It is earnestly hoped that this book may serve as an additional incentive to such research. The author desires to acknowledge his indebtedness to the writings of Heaviside, Kelvin, J. A. Fleming, C. P. Steinmetz, and many others. A necessarily imperfect bibliography of the subject, in order of date, is offered in an Appendix. He is also indebted to the Engineering Departments of the British Post Office, the National Telephone Company and Mr. B. S. Cohen, the Eastern Telegraph Company and Mr. Walter Judd, also the American Telegraph and Telephone Company and Dr. F. B. ; Jewett, for data and information ; likewise to Mr. Robert Herne, Superintendent of the Commercial Cable Company, in Rockport. Massachusetts, for kind assistance in obtaining measured cable signals. He also has to thank Professor John Perry, Professor Silvanus P. Thompson, and Mr. W. Duddell for valued sug- gestions. In particular, he is indebted to the great help and courtesy of Dr. R. Mullineux Walmsley, in the presentation of the lectures, and in the publication of this volume. Although care has been taken to secure accuracy in the mathematics, yet errors, by oversight, may have crept in. If any should be detected by the reader, the author will be grateful for criticisms or suggestions. A. E. K. Cambridge, Mass. (U.S.A.), December 1911. TABLE OF CONTENTS CHAP. FAOB I ANGLES IN CIRCULAR AND HYPERBOLIC TRIGONO- METRY . 1 II APPLICATIONS OF HYPERBOLIC FUNCTIONS TO CONTI- NUOUS-CURRENT LINES OF UNIFORM RESISTANCE AND LEAKANCE IN THE' STEADY STATE . . 10 III EQUIVALENT CIRCUITS OF CONDUCTING LINES IN THE STEADY STATE . . ... . .28 IV REGULARLY LOADED UNIFORM LINES . . . 42 V COMPLEX QUANTITIES - . . . . . .49 VI THE PROCESS OF BUILDING UP THE POTENTIAL AND CURRENT DISTRIBUTION IN A SIMPLE UNIFORM ALTERNATING-CURRENT LINE . ... .69 VII THE APPLICATION OF HYPERBOLIC FUNCTIONS TO ALTERNATING - CURRENT POWER - TRANSMISSION LINES . . . .... . .86 VIII THE APPLICATION OF HYPERBOLIC FUNCTIONS TO WIRE TELEPHONY . . . . . .112 IX THE APPLICATION OF HYPERBOLIC FUNCTIONS TO WIRE TELEGRAPHY ...... 179 X MISCELLANEOUS APPLICATIONS OF HYPERBOLIC FUNCTIONS TO ELECTRICAL ENGINEERING PROB- LEMS . . 202 APPENDIX A Transformation of Circular into Hyperbolic Trigono- metrical Formulas . . . .' 213 vii viii CONTENTS APPENDIX B PAGE Short List of Important Trigonometrical Formulas showing the Hyperbolic and Circular Equivalents 215 APPENDIX C Fundamental Relations of Voltage and Current at any Point along a Uniform Line in the Steady State 216 APPENDIX D Algebraic Proof of Equivalence between a Uniform Line and its T Conductor, both at the Sending and Receiving Ends . 220 APPENDIX E Equivalence of a Line H and a Line T 222 APPENDIX F Analysis of Artificial Lines in Terms of Continued Fractions ........ 225 APPENDIX G A Brief Method of Deriving Campbell's Formula . 240 APPENDIX H Analysis of the Influence of Additional Distributed Lecikance on a Loaded as compared with an Unloaded Line ....... 244 APPENDIX J To find the Best Resistance of an Electromagnetic Receiving Instrument employed on a Long Alternating-Current Circuit .... 245 CONTENTS ix APPENDIX K PAOB On the Identity of the Instrument Receiving-end Im- pedance of a Duplex Submarine Cable, whether the Apex of the Duplex Bridge is Freed or Grounded 248 APPENDIX L To Demonstrate the Proposition of Formula (7), page 4 250 List of Symbols employed and tlieir Brief Definitions 253 Bibliography , . . 266 INDEX 275 THE APPLICATION OF HYPERBOLIC FUNCTIONS TO ELECTRICAL ENGINEERING PROBLEMS CHAPTER I ANGLES IN CIRCULAR AND HYPERBOLIC TRIGONOMETRY Generation of Circular Angles. If we plot to Cartesian co-ordinates the locus of y ordinates for varying values of & abscissas in the equation y 2 + x 2 = 1 . . (cm., or units of length) 2 (1) we obtain the familiar graph of a circle, as indicated in Fig. 1 ; where O is both the origin of co-ordinates x, y, and the centre of the portion of a circle /'A#. The radius OA, on the axis of abscissas, is taken as of unit length. As x diminishes from + 1 to 0, y increases from to +1, and the radius-vector OE moves its terminal E over the circular arc AE^. At any position such as OE, the tangent E/ to the path of the moving terminal is perpendicular to the radius- vector. As the radius- vector rotates * about the centre O, it describes a circular sector AOE and a circular angle, /? = AOE. The magnitude of this -circular angle may be defined in either of two ways, namely (1) By the ratio of the circular arc length s described, during the motion, by the terminal E, to the length p of the radius- vector ; * Only the positive root of equation (1) is here considered, with the corresponding positive or counter-clockwise rotation of the -radius-vector. In what follows, a hyperbola may be understood to be in all cases a rectangular hyperbola. 1 B 2 APPLICATION ; OF HYPERBOLIC FUNCTIONS (2) By the area of the circular sector AOE swept out by the radius- vector during the motion. Generation of Hyperbolic Angles. If we plot to Cartesian co-ordinates the locus of y ordinates for varying values of x- abscissas in the equation x - ' . (cm., or units of length) 2 (2) 2c x - %. * r / we obtain the familiar graph of a rectangular hyperbola, as- indicated in Fig. 2 ; where O is both the origin of co-ordinates. s FIG. 1. Circular Sector and Circular Functions. FIG. 2. Hyperbolic Sector and Hyperbolic Functions. x, y, and the centre of the hyperbola branch /'A/. The radius or semi-axis OA, on the axis of abscissas, is taken as of unit length. As # increases from 1 to oc , y increases from to oc , and the radius-vector OE moves its terminal E over the hyperbolic arc AE/. At any position, such as OE, at which the radius-vector makes a circular angle /? with the X axis, the tangent E/ to the path of the moving terminal makes a cir- cular angle /? with the Y axis ; or a circular angle of 2/5 with a perpendicular to the radius-vector. As the radius-vector TO ELECTRICAL ENGINEERING PROBLEMS 3 rotates about the centre O, it describes * a hyperbolic sector AOE, a circular angle /?=AOE, and also a hyperbolic angle AOE. The magnitude of this hyperbolic angle may be defined in either of two ways, namely (n) By the ratio of the hyperbolic arc distance s described. during the motion, by the terminal E, to the length p of the radius-vector ; (2) By the area of the hyperbolic sector AOE swept out by the radius-vector during the motion. [ Algebraic Definition of any Angle, Circular w Hyperbolic. In the circular locus AE# of Fig. 1, or in the hyperbolic locus AE/ of Fig. 2, let the rotating radius- vector OE generate at any time an element of arc of length ds= ,J(dyyt + (dx^ ..... cm. (3) and let p = be the corresponding instantaneous value of the radius-vector length. Then the element of angle described during the motion will be djj = d6 = - circular or hyperbolic radian (numeric) (4) That is, the element of angle described in the circular locus of Fig. 1 will be a circular angle element d/3, and will be express- ible in units of circular radians; while the element of angle described in the hyperbolic locus of Fig. 2 will be a hyperbolic angle element dO, and will be expressible in units of hyperbolic As the motion proceeds in Figs. 1 and 2 from an initial to a final position of the radius-vector, the total angle described during the motion will be /? = =f circular or hyperbolic radians (5) In the case of the circular locus of Fig. 1, the radius- * Only the positive root of equation (2) is here considered, with the corresponding positive or counter-clockwise rotation of the radius-vector. B2 4 APPLICATION OF HYPERBOLIC FUNCTIONS vector p is a constant, and equal, by assumption, to unity; consequently, equation (5) becomes for the circular case *2 ft / y- = s circular radians (numeric) * (6) SI where s is the length of the circular arc described between the limits ^ and s 2 , while ft is the corresponding angle in circular radians. In the case of the hyperbolic locus of Fig. 2, the radius- vector p varies. Consequently, equation (5) becomes for the hyperbolic case S2 /ds s - = , hyperbolic radians (numeric) f (7) s i where s is the length of the hyperbolic arc described between the limits s l and s 2 ; while p' is the integrated mean value o^ p during the motion, as defined by (7), and 6 is the corre^ spending angle in hyperbolic radians. Anyles in Terms of Sector Area. In the circular sector of Fig. 1, or the hyperbolic sector of Fig. 2, the magnitude of the angle described by the radius-vector OE, between an initial and a final position, is numerically twice the area of the sector swept out by the radius-vector during the motion. Thus in Fig. 1, with the radius OA 1 cm., if the radius-vector describes the heavy arc A, &, c, d, E, then the circular angle ft will be, in circular radians, double the sector area AOE in sq. cm. ; or will be equal to the shaded double-sector area EOE' in sq. cm. Similarly, in Fig. 2, with the radius OA = 1 cm., if the radius-vector describes the heavy arc A, b, c, d, E, then d will be, in hyperbolic radians, double the sector area AOE in sq. cm.; or will be equal to the shaded double-sector area EOE', in sq. cm. In Fig. 1, the circular angle AOE = ft is * The dimensions of all angles, whether circular or hyperbolic, are assumed in this discussion to be zero ; so that an angle is accepted as a numerical quantity, notwithstanding the fact that the arc ds has a different direction in the Cartesian plane from that of the radius-vector p. f See Appendix L. TO ELECTRICAL ENGINEERING PROBLEMS 5 represented as 1 circular radian, and the double -shaded sector area is 1 sq. cm. if OA = 1 cm. Similarly, in Fig. 2, the hyperbolic angle AOE = 6 is represented as 1 hyperbolic V" X?. o- X V X \ X X X . V _ i-o FIG. 3. A Circular Angle of 1 circular radian, in five sections of 0*2 radian each, expressed as ft J- radian, and the double-shaded sector area is 1 sq. cm. if OA = 1 cm. It is evident that the hyperbolic angle 6 of the sector AOE in Fig. 2 must be carefully distinguished from the cir- cular angle /? of the same sector. In the case represented 6=1 hyperbolic radian ; whereas ft = 0'65087 circular radian (37 17' 33"). 6 APPLICATION OF HYPERBOLIC FUNCTIONS The preceding algebraic relations between arc and radius- vector ratios of circular and hyperbolic angles are illustrated in greater detail by Figs. 3 and 4. In Fig. 3 each of the circular arcs AB, BC, CD, DE, EF possesses a length of 0'2, if the radius OA be taken as of unit length. Consequently, each of the circular angles in the sectors AOB, BOC, COD, FIG. 4. A Hyperbolic Angle of 1 hyperbolic radian, in five sections of 0'2 radian each, expressed as 8 =./". P DOE, and EOF is 0-2} circular radian. The total circular angle AOF of the sector AOF is thus 1 circular radian. In Fig. 4 each of the hyperbolic segments AOB, BOC, COD, DOE, and EOF contains a hyperbolic angle of 0'2 hyperbolic radian, the length of the arcs AB, BC, CD, DE and EF, increas- ing as the hyperbolic angle increases, and also the lengths of the integrated mean radii-vectores p which are indicated in Fig. 4 for each sector. Consequently, the total hyperbolic angle of the sector AOF is 1 hyperbolic radian, the arc ABCDEF having a total length of 1'3167 units, if the radius OA be taken TO ELECTRICAL ENGINEERING PROBLEMS 7 as of unit length. The integrated mean radius-vector p for the total hyperbolic angle of the sector AOF intersects the curve at /. For brevity, we may use the term " hyp." as an abbreviation for the unit hyperbolic radian ; so that in Fig. 4 we may say that each of the sectors contains, and each of the arcs subtends, a hyperbolic angle of 0*2 hyp.; while the total sector AOF contains, and the arc ABCDEF subtends, a hyperbolic angle of 1 hyp. Hyperbolic angles and hyperbolic trigonometry are of great importance in the theory of electric conductors as used in electric engineering. TRIGONOMETRIC FUNCTIONS OF CIRCULAR AND HYPERBOLIC ANGLES. Trigonometry recognizes certain functions or ratios of lengths in connection with circular and hyperbolic angles. If we retain the initial radius as of unit length, the ratios become simplified into the numerical lengths of certain straight lines. In Figs. 1 and 2, XE is the sine, OX is the cosine, and At the tangent of the angle of the sector, circular or hyperbolic.* It is evident that when the angle is very small, both the hyperbolic and circular sines are likewise very small; the hyperbolic and circular tangents are likewise very small, while the hyperbolic and circular cosines are very nearly unity. As the angle increases through many radians, the circular sine periodically fluctuates between the limits + 1 and 1, while_ the hyperbolic sine increases steadily from to cc . The circular cosine periodically fluctuates between + 1 an d 1, while thehyperbolic cosine increases^gteadilj from 1 to oc . The circular tangenT~perTodically fluctuates disconlmuously between -f- oc and oc , while the hyperbolic tangent steadily * Reference is here made only to the numerical lengths of these functions ; and their proper direction in the plane, real or imaginary, is ignored. 8 APPLICATION OF HYPERBOLIC FUNCTIONS increases from to 1. Fig. 5 shows the graphs of the hyper- bolic functions for the first few hyps., the hyperbolic angle FIG. 5. Diagram showing the graphs of the sine, cosine, tangent, cotangent, secant and cosecant of a hyperbolic angle. Ordinates :- Numerical value of the trigonometrical function Abscissas :- Numerical value of the hyperbolic angle, HYPERBOLIC ANGLE 0.9 I 1-5 2 2.5 3 3.5 4 4.5 HYPS. 5.5 6 6.5 7.5. being marked along the axis of abscissas, and the numerical value of the function along the axis of ordinates. In order to distinguish between hyperbolic and circular TO ELECTRICAL ENGINEERING PROBLEMS 9 functions, the letter h is affixed to the function when the hyperbolic function is denoted ; thus, the sine, cosine, versine, tangent, secant, cosecant, and cotangent of a hyperbolic angle & are respectively indicated by the customary notation sinh 0, cosh 6, versh 6, tanh 0, sech 6, cosech 6, coth 6. By the process described in Appendix A, the standard formulas of circular trigonometry may be readily transformed into corresponding formulas of hyperbolic trigonometry. .Jt will be found that circular function formulas involving only first powers, in general transform into corresponding hyperbolic oiit" change. Thus the formula sin 2/? = 2 sin /? cos {$ . . . numeric (8) transforms directly into sinh 26 = 2 sinh 6 cosh 6 . . (9X But circular function formulas involving squares, or second powers of functions, usually involve one or more changes of sign in hyperbolic transformation. Thus cos 2 ^ + sin 2 = 1 . . . . numeric (10) becomes cosh' 2 - sinh 2 = 1 ..... (11) With this reservation in mind, it is not worth while pre- paring a special list of hyperbolic trigonometric formulas. They may be obtained from the corresponding circular trigono- metric formulas by transformation on inspection. Conse- quently, no appreciable additional mental labour is needed for memorizing formulas when learning to apply hyperbolic trigonometry, after the student has learned to apply circular trigonometry. The formulas already learned with the latter suffice for both. A short list of comparative formulas in circu- lar and hyperbolic trigonometry is given in Appendix B. CHAPTER II APPLICATIONS OF HYPERBOLIC FUNCTIONS TO CON- TINUOUS-CURRENT LINES OF UNIFORM RESIST- ANCE AND LEAKANCE IN THE STEADY STATE Perfectly Insulated Lines. Rectilinear Graphs. Let us first consider a uniform conducting line such as a telegraph line L kilometers long, but perfectly insulated from the ground and from all other conductors. Such a line will have a uniform linear conductor-resistance of r ohms per km., and its total conductor-resistance will be Lr ohms ; but it will, by assumption, be devoid of leakance. If we free the distant end B of this line AB, Fig. 6, and apply an e.m.f. of EA volts to the home end A, as by means of the battery shown, it is evident that all parts of the line conductor will take the same electric potential, and the graph of this potential as ordinates, to distance along the line as abscissas, will be the straight line AB parallel to the axis of abscissas. Again, if we ground the distant end B of the line, as in Figs. 7 and 8, the graph of electric potential will be the inclined straight line AB, Fig. 7, falling from E A volts at A to zero at B ; while at any distance L x km. from A the potential will be e = E A - IL : r volts (12) Moreover, since there is no current leakage along the line, the current strength will be the same at all points, and the current graph will be the straight line AB, Fig. 8, parallel to the axis of abscissas. Similarly, if the line instead of being either freed or grounded at B, is grounded there through some constant resistance, it is evident that the graph of potential along 10 APPLICATION OF HYPERBOLIC FUNCTIONS 11 the line under an impressed e.m.f. E A at A, would still be a straight line, but a straight line of lesser inclination than AB in Fig. 7 ; while the graph of current would still be a straight line parallel to the axis of abscissas, but a straight line lower than AB in Fig. 8. Similar reasoning applies when an e.m.f. is applied at B, either alone, or in conjunction with an e.m.f. at A. A B FIG. 6 CURVE OF POTENTIAL ALONG A PERFECTLY INSULATED LINE, FREE AT THE DISTANT END -d=- CURVE OF POTENTIAL =- ALONG A PERFECTLY INSULATED LINE. -=- GROUNDED AT THE FAR END. FIG. B CURVE OF CURRENT - STRENGTH ALONG A PERFECTLY INSULATED LINE, GROUNDED AT THE FAR END. Consequently we may include all possible conditions under the statement that the graphs of potential and current over any uniform perfectly insulated conductor, in the steady state, are straight lines.* . Lines of Uniform Resistance and Leakance. If now the line, instead of being perfectly insulated, has a uniform linear leakance of y mhos per km. ; then if we free the distant end B, and apply an e.m.f. E A to the home end A, as in Fig. 9, the graph of electric potential along the line will become * The steady state of current How will be assumed to have been established in all cases considered. 12 APPLICATION OF HYPERBOLIC FUNCTIONS a catenary, or curve of hyperbolic cosines, such as AB r and the graph of current along the line will be a catenary- slope curve, or curve of hyperbolic sines, such as the dotted line Ab, Fig. 9. In the case there represented L = 500 km., / = 10 ohms per km., and g = 0*5 x 10~ 6 mho per km., or half a micromho per km. corresponding to a linear insulation-resistance of 2 megohm-kilometers. A 200 150- 100- 50- 0.03- 0.02 0.01 Q. Kilometers. T FIG. 9. Fall of Pressure and Current along a uniformly leaky line, free at the far end. Under these conditions, as shown in Appendix C, we obtain the following fundamental equations for potential and current. With no leakage E A - IA!^- = e = E B + IB V volts (13) I A = i = IB . . . . amperes (14) With uniform leakage EA cosh L x a IA.T O sinh L x a = e EB cosh L 2 a -f IB?' O sinh L 2 a volts (15) TT TT I A cosh LjCt sinh LjCt = i = IB cosh L 2 a + B sinh L 2 a ' o o amperes (16) TO ELECTRICAL ENGINEERING PROBLEMS 13 where E A and I A are the e.m.f. and current at A E B IB B e i some intermedi- ate point, distant L x km. from A, and L 2 km. from B (Fig. 10) also a = *Jrg .... hyp. per km. (17) *o = \ 7 ~ ohms (18) y The constant a is to be considered as a hyperbolic angle subtended by unit length of line. It is called the attenuation- constant of the line. Its dimensions are (= T \ or a numeric Mength/' divided by a length. ( _. FIG. 10. Diagram of Simple Ground-Return Circuit : A, Generator End; B, Motor or Receiving End ; P, Intermediate Point. The constant r is to be considered as a characteristic resistance pertaining to the line. It is the resistance which an indefinitely long line, of the given linear constants r and g, would offer at either end say A, as measured to ground, whether the other end were freed, grounded, or left in any intermediate condition of ground through resistance. It is called the surge-resistance of the line. In the case of the line considered with Fig. 9, the surge-resistance is r = ^/10 X 2,000,000 = 3472 ohms. The attenuation-constant for the case indicated in Fig. 9 is a = ^/10 X 0-5 X 10- 6 = 0-002236 hyp. per km. The physical meaning of the constant is that when an impressed e.m.f. E 14 APPLICATION OF HYPERBOLIC FUNCTIONS volts is applied to one end of a line, which is either indefinitely long, or is grounded at the distant end through a resistance equal to the surge-resistance of the line, the potential, at a distance of one km. from the home end will have fallen from E to Ee~ a volts, where s is the base of Naperian logarithms, or 2'71828. In each and every unit length of line the potential will fall by the factor ~ a . Consequently, after Lj km. the potential will have fallen to Ee~ L i a volts. The factor ~ Lllls = 200 x 0-3269 = 65*38 volts. The normal attenuation-factor for 500 km. of this line is therefore 0'3269. If the line were grounded at B through more or less than r ohms, the attenuation-factor would be greater or less than the normal. Angle subtended ly a Uniform Line. A uniform line pos- sesses, or may be said to subtend, a hyperbolic angle 6 = La = LV^= V'RG . . . hyps. (19) where R Lr is the total conductor-resistance of the line in ohms, and G = L# is the total dielectric conductance of the line in mhos. That is, the angle of the line in hyps, is the geometric mean of the conductor-resistance and dielectric conductance. The angle of a uniform line increases directly with its length. The attenuation-constant a, in overland tele- graph lines, varies between the approximate limits of 10 ~ 5 and 10 ~ 2 hyp. per km., according to the condition of insulation. If we take 1000 km. as the greatest length of telegraph line likely to be operated in a single section, without repeaters, the angle of such a line may vary between the limits of 0*01 and 10 hyps. In practice, however, the line would probably cease to be workable telegraphically when the leakance became TO ELECTRICAL ENGINEERING PROBLEMS 15 sufficiently great to bring the line angle to 4 hyps., for which the normal attenuation-factor is e" 4 or 0*018; so that the received current would be only 1*8 per cent, of the current at the sending end, if the receiving end were grounded through a resistance equal to the surge-resistance of the line. A uniform line is a more efficient transmitter of current from the generating to the receiving end as its line angle is reduced ; although not in simple proportion. Trigonometrical Properties of a Simple Uniform Line in Relation to its Angle. Distant End freed. It is easily shown from equations (15) and (16), substituting the proper terminal values of potential and current, that when a line of angle is freed at the distant end, with a steady e.m.f. E A applied at the home end, the resistance offered by the line at the home end (see Appendix C) is R r = TO coth . . , . . ohms (20) the current entering the line is E F I = A tanh = - 4 o . . amperes (21) r r Q coth and the potential at the distant free end is E B = E A sech = EA a . . . volts (22) cosh u Table I gives the potential at the distant free end of a line as a decimal fraction of the e.m.f. impressed at the home end, for different lengths of line and various attenuation-constants. Thus, with a line of attenuation-constant 0*0025 hyp. per mile or km., and a length of 500 miles or km., the line angle would be 1-25 hyps., and the Table shows that the voltage at the distant free end would be 0'53, or 53 per cent, of the impressed voltage. Distant End grounded. Similarly operating upon the funda- mental formulas (15) (16) we have, with the distant end of the line grounded, the line resistance offered at the home end R^ = r Q tanh ohms (23) 16 APPLICATION OF HYPERBOLIC FUNCTIONS TABLE I POTENTIAL AT DISTANT FREE END, WITH UNIT E.M.F. IMPRESSED ON LINE AT HOME END. CONTINUOUS-CURRENT CASE. a = o-oooi 0-0005 o-ooi 0-0025 0-005 0-0075 50 100 200 0-9999 0-9999 0-9998 0-9997 0-9987 0-9950 0-9988 0-9950 0-9803 0-992 0-970 0-970 0-887 0-887 0-648 0-930 0-772 0-425 300 js 400 t* 500 0-9995 0-9992 0-9988 0-9888 0-9803 0-9695 0-9566 0-9250 0-8868 0-772 0-425 0-648 0-266 0-530 0-163 0-209 0-094 0-048 * 600 700 800 0-9982 0-9975 0-9968 0-9566 0-9416 0-9250 0-8435 0-7967 0-7477 0-425 0-094 0-337 0-060 0-266 0-037 0-022 0-011 0-005 900 1000 0-9960 0-9950 0-9066 0-8868 0-6978 0-6480 0-209 0-022 0-163 0-014 0-002 o-ooi o-oi o- 0-6481 0-2658 0-094 0-037 0-014 0-005 0-002 o-ooi 0-0003 0-0001 The current entering the line at the home end is E A la = - COth 6 = r tanh 6 amperes (24) The current escaping to ground at the distant end is I amperes (25) r sinh 6 So that the line behaves at the distant end as though it had a line resistance r sinh 6 ohms without leakance. This ap- parent resistance of the line, as j udged at the distant grounded end, is called the receiving-end resistance grounded. It is ft, = r o sinh 6 ohms (26) Apparent Home-End Resistance of a Uniform Line. It is evident from formulas (20) and (23) that ohms (27) -R, = y r. =/ y Q - or, that the surge-resistance of a uniformly leaky line is the geometrical mean of its apparent resistances when freed and grounded, respectively, at the distant end. When the line is TO ELECTRICAL ENGINEERING PROBLEMS 17 electrically long, i. e. when 6 is over 2'5 hyps., tanh 6 ascend- ingly approaches unity within less than 0*5 per cent., and coth 6 descendingly approaches unity within less than 0*5 per cent. ; so that the apparent resistance offered by an electrically long line converges to the surge -resistance r , whatever the condition of the distant end. Moreover, dividing (23) by (20) obtain R so that and Consequently, if the apparent resistances R/, R^, of the line are correctly measured at either end, the values of r and a are determined with the aid of tables of hyperbolic functions,* and from these the corrected linear constants of the line are found by the relations r = a/' .... ohms per km. (31) and a = ..... mhos per km. (32) ?*0 Thus, if a telegraph line, when freed at the distant end, was observed to offer a resistance R/=5912 ohms, and when grounded at the distant end, a resistance R^ = 4434 ohms, the surge-resistance of the line would be r = x/5912 x 443^ = \R ; 6 = tanh-'A/f? \K/ .... hyps. (29) 6 a ^ hvp. per km. (30) 5120 ohms, and the angle of the line 6 = tanh - 1 = 0*7 A L. tanh- 1 0-86603 = 1*317 hyps. If the line is known to have^a length L of 800 km., the attenuation-constant, or linear angle, 1'Sl 7 is = 0-001646 hyp. per km. Consequently, the inferred * The best tables of hyperbolic functions of real hyperbolic angles are probably Hyperbolic Functions, by G. F. Becker and C. E. van Orstrand, 18 APPLICATION OF HYPERBOLIC FUNCTIONS true linear resistance of the line is r = 5120 x 0*001646 = 8'428 ohms per km., and the inferred true linear leakance is 0'001646 nnn-n mho /0 . ,. g = = 0'321o X 10- 6 -j_ (3,110,600 ohm-km. linear insulation resistance). The apparent or uncorrected values 4434 would have been r l = -T^TT- = 5'543 ohms per km., and oUU mho -4 KM FIG. 11. Diagram of four kilo- meters of line with 10o> per loop kilometer and 2 megohm-kilometers. FIG. 12. Diagram of four kilo- meters composed of two separate cir- cuits with ground return, each having 5oo per kilometer and 1 megohm- kilometer. The constants a and r are more important in the theory of long uniform electric conductors than the constants r and y. The former may therefore be called the characteristic constants or characteristics of a line; while the latter are the secondary,' constants. Characteristics of Loop-Lines and of Wire-Lines. We have hitherto discussed the characteristic constants a and r o of single-wire lines with ground-return circuit, such as are used in wire telegraphy. We now proceed to discuss the characteristics- of loop-lines such as are used in wire telephony. In Fig. 11 a loop-line or metallic circuit is indicated, 4 km. in length, with an e.m.f. of E /y = 80 volts impressed at the sending end, and a load of o,, = 200 ohms resistance at the receiving end. The linear conductor-resistance is r fl 10 W per loop-km., and the linear insulation-resistance is 2 megohm-kilometers TO ELECTRICAL ENGINEERING PROBLEMS 19 representing a linear dielectric leakance g /t = 0'5 x 10 - 6 mho per loop-km. The same system is represented in Fig. 12 with respect to a symmetrical dividing line of zero electric potential, or neutral plane of ground potential. No change in the distribution of potential, current, resistances, or power would be made by connecting the dividing line to ground, or by separating the two halves of the system, and completing each portion by a perfectly conducting ground return. In each half of Fig. 12, we have then an impressed e.m.f. of E y = E y/ /2 = 40 volts, a load resistance With B grounded, UP = U A ...... volts (40) ... cosh 6 < 42 > Particular Case of Very Short Lines. Approximate Formulas. When a uniform conducting line is electrically very short, i. e. when its angle 6 is very small, say not exceeding 0*1 hyp. t we may without much error substitute 6 for sinh 6 1 cosh 6 6 tanh 6 TJ cosech 6 u 1 sech 6 , coth 6 u We" then have with the distant end free, by (20), (21) and (22) R / = ^=l . .. . . . ohms (43) If = E A G . . . . . amperes (44) E B = E A . ...-**. volts (44a) 22 APPLICATION OF HYPERBOLIC FUNCTIONS and, with the distant end grounded, by (23), (24) and (26) E,,j = r 6 = R ohms (45) I y = E A /R amperes (46) R, = r o = R ohms (47) Or the conditions become equivalent to those of a very short line of resistance R ohms, and of very small leakage conduct- ance G mhos. Particular Case of Short Lines. Approximate Formulas. We may regard a line as a short line, although not a very short line, when its angle lies between O'l and 0'5 hyp. In the case of short lines, we may use two terms in the expansions by series of the trigonometric functions (Appendix B) and substitute + ~ for sinh 6 1 + 9-, cosh 6 fl3 , tanh 6 , cosech 1 - ~ sech d Q + ; coth 6 We then have with the distant end free, by (20), (21) and (22) R/= (K 1H -f) = -| + (I ' ' ohms (48) I / =E A G(I -^ +G am P eres - . volts (50) That is, a short line offers a resistance at the home end, when freed at the distant end, as though all its leakance were applied as a single leak one-third of the line length away from TO ELECTRICAL ENGINEERING PROBLEMS 23 the home end. The potential at the far end also behaves as though the leakance were lumped and applied as a single leak half-way along the line, the drop of pressure in the line being R then E A G-- 9 - volts. Similarly, we have with the distant end grounded, by (23), (24), and (26)- R, = R(l -- ^) =R(l -^ G ) . . ohms (51) * = %( x + ?) = EA (E + 1) amperes (52) R, = R(I + ^) = R(l + G 6 R ) . - ohms (53) That is, a short line offers a resistance at the home end, when grounded at the distant end, as though the leakance were withdrawn and one-third of it were applied as a single leak at the home end. The current escaping to ground at the far end behaves as though two-thirds of the lumped leakance were applied as a single leak at the middle of the line. Angle subtended by a Terminal Load. If instead of grounding a line at the distant end directly, we ground it through a resistance a ohms, the effect is the same as though -a certain angle were added to the line at the distant end. Thus, Fig. 14 represents a uniform line AB of angle 6, grounded at B through a terminal load resistance CD = a ohms. The angle 6' subtended by this load is such that tanh 6' = - ........ numeric (54) To or 6' = tanh- 1 ^) ..... hyp. (55) \T / This load angle therefore depends not upon the absolute value of the load-resistance, but upon the ratio which the load- resistance bears to the surge-resistance of the line to which it 24 APPLICATION OF HYPERBOLIC FUNCTIONS is applied. Three cases have to be distinguished with con- tinuous-current lines, according as is less than, greater than, TO or equal to, unity. With alternating-current lines, the dis- tinction becomes unnecessary. First Case. Terminal Load- Resistance less than Surge-Resist- ance. If a is less than r , the equivalent terminal load angle- 6' is easily found from a table of hyp. tangents. Thus, if a & 6 ) = 1 i KVVVVVVA/*- A R. BC A^ 3 ^ f * f , f > / > f ^ r ^ ( ^ r > r > r ^ ' ^ FIG. 14. Uniform Line grounded through a terminal load with an increase in. line-angle. line AB, 500 km. long, has r = 4 ohms per km., and g = 10~ 6 mho per km., its conductor-resistance R will be 2000 ohms and its dielectric leakance G will be 5 X 10" 4 mho. Its angle will be = ^2000 x5 xlO" 4 = 1 hyp., and its surge-resistance r = A/ K , A _ 4 = 2000 ohms. If this line is grounded through \ o X 10 a terminal load-resistance (CD, Fig. 14) of, say, a = 500 ohms,, the angle of this load will have as its tangent a/r = 500/2000- = 0*25. The angle is thus found by tables to be 0*25542 hyp. The angle at A subtended by the terminally loaded line is <5 A = 6 + 6' = 1-25542 hyps. The resistance offered by the terminally loaded line at A is therefore by (23) through extension of the line angle RP = r tanh <5 A = r tanh (6 + 0') . ohms (56) TO ELECTRICAL ENGINEERING PROBLEMS 25 The current escaping to ground at distant end is by (25) (41) E A cosh ff E A cosh 6' IB = -- u * = -- u /a ,, /wv amperes (o7) r smh 6 A r smh (6 -f 6 ) so that by inserting the resistance o in the ground path the receiving-end resistance of the line is increased from R f = r c sinh 6 to r sinh (6 + V} RZ = // .... ohms (08) cosh = r sinh 6 -f a cosh ... ohms (59) Formulas (40), (41), and (42) will be found to apply for aify point P of the line AB ; that is, for any angle d between 6' and (6 + 0'). The terminal load a has no angle of its own,. but gives to the end B of the line a virtual angle of 6'. Formulas (40) to (42) will therefore not hold for values of & less than 6'. The potential at B is by (40) In the case considered, the terminally loaded line would offer at A a resistance of 2000 x 0'84979 = 1699'58 ohms. The current which would escape to ground would be the current escaping through the junction BC, and by (41) or would be, with 100 volts applied at A: 16 1 216 = ' 32 ampere, and the potential at B, 100 x y ^ = 16*0 volts. -Lul^lu Second Case. Terminal Load- Resistance greater than Surge- Resistance. If o is greater than r , we apply formulas (37), (38), and (39), instead of (40), (41), and (42). That is, the line condition is considered as though modified from the freed state. The angle of the terminal load at B is now ' . hyp. (61) which is found by a table of tangents. The angle at the home end A is then <5 A = 6 + 0' as before. 26 APPLICATION OF HYPERBOLIC FUNCTIONS The resistance offered by the line as measured at A is by (20)- R,- = r a coth <5 A = r coth (6 + 6'). . ohms (62) The potential at B is by (37) I cosh 6' ,, volts < 63 > The current escaping at B through the load to ground is by (38)- E A sinh 0' E A That is, the formulas of (37), (38) and (39) apply between the limits 6=0' and 6 = 6 + 6' ; but fail for lower values, or in attempting to apply an angle within the load-resistance CD. Third Case. Terminal Load equal to Surge- Resistance. In this case the angle given to the end B by the terminal load is infinite, either by (55) or (61). This means that the voltage and current fall off exponentially. The resistance offered by the line at A, or at any point on the line beyond A, is r ohms, as already noticed on page 14. The potential at any point P, whose angular distance from A is hyps., is U P = U A - ..... volts (65) the current at the same point is also IP = I A e-f . . . . amperes (66) the current at the sending end is E I A = f> A ..... amperes (67) 1 and at the receiving end IB = I A ~ 6 = I A ' La . . amperes (68) This is the case of normal attenuation referred to on TO ELECTRICAL ENGINEERING PROBLEMS 27 page 14. In the case of a very long line, when 6 is over e 4 hyps., sinh 6 becomes very nearly ^ , and the receiving-end /' impedance direct to ground -^ . The current escaping to 2i ground over a very long line is thus I B = 2I A e-. . . . amperes (69) or double the current through a terminal load equal to the surge-resistance. CHAPTER III EQUIVALENT CIRCUITS OF CONDUCTING LINES IN THE STEADY STATE. IT is evident, and has been quantitatively demonstrated in the last chapter, that any given uniform conducting line in the steady state offers a certain resistance at the sending end, and also offers a certain receiving-end resistance at the re- ceiving end. That is, the line system, with its distributed resistance and leakance, may be replaced by a certain equivalent resistance, or group of resistances, at the sending end, and the same proposition applies also to the conditions at the receiving end. Pursuing this inquiry, it may be proved that there exist an infinite number of groups of resistances which, in the steady state, may replace the actual uniform line, with its distributed constants, both at the sending end and at the receiving end. So that, if we select any one of this infinite number of resist- ance groups and substitute it for the line, there will be no change made, by the substitution, in the distribution of poten- tials, currents, or powers external to the group or equivalent line conductor. Such an equivalent group of resistances, capable of being substituted for the line without disturbing the electrical conditions outside the line, is called an equivalent circuit of the line. Although an infinite number of equivalent circuits, made up of four resistances, exist for any given line, and a fortiori of more than four resistances, yet there are only two equivalent circuits that can be made up of three resistances, and none of less than three. In the case of a single uniform line, either unloaded or symmetrically loaded, one of these triple groups of resistances is a star group of three branches, or a T of resistances, two arms of the T, or two branches, being equal, 28 APPLICATION OF HYPERBOLIC FUNCTIONS 29 and disposed in series to represent the line resistance, and the third branch, or staff of the T, a resistance in derivation, acting as a leak, or leakage conductance. The other triple group of resistances is a delta group, or triangle group, of three branches, or a 77, one branch, the architrave, being disposed in series, to represent the line resistance, and the two other resistances, which are equal, are in derivation, and form the pillars of the 77, acting as equal leaks or leakage conductances. In Fig. 15 we have at AB a diagram of a single uniform line of total actual conductor-resistance R ohms and total actual leakance G mhos. The angle of the line is 6 hyps. The surge -resist- ance of the line is z ohms.* The surge admittance of the line is y o =l/z o mhos. At A'B' (Fig. 15) is shown the equivalent T of the actual line AB, consisting of the two arms A'O and OB', each equal to a certain resistance of p ohms, and the staff OG' of a certain resistance R', with corresponding conductance and in the staff, the actual total leakance G of the uniform line. This nominal T will, however, fail to be equivalent to the actual uniform line; because, although the total conductor resistance and the total leakance of the nominal T are respect- ively the same as the total conductor-resistance and leakance of the line ; yet in the nominal T the leakance is collected into- one lump and placed at the middle of the line; whereas in the line the leakance is distributed. The nominal T has, there- fore, an error due to the lumpiness of the leakance ; whereas in the equivalent T this error is eliminated. The correcting factor which must be applied to each arm of the nominal T 7 , 32 APPLICATION OF HYPERBOLIC FUNCTIONS in order to convert it into the corrected value, or arm of the equivalent T, is tanh ' ' ' ' numenc \2 This factor approaches the value 1 as 6 becomes very small. It tends to diminish with increasing real values of 6. That is, the arms of an equivalent T are always smaller than those of the corresponding nominal T of a continuous-current line, and tend to become smaller as the angle of the line increases. The numerical values of k fi are, however, subject to great fluctuations in the alternating-current case. Similarly, the correcting factor which must be applied to the conductance in the leak of the nominal T is , sinh 6 . /h __, k g/ = . ...... numeric (75) a quantity which commences at 1 when 6 = 0, and increases indefinitely with 6. Consequently, the correcting factor of both leak and arms is unity for a very short line; or the equivalent T degrades into the nominal T for 6 = : but as the line angle increases, the equivalent T diverges from the nominal T, the equivalent T arms diminishing relatively in resistance, and the equivalent T leak increasing relatively in conductance, in the case of a continuous-current system. Equivalent 17. As indicated in Fig. 15, the following values must be assigned to the parts of the 77 in order that it may be externally equivalent to the uniform line AB. Calling p" the value of the architrave-resistance, and g" the value of the conductance in each pillar p" = z o sinh 6 ...... ohms (75a) f\ there corresponds one, and only one, uniform line with distributed constants. Suppose a symmetrical T given, as in Fig. 15, A'OB'G', with two equal arms of p ohms, and a leak of g mhos. Then the apparent angle of this line, treating it as a nominal T of total TO ELECTRICAL ENGINEERING PROBLEMS 35 line resistance R = 2p' ohms, and total line leakance G = g mhos, would be by (19) 6 = x/2/. g 1 . . . apparent hyps. (79) and the apparent semi-angle or apparent angle of half the line would be 2~V~ 2 ' ' ' a PP arenfc hyps. (80) But this semi-angle cannot be correct, because it is based on a lumped leakance instead of a distributed leakance as assumed in (19). The correction-factor by which the left-hand member of (80) must be multiplied, in order to eliminate this error of assumption, is sinh (|) . . . numeric (81) That is, the formula (80) becomes on correction ^ .... numeric (82) or the angle of the required equivalent uniform line is = 2sinh- 1 ^ P --. - -V. hyps. (83) = (84) if R' = is the resistance of the staff-leak in ohms. 9 Again, if we proceed to form the surge-resistance of the line corresponding to a nominal T, we have, by (27) V~n -g . ohms (85) But applying this process to the equivalent T, Fig. 15, with 2p' instead of R and g instead of G, we obtain an apparent surge- resistance r a ' ; namely fW r ' *A r~ = ^2/o'R' . . apparent ohms (86) which contains a lumpiness error. The correcting factor which D 2 36 APPLICATION OF HYPERBOLIC FUNCTIONS must be applied to the left-hand member of (86), in order to arrive at the true surge-resistance r of the equivalent uniform line, is _ -,.' = cosh 2 = A/ 1 + if ' ' ' numeric ( 87 ) where 6 is obtained from (83) or (84). Consequently r o = r' Q cosh = cosh s A/-T- = R' sinh 6 ohms (88) + 2 ^'= P 'coth| (89) Reversion of Equivalent 77. If a symmetrical 77 is given, like that in Fig. 15, we may proceed to determine the uniform line which is its external equivalent. If the 77 were a nominal 77, we should have R = /o", and G = 2g". In that case, the angle of the line would be, by (19) 6 = *Jp"ty" .... apparent hyps. (90) and the semi-angle of the line would be ' ' a pp arent h yp s - ( 91 ) But owing to the fact that the 77 is an equivalent, and not a nominal 77, this equation contains a lumpiness error. The cor- rection-factor which must be applied to the left-hand member of (91), in order to eliminate this error, is sinh numeric (92) 7W\ (T) That is, the formula (91) becomes on correction f) I " " sinh-x = A/' .... numeric (93) or the angle of the required equivalent uniform line = 2sinh-i // ^ L =R/. . . . . (104) P'/9" = P"// = R' V = R V ' = R/ R, = r/^r = R/G = r 2 ohms 2 (105) P y = p "^" = cosh - 1 = versh = 2 sinh 2 ^ numeric (106) 38 APPLICATION OF HYPERBOLIC FUNCTIONS Artificial Lines. Artificial lines are either single- or double- wire lines, according as they are of the telegraphic or tele- phonic type. All double-wire artificial lines may be divided symmetrically into a pair of single-wire lines, each operated to neutral or zero-potential plane in the manner indicated in Figs. 11 and 12. Consequently, it is only necessary to discuss the behaviour of single-wire artificial lines. In the continuous- current case, such lines are made up of uniform sections of line resistance in series, and leakance in derivation. A par- ticular case of such an artificial line of five sections is illustrated in Figs. 17, 18 and 19. Each section consists of a symmetrical T having 250 ohms in each arm, and a leak of 250 micromhos in the staff. Consequently, by what has been seen in relation to the equivalence of a uniform line to a given symmetrical T, it follows that each such section is externally equivalent to a corresponding section of uniform line with distributed resist- ance and leakance. Applying formulas (83) and (88) to the case of Fig. 17, we find for the equivalent line section 6 = 0'35172 hyp. and r = 1436*13 ohms. The whole line, therefore, subtends an angle of 1'7586 hyps, at A. The secondary constants of the equivalent uniform line are by (72) and (73), R = 2525*5 ohms, and G = 1 '22495 millimho. In the artificial line there are actually 2500 ohms in series-resistance and 1*25 millimhos of total leakance. The artificial line, therefore, behaves externally in the steady state exactly like a uniform distributed-constant line of 17586 hyps, and 143613 ohms surge-resistance. In Fig. 17 the arti- ficial line is grounded at the distant end B, and 100 volts potential is applied at A. The currents and potentials are indicated in the various elements of the line by Ohm's law computation; but they also conform to the hyperbolic trigo- nometry of a single uniform line at the ends A and B. Moreover, at each junction between sections, marked respect- ively 1, 2, 3 and 4 in Fig. 17, the hyperbolic angle is indicated, and corresponds to like symmetrical points along the equivalent uniform line; so that the potential and current distributions along the artificial line conform precisely to those on the TO ELECTRICAL ENGINEERING PROBLEMS 39 -o- 00 001 e-oir.yoti 1 ^ | -O- Jl f*6Lr-o*Q J.7s.r-o*9 2, HlfyL'O OffC/ c j < s^?: +t SA/VN WOi 66yot-/ *& If-- . I'^r*?-/-^ ! *,fel'SL<)-9 ft'ltf/ -co Jfl .93 ^t9'6SI/ +S1ZW ,1-vt */?<> WIOSU* jro+ftoL'O jrff'6o+i *00'OOI ^ i%r < * tfr ? ^1 S 2> 40 APPLICATION OF HYPERBOLIC FUNCTIONS equivalent uniform line, not merely at the two ends A and B, but also at the four section-junctions; or at six points in alL In order, therefore, to compute trigonometrically the distri- bution of current or potential at any point on the artificial line, say at the second leak, it is only necessary to compute the corresponding condition at the nearest equiangular point on the uniform equivalent line, by formulas already given, and then to apply Ohm's law over the intervening semi-section of line-resistance or arm of a T. This trigonometrical method of computing the distribution is, in general, much less laborious than that of working along the artificial line from either end by Ohm's law that is, the trigonometrical method is a labour- saving device in the continuous-current case, and would also be so in any alternating-current case, if the proper hyperbolic- function Tables were available. In Fig. 18 the artificial line is freed at the end B with an impressed potential of 100 volts at A. The distribution of potential and current over the artificial line agrees with that of the above-mentioned equivalent distributed-constant line at the ends, and at the four section-junctions. If we plot the potential along the line as ordinates to line angle or length as abscissas, we shall have a true catenary falling from 100 to 33*461 volts, for the graph in the case of the real line ; whereas we shall have a broken line or series of descending straight lines falling from 100 to 33*461 volts for the graph in the case of the artificial line. The true catenary will cut these straight lines at points corresponding to section- junction points. (See Appendix F.) The broken straight line is, in fact, the funicular polygon obtained by lumping the masses symmetrically at the centres of the equivalent catenary-sections. In Fig. 19 the artificial line is grounded at the end B through a resistance of 750 ohms. The effect is just the same as though the equivalent real line were so grounded. The equivalent angle subtended by the terminal load resistance is 0' =0*57941 hyp., and the angles of all equivalent points along TO ELECTRICAL ENGINEERING PROBLEMS 41 the artificial and equivalent real lines are increased by this amount in the manner indicated. An alternative method of dealing trigonometrically with the distribution of potential and current over an artificial line, by the use of continued fractions, is discussed in Appendix F. CHAPTER IV REGULARLY LOADED UNIFORM LINES A LOAD in a line may be defined as an element of resistance Inserted in the line, or an element of leakance inserted in derivation on the line; or a combination of both; in such a manner as to introduce a discontinuity into the line, or break up its uniform distribution of line constants. Loads inserted in series into a line may be called series loads, or impedance loads ; while those added in derivation may be called leak loads. Loads at the end of a line are called terminal loads; while those elsewhere are called intermediate loads. When loads are inserted at regular intervals along a line, or according to some assigned law of succession, they are called regular loads. When inserted otherwise, they are called either irregular loads or casual loads. Regular Series Loading. W T e may first consider the effect of loading a line at regular intervals with equal resistances. This is the simplest and most important case of regular loading. On the first row of Fig. 20, we have a line indicated which at uniform angular intervals of 6 = 0*35172 hyp. has series resistances inserted, each of 2 = 200 ohms. The surge- resistance of the line before the loading is r = 1436*13 ohms. Each section of line between adjacent loads might be 100 km. It is required to find the characteristics of the loaded line. In the process represented by Fig. 20, the first step is to find the equivalent T of a line section. This is done with the aid of formulas (70) and (71). This gives us a resistance p of 249*985 ohms in each arm of the T, and a resistance R' of 4000*22 ohms in the staff of the T, corresponding to a leakance g' = 249*92 micromhos. The angle subtended by this- 42 APPLICATION OF HYPERBOLIC FUNCTIONS 43 T will be the same as the angle of the line-section to which it corresponds (6 = 0'35l72 hyp.). The actual resistance R in ! o-snjz 1 0- -^$r-. ?w ^ v- V Hmt/ffv" ^ ^* ' 9* ?! * **fe r H \ 1 -v^ _ ' / 7WW 1 t TWflP otx ^ T z , ^ N 7 >r y > i if VO /r' = 242*94 micromhos, or a virtual reduction of 2 '05 micromhos. It is self-evident that if the 200 additional ohms of load- resistance per section had been inserted by uniform distribution instead of in lumps, there would have been only 200 ohms increase in the section B, with no change in the section G. Consequently, the effect of inserting the added resistance in lumps at 100 km. intervals is a virtual increase of 4*96 ohms of line-resistance per section, together with a slight virtual reduction in leakance. The steps of the process also indicate that for a given amount of additional resistance to be inserted at regular intervals into a line, the shorter the interval, the smaller is the inserted load, and the smaller the change in the extension of the equivalent J's; whereas with long sections, large loads must be inserted at their junctions. The extensions of the equivalent T's are then correspondingly enlarged, with increased effect on the line angle, and on the virtual extra resistance of the loads due to their lumpiness. If 6 be the angle of a section before loading (hyps.) 0, ,, after z surge-resistance of the line before loading (ohms) o after 2 2a be the resistance of each series load (ohms), TO ELECTRICAL ENGINEERING PROBLEMS 45 then the process above outlined leads to the following results numeric (107) u , sinh -~ sir / 1 2 V o coth - Z o sh e \l 1 f\ o tanh I * n |^y j n - - - o f OTI n ' f-0 nh / Z Q , let = coth d. Then coth ^ = j'coth | . coth (4 + ^) . numeric (110) Also sinh 0, = sinh 6 J 1 + : coth -f f ~V . (111) cosh 7 = cosh + ^- sinh (112)* + f coth0 + (Jj .... (114) 2' sinh 0. or - = . . ' numenc (Ho) z sinh Consequently, the characteristics 0, and z' of a section after loading can always be computed when the value 2 of each * Formula 112 was first published by Dr. G. A. Campbell. See Bibliography, 27. 46 APPLICATION OF HYPERBOLIC FUNCTIONS resistance-load is given, as well as the characteristic constants 6 and z o of the section before loading. On the other hand, if the resistance of the load were distributed uniformly over the section, the effect would evidently be to increase 6 and z in the same ratio ; that is Regular Leak Loading. If as indicated on the first row of Fig. 21, equal leakances of conductance T mhos are inserted at uniform angular distances 6 hyps, along a line, the surge- resistance of which is z o ohms ; we may proceed to determine the effect of this loading on the line characteristics. The first step is to break each load into halves of conductance y mhos (F = 2y), as on the second row of Fig. 21. The next step is to find the equivalent U of each unloaded section of line, as on the third row of Fig. 21, with the aid of formulas (75) and (76). We now extend each section U by adding the leak y to each pillar conductance, as indicated in Fig. 21, at the fourth row; where g /t = g" + y mhos (117) The extended /7's will now subtend a new angle 6 f , which we proceed to find by reverting them to their corresponding uniform line-sections, using (95) and (99). In the case represented in Fig. 21, each leak T has a conductance of 0'09697l millimho, and the line-sections are the same as in Fig. 20. The final equivalent line-section angle is Q t == 0*41532 hyp., and the surge-resistance 1206'4 ohms. With the notation of (107) we find . i v . c 2 /i i o\ TJ- . numeric (Ho) -f yz tanh ^ = A /tanh|-tanh(| + d) . . (119) TO ELECTRICAL ENGINEERING PROBLEMS 47 if yz = tanh d, and z o ohms (120) y ( /0 , l + yz tanh 1 + y*o coth B -A -, FK;. 21. Reduction of a Uniform Actual Line with loads in derivation to an Equivalent Unloaded Actual Line. Regular leak loads y produce the same effect on the line section-angle as regular series loads a, when - = z 2 ohms (121) The final surge-resistances in the two cases will not, however, 48 APPLICATION OF HYPERBOLIC FUNCTIONS be the same, but their geometrical mean will be the original or unloaded surge-resistance. It follows from the preceding formulas that every regularly loaded line can be replaced by an equivalent uniform unloaded line of the same length, which, in turn, can be completely replaced for all external purposes by a certain equivalent T, and corresponding equivalent 77. The effect of loading a line at regular intervals, either with series resistances, or with leaks, is always to increase the angle 6 of the line, and reduce its normal attenuation-factor e~ e , in the continuous-current case ; besides the effect of long intervals between loads, or lumpiness, which exaggerates the influence. In the alternating-current case, it will be seen that loading always increases the numerical hyperbolic angle of the line ; but by no means always reduces the normal attenuation- factor. CHAPTER V COMPLEX QUANTITIES IN the discussion which has preceded, all the quantities employed have been simple "real" numerical quantities, comprised between the limits of oc and + oc ; so that they are capable of being represented geometrically by their positions on a single graduated straight line ; i. e. by their assignment in one dimensional space. It is, however, a seemingly universal and a wonderful law, that all the numerical formulas and rules of quantitative behaviour for continuous-current circuits, or conductors, are exactly the same for single frequency alternating-current circuits or conductors, in respect to potentials and currents as also (with minor reservations) to power and energy, if these formulas and rules are interpreted as relating to complex numbers ; or such numbers as are represented by their positions on a single graduated plane ; i. e. by their assignment in two- dimensional space.* The importance of this law will be evident, when it is recognized that each and all of the formulas hitherto discussed in relation to continuous-current lines and systems, are immediately applicable, without any change in notation, to alternating-current lines and systems, provided that we extend the meaning of the notation to include two-dimensional numbers instead of one-dimensional numbers. From this standpoint, it will be seen that we have already dealt with alternating-current lines and systems unawares, and that the continuous-current case is merely the particular case, in each formula, when the numerical quantities it employs degrade into ordinary one-dimensional numbers. * See Bibliography, 6, 9, 9a, 10, 17, 70. 49 K 50 APPLICATION OF HYPERBOLIC FUNCTIONS There is, moreover, a great advantage in dealing first with the hyperbolic trigonometry of continuous- current lines ; because one-dimensional arithmetic is easier to grasp and follow than two-dimensional arithmetic. On this account, our ideas are limned more clearly, and our advance proceeds more surely, when we deal first with the problem on the one -dimensional aspect which the continuous-current case supplies. After having mastered the subject in one dimension, we are in a strong position to attack the vastly wider two-dimensional fields,, which alternating-current cases offer successively to our mental vision. We need never have doubts or fears as to the safe road 1.5 >c 3 FIG. 22. Geometrical Representation of Simple Numbers. to pursue in our advance on a two-dimensional problem, if our formulas and weapons of attack have been forged on the one- dimensional hearth. Whenever a new problem arises in alternating-current technology, whether it is more conveniently dealt with by hyperbolic trigonometry or not, the safe rule is to find the corresponding problem in continuous-current technology, and solve it there by simple one-dimensional arithmetic. The equations and formulas of the continuous-current solution then apply to the alternating-current solution, by extending their meaning into two-dimensional arithmetic. Simple and Complex Numbers. In Fig. 22, three simple numbers are represented geometrically, namely Oa = 2, Ob = +1*5, and Oc = -f- 3. These numbers all lie in one TO ELECTRICAL ENGINEERING PROBLEMS 51 direction, across the page, in the plane of the paper, and positive numbers are directed towards the right hand. All simple arithmetical operations upon such one-dimensional numbers, such as addition, subtraction, multiplication, division, powers, roots, etc., beget other one-dimensional numbers, so that all simple arithmetic belongs to the one-dimensional system, or to a single straight line in space. Any convenient straight line in space will serve as the line of reference. It is convenient to At & ^ ^Z titC J- , o u- *\ B O FIG. 23. Geometrical Representation of Complex Numbers. take the line across the page ; but the selected line might be, say, at right angles to this, or up and down the page. Three complex numbers are also represented in Fig. 23, namely OA = 2 /45, OB = 1 ^60^ and OC = 3'0 X 12Q.* It will be seen that each number has appended to it a circular angle, which defines the direction of the line representing the number in the single plane of reference, with respect to an initial direction of reference therein. We are at liberty to choose any convenient plane, and any convenient direction of * In what follows, A / ft has the same meaning as A cis ft or Ae# in regular mathematical notations, and may be regarded as a convenient abbreviation for either of these forms of expression. The modulus A is the length-factor, or tensor, and the argument ft is the circular-angle- factor, or versor, of the complex number. E 2 52 APPLICATION OF HYPERBOLIC FUNCTIONS reference in the same. It is convenient to select the plane of the page, and the direction parallel to the lines across the page as the direction of reference. From this standpoint, all simple numbers, as in Fig. 22, are numbers having the angle /0. Complex numbers are sometimes called plane- vectors or, for brevity, vectors. They must, however, be carefully distinguished from three-dimensional vectors. It is also to be noted that negative signs are not indispensable when writing or specifying individual complex numbers. The 1-9H B FIG. 24. Vector Sum of Two Complex Numbers. angle inseparably attached to each number is sufficient to define a negative direction. Thus 10\180 isthe same as -10 /O^. We may, therefore, use negative signs or not, as may be convenient. In Fig. 23, we might express O C as 3'0 x O B, if desired. Addition of Complex Numbers. When two complex numbers are added together geometrically, one of them is transferred to the end of the other, and the new number, or sum, is that corresponding to a line drawn from the origin of the latter to the free end of the former so transferred. Thus in Fig. 24 2/45^+1 \W = OB = 1-991 /ICT. Again 1 \60 + 3'0 \120 = OC = 2'0\120. This process of adding on one vector to the end of another is aptly described as geometrical addition. TO ELECTRICAL ENGINEERING PROBLEMS 53 By successive additions of complex numbers, any or all parts of the plane of reference may be invaded. The process of vector addition, or geometrical addition, of complex numbers is easily conducted by a draughtsman at the drawing-board ; but in order to be carried out numerically, it is desirable to analyse the complex numbers into components. A complex number is most conveniently analysed into two mutually perpendicular components, one in the direction of reference, or across the page, and the other up and down the page. These components are called respectively the real and imaginary components of the complex number, the terms having been bestowed by algebraists from the standpoint of one- dimensional arithmetic. The rule for the analysis of any number of length or modulus A and angle or argument $ is by circular trigonometry A //? = Acos0-K;*A sin . \ ; '. cm. /_ (122) it being understood that we are discussing the complex number in geometrical terms. Thus, in Fig. 23, the number OA is 1*414 + y 1-414; OB is 0'5 -/0'866; OC is - T50 -h/2'6. The symbol/ here indicates that the quantity to which it is prefixed is to be measured upwards along the " imaginary " axis. The sign j prefixed to a number means that it is to be measured downwards along the imaginary axis. The/ " operator " is thus an operator which, when applied to a line, representing a number, rotates it in the positive or counter-clockwise direction through 90, or n/% circular radians, in the plane of reference. Two successive applications of the j operator thus reverse the direction of a line, or rotate it through 180 ; so that j x j or j- is equivalent to giving the negative sign to a complex number without changing its angle. Thus we have the well- known relation /= V 77 ! numeric /. (123) and j 2 = 1, j* = + 1, y 3 = j, and so on. To find the sum of a plurality of complex numbers, we add their real components, and also their imaginary components. * j is used in electrotechnics, instead of i as in mathematics, to avoid confusion with currents. 54 APPLICATION OF HYPERBOLIC FUNCTIONS Thus in Fig. 23, OA + OB + OC, becomes in Fig. 25, OA + AB -f BC = (1-414 + 0-5 - 1-50) +/(T414 - 0'866 + 2-60) = 0-414 + /3148 = 00 = 3175/82^, where OC is the vector sum of OA, OB, and OC, the three complex numbers in Fig. 23. In order to resolve the components of a complex quantity into a plane vector, let + # be the real component and 0-4-14- FIG. 25. Vector Sum of Three Complex Numbers. the imaginary component, A / /?, the resultant complex number. Then by circular trigonometry and (122) A = V^M-1/ 2 - modulus, cm. (124) and /? = tan " l (^~^) argument, radians (125) Thus in Fig. 25, with x = + 0'414, and y = +/3'148, we have A = ^/G-414 2 + 3-148 2 = ^01714 + 9-9099 = V 10 ' 0813 = 31751, and = tan - 1 ( + 3148 /+ 0414) = tan ~ l (+ 7'604) = 82 30' or 82-5. TO ELECTRICAL ENGINEERING PROBLEMS 55 Subtraction of Complex Numbers. Subtraction of one complex number A 2 / ft from another A 1 / ft merely requires that the negative sign be given to the former, or that, without change of sign, its angle / /ft., be changed by 180, and addition then follows. On the drawing-board, this is performed by first laying off A! / ft and then adding A 9 / ft to the end of it. In order to perform the operation arithmetically, each vector is analysed into its components, and the subtraction then proceeds along ^ach axis, as in one-dimensional arithmetic. Thus if A x / ft = *i J!/ and A 2 /ft = x z jy v Then A x /_ft - A 2 /ft = (*i -* 2 )/(yi-y 2 )= X/Y. Multiplication of Complex Numbers. The multiplication of complex numbers in any order is effected by multiplying together their lengths or moduli, as in one-dimensional arithmetic, and adding their angles. Thus if A 1 /ft and A 2 /ft are the two numbers to be multiplied together, and A //ft the product, we have A = A, /ft x A 2 /ft = A x A 2 /ft + ft numeric /_ (126) Thus if A! /ft = OA Fig. 23 = 2 /45, and A 2 /ft = OB = 1 \60, then the product A / is 2 \15. If the two complex numbers to be multiplied are analysed into components, the multiplication may be effected, although less conveniently, by the rules of algebra. Thus if ( x \ i/^i) and ( + ,r 2 /y 2 ) b e tne two numbers, their product will be (i2 - y\ 2/2) J( x 2 q Ji + x zv3 = x y Y - Reciprocal of a Complex Number. The reciprocal of a complex number has for its modulus the arithmetical reciprocal of the modulus of the number, and for its argument the negative value of the argument of the number. Thus, if A //? is the complex number, its reciprocal will be - \/ft = / ft 56 APPLICATION OF HYPERBOLIC FUNCTIONS If the complex number is say 10 /20, its reciprocal will be 0-1 \W*. If the number whose reciprocal is required, be analysed into components in the form (#+y?/), its reciprocal will be . . This may be reduced to the original form, by vjy multiplying both numerator and denominator by (+x+jy\ The result is ^ -t/ v - - Division of Complex Numbers. Division of complex numbers- can always be effected by taking the reciprocal of the divisor according to the last preceding rule, and then multiplying this reciprocal into the dividend. Thus if Aj_ /ft has to be divided by A 2 /ft the quotient is ( ^ \ /ftj-_ft = A /ft If A l /ft A 2 /ft are OA and OB of Fig. 23 respectively, the quotient is 2\10o. Powers and Roots of Complex Numbers. The nth power of a. complex number is formed by the nth arithmetical power of the modulus, and multiplying the argument by n. That is (A //?) n = A n /nfi . . . numeric /_ (127) Similarly, the nth root of a complex number is formed by taking the nth arithmetical root of the modulus, and dividing the argument by n. That is 1 - iF /ft) = (A /ft) n = A n / 7- . numeric /_ (128). Summing up, we may say that two-dimensional arithmetic is performed by rules which degrade into those of ordinary or one-dimensional arithmetic when the arguments are all zero. When adding or subtracting complex numbers numerically, it is desirable to analyse them into components ; but when multiply- ing and dividing them, or when taking powers and roots, it is desirable to express them in angular form. Trigonometrical functions of Complex Angles. We have- TO ELECTRICAL ENGINEERING PROBLEMS 57 already seen that in the case of a simple angle in generalized trigonometry (Figs. 1 and 2), the circular functions can be read from a circle diagram (Fig. 1), and the hyperbolic functions from corresponding elements of a hyperbola diagram (Fig. 2). When the angle to be dealt with is complex, or of the type (x + jy) radians, both the circular and the hyperbolic functions can be derived from a mixed circle and hyperbola diagram. Circular Functions of a Complex Angle. Construction for sin (x +jy), Fig. 26.* Take OA = 1 along the negative end of FIG. 26. Construction for Sin (x jy] and Sin- 1 (a + jy). the Y-axis. From OA as initial line, mark off the circular angle x and sector area AOB From OB as initial line, mark off 77 the hyperbolic angle y and sector area BOD = ^ Let C be the foot of the perpendicular from D on OB produced. Drop perpendiculars from C and D on the axis of reals OX, at c and d respectively. About c as centre, rotate cd positively through 90 to cZ. Then will the complex vector OZ = Qc+jcd be the required circular sine of the complex angle x +jy radians. * See Bibliography, 31. APPLICATION OF HYPERBOLIC FUNCTIONS In the case represented, sin (1 -f/1) == 1'299 -j- /O635 = 1'446 /26'05. As y varies, Z moves along the hyperbola JX 2 Y 2 _ sin 2 x cos 2 x as x varies, Z moves along the ellipse Y 2 V2 cm. 2 /_ (129) cosh 2 y sinh ^y Y 1. cm.- (130) FIG. 27. Construction for Cos (x jy} and Cos-i (x jy). From the same figure we have also, if Oc = u and cZ =jv sin' 1 OZ = sin- 1 ^ jv) = sm~ l Ob jcosh~ l OE = cn Construction for Cos (x -\-jy). Fig. 27. Take OA = 1 along the positive end of the axis of reals. From OA as initial line, mark off the circular angle x or the sector area AOB = -- etc., 2t precisely as in the preceding paragraph. The complex vector TO ELECTRICAL ENGINEERING PROBLEMS 59 OZ = Oc -\-jcd thus obtained will be the required circular cosine of the complex angle (x +jy) radians. In the case represented cos (1 +/1) = 0-834 -yO-989 = T293 V 49'866. As y varies, Z moves along the hyperbola < 132 > As x varies, Z moves along the ellipse (130). From the figure cos- 1 OZ = cos- 1 (u jv) = cos' 1 Ob + cosh- 1 OE _ cos -i I MVC (133) Hyperbolic Functions of a Complex Angle. Constmiction for Sink (x jy). Fig. 28. Take OA = 1 along the positive end of the axis of reals. From OA as initial line, mark off the circular angle y and sector area AOB = |. From OB as initial line, mark off the hyperbolic angle x and sector area BOD 9! =- Let C be the foot of the perpendicular from D on OB produced. Drop perpendiculars from C and D on the axis of imaginary OY, at c and d respectively. About c as centre, rotate cd negatively, or clockwise, through 90 to cZ. Then will the complex vector OZ be the required hyperbolic sine of the complex angle x -{-jy radians. In the case presented in Fig. 28, sinh(l +/1) = 0'635 +/1-2985 = 1-446/63-95 . As x varies, Z moves along the hyperbola Zbz. As y varies Z moves along the ellipse XE#y. Both the ellipse and the hyperbola have foci at F and /, points situated at unit distances from O on the Y axis. 60 APPLICATION OF HYPERBOLIC FUNCTIONS From the same figure, if cZ = u, and Oc = jv, we have +^ + sinh- (u fi) = cosh- . = x jy ...... hyp. radians /_ (134) FIG. 28. Graphical Constructions for Sinh (a; jy) and Sinh-i (x jy}. Construction for Cosh (x -{-jy). Fig. 29. Take OA = 1 along the positive end of the axis of reals. From OA as initial line, mark off the circular angle y, or the sector area AOB = ^. TO ELECTRICAL ENGINEERING PROBLEMS 61 From OB as initial line, mark off the hyperbolic angle x and the sector area BOD = - Let C be the foot of the perpendi- cular from D on OB produced. Drop perpendiculars from C and D on the axis of reals OX, at c and d, respectively, About FIG. 29. Graphical Constructions for Cosh (x jy) and Cosh-l (x jy). c as centre, rotate cd negatively, or clockwise, through 90 to cZ. Then will the complex vector OZ = Oc +jcd be the required hyperbolic cosine of the complex angle x -\-jy radians. In the case represented, cosh (1 +/1) = 0'834 -f yO'989 = 1-293/49-866 . As x varies, Z moves along the hyperbola .Zbz. As y varies, Z moves along the ellipse TZZez. Both ellipse and hyperbola have foci at a and A, points on the real -axis equally remote from O. 62 APPLICATION OF HYPERBOLIC FUNCTIONS From the same figure, if Oc = u and cZ =jv, we have cosh~ l (u jv) = ) cos I ~2 xjy hyp. radians /_ (135) Construction for Tanh (x -\-jy). Fig. 30. Mark off on the axis of reals &OX, two points T and X, such that the former is distant by tanh x and the latter by coth x from the origin 0. Find the point C midway between T and X. FIG. 30. Graphical Constructions for Tanh (x jy) and Tanh-l (a; jy). This point will incidentally be distant coth %x from O. With centre C and radius CT = cosech 2a?, draw the circle TXZ. Mark off on the axis of imaginaries 2/OY two points t and y, such that the former is distant by tan y and the latter cot y from the origin O. Find the point c midway between t and y. This point will incidentally be distant cot 2y from O. With centre c and radius d = cosec 2?/, draw the circle ~ByAt. This TO ELECTRICAL ENGINEERING PROBLEMS 63: FIG. 31. Loci of Sinli for the Imaginary-Keal Ratios 1, 2, 3, 4 and 10. 90 30 15 : FIG. 32. Loci of Cosh for the Imaginary-Real Ratios 1, 2, 3, 4 and 10. 64 APPLICATION OF HYPERBOLIC FUNCTIONS circle will cut the axis of reals at two points A and B distant each one unit from O. It will also intersect the circle TXZ orthogonally in Z. Connect OZ. This vector OZ is the required hyperbolic tangent of the angle x -\- jy radians. In the case presented in Fig. 30, tanh (1 4- /I) = 1*084 -f/0-2718 = 1118/14-583 . FIG. 33. Loci of Tanh 6 for the Imaginary-Real Ratios 1, 2, 3, 4 and 10. As x varies, Z moves along the circle AB. As y varies, ZA moves along the circle TZX, performing one complete revolution for each n units of increase in y. From the same figure, if OZ = u + jv, jv) = * v . radians /. (136) TO ELECTRICAL ENGINEERING PROBLEMS 65 Principal Formulas for Deriving the Hyperbolic Functions of Complex Angles. As distinguished from constructions for the 1.3 1.4 FIG. 34. Loci of Sech 6 for the Imaginary-Real Ratios 1, 2, 3, 4 and 10. hyperbolic functions of complex angles, the following are among the most important formulas for computing them sinh (x jy) = sinh x coshjy cosh x smhjy . . . (137) = sinh x cos y + j cosh x sin y .... (138) = ^sinh 2 x + sin 2 y /+ tan" 1 (coth x tan ?/) (139) = ^/cosh 2 x cos 2 y /+ tan" 1 (coth x tan y) (140) jy} = cosh x coshjy + sinh x sinty/y . . . (141) = cosh x cos y _ j sinh x sin y) .... (142) = x /cosh 2 x - sin 2 y /+ tan" 1 (tanh x tan y) (143) = ^/sinh 2 x + cos 2 y / tan" 1 (tanh a? tan y) (144) cosh (x 66 APPLICATION OF HYPERBOLIC FUNCTIONS It is evident that the values of the complex hyperbolic functions may be computed and tabulated either for varying values of x and y * or for varying values of A and {3, the modulus and argument of the angle expressed as a vector. For most electrotechnical purposes the latter are the more convenient^ / i FIG. 35. Loci of Cosech for the Imaginary-Real Ratios 1, 2, 3, 4 and 10. Tables of these functions are still greatly needed. They have- been published for the particular case of /? = 45. t Other tables are in course of preparation. Since sinh 6 = - s'mjd J (146) * Tables of sinh (x + jy) and cosh (x + jy) over a limited range have been published by Prof. James McMahon and by the General Electric- Co., Schenectady (N.Y.). See Bibliography, 15 and 64. f See Bibliography, 33 and 63. TO ELECTRICAL ENGINEERING PROBLEMS 67 sin 6 = sinh/0 ..... (147) cosh 6 = cos/0 (148) cos = cosh jO (149) where 6 is any angle, simple or complex, it follows that tables of complex hyperbolic functions can be used, with a little ~~~ 2 FIG. 36. Loci of Goth 6 for the Imaginary-Real Ratios 1, 2, 3, 4 and 10. extra trouble, as tables of complex circular functions, and reciprocally. Figs. 31 to 36 indicate the vector values of complex hyper- bolic functions for the five values of f$ whose tangents are 1, 2, 3, 4 and 10 respectively (ft = 45, 63'43, 71-57, 75'97, and 84-28') up to A = 1'5. Thus taking Fig. 31 the hyp. sine of a complex angle 6 = 1-4/45 or 1'4/tan- 1 ! is found by following the heavy curve marked 1 to its intersection with F2 68 APPLICATION OF HYPERBOLIC FUNCTIONS the dotted line 1*4. The intersection marks the complex vector required. It is r43/63'57. Curve-sheets like those of Figs. 31 to 36 can be drawn on suitable scales for, say, each degree of {3 and each step of O'l or less in modulus A. Such curves extend, theoretically, to infinity or cover the XY plane, not only once, but many times in succession, if A be taken large enough. CHAPTER VI THE PROCESS OF BUILDING-UP THE POTENTIAL AND CURRENT DISTRIBUTION IN A SIMPLE UNIFORM ALTERNATING-CURRENT LINE Ix order to present the application of hyperbolic functions to the analysis of alternating-current lines from an alternative and illuminating view -point, we shall here discuss the simplest elements of electromagnetic wave motion over such lines during G G- FIG 37. Simple Alternating-Current Circuit. the construction period which precedes the steady state. Since, however, the steady-state distribution is the object of our investigation, and the preceding unsteady state demands a like share of analysis for its investigation, we shall pass over the latter very briefly, and make certain assumptions as postulates which may be readily verified. Fig. 37 presents the essential elements of a simple a.c. (alter- 69 70 APPLICATION OF HYPERBOLIC FUNCTIONS nating-current) circuit. The line AB is uniform, and has uniformly distributed secondary constants ; namely r = linear resistance (ohms per wire-km.) / = inductance (henrys ) g = conductance (mhos ) c = capacitance (farads ) f = the frequency of the impressed e.m.f. (cycles per second) co the angular velocity (radians per second) z = (r +//ft>) linear conductor impedance (ohms per wire-km.) y = (g 4-jica)) dielectric admittance (mhos ) a = ^/z.y = ^/(r -\-jlcj) (g -\-jc(o) . hyps, per wire-km. (150) will then be the attenuation-constant or linear hyp. angle of FIG. 38. Analysis of the Vector Attenuation- Constant OA = a into the real part OB = of a hyperbolic angle a has the properties of a circular angle. Again the surge-impedance of the line is (152) The hyperbolic angle of the line will be e/l = La H = L (a, +>,) =J~Z.T= 6, +/0 2 . hyps. (153) Where L is the line-length in km,, Z is the total conductor TO ELECTRICAL ENGINEERING PROBLEMS 71 impedance of the line or L (r + jlw) Lz ohms and Y is the total dielectric admittance of the line, or L (g -\-jca*) = Ly mhos. The real part 6 l of the complex angle is expressible in hyperbolic radians, and the imaginary part in circular radians. It will be seen that (153) is merely a restatement of (19), <152) of (18), and (150) of (17) with two-dimensional signifi- cation. In other words, formulas (17), (18), and (19) for continuous-current circuits apply to the alternating-current case here to be discussed, when interpreted as involving com- plex resistances or impedances, and complex conductances or admittances. We use, therefore, z for r, Z for R, y for g, and Y for G, in order to emphasize the two-dimensional meaning ; but we may use all the preceding continuous-current formulas unchanged, if we keep the vectorial interpretation before the mind. The significance attached to the linear angle or vector attenuation-constant a, is that any wave of potential or current running along the uniform line, shrinks or attenuates by the linear attenuation-factor e~ a in each unit of length (miles or km.), for the particular a.c. frequency under consideration. If then, when starting, at end A, a wave of potential or current has a value taken as unity, it will, after having run 1 km., have dwindled to -*, after 2 km. to e~ a .e~ a =e~- a t after n km. to -"*, and just before arriving at B, to e~ La = e" , where 6 is the line angle. numeric But ~ a = -< a l-r./a 2 ) = _ g-O! X -jo* . < -- j_ (154) so that -* is the product of two factors, namely, e -% which is a real numeric, and s~^ t which is an angle, and may be written ->2 = / a., = \a7 . . . radians (155) Consequently, we may write . numeric . ,- ^ N - = -v 2 .- "" L ( } and -* = -#! ^d^=- La i La., . numeric/. (157) 72 APPLICATION OF HYPERBOLIC FUNCTIONS The normal attenuation-factor of the line s ~ 9 , which is also the inevitable attenuation-factor for any one wave of potential or current running over the line, comprises a real component e~ 9 \ and an imaginary component e~J 2 . The former is the actual numerical attenuation-factor which applies to the amplitude of the wave ; while the latter is an angle, and applies only to the phase of the wave. If, for the moment, we leave phase angles out of consideration, and consider only the shrinkage or attenuation of the waves running over the line, then we only consider the real component a 1 of the attenuation-con* stant a, and the real or hyperbolic component 1 of the line complex angle 0. In practice, an alternating-current line, whether used for signal transmission, telegraphy, telephony, etc., may be pro- visionally regarded electrically as a very short line if 6 l is less than O'l, e -0i being greater than 0'905 a short line if ,, is between O'l and '5, e -^i ,, ,, 0'607 a line of moderate length if ,, ,, 0'5 ,, 2 "5, e -i ,, ,, O'lOO a long line if,, ,, 2'5 ,, 4'0, e -i ,, ,, 0'020 a very long line if,, isovt-r 4'0, e -#i being less than 0'020' it being understood that 1 depends upon the frequency of alternation as well as upon the line-length and secondary constants. A given line when used, say, for power transmis- sion, is ordinarily very short with respect to the fundamental frequency of the generator applied to it, but perhaps long to- some higher harmonic frequency such as the llth-frequency harmonic that the generator may produce. The essential difference with respect to attenuation between the line-angle 6=^/ZY of an a.c. line and that of a c.c. (continuous-current) line 0=^/11.0., is that, in the former,, only the real component cos /?, or 6 V counts, while the latter is all real, and all counts. An a.c. line may have a complex angle of 50 hyperbolic radians, and yet, if the real component is only, say, 2 radians, the line is only of moderate lengthy whereas a c.c. line of 50 radians would be of enormous elec- trical length. As above defined, a very short line attenuates TO ELECTRICAL ENGINEERING PROBLEMS 73 waves less than 10%, so that they arrive at the distant end of the line with over 90% of their amplitude at the home end. A short line attenuates less than 40 %, so that the wave attenuation-factor for one run over the line is over 0'6. A moderately long line reduces the wave attenuation-factor to something between 0*6 and 01. A long line may bring it down as low as 2 / Qt and a very long line to yet lower values. These will also be the normal attenuation-factors of the lines in their steady a.c. state, i. e. if the terminal-load impedance at B is equal to their surge-impedance 2 . In Fig. 37, let the a.c. generator have negligible impedance, and produce a sinusoidal e.m.f., or pure sine wave. If the terminal-load impedance Z>. is made infinite, the line is freed at B, if it is made zero the line is grounded at B. If it is made equal to z the line is in the normal state as to attenua- tion, the attenuation-factor of the steady state being then the same as the attenuation-factor for any single wave in the unsteady state. The physical significance of the surge-impedance Z Q is that, at any point, the line offers this impedance to an advancing wave of the frequency considered. That is, at any point volts where e and i are the instantaneous values of the potential and current strength at the point. Consequently, the surge- impedance of the line is not only the natural impedance which it offers everywhere to surges of the frequency considered, but it is also the initial impedance of the line at the sending end. It is, therefore, also the initial sencling-end impedance of the line, as distinguished from the impedance which the line offers at the sending end in the steady state, when a number of waves are merged together. The velocity with which any wave of the frequency / runs over the line considered, is determined by the relation v = - . . . . km. per second (159) 2 74 APPLICATION OF HYPERBOLIC FUNCTIONS When the linear conductor resistance r and dielectric con- ductance g are both made equal to zero in (150) we obtain a = ja. 2 = jo)^/d . . . hyps, per km. (160) and the velocity over such a line would become v = . km. per second (161) \/Cl' which for a free uniform line, surrounded by air at all points, and ignoring inductance within the substance of the wire, is known to be the velocity v of " light " in air, or 300,000 km. per second, owing to the fundamental relations between the inductance and capacity of such a wire. Owing, however, to the presence of effective linear resistance of all kinds that dis- sipate electromagnetic wave energy into heat in the conductor, and to effective linear conductance g, of all kinds that dissipate such energy into heat in the dielectric, the speed of the waves drops, even in an aerial line. Moreover, when the dielectric is a solid, the free limiting speed of waves is reduced inversely as the square root of the specific inductive capacity of the material ; so that with solid-insulated conductors, the speed of wave- advance may be only a small fraction of 300,000 km. /sec. On loaded lines, the wave speed, v = co/a. 7 , is artificially reduced still more. Let us assume a line having at a certain frequency an attenuation-constant of a 0*07675 +^0*7854 hyps, per km. Then, in running 1 km. over this line, a wave shrinks in am- plitude by e-O' - 6 ? 5 or 0*9259, i. e. to 92*59 %. It also shrinks in phase by 0*7854 radians (n/4> = 45). This means that it loses this phase angle with respect to the phase of the wave then being delivered by the generator at A. When a wave of either potential or current is delivered to the line, that wave goes on with its phase unchanged with respect to its own parts. The crest of the wave remains a crest, and a zero-point on the wave remains a zero-point. But the generator keeps on changing its phase and the phase of the next wave that is emitted. Consequently, the wave which has been released and is travelling along the line is continually falling further and TO ELECTRICAL ENGINEERING PROBLEMS 75 further behind the instantaneous phase of the generator end. It loses co radians per second, and since the velocity of wave transmission is v =a)/a. 2 km. per second, the advancing wave must lose co/v = a 2 radians per km. The magnitude and phase relation of a released wave of either potential or current as it runs over the first 10 km. of the circuit here considered, are indicated in Fig. 39. Assum- ing the initial magnitude and phase at start to be represented FIG. 39. Diagram of Relative Magnitudes and Phases of Outgoing "Wave over the Line of Fig. 37, for the first ten kilometers. by the vector OE, these will have changed, after 1 km., to O, 1', an amplitude of 0'926, and a phase lag of 45. After 2 km. the vector has become O, 2', with an amplitude of (0-926)'- = 0-857, and a lag of 2(45) = 90. After running 10 km. the amplitude vector is O, 10', with an amplitude of 0'464 and a phase lag of 450, corresponding to the attenuation- factor g-io(i+/2) = - 10 . Another view of the condition is indicated in Fig. 40, where it will be seen that at the point BB', 30 km. from A, the wave has fallen 3f complete cycles 76 APPLICATION OF HYPERBOLIC FUNCTIONS or 1350 in space-phase behind the generator at A, and the wave then being emitted. The length of the wave on the line must in all cases be 1 = ~ km/; (162) a 2 a relation which holds either for the unsteady or steady state. The wave-length may be regarded as the distance through which a wave must run in order to lose 1 cycle, 2yr radians, or 360 with respect to the generator phase. The distance L e to which a wave must run over a uniform line in order that its amplitude may shrink to Vth part (1/2-71828 or 0'3679) will evidently be such that L^ = 1 or L e = l/c^ .... km. (163) Similarly the distance L^ to which a wave must run in order to be at- tenuated down to J amplitude or lose 50 per cent, will be 0-69315 L 4 = . . km. (164) a i In the case considered the waves- would fall to J in 9*03 km. and to Yeth in 13'03 km. A mechanical model might be constructed to illustrate the preced- ing principles, in the manner in- dicated in Fig. 41.* A wooden shaft 00' is mounted in a long wooden box so as to rotate in bearings at opposite ends, and so as to be rotated by the external handle H. The shutter ss, or lid of the box, is arranged to slide in a groove at the top, and is geared with the * This idea lias evidently no novelty. See Bibliography, 34 and 44. FIG. 40. Curve of wave-at- tenuation for the first 62 kilometers of a circuit of attenuation-constant a = 0-07675 +./0 7854 per kilo- meter. TO ELECTRICAL ENGINEERING PROBLEMS 77 handle iu such a manner that starting closed with the handle at the top, it slides, to open the lid, by one wave-length for each turn of the handle. Radial pins are then permanently inserted in the shaft according to the plan of Figs. 39 and 40 : i.e. so spaced angularly as to correspond to the circular component a. 2 of the linear complex angle or attenuation- constant, and so spaced longitudinally as to correspond to the hyperbolic component a r If now light falls vertically upon the closed lid of the box, and the handle H is slowly turned, the lid begins to open and the vertically falling shadow of FIG. 41. Diagram of Model for exhibiting the propagation of the first outgoing wave-train along a uniform line. each pin begins to execute a simple harmonic motion on the floor of the box. The phase and amplitude of this shadow- motion at any instant and distance along the model correspond to the phase and amplitude of the wave considered in motion over the line, assuming that no reflections occur. Terminal Eeflectiotis. In what follows it will be assumed that when a potential wave, or wave of electric flux, comes to the open end of a freed line, it is reflected backwards without loss of amplitude, of time, or of velocity, and without any change of sign. When, however, it comes to a grounded end, it is reflected back with reversed sign, or 180 change in phase. 78 APPLICATION OF HYPEKBOLIC FUNCTIONS Also, when a current wave, or wave of magnetic flux, comes to a grounded end, it is reflected back without any change of sign ; but when it comes to a freed end, it is reflected back with a reversal of sign, or 180 change in phase. The presence of the impedanceless generator E in the circuit, Fig. 37, does not change these conditions of reflection. The only action of the generator, which is significant in this discussion, is that it continually generates an electric disturbance at the home end, and seeks to send electromagnetic waves over the line. Initial Disturbances. If we close the generator switch S at the peak of a positive impulse, a wave will immediately be urged along the line under the instantaneously applied full e.m.f. Along with the normal wave, there will however be an exponentially decaying wave which dies out faster than the accompanying normal wave element. The first few waves in the advancing train will therefore not be sinusoidal ; although they will tend to recover the sinusoidal form as they advance. If, however, we close the generator switch at the moment when the generated e.m.f. is passing through zero, there will be, in the simplest condition, no abrupt disturbance to the system, and the outgoing waves are all sinusoidal. We shall assume that the switch is closed at the proper instant to avoid initial disturbance, as is always theoretically possible. Line Freed at the Distant End. If the aerial line-length is, say, just 300 km., a wave would traverse it in one millisecond, if there were no retardation due to attrition in conductor and dielectric (r = o, g o). We may suppose, for simplicity, that the line-length is so chosen that with the actual velocity co/a 2 , the time of one passage, or traverse-time, is just one milli- second. We now close the switch at A, and the first potential wave of the entering wave-train runs along the line undergoing attenuation as it runs. Nothing happens at B until after the lapse of one millisecond. Then the leading wave arrives, and instantly retreats backwards, as though the return were a prolongation of the wire that happened to be bent back parallel to the actual line. The amplitude of the wave at the TO ELECTRICAL ENGINEERING PROBLEMS 79 start from A was E volts. On arriving at B it had become Ee~ , where 6 is the complex line-angle. But the returning wave after reflection also has an amplitude of Ee- ; so that after the wave strikes the distant end B, it immediately doubles up, or produces an amplitude of 2 Ee~ 9 volts, which continues to undergo simple harmonic variation at B. When the head-wave gets back to A, it goes to ground, and returns reflected on the line as Es~ 2e volts. It runs back again to B, getting there in three milliseconds from the start. Its condition is now 2 Ee- 3 *, allowing for the doubling on reflection. It returns to A after the total lapse of four milliseconds, in the condition Ee~ 40 . It goes to ground and comes back instantly reversed, or as Ee~ 40 , running back to B, where it arrives after a total lapse of five milliseconds, in the condition Ee~ be , when it doubles up on itself and returns again to A. This state of to-and-fro activity continues theoretically for eternity ; but in any practical case the residue is insignificant after a compara- tively few traverses and milliseconds, owing to the continual attrition and attenuation. It is interesting to watch the process of accumulation at B. We have after the total lapse of 1 millisecond 2 Ee~ e volts [_ 3 milliseconds -2 Eg- 30 5 4-2Ee-*> 7 -2E -'> and so on. Each new term is added on to the general accumulation, and since each term is a vector, the addition must be made vectorially. The total accumulation at B becomes then E B = 2E(e-* - s~ B0 -f E- - e-' +....) . volts /_ (165) -**-'' rn= < 167 > = . f E r ^_J|_Egecb0 .... (168) e + e~ 6 cosh 6 a result which agrees with (22) when that formula is inter- preted vectorially. That is, the hyperbolic function final $0 APPLICATION OF HYPERBOLIC FUNCTIONS potential at B in the steady state is due to the superposition of successive jumps of potential, each smaller than its pre- decessor, which arrive at different amplitudes and phases according to a simple exponential law. As an example, we may take the case of a submarine cable having a linear resistance r = 10 ohms per naut, I = o ; c = J X 10~ 6 farad per naut, g o,f= 100 cycles per second, co = 628'3 radians per second. If an e.m.f. of 100 volts maxi- mum cyclic amplitude is applied without initial disturbance at A, required to find the potential at the distant free end when L, the length of the cable, is 24*906 nauts. Here z = 10/0^, y = 2-094 x 10- 4 /W, a = ^20-94 |90_ X 10- 4 = 0-04576 /45^ = 0'03236 +/0-03236 hyps, per naut ; z = 218\45 ohms; = 0'8059 +y0'8059 = M4/4T hyp. The amplitude of the first wave reaching the end B is 100 - <8059 \0-8059 radian = 100 e- ' 8059 \4CF2 5 = 44'67 \46'2. This doubles on arrival and becomes 89'34 \46'2, as indicated at 01 in Fig. 42. That is, the first potential wave would build up to 89.34 volts amplitude in a simple harmonic motion, and if no other waves arrived, the voltage at B would be 89*34/^/2 volts by static voltmeter. The phase of this voltage would be 46*2 behind that of the generator at A. The next return of the leading wave would be in the condition -100 e- 2 ' 4177 \F4T77 = -8-9/l38 T 5~= +8-9/41-5. The doubling up of this makes a rise l7'8/41-5 volts as indicated in the figure at O2. The total harmonic e.m.f. at B is now the vector sum of Ol and O2 or O2 1 , and if no other reflected waves arrived, the potential would perform this harmonic motion at B with a frequency of 100 cycles per second. The third return of the leading wave is in the condition 4-100 c - 4-0295 x 4-0295. = l'78/230 : r, which contributes on doubling, the element O3 = 3'56 /230'9. At the fourth return the condition of the leading wave is 100 - 5 ' 641 \5-641 = - 0-356 /323-3 , which contributes 0712 /143-3 volts. The TO ELECTRICAL ENGINEERING PROBLEMS 81 vector addition of all these components is OB = 88'17 \35'0. But by formula (22) the voltage in the steady state at B is E sech 6 = 100 sech (1-14/45) = 100 X 0'882 \3MT = S8*2 \35'0. Consequently, neglecting all terms after the 270 240 2(0 330 FIG. 42. Successive Leaps of the Potential at free end of a particular alternating- current line during the constructive stage prior to the stead}' state. fourth, the total increment of voltage agrees satisfactorily with the value determined by the steady-state formula. In the case considered, 6. 2 the j component of the hyp. angle of the line, i. e. the circular angle component, is 0*8059 radians, or a little more than 45 (46'16). Consequently the successive 82 APPLICATION OF HYPERBOLIC FUNCTIONS increments of voltage, which come at phase intervals of 20 2> are nearly perpendicular to each other. If, however, the / component of the line angle happens to approximate one right angle, or any odd number of right angles, the successive increments will be nearly in the same phase, so that the final distant end voltage will build up; or sech 6 will be greater than e~ e . This will be particularly the case when 2 is just 90, or the line has one quarter wave-length. If the attenu- ation is small, the successive increments do not rapidly dwindle, and being in the same phase, they may build up to a voltage greatly in excess of that impressed by the generator on the sending end of the line. This effect is called the Ferranti- effeet. While it must occur whenever the line is an odd number of quarter wave-lengths,- it will only be noticeable as an actual increase of potential towards the distant free end when the real component 6 l of the line angle is distinctly less than the imaginary component 6. 2 , i. e. when the attenuation is relatively small. No rise of potential can occur towards the free distant end of a line in the steady state when 6 l is greater than 6. 2 . On the other hand when 6 l is made sufficiently small, and 0. 2 = n/'2, the Ferranti-effect of potential rise can theoretically be made indefinitely great, and practicallv the free-end potential can be many times the impressed potential at the home end. A small terminal load usually suffices to- destroy the effect. (See Chapter VII.) If on the other hand the line has half-wave length, or any integral multiple thereof (0 2 = nit), where n is any real integer, the successive increments of potential during the constructive state arrive in opposite phases; so that the distant end potential does not tend bo build up, even with low attenuation. By a similar summation of outgoing and reflected waves at A, we should find the total amount to E A volts, or merely that impressed by the generator; because each reflection from ground at A takes the negative sign and cancels the effect of the arriving potential wave. Similarly, if we take the current-wave at B, with the line- TO ELECTRICAL ENGINEERING PROBLEMS 83 free : on the first arrival, the amplitude is I s~ amperes, where I = ^ ..... amperes /_ (169) o is the initial outgoing current at A. But the reflected current- wave from the open end at B is immediately I e~ e , which cancels the arrival ; so that the resultant rise of current is nil. The same action occurs at each return of the current- wave* Hence the current remains at zero throughout the steady state, as is, of course, the inevitable condition at an open end. Again, with the distant end B still free, let us trace the building-up of current-waves at A. The outgoing wave, as we have seen, is I . On first return from B it has become I e~ 20 y which on being reflected from ground at A, takes no change of sign, and so doubles the increment to - 2I e~ 20 . On second return from B it is + I _40, which likewise doubles at A. Continuing this process, the summation at A is I A = I - 2I c e-* + 2I - 4 ' - 2I - 6 ' + . . . amperes /_ (170) = L {l -2- 2 '(l-- 2 ' + - 4 '- . . . (171) {9c-20 ^| i-ITF" .... > < 172 > amperes /. (173) " " (174) which agrees with (21) when that formula is interpreted in complex numbers. Line Grounded at Distant End. If we ground the line at B, the series of reflected current- waves returning to the sending end A is the same as in the preceding case (170) except there is no change of sign in the successive elements. The sum- mation of current at A is then G 2 I 84 APPLICATION OF HYPERBOLIC FUNCTIONS I A = I o + 2I - 2 ' + 2I - 4 ' + 2I o -<* + . . . amperes L (175) = I o {l + 2 - 2 ' (1+e--' + - 4 '+ . . . (176) = I o coth d (177) E, z c tanh 6 ' ' " which agrees with (24) when that formula is interpreted in complex numbers. Again, if we ground the line at B, and sum the current-waves arriving at B, their conditions are I ~ d , I ~ 3e , I e~ 5fl , etc. Being reflected at B without change of sign, they contribute doubled increments at B. Hence the summation at B is IB- 2(I e- + I e- 3 * + I o e- 3 + ..... amperes /_ (179) = 2I - e (l + - 20 + e-^+ ...... (180) nan which agrees with (25) when that formula is interpreted in complex numbers. With the line grounded at B, the potential waves cancel their arrivals on each reflection and at each end of the line ; so that the summation of potential at the sending end is always E A , and at the receiving end always zero. If now the line is grounded at B through an impedance z r ohms, any current- wave on arriving at B is split into two parts ; namely, a transmitted part which is absorbed to ground without further reflection, and a reflected part which goes back as though from an open end. Let m be the fraction of the wave that is transmitted or the transmission coefficient, and \ m the fraction that is reflected or the reflection coefficient. It was shown by Heaviside that in the symbols here used 2~ m = . . numeric /_ (183) ~ i Z r TO ELECTRICAL ENGINEERING PROBLEMS 85 the reflected current-wave retreats with its sign reversed, or as with a coefficient m 1. The current- wave to ground at B on first arrival is mI - d , and (m 1)I ~* goes back to A. It reaches A in the condition (m 1)I ~ 2 *, and is reflected back to B without change of sign. It arrives at B for the second time in the condition (m l)I o e~ M . Of this m (m l)I e~ 3 * is absorbed to ground, and the remainder (m l) 2 I e~ M retreats to A. The final summation current absorbed to ground at B is I B = wI c -'+m (?tt-l)I - 3 * + w(w-l) 2 I e- M + - - . amperes L (185) -l>- 2 +(w-l)%-^+ . (186) mI e /m-1 in V ni I + s i n h cosh 6 + sinh (9 m ^ =- -0 amperes /. (188) sinh + P cosh 6 Z Q sinh + ^ cosh (9 ''//' which agrees with (59) when that formula is interpreted in complex numbers. It can readily be seen that the potential at B in the steady state is IB^V volts, and that potential reflections at A cancel off. We might similarly sum the potential or current waves at any intermediate point on the line and derive formulas (37) to (42). CHAPTER VII THE APPLICATION OF HYPERBOLIC FUNCTIONS TO ALTERNATING-CURRENT POWER-TRANSMISSION LINES. ALTERNATING-CURRENT power-transmission lines differ from alternating-current power-distribution lines in having only terminal loads applied to them, as distinguished from a num- ber, usually a large number, of intermediate distributed loads. They are ordinarily of the three-phase type, as indicated in Fig. 43, and consist of three line conductors. The system may be regarded as being divisible into three independent single-phase lines, AB, A'B', A"B", each operated, at star voltage, to FIG. 43. Set of Three-phase Transmission Wires. Nominal circuit of three- phase set of wires. Three equal condensers connected in star between mid- length points 0, 0', 0". ground, or neutral-potential surface. If the three wires are symmetrically disposed, the three individual single-phase lines will have equal wire inductances, and also equal wire capacities, acting like a star group of condensers, with zero potential at the neutral point (Fig. 43). The corresponding conditions for a two-wire system are indicated in Figs. 44, 45 and 46. If the geometrical disposition of the three wires in the system of Fig. 43 is dissymmetrical, the three individual inductances 86 APPLICATION OF HYPERBOLIC FUNCTIONS 87 and capacities of the independent single-phase wires will be unequal, but can ordinarily be computed from the geometrical data. FIG. 44. Diagram of an Alternating-current Circuit with the entire Line Capa- city centered at B, and the resistance and inductance divided between the four choking-coils L, L, L, L. Every alternating-current power-transmission system may therefore be analysed into a group of parallel wires, each oper- ating independently to ground potential. Strictly speaking, FIG. 45. Division of the Circuit of Fig. 44 into two equal and symmetrical portions about the neutral mid-plane 00 of zero potential. the capacity of each line is uniformly distributed, and it is the recognition of this condition that introduces hyperbolic func- tions as the natural key to the true behaviour of such lines '.f i A 1 B' L' C' FIG, 46. 46. Analysis of the Double-wire Circuit of Fig. 44 into two equivalent single-wire circuits A, B, C and A', B', C', each having twice the condenser capacity of the circuit in Fig. 44. in the steady state ; but, as a first approximation, which is suf- ficiently good for all but very long lines, at ordinary operating 88 APPLICATION OF HYPERBOLIC FUNCTIONS frequencies, the capacity may be either lumped into a single condenser at the middle of the line, as in Figs. 45 to 47, thus forming the nominal T of the line; or, as is usually more convenient for the purposes of computation, all the capacity may be collected into two equal condensers, and these applied one at each end of the wire, as shown in Fig. 48, thus forming the nominal II of the line. This method has been called the G FIG. 47. T-Conductor Equivalent to an alternating-current transmission wire. " split condenser " method of analysing approximately the electrical behaviour of a transmission line.* One wire AB of a transmission system is represented, in Fig. 49, as operated to neutral potential. The wire has a vector impedance R -f-;/X = Z /_/? ohms. At the receiving end B, is a motor M, or other load, of known magnitude and power-factor, at a given voltage EB. At the sending end A, a single-phase generator G,. Gr <* FIG. 48. n-Conductor Equivalent to an alternating-current transmission wire- delivers such a voltage E A as will maintain the given voltage EB at B. Half of the capacity of the wire to neutral surface is applied as a condenser at A, and the other half at B. Each such condenser offers an admittance of Y A == Y B mhos. The admittance Y B receives a current I' B from the voltage E B ;. while Y A similarly receives a current I' A from the voltage E A . The line current I has the same strength at all points between A and B. It is the vector sum of the load current I B and the condenser current I' B . All voltages are r.m.s. vector star- * " Calculation of the High-Tension Line, " by P. B. Thomas : Am. Inst. Electrical Engineers, Part I, vol. xxviii, pp. 641-68(5,. June 1909. TO ELECTRICAL ENGINEERING PROBLEMS 89> voltages, in the case of a three-phase system, and all currents are r.m.s. vector amperes. We take the phase of the voltage EB as standard, and refer all other voltages to this phase. M FIG. 49. Circuit Connections and Vector Power Diagram for one wire of a power- traosmission system. Nominal IT. The diagram O a I g e, Fig. 49, is the stationary vector power diagram relating to this wire. Let the vector O6 represent, to scale, the power load delivered at B in the branch M to the phase of current I B as standard. Denote this load by PB=E B . IB LP - - watts L ( 19 ) or volt-amperes /__ 90 APPLICATION OF HYPERBOLIC FUNCTIONS Then the horizontal component Oa of this load is the effective power delivered ; or P /B = E B . IB . cos ft . . watts (191) and the vertical component ab is the reactive power; or Ptu =/E B . IB - sin ft . ./watts (192) The power-factor of the load is cos /?. Since the receiving end of the line is assumed to be main- tained at a steady star-voltage E B , the charging current in the admittance Y B is I' B =yE B . Y B . . amperes /_ (193) The power absorbed by this admittance is p B = E B I'i3 = /E 2 B . Y B . j watts (194) This is -\- j reactive power, to voltage standard phase, but must be reckoned as j reactive power with respect to cur- rent standard phase. It is, therefore, measured along ba, and is indicated at be. The power supplied at the B end of the line including the admittance Y B , or half-line condenser, is Or watts or volt-amperes. In other words, part of the reactive power in the load is supplied by the end-condenser YB. The line current is the vector-sum of the load current I B and the condenser current I' B ; or I = I B -f- I' B r.m.s. amperes {_ (195) The power expended in the line is I 2 Z = PR +/PX . watts or volt-amperes /_ (196) where the phase of the line current must be taken as standard. This power is represented by the vector ce, cd being the effective or dissipated power, and de the reactive or non-dissipated power. Consequently, the power supplied to the line at A, beyond the condenser Y A , is represented, to scale, by the vector Oe = P' A . The power developed in the condenser admittance Y A is /E A 2 Y A watts, or p A on the diagram. For this item, we find the value of E A ; namely E A = E B + IZ . volts L (196) TO ELECTRICAL ENGINEERING PROBLEMS 91 Finally, the power supplied by the generator G to the end A of the single wire considered, including the condenser Y A , is Of = P A vector watts or volt-amperes ; of which the horizontal component Og = P /A = E A I A cos y is the effective component, and Pjt A = gf is the reactive component. The power-factor at the generator is cos 7. The electrical efficiency of the line is Oa/Og. The simplest method of taking distributed capacity into account in such a problem is to substitute, by formulas (77) and (78), the equivalent 77 for the nominal II. We may take the example of a three-phase transmission line having a length L = 250 km. (155*34 English statute miles), consisting of three No. 000 A.W.G. copper wires, 1*041 cm. in diam. (0*41"), supported symmetrically, on pole insulators, at a uniform interaxial distance of 193 cm. (72"). The following values of linear resistance, inductance and capacitance are taken for each of these three wires, to neutral potential surface r = 0'206 ohm / wire km. = 0*33 ohm / wire mile. I = 1*22229 millihenry / wire km. = 1*967 millihenry / \vire mile. c = 0*0094828 microfarad / wire km. = 0*01526 microfarad / wire mile. The linear leakance is taken as negligible. The frequency of operation is /== 25 cycles per second; or o> = 157*08 radians per second. With the above linear secondary constants we obtain for the total line constants Lr = R = 51*5 ohms, LI = 0'30557 henry, Lc = C = 2-3707 microfarads. We may assume that the star voltage at the receiving end of the line is 50 kilo volts r.m.s. or 86*6 kv. between any pair of the three wires. At the above fundamental frequency, the linear reactance of each wire will be jx =jla> = /l*22229 X 0*15708 = /0*191996 ohms per km. The total line reactance jLx =/47*999 ohms. 92 APPLICATION OF HYPERBOLIC FUNCTIONS The linear dielectric admittance jcco =/l '48955 X 10 ~ 6 rnho per km., and the total dielectric admittance jY =j"Lcco = ^3-72390 X 10 ~ 4 mho. The total wire impedance is therefore Z = 51-5 -f y47D99 = 70'40 /42 59' 05" ohms. Fig. 43 shows, at AB, the nominal H of this line for the above-mentioned frequency and conditions. It consists of a line impedance with half the capacity susceptance at each end ; i.e. jl '86194 X 10 " 4 mho corresponding to a reactance of 537074 [90 5 ohms. The hyperbolic angle 6 subtended by the line will be, by (19), 6 = v'ZY = x/70'40/42~59 / ~05 // x 372390 X 10 - 4 90~ = V 0-0262167/132' 59'" 05" = Q'161915/66 29' 32" = 0-064583 + yO-148476 hyp. The nominal /7 of this line is presented in Fig. 50, at ABGG, for the frequency of 25 ~. The architrave impedance is the line impedance Z above mentioned, and half of the dielectric admittance is placed in each pillar. In order to form the equivalent II of the line, we require to f\ . , tanh -- form and apply the ratios ~ and ^ .* By the help of 2 (139) and (145) we find sinh 6 = 0-161431 /66 40 7 32" r\ and tanh - = 0'0810865 /66 23' 17"; consequently- fy, = = 0'99703 /O^ir 00" = = 1-0016 \0 06' 15". * Tables of fland tanh . with five significant digits have been 0/Z computed by the writer for each degree between 60 and 90 of argument and each O'l of modulus in up to 0'5. These tables are shortly to be published (Bibliography, 73). TO ELECTRICAL ENGINEERING PROBLEMS 93 That is, the correction-factor which transmutes the nominal into the equivalent 77 for this 250 km. line differs from unity 6 = 0-16)91 fl-fO + jLj.oao-=.'r/>-LO-/W'f9''Of* w B r X o 0-99703 /o'-it'-oo' /OCt6\o:o6' =70-192 x 0: II g FIG. 50. Nominal and Equivalent n for the particular transmission line at 25 ~. by only 0*3 per cent, for the architrave and 016 per cent, for the pillars. In other words, the correction for distributed capacity is negligible for such a line. 94 APPLICATION OF HYPERBOLIC FUNCTIONS In Fig. 50, when we multiply the architrave impedance AB of the nominal 77 by 0*99703 /0 11' 00", we obtain the architrave impedance 70192 /43 10' 05" ohms, a, I, of the equivalent 77. This means that the line behaves in the steady state, at this frequency, as though its conductor resistance were reduced from 51*50 to 51 '21 ohms, and its inductive reactance increased from 47*999 to 48*01 ohms. Similarly, multiplying the pillar admittance AG or BG, of the nominal 77, by 1*0016 \0 "~0()' 15", we obtain the pillar admittances 1 '86492 X 10 ~ 4 /89__53' 45" mho, a,g, or I, g, of the equivalent 77.* This is equivalent to assuming that either a certain small resistance (9*73 ohms) is inserted in series with a slightly increased condenser (11873 ///) ; or, that a non-inductive leak of 0'339 micromho, has been applied to each condenser in shunt. We shall retain the latter conception for convenience. The circuit-connections and the vector power diagram, for one wire of the line considered, are given in Fig. 51, under an assumed load of 4000 kw. (4 megawatts) of effective power (12 megawatts for the entire three-phase system), at a power factor of 0*8. The apparent or resultant power delivered at B is, therefore, P B = 5*0 /36 52' 12" megawatts, to standard current phase, and the inductively reactive power j/'3 megawatts. The current received through the load, under 50 kilovolts at B, is thus 80 - jW = 100 \36 52' 12" amperes to B- voltage phase. That is, the load current lags by this angle behind the voltage at the receiving end of the line. The current in the leaky condenser at B is 0*01695 +^'9*3245 amperes, carrying a power of 0*8475 -f ^466*23 kw. to B-voltage phase, or 0*8475 /4G6*23 kw. with respect to current phase. The total power delivered at B, including the terminal condenser, is thus : 4*000848 +y2*53377 megawatts. All pressures and currents are expressed in r.m.s. values. The current in the line is 80'017 /50*676 amperes to B- * It will be understood that the degree of arithmetical precision aimed at in these examples, for the sake of thoroughness, is much greater than is necessary in ordinary transmission-engineer icg computation. TO ELECTRICAL ENGINEERING PROBLEMS 95 voltage phase = 94'714\32 20' 47". The IZ drop in the line is 94-714 32 20' 47" X 70192/43 10' 05"=6648 /10 49' 18" = 6530 + yi248 volts. The I 2 Z power is (94714 /OT X 70192 /43 10' 05" = 629686 /43 10' 05" watts = 0-6297 p 4 . 1-44* FIG. 51. Circuit Connections and Power Vector Diagram for one wire of the particular transmission line at 25 ~. /43 IP 7 05" megawatt =- 0*45926 + /043080. In this power computation, the current must be taken as of standard phase, as all power products P = E I watts /_ , require one of the vectors E or I to be taken at standard phase, or zero argument. The voltage at A is the vector sum of the B-voltage and 96 APPLICATION OF HYPERBOLIC FUNCTIONS the IZ line drop. It amounts to 56,530 +/ 1248 = 56543 /1 15' 54" volts. At this voltage, the current in the leaky condenser at A is 56543 /1 15' 54" x 1*86492 x 10 ~ 4 /89 53' 45" = 10-5449 /91 09' 39" amp. = - 0*2136 + j 10*5427. The power delivered to this condenser is 56543 /0 X 10-5449 /89 53' 45" - 596,247 /89 53' 45" watts = 1084 -\- j 596,240 watts with reference to voltage phase, or 1084 y 596,240 watts with reference to current phase. Adding this power vectorially on the diagram, by the vector ef, we arrive at /, and Of is the vector power delivered by the generator at A = 5'051 /27 57' 45" megawatts = 4'461 + j'2'368 megawatts at a power-factor of 0'8833. The current delivered by the generator is the vector sum of the line current and the A-leak current ; or 79*803 / 40*133 = 89-322 \26 41' 52" amperes, under a pressure of 56,543 /1 15' 54" volts, with a power of 89.322 /0^_ X 56,543 /27 57' 46" = 5,051,000 /27 57' 46" watts (current phase) which checks the preceding result. The efficiency of the line under this load is 4*0/4*461 = 0'8967. It is evident that the power diagram and line computations would be only slightly modified, in this case, if we employed the nominal 77 of the line, instead of the equivalent II. The com- putations would also be simplified ; because the leaks at A and B are pure reactances or /-quantities in the nominal II, and pertain to pure condensers ; whereas, in the equivalent 77, they are complex quantities, or pertain to leaky condensers. Relation of Line Angle to Length and Frequency. The hyper- bolic angle 6 subtended by a uniform alternating-current line manifestly increases with the length of the line. If the line had no conductor resistance, or dielectric leakance, the angle would be, by (150) 6 =jLco ^/lc . . hyp. radians (197) = Leo *Jlc, . circular radians (198) which shows that it increases directly in proportion to the frequency. To a first approximation, therefore, the line angle TO ELECTRICAL ENGINEERING PROBLEMS 97 increases with the frequency, provided that the dissipative linear constants of the line (g and ?) are relatively small, and this is true for power-transmission lines. As a rough rule, we may say that 1000 km. of wire such as is used in power- transmission, operated at a frequency of 50 ~, has a line-angle of about 1 hyp. at an argument usually between 60 and 80. Consequently, to the same low degree of precision, the modulus of the hyp. angle subtended by a wire of L km. operated at a frequency of / ~ is roughly Thus, although the hyperbolic angle subtended by a trans- mission line, at its fundamental working frequency, may be so small that there is very little difference between its nominal and its equivalent T or 77, yet steadily increasing differences will develop with the ascending harmonics in the impressed e.m.f., if such harmonics are present. In a properly constructed three-phase system it is well known that no harmonic frequencies can exist of three times, or of 3n times, the fundamental frequency. Such harmonics as exist must be of 5, 7, 11, etc. times the fundamental frequency. In Fig. 52, the nominal and equivalent 77's are presented for the quintuple frequency of 125^ in the case of the 250-km. line al ready .considered. The nominal U of the line differs only from the nominal 77 at 25 ~, in having five times the inductive reactance in the architrave, and five times the condenser sus- ceptance in the pillars. The hyperbolic angle of the line is 6 = ^245-458/77 53' 19" X T86194 x lQ-^90 = 0'676039 /83 56' 40" = 0-071318 + ./Q-672267 hyp. The correcting ratios for this angle are k pti = 0'92722 /0 56' 40", and ky a = 1-03893 \0 28' 50". Applying these factors to the architrave and pillars of the nominal 77 respectively, we obtain 227-595 /78 49' 59" ohms for the architrave, and 9'6721 x 10~ 4 /89 31' 10" mho for the pillars of the equivalent 77. The corresponding conditions for the septuple-frequency 98 APPLICATION OF HYPERBOLIC FUNCTIONS /cm,. 0= B = 0-92722 ^Wuf a 44. FJG. 52. Nominal and Equivalent n for the particular transmission line at 125 ~. harmonic 175~ are indicated in Fig. 53. The hyperbolic angle has reached 0'941311 /8o 38' 84" hyp. The correcting TO ELECTRICAL ENGINEERING PROBLEMS 99 2 SO at = O> u c- V S G 0= 0- 335"-993= 339- 9f 7 /8l og' * 4v FIG. 53. Nominal and Equivalent n for the particular transmission line at 175 ~. factor k p has a modulus of 0'86039, and l> 9ii 1-08088. The equivalent 77 has not only less line reactance, but also less H 2 100 APPLICATION OF HYPERBOLIC FUNCTIONS line resistance than the nominal II, which means that a given current strength delivered over the line dissipates less power in transmission, by reason of the uniform distribution of capacity, compared with that which would be dissipated with the capacity in two terminal lumps. Graphical Method of Combining Harmonic Maximum or R.M.8. Values of Voltage or Current into a Resultant Maximum or R.M.S. Value, If the impressed e.m.f. at the generating end of the line is impure, and the magnitudes of the various harmonics are known, then, in order to determine completely the distribution of voltage and current over the line, taking distributed capacity into account, it is necessary to compute the equivalent H or T of the line for each harmonic frequency, as well as for the fundamental, to ascertain the impedance which the load at B offers to each frequency respectively, and to compute the voltage-current distribution for each frequency independently, in the manner indicated at AB, Fig. 51. Finally, knowing the components of r.m.s. voltage, or current, at any point in the system, the resultant r.m.s. value is found by the process of " crab addition," or the successive addition of components, each added perpendicularly to the last resultant. Thus, in Fig. 54, let OA be to scale, the r.m.s. value of a fundamental frequency component of voltage, or of current, at a given point in the system, AjB, B 1 C, Cj_D, other-frequency r.m.s. com- ponents of the voltage, or of the current, at the same point, in any order of selection. In practice A X B might be a 5th harmonic (quintuple-frequency harmonic), B X C a 7th har- monic, and CjD a 13th harmonic, and so on, but the proposition applies equally well to components of any different frequencies, whether harmonic or not. Then, no matter what the relative phases of the different components may be, the resultant r.m.s. value of all the components may be found graphically by adding them rectangularly, and successively, in any order. Thus, taking OA as TO, repre- senting say 1000 volts r.m.s. as the voltmeter value of a fundamental e.m.f., associated with a 5th harmonic of 0*333, or 333 volts r.m.s., also with a 7th harmonic of 0*111, or o TO ELECTRICAL ENGINEERING 101 111 volts r.m.s., also with an llth harmonic of 037 or 37 volts r.m.s., the resultant of all would be Od = 1'0605, or 1060'5 volts r.m.s. The same reasoning and process evidently applies if each component is expressed in terms of its maximum cyclic, instead 1-0 B( ^IAU C 0.037 FIG. 54. Composition of Fundamental and Harmonic Frequency R.M.S. com- ponents of voltage or current into a resultant R.M.S. value by tlie process of *' crab addition " or perpendicular summation. of its r.m.s. value. In such a case, the resultant would also be a maximum cyclic value. Moreover, if each harmonic component is analysed into two quadrature sub-components, of the sine and cosine type respect- ively, such sub-components, although of the same frequency, may be included correctly in the rectangular summation process. In other words, quadrature sub-components of any harmonic component of voltage, or current, act, in this particular, as though they had different frequencies. Expressing the same 102 / .^JPL5Ci\TI0ffiir t ; 0? * HYPERBOLIC FUNCTIONS proposition algebraically, if a complex harmonic quantity be analysed into the Fourier series A + B sin tat + C sin 3 at + D sin 5 ut + E sin 7 ut + + /> COS tat + C COS 3 + d COS 5 cat + e COS 7 atf + where the constants ABC, etc., are maximum cyclic values. Then the resultant modulus is well known to be 2 + BM- + C 2 + e* + D 2 + # + E* + e 2 + which is obviously the value given geometrically by the rect- angular summation process. Moreover, since the maximum cyclic value of any single harmonic component is ^'2 times its r.m.s. value, the proposition must be capable of application either to maximum cyclic, or to r.m.s. values, throughout. Summing up, then, the conclusions reached in this chapter; we may say that power- transmission lines of the greatest lengths in ordinary industrial service to-day, operated at ordinary frequencies, do not require correction for- distributed electrostatic capacity, if analysed on the basis of the nominal /7> or split-condenser method, unless great precision is required ; because the hyperbolic angle subtended by such lines is usually less than 0'5 in modulus. If, however, high harmonics have to be taken into consideration, the hyperbolic angle subtended by the line may be over 1 hyp., and the correction for dis- tributed capacity in such cases may be material. In every case where a correction for distributed capacity is required, the simplest method of effecting it is to substitute the equivalent for the nominal 77 of the line, at the frequency considered. Fermnti- Effect. The property of an alternating-current power-transmission line to develop a higher voltage at the receiving end than at the sending end, or to possess a negative drop of potential, has been called the Ferranti-effect, having been first pointed out in connection with the Deptford-London power transmission cables in 1890.* ' * " Capacity and Self-induction in Alternate Current Working," by Gisbert Kapp,~ The Electrician, Dec. 19, 1890, p. 197, and Dec. 26, 1890, p. 229. " On the Rise of Electromotive Force observed in the Deptford Mains, 1 ' by R. T. Glazebrook, The Electrician, Dec. 26, 1890, pp. 232-233. TO ELECTRICAL ENGINEERING PROBLEMS 103 The Ferranti-eftect is observable on such lines only at or near no-load. It usually disappears with a very small load on the line. It is commonly supposed to depend upon the influence of distributed electrostatic capacity in the line ; but it is pro- duced by the charging current of the line passing through the inductive reactance of the wire, and this charging current may be regarded as due to the total' line capacity lumped in a single condenser at the middle point, as in the nominal T ; or lumped in two split condensers, one at each end, as in the nominal II. The only influence exerted by the distribution of capacity on the Ferranti-eftect is that due in detail to the substitution of the equiva- lent T or II for the nominal T or n. Referring to the nominal II of such a line-wire as is represented at AB, in Fig. 49, let E A be the r.m.s. vector voltage impressed at A, in the steady state. Let the end B of the line be freed, so that there is no load on the line, and let E B be the vector r.m.s. voltage developed at B. Then, if Z = R +/X is the vector impedance of the line and Z,. = yX c the vector impedance of the semi-line condenser at B, we have, as in the continuous-current circuit E B = E A ^-^ volts L (200) FIG. 55. Vector Diagram indicating the Limitation of the Ferranti-effect as deter- mined by the line-resistance. 104 APPLICATION OF HYPERBOLIC FUNCTIONS That is, the B-voltage is the A-voltage multiplied by the 2 vector fraction ^ c -=- In Fig. 55, let ob represent, to a b -\- L c scale of ohms, the line nominal impedance Z, as the vector sum of the line resistance R and reactance /X. Let ~bc be the impedance Z ( , to the same scale, of the semi-line condenser at B. Then oc will be the vector sum Z+Z C . Consequently, in (200), the B-voltage is the A-voltage multiplied by the ratio be . This ratio will always be greater than unity, if the OC condenser impedance be is greater than the line reactance ab, provided that the line resistance ca is less than da, the point d being on the circle drawn with centre c and radius cbjZ c ohms. In practice, the line reactance j"X. is always small compared with the semi-line condenser reactance jZ c ; so that the B-voltage in the steady state must exceed the A-voltage at no load ; i.e. the Ferranti-effect must occur on any normal well insulated line, unless the line resistance R is less than the critical value da ; or algebraically, unless R < x/X (2Z, ; - X) ... ohms (201) Thus, in the case presented by the nominal /7, AB, of Fig. 51), where R = 51-50, X = 48, Z c = 5370'7 ohms, the Ferranti- erfect occurs, and must occur, until R is not less than /48 (10741-4 - 48) = 714'8 ohms. The equivalent 77 of this line (ab t Fig. 50) differs so little from the nominal 77, that this deduction is scarcely affected by the uniform distribution of electrostatic capacitance. Since the linear resistance of all power-transmission lines must be kept relatively low, their line resistances are, in practice, well below the above critical value ; so that they exhibit the Ferranti-effect, at no load, almost invariably. But the magni- tude of the effect is ordinarily very small, although it increases when the length of the line is increased. Thus, if we repeat the diagram of Fig. 55 to the scale pertaining to the nominal 77 of Fig. 50, and the 250 km. power wire operated at 25 ~, TO ELECTRICAL ENGINEERING PROBLEMS 105 we obtain the diagram of Fig. 56, where ob is the line-wire pedance of 51'5 +./4S ohms, and Ic the split-condenser im- pedance of y537074 ohms. The vector oc is therefore 5322-99 ohms Ferranti - effect =_ = 1-00897 \0 33' 16 O and the factor be = 5370-74 90 oc~ 5322-99 \ 89 26' 44 which means that the voltage at the B-end of the line exceeds the impressed voltage at the A-end by 0'897 per cent., and lags in phase by 33' 16". The substitution of the equiva- lent 77 for the nominal II of the wire barely affects this result. If, however, we increase the length of the line : or if, leaving the line-length unchanged, we increase the frequency of operation ; then the line reactance ab increases, while the capacity reactance Ic diminishes, so that the Ferranti-effect factor increases. At the same time, the change from the nominal to the equivalent 77 becomes more marked. Conse- quently, the Ferranti-effect, which is in- significant on ordinary aerial lines, at ordinary low frequencies, may become very large at ex- traordinary lengths of line, and especially at upper harmonic frequencies. In order to determine the maximum Fer- ranti-effect factor that can be produced on a given line, it is expedient to refer to Fig. 42, which shows the successive vector additions to the receiving, or B-end, free voltage, as built up by reflections during the unsteady state It is evident from an inspection of that diagram, that in order to build up the maximum B-voltage it is necessary that there shall not only be small attenuation on a FIG. 56. Vector Diagram indicat- ing the magni- tude of the Fer- ranti-effect in the case of the particular line at no load. 106 APPLICATION OF HYPERBOLIC FUNCTIONS the line ; so that the successive modulus additions may be large, and numerous ; but also that they should arrive in the same phase-direction. This means that the argument of the angle 6 shall be j3 = n/2 radians, or 90 ; because the phases of the successive increments are 2/? apart. But we have seen that this requires the /-component of the hyperbolic line angle 6 to be 71/2, and the wave-length A of the line to be just four times the length of the line. In other words, the line must be a quarter-wave in length. When a line is operated at such a frequency as makes it a quarter wave-length, then the successive voltage increments _ in the preliminary unsteady state fall vectorially into line with each other, and build up the maximum Ferranti-eftect multiplier that the attenuation over the line will permit. On the contrary, if the frequency of operation and line-length are such that the line has half a wave-length, then the successive increments of voltage in the unsteady state arrive in alternate directions and produce a minimum Ferranti- eftect factor. The same proposition applies with reduced force to all quarter and all half wave-lengths, as the length of line is increased. In the case of the 250-km. power-transmission line above considered, it can be readily found that the frequency of f= 293*424 ~, with CD = 1843*64 radians per second, brings the line into resonance at one quarter wave-length. For the line angle at this frequency is 6 = O071650 + /1/5708 - 0-071650 +j n/2 = 1'5724338 /87 23' 18" hyps. The at- tenuation-constant is also a = (2-86599 -f/62-832) 10" 4 hyp. per km. The wave-length is then by (162) A = 2^/(62'832 X 10~ 4 =1000 km. and the line-length is just one-fourth of this. The velocity of propagation is also, by (159), v = co/o, = 1843-64/6-2832 x 10- 3 = 293424 km/sec. This frequency is 11737 times the fundamental frequency of operation, and is, therefore, not exactly a harmonic frequency ; but if the fundamental frequency were increased from 25 to 26"675 ~ or by 6"7 per cent., it would then become exactly the llth harmonic frequency. Under these special circum- stances of an llth harmonic frequency, we should expect to TO ELECTRICAL ENGINEERING PROBLEMS 107 derive the maximum Ferranti-effect possible on this type of line- wire. The nominal and equivalent 77 of the line-wire for this resonant frequency are indicated in Fig. 57. It will be noticed that the equivalent line resistance has fallen from 51*5 to 16 - 435 ohms, ignoring "skin-effect" or extra resistance due to imperfect conductor penetration, which begins to be appreciable at this frequency for the degree of precision under considera- tion. The line reactance has fallen to /360'31 ohms ; while the semi-line condenser reactance has fallen to 9*355 /359t>4 ohms ; leaving in circuit a total impedance of 25*790 -j-/0'67 = 2-V7985 /I 29' 24" ohms. The vector-diagram of this case is presented in Fig. 58. The FerranU-effect factor i, the ratio = 13*945 90, which means that the B- voltage at the receiving end lags 90 behind the impressed voltage at A, and is 13*945 times as large. This checks the result of formula (22) ; because cosh 1-57243 /87 23' 18" = 0*0717109 |90, and the voltage at the distant free end of the line is E A /0*07l7l09 90 = 13*945 E A 90 5 . No rise of voltage nearly so great as 13*945 -fold has yet been reported upon any actual transmission line. The con- ditions are, however, very special, since a 250-km. line, perfectly insulated, is assumed to be operated at the relatively high frequency of 293*424 ~. Nevertheless, tests made in the laboratory with an artificial power-transmission line at a similar frequency have produced a resonant rise of potential of the same order, in good agreement with the results obtained by hyperbolic formulas, and if an actual line of the length and conditions here considered failed to develop so large a Ferranti- effect factor, it would be owing to attenuation and energy- dissipation due to extraneous causes omitted from the preceding calculations, such as imperfect insulation, dielectric loss, or the like. In practice, on actual long lines, no such large Ferranti-effect T 108 APPLICATION OF HYPERBOLIC FUNCTIONS zyo 0=0.0] M6:f -fji-5"3os = 1-572,43/875 ^3-/8"' A *i m *+] b%5-3^2. = 5%5'-1M /84- 4-6'.36" o> ^ - ?o 1 i C v_S CK> Wl <^J o> ^5 VJ tf; vo V, X -^ X 6 t VD! O V o| * i r No g h. 1 ; 3 3 T 00 G uxrvrt ~ ~T" - J'ij/91 y-W-m J360-3J = 8g8= B Fi(i. 57. Nominal and Ecjuivalent n for the particular transmission line and the resonant frequency 293 '424-'. factor is likely to be encountered at fundamental operating frequencies ; because as the line is increased in length to TO ELECTRICAL ENGINEERING PROBLEMS 109 develop quarter-wave length, the line resistance increases, and so adds to the base oc of the vector-diagram, Fig. 58. More- over, even if the line had such a length as corresponded to the resonant condition, the effect disappears very rapidly as load is applied at B ; because the large resonant rise clearly depends upon a somewhat sensitive adjustment of posi- tive line-reactance in opposition to a nearly equal negative condenser-reactance in the pillar of the equivalent II. Shunting the pillar by a load would tend to destroy the balance, to take the resonant load off the generator, and to keep down the excessive voltage at B. It would seem, therefore, that the danger of resonance could be avoided by keeping the line always under load, either at the distant end, or at intermediate points, or at both. Corona losses along the line would also prob- ably assist automatically in keeping down the excessive potential. But if the generator delivered voltage to the 250-km. line at a fundamental frequency of 26*675 ~, and happened to possess an appreci- able llth harmonic, say 5 per cent, of the amplitude of the fundamental; then, with no load on the line, there would be a distant-end harmonic component of about 70 per cent, of the fundamental, and the resultant B- voltage would be v/1 + 0'7 2 = 1 -22 times the A-voltage, or a Ferranti-effect of 22 per cent, due almost wholly to harmonic resonance. This effect would speedily disappear with the application of load. There- fore, when an aerial line of such a length as 250 km., operated at as low a frequency as 25 ~, displays an evident Ferranti- effect, the presence of an upper harmonic nearly in resonance with the line is to be suspected. For convenience of reference, in connection with the hyper- bolic theory of transmission lines, the data concerning the . Vector indicat- the Ferranti- eflect in the case of the particular line at no load and resonance. 110 APPLICATION OF HYPERBOLIC FUNCTIONS particular 250-km. wire above considered are collected in the following Tables Table of Fundamental Data. Frequency Angular Velocity rad/sec. Hvp. Angle 9 Hyp. Radians. Modulus Argument. Snrge-lmpedance ohms __ Modulus Argument. Transmis- Velocity i) km/sec. Attenuation- Constant Ct| Cto Hyp/km.Rad/km. X 10" 4 X 10" 4 2-58332 5-93904 2-85272 26-89066 2-86057 37-54362 2-86599 62-832 25 125 175 293-424 157-08 785-398 1099-556 1843-635 0-1 61 91 4/66 29' 32" 264,4-7 292 070 292,875 293,424 434^0 \23 3d' 28" 0-676039 /8356'40" 0-941311/>'538'34" 1 -572434^7 23' 18" 363 -083 \ 6 03' 20" 361-1 10 \4 21' 26" 359-768 \2 36' 42" Table of Secondary Data. Frequency. 125 175 293-424 Sinh 9. Modulus Argument. Cosh 9. Modulus Argument. TanhA. Modulus Argument. Ferranti-effect Factor.* Modulus Argmnt. 0-161431 /66 40' 32" 0-9911077 /0 33' 10" 0-0810865 /66 23' 17" 1-00897 \033' 0-626835 /84 53' 20" 0-78566 /3 14' 37" 0-351178 /8327'55" 1-2728 \3 15' 0-S09S95 /87 00' 23" 0-595244 /5 34' 00" 0-5082545 /84 56' 21" 1-6800 \534' 1-002557 '90 0-0717109 90 1 -000 /85 53' 54" 13-945 fiKF Table of Correcting Factors. Frequency. Sinh 9 Tanh ( 02 > ' 9 (92) 125 1-0016 VJ'Otf' h' 175 1-03693 \0"2o- uo ' 93*4 9 4 1 -080*8 \0 4-2' 13" 1-27191 \r 29' -'4" * In the case of an actual line in California, 248 km. long, with a total single-wire resistance of 50 ohms, a wire capacity to neutral surface of 2' 2 microfarads and a single-wire inductance of 0'323 henry, operated at 60 cycles per second, by an alternator giving a fairly pure sine wave, the observed Ferranti-etfect factor was 1-238, as reported by G. Faccioli ("Electric Line Oscillations": Proc. Am. Inst. Elect. Enyrs., July 1911, pp. 1621-1668). As the fundamental Ferranti-effect factor on this nominal n would only be T053, the assistance of harmonics is suggested. TO ELECTRICAL ENGINEERING PROBLEMS 111 Equivalent T of One Line Wire at Different Frequencies. Freqrency. Total Wire Impedance. Ohms. Equivalent Wire Inductance. Henrys. St;:ff Impedance. Ohms. Equivalent Capacity. Microfarads. /51-50 +JO {_ 51-50 /0 |5r669 +J47-952 0-30557 0-30564 - oc r-S-649 -J2693-3 2-3707 '3636 125 ; 55-594 +J248-8S 0-31688 ( 2G93-3 /90 = 11'03" /- 9-545 -J579-15 9-Jl)g5 X 255*014 /< 7 24' 21" I 579-23 /90 5 of which the real hyperbolic component a v affects the attenuation of the waves of that frequency, and the circular component a 2 the phase. A wave of current, starting with unit amplitude over the line, from a given point, becomes attenuated, after 1 kilo- meter, to the amplitude e - a = e - <' +^2) = e - i . - >2 = - *i\a. 2 . That is, it has shrunk in amplitude, from 1 to e ~ ai , and has retarded in phase 2 radians with respect to the phase of waves starting at the instant of arrival. For this reason, the linear hyperbolic angle of the line is called the attenuation-constant of the line. We have already seen that the attenuation-con- stant has the same numerical value in hyps, per mile or km., whether we take the linear resistance, inductance, capacitance, and leakance per wire-km. or per loop-km. In what follows, we shall use the wire-km. constants consistently. Normal and Actual Attenuation- Factors. If a line-wire were indefinitely long, the attenuation-factor over a given length L km. would be L = e ~ La] . . . numeric /_ (202) But if the line, instead of continuing indefinitely beyond L, stops and connects to terminal apparatus, the reflection of the waves at this terminal load alters, in general, the attenuation- factor. If, however, the impedance of the terminal load happens to be the same as the surge-impedance z o of the line, then the attenuation-factor of the line is the same as is given in (202) for a length L of an indefinitely continuing line. This value may therefore be called the normal attenuation-factor of the length L, to distinguish it from the actual attenuation-factor in the presence of a particular terminal load. TO ELECTRICAL ENGINEERING PROBLEMS 117 Thus, we have by (59) the amplitude of current received through the terminal apparatus E . max. cyclic amperes /_ (202) where E wA is the maximum cyclic e.m.f. of the simple har- monic frequency considered, impressed on the sending end of the Jine at A, z 3 is the vector surge-impedance of the line, L is the length of line in km., a the vector attenuation-constant, La the hyperbolic angle 6 of the line, z- f the vector impedance of the terminal apparatus to ground potential at B ; or half the vector impedance of the terminal apparatus between the two wires of the loop line. The equation must be worked out by the rules of two-dimensional arithmetic, or plane-vectors, as explained in Chapter V. The current at the sending end of the line is also by (56) -p L "' A = *; tanh (LcT+flT) ' max " c y clic am P eres L (20) where 6' is the auxiliary hyperbolic angle subtended by the receiving apparatus to ground when connected to this particular type of line, such that tanh 6' = ~ . . numeric /_ (202c) o The actual attenuation-factor of the line at this frequency is therefore * t = I"' 8 = y , ta T nh -- nu m eric L (202rf) ImA 2 S1Dn La z COsh La If, however, the impedance z of the receiving apparatus to ground, happens to be identical with z , the surge-impedance of the line, we obtain , IB tanh (La + 6') tanh (La + 0') L= T = ,- T ,-C = numeric/ (202^) I mA smh La -f cosh La s 1 ^ At the same time, however, tanh 6' = 1 or 6' = oc . Conse- quently, tanh (La -f 6') = 1, and we conclude that -^XaT . numeric L (202/) 118 APPLICATION OF HYPERBOLIC FUNCTIONS or the actual attenuation -factor becomes the normal attenuation- factor. Values of Attenuation-Constants. If we consider an aerial telephone line consisting of a pair of No. 10 A.W.G. (American Wire Gauge) copper wires, of diameter 0'2589 cm. (0'1019"), interaxially separated by one foot (30'48 cm.), we may take the following linear constants /" = 10'6 ohms per loop mile. /" = 3-G76 X 10 - 3 henrys per loop mile. ,." = 0-8018 X 10 - s farad per loop mile. g"= o. r = 3'293 ohms per wire km. /= 1142 X 10 - 3 henry per wire km. r = 0-9964 X 10 - 8 farad per wire km. ff = 0. Then using formula (150), we have for the attenuation-constant at various frequencies up to 15920 ~, the data in Table I. TaUc I For single-line copper wires No. 10 A.W.G. 0"2589 cm. diam. at interaxial distance of 30'48 cms. r = 3-293, / = 0'001142, # = 0,^ = 0-9964 X 10 - s kilometer units. / b) a 1 2 A. V % 14 Cycles per Second. Radians per Second. Vector Attenuation- constant, hyp. per Kilometer. Real Attenu- ation-con- stant hyp. per Kilometer. Radians per Kilo- meter. Kilo- meters. Kilo- meters per Second. Free Air Velo- city. Kilo- meters. 7-96 50 i 0-001281 1 45 30' 0-000898 0-0009134 6880 54,750 18-25 772-2 15'92 100 0-001812 ' 46" 0-001259 0-001303 4822 76,750 25-6 550-5 39-80 250 0-0028(59 47 29' 0-001939 0-002114 2972 118,300 39-4 357-4 79-60 500 0-004080 49*55' 0-002627 0-003122 2012 160,154 53-4 263-8 159-2 1,000 0-005892 54 34' 0-003417 0-00480 1309 208,330 69-4 202-8 200 1,257 0-006706 56 47' 0-003673 0-00561 1120 224,000 74-7 188-7 398 2,500 0-01042 65 28' 0-004326 0-00948 662-8 263,700 87-9 160-2 796 5,000 0-01812 75 01' 0-004684 0-01751 358-8 285, 00 95-5 148-0 1,592 10,000 0-03442 81 57' 0-004819 0-03408 184-3 293,430 97-8 143-8 15,920 100,000 0-3374 89 11' 0-004859 0-3373 18-62 296,470 98-8 14-2-6 The Table shows that the real attenuation-constant a x in- creases very slowly, beyond the frequency of 400 ~. Thus, taking the frequency of 796 ~ (to = 5000), as the reference frequency, the real attenuation-constant c^ is 0'004684 hyps. TO ELECTRICAL ENGINEERING PROBLEMS 119 per km., so that at this frequency a wave would diminish in amplitude by e - 0-00*68^ or 047%, after running one km. At the frequency of 1592 ~ (co = 10,000) or one octave higher, the real attenuation-constant has only increased to 0'004819 ; so that the normal distortion-ratio is only s ' O' 000135 for 1 km. and this octave. The wave-length 1 is obtained by formula (162), and becomes shorter the higher the frequency. The velocity of propagation v is obtained by formula (159). As the frequency increases, it approaches the velocity of light in air (3 X 10 km./sec.). It falls short of that value ; first, because there is some internal inductance within the substance of the wire, and this constitutes a load distributed along the line. Only when a wire has no internal inductance can the velocity of propagation attain that of light in the dielectric ; and, second, owing to loss of energy into the substance of the wire, the speed of propagation falls short of the speed of disturbances in the external medium. Only when there is no loss of energy either in the conductor or in the dielectric, can the velocity v attain that of a disturbance in the medium. The last column gives the semi-amplitude range, or the distance in km. to which the waves can run, at each frequency, before being normally attenuated to one half of their amplitude at the start, as obtained from the equation e - '! = 0'5 numeric (203) or since -0-69315 = -5 ; x = 0'69315/a! km. In comparison with the above results, let us consider the attenuation-constant at different frequencies of cabled copper wires, No. 19 A.W.G., 0'0912 cm. in diameter (0'03589"), paper insulated in twisted pairs, with the following linear constants /" = 90 ohms per loop mile. r = 27*96 ohms per wire km. ,-" = 0'08 X 10- farad per c= 0'994 X 10 ~ r farad per loop mile. wire km. 1-126 X 10 - 3 henry per /= 0'35 X 10 ~ 3 henry per loop mile. wire km. 0. g= 0. 120 APPLICATION OF HYPERBOLIC FUNCTIONS The results are given in the following Table Table II For single-line copper wires in twisted pair cables No. 19 A.W.G. 0-0912 cm. diam. / a) a "l a 2 \ V % Li Cycles Radians Attenuation- constant Real Attenu- ation-con- Radians g.. Kilo- meters Free Air Kilo- per Second. pei- Second. hyp. per Kilometer. stant hyp. per i Kilometer. I>r '' eS " "' ete ' s - per Second. Velo- city. meters. 9-95 62-5 0-01318 45 0' 0-00932 0-00932 674-1 6,70(5 0.9 74-38 19-9 125 0-01862 45 0' 0-01317 0-01317 477-1 9,492 3-2 52-63 39-8 250 0-02(536 45 06' G'01790 0-01796 349-8 13,920 4-6 38-72 79-6 500 0-03724 45 10' 0-02626 0-026*1 237-9 18,930 6-3 26-39 159-2 1,000 0-05272 45 22' 0-03704 0-03752 167-5 26,650 8-9 18'72 397-9 2,500 0-08335 45 54' 0-0580 0-05986 105-0 41,760 18-9 11-95 796 5,000 0-1180 46 47' 0-08079 0-08602 73-04 58,130 19-4 S'58 1592 10,000 0-1672 48 34' 0-1106 0-1253 50-14 79,810 26-6 6-26 It will be seen that the real component of the attenuation- constant increases with the frequency. Thus, at w 5000, QJ = 0-08079, and at oj = 10,000, a x = 0-1106. The normal linear distortion-ratio or the distortion-ratio for 1 km., and this octave, is therefore, - (o-noe - o-osos) = - 0-020^ After passing over 50 km. (31'07 miles) the distortion-ratio would be ~ 1>4!> = 0-2254, so that the amplitude of the 10,000 rad-per-sec. waves would be 22 '5 per cent, of the amplitude of the 5000 rad-per-sec. waves, assuming normal attenuation, and that they started with the same initial amplitude. This is, in the present state of information, an approximate limit to the distortion- ratio for this octave in satisfactory commercial telephony. That is, if the normal distortion-ratio falls below 1/e 1 ' 5 for this octave, the articulation is unsatisfactory. Further measure- ments are needed, however, in this direction. It should be pointed out that the real attenuation-constant is somewhat larger in the cable of Table II than is obtainable in practice ; for the reason that the linear capacity is taken as 0'08 microfarad per loop mile (0'05 /{/per loop km.), whereas in good practice, such cables are made with a linear capacity of 0-072 fjf per loop-mile (0'046 yf per loop km.) corresponding TO ELECTRICAL ENGINEERING PROBLEMS 121 to 0*092 fjf per wire-km. or c = 0'92 x 10 - 7 . On the other hand, however, the most recent measurements reported in the United States give an effective leakance of g = 1*73 X 10 ~ 6 mho per loop mile (3'46 X 10 - 6 mho per wire mile, = 2'15 X 10 - 6 mho per wire km.), and in Great Britain * 5 x 10 ~ 6 mho per loop mile (3'1 X 10 ~ 6 mho per loop km) = 10 ~ 5 mho per wire mile; or g = 6'2 X 10 ~ 6 mho per wire km. This effective leakance is probably due to dielectric hysteresis aided by the presence of residual moisture, rather than to leakage con- ductance. In either case, however, it represents loss of energy in th^ dielectric, increasing as the length, and as the square of the voltage. This small effective leakance tends to increase the real attenuation-constant a v by reducing the argument of a. It is known that a relatively very small amount of moisture resident in the paper dielectric of a lead-covered cable will bring about an increase both of capacitance and leakance. Moreover, paper absorbs moisture so readily that it is difficult to secure and maintain the cable insulating material moisture-free. The changes in c^ effected by the above amendments are, however, relatively small. In comparisons of electric or acoustic properties of lines, telephone engineers frequently employ as their standard a telephone cable of the following constants. Copper wires No. 19 A.W.G. paper-and-air insulated in twisted pairs, dry, within a leaden sheath. Diameter 0'03589" (0'0912 cm.). r" 88 ohms per loop mile = 54'68 ohms per loop km. I" 10 ~ 3 henry per loop mile = 0-6213 x 10 - 3 henry per loop km. g" 5 x 10 - 6 mho per loop mile = 3*107 x 10 - 6 mho per loop km. c" 0'54 x 10 ~ 7 farad per loop mile = 0-3355 x 10 " 7 farad per loop km. Referring these to the wire kilometer we have /== 27-34; /=0-3107xlO- 3 ; #=6'214x 10' 6 ; t=0-6711xl0- 7 . With the preceding data we obtain the following values * For these data, the author is indebted to the courtesy of the Engineer- ing Departments of the American Telegraph & Telephone Co., and of the National Telephone Co. Ltd., of Great Britain. 122 APPLICATION OF HYPERBOLIC FUNCTIONS Table III For single-line copper wires in twisted-pair " standard " cable. Cycles per Second. I u> a o ,,:,. Attenuation-constant per ^yP- l' er Kilometer. Second. Modulus. Argument. l Real Attenu- ation- con - stant. I et.2 A Radians Waye . K P no- -i-g^ meter - n^er's. Kilo- meters per Second. Free Air. 14 Semi- ampli- tude Range. Kilo- meters. 99'5 199 398 796 1592 625 ! 0-03405 1,250 j 0-04796 2,500 ; 0-06776 5,000 I 0-09586 10,000 ! 0-13590 40 54' 44" 0-02573 0-03491 0-04803 0-06647 0-09098 0-02230 281 -8 0-03288 191-1 0-04775 j 131-6 0-06907 90-97 0-10096 62-24 23,029 38,013 52,357 72,395 99,051 9-3 12-7 17-5 24-1* 26-94 19-86 14-42 10-43 7-62 /43 17' 19" ,'44 48' 03" 46 05' 44" ,47 58' 35" 33 The following is an example from Table III of the method of computation. It assumes co = 5000, r = 27%34, / = 0'3107 X 10~ 3 . g = 6-214 x 10-, and c = 0'67ll X 10 ~ 7 . a = '! 5535)(6214 + /335 7 5)10 - . ^) (335-56 X 10 -* /88 56 20 = V9188-91 X 10- 6 V 92 TT^8 /7 = 95'859 x 10 - 3 /46 05' 44". = 0-095859 /46 05' 44" = : 066474 + yO-069066 hyp7per"km.^ sec. ^ = oSS6 = 9 ' )74km - = o> = 5000 = ~ a, ~ 0-069066 ~ L = Q : 69315 __ 0'69315_ - c^ . 0-066474 It will be seen, from an. examination of Table III, that the normal linear distortion-ratios of this line are in four succes- , and ~' 024 , showing that the sive octaves, e' " 009 , - 0>013 , - ' 018 distortion increases as the frequency is increased. * These arithmetical steps, ordinarily taken to a lower degree of numerical precision, are expedited by certain tables of interconversion between a complex quantity (a + jb) and its plane vector p/, which have been completed by Herr Gati Bela ; but not yet published. They are a type of Tables resembling what are called in the Science of Navigation 4 Traverse Tables." TO ELECTRICAL ENGINEERING PROBLEMS 123 The attenuation-constant per English statute mile of cable is always obtainable from the attenuation-constant per kilometer, by multiplying each of the components a x and a 2 , by 1 '60933. The vector attenuation-constants recorded in Tables I and II are shown graphically in Fig. 59. The origin is at O. The J 0.12 ^ o.n "t O.JO ^1 | 0.09 b 0.08 1 C 0.07 0.06 Is s 1 0.05 ! 0.04 m 1 0.03 02 A ^i 00 ^ K / Q AAfor up CC for an aen< to 560 a cable to-H6O tl wire 0/y wire < at free t freqi uencie encies 7 / / '/ 800 y/ ^ 400^4 /' / 16, )0 \60f A >X ) o.oi O 80 ) / /20 40 I y \ t \ - x 3 6 Real Attenuation-Constant Hyperbolic Radians per kilometer FIG. 59. Curves showing the Loci of Vector Attenuation-Constants. curve OCC indicates the locus of the attenuation-constant for the cabled wires of Table II, and OAA the locus of that for the aerial wires of Table I. The real components a x extend along OX, and the circular components a. 2 along OJ. The dotted line OK at 45 with each axis, marks the locus of the attenua- tion-constants in lines of such small linear inductance, with 124 APPLICATION OF HYPERBOLIC FUNCTIONS respect to capacity, that the former may be neglected. The small amount of linear inductance in the cabled wires of Table II causes the locus to bend upwards, and to leave the dotted line perceptibly at the frequency of 160 ~ ; while the relatively greater linear inductance and smaller linear capacity of the aerial wires in Table I cause the locus OAA to bend up sharply, and to become nearly parallel to the / axis. The real attenuation -constant a x is, therefore, roughly the same for all frequencies on the aerial line above 400 ~ ; whereas it con- tinually increases with the frequency in the cabled wires. Thus, not only is the electric distortion at telephone frequencies much more marked on cabled wires than on aerial wires, but the attenuation is also much greater at all telephonic frequen- cies on ordinary sizes of cabled wires than on ordinary sizes of aerial wires, so that about 50 km. of such cable, as is referred to in Table II, would be approximately the commercial limiting telephonic range, as against about 670 km. of such aerial wire as is referred to in Table I. Particular Values of the Attenuation- Const ant. In the parti- cular case when / = and g = 0, that is, when both the linear inductance and leakance are negligible, a case closely approached by well insulated cabled wires, the argument of the conductor impedance, /? x = 0, and the argument of the dielectric admit- tance fi z = 90 ; so that the argument of the attenuation- constant is 45, or may be called a semi-imaginary quantity, having as large a real as imaginary component. In such a case I era) /a^/n - P er km - ( 204 > Here the real attenuation-constant a 1 increases as the square root of the frequency. When a considerable amount of linear inductance exists, as in aerial lines, the real attenuation-constant c^ tends to a limiting value as the frequency increases. This value is r r J2_ _ 2 . . hyps, per km. (205) 0^1 i r TO ELECTRICAL ENGINEERING PROBLEMS 125 where z 00 is the limiting value of the surge-impedance in (152), or ^/icohms. Thus, for the aerial line-wire referred to in Table I, z tends to the limit 328*6 /(T ohms, and -!J- = 1*647 ohms per semikilometer. Consequently c^ tends to the limit 1*647/338*6 = 0*004864 hyps, per km., as is indicated in Table I. When an appreciable amount of linear leakance exists, an approximate value for a l can be obtained by adding a correction factor to (205), thus i--l+ hyps, per km. (206) Otherwise, the full formula for the real attenuation-constant is hyps, per km. (207) and the imaginary component, or wave-length constant, is radians per km. (208) As a general rule, it is easier and more convenient to employ the vector formula (150), which is also readily remembered, than to employ the cumbersome non-vector, or scalar, formulas (207) and (208). Moreover, the results obtained with the scalar formulas are more liable to be vitiated by slight arithmetical errors than if the vector formula (150) is used. Sn rye-Impedance* of Telephone Lines. The surge-impedance z of a telephone line varies considerably with the frequency, as is shown by formula (152). When r, the linear conductance of the conductor, and y, the linear leakance of the dielectic, are relatively very small ; or, in any case, when the angular velocity is high, the surge-impedance approximates to the value ... ohms (209) 126 APPLICATION OF HYPERBOLIC FUNCTIONS and is therefore greater in aerial lines than in cabled lines. This is the impedance * which a line tends to offer to its own surges in the unsteady state. The limiting value of (209) is a pure resistance, or has no reactive component ; so that in the limiting case there is no difference in phase between a wave of e.m.f. and its accompanying wave of current travelling along the line. If there be relatively greater conductor reactance, ko, than dielectric susceptance ceo ; so that the argument of the con- ductor impedance /^ = tan- 1 , exceeds the argument of the dielectric admittance /?., = tan-M ), then the actual surge- \ g R ft impedance has a positive argument ~ ' 2 , and the line at and beyond any point behaves like a positive reactance or induction coil. In practice, however, the reverse is the case, and the dielectric admittance argument /? 2 always exceeds the conductor impedance argument /^ ; so that the argument of the surge- impedance is ( 2 ^M, a negative angle, and the line behaves \ Zi / as a condenser associated with a resistance. If the surge- impedance of the line had a positive argument, it would mean that any wave of current travelling along the line would stow away more energy of magnetic form in the dielectric than is stored electrically; while, in practice, the fact that the surge-impedance of either unloaded aerial or unloaded cabled conductors has a negative argument indicates that any wave travelling over the line stows away more energy of electric form in the dielectric than is stored magnetically. Table IV gives the surge-impedance of the aerial line above considered in Table I for various frequencies between 8 and 16,000 cycles per second, both as a vector, and as a complex number, of ohms per wire. If we consider the surge-impedance of the circuit, we know that it will be twice the surge-impedance per wire (36). * Dr. Steinmetz has recently suggested the name "natural impedance" for this important quantity. See Journal of the Franklin Institute, July 1911, "Electric Transients," by Charles P. Steinmetz, p. 46. TO ELECTRICAL ENGINEERING PROBLEMS 127 Table IV Initial Sending-end Impedance per Single Wire for a pair of No. 10 A.W.G. Copper Wires (0'1019" or 0-2589 cm.) inter- axially separated 1 ft. (30*48 cms.) at different impressed frequencies. Cycles per Second. H Radians per Second. , Vector Ohins. Modulus. Argument, i Complex Quantity. Ohms. 7-95 50 2570- - 44 30' 1833- 1 -;isoi- 15-9 100 1818- -4400' 1308- -J1216- 39-8 250 1152- - 42 5:2' 849- -j 778-8 79-6 500 819-0 - 40 05' 626-7 -; 527-4 159-2 1,000 591-4 - 35 26' 481-9 -j 342-9 397-9 2,500 418-4 - 24 32' 380-6 -j 173-7 795-8 5,000 363-7 - 14 59' 351-3 -j 94-02 1,592- 10,000 345-4 - 8 03' 342-0 -j 48-36 7,058- 50,000 33S-S - 139' 338-6 -j 9-76 15,920- 100,000 338-7 - 050' 338-5 : -j 4-93 At the infra- telephonic frequency of 7 '95 ~ the surge-im- pedance of this wire is 2570 \44 30' ohms, and it diminishes to 338'7 050' ohms at 15,920 ~; but undergoes very little change after reaching 800 ~. The limiting value (209) is, in fact, 338-5 XO 5 " ohms. The wire surge-impedances of the cable circuit considered in Table II are also presented for various frequencies in Table V. At 10 ~ the impedance commences at 2122 \45 ohms, and at 15,916 ~ it has fallen to 6712 \T if the line is grounded through an impedance which subtends with the line an auxiliary hyperbolic angle 6'. Transmission over an Indefinitely Long Line. We have already seen [(15) (16)] that the pressure and current along a 130 APPLICATION OF HYPERBOLIC FUNCTIONS line of uniform electric constants in the steady state are subject to the following conditions E P = E A cosh L^ IA^O sinb L x a . r.m.s. volts /_ (211) p and IP = I A cosh L x a - -^sinh ^i a r.m.s. -amperes /__ (212) where E P and I P are respectively the voltage and current at the point considered, L x km. (or miles) from the sending end ;: while E A and I A are the impressed r.m.s. voltage and the enter- ing r.m.s. current at the sending end of the line. z is the vector surge-impedance and a the vector attenuation-constant of the line at the frequency considered ; so that L x a is the hyperbolic angle of the line, as measured from the sending end to the point considered. When the line is of such great length that the returning waves reflected from the distant end of the- line during the unsteady state can be ignored, the final sending- end impedance remains equal to the initial sending-end imped- ance z , and the final outgoing current in the steady state remains the same as the initial outgoing current ; viz. E I A ^ A r.m.s. amperes /_ (213) ^ '"o Consequently, for such long lines (211) and (212) become E P = E A (cosh L^ sinh L x a) = E A e - L i a . r.m.s. volts /_ (214) IP = (cosh LjCt sinh L x a) = -^ '~0 ~0 E A e - L i a i ^a. 2 r.m.s. amperes /_ (215) "O Thus the r.m.s. voltage and current, at a distance ^ km. from the sending end, are the normally attenuated values of the initial values impressed at the sending end. The ratio of the voltage and current remains constant at z ohms for all points along the line. As an example, we may consider a circuit length of 100 miles (160-9 km.) of the twisted pair cable of No. 19 A.W.G. copper wires already referred to, subjected to an impressed e.m.f. TO ELECTRICAL ENGINEERING PROBLEMS 131 of 4 volts, at a frequency of 796 cycles per second. This will correspond to 2 volts on each wire. Table II gives the attenua- tion-constant at 0118 /46 47' = 0*08079 +/0*08602 hyp. per km.; while Table V give the initial sending-end impedance at 237*4 \43 13' ohms. The initially outgoing current on each wire will therefore be ^ = 0*008425 /43 13' amperes; or 8*425 milliamperes, leading the impressed e.m.f. by 43 13', or nearly one-eighth of a cycle. Because the cable chosen is so long, and the waves that return reflected from the distant end are so minute, the outgoing current in the steady state has the same strength as the initially outgoing current. At a distance of L x = 30 miles say (48*28 km.), the hyp. line angle L lttl will be 48*28 X 0*08079 == 3*901. The real attenuation-coefficient will be - 3-901 = = 0*02023. The voltage will have fallen to 2 x 0*02023 = 0*04046 volt. The current strength will have fallen to 8*425 /43 13' x 0-02023 = 0*1704 /43 13' milliamperes, the current still lead- ing the local voltage by this phase. Both the current and pressure will, however, have been retarded in the transmission by 48-28 x 0*08602 = 4'153 radians, or 238; so that the full expression of voltage and current for the point considered, with reference to the phase of the e.m.f. impressed at the sending end is E P = 0*04046 \238^ volt IP = 01704 \194 47' milliampere the ratio of which is 237'4 \43 13' ohms, or z . Or we may express the same result by saying that the vector attenuation- coefficient is e- L i a = e- 3 ' 901 . e -;*-i5s = 0'0203\238 ; so that the voltage at P is E P = 0'0406 V238^ and the current at P is IP = 01704 \ 194 47' milliampere. Table VII gives the voltage and current in each wire of the telephone circuit considered, for varying distances L x miles from the sending end. K 2 132 APPLICATION OF HYPERBOLIC FUNCTIONS Table VII Hyperbolic Line Angles and Attenuation-Factors for Cable Circuit at the frequency of 796 or co = 5000. L! L a a e - L I( X = EP IP Kilometric Units. e - L iai . e - Li2 Attenu- Miles. meters. Ljax. Liag. ation- Factor. Lag. Volts. Millianrperes. I 1-0 2-0 8-425 /43 13' 1 1-609 0-13 0-1384 0-8781 7 55' l-7562\755' 7-398 /35 18' <2 3-219 0-26 0-2769 0-7711 15 50' 1 -542 \ 15 50' 6-498 /27 23' 5 8-040 0-05 0-6922 0-5221 39 40 D 1-044\39 40' 4-399 /3 33' 10-00 ; 1-30 1-384 0-2725 79 20' 0-545\79* 20' 2-290\367' 20 32-19 2 -CO 2-769 0-07431 158 40' -149 \ 158 40' 0-626 \ 115" 27' 30 48-28 3-90 4-153 0-02023 238 0' 0-0405 \238 -170 \ 194 47' 50 80-46 0-50 6-922 0-001503 396 36' -003 \ 396 36' 0-013 \S63 23' It is evident from what has already been considered in connection with normal attenuation, that if instead of finding the current and voltage at different distances along an in- definitely extending line, we cut the line at "successive distances, and connect each wire to ground potential through an impedance Z L ohms, we obtain the same results. Thus, if we cut the line at 30 miles (48'28 km.) from the sending end, and bridged the ends through 474'8 \43 13'"; or 237'4 \43 T3' between each wire and ground, the current through this terminal impedance would be 0*1704 \194 47' r.m.s. milliarnpere, while the voltage to ground at each wire end would be 0'0405 \238 volt (0'081 \238 volt between the wires), corresponding to normal attenuation at this distance from the sending end. If, however, instead of grounding each wire through z ohms., at the point whei*e we cut the line, we ground through some other impedance ; then the steady state will be disturbed by reason of the reflections set up from the terminal impedance during the unsteady state, and formulas (15) and (16) must be used, in order to determine the terminal voltage and current. In general, if the terminal impedance z. c is greater than z , the TO ELECTRICAL ENGINEERING PROBLEMS 133 current to ground will be reduced, and if z r is less than z , the current to ground will be increased, with respect to the current of normal attenuation ; but the conditions evidently depend on the argument as well as on the modulus of the terminal impedance used. In the particular case, however, when the distance L x from the sending end is so great that the current wave reflected from the grounded end makes no appreciable reappearance at the sending end, we have by (169). Tj 1 I A = -^ r.m.s. amperes /_ (216) (218) 2E A 2 ,, (219) e But when L x is large, e ~ LI(X becomes very small by comparison with e L i a , and may be ignored. Consequently 9TT 91? 9T? IB = - ^ = ^ e - L ' = A s - ^i \L ia , r.m.s. amperes (220) which received current is just double that which flows to ground through a terminal impedance equal to the surge-impedance, or the normally attenuated current at the distance L r This is for the reason that the effect of grounding a line is to reflect the arriving voltage wave with 180 change of phase, or to annul that wave locally ; whereas the arriving current wave is reflected with no change of phase, or is doubled in amplitude. The same proposition applies to a composite telephone line, or line of several different sections in series, provided that the length of the last section is so great that waves reflected from the grounded end do not appreciably disturb the waves as they enter that section on the sending side. That is, if the normally attenuated current at the distant point would be I B L r.m.s. amperes, the 134 APPLICATION OF HYPERBOLIC FUNCTIONS current flowing direct to ground at that point will be 2I B L r.m.s. amperes. Fig. 61 is a reproduction of a curve-sheet (Fig. 1) accom- panying the paper on " Loaded Telephone Lines in Practice/* Length of Line (kilometers) \ \ 10 20 50 40 50 60 70 80 Length of Li tie (milfx) JfL_ 100 110 120 150 UO ISO FIG. 61. Observed Attenuation Factors and Relative Telephone Currents on unloaded arid loaded cable circuits. read by Dr. Hammond V. Hayes before the International Electrical Congress of St. Louis.* Curve 1 gives the observed attenuation- factor on an unloaded telephone cable circuit of * Trans. Am. Int. Elect. Congress of St. Louis (1904), vol. iii. pp. 643, 649. TO ELECTRICAL ENGINEERING PROBLEMS 135 88 ohms per loop-mile (27*34 ohms per wire-km.) and 0*068 microfarad per loop-mile (0'0845 X 10 ~ 6 farad per wire-km.) in -actual tests with standard terminal apparatus. It corresponds, therefore, to the resultant attenuation-factor of all the range of frequencies entering into telephonic transmission. If we replot Curve 1 on semi-logarithm paper, i.e. paper ruled with ordinary equidistant abscissas, but with logarithmic ordinates, like the distances along a slide-rule, we obtain the wavy line bbb (Fig. 62). This corresponds substantially with the straight line I'O, V. If the attenuation were normal for a single frequency, it would follow such a straight line. Thus the attenuation-factors in Table VII, plotted in Fig. 62, give the broken line TO, B. The straight line I'O, V falls to O'Ol in 36 miles, or 58 km. Consequently the real attenuation-constant j on this actual circuit was substantially -5Sa l = Q. 01 = g - 4-605 or a 1 =0'0795 hyp. per km. (0128 hyp. per mile). But there is one and only one simple frequency which would develop this attenuation-constant on such a cable, and it is determined by the semi-imaginary relation v /( 2 7'34 + jO)(0 + jO-0845 x 10- 6 o>) = 0-0795 + jO-0795 . (221) 90) = 0'1124 /45 V2-31 x 10- 6 JJK) = VO'012634 | 90 <> r . i y^L = 5469 . . . rad/sec. corresponding to the single frequency /= 5469/6-283 = 870'6 ~. The line therefore behaved substantially as though a single frequency existed in the voice ; or as though the pitch of the acoustic vibrations were between g" and a", at the top of the treble clef. Assuming that such a single frequency were impressed on the circuit at the sending end, we have seen that the attenuation would not be normal as the circuit increased in length ; because the impedance of the receiving apparatus is not the same as the 136 APPLICATION OF HYPERBOLIC FUNCTIONS surge-impedance. The deviations of the curve b I I (Fig. 62) from the straight line Oft' might possibly be explained in this way. 1.0 0.9 0-8 0-7 0-6 0-5 0-4 0-3 0-2 0-1 0*07 0'06 0-05 0-04 003 0-02 0-01 Km. 10 20 30 4 , 1,1, 3 50 60 70 80 , I i I 1 i \\ \\ V . \ 3 \\ \ \ V \\ V _>r ^s \ ^" X \J\ \"^b x ^ \ \ A v Mil i M 1 MM MM MM MM 1 1 1 Ifi W ' 1 'Ml 1 I i i , 10 20 30 bD 40 5C Miles. FIG. 62. Curves showing the observed Telephonic Attenuation-Factor on an Actual Cable as compared with Normal Attenuation for a single frequency. Semilogarithm Paper, Ordinates, Logarithm^ of Attenuation-Factors. AbscissHS, Distances from Sending end, Miles and Km. From many such measurements, telephone engineers have ^J as the mean TO ELECTRICAL ENGINEERING PROBLEMS 137 angular velocity of telephony, corresponding to the frequency f= 796 ~, which is taken as the standard telephonic fre- quency. The reason for this remarkable apparent singularity of frequency dominant in a telephone circuit is perhaps con- nected with the fundamental tone of the transmitter and receiver diaphragms. Fifty kilometers (31 miles) of such cable is approximately a moderate commercial limiting telephonic range, while Figs. 61 and 62 show that the attenuation-factor is about O'OIS at this range, or ~ 4 . Consequently, a first approximation to a moderate commercial limiting range on any circuit is given by the formula La, = 4 . hyp. (222) or L = 4 km. (223) a i where a : is the real attenuation- constant at co = 5000. Applying this rough rule to the aerial circuit taken in connection with Table I, we find that L = 4 0*004684 = 786*8 km. (489 miles). In the case of the cable circuit taken in connection with Table II, L - 4 0'08079 = 49*52 km. (30'77 miles); and in the case of the standard cable circuit of Table III, L = 60*17 km. (37*4 m.). At this range, the normal distortion-ratio for the octave above the standard frequency would be e' 1 " 06 for the aerial circuit, s " 1 ' 470 for the cable circuit of Table II, and -1-474 f or the standard cable circuit of Table III. This explains why the articulation is ordinarily better over long aerial lines than over cable lines of these types, when the sound in the telephone is reduced to the commercial limit of volume. Although the value of c^ for the standard telephonic frequency is, in (222) a criterion of the telephonic limiting range over a given uniform line ; yet this can only be a first approximation because it makes no allowance for the reflection of waves from the terminal receiving apparatus in the unsteady state, and the reflections are different with different types of line. Conse- quently neither the real component of the hyperbolic angle subtended alone by a line, nor the value of the real attenuation- constant, is an accurate criterion of the telephonic range of the 138 APPLICATION OF HYPERBOLIC FUNCTIONS line, even with standard terminal apparatus, and when the line is uniform throughout. Still less are the hyperbolic angles of the sections of a composite telephone line, or their simple vector sum, a proper criterion of the limiting range ; because the angle of a line section depends, as we shall see, upon the constants of the sections with which it is connected. Strictly speaking, the criterion of telephonic range is determined by formula (59) or TT IB = . u a , A T- ^ . r.m.s. amperes (224) z smh 6 + z r cosh 6 where E A is the r.m.s. voltage of standard frequency impressed on the uniform line at the sending end, z is the vector surge- impedance of the line, z r is the vector impedance of the receiving apparatus, and 6 is the vector hyp. angle of the line, all at standard frequency. If the distortion-ratio of the circuit in the upper necessary frequencies is not too low, the circuit will fail to transmit satisfactory speech only when I B falls below a certain limit. Taking E A at an average standard value, this means that the circuit will fail when the receiving-end impedance z sinh 6 + z,. cosh 6 exceeds a certain value. In practice, it appears that when this impedance exceeds 100,000 ohms per wire, or 200,000 per loop, even expert telephonists are unable to communicate, but when it does not exceed 12,500 ohms per wire, or 25,000 ohms per loop, commercial telephony is readily possible. The receiving-end impedance of a simple non-composite telephone circuit may, then, be written Z t = ;. sinh 6 -f z.,. cosh 6 = z sinh 6 ( 1 + coth 6 \ z / ohms /_ (225) The proportional increase in the impedance of a circuit due to the receiving apparatus depends thus upon the ratio z,./z and upon the cotangent of the line angle 6. As an example, let us consider a " Standard " cable of the type discussed in connection with Tables III and VI, 60*174 km. (37 '4 miles) in loop-length, with a terminal receiving appara- tus of 100 ohms effective resistance, and 500 ohms effective TO ELECTRICAL ENGINEERING PROBLEMS 139 reactance, at the angular velocity of CD = 5000. The angle subtended by the line alone is = 57682 /46 05' 44" = 4*000 -j-/4*15597 hyps, (see Table III). The wire surge- impedance at the standard frequency is, by Table VI, 285*65 \42 50' 36". If the receiving instrument is short-circuited, the receiving-end impedance per wire of the circuit is by (26) and (225) z sinh 6 = 285*65 \42 50' 36" sinh (57682 /46 05' 44") ohms. = 285-65 \42 50' 36" x 27*3214 /238 08' 12" = 7804-40 /195 17' 36" ohms, and the receiving-end impedance of the loop would be double this quantity. Now, removing the short-circuit, and inserting the instrument impedance of 100 + /500 = 509*90 /78 4,1' 24" ohms into the loop at the receiving end, or z r = 254*95 /78 41' 24" ohms to ground potential in each wire, we have, by (225), the total receiving-end impedance per wire 254-95/78 41' 24" Z,=7804*43/19517'36"(l + -cothfl] ohms. 285-65 \42 50' 36" = 7804-43 /19517' 36" (1+ 0'892517\121 32' 00" x 1*00097 /0 02' 04") (1 +0-89338 \121 34' 04") (1- 0-46769 +yo-76il8)~ (0-53231 +/076118) = 7804-43 /195 17' 36" x 0-92884 /55 02' 03" = 7249-09 /250 19' 39" ohms, which shows that, in this particular case, the insertion of the receiving instrument diminishes the receiving-end impedance per wire from 7804*4 to 7249*1 ohms; so that inserting this particular receiving apparatus would increase the strength of the current received at B by 7*65 per cent. But if the receiving apparatus, keeping an impedance of 254*95 ohms per wire, happened to possess an argument of \42 48' 32" instead of /78 41' 24", the insertion of the instrument would add 89*25 per cent, to the receiving-end impedance, or increase it to 14770 140 APPLICATION. OF HYPERBOLIC FUNCTIONS /195 IV 36" ohms per wire (29,540 ohms per loop). More- over, it is easy to see that the effect of inserting similar receiving apparatus into a cable circuit on the one hand, and into an FIG. 63. Equivalent Circuits of Lines with ground return and metallic return. r = 27-90 ohms>.fc)H. ; i = 0'35 X 10-3 h'tc.i-m. ; g = ; c = 0'0994 X IQ-l aerial circuit on the other, will, in general, make a considerable relative difference in the receiving- end impedance, owing to the difference in argument of z o for the two circuits. Thus, if we insert the instrument above considered into the receiving end TO ELECTRICAL ENGINEERING PROBLEMS 141 of an aerial line of 786'8 km. of loop-length, and of the con- stants discussed in connection with Tables I and IV, the effect would be to increase the receiving-end impedance nearly 20 per cent. In both circuits the real component of the line hyperbolic angle would be 4'0 hyps., and their normal attenuation-factors would each be 0'0183 ; but the insertion of this particulai receiving apparatus would change their relative actual attenu- ation-factors considerably. It should be noticed, moreover, that although the impedance of an ordinary receiving sub-station set has a modulus of about 250 ohms per wire ; yet its argument is ordinarily more nearly 30 than 78 41' 24'', as above assumed. Equivalent Circuits of Telephone Lines. The simplest types of fixed-impedance conductors capable of replacing, in all external relations, a given telephone line, at a given single frequency, are, as already observed in relation to (70) and (75a), the equivalent T and 77. As an example, we may take the case of a cabled line 50 km. (31'068 m.) in length, with the following linear constants Per Loop Mile. Per Wire Mile. Per Loop km. Per Wire km. Ohms . . . Henrys . . Mhos . I . Farads . . . r" = 90 r = 1-126 x 10-3 g" = c" = 0-08 X 10-6 r> = 45 ' V = 0-563 X 10-6 ^ = c' = 0'16x 10-6 r,, - 55-92 = 0-70x10 - tf,,=0 e,, = 0-0497 X 10-6 r = 27-96 1 = 0-35 X 10-3 g = Q c = 0-0994 X 10-6 We have already seen (35) that we obtain the same value of attenuation-constant, whether we use loop- or wire- constants. With the above values, and co = 5000, we find a tt = a = 01179766 /46 47' 26" hyp. per loop-krn., or wire-km. O tt = e = 5-89883 /46 47' 26" hyps, for the loop or either wire. z oll = 474755 \43T2'1*4" ohms for the loop circuit. * = 237-3775 \43 12' 34" ohms for either wire. The simplest elements of a telephone circuit are indicated at the top of Fig. 63, both for the loop, and for one wire to neutral potential surface. 142 APPLICATION OF HYPERBOLIC FUNCTIONS The equivalent U and T of one line wire are indicated at ABGG' and AOBG in Fig. 63. The architrave impedance is- G736-96/ 156 51' 15" ohms, which is also the receiving-end impedance of each line, excluding the receiving instrument z r ; because, if we ground the line at B, the current which will flow to ground at B will be the impressed potential at A, divided by this architrave impedance. The equivalent circuits of the loop line are indicated at ABB"A" and AOBB'O'A' in Fig. 63. The former is a rectangle of impedances, the latter an / of impedances. It will be seen that the rectangle ABB"A" is merely a doublet of the single line 77, ABG'G ; while the 7, AOBB'OA' is merely a doublet of the single-line T, AOBG. The receiving-end impedance of the loop circuit is evidently 2 x 6736'96 /TseFlT' 15" = 13473*92 / 156 51' 15" ohms, excluding the receiving instrument z r . Since, then, the equivalent circuits of metallic-circuit, or loop lines, are mere doublets of those for their component single wires, and the latter are easier to think about and discuss, we will confine our attention to the latter. Artificial Lines for Telephony. It would appear from Fig. 6S that, at the frequency considered, either the single rectangle ABB^A", or the single / AOB, A'O'B', is the complete equiva- lent externally of the actual line. This being the case, it is theoretically unnecessary to employ an artificial line divided into numerous sections to represent the behaviour of the actual line. Either one of these equivalent circuits is sufficient at and for this single frequency. It is to be noted, moreover, that the rectangle is not capable of being constructed in impedances without the aid of transformers, because the argument of the architrave exceeds 90. The / is capable, however, of being constructed of resistance, inductance, and capacitance, without transformers. As a general rule, either the T or the 77 of a line is capable of being constructed, sometimes the one, and some- times the other, according to the length of the line, the linear constants, and the frequency. Since the T and 77 of a line vary with the frequency, it is TO ELECTRICAL ENGINEERING PROBLEMS 14S evident that the T or U which represents a line at co = 5000 r fails to represent it correctly for other telephonic frequencies^ and that the discrepancy depends for its amount, among other things, on the length of the line. Consequently, it is unsafe to- assume that a single-section artificial line its T or U which happens to represent it correctly at the standard telephonic frequency, represents it adequately for all telephonic purposes. Experiments have been reported to show that 10-mile (16*1 km.) sections of artificial standard cable adjusted to the equivalent 77 at (o = 5000, satisfactorily imitate such a cable, so far as acoustic behaviour is concerned ; but the differences between the U of a 10-mile section at co = 5000 and co = 10,000 are not very great. Further published research is needed, in critical com- pany with the theory, to show how few sections of artificial line can be used successfully to represent a full-range actual line for all acoustic purposes. The answer to the question seems to depend upon the relative importance and prominence of the upper frequencies. Influence of Increasing the Distributed Linear Inductance of a Line. As was first pointed out by Heaviside, the effect of dis- tributed inductance in a telephone line is to diminish the attenuation of the telephonic current. The influence of increas- ing linear inductance on the attenuation-constant is clearly shown geometrically in Fig. 64, where O I is the vector linear impedance of the conductor, z = r -\- jx ohms per km. The linear admit- tance of the dielectric, y= g +jb mhos per km. is shown similarly at O A. The argument of z is fi lt and of y, f} 2 , radians. The product z y, or a 2 , is then indicated at O B, whose modulus is the product of the moduli z and y, and whose argument is A + A>> ^ e sum f th e arguments of z and y. At O C is indi- cated the square root of O B, or the vector attenuation- constant a in hyps, per km., where the argument is half the argument of O B. The real part Oc is the hyperbolic component, and the imaginary or^-part cC is the circular component. As the linear inductance I of the line is increased, the linear reactance x = Ico increases proportionately. This has the im- mediate effect of increasing the modulus or length O I, and also 144 APPLICATION OF HYPERBOLIC FUNCTIONS FIG. 64, Vector Development of an Attenuation-Constant. TO ELECTRICAL ENGINEERING PROBLEMS 145 of increasing the argument f} r The secondary effect of the change is to increase the modulus O C in the square-root ratio, and to increase the argument of O C by half the increase of f$ v The vector attenuation-constant a is thus somewhat increased ; but, owing to the increase in argument, the real component a t is markedly diminished ; while the imaginary component a 2 is increased. The increase in a. 2 merely diminishes the velocity of propagation, and shortens the wave-length ; while the reduction in a x reduces the attenuation. Limit to the Reduction in Attenuation with Increasing Dis- tributed Inductance. If, as generally happens, there is dissipation of power in the dielectric ; so that g is not zero, and the argu- ment /? 2 of O A, Fig. 64, is less than 90 ; then there is a limit to the benefit that can be obtained in lessening attenuation by increasing the distributed linear inductance. It can be demon- strated, either algebraically, or geometrically, that increasing jx in Fig. 64 diminishes a x until ^ becomes equal to /? 2 . Beyond this critical point, further increase in jx increases a x instead of diminishing it. This result is indicated geometrically in Fig. 65, which con- tains a direct geometrical construction for a, having given the vector linear conductor impedance oz', and the linear dielectric admittance oy '. Construct the triangle Qgy to the proper scale of linear admittance, with the base Og on the axis OX. Then the angle yOg = /L On Oy as base, construct the triangle Orz, to the proper scale of linear impedance. Then the angle zOy = ft l and the angle sOX = & + /? 2 . Take Ox = Oz. With centres x and z, and equal radii, draw intersecting arcs at A, to bisect the angle zO X. Join O A, which is the bisecting line. Through O draw the dotted line aa perpendicular to OA. With centre O, carry circular arcs from z and y, to intersect aa at the points Z and Y respectively. On ZY as diameter, construct the semi- circle ZpY, intersecting OA in p. Then Op is, to scale, the vector attenuation-constant a. Its projection O^ on the axis OX is a lt the real or hyperbolic component ; while pq is a 2 , the imaginary, or circular component. As r'z', the linear conductor reactance is increased, by 146 APPLICATION OF HYPERBOLIC FUNCTIONS --x FIG. 65. Geometrical Construction of Vector Attenuation-Con tant showing the- devolopment of a minimum real component. TO ELECTRICAL ENGINEERING PROBLEMS 147 increasing distributed inductance on the line, the triangle Orz extends, in succession, to the points 123 ... 8, along the dotted line rS. By repeating the above-described construction, it will be found that the locus of the vector attenuation-constant pursues the corresponding curved path 012 ... 8. The minimum real attenuation-constant, O5', occurs at the point 5, where the argument oOr of /^ is equal to the constant argument /? 2 of the dielectric admittance. If, then, there be no dissipation of power in the dielectric, or (I = o, there will be theoretically no limit to the reduction in a obtained by increasing x, assuming r constant; although the proportional reduction becomes less and less as x is increased. But when dielectric dissipation exists, no benefit of reduced I c attenuation is obtained when /^ overtakes /? 2 ; i. e. when = ; ft/ or when the time-constant of the conductor is equal to the time- constant of the dielectric. In such a case, the argument of z is zero ; or the line behaves like a non-inductive resistance to all frequencies, and the real attenuation-constant a x is the same at all frequencies, so that the normal distortion-ratio is 1 for all ranges and the line is distortionless. The same proposition would hold, by symmetry, if /^ were greater than/? 2 . In that case, the increase of linear capacitance c would reduce the real attenuation-constant until the argu- ment /? 2 overtook the argument /? r In either case, the maximum benefit is obtainable when /^ = /?. both as to minimum attenua- tion for the frequency considered and as to the same attenuation for all frequencies or absence of distortion. It is also evident that no circuit can theoretically be distortionless which has no dielectric dissipation ; because such a perfect dielectric could only make ft=/? 2 at infinite linear reactance x, since r cannot be made zero. In practice, with actual telephone circuits, /? 2 is always greater than f$ v which means that inductance has to be added to the line to diminish attenuation and distortion. This is true even for aerial lines, and it is markedly true for cabled lines. Separating the wires of an aerial loop, as suggested by Heaviside, helps to reduce attenuation and distortion, by increasing I while L 2 148 APPLICATION OF HYPERBOLIC FUNCTIONS diminishing c ; but since I only increases as the logarithm of the distance separating the wires, the practical benefit obtainable in this way is comparatively small, and no marked benefit was obtained .until the Pupin system was adopted of artificial in- ductance coils, inserted in the line at suitable regular intervals ; i. e. until lioes became regularly loaded with inductance. Loaded Lines. A regularly loaded line differs from the same line, with the same total inductance uniformly distributed, owing to the effects of lumping or imperfect distribution. The principal formulas dealing with the effects of regular inductance loading are (107) to (115). From a theoretical standpoint, we may either analyse the behaviour of the loads, as Campbell* has done, by analogy with the propagation of waves along a periodically loaded string,t studied by Godfrey; or by con- sidering line sections replaced by their equivalent J"s as in Fig. 20, and (107) (Bibliography, 48) ; or by grounding the line at the middle of each successive load and comparing the received current in each section with that received over an equal length of smooth line. (Appendix G.) (Bibliography, 29.) The following particular case may be taken from actual practice, as an example. A cabled line has the following linear constants Per Loop-mile. Per Wire-mile. Per Loop-km. Per Wire-km. Ohms . . . Henrys . . Mhos . . . Farads . . . r" = 88 /" = 0-65 X 10-3 c/" = 173 X 10-6 c" = 0-072 x 10-6 r' = 44 1' = 0-325 X 10-3 #' = 3-46 X 10-6 c' = 0-144 + 10-6 } = 54-68 l it = Q-404 X 10- 3 g,, = 1-075 x 10-6 c,, = 0-04724 X 10-6 r = 27-34 1 =0-202 X 10- 3 7 = 2-15 X 10-6 c = 0-08948 xlO-6 At successive distances of 2*607 km. (1'62 m.), inductance coils of 9*07 ohms effective loop resistance, and 01766 henrys loop inductance, are inserted in the cable. The attenuation-constant of this cable, unloaded, at the standard telephonic frequency (co = 5000), is a = %/(27-34 +/1-010) (215 +/447-4)10 - 6 - x/(27%359 /2~ 06' 57") (4474789~43 > '30 // y^ri0^ i = 011064 /45^ 55' 14 77 = 0-07697 +/0-07948 hyp. per km. * See Bibliography, 27. t See Bibliography, 19. TO ELECTRICAL ENGINEERING PROBLEMS 149 The surge-impedance, unloaded, is z = x/(27'359 /2 06' 57")/(447*4 x IP" 6 /89 43' 30") = 247-284 4^T'~l6" ohms. The hyp. angle subtended by a section of 2*607 km., unloaded, is 6 = 0-28843 /45 55' 14" = 0*20065 +yO*20720 hyps. From which sinh 6 = 0*28831 /46 42' 52", and sinh 6/9 = 0*99957 /0 47' 38". The load 2 per wire, at oj = 5000, is 4*535 +^'441*5 ohms. The semi-load o per wire, at co = 5000, is 2*268 + /22075 = 220*76 /89 24' 41* Fig. 66 shows, at AB, a section of this line wire before loading, with its length, angle, and surge-impedance. The nominal T of f\ this section is shown at aob ; where ao = ob = -^Z Q ohms, and /\ 0G, the staff of the T, or nominal total admittance, is The o equivalent T of the section is then determined by (74) and (75). It is shown at a'oG'. A semi-load o is now added to each arm of the equivalent T, producing the extended T, A'oB'. The arm of this extended T is indicated, at oB', as 37*90 -f- ^'222*56 ohms. The next step is to revert from the extended T, which includes the loads, to the smooth line A"B", its equivalent by (84) and (86). The angle of one section of loaded line is thus 0*0621 -f-/0*7384 hyps., and the loading has reduced the real, or hyperbolic, component of the line angle, from 0*20065 to 0*0621 hyp. It is to be observed that on the short section of 2*6 km. here considered, there is very little difference between the nominal T, aolG, and the equivalent T, a'ob'Q'. The great change is brought about by the addition of o to each arm of the T. An examination into the effect of this extension shows that there is a certain amount of reactance in a which the T will stand without unduly increasing the real part of 7 , the equivalent 150 APPLICATION OF HYPERBOLIC FUNCTIONS angle of the extended T. Beyond this critical value, the real part of 6' runs up, at first slowly, and then very rapidly, until the real part becomes enormous. L'= 2-607 ^ /m - 0'Z006f+j 0-20720 r.zo 09 -- - CT* a' I/S 2-607 ku - Q'0&2\ + j 0-738-4 FIG. 66. Section of a Loaded Line developed through the Equivalent T. If we assume that the additional resistance and inductance of the loads in the case considered is distributed uniformly along the line, we have the modified linear constants r + /' = 29-08, / + /" = 0-034072, rf = 2'15 x 10 ~ 6 , <" =0-08948 x 10 -\ and we obtain, for at = 5000, by (33) or (150) the corresponding smoothed attenuation-constant a" = 0-02414 +/0-27702 = 0-27807 /85 01' 08" hyp. per km. TO ELECTRICAL ENGINEERING PROBLEMS 151 and the smoothed angle subtended by a loaded section from mid-coil to mid-coil 0" = 0-06294 + yO-72213 = 0'72493 /85 01' 08" hyp. with a smoothed surge-impedance s o " = 621-518 \4"42'~20" ohms, whereas the actual attenuation-constant of the loaded line is ' = 0-02382 + yO-28323 = 0-28423 /85 II 7 30" hyp. per km. This means that the real attenuation-constant at to =5000, of the loaded line, with its inductance added in lumps every 2'607 km., is less by 1*4 per cent, than it would have been if the same extra resistance and inductance were distributed uniformly. The possibility of this unexpected result was first demonstrated mathematically by Campbell. In general, however, if the lumps of inductance are relatively large, and are too far apart, the reverse condition sets in ; namely, that the real attenuation-constant a\ of the lumpy loaded line is greater than the real attenuation-constant a/' of the equally loaded smoothed line, and in some cases enormously greater. We obtain the same results as those above stated, and pointed out in Fig. 65, if we employ the Campbell formula (112) for deriving the actual loaded section-angle 6', or attenuation- constant a. But whichever formula we select from (107) to (112), for this purpose, we find in it a high degree of sensi- tiveness. That is, a relatively very small error in the steps of the computation, or in the values of the hyperbolic functions, may involve a considerable error in the result. Consequently, more than ordinary care is necessary in working with these formulas, and a higher degree of precision is needed in the tabular values of sinh (6 L. ) cosh (6 L ) and tanh (6 L ) than is ordinarily required. Formula (112) has, however, been carefully investigated by Campbell, who has shown* that for the case of an extra effective resistance in the load coils, amounting to half the unloaded line resistance, and negligible inductance in the * See Bibliography, 27. 152 APPLICATION OF HYPERBOLIC FUNCTIONS unloaded line, there is the following percentage of excess in a/ over a 1 " For n" 3 coils per smoothed wave-length, a \ is over 500% greater than o'V 4 , , 16% greater than a'\. 5 6 These results are affected to some extent by the ratios r rf jr and I" /I. With more than 8 coils per smoothed wave-length, the difference between a\ and a!\ becomes insignificant. The " smoothed wave-length " is defined by the formula co f km. (226) where I" is the extra linear inductance of the loading assumed distributed. Thus, the smooth wave-length in the case con- sidered is ~ 795-8V6-034072 x 0-08948 x 1(F so that, since there is one coil per 2'607 km., there are 8'73 coils per smoothed wave-length. In the case considered, the extra linear resistance of the loads instead of being 50 per cent. of the unloaded linear resistance, is only 6*4 per cent, thereof, which changes a\ to slightly less than, instead of slightly greater than, a" r Strictly speaking, the smooth wave-length 1" should be obtained by the full formulas (151) and (162) ; but in the case of a loaded line, r and g become so much smaller than I and c- respectively, that the shorter formula (226) suffices. For the same reason, the smoothed velocity of propagation becomes by (159) + /"), ' k - Perse, (227) which is, therefore, to the degree of precision under discussion, constant for all frequencies impressed on the line. The 0) TO ELECTRICAL ENGINEERING PROBLEMS 153 number of coils which advancing (smoothed) waves of any frequency will encounter per second must therefore be " " C ils persea (228) In the case considered, the actual velocity at co = 5000, over the unloaded line, is v = 62,910 km/sec. The actual velocity over the loaded line is v' = 1 7,654 km/sec. The smoothed velocity,. by (227), is v" = 18,111 km/sec. The smoothed number of coils struck per second by advancing waves is then, by (228),. N" = 6947. The number of coils per smoothed wave at any frequency/ is /" N" n" = p = j- . coils per smooth wave-length (229) Thus, in the case considered at co = 5,000 or/ = 795-8, n" = 8'73 a) = 10,000 /= 1591-6, n" = 4'365 co == 12,566 /= 2000, n" = 3'473 to = 15,000 /= 2387-4, n" = 2'91 We have already seen, however, that at n" = 3, a\ is more than six times a/', which means that there would be heavy attenuation at co = 15,000, and the attenuation commences to- rise rapidly at co = 10,000. But it is found in practice that N" = 6947, or roughly 7000 coils struck per second, is a satis- factory condition for commercial telephony. Hence we may consider it demonstrated that /= 2000 (co = 12,566) is ap- proximately the highest frequency that has to be preserved for intelligible speech. A loaded cable line is, in fact, a wave sieve, that rapidly damps out and extinguishes currents of frequency higher than N"/ft cycles per second. (See Appendix G.) It should be remembered that, in the preceding discussion, we have referred for convenience all our results to v rf , N", and A", the smoothed-line conditions. Near the limit of n coils per smoothed wave-length, the actual wave-length /' shortens con- siderably, and the actual velocity v' t diminishes in like manner ; so that there are actually n' = 2 coils per actual wave-length, when there are n" = n coils per smoothed wave-length. 154 APPLICATION OF HYPERBOLIC FUNCTIONS In designing loads for a line, it is sufficient, according to the above principles, to provide sufficient extra inductance, assumed uniformly distributed, for the required reduction of the real attenuation-constant a L to a/'. Then the spacing of the load coils must be such that 7000 are struck by advancing waves per second ; or, in other words, that there shall be more than .n coils per smoothed wave-length A", at the highest frequency (about 2000 ~) which has to be preserved. There will then be 2jr coils per wave at/= 1000 ~ t 4jr coils per wave at f = 500 ~, and so on. For accurate results, however, recourse must be had to some one of the full formulas (107) to (115). Effect of Leakage on Loaded Lines. It is observed in practice, and is noticeable in the arithmetical theory here under discus- sion, that when a line is heavily loaded, it is more subject to disturbance from casual extra distributed leakage than when it is in the unloaded state. Accidental leaks along the line, and -defective insulation during storms, influence a loaded line more prejudicially than a similar line unloaded. An explanation of this behaviour is found in the relation of a change in the dielectric conductance argument on the attenuation-constant argument in the two cases. Thus, referring to Fig. 64, if, by reason of linear leakance g, the argument of y falls from 90 to 88, the result will be that the argument of a will fall 1. In an unloaded cable the argument /^ of the linear impedance is small, and the argument ^ 1 "t ^ 2 of the attenuation-constant is in the neighbourhood of 45 ; whereas in a heavily loaded cable, the argument of the linear impedance is nearly 90, and that of the attenuation-constant in the neighbourhood of 85. If now the argument of a for the unloaded line drops from 45 to 44, the change in a is an increase of about 1'7 per cent. ; but if the argument of a for the loaded line drops from 85 to 84, the change in a L is an increase of nearly 20 per cent. Leakance in a loaded line has, therefore, to be restricted and avoided more carefully than in an unloaded line. (For a more formal demonstration, see Appendix H.) Influence of Loading on the Normal Attenuation. Since the TO ELECTRICAL ENGINEERING PROBLEMS 155 immediate effect of loading a line is the same as that of adding distributed inductance, except in so far as the spacing of the loads i.e. the lumpiness may affect the result, the loading of a line reduces the real part of its hyperbolic angle, and so reduces in the same proportion the exponent La/' of its normal attenuation-factor ~ Lai ". As has already been shown, there is no theoretical advantage obtainable, however, in carrying the loading so far that the argument of the linear conductor impedance ft" overtakes the average argument of the dielectric admittance ft". Moreover, it usually happens that the expense of loading, both in cost of coils and in the cost of space to accommodate the coils, makes it inexpedient to carry the loading up to this limit, especially because the advantage obtainable from any increase in I" is necessarily partly offset by the corresponding accompanying increase in r". That is, economy usually demands that the loading falls considerably short of the theoretical limit ft" = ft" . In the case above considered, for example, the loaded linear conductor impedance is = 172'826 /80 18' 48" ; while the loaded linear dielectric admittance is = 447 '405 x !Q- 6 /8943 / 28". Influence of Loading on the Surge-Impedance. An important and inevitable effect of loading a line with regular inductances is to increase its surge-impedance of Z Q . Thus, in the case considered, the surge-impedance of the unloaded circuit was 247-284 \43 48' 16" ohms per wire ; while in the loaded circuit it was 579'77 \4 25' 15" ohms per wire. But since the receiv- ing-end impedance is by (225), z sinh 6 ( 1 -f ~ coth 6 \ it is evident that, neglecting the impedance z r of the receiving apparatus, the receiving-end impedance is increased in direct proportion to z . This means that for very short lines, in which the reduction in sinh 6 due to the loading has had no oppor- tunity to develop, loading a line makes the received current less instead of greater. This result is indicated in curve 2 of Fig. 61, which shows the received current on a loaded cable, as com- pared with curve 1 for the same cable unloaded. It is evident 156 APPLICATION OF HYPERBOLIC FUNCTIONS that the current received on a loaded cable of very short length is only 25 per cent, of that received on the same cable unloaded, and it is not until the cable is about 19 km. (12 m.) long that the current received over the loaded cable overtakes that received over the unloaded cable. Beyond this distance, the diminished real component of the loaded line-angle more than compensates for the increase in surge-impedance. Neverthe- less, it is evident that the increase in z is a heavy drawback on the advantage secured by loading the line. The physical reason for the increase in z with loading may be found in the terminal reflections of the waves running over the line in the unsteady or building-up state. It would be a tedious and complicated task to compute these reflections, sum them, and find their total effect ; but in the hyperbolic theory, they are automatically and accurately integrated by the change in z of the steady state. Another way of looking at the matter is borrowed from the concepts associated with a power-transmission circuit. The loading of the line with reactance increases the impedance of the line, and diminishes the current flow. This reduces the PR loss along the line, and enables the energy to be carried further without being absorbed ; but the line has become raised in voltage, and calls for a change in the winding of the generator- and motor-apparatus at the terminals. These should be wound with finer wire, in more numerous turns, so as to generate and absorb less current, but at a higher voltage, than on ordinary unloaded lines. Theoretically, then, if the apparatus remained unchanged in efficiency after re- winding, such a modification would be capable of avoiding the excessive terminal reflections, and of preventing the change in Z Q from reducing the received current, by operating on the term coth 6 in (220). But even if this plan could be satisfactorily o carried out in practice, it would involve the use of two sets of terminal telephone apparatus, one for unloaded and the other for loaded lines, a very objectionable differentiation. A better partial solution of the difficulty has been obtained by the use of TO ELECTRICAL ENGINEERING PROBLEMS 157 4< terminal tapers " ; i.e. graded reductions in the loading near the nds of a loaded line, whereby the terminal surge-impedance is reduced, at some sacrifice in the line-angle. The effect of such terminal papers is shown in curve 3 of Fig. 61 from. Hayes's paper. Here the initial current loaded is raised to 67 per cent, of the current unloaded, without appreciable detriment to the gain in attenuation. Another expedient in extended use at the present time for reducing the effect of the rise in surge-impedance with the loading of a line is the use of terminal transformers. The theory of such transformer reduction in z is presented in Fig. 67. At AA' is a loaded telephone circuit in its simplest elements. At the frequency considered, the loaded line sub- tends an angle of 6 hyps., and has an excessive surge-impedance f Z L nms P er wire, or 2z /_ ohms per loop. The transmitter operates under an e.m.f. indicated as double, and the receiver has a loop-impedance of2z,./_ ohms. At each end of the line is a transformer with the higher-tension side to line. Let v be the ratio of transformation ; i. e. the ratio of the e.m.f. generated in the higher-tension winding to that generated in the lower- tension winding. Neglecting magnetic leakage, we know that v is the ratio of the turns in the higher- to those in the lower- tension winding. In the presence of actual magnetic leakage, v will vary slightly with the frequency. At BB' one wire only of the circuit is shown, worked to neutral-potential plane, and the loaded line-wire is shown re- placed by its equivalent T, with p /__ ohms in each arm, and y' l_ mhos in the staff, in the manner represented in Fig. 66. At CC' the two transformers are supposed to have been changed to level transformers, by the imaginary process of removing the higher-tension winding, and replacing it with a winding of the same number of turns as the lower-tension winding, keeping the same volume of copper and of insulation. The e.m.f. induced in these windings will now be reduced in the ratio of \\v for the same magnetic flux in the core as before. Nevertheless, the power in the line system will be the same as before, provided that all impedances, both in- and out-side the 158 APPLICATION OF HYPERBOLIC FUNCTIONS changed winding, are reduced in the ratio 1/v 2 , which means that admittances must be increased, in the ratio v 2 . Let, then, each of the arms of the T be reduced in impedance to p i /v 2 [_ B e E --*-- --t ___*_. _.i___jt___.t_._.t_..._...t.__jL. A !V. Fi<;. 67. Diagrams illustrating Effects of Terminal Transformers in loaded lines. ohms, and the admittance in the staff be increased to y l v 2 /_ mhos. The impedance inside the coil will be automatically changed in the proper ratio by performing the substitution above described. A level transformer in any circuit is known to be, in all TO ELECTRICAL ENGINEERING PROBLEMS 159- respects, equivalent to a conductive connection zfa plus a leak z s , as shown at DD'. The magnitudes of the impedances z l and z.> will be the same if the two level coils are symmetrical in all respects, and the magnitude of the leak z 3 is determined by the excitation losses of the transformer with its secondary circuit open. Consequently, at DD', the two transformers have been virtually replaced by an impedance- T at each end of the line, the line itself being modified in the manner indicated.* Finally, replace the modified T of CC' and DD' by its equiva- lent smoothed line, by (84) and (89). Then, it will be evident that the line-angle 6 will be the same as at AA' before the conversion, and the new surge- impedance will be zjv- /_ ohms. The circuit AA' is therefore equivalent to a circuit without terminal transformers, but with the surge-impedance of the line reduced in the ratio 1/V 2 , and with a certain impedance- 7* injected at each end of the line. The power losses in these 7"s will offset, to a certain extent, the benefit of the change in 2 ; but, theoretically, if the terminal transformers had no losses and perfect efficiency, they would secure the required reduction in surge-impedance to the original unloaded value, without any detrimental effects. Precisely similar reasoning would apply if instead of employ- ing the equivalent T of the line, at BB', we employed the equivalent II. Cojiiposite Lines. A conducting line formed of two or more successive sections, each section having its own length, and uniformly distributed constants, may be called a composite line, as distinguished from a simple uniform line, which may be called a single line. A composite line is, therefore, made up of a series of successive single lines. In practical telephony, most long lines are composite, since in every large city the wires must go underground, and interurban lines are ordinarily aerial lines. Consequently, a very simple circuit, connecting two subscribers in different cities, would be a three-section compo- site line, consisting of a central aerial section, and two terminal underground sections. In practice, a composite line may * See Bibliography, 16, 1 7, and 21. 160 APPLICATION OF HYPERBOLIC FUNCTIONS include many sections. If the hyperbolic -theory of telephony is to have general and useful application, it must embrace composite lines with a satisfactory degree of simplicity. As the theory is extensive, has already been worked out to a con- siderable extent,* and is likely to be worked out much further in future, only an abstract can be given here. An obvious method of dealing arithmetically with a composite line, in order to arrive at a quantitative knowledge of its properties, is to find either the equivalent T or the equivalent JJ, for each successive section separately, connect these equiva- lent sectional conductors together, in the proper series order, and compute an equivalent T or 17 of the combination, by repeated use of the star-delta theorem (Appendix E). Such a final equivalent T or II may be called a merger T or 77; because it is arrived at by merging successive T's or 77's. It is always possible, theoretically, to arrive at the merger T or U of .a composite line in this manner ; although, in the case of a line of many sections, the process is long, tedious, and liable to arithmetical mistakes. In general, the final T or 77 of a com- posite line is dissymmetrical ; whereas the equivalent T or 77 of a single line is symmetrical. That is to say, the equivalent T of a composite line has its two arms unequal, and the equiva- lent 77 of a composite line has its two pillars unequal. Any composite line, no matter how numerous may be its component sections, and no matter how many casual loads may be applied to it, at its junctions, either as impedances inserted in the line, or as leaks to ground, must be capable of representation by one and only one T, or by one and only one 77 ; t so that one * See Bibliography, 52 and 61. t If we consider a composite telephone line as having a dissymmetrical n, then referring to Fig. 63, it is evident that, in general, such a line should present some telephonic dissymmetry ; because the pillars of the n, acting as shunts to the terminal apparatus, are unequal at the two ends of the line. Consequently, a markedly dissymmetrical line, having, say, a long cabled section at one end and a long aerial section at the other might be expected to show some dissymmetry in telephonic operation. Although such dis- symmetry has long been known in telegraphy, over dissymmetrical compo- site lines worked at high speeds with Wheatstone apparatus, yet the condition of dissymmetrical telephony does not seem to have been reported in any publication. TO ELECTRICAL ENGINEERING PROBLEMS 161 volt applied at each end, in turn, will send the same strength of current to ground at the other end. This proposition assumes that the composite line is connected directly to ground at the receiving end ; also that all leaks are of constant resist- ance and are devoid of any variable e.m.f. such as might be caused by polarization. It is easy to demonstrate that, by the use of hyperbolic position-angles, assigned by definite law to the successive junctions of a composite line, the resultant distribution of potential, current, impedance and power over the line, as well as the final equivalent T or II of the composite line, may be determined by relatively simple single formulas, that involve much less time and labour to work out than do the successive steps of the merging process. The T or U computed by hyper- bolic trigonometry may be called the hyperbolic T or /7, io contradistinction to the merger T or II. Nevertheless, the hyperbolic method requires frequent references to tables of functions of complex hyperbolic angles, and is only a swift method by reason of the existence of such tables. When such tables are not available, the labour of the hyperbolic method becomes increased by the labour of computing the needed sines, cosines and tangents, or their inverse functions ; so that in the existing absence of proper tables, the merger method is usually less onerous than the hyperbolic method. Suitable tables of complex hyperbolic functions are not yet available, but are in process of formation ; so that the hyperbolic theory may advantageously be studied, even though, for theipresent, its application may have to be deferred. FIRST CASE. Section* of the Same Attenuation-Constant and of the Same Surge-Impedance. If a line AB (Fig. 68) of Lj km. is connected to a line CD of L 2 km., and each has the same attenuation- constant a, and the same surge-resistance z ohms * (conditions which imply the same linear constants), the line-angles will be 18 We abbreviate z to z, and i/ to y, for convenience, when discussing that branch of the theory which relates to composite lines. M 162 APPLICATION OF HYPERBOLIC FUNCTIONS 0j = L 1 a and 6. 2 = L 2 a hyps, respectively. Then, if we free the composite line at D, the resistance at A is E f = 3 coth (6 1 + 2 ) . . ohms (230) while, if the composite line be grounded at D, the resistance at A is ~R g = z tanh (19, + 2 ) . . ohms (231) Reciprocally, freeing and grounding the composite line at A, we get resistances R/ and R^ at D, respectively, of the same values as in (230) and (231). A; ft S & ,D Z X FIG. 68. Composite Line with sections of the same Attenuation-Constant and Surge-Resistance. It is evident, then, that the composite line differs in no way, except in length, from either of the component sections. The angle subtended by the whole line AD is the sum of the component section line-angles. SECOND CASE. Sections of Different Attenuation-Constant but of the Same Surge-Impedance. If a section CD (Fig. 68) of L 2 km. be connected to a section AB of L x km., and their respective linear constants r z , g 2 and r v g^ are such that their attenuation-con- stants a v a. 2 differ ; while their surge-resistances z are the same, we assign the angles subtended by the sections 6 l = L^ and 0., = L 2 a 2 hyps. The angle subtended by the whole line will then be 6 l + 0. 2 , as in the preceding case. That is, except for a disproportionality between the section-angles and their line- lengths, two sections of different attenuation-constant, but of the same surge-resistance, connect together like two sections of one and the same type of line. This is for the reason that in the unsteady state, or period of current building prior to the forma- TO ELECTRICAL ENGINEERING PROBLEMS 163 tion of the steady state here discussed, there is neither wave reflection nor discontinuity of wave propagation at the junction BC, when the surge resistance or impedance z is the same on each side thereof. In order, however, to simplify the transition to more complex A a = 2 B ft = i p G' G' G' Q" O" G" G' J6i6-86- BC "O J223-709- O'' s ? to >* ^ ? ^ 2 S i = 9S X c > c S 1* G G A ooiT7- D 0-998 ZIZfKtO'V 1 ! 1 i a X ^ s C < r C i G' ^D* G' FIG. 69. Composition of two sections with the same Surge-Resistance, but with different Attenuation-Constants. cases later on, we may pause 4 to consider the following case of two sections, with different a but the same z. L! = 100 km., r : = 20 ohms/km., g l = 2 x 10~ 5 mho/km. L 2 = 100 km., r. 2 = 10 ohms/km., g 2 = 10 ~ 5 mho/km. Whence c^ = 0'02 hyp/km., ^ = 1000 ohms ; a, = 0-01 hyp/km., ^ 2 = 1000 ohms. M 2 164 APPLICATION OF HYPERBOLIC FUNCTIONS Fig. 69 shows the two lines at AB and CD respectively. It shows the 77 and T equivalent circuits of AB, at A"B"G"G" and A'OB'G', likewise of CD, at C"D"G"G" and C'OD'G'. If we connect the sections together at BC, into a composite line AD, we virtually connect together some one pair of the combina- tions of equivalent circuits 77^77^, T AB T C D > n AB T C D, T AB n C D. The first two combinations are shown at ABCDGGG and A'OBCOD'G'G'. If we merge together the two elements of any such pair by the formulas of Appendix E, we arrive either at the equivalent 77, ADGG ; or the equivalent T, AODG, of the composite line. In all of the examples to be considered, the equivalent 77 and T of the various composite lines have been derived hyperboli- cally ; but have also been checked by the merging process. Equivalent 77. In order to compute hyperbolically the equivalent 77 of the composite line AD (Fig. 69), we proceed as follows Ground either end of the composite line AD, say the end D. Assign the junction-angle 2 at BC. Then the angle subtended by the composite line at A will be 6 A = O l -f- # 2 n yp s - ^ ne sending-end resistance of the composite line at A is, by (23) ^>gA = 2i tanh (5^ .... ohms (232) = 1000 tanh 3 = 995'055 ohms. G gA = lfR 0A = y^ coth d A . mhos (233) = O'OOl X coth 3 = 10-049,7 x 10 ~ 4 mho. Then the architrave resistance AD of the composite 77 will be p" = z l sinh (5 A . . . . ohms (234) = 1000 sinh 3 = 10017*87 ohms. y" = 1/p" = 0-9982125 X 10- 4 mho. The conductance g" A of the leak at A is by (23) 9'\ = y x coth di y". . . mho (235) = 9-05149 X 10- 4 mho. If we ground the composite line at A instead of at D, the angle subtended by the whole line at D will be <5 D = X + 2 = <5 A - The architrave resistance DA will be the same as that given in TO ELECTRICAL ENGINEERING PROBLEMS 165 (234). The sending-end resistance R^ D and conductance G gD will be identical with R^ A and G g respectively, by (232) and (233) ; so that the leak -conductance /' D at D will be identical with #" A by (235). This completes the hyperbolic 77, ADGG of the composite line. Equivalent T. To find the hyperbolic equivalent T of the composite line AD, Fig. 69, free the line at one end, say D. Then the angle subtended by the line at A will be, as before, ^ hyps. Again, if an ammeter or other device, of resistance R r , is inserted in the circuit to ground at B, the angle subtended by the instrument will be, assuming r > R r O^tamV 1 ^- .... hyps. (271) o * The Propagation of Electric Currents in Telephone and Telegraph Conductors, by J. A. Fleming, F.K.S., chap. v. TO ELECTRICAL ENGINEERING PROBLEMS 181 and the angle of the line and instrument together becomes (6 + 6') hyps. The apparent resistance of A under these conditions becomes R',, = r tanh (0 + 0') . . ohms (272) The receiving-end resistance, as judged from the current in the ammeter at B, and the e.m.f. impressed at A, is the line being assumed devoid of all earth- currents or polariza- tion. If 0' is not over 01, cosh 0' may be taken as unity, without much error, and we have approximately RI = r o sinh (0 + 0') . . . ohms (274) Consequently, if instead of having the line freed and grounded at B, making two successive observations at A, we have the line merely grounded at B through an ammeter, and insert an ammeter aud voltmeter at A, thus making a single test, with simultaneous observations at both ends, we obtain from (56) and (58) (0 + 00 = cosh-i(j^-) . . . hyps. (275) from which, r 0> 0', and can obviously be successively deduced. As a simple arithmetical case, let us take a uniform single line 300 km. long, with the linear constants r = 6 ohms per km., and g = To x 10 ~ 6 mho per km. The attenuation- constant of the line is ^9 x 10 ~ 6 = 0*003 hyp. per km., and its surge-resistance ^6000000/1-5 = 2000 ohms. Its line angle is 0'9 hyp. With the line first freed, and then grounded at B, and both tests taken successively at A, we should have R/ = 2000 coth 0-9 = 2792-2 ohms and R^ = 2000 tanh 0'9 = 1432'6 ohms. The surge-resistance of the line is therefore computed to be r = ^792-2 x 1432*6 = 2000 ohms, while the angle of the line is computed to be 6 = tenh- = tanh-VO'51307 = tanh- 1 07 1630 = 0-9 hyp. 182 APPLICATION OF HYPERBOLIC FUNCTIONS From these results we obtain at once, E, = r 6 = 1800 ohms in conductor-resistance, or 6 ohms per km., and g = 6/r = 0'45 X 10 ~ 3 = 450 X 10 ~ 6 mho in dielectric leakance, or 1/5 micromho per km. In the second case, with the line grounded at B through a milliammeter of 200 ohms, the angle subtended by the instru- ment is tanh- 1 (01) = 010033. The total angle of line and instrument subtended at A is 1 '00033 hyps. If 50 volts were applied at the A-end of the line, a current of 32'82 milli- amperes might be expected at A, and 21 '37 milliamperes at B. The sending-end resistance would then be U' g = 50/0'03282 = 1523*5 ohms, and the receiving-end resistance R^ = 50/0*02137 = 2339*7 ohms. From these observations, exchanged tele- graphically, the total angle of the line would be, approximately, /2339'7\ by (275): (0 + 0') = cosh -i(^l) = cosh - 1 1 '53578 -OD9374 MoZo O/ hyp., as against the correct value of 1 '00033. The inferred value of the surge-resistance is then 1523'5/tanh 0'99374 = 2007-3 ohms, as against 2000. The inferred value of 6 r is also tanh- 1 (200/2007'3) = 0*09997 hyp. as against 0100 hyp., leaving the line-angle 6 = 0'89377 hyp. as against the correct value of 0*9. With alternating- current testing, the first case of measure- ments at A with the B end freed and grounded successively, is perfectly applicable, by extending the formulas into two dimensions. The second case is not strictly applicable, because the phase of the received current with respect to the impressed e.m.f. at A is not measured. Class 2 : Steady -state Signalling. Best Resistance of a Receiv- ing Instrument. Electromagnetic receiving instruments in wire telegraphy may be divided into two classes, namely : (a) those, as of the D'Arsonval movable-coil type, in which the magneto- mechanical force, or torque, is directly proportional to the ampere-turns in the coil ; and (b) those, like simple polarized or non-polarized relays, in which the magneto-mechanical force, or torque, may be nearly proportional to the square of the ampere-turns at low magnetic saturation, but, as the saturation TO ELECTRICAL ENGINEERING PROBLEMS 183 increases, may fall to perhaps a lower power than the first. In either case, the magneto-mechanical force may be expressed by- F = a (IB^K .... dynes or dyne -perp. cm. (276) where F is the force in dynes, or the torque in dynes acting perpendicularly to a radius of 1 cm., I B is the received current strength in amperes, /^ is the number of turns of wire in the winding, a a constant of the instrument depending on its con- struction, and p some real exponent not greater than 2. The received current I B is found by (57). The number of turns n^ in a given winding-space is well known to be sensibly propor- tional to ^Rr, the square root of the resistance in ohms of the winding, provided that the size of insulated copper wire selected lies within the fairly wide range where the ratio of covered diameter to the bare diameter of wire is sensibly constant. Consequently we have, approximately, with a' a modified instrument constant ^iaarlq^^)* d y ne80r * n radians per second, the angle subtended by the cable taken separately is, by (19) and (150) L (282) and its surge-impedance is, by (152) assuming, as we safely may, that the ordinary submarine cable has relatively negligible linear inductance and leakance at low frequencies. 186 APPLICATION OF HYPERBOLIC FUNCTIONS The receiving-end impedance of the cable and receiving* apparatus is, by (59) Zi = z shin 6 + z r cosh 6 . . ohms /_ (284) But on any cable worked nearly to its speed-limit, 6 is so large that sinh 6 = cosh 6 very nearly ; so that Z l = (z +z r )smhd (285) If now the receiving instrument is of such a nature and degree of sensitiveness that pure alternating- current signals can be read satisfactorily when the received current strength has a maximum cyclic strength of I mB amperes ; then amperes /_ (286) where E mA is the maximum cyclic e.m.f. impressed on the send- ing end of the cable at A. This will differ from the maximum cyclic e.m.f. E m at alternator terminals, owing to the IZ drop of pressure in the inserted impedance z s , which is determined as follows The final sending-end impedance of the cable is by (202b) Z A = z tanh (0 + 0') . . ohms L (287) where 6' is the angle subtended by the receiving apparatus. But is so large that tanh 0=1, and therefore, a fortiori, tanh (0 + 0') = 1. Consequently, on long cables Z A = z .... ohms i_ (288) The maximum cyclic e.m.f. impressed on the cable is E mA = E m - max. cy. volts [_ (289) Or the maximum cyclic current at B is TT ImB = - - max. cy. amperes /_ (290) If, then, we assign a limiting value to E m , such as 50 volts, it is evident that the speed and frequency of the alternator at A TO ELECTRICAL ENGINEERING PROBLEMS 187 can be increased until the current I TO B just falls to the lowest satisfactory strength agreed upon. Increasing the frequency will alter z t to some extent, and will also alter z inversely as the square root, by (283) ; but sinh 6 will increase much more rapidly. When 6 is large _4 0_ sinh 6 = - l*/jL .... numeric L (291) so that sinh 6 increases with 6 exponentially. For stereotyped terminal values of E m , z tt and z r , it is evident that on any cable the limiting alternating-current frequency and speed are fixed, to a first approximation, by _0 sinh 0, or ^ 2 ; so that, under such conditions, all cables must have o approximately the same value of V2 , and, therefore, the same value of 0. But, by (282), = */ TO> /45 hyps. Consequently, if we call this limiting value of 0, we have, with o> the associated limiting angular velocity - . . . hyps. /45 (292) and o> = -- radians per sec. (293) _ o 2 !] _ 2 1 2 71 ^T 2 JT ^ O H where / is the corresponding limiting impressed frequency. That is, the pure alternating-current frequency on any long cable of negligible linear inductance and leakance, under stereotyped conditions of terminal apparatus, and neglecting relatively small variations in z , is inversely proportional to the time-constant T = C R of the cable, a deduction in accord- ance with Kelvin's law. Our tentative alternating-current signalling theory agrees, therefore, with the fundamental theory, and with practical observations of alphabet-signalling, at least to this extent. The next question is how properly to connect the frequency 188 APPLICATION OF HYPERBOLIC FUNCTIONS of the hypothetical alternator with the frequency of alphabetical signalling. In Fig. 74, the two particular words Submarine Telegraph are analysed in terms of their international cable- alphabet signals. Each black rectangle, above or below the zero line, represents the impression of continuous e.m.f. on the sending end of the line, assuming the simple case of a sending- key directly connected to the cable. Ordinates then represent impressed voltage, and abscissas elapsed time. Dot- and dash- elements occupy equal intervals, so that each signal occupies one dot-element. The canonical interval between signals in a letter is one dot-element, and that between successive letters three dot-elements. Between successive words there may be six dot- elements. The word submarine thus covers 59 dot- elements, and telegraph 61. The two together 59 + 61 + 6 = 126. The signals as received over a cable of 1'47 seconds time-constant are shown beneath, at CC. Dot-Frequency and Reversal- Frequency. There are two hypo- thetical uniform signalling frequencies. One is indicated at A A or A' A' (Fig. 74), the other at BB. The A A type may be called pure dot-signalling. In this type of signalling, the impulses have all the same sign, and the complete period is two dot-elements. The A A type would be equivalent to a certain complex alternating-current e.m.f., superposed on a positive continuous e.m.f. of half the A amplitude. The A' A' type would be equivalent to the same alternating current e.m.f, superposed on a negative continuous e.m.f. of half the A' ampli- tude. The B B type may be called pure reversal-signalling. It corresponds to a complex alternating e.m.f. of period equal to four dot-elements. Reversal-signalling has, therefore, just half the frequency of dot-signalling, and requires no associated continuous e.m.f. There has been some debate as to which of these hypothetical types of rhythmic signalling more nearly corresponds to practical alphabet-signalling. On behalf of dot-signalling, it may be urged that it employs the actual frequency of the sending keys, disregarding the direction of current. On behalf of reversal- signalling, it may be urged that on long submarine cables the TO ELECTRICAL ENGINEERING PROBLEMS 189 dots do not appear. They are smoothed out by retardation, and can only be inserted in their proper places by the trained intelligence of the receiving operator; whereas the reversals actually show on the record, and form the sign-posts, so to speak, by which the operator is guided to interpretation. L- u b marine Eifc I e 1 g _ eg r*p t B III I I I I I A' A a. b _ . -nJ\ /V-^-/V-/\ -A A /^- f\ f\ f\ f\ ^ a' a' FIG. 74. Signals of Impressed E.M.F. at Sending and Receiving Ends in the two particular words Submarine Telegraph, as compared with either "dot- signals" or "reversals." As for the actual comparison of the sending record with either dot-signals or reversals, there is no marked preponderance of evidence. Thus, in the two particular words of Fig. 74, which have been taken at random, compared with dot-signals, there are 8 cases of 2 successive dots or dashes, in s, u, b, m, i, I, g, p, h. 3 3 s, I, h. 1 ^ ^ h- Compared with reversals, there are 11 cases of a cycle or complete reversal, in u, 6, a, r, n, /, g, r, a,p,p. 3 cycle and a half r, I, r. ,, 2 cycles. 190 APPLICATION OF HYPERBOLIC FUNCTIONS We shall see that so far as concern's the cable alone, it is a matter of indifference whether we refer alphabetical signalling to reversal- frequency, or to dot-frequency ; but, in regard to the influence of the terminal apparatus, there is a considerable difference. We shall, therefore, discuss both standards, but lay emphasis on the reversal standard, since cable engineers are believed to prefer the latter.* In the two words of Fig. 74 there happens to be 18 letters, and 126 dot-elements, in all, or, allowing a space before the next word, 132 dot-elements. This is at the rate of 7 '35 dot- elements per average letter. In sentences taken from an English newspaper, an average letter occupies about 7'3 dot-elements including the average necessary spacing. In unintelligible 10-letter code words, with a diminished number of vowels, the average rises to about 8'0 dot-elements. We may take as a working mean 7'7 dot-elements per letter. This cor- responds to 3*85 cycles in pure dot-signalling, or 1*925 cycles in pure reversals. If, therefore, we have as a speed of signalling n letters per second, the corresponding hypothetical frequencies will be for dot-signalling, f'=S'8on cycles per sec. (295) for reversal-signalling, /" = T925 ^ (296) We have already seen in (281) that a well-known empirical formula for cable-speeds under stereotyped terminal conditions is that n = 10/T. Consequently, the limiting frequency / is OQ.^ for dot-signalling, / ' = - cycles per sec. (297) 1Q-9^ for reversal-signalling, / " = --- (298) Substituting the value of frequency in terms of a limiting cable angle in (292), we find for dot-signalling, G f = x2rc x 38*5/45^ =15-55 /45 = 11 +/11 . hyps./. (299) for reversal signalling, " = x/2?TX 19-25/45 =11-0/45 =7-777 + y7'777 hyps./. (300) * See Bibliography, 46. TO ELECTRICAL ENGINEERING PROBLEMS 191 That is, at dot-signalling frequency, the normal attenuation- factor is - n = 0-000,0167, or to nearly 1/600 of 1 per cent. ; while at reversals-frequency, it is e ~ 7 ' 7T: = 0'000,42, or to less than 1/20 of 1 per cent. These values of may be regarded as the equivalents of formula (281) in alternating-current cable theory. Influence of Impedance at the Sending End. We may now consider the effect of modifications in the terminal apparatus upon the speed of signalling according to our tentative alter- nating-current theory. In Fig. 73 let the cable have a linear conductor-resistance r = 67 ohms per nautical mile (naut) and a linear capacitance = 0'42 x 10 ~ 6 farad per naut. Then, by (283), the surge- impedance of the cable is 3994/>v/co. With a length of 1229 nauts, the time-constant of the cable is x = 4 '25 seconds. When working at the rate of 150 letters per minute = 2'5 letters per second, the corresponding reversals-frequency would be 4'812 cycles per second by (296), and the angular velocity co" = 30'23 radians per second. The angle subtended by the <;able would then be, by (282), " = *Jl28 T o /45 = 11-35 /45 hyps The surge-impedance is then 726'4 \45 ohms. If the terminal impedance z s happened to be a condenser of, say, 40 microfarads capacitance, its impedance would be j/caj" = /827'1 ohms. The total impedance offered to the alternator at the sending end is, therefore, z s + z = 513'7 /(513'7 + 827-1) = 1436\69~~02"' ohms. The ratio of the potential at the end of the cable at A to the potential at the terminal of the alternator would be, by (289) 1^7264X45: = 0.506/24 02'. E m 1436 \ 69 02' That is, the potential impressed on the cable would be nearly 50 per cent, less than that which would be impressed if the condenser z t were short-circuited. The lowered potential would also lead the potential at alternator-terminal a by 24 02'. The reduction in impressed potential would either require a compensating increase in the generated e.m.f. of the alternator, 192 APPLICATION OF HYPERBOLIC FUNCTIONS or a reduction in the frequency and equivalent signalling-speed, in order to restore the received current to its proper strength. If, however, in place of a condenser, we insert a reactance-coil at the sending end, with an effective resistance of 15 ohms, and an inductance of 12*5 henrys, the impedance of the coil at this frequency will be 2 s =15+/378 ohms. The total impedance at the sending end will then be 528*7 /135'7 = 545-8 \ 14 24A The ratio of the impressed voltage at A to the generated voltage at , is then -.~ = 4 n E m 545-8 \ 14 24' = 1"33 x 30 36' or the effect of inserting a magnetic reactance in z s , instead of a condensive reactance, is to raise the im- pressed potential 33 per cent., instead of lowering it 50 per cent. This would mean increasing the received current at B, and the speed of signalling could be slightly but distinctly increased, to bring up sinh 9, and restore the original current strength agreed upon. Best Resistance of the Receiving Instrument. It has already been shown, in connection with (278), that the best resistance for an electromagnetic receiver winding to possess, when operated by continuous currents in the steady state is z tanh 0. It is shown in Appendix I, that whether the mechanical force, or mechanical torque, exerted in the receiver varies directly with the ampere-turns, or the square of the ampere-turns, or any intermediate power thereof, the largest maximum cyclic force, or torque, exerted by the receiving instrument will be obtained when the reactance of the receiving apparatus balances and annuls the surge-reactance or reactance com- ponent of the surge-impedance, and when, moreover, the resist- ance of the winding in the receiver is equal to the surge- resistance or resistance-component of the surge-impedance, increased by any other receiving-apparatus resistance present. That is, the most powerful reversal-signals or dot-signals will be respectively obtained when a = r + R' r ohms (301) where a is the resistance of the receiving-coil or coils, as TO ELECTRICAL ENGINEERING PROBLEMS 193 measured with continuous currents or with alternating currents of signalling frequency, r is the real, or resistance- component of the surge-impedance 2 , and R',. is the resistance-component of any reactive apparatus in the receiving circuit for balancing the surge-reactance. If R',. is so small that it may be neglected, then the receiving-instrument reactance should be equal and opposite to the surge-resistance of the cable, and the receiving instrument resistance should be equal to the surge-resistance of the cable at signalling frequency. In other words, the receiving-circuit impedance should be equal to the surge- impedance in modulus, and have the equal but opposite argument. As an illustration of these principles to dot- and to reversal- alternating-current signalling, the following Table gives, in parallel columns, the receiving-end impedance of the 4"25- second cable previously discussed, both for the pure reversal- frequency of a)" = 30-23, and the dot-frequency of a/ = 6046 radians per second, corresponding to the signalling frequency of 2'5 letters per second ; on the supposition that the receiving instrument is connected directly between cable and ground. Reversals Dots Frequency/, cycles/sec. Angular Velocity ca, radians/sec. 4-81 30-23 9-62 60-46 Surge-Impedance z ohms. ,, r j-Xo Receiving inst. res. a, ohms. (~~o + *,) 726 '4 \ 45 513 -7-./513 -7 513.7 1027-4-^513 7 513-7 \ 45 363-2 ./363-2 363-2 726-4-/363-2 Angle of Cable 0, hyps. >5 J> Sinh 6, numeric. 1148-5 \ 26 34' 8-014 + ./8-014 11-334/45 1510 /45rTuT 812 \2634' 1 T334 +jll -334 16-03/45 41900 /649 30' Receiving-end Impedance, ohms. 1,734,000/432 36' 34,030,000 /622 56' The above Table indicates that doubling the frequency impressed on the cable has had the effect of increasing sinh 28 times, and of increasing the receiving-end impedance 20 times. If we assume that the maximum cyclic amplitude of current 194 APPLICATION OF HYPERBOLIC FUNCTIONS required for effective reversal-frequency signalling is 15 micro- amperes, and for effective dot-frequency signalling 0'7o micro- ampere ; then the maximum cyclic potential E, (iA impressed on the sending end of the cable would be 26'01 volts with reversals, and 25*52 volts with dots. It must be remembered, however, that 15 microamperes on each side of zero would be able to give a distinct record on a sensitive siphon recorder in good adjust- ment ; whereas 0'75 microampere could not be expected to give any perceptible amplitude. A condenser would be needed in the circuit, either at the sending or receiving end, with dot-signal- ling, to shut off the continuous-current component mentioned on page 188. In pure reversal-signalling there is also need for such a condenser owing to the constant departure, towards dots or towards dashes in succession, from the pure reversal regime. If now we insert a reactance into the receiving circuit, equal to the surge-reactance, without appreciably increasing the resistance, then we obtain the following conditions- Reversals Dots Frequency/, cycles/sec, as before Surge Impedance r jx ohms. Receiving Apparatus Impedance a- +jx r ohms. Extra Inductance in receiving cir- cuit, henry s. (z -f z r ) ohms. Sinli e 4-81 513-7 /513-7 513.7 + ./513-7 17-0 1027-4 /0 1510 /45't + i sin 3 a>'t + \ sin 5 co't + . volts (303) 2 7i{ 3 5 where co' is the angular- velocity of the dot-frequency f',t is the time, and n = 3*141 59. The first term only, or funda- mental sinusoid, is represented at oaaa, Fig. 76. Its amplitude is 2/n of the battery voltage OA. Consequently, if a battery of 10 volts is applied at regular make-and-break intervals, the effect will be the same as though 5 volts were steadily applied, 198 APPLICATION OF HYPERBOLIC FUNCTIONS and on this a series of Fourier harmonic e.m.f.'s, the first term of Avhich would have a maximum cyclic value of 6'365 volts, of frequency equal to the dot-frequency/'. The next term would have a maximum cyclic e.m.f. of 2'12 volts, and a frequency of 3/', and so on. All these theoretical alternating e.m.f.'s would be in steady operation at the sending end, and each would send its own current into the cable, independently of the rest. That is, the various hypothetical harmonic currents of the FIGS. 76 and 77. Dot Signals and Dash Signals with their Equivalent Fundamental Alternating E.M.F. Fourier series, having different frequencies, would not interfere with one another. The total maximum cyclic amplitude would be the rectangular summation of the respective individual amplitudes (Fig. 54). It can easily be shown, however, that when the angle subtended by the cable exceeds 4 hyps., as must happen on a long cable operated at or near full speed, the current received to ground at the distant end of the line from the triple-harmonic frequency component is insignificantly small, and quite negligible. In other words, only the funda- mental frequency component arrives at the receiving end. All TO ELECTRICAL ENGINEERING PROBLEMS 109 the currents of higher frequencies are absorbed in transmission. The higher harmonic currents have very appreciable amplitude at the sending end of the line, but are regarded, in the ten- tative alternating-current theory, as mere idle currents, that neither help nor hinder the fundamental working-currents. The theoretical condition in sending a regular succession of dashes is indicated at A' A', Fig. 77. Here the same maximum cyclic fundamental e.m.f. o a'a' is operating at dot-frequency, in opposite phase to that at a a ; but the continuous component 1-0 e-i o-u. E .04 -4 -'i FIG. 78. Reversals and their Equivalent Fundamental Alternating E.M.F. o' o' has reversed its sign from that at o o, Fig. 76. Conse- quently, in changing from a series of dots to a series of dashes, the equivalent alternating e.m.f. is not altered in amplitude, but both it and the continuous component E/2 are reversed. The corresponding theoretical condition or reversal-signalling is indicated in Fig. 78. Here there is no continuous e.m.f., and the fundamental alternating component of reversals-frequency has a maximum cyclic amplitude of 2^/2 re = O90, with respect to the impressed e.m.f. of the sending-battery. If, therefore, the sending-battery has, say, 50 volts e.m.f.. and is applied in 200 APPLICATION OF HYPERBOLIC FUNCTIONS a uniform succession of reversals, the maximum cyclic ampli- tude of the fundamental reversals-frequency is 45 volts. If, then, in the arrangement of Fig. 75, the sending-battery E had a terminal e.m.f. under working conditions of 50 volts, the equivalent reversals alternator would deliver 45 maximum cyclic volts at the bridge apex , and 60 at the end A of the cable. The strength of current received through the instru- ment at B would be 60/2715000 = 22'1 microamperes maximum cyclic current. These principles are well illustrated and checked in the records aa, a'a', and Ib of Fig. 74, which were received over the Canso-Rockport cable. The frequency of the dot series was 5'34 dot-cycles per second, representing co r = 33'55. The frequency of the reversals bb was 2'67 reversal cycles per second, representing co" =16*75. The maximum cyclic amplitude of the dot-signals is approximately 60 microamperes, and that of the reversal -signals 200 microamperes. Each series consists of sine-waves as nearly as the eye can detect. The sending e.m.f. was 30 volts, and in the duplex connections, double-block condensers of 50 microfarads each occupied the bridge-arms. The shunted siphon recorder had a resistance of about 220 ohms, and was in circuit with 30 microfarads. The cable has a time constant of T47 seconds. At the dot-frequency, its angle would be 7*0 /_ 45 hyps, and its surge-impedance 865 /865 ohms. At the reversals-frequency, its angle would be 4'96 /45 hyps, and its surge-impedance 1225 /1 225 ohms. The ob- served amplitudes of the dot and reversal signals are in fair accordance with the formulas above given. Summing up, then, the facts concerning the tentative hyper- bolic theory of long submarine cable-signalling, that theory is in accordance with the'simple C.R. law, when no consideration is given to terminal apparatus. The theory undertakes to define the amplitudes of received signals when those signals consist either of uniform dots, dashes, or reversals. On the assumption that alphabetical signalling follows the conditions of reversal- signalling of the same impulse frequency, the limiting alpha- betical frequency with 50 volts of battery e.m.f. is found when TO ELECTRICAL ENGINEERING PROBLEMS 201 the receiving-end impedance rises to 27 megohms, reckoned from the sending end of the cable. A wide field is, however, open to experimental research for determining the actual relation of alphabetical signalling to alternating current-signalling. Herr Bela Gati has undertaken to show * that a compara- tively small extra distributed inductance in a submarine cable is capable of producing a relatively large reduction in the real component of the angle subtended by the cable at frequencies near 1600 cycles per second, assuming no material increase in linear leakance. He has also published experiments and measurements on telegraph and telephone lines confirming the hyperbolic theory. * See Bibliography, 57, 65. CHAPTER X MISCELLANEOUS APPLICATIONS OF HYPERBOLIC FUNCTIONS TO ELECTRICAL ENGINEERING PROBLEMS WE propose in this chapter to consider two applications of hyperbolic functions to problems in electrical engineering, quite different from those we have studied, without attempting to discuss them in detail. When a condenser is connected in series with resistance and inductance in such a manner that both the resistance and the inductance have to be taken into consideration, it is well known that the conditions of current flow when the circuit is energized or de- energized divide themselves into three classes according to the amount of resistance present (1) Oscillatory current flow ; (2) Aperiodic current flow ; (3) Ultraperiodic current flow. If we denote the dissipative ohmic resistance of the circuit by r = 2p, and the surge-resistance A/ by z, then if p is less than z the circuit is oscillatory. If p is greater than z the circuit is non-oscillatory and ultraperiodic. If p=z, the circuit is in the intermediate condition, being aperiodic. Considering first the periodic case, if we construct an impe- dance triangle, Fig. 79, on a base p and with hypothenuse z, the perpendicular side will be the reactance x = Ico. Then the discharge frequency in the presence of the resistance will be this reactance divided by the inductance /, and will be less than the frequency in the absence of resistance, in the ratio x/z. The triangle will have, at the base, an angle cp = tan" 1 f Y 202 APPLICATION OF HYPERBOLIC FUNCTIONS 203 To find the conditions during discharge, let OU , Fig. 81, drawn to scale from the original O, along the X-axis, represent the initial voltage of the condenser that is about to send dis- charging current through the circuit. Draw OU such that UOU = 90 - (p. Draw OE reverse to OU and OE related in the same manner to OE that OU is to OU . Then OU will be the initial vector discharging e.m.f. and OE the initial vector e.m.f. of self-induction in ,the circuit. The projections of FIG. 79. Stationary Vector Diagrams of Impedance in Periodic and Ultraperiodic Circuits respectively. these vectors at any instant on the axis XOX give the respec- tive instantaneous values existing in the circuit. Midway between OU and OE lies the current vector OI drawn to current scale, the ratio ^ being made equal to z. Multi- plying I by the resistance 2p = r in the circuit, we obtain the vector OR, which being reversed in direction to OI represents 1 7'. Now let all four vectors rotate positively, or counterclock- wise, at the angular velocity a> which exists in the presence of resistance. The projections of these various vectors on XOX reveal at any moment the quantities that would exist in the circuit except for the attenuation or damping. Each projected quantity must therefore be subjected to the damping coefficient 204 APPLICATION OF HYPERBOLIC FUNCTIONS tp s ~ ~i , where / is the inductance in the circuit and t the time in seconds elapsed from the commencement of discharge. Or each vector, instead of rotating in a pure circle, and independently subjected to damping, may be allowed to rotate in an equi- angular spiral as indicated in Fig. 81. FIG. 81. Vector Diagram of a Discharging Oscillatory Circuit, the vectors rotating on circles if subjected to independent damping. If the circuit is ultraperiodic, so that p > z, construct art impedance triangle, Fig. 80, with z as base and p as hypothenuse. The perpendicular will be a hyperbolic reactance IQ, such that dividing this reactance by the inductance I we obtain the hyper- bolic angular velocity Q. The angle \ff at the base of the triangle corresponds to the angle (p in the periodic case. In Fig. 82 we TO ELECTRICAL ENGINEERING PROBLEMS 205 have the analogue of Fig. 81 with respect to rectangular hyper- bolas instead of to circles. OU is the initial vector discharging volts- -- 3, 2. 1, p- \ FIG. 82. Vector Diagram of a Discharging Ultraperiodic Circuit, the vectors rotating on rectangular hyperbolas if subjected to independent damping/ e.m.f. the projection OU of which on the XOX-axis, is the ini- tial discharging e.m.f. actually existing in the condenser. The hyperbolic angle of the sector OI U has its Gudermannian 206 APPLICATION OF HYPERBOLIC FUNCTIONS angle the circular angle x//- of the impedance triangle. The vector current OI lies midway between the vector discharging e.m.f. OU and the vector e.m.f. of self-induction OE . It is drawn to current scale, the ratio ^ , being made equal to z. Multiplying I by the resistance r of the circuit, we obtain the vector Or, which should be reversed in direction if there were room on the diagram, in order to represent the vector -I r drop. As in the circular case, the three vector e.m.f.'s OU , OE or Or have always zero as their vector sum. In the periodic case they all rotate in circles with uniform circular angular velocity co. In the ultraperiodic case, they all rotate in rec- tangular hyperbolas with uniform hyperbolic angular velocity Q. The damping coefficient may be applied to each projected quantity separately, or the vectors may instead be considered to rotate in the spiral curves shown. A considerable number of trigonometric formulas which apply to the rotatory vector diagram of Fig. 81, also apply to that of Fig. 82 when hyperbolic functions are substituted for the corresponding circular functions (Bibliography, 70). The important theorem that a hyperbola takes the place of the circle when the discharge changes from the periodic to the ultraperiodic type was first published by Dr. Alexander Macfarlane (Bibliography, 18). Inverse Hyperbolic Functions. Let an indefinitely long per- fectly conducting cylinder be supported parallel to an indefi- nitely extending perfectly conducting plane, but separated therefrom by a uniform conducting medium of resistivity p absohm-cm. (or C.G.S. absolute magnetic units of resistance in a cube of 1 cm. between opposed faces). A cross-section of the system is indicated in Fig. 83, where DEF is the conducting cylinder of radius cr cm., with its centre at C, which is per- pendicularly distant d cm. from the conducting plane Z'OZ. Let us take in imagination two such sections at a distance of 1 cm. apart along the cylinder, so as to comprise between them a slab of the system 1 cm. thick. Then the resistance of the TO ELECTRICAL ENGINEERING PROBLEMS 207 medium in this 1 cm. slab between the cylinder and the plane will be equal to that of a 1 cm. slab of the rectilinear system shown in Fig. 84, where EF, a conducting strip D = 2rra cm. FIG. 83. Section of a Conducting Cylinder DEF parallel to an indefinitely extending plane ~'a~. wide, is placed parallel to a similar strip Z'Z, but separated ~ l id\ therefrom by the distance a cosh ( J cm. between insulating walls shown shaded in the diagram. Thus the linear resistance of the system in Fig. 83, is the same as that of the system in '6 E ) D i 1 S I Z' o z FIG. 84. Equivalent Slab Section corresponding to infinite plane and parallel cylinder of Fig. 83. Fig. 84. In the diagram of Fig. 83, all the stream lines of current are circles and the equipotential lines are orthogonally intersecting circles. In the diagram of Fig. 84 the stream lines are all parallel straight lines and the equipotential lines likewise. From the linear resistance of a conducting medium we may pass by a well-known transition to the linear capacitance of the corresponding dielectric medium. The linear capacitance of two 208 APPLICATION OF HYPERBOLIC FUNCTIONS parallel cylinders, whether of the same diameter or of different -diameters, then follows at once in terms of inverse hyperbolic functions or anti-hyperbolic functions. These formulas are rigid at all distances, whereas the ordinary linear capacitance formulas are only approximate formulas, in which the error is practically insignificant when the distance between them is more than say 25 times the diameter of each cylinder (Bibliography, 55). rr ___ __ __ __ __ .. . * t w ___ . . . ___ _____ ___ _ FIG. 85. Two Parallel Eccentric Cylinders, one enclosing the other, and the inferred common zero-potential plane. Inverse hyperbolic functions apply very conveniently also to eccentric parallel cylinders, as indicated in Fig. 85. The geometrical transformation of a rectilinear area into a conjugate circular area is represented in Fig. 86. Here in v w plane we have abscissas from v= 1 up to # = 1, and ordinates from w = ^ to w = + ~-. If we apply to this plane the -. tanh function in the form y + fe = tanh (v + jiv) we obtain the curvilinear diagram shown. Corresponding portions of these two diagrams are shaded alike. Thus if TO ELECTRICAL ENGINEERING PROBLEMS 209 pqm and tuvx are the sections of two parallel wires in the yz plane, then the flow lines of current in a conducting medium or of displacement in a dielectric medium follow the circular -63- -^6- -e4 &- mm* 0^4 .Q.(3 ^6 Ms FIG. 86. Graphical Comparison of (v + w;^/ - 1) and of tanh (v 4- ?r N / - 1). segments such as c'b'a'd'e'f and the equipotential lines are the circles intersecting these. The total flow would be the same as between the flat surfaces pqrs and tuxx in the recti- linear diagram. Corresponding amounts of the flow occur in 210 APPLICATION OF HYPERBOLIC FUNCTIONS corresponding shaded areas and corresponding potential drops can be traced in the same way. TO ELECTRICAL ENGINEERING PROBLEMS 211 00000 0, oj J - -,-. _ . / ^ol" J / / ; 1 / 7 / 8000 ,| 7000 C i > > i / / r .s " > / / [7 i s ; 1 / / y 5000 -g c > / 1 / ^ G r ' 1 > / ' / / 0000 g ^ y o & ! B - / / rr- MW o g ^ -2 ~t "? 2 500 S g ,..[ oooo ^55 5 f i ' 7 ^ wo x - g 9* ' 15CO * 11 o 1 || / I 1 i!J .- > i-i > ^ / / / !vi^ ^^ a > c [_.'.! / / ^ 1 8 [ I ' _J 5o ^ / B r jj Q -^ *" II W \....$t -a S| ^ > j f j ^ ' ^ t ~C -^t! "jt IS I 40 - 2 s D . ' 350 <3 -f^ * U I/I J 7 32 -2 T x - -^nn '" = P it I MI - - ;3j- i - + ouu g ^ c j ^ o 2fio r* ^ " I I A 1 M .^ asl 3J T if' IT Ifs 31 .s i || ta? ^ ill / 5 So'rf 2 1 - S, 2 i -sjs IftO H v ~ < II II / / * ^i 1 / i ! ' :- 90 -g an o cS , __^ 70 g -T---- / ----- fin ** "* w - - 5? o If 1 50 go Y~ rr F :]/: ir -2 r-T p ' 4--f--- 36 r; ^ _ 30 > g. = =!!!:! = = -:.-- J- = -i = ==26 | *> OO Or. aOt-;-oo^.}',m> t ,vn MM WQW umipq fyiondvo .nrji^j // 212 APPLICATION OF HYPERBOLIC FUNCTIONS The application of these inverse hyperbolic formulas to linear capacities between equal parallel wires, when the wires are very close together, is illustrated in Fig. 87, which gives the linear capacitance per wire-meter and wire-foot for various interaxial distances up to 25 diameters.* The linear capacitance per loop-meter or loop-foot will be respectively half of the amounts shown. Finally, Fig. 88 gives by inspection the linear capacitance of pairs of parallel wires for various separations greater than 25 diameters, and at which the error in the ordinary formulas becomes negligible for practical purposes. Since the external linear inductance of any uniform linear system of insulated conductors expressed in abhenrys per cm. must be the reciprocal of the linear capacitance of the same system of conductors expressed in statfarads t per cm., it is evident that inverse hyperbolic functions may also be applied to computing the linear inductances of cylindrical conductors in close mutual proximity. Dr. C. V. Drysdale has devised and published an interesting application of hyperbolic functions to the theory of the magnetism of linear magnets. (See Bibliography, 71.) Hyperbolic functions have also been applied for a long time to the problem of eddy currents in conducting laminae sub- jected to simple harmonic magnetic forces in their plane. (See Bibliography, 4, 62.) Hyperbolic functions have also been applied by Professors J. J. Thomson, Alexander Russell, and others, to several elec- trical problems other than those discussed in the preceding pages. (See Bibliography, 4, 36, 56.) * See Bibliography, 55, 67, 68. t The abhenry is an un authoritative name provisionally suggested for the absolute unit of inductance in the C.G.S. magnetic system, and the statfarad is a similarly provisional name suggested for the absolute unit of capacitance in the C.G.S. electrostatic system. APPENDIX A Transformation of Circular into Hyperbolic Trigonometrical Formulas. THE following identities are well known JX _ ~JX sin x = - ^ . .... . numeric (1) .>'* _1_ -jX cos x = .... (2) g.r _ c-X sinh x = - 2 ..... (4) ' -4- s~ x cosh x = g - ..... (5) tanh x = I +*-* ..-.. (6) where x is any real quantity andy = ^/ 1. In formulas (1), (2) and (3) substitute/?/ for x. Thus sin x = smjtj = -j =.; . -- - =J smh 2/ numeric (7) cos X = cos jy = "" -+ ?/ = 1^! = cosh y (8) tan . = tan^ = j-g ~ J = y J^ =y tanh y (9) Consequently in any transformation formula of simple circular trigonometry containing sines, cosines, or tangents of any real numerical argument x, we may substitute j sinh y or also j sinh x for sin x . . numeric (10) cosh?/ cosh re cos x . . (11) j tanh y y tanh a; tana? . . (12) since a transformation equation in terms of y will hold true 213 214 APPENDIX A generally. We may thus derive the corresponding formula of hyperbolic trigonometry. For example, taking the well-known formula sin x . /10X tan x = - - . . . . numeric (13) cos x , , . . , , j sinh x we obtain /tanh#=^ . . . . (14) coshx sinh x . ,-.,,, or tanh x = f ... (15) coshx Again taking cos 2 x + sin 2 x = 1 . . . (16) we obtain cosh 2 x + j 2 sinh 2 ^ = 1 . . . ,, (17) or cosh 2 x sinh 2 x = 1 . (18) Thus all the regular formulas of circular trigonometry may be transformed. APPENDIX B Short List of Important Trigonometrical Formulas showing the Hyperbolic and Circular Equivalents. HYl'EKliOLIC cosech sech = Coth 6 = 1 sinh 1 cosh . 1 tank cosh 2 B siuh 2 6=1 sech 2 0=1 tauli' 2 6 sinh 20 = 2 sinh cosh 6 cosh 26 = cosh 2 + sinh 2 2 tanh 2 coth _0 sinh cosh 1 t"" 1 2 = 1 + cosh = sinh . /cosh - 1 cosh 0+1 siuh (0J 2 ) = sinh 6 l cosh 2 cosh 6 l sinh 2 ^ cosh (0j 2 ) = cosh 6 l cosh? sinh X sinh 2 . tanh 0, tanli 0. 2 tanh (6, J . coth 0j coth 0o othtf.J- eoth The solutions of (5) and (6) are known to be (Biblio- graphy, 7) e = A cosh (njgr . x) + B sinh (^/gr . x) . volts (7) i = C cosh (Jgr . x) + D sinh (^/gr . x) amperes (8) The bracketed quantities in formulas (7) and (8) are hyper- bolic angles ; while A, B, C and D are constants depending on the terminal conditions of the line. If the steady conditions at the sending end are known to be an impressed e.m.f. of E A volts and an entering current of I A ' amperes ; then e = E A cosh L L a Ir sinh LjCt . . volts (9) p and i = I A cosh L x a sinh L x a . . amperes (10) ' TO where L^ is the distance x from the sending end, a = ^/gr and r = J'-. If, on the other hand, the conditions at the receiving end, B, are known to be a terminal e.m.f. of E B volts and an escaping current of I B amperes ; then (7) and (8) become e = E B cosh L 2 a + I B r sinh L. 2 a . . volts (11) 218 APPENDIX C ip and i = I B cosh L.,a + ~ sinh L 2 a . . amperes (12) ^0 where L 2 is the distance of the point x from the receiving end of the line. Thus, if the distant end B of the line is freed (Fig. 10), then in (11) and (12) we know that the current to ground at that ^nd vanishes; so that I B = 0, and those equations become c = E B cosh L.,a .... volts (lla) TT i = - sinh L 2 a . . . amperes (12#) ^ o where e and i are respectively the e.m.f. and current at the sending end of the line of length L 2 km. Consequently, the resistance offered by this line at P is f> R, = -. = r coth L 2 a . . . ohms (13) But although (lla), (12a) and (13) are arrived at over a length L 2 km. between the point P and the distant free end B, yet the reasoning they present is general, and applies if the point P be moved back to A (Fig. 10). That is, they apply to any line of length L and angle 6 ; so that, substituting EA for e y and I A for i, we have E B = E ^ -> = E A sech . volts (14) cosh (J R y . = TO coth 6 ohms (15) Again, if the distant end B of the line is grounded, then in (11), the terminal e.m.f. E B =0, and this equation becomes c = I B r sinh L 2 a . . . volts (lift) -or the current IB, received to ground at B, is IB = - - ... amperes (16) r sinh L 2 a where e is the e.m.f. impressed on the section L 2 at P. If we move P back to A, the length becomes L, and the angle of the line becomes La = 6 ; while e becomes E A . Hence .... amperes (17) APPENDIX C 219 To find the sending-end current at A with the line grounded at B, we may take (12), make E B = 0, and substitute the value of IB from (16) ; or, we may take (9), and make e = therein. In either case, we find I A = - ^ x = ?^coth 6 . amperes (18) - tanh 6 r^ c In the same manner, if we wish to find the potential and current under any assigned terminal conditions, we have only to apply those conditions to the proper equation in (9), (10), (11) or (12), and arrive at the required result. APPENDIX D Algebraic Proof of Equivalence between a Uniform Line and its T Conductor, both at the Sending and Receiving Ends. LET A'O'B'G', Fig. D, be the equivalent T of a line having an angle 6 hyps, and a surge-resistance z o ohms or surge-con- ductance ?/ c = l/z o mhos. Let the two branch resistances be t\ each made equal to // = z o tanh-^ ohms, and the conductance of the leak be made g = y sinh 6 mhos. Then if the T be grounded at B through a resistance a ohms, the apparent resist- ance of the line at A will be B' FK;. D. Equivalent T circuit with a terminal load a- at the receiving end and, an e.m.f. E impressed at the sending end. = P ' + . __. = P > + py 1 . ohms (1) Substituting z c tanh - for /', y Q sinh 6 for //', and z tanh 6' for a, by (54) we have (j - + 2, tanh f/ = z tanh -= -f- tanh sinh + tanh 0' sinh z 220 ohms (2) APPENDIX D 221 tanh | + tanh 6' } 5+_ - ohms (3) 2 ^ cos h e + sinh 6 tanh V } ( sinh i, />/ 1 i /i f ^ + tanh u ,.^ J sinh 1 + cosh 6 (4) [l + cosh 6 **" cosh + sinh tanh 0' J /sinh cosh + sinh 2 e tanh fl' + sinh 6 + tanh tf (1 + cosh e)} ohms X (I + cosh 0) (cosh + sinh tanh J) f (5) J(l + cosh_0)_(sinh 6 + cosh tanh ^V + sinh tanh ' = ^l_ = s. tanh (0 + 0') I+ tanh tanh This result is in accordance with formula (56) for a terminally loaded uniform line. The current entering the line at A will therefore be, if E A is the impressed e.m.f.,I A = tanh ^ , ^ amperes, and the cur- rent reaching ground at B will be % IB = I A . 5nO_ ampereg (?) , o tanh +* c tanh 9' + /I sinh tanh ^ + sinh tanh & + 1 (8) = AA. 1 + (cosh - 1) + sinh tanh 0' *' cosh + sinh tanh 0' * = l v cosh 6' (u . cosh cosh r + sinh sinh 0' = lA.^sh0_ . . . /. (12) cosh (0 + ) = EA cosh 0' /i f\\ . k 1 ^ z o sinh (0 + 0') which is in accordance with formula (57). APPENDIX E Equivalence of a Line II and a Line T. LET 2 A *B 2 G (Fig. E) be the three impedances of the star, 2/A 2/B 2/G be the three corresponding admittances , - , -7 respectively. E. Externally equivalent star and delta of resistances or impedances. 2-22 APPENDIX E Let z l z 2 2 3 be the three impedances of the delta, y y 9 y 3 be the three corresponding admittances , ~ , respectively. Then the star will be the external equivalent of the delta * if _?2- ohms (1) A z, 4- z a 4- :.. Zz :./... ^r g XQ\ ^ = :^~rv _i_ - y^ .... ~1 + -2 "r 2 3 Zz where Zz represents the sum of the three delta-impedances. The delta will likewise be the external equivalent of the star if or , = ^ ohms (4) (5) where Zi/ represents the sum of the three star-admittances. f\ f\ If yA = 2/c/ tanh "2 2/ B = 2/o/ tann ^ ? /G = 2/<> sinh ; then by (4) y (-^e \tanh tanh (2 + sinh 8 tanh | 2/o tanh 2 g ohms (7) * See Bibliography, 22. 224 APPENDIX E , 0/ sinh 2 6 \ . 0/_ = z tanh -(2 + 1+CQsh J = * tanh ^(l + cosh 6 ohms (8) = z o sinh ....... ...... (9) which is the impedance to be inserted in the architrave of the 77. Again by (5) o sinh 6 2/c-- e tanh -7: i A, y A ?/ G . _2_ smh_0_ ~ u A I i e " o , sinh2 ^ + sinhOtanh- 2 + 1 + sinh 6 , T = y t ann -^ 1 + cosh 2 which is the admittance to be inserted in the A pillar of the 77, and, by symmetry, also in the B pillar. APPENDIX F Analysis of Artificial Lines in Terms of Continued Fractions. BEFORE taking up the analysis of an artificial line, it is desirable to consider two indispensable propositions in the algebra of continued fractions. A continued fraction of the type F n (a, b) = 1 numeric /_ (1) a -f- 1 'X^TT ** _ is called an alternating continued fraction ; because the two terms a and b appear alternately in the successive denominators. On the other hand, a continued fraction of the type F n (c) = 1 numeric _ (2) may be called a constant continued fraction. The quantities a, b, and c, here considered, are numerical constants, positive or negative, integral or fractional, real, imaginary or complex.* If we multiply the four-stage continued fraction (1) by any constant factor d, we obtain by successive steps d 1 1 . . numeric L (3) = T? \d, (" u, * Certain limiting cases of the propositions, including Strehlke's theorem (Bibliography, 2), are denned in the original publication. (See Biblio- graphy, 47.) 225 226 APPENDIX F This process is perfectly general, and may be carried on to any number of stages. It leads to the following conclusions, which may be also demonstrated in other ways (a) The effect of multiplying a continued fraction by a constant d is to divide all the odd denominators by d, and to multiply all the even denominators by d. (b) The effect of dividing a continued fraction by a constant d is to multiply all the odd denominators by d, and to divide all the even denominators loy'd. (c) The effect of multiplying an alternate continued fraction F n (a, b), by a constant d is to produce a new alternate continued fraction, in which the odd denominators are a/d, and the even denominators are Id. (d) The effect of multiplying an alternate continued fraction F n (a, b), such as (1), by the particular constant d = */ - is, by the last preceding proposition, to produce a new alter- nate continued fraction, in which the odd denominators are - = Jab and the even denominators are &A/ -7 Jab. Thus the new alternate continued fraction reduces to a constant continued fraction Y n (Jab) for the particular case C=A/T-. or- F n (, b) = F M (Jab) . . numeric /. (4) and- F.(, &)=y- F. (/&) . ,,(5) so that ,, (6) "a + 1 = Ib 1_ \ a ^6 + 1 a + 1 That is, any alternate continued fraction may be expressed as a APPENDIX F 227 factor multiplied by a constant continued fraction of the same number of stages, the constant term of the latter being the geometrical mean *Jab of the two alternate terms of the former, and the factor being A/ the square-root of the ratio of the \ a alternates. Terminally Loaded Alternate Continued Fractions. If an alter- nate continued fraction of n stages in a and b terminates in a denominator m, thus F (, &)i_ = 1 numeric /_ (7) m a + I rc + 1 m it is called a terminally loaded alternate continued fraction, and the final fraction l/m is called the terminal load of the continued fraction. Expression of a Constant Continued Fraction in Hyperbolic Functions of an Auxiliary Angle. It may be proved * that -p, / x -, sinh nu . . . = 1 or *** cosh nu . r . , , mN = . , , . if n is odd (9) (* 1 / ^> \ where sinh u = -^ or u = sinh ( - ) numeric /_ (10) Consequently, any constant continued fraction n stages may be expressed as a sine-cosine ratio of n and n -j- 1 times a certain auxiliary hyperbolic angle u, defined by (10). It follows, * See Bibliography, 47. Q 2 228 APPENDIX F therefore, from the preceding that any alternate continued fraction of n stages may be expressed as a factor, multiplied by sine-cosine ratio of n and w + 1 times a certain auxiliary hyperbolic angle u. Example. Consider the particular 3-stage alternate continued fraction F 3 (af) = ^02T+T 500 + 1 0-00025 Here a = 0'00025 and 6=500. This can be transformed by pro- position (d) formula (6) into the constant continued fraction 50C 0-00025 ^6125 x/0'125 + 1 /v/0125. This, again, becomes, by (9), x/2,000,000 x COS ^~ sinn ^ VO'125 0-353554 where smh u = = ---- 2 ~- = 0176777 or, by tables, n = 017586 .... hyp. , , -r, . cosh 0-52758 so that F, (of) =- 1414-2138 x -. = 2117-7, a result easily checked by direct arithmetical solution of F 3 (,&). In this comparatively simple case, it is easier to solve the alternate by direct arithmetic ; but in the case of an alternate continued fraction of unwieldy constants, and many stages, it is much easier to obtain the solution by the hyperbolic process. Terminally Loaded Alternate Continued Fractions Expressed in Terms of Hyperbolic Functions. Considering the following notation for an ascending series of terminally loaded alternate continued fractions t -f- 1 m numeric /_ (11) APPENDIX F 229 / numeric /_ (12) ~~ F 3 (0,6)1. = . (13) a + I_ m etc. ; we may write consistently with the above notation F (M)jL = : numeric /_ (14) m m It is easy to show that if n t is an odd value, and ii tl an even value of n I b cosh (nu 4- u') , /-, ~\ F., (a, 6)ji_ = yLj . Sfc j-^T-ji-pj numeric A (15) sinh ()iu + ?Q / 1 rN (/-, I 1 \ X,. I .') " " \ / where, as before = sinh -i ^j = sinh -i ^| . |j (17) rcosh u \ ~W~ } . . (18) - f= smh u l When a, &, and ??t are positive real quantities, (18) becomes uninterpre table if the denominator becomes smaller than the numerator. That is, m must not be less than e u /*/ u lb~ If m is less than e ^1, we may write u' =j _ + w" . . . . . niimeric (19) whence u" = tanh - 1 \VjT_ ']. . . (20) \ cosh u J 230 APPENDIX F where u is positive and real. Formulas (15) and (16) then become ^ Ib sinh (mi 4- u fr ) F n (a,b}i = J- - - -77- numeric (21) 7 \ a cosh {(w 4- 1) u 4- w } /a cosh (?m + ?//') sinh {(n 4- 1) u 4- u"} In the particular case where m = 5/2, u" ; so that ,,, , 7X Ib sinh nu *-. M>./=Vff cosh7^TT) ' ' nmnerio(88) / 2 cosh , 'A' smhOn-J) Moreover, if we add - to F rt/ (V^, b) as in the particular case numeric (25) 6+1 we have by (6) and (9) Ib f^/ccb cosh nu ) . , . - ( "' &) = V I 2 + smh . + 1 nUmenC (26) 2 + F - ' 2 smh (., + 1) ,- . sinh w = J-{cosh % . coth (n f + 1) w} (28) \ ^ Again, if we add -^ to F Hy (a, b\ b as in the particular case /2 /a APPENDIX F we have, by (23) & - -ri / 7\ /& ( +/ab sinh /t;& | .C ,j ( rt C? ) 1 7> ~ ,v / "\ ' ' 231 * f \a( 2 V< csinVi ' r cosh (?i / + sinh n \ ra / M ) CQI-V ( = J-{cosh u cosli (?i y tanh (w / - + 1) J " f 1) } (32) Application of Foregoing Formulas to Artificial Lines. There are two types of artificial line, the single-conductor and the double-conductor line, as shown in Figs. 1 and 2 respectively. D Z /WvW FIG. Fl. Single-Conductor Type of Artificial Line. These are electrically equivalent, ignoring questions of lumpiness, circuit balancing, and circuit symmetry, if in each section AB of A' B' C' D' E' vww B" C" D" FIG. F2. Double-Conductor Type of Artificial Line. Fig. 1, there is the same conductor-resistance as in - Zi of Fig. 2, and that the capacity in each section of Fig. 1 should be twice the capacity in each section of Fig. 2 ; so that the OR 232 APPENDIX F product, i. e. the total resistance R in the line circuit, and the total capacity C across the circuit shall be the same. It is thus evident from what has been considered in earlier chapters that it is sufficient to discuss single-conductor lines only, with the understanding that the discussion applies immediately to corre- sponding double-conductor lines. It is also evident, from what has preceded, that we need only consider single-conductor lines of the continuous-current type, with sections of conductor -resistance and leaks in derivation as shown in Fig. 3. The formulas which we shall <___,. r, -x ---- r, ---- x ..... T, ..... x ..... r, ..... > FIG. F3. Artificial Line of Four Sections. use will then apply to alternating-current artificial lines with condensers, by extension from real to complex numbers. Fig. 3 shows a four-section artificial line AB. The four sections are alike. Each consists of a conductor-resistance i\ ohms, with a leak of g^ mhos at the centre. In other words, the artificial line is a simple succession of uniform T's. The junctions are numbered from (0) at B to (4) at A, and the leaks are marked with corresponding Roman numerals. Sending-end Resistance of Artificial Line when freed at Distant End. Let it be required to find the sending-end resistance of the artificial line at A, when freed at B. Commencing at leak I nearest to B, the free end, the con- ductance of this leak is y l mhos. Consequently, the resistance at and beyond point I on the line is -- ohms. The resistance y i APPENDIX F 233 in the line between leaks is 1\ ohms. Therefore the resistance beyond leak II is R/II = >i + - - . ohms (_ (33) i/i Expressing this as a conductance beyond point II G /ri = - = F 2 ^*0 ohms (37) Expressing this as a conductance mhos Adding the conductance of leak III to obtain the total con- ductance at and beyond this leak G'/III = ff 1 + I = ft + F 4 0i,ft) mhos L (39) y i "t ft + 1 234 APPENDIX F Expressing this as a resistance B/ni-r- =F 6 ^i^i) ohms (40) r, + 1 //i + Transferring attention to leak IV, the resistance beyond this leak is R/iv = ^ + *- = ^ + F 6 (ft, r x ) ohms L (41) V i <7i Expressing this as a conductance 0^v=.^- = F (- 1I ft) mhos/. (42) 'l I - 1 /'! + Adding in the conductance of leak IV, the total conductance at and beyond IV is G'/iv = 'i) i ..... ohms L (51) * *+! 'i/a Proceeding from leak to leak as before, we find for the total sending-end resistance at A = + F ? toi. ? i> i ' hms L (52) ft+1 '>! + 1 ft + By (32) this becomes R y/A = J- 1 - . cosh u . tanh 8?/ . ohms /_ (53) = Z Q ' cosh u . tanh 8/t . . . ,, (54) = z tanh 8?i ...... (55) APPENDIX F 237 In the case considered, with 7^ =500 ohms, and # 1 = 0'00025 mho, 2 = 1436'1 ohms, and ^=0*17586 hyp. the artificial line behaves like an actual smooth line of surge-resistance 1436'! ohms, and angle 1 '40688 hyps., of which by (23), page 15, the sending-end resistance at A when grounded at B is R M = z o tanh 140688 = 1273'633 ohms . . . (56) Proceeding in this way, we should find that in every respect the artificial line would behave in the steady state exactly like a real smooth line of surge-resistance z and angle Su hyps. Along and within the artificial line, the distribution of potential, current, resistance, power and energy would not be the same as in the corresponding smooth line; except at section junctions, where the agreement would be complete, each section being the external counterpart of a smooth line having a surge-resistance V 1 + 1= 2 J V^fp ohms L (57) and an angle = 2 u = 2 sinh-' ( 9 >) = 2 sinh-' ('J - ')hyps. L (58) which means that each section of artificial line is the equivalent T of the corresponding section of actual smooth line. See page 35, where // = r 1 /2. If each T-section of the artificial line, in the case considered, represented 100 km. of actual smooth line ; then the actual line of 400 km., having a surge-resistance of 1436*1 ohms, and an angle of 1 '40688 hyps., would have a conductor-resistance of 14361 X 1'40688 =20204 ohms, and a total dielectric leakance of 1-40688/1436-1 =0-979956 xlO- 3 mho. To these values correspond linear constants of r=5'051 ohms per km. and g= 2-44989 X 10~ 6 mho per km. ' If the leak in each section of artificial line is not applied at the middle of each section, the computation is only slightly altered, provided all the sections are alike. The value of u, the semi-section angle, remains unchanged and z only is affected. 238 APPENDIX F If the leak is applied half at each end of a section as in a symmetrical 77; then if z " = \fi\j (^ z = 2 "/cosh u .... ohms /_ (59) so that the smooth line corresponding to the artificial line is readily found. In the case of alternating-current artificial lines in the steady state, with condensers in the section leaks, the problem is the same ; namely, to find the equivalent section of smooth line represented by each symmetrical ^-section. The same formulas apply, but must be interpreted vectorially or in two dimensions. This means that the smooth actual line which is the external counterpart of a given section of artificial line comprising line and leakage impedances, varies with the frequency of operation. Fig. 4 gives the graph of potential distribution over a five- section line of the type above considered, with the distant end free, and 100 volts applied steadily at the home end, as represented in Fig. 18, page 39. The heavy broken line repre- sents the fall of potential in the artificial line, and the dotted curve that which would be found in the corresponding uniform smooth line. The heavy broken line is therefore a funicular polygon, or inelastic string loaded with equal weights at the equidistant points I, II, III, IV, and V; while the dotted curve is a simple catenary between the same terminal points. The catenary coincides with the polygon at the section-junctions 1, 2, 3 and 4 ; so that each section of the polygon is externally equivalent to the corresponding section of the catenary. It follows from the foregoing, that the distribution of resistance potentials, currents, power and energy at the section-junction of a J-section artificial line, operated either with a continuous current, or with a single-frequency alternating -current, is identical with that of the corresponding points on the equivalent smooth line. Inside each ^-section, the electric distribution is evidently different from that within the corresponding sections of the equivalent smooth line. The actual distribution at any point inside each T-section may, however, be readily found by Ohm's law deduction, from the distribution at the adjacent -th 35: 100 So So 44 Xra Flo. F4. Curves of Potential along five-section artificial lino, and al the corresponding smooth uniform actual lino funicular polygon and catenary. [Tofaa /,";/' -2^ APPENDIX F 239 section-junction. The conditions at any given leak may also be readily determined in hyperbolic functions. Thus, with the distant end free, the ratio of the resistance beyond the (N-f-l)th leak, to that at and beyond, i. e. including the Nth leak, is S^-^rS^ ic ^ (eo) R j N cosh (2 N 1) u With the distant end of the line grounded, the ratio of the resistance beyond the (N-fl)th leak, to that at and beyond, i. e. including the Nth leak, is sinh(2N + I) u , ; OXT ^ . . . numeric / (61) smh (2 N 1) u The line current at one side of a leak is obviously the same as the line current at the next adjacent junction on that side, whether the distant end is free, grounded, or in any inter- mediate state. Moreover, whether the distant end is free or grounded, if EX is the potential on the equivalent smooth line at the position corresponding to leak N (Figs. 3 and 4) ; then the potential at this leak on the artificial line is E' N = E N sech u .... volts /. (62) APPENDIX G A Brief Method of Deriving Campbell's Formula : Cosh L'a' = Cosh L'a -\ sinh L'a . . numeric. IN Fig. Gl, let AB be a line loaded at regular distances L' km. with series loads of 2 = 2 o ohms. In the continuous-current case 2 is a real numeric. In the single-frequency alternating- current case, Z is a complex numeric. A'B' represents a second line which is without loads, but which has such linear constants as to be electrically equivalent to AB from any one mid -load point a to any other such as b. A 2 5 L - z B oj /r BIT n& an /rrn\ *^ r f y sinh L'a . amperes /_ (2) where a is the attenuation-constant of the line AB without loads. Now referring to the equal length L' km. between the 240 APPENDIX G 241 corresponding points a'b' on the companion line, A'B', grounded at b' but without loads, we have, since EV = I' n - = IV cosh L'a' . . amperes /_ (3) where a' is the attenuation-constant of the line A'B'. Coll g per smooth wave-length. = * r 8.50- 2 40- I I 18 1 I 5 1 1C i a ! *> > *r i III 1 T^ Z S 11 i i I 2.30 2.80 2.00 1.90 1.50 i.ro l.CO 1.50 1.40 1 30 f 1 rTTENUATION CONSTAN1 IAGRAM FOR LOADED LIN1 tflTH NO LEAK AGE. NO LIN1 JDUCTANCE.AND VANISHIF )"< ALSO DOTTED CURVE FOR 0*1= ^10 AND UNIFOR1V LINES OF THE SAME . B 2S 1 If . i -D - ^ -r / 1 j / III * j sC S 1 1 If fill _> ^ / / 7 * / Hi / " s y /// 7 -fl -a / /fill / ^ "3 X (5 " J 4 / , '/ / -S >" s^" ^Vj- / // ffl i 20 X ? (J- * ^ *<\A / n ^7 1.10 1.00 .90 .50 ^ "" _^ ^ ^ *3& A /\ 11 1 ^^ ^ . - 5?S ^-T-^- 1 ^ ^~i ^: _^ =: == ^ '. ~~ S - . : - =s ^ ' -- -*1 ^ X c5 f Gs s'lF OTt? i / \ LINES < 1 ? r***"* 1 ^^* / 3 ff i ^ 8 <' / ! .50 40- ^/ i I/ * .30 .20 .10 /|7 i = ^ *?-\/ ;|f 1 J $ ..;.. ----- ., s 5= ^ v.y. -. ~~ d --/y I u^ irORN L NK 5 .-'""' "1 1 1 .10 .20 .30 .40 .50 .60 .70 .SO .90 LOO LI L0 L30 1. LSO FIG. G2. Curve Sheet taken from Campbell's Paper showing the increase in ^ \, for different numbers of coils (n") per smooth wave-length in a loaded cable, for various values of b = r"/r. Assume that IV = I&, and I' a > = !, so that the two lines are identical in regard to the entering and leaving currents; then equating (2) and (3), we have- cosh L'a' = cosh L'a + sinh L'a numeric / (4) 242 APPENDIX G Figs. G2 and G3 have been taken from Dr. Campbell's paper on "Loaded Lines"*, with the notation altered to conform to that here used, and with an additional scale of abscissas added. These figures relate to the properties of formula (4) as applied to lines of neglible inductance in the unloaded state. Referring to Fig. G2, and to the upper scale of abscissas, it will be seen that at n coils per smooth wave-length A", as obtained by Coils per smooth wave-length. SO ^ 15" II 10 9 g f 6 TI 3-0 VELOCITY DIAGRAM FOR LOADED LINES WITH NO ^ LEAKAGE. NO LINE INDUCTANCE, AND VANISHING Ol ALSO DOTTED CURVES FOR 0"l = '/, AND UNIFORM LINES OF THE SAME 82 i.o 1.2 1.3 1.4 1.5 FIG. G3. Curve Sheet taken from Campbell's Paper showing the decrease in velocity of wave propagation and the ratio -,-> for a loaded cable with different numbers of coils (n") smooth wave-length, and various values of b = - formula (226), the real attenuation -constant a\ as found from (4) is more than 2'5 times the smooth value a'\ determined by (150), in which the loading is imagined to be distributed smoothly. The real attenuation-constant a\ is rising very rapidly at this limit. The different heavy-line curves refer to r" different values of b = -. the ratio of the linear smooth load T resistance to the linear unloaded conductor-resistance. Thus in the case worked out on p. 150, the value of r" is 1'74 ohms per wire- km. in the load coils and r 27'34 ohms per wire- km. ; so that b = 0*064. Interpolating in Fig. G2 between the * See Bibliography, 27. APPENDIX G 243 heavy lines of b = and b = 0*1 at n" = 8*73 coils per smooth / wave-length, and we find ~ = 0*986 approximately, or the real attenuation-constant is actually 1*4 per cent, less than if the loads had been distributed smoothly, the result already found on p. 151. The dotted lines closely conforming to the heavy lines indicate the changes in the curves when the real components of the loaded section angles reach 0*1 hyp. In the case considered they only reached 0*06 hyp. In the lower part of the figure are curves which to the right- hand scale show the ratio 1 / /^i" f the real components of loaded section angles at different coil spacings. Thus at n" = 2*5 coils per smooth wave-length A", the real component of the loaded section angle would be over fourteen times greater than that computed by (226), on the basis of smoothly distri- buted extra inductance. Fig. G3 shows the corresponding ratio of actual velocity v' to smooth velocity v" (227), with different spacings of the coils in cabled wires. Thus for the case discussed on p. 152 with 8*73 coils per smooth wave-length this ratio on the curve sheet G3 is about 0*975, which agrees with that found near (229) ; namely, 17654/18111. It will be seen that at the critical value of n coils per smooth wave-length, the actual velocity becomes from 0*638 to 0*73 of the smooth velocity, according to the value of b. That is, for b0, or negligible resistance in the load coils, the velocity v' is reduced to 0*638 v", and the wave- length /' is therefore reduced in the same ratio to A" X 0*638. Consequently, when there are apparently n coils per smooth wave-length there are only two coils per actual wave-length. R 2 APPENDIX H Analysis of the Influence of Additional Distributed Leakance on a Loaded as compared with an Unloaded Line. WITH reference to Fig. 64, for any given line loaded or unloaded Let /^ be the argument of the vector conductor impedance z. Let /? 2 be the argument of the vector dielectric admittance y. Then the real component of the attenuation -constant of the line is by (150) / /-? | O \ i = v% cos ( Pl 9 P2 ) . hyp. per km. (1) Differentiating with respect to f$ 2 , we have da,= _sin<% . hyp. per km. (2) numerc I When a cabled line is unloaded, - l -~-&, at co = 5000, is near 2i 45 and - -~ l = -- . approximately. When such a line is loaded, & + & is usua n y nea r to 85 and tan ^+ ~ = n ' 4 approximately; so that l = 11*4 -^ 2 ; that is, the per- a 1 centage change in a 1 for a given small reduction in f$ z due to extra distributed leakance is more than ten times as great for the loaded as the unloaded line. o I n When an aerial line is unloaded, ^ ^' 2 , at w 5000, is usually near to 75, and ( ttl = 3'7 - 2 . When such a line 1 is loaded, ^+A approximates 85, and tan ^^&- H-4. Consequently, the percentage change in c^ for a given small reduction in /? 2 , due to extra distributed leakance, is some three times as great for the loaded as the unloaded line. 244 APPENDIX J To Find the Best Resistance of an Electromagnetic Receiving Instrument Employed on a Long Alternating-Current Circuit. THE receiving instrument is assumed to be of the electro- magnetic type, employing a coil or coils of fine insulated copper wire in the receiving circuit to ground. Let o be the resistance of this winding in ohms. Let the inductance of the receiving apparatus be considered as independent of the resistance, i. e. the inductance of the winding is either small relatively to the resistance ; or the inductance accompanying the resistance of the winding can be modified, compensated for, or overshadowed by independent adjustments of other inductive elements in the receiving circuit. Then following the reasoning explained in connection with (276), either the magneto-mechanical force in dynes or the magneto-mechanical torque in dyne perp. cms., exerted by the apparatus under the excitation of a certain maximum cyclic received current I HlB , is either proportional to the ampere-turns %I rrtB , or to some power of the ampere-turns (w 1 l m B) p where p is a positive real exponent not greater than 2. If the receiver winding has fixed dimensions and volume, a winding of very fine wire will make a coil of numerous turns and, therefore, great sensibility, but possessing very high resistance. On the contrary, if the coil is formed of coarse wire, the resistance will be small, but the sensibility will be low. Evidently it is advantageous to reduce the size of the insulated wire until the increase of resistance more than offsets the increase of sensibility. If throughout the range of working sizes of wire, the ratio of bare to covered wire diameter is the same, so that the same total weights of copper and of insulating material will enter the winding with each size ; then halving the diameter of the wire will quadruple the number of turns, if the winding is carefully executed, but will increase the resistance sixteen-fold. Following this reasoning, the number of turns in 245 246 APPENDIX J a winding of fixed volume and dimensions varies as the square root of its resistance to continuous currents and also to low- frequency alternating-currents. Consequently formula (276) becomes F = a (I m v*/o) p dynes or dyne j_ cm. (1) where a' is a constant of the apparatus, depending upon its construction. The value of the received current by (286) is max - <* am p eres L (2) a vector equation. Let- * = ><> + A ..... ohms/. (3) and z r = o+W r +jx r ... L (4) where R' r is the extra resistance and z r the total reactance in circuit with a. The current I mB will always be a maximum for any given values of z , o, and R' r when jx r +jx =0 ...... ohms (5) or x r = x ...... (6) Since X Q is always a negative reactance, x r must be positive or magnetic reactance. Substituting in (2) we have cy. amperes (7) where [sinh 6] means the modulus of sinh 6 or the numerical value of sinh 6 with its argument suppressed. Formula (7) is now reduced to real numerical form and may be substituted in (1)- raax - In order to find the value of a, which will make F a maximum, we differentiate F with respect to o and equate to zero in the usual way ^L- 'J_ E ^Va V 1 - 1 do~ P \(r + o + R"';)[sinh 6>]J (r + a + R' r )[sinb 6] - E mA Jo [sinh 6] (r + a + R' r ") 2 [sinh 0] 2 max. cy. dynes dyne cm. ~ ohm ohm APPENDIX J 247 In order that this expression shall reduce to zero, it is necessary that r + a -h R'r - 2o = ohms (10) or o = /- + R' r . . (11) If the extra resistance-component of impedance in circuit with a is negligible ; then o = r ohms (12) Consequently, the maximum cyclic magneto-mechanical force, or torque, developed in the receiving instrument in the steady alternating-current state, will be a maximum if the reactance x r in the receiving circuit is equal and opposite to the reactance- component in the surge-impedance, and if the resistance a of the winding is equal to the resistance-component of the surge- resistance increased by any extra resistance R',. in the receiving circuit. APPENDIX K On the Identity of the Instrument Receiving-end Impedance of a Duplex Submarine Cable, whether the Apex of the Duplex Bridge is Freed or Grounded. IN Fig. K, let BaDG be the receiving connections at either end of the cable, with the apex a freed, and B'D'G' the corre- B B' FIG. K. Diagram of Impedances at either receiving-end in duplex connections, with the apex of the bridge disconnected in one case and connected to ground in the other. spending connections with the apex a grounded. It is required to show that in either case the expression (Z Q -\- z r ) ( : - ) has the same value, where Z Q is the surge-impedance, at the frequency selected, of both the real and artificial cables, z r is the impedance of the receiving apparatus to ground, as a whole, z l is the impedance of the instrument branch, z. 2 the imped- ance shunting the instrument. Apex Freed. With the apex free as at BaDG, we have *r=oirr^ + *o. ohms^(i) 248 APPENDIX K 249 where h is the impedance of each arm in the bridge. Also 2 X = g and z. 2 = 2h. Consequently ... (4) Apex Grounded. With the apex grounded, as at B'G'D'G', we have hz \ A+ *+ 9 + li 4-z \ _i_ i uuiiis i_ \r>) hz Also z 04- (Q\ ^ h + z and y ^ (7} 2 v / z l 4- z. 2 g (i ^ 4- ) 4- ^ H~ ^(^ ~J" 2 o) /q\ 7/7 I /v \ -.-... () V ~T O/ _(h + . (9) Hence ohms /. (10) . (12) APPENDIX L To Demonstrate the Proposition of Formula (7), page 4, 2 6 = I . . hyperbolic radians. i WE assume the well-known proposition that the magnitude of a hyperbolic angle is twice the sectorial area swept out by A X FIG. L. Diagram for the demonstration of the Theorem 6 = f ' 81 the describing radius-vector over a rectangular hyperbola of unit radius. In Fig. L, let the curve APB be a segment of a rectangular hyperbola whose radius OA is taken as unity. To a point P on the curve, whose cartesian co-ordinates are x and y, draw the radius-vector from the origin OP = p, making a circular angle 250 APPENDIX L 251 ft with the initial line and radius OA. Draw PT the tangent to the curve at P. Then the equation to the curve is r=^ 2 -i ........ (i) Differentiating both sides 2ydy = ( 2xdx ........ (2) ^=*=cot/*. (3) dx y If then we move the point P through a differential distance ds over the curve to P', and draw P'c and PC parallel to the X and Y axes respectively, cP' = dx and PC = dy, so that by (3) the angle cPF = YPT = ft. Join OP', and draw PQ perpendicular thereto, intersecting the same at p. Then the angle QPY = /?, because QP is per- pendicular to OP, and YP is perpendicular to OX. Therefore the angle QPP' = 20. Dividing (1) by p 2 we have ? 2 ./- 1 or sin 2 ft = cos 2 ft - -\ ....... (5) i = cos 2 ft - sin- ft ....... (6) -cos 2/5 ......... . (7) Representing the differential angle POP' by df$ Pp=p.dp .......... (9) and- = ?>. (10 ......... (12) 252 APPENDIX L But the differential area dA. enclosed between OP, OP' and the element ds is ^A=> 2 (18) /. - =2dA . . (14) P but if we denote the hyperbolic angle of PP' by d6 (W=2dA (15) /. (IB = - (16) P Proceeding in this manner from an initial point s : whose distance along the curve from A is s l units up to a final point S 2 whose distance along the curve from A is s 2 units, we have *2 6 ' d * - f ( l^ J 7 and this hyperbolic angle must be equal to the arc length s=s 9 ,\ divided by a certain integrated mean radius-vector p'or- LIST OF SYMBOLS EMPLOYED AND THEIR BRIEF DEFINITIONS A . . Area of a hyperbolic sector (cm. 2 ). A, a . . " . Fourier-series constants (volts). A, Aj, A 2 . . Moduli of complex numbers (cm. or length units). a . . .In diagrams, an abbreviation for current- strength in amperes /_ . a, a' . . . Torque constants of receiving instruments (dyne _L cm. per ampere). ti , . . Silence-interval in Kelvin theory of submarine cables (seconds). a.-c. . . . An abbreviation for alternating-current. a . .A constant in the hyp. theory of continued fractions (numeric /_ ). a . . . Attenuation-constant of a line (hyps, per wire-km.). a, Attenuation-constant of one line of a loop circuit (hyps, per wire-km. /_ ). a n . . . Attenuation-constant of a loop line (hyps, per loop-km. /_). a p a.y . . Real and imaginary components respectively of an attenuation-constant (hyps, per wire-km.). a v a. 2 , a^ . . In hyp. theory of composite lines, the attenua- tion-constants of successive sections (hyps. per wire-km. /_ ). a', a v a., . . Attenuation-constant and its components for a loaded line including lumpiness effects (hyps, per wire-km. /_ ). 253 254 LIST OF SYMBOLS a", a" v a" 2 . Attenuation-constant and its components for a smoothed loaded line (hyps, per wire- km. {_) B, b . . . Fourier-series constants (volts). b . .A constant in the hyp. theory of continued fractions (numeric /_ ). Also in the theory of loaded lines, the ratio T" IT (numeric). /?, j^, /? 2 . . Circular angles (radians or degrees). C, c . . . Fourier-series constants (volts). C Total capacitance of a line (farads). c.c. . . . An abbreviation for continuous current. c Linear capacitance of a line (farads per wire-km.). c t . . . Linear capacitance of one line of a loop (farads per wire-km.). c tt . . Linear capacitance of a loop-line (farads per loop-km.). c' Linear capacitance of ooe line of a loop (farads per wire-mile). c" . . Linear capacitance of a loop-line (farads per loop-mile). F Admittance of a leak load (mhos /_ ). y = F/2 . . Admittance of a semi-leak load (mhos /_ ). y Also the power-factor circular angle at the sending end of a transmission-line (degrees or radians. D, d . . . Fourier-series constants (volts). D Distance between the axes of two parallel eccentric cylinders (cm.). d Distance between a plane and the axis of a parallel cylinder (cm.). Also a constant in the hyp. theory of continued fractions (numeric /_ ). Also sign of differentiation. LIST OF SYMBOLS 255 d v d 2 . . Distances between an inferred plane and the axes of eccentric cylinders (cm.). A = GI o. 2 . Difference of radii of two conducting cylin- ders (cm.). ^A> <5fi> & Hyperbolic angles of junctions of a series- load, and of a selected point on a line (hyps.). E, e . . . Fourier-series constants (volts). E . E.M.F. (volts /.). E A , E B , E p . E.M.F. at the sending end, receiving end, and intermediate point of a line (r.m.s. volts /_ ). E'x, EX . . E.M.F. ataleak of an artificial line, and at the corresponding point of the equivalent uni- form line (r.m.s. volts j_ ). E y/ , E, . . E.M.F. impressed on a pair of looped lines, and on either component single line (r.m.s. volts l_ ). E m , E, n A .-,- . Maximum cyclic e.m.f. at alternator terminals, and at sending terminal of a submarine cable (max. cy. volts [_ ). E . . .In oscillating-current theory, the initial value of a vector e.m.f. of self-induction (volts /_ ) e Instantaneous value of an e.m.f. (volts /_ ). Also the value of e.m.f. at an intermediate point on a line (volts /_ ). e = 271828 . . . (numeric). e.m.f. . . An abbreviation for electromotive force. F Magneto-mechanical force, or torque,developed in a receiving instrument (dynes or dyne _L cm.). F ( ) . . Expression for an alternate or constant con- tinued fraction of n stages. / . . . Frequency of an alternating- current or e.m.f. (cycles / sec.). f = 2/" . . Frequency of dot-signalling on a submarine cable (dot-cycles / sec.). 256 LIST OF SYMBOLS ,/" . . . Frequency of reversal-signalling on a sub- marine cable (reversal-cycles / sec.). f Q . . Limiting maximum frequency of alternation on a submarine cable (cycles per sec.). G . Total dielectric leakance of a line (mhos {_ ). Gp A . . . Admittance of a line at sending end when grounded at receiving end (mhos /_ ). G/p . Admittance at and beyond a point on an artificial line, free at far end (mhos /_ ). G' rP . . Admittance at and including a leak on an artificial line, free at far end (mhos J_ ). g . Linear leakance of a line (mhos per wire- km. /_). Also, impedance of a receiving instrument in submarine cable duplex theory (ohms /_ ). ffi Admittance of the leak in an artificial-line section (mhos /_). g, Linear leakance of one of a pair of looped lines (mhos per wire-km. /_ ). g' - - Linear leakance of one of a pair of looped lines (mhos per wire-mile /_ ). 9,i - Linear leakance of a loop-line (mhos per loop-km. /_). Also the admittance of a 17 pillar after adding a leak load (mhos /_ ). 9" - Linear leakance of a loop-line (mhos per loop- mile l_ ). g' . Admittance of the leak in the staff of a T (mhos /_). g" - Admittance of the leak in the pillar of a 77 (mhos /_). Also in the theory of loaded lines, the total linear leakance of a smoothed loaded line (mhos per vvire-km.). 9 1 Apparent linear leakance of a line, uncorrected for line resistance (mhos per km. /_ ). LIST OF SYMBOLS 257 h Impedance of a duplex telegraph bridge arm (ohms /,). hyp.. . . An abbreviation for hyperbolic radian (numeric /_). C . . . Hyperbolic angle of a line, or section (hyps. /_). 1f .1 . . Auxiliary hyperbolic angle of an appended impedance or line section (hyps. [_ ). O f v . . Hyperbolic angle of a line-section after being loaded (hyps. /_). O tt . .. . Hyperbolic angle of a looped line (hyps. /,). OR ,. . * . Hyperbolic angle of a loaded and smoothed line-section (hyps. /_). p . . Real and imaginary components of a hyper- bolic angle (hyps.). O v , 3 . . In hyp. theory of composite lines, hyp. angles of successive sections (hyps. /_ ). . . . . Limiting hyperbolic angle subtended by a submarine cable at maximum working speed (hyps.). $' o , 0" o ... ? . Values of for dot-signalling and reversal- signalling respectively (hyps.). IiA, IiB . Maximum cyclic current-strengths at sending and receiving ends of line (amperes /_ ). IA, IB, IP Current-strengths at sending end, and receiving end, and assigned point of a line (r.m.s. amperes /_ ). I' Aj I'B - . Condenser currents at sending and receiving ends of a line 77 (r.m.s. amperes /.). 1^ . ; . ' . Current-strength at sending end of line with far end free (r.m.s. amperes /_ ). I p . . . Current-strength at sending end of line with far end grounded (r.m.s. amperes /.). 1 .. - . . Instantaneous value of an alternating current (amperes /J. Also the current-strength at an intermediate point on a line (amperes /_). 258 LIST OF SYMBOLS T . .In oscillating-current theory, the initial value of the vector discharging current (am- peres /_). j = ^7 1 . The quadrantal operator (quadrantal versor). kg . . . Correcting factor for reducing a nominal T staff to an equivalent T staff (numeric [_ \ k :// . . . Correcting factor for reducing a nominal H pillar to an equivalent //pillar (numeric [_ \ Jc e . . . Correcting factor for correcting the line angle pertaining to a T 7 (numeric /_ ). k & . . . Correcting factor for correcting the line angle pertaining to a II (numeric /_ ). k,.' . . . Correcting factor for correcting the surge- impedance to a T (numeric /_ \ k,. r " . . . Correcting factor for correcting the surge- impedance to a II (numeric /_ ). kp . . . Correcting factor for reducing a nominal T arm to an equivalent T arm (numeric /_ ). kp . Correcting factor for reducing a nominal 77 architrave to an equivalent II architrave (numeric /_). &L Normal attenuation-factor of a line of L km. (numeric /_). km. . . . An abbreviation for kilometer. L Length of a line or section (km.). L' . . . Length of a section of loaded line, or distance between loads (km.). L p L 2 . . Distances along a line as measured from send- ing and receiving ends respectively (km.). L p L.,, L.< . . In the theory of composite lines, lengths of successive sections (km.). L e . . Length of line in which the normal attenua- tion factor is l/e (km.). Lj . . . Length of line in which the normal attenua- tion factor is 1/2 (km.). LIST OF SYMBOLS 259 . ; _ . . Linear inductance of a line (henrys per wire-km.). . , . Linear inductance of one line of a loop (Henrys per wire-km.). .;.,. . Linear inductance of one line of a loop (henrys per wire-mile). . Linear inductance of a loop line (henrys per loop-km.) . Extra linear inductance in the smoothed loads of a line (henrys per wire-km.) X . ..... Wave-length on a line (km.). /." . ... Wave-length on a loaded smoothed line (km.). i f . . i. Wave-length on a loaded line including lumpiness effects (km.). iri . . Transmission coefficient of a current wave at a point of discontinuity (numeric /_ ). Also a numerical constant in the hyp. theory of continued fractions (numeric /_ ). fjtf . .. An abbreviation for microfarad. X" . ...'.;, Number of load coils encountered by an advancing wave per second on a single loaded smoothed line (numeric / sec.). . . . Number of load coils encountered by an advancing wave per second on a single loaded line, including lumpiness effects (numeric / sec.). X , In the theory of artificial lines, the number of a leak counting from the far end of an artificial line (numeric). u . . . Number of letters per second in maximum working speed of a submarine cable (numeric / sec.). Also in the hyp. theory of continued fractions, the number of stages of such a fraction (numeric). Also in the theory of complex numbers and of the Ferranti-effect, any integer (numeric). s 2 260 LIST OF SYMBOLS . n^ . . . Number of turns in a receiving-instrument winding (numeric). >n" . . . Number of load coils per smoothed wave- length of a loaded line (numeric / km.). 'ii' . . . Number of load coils per wave-length of a loaded line, including lumpiness effects (numeric / km.). v Ratio of transformation of a transformer (numeric). ... Hyperbolic angle of a point on a line, measured from sending end (hyps. /_ ). P A) P B . . Alternating-current power, to sending, and at receiving, end of line (watts /_ ). P fA) P/B . . Effective components of PA and PB respec- tively (watts). PfcA, PJB Reactive components of P A and PB respec- tively (j watts). p . .A real exponent between and 2 (numeric). II . .A delta connection of three impedances externally equivalent to a line (ohms /_ ). 7t = 314159 (numeric). R Total conductor resistance of a line (ohms /_ ). R A) R P . . Resistance of a line at its sending end, and at a selected point (ohms /_ ). R?A, R/B Resistance of a line at A or B when free at far end (ohms /_ ). R/ . . Resistance of a line when free at far end (ohms l_). R,, . . . Resistance of a line when grounded at far end (ohms /_). R^A, RJ/B . . Resistance of a line at A or B when grounded at far end (ohms /_). R>. . . . Resistance between receiving end of a line and ground (ohms /_ ). R' r .- . . Additional resistance inserted in receiving- end circuit of a submarine cable (ohms {_ ). LIST OF SYMBOLS 261 v \ . Resistance offered by a line when grounded through apparatus at far end (ohms /_ ). . . Resistance at and beyond a point on an arti- ficial line free at far end (ohms /_ ). R' /P . . Resistance at and including a point on an artificial line free at far end (ohms /_ ). R/ . . Receiving-end resistance of a line (ohms /_ ). R' . , ; g > Resistance in the staff of a T (ohms /_). R" . - . . Resistance in the pillar of a 77 (ohms /_). r . '-...' . Linear resistance of a line (ohms per wire- kin.). r 1 v _. ... Apparent linear resistance of a line, uricor- rected for leakance (ohms per wire-km. /_ ). f-j '. . . . Linear resistance of single line in a loop (ohms per wire-km. [_ ). t\ . . .In hyp. artificial-line theory, the resistance of a conductor section of artificial line (ohms l_\ r" . . .In the theory of loaded lines, the extra linear conductor-resistance due to the load coils smoothed (ohms per wire-km.). r tl . . - Linear resistance of loop line (ohms per loop-km. /_). r . . . Surge-resistance of a line (ohms /_ ). r cf . . ' . Surge-resistance of one wire of a loop (ohms /_). r 0// . ; . Surge-resistance of a loop-line (ohms /_ ). r ' . * . Apparent surge-resistance of a T (ohms /.). r ". . . . Apparent sur ge-resistance of a 77 (ohms /_ ). r.m.s. . . Abbreviation of root-mean-square. * Radius-vector to a point on a curve (cm.). Also Resistance in each arm of a nominal T, or in architrave of nominal 77 (ohms /__ ). p . ' . . Resistance in each arm of a T (ohms /_). Also, in the theory of hyperbolic angles, the integrated mean value of a radius-vector (cm.). 262 LIST OF SYMBOLS p" . . Resistance in architrave of a 77 (ohms /_ ). p . .In oscillating-current theory, half the resist- ance of the circuit (ohms). Also in plane-cylinder theory, the resistivity of the environing medium (absohm-cm.). S . .A Fourier-series, equivalent to an impressed rectangular e.m.f. (volts /_.). ,s, Xj, >., . . Distances measured along a curve (cm.). 2 .. . Resistance, in one line, of a regular series line- load. (ohms /_). 2 . . .-In plane-cylinder theory, the sum of the radii of two parallel eccentric cylinders (cm.). 2 y . . . Sum of the three admittances of a triple star (mhos /_). 2 Z . . . Sum of the three impedances of a delta (ohms /_). o Resistance inserted between line and ground at receiving end of line (ohms /_ ). o = 2/2 . . In theory of loaded lines, half of the resistance of a regular load inserted in each line (ohms /_}. o . Resistance inserted between line and ground in each wire of a loop (ohms /_ ). o /y . . . Resistance inserted between .the receiving ends of a loop-line (ohms /_ ). o . .In plane-cylinder theory, the radius of a cylinder (cm.). T . .A triple star, or Y, connection of three im- pedances, externally equivalent to a line (ohms /_). . Time elapsed since a certain epoch (seconds). r = en . . Time-constant of a submarine cable (seconds or farad-ohms). U . . . In oscillating-current theory, the initial value of a vector discharging e.m.f. (volts /_ \ LIST OF SYMBOLS 263 U A , UP . . Potential at sending end and at a selected point on a line (volts /_). u t ', n" . . Auxiliary hyperbolic angles in theory of artificial lines and continued fractions hyps. L}- u, v . -. . Real and imaginary components of a complex number (numeric). t', W . ^ - . Cartesian co-ordinates of a point in a plane (cm.). v . . ..In diagrams, an abbreviation for potential in volts l_. T Velocity of propagation of waves on a line (km/sec.). v" . ' . . Velocity of propagation of waves on a smoothed loaded line (km/sec.). v f . . Velocity of propagation of loaded line, in- cluding lumpiness effects (km/sec.). X . . . Total reactance of a line- wire (/-ohms). x Distance from the sending end of a line (km.). Also the linear reactance of a line (/-ohms/km.). Also the inductive reactance in an oscillating- current circuit (ohms). X, Y, jc, y . Cartesian co-ordinates of a point in a plane (cm.). Y . Total admittance of a line (mhos /_ ). Y A , YB . . Admittance of semi-line condensers at ends of a II (mhos /_ ). #A, #B, yc - - Admittances of the branches of a star (mhos /_ \ y\> y& #3 - .In composite- line theory, the surge-admit- tances of successive sections (mhos /_). Also admittances of the sides of a delta (mhos l_). y = I/z . . Surge-admittance of a line per wire (mhos /_). y" I /p" . Admittance of the architrave of a /7 (mhos /_ ). y = 9 +j r <*> - Linear admittance of a conductor (mhos per km. L )- 264 LIST OF SYMBOLS Z Total impedance of a line- wire (ohms /_ ). Z c . . Impedance of a half-line condenser in a U (ohms /_). Z{ . . -Receiving-end impedance of a line (ohms /_). Zi'i . . Receiving-end impedance of a line corrected for the shunting of the receiving instru- ment (ohms /_). & ' . . . In oscillating-current theory, the surge-impe- dance of the circuit, /^///r, in the absence of resistance (ohms). z = r -\-jcco . Linear impedance of a conductor (ohms per km. /_). z . . . Surge-impedance of a line-wire (ohms /_ ). z oll . . . Surge-impedance of a looped line (ohms /_ ). 00 = /V ///V < . Surge-impedance of a line- wire with no dissi- pation (ohms). ' . . Apparent surge-resistance of a 7 7 -section artificial line (ohms /_). z c " - Apparent surge-resistance of a //-section artificial line (ohms /_). i'i . . Impedance of a receiving instrument when shunted (ohms /_ ). 2 . Impedance of the shunt to a receiving instru- ment (ohms f_ ). . Impedances of the branches of a star (ohms /_). I n composite-line theory, the surge-impe- dances of successive line-sections (ohms /_ ). Also impedances in the three branches of a delta (ohms /_ ). . Impedance of receiving apparatus to ground (ohms l_ ). . Impedance in the sending-circuit of a line (ohms l_ }. LIST OF SYMBOLS 265 Q . Hyperbolic angular velocity in an ultra- periodic circuit (hyps. / sec.). co = '2nf . . Angular velocity of an alternating e.m.f. or current (radians / sec.). Also, in diagrams, a sign for ohms /_ , co', a>" . . Angular velocities of dot- and reversal-fre- quencies impressed on a long submarine cable (radians / sec.). Q> O . :. . Limiting maximum angular velocity impressed on a submarine cable (radians / sec.). (o . ^ . In diagrams, a sign for mhos /_ . = . . ' . " Approximately equals." l_ . Sign of an angle, to indicate the existence of a vector or complex quantity, either actually or potentially. < . ... . " Is not less than." 0-5" . . 0-5 inch. ~ . .'-" . Cycles per second. BIBLIOGRAPHY PUBLICATIONS DEALING WITH HYPERBOLIC FUNCTIONS OR WITH COGNATE SUBJECTS 1. LORD KELVIN " On the Theory of the Electric Telegraph." Proc. Roy. Soc. London, May 1855. " Mathematical and Physical Papers," Vol. II, p. 71. "2. A. B. STREHLKE " Uber Periodische Kettenbriiche." Grunert's Archiv der Mathematik und Physik, Vol. XLII, p. 341. 1864. 3. H. R. KEMPE "On Testing by Received Currents." Journ. Soc, of Tel. Engrs., Vol. IX, pp. 222-231. 1880. 3A. W. LIGOWSKI -"Tafeln der Hyperbelfunctionen." Berlin: Ernst & Korn, 1890. 4. J. J. THOMSON " On the Heat produced by Eddy Currents in an Iron Plate exposed to an Alternat- ing Magnetic Field." The Mectrician, Vol. XXVIII, p. 599. 1891. 5. O. HEAVISIDE Reprinted " Electrical Papers." London. Vol. II, p. 247. 1892. (J. A. E. KENNELLY "Impedance." Tram. Am. List. El. Eiigt'*, Vol. X, p. 175. April 1893. 7. O. HEAVISIDE " Electromagnetic Theory." London. Vol. I, p. 450. 1893. 8. F. BEDELL and A. C. CREHORE " Alternating Currents." New York. Chap. xiii. 1893. 9. C. P. STEINMETZ " Complex Quantities and their Use in Electrical Engineering." Proceeding* of the -Int. El. Congress, Chicago, pp. 33-76. August 1893. 266 BIBLIOGRAPHY 267 9 A. A. E. KENNELLY "Impedance of Mutually Inductive Circuits." The, Electrician, Vol. XXXI, pp. 699-700. Oct. 27, 1893. 10. A. E. KENNELLY " On the Fall of Pressure in Long- Distance Alternating-Current Conductors." Eke. World, Vol. XXIII, No. 1, p. 17. Jan. 1894. 11. A. E. KENNELLY "A Contribution to the Theory of Telephony." Eke. World, Vol. XXIII, No. 7, p. 208. Feb. 1894. 12. W. E. AYRTON and C. S. WHITEHEAD " The Best Resistance for the Receiving Instrument on a Leaky Telegraph Line." Journ. Inst. of EL Enyrs. London. Vol. XXIII, Part 3, p. 327. March 1 894. 1 3. A. BLONDEL " Inductance des Lignes Ariennes pour Courants Alternatifs." L'Eclairagc Elcctrique, Oct.- Nov. 1894. 14. E. J. HOUSTON and A. E. KENNELLY " Resonance in Alternating-Current Lines." Trans. Am. Inst. El. Engrs, Vol. XII, pp. 133-169. April 1895. 15. J. McMAHON "Hyperbolic Functions." Chap. IV. of " Higher Mathematics," by Merriman and Woodward, pp. 107-168. Tables of sinh and cosh (x + j y) to x = 1-5, y = 1-5. Wiley & Sons, New York. 1896. 16. E. J. HOUSTON and A. E. KENNELLY "Alternating- Current Machinery." El. World and Engineer. Oct. 30, 1897. " Electrical Engineering Leaflets." Advanced Grade. The Electrical Engineer, N.Y. 1897. 17. C. P. STEINMETZ " Theory and Calculation of Alternating- Current Phenomena." W. J. Johnston Co., New York. 1897. 18. A. MACFARLANE " Application of Hyperbolic Analysis to the Discharge of a Condenser." Trans. Am. Imt. El. A',////-*, Vol. XIV, p. 163. 1897. 19. C. GODFREY "W r ave Propagation along a Periodically Loaded String." Phil. Mag., Vol. XVI, p. 356. 1898. 268 BIBLIOGRAPHY 20. M. I. PUPIN-T-" Propagation of Long Electrical Waves." Trans. Am, lust. EL Eiigrx., Vol. XVI, pp. 93-142. March 1899. 21. A. E. KENNELLY "On the Predetermination of the Regulation of Alternating-Current Transformers. El a: World and Engineer, N.Y. ' Vol. XXXIV, p. 343. Sept. 2, 1899. 22. A. E. KENNELLY " The Equivalence of Triangles and Three-Pointed Stars in Conducting Networks." EL World and Engineer, N.Y. Vol. XXXIV, No. .12, pp. 413-414. Sept. 16, 1899. 23. A. C. CREHORE and G. O. SQUIER " A Practical Trans- mitter using the Sine Wave for Cable Telegraphy and Measurements with Alternating-Currents upon an Atlantic Cable." Trans. Am. List. EL Engrs., Vol. XVII, p. 385. May 18, 1900. 24. M. I. PUPIN '' Wave-Transmission over Non-Uniform Cables, and Long-Distance Air-Lines." Trans. Am. List. EL Engrs., Vol. XVII, pp. 445-513. May 1900. 25. M. I. PUPIN " Wave Propagation over Non-Uniform Conductors." Trans. Am. Math. Soc., Vol. I, No. 3, pp. 259-286. July 1900. 25A. A. E. KENNELLY " The Reactance Drop and Reactance- Factor of Transformers." El. World and Engineer, Vol. XXXVIII, No. 3, pp. 92-94. July 20, 1901. 25B. M. I. PUPIN " A Note on Loaded Conductors." EL World and Engineer, Vol. XXXVIII, No. 15, pp. 587-588. Oct. 12, 1901. 25c. A. E. KENNELLY " Surges in Transmission Circuits." El. World and Engineer, Vol. XXXVIII, No. 21, pp. 847-849. Nov. 23, 1901. 26. M. LEBLANC " Formula for Calculating the Electromotive Force at any Point of a Transmission Line for Alter- nating-Current." Trans. Am. Inst. EL Engrs., Vol. XIX, pp. 759-768. June 1902. BIBLIOGRAPHY 269 "27. G. A. CAMPBELL "On Loaded Lines in Telephonic Transmissions." Phil. Mag., Series VI, VoL V, p. 313. March 1903. 28. J. HERZOG and C. FELDMAN "Die Berechnung Elek- trischer Leitungsnetze." Julius Springer, Berlin. Vol. I, chap. v. April 1903. 29. A. E. KENNELLY" On Electric Conducting Lines of Uniform Conductor and Insulation Resistance in the Steady State." Harvard Engg. Journal, pp. 135-168. May 1903. 30. A. E. KENNELLY " On the Mechanism of Electric Power Transmission." Elec. World, N.Y. Vol. XLII, p. 673. Oct. 24, 1903. 31. A. E. KENNELLY "Two Elementary Constructions in Complex Trigonometry." Am. Annals of Mathematics, Salem Press, 2nd Series. Vol. V, No. 4, pp. 181-184. July 1904. 32. H. V. HAYES "Loaded Telephone Lines in Practice." Proc. Int. Elec. Congress, St. Louis, Sec. G., p. 638. Vol. Ill, 1904. 33. A. E. KENNELLY "The Alternating-Current Theory of Transmission Speed over Submarine Telegraph Cables." Proc. Int. El. Congress, St. Louis, Sec. A. Vol. I, pp. 68-105, with Table of sinh, cosh, tanh, coth, sech, cosech P /45^to p = 20'50. 1904. 34. J. A. FLEMING " A Model illustrating the Propagation of a Periodic Current in a Telephone Cable, and the Simple Theory of its Operation." Phil. Mag., Aug. 1904, and Proc. Phys. Soc. London. Vol. XIX. 1904. 55. A. E. KENNELLY "High-Frequency Telephone Circuit Tests." Prop. Int. El. Congress, St. Louis, Sec. G. Vol. Ill, pp. 414-437. 1904. 36. A. RUSSELL" A Treatise on the Theory of Alternating Currents." Cambridge University Press. 1904. 37, J. HERZOG and C. FELDMANN "Die Berechnung Elektrischer Leitungsnetze." Julius Springer, Berlin. Vol. II, chap. vii. Sept. 1904. 270 BIBLIOGRAPHY 38. G. ROESSLER "Die Fernleitung von Wechselstromen. J. Springer, Berlin. 1905. 39. A. E. KENNELLY "The Distribution of Pressure and Current over Alternating-Current Circuits." Harvard Engineering Journal. Vol. IV, No. 3, pp. 149-165, No. 4, Oct. 1905, pp. 206-225, Jan. 1906; Yol. V, No. 1, pp. 30-56, April 1906. 40. C. V. DRYSDALE " The Measurement of Phase Differences." The Electrician, Vol. LVII, pp. 726-783. 1906. 41. BELA, GATI u On the Measurement of the Constants of Telephone Lines." The Electrician. Nov. 2, 1906. 42. C. V. DRYSDALE "Some Measurements on Phase Displacements in Resistances and Transformers.'! The Electrician, Vol. LVII. Nov. 16 and 23, 1906. 43. B. S. COHEN and G. M. SHEPHERD Telephonic Trans- mission Measurements." Jo'urn. of Proc. lust. Elec. Engrs. London. Vol. XXXIX, p. 503. 1907. 44. A. E. KENNELLY " The Process of Building-up the Voltage and Current in a Long Alternating- Current Circuit." Proc. Am. Ac. Arts & Sc, t Vol. XLII, No. 27, pp. 701-715. May 1907. 45. C. V. DRYSDALE " The Theory of Alternate-Current Transmission -in Cables." The Electrician. Dec. (>, 13, 20, 27, 1907, and Jan. 10, 1908. 46. O. LODGE and B. DAVIES " On the Measurement of Large Inductances containing Iron." Journ. List. EL Engrs., London. Vol. XLI, pp. 515-526. March 1908. 47. A. E. KENNELLY " The Expression of Constant and of Alternating Continued Fractions in Hyperbolic Func- tions." Am. Annals ~of Mathematics, Salem Press. Vol. IX, No. 2, pp. 85-96. Jan. 1908. 48. A. E. KENNELLY "Artificial Lines for Continuous Currents in the Steady State." Proc. Am. Ac. of Arts & Sc., Vol. XLIV, No. 4, pp. 97-130. Aug. 26, 1908. BIBLIOGRAPHY 271 49. B. S. COHEN" On the Production of Small Variable- Frequency Alternating Currents." Phil. Mag., Sept. 1908 ; or, Proc. Phys. Soc. London. Vol. XXI, p. 283. 1909. 50. BELA, GATI " Description et Utilisation de la Methode pour la Mesure des Constants de Ligne au Moyen du Barretter." Oct. 1908. 51. C. V. DRYSDALE- " The Use of a Phase-shifting Trans- former for Wattmeter and Supply Meter Testing." The Electrician, Vol. LXII, p. 341. Dec. 11, 1908. 52. G. Di PiRRO "Sui Circuit! non uniformi." Atti dclF Assoc. Elettrotecn, Ital, Vol. XII, No. 6, 1909 ; also La Lumterc Elect rique, Series 2, Vol. VII, p. 227. 1909. 53. A. E. KENNELLY " The Influence of Frequency on the Equivalent Circuits of Alternating-Current Lines." The Elec. World, Vol. Mil, p. 211. Jan. 21, 1909. 54. C. V. DRYSDALE "The Use of the Potentiometer on Alternating-Current Circuits." Phil. Mag., Vol. XVII, p. 402, March 1909; also Proc. Phys. Soc., London, Vol. XXI, p. 561, 1909 ; also TJie Electnciau, Vol. LXIII, p. 8, April 16, 1909. 55. A. E. KENNELLY " The Linear Resistance between Parallel Conducting Cylinders in a Medium of Uniform Conductivity." Proc. Am. Phil. Soc., Vol. XLVIII, pp, 142-165. April 1909. 55A. G. F. BECKER and C. E. VAN ORSTRAXD " Hyper- bolic Functions. Tables of Real Values, Smithsonian Mathematical Tables." Smithsonian Institution, Wash- nitfon, D.C. 1909. 56. A. RUSSELL "The Coefficients of Capacity and the Mutual Attractions or Repulsions of Two Electrified Spherical Conductors when Close Together." Proc. Roy. Soc. A., Vol. LXXXIJ. June 1909. 57. BELA, GATI "Das C. R. Gesetz und die Kabelschnell- telegraphie." Elcktrotechnik und Maschinenlav. Heft 37. 1909. 272 BIBLIOGRAPHY 58. BtiLA, GATI " Wechselstrom als Trager von Telephon- stromen." Elektrotechnische Zeitschrift. Heft 39. 1909. 59. B. S. COHEN "The Impedance of Telephone Apparatus." National Telephone Journal, p. 113. Sept. 1909. 60. A. BLONDEL and C. LE ROY " Calcul des Lignes de Transport d'Energie a Courants Alternatifs et tenant compte de la Capacite et de la Perditance Reparties." Rerue (^ Electricity Sept. 18 and 25 ; Oct. 23 and 30. 1909. 61. A. E. KENNELLY "The Equivalent Circuits of Composite Lines in the Steady State." Proc. Am. Ac. Arts & Sc. t Vol. XLV, No. 3, pp. 31-75. Sept. 1909. 62. F. RUSCH " Uber die Wirbelstromverluste im Leitungs- kupfer der Wechselstromarmaturen." Elcktrotcchnik und Masehincnbau. Jan. 23 and 30, 1910: 63. J. PERRY" Telephone Circuits." Proc. of the Phys. Soc. London. Vol. XXII, pp. 674-684. ' Feb. 25, 1910. Phil. Mag. May 1910. With Table of sinh and cosh, p /45, up to p = 1, in small steps. 64. W. E. MILLER "Formulae, Constants and Hj^perbolic Functions for Transmission-Line Problems." Table of sinh and cosh (x+jy) up to # = 1, y = l. General Elec- tric Review, Supplement, Scheneetady,N.Y. May 1910. 65. BtiLA, GATI " Uber die Anwendung hyperbolischer Func- tionen auf weite Entfernungen wirkenden Telegraphers und Telephonstromen." Mektrotechnik und Masctiinen- bau. Heft 33. 1910. 66. A. E. KENNELLY " Vector Power in Alternating-Current Circuits." Proc. Am. lust. El. Enyrs., pp. 1023-1057. June 27, 1910. 67. H. PENDER and H. S. OSBORNE "The Electrostatic Capacity between Equal Parallel Wires." Vol. LVI, No. 12, pp. 667-670. Sept. 12, 1910. 68. A. E. KENNELLY " Graphic Representations of the Linear Electrostatic Capacity between Equal Parallel Wires." Eke. World. Oct. 27, 1910. BIBLIOGRAPHY 273 69. W. A. J. O'MEARA " Submarine Cables for Long-Distance Telephone Circuits," and Discussion on same. Jouni. Proc. List. El. Engrs. Part 206, Vol. XLVL Dec. 1910. 70. A. E. KENNELLY " Vector Diagrams of Oscillating-Current Circuits." Proc. Am. Ac. Arts & Sc. Vol. XLVL No. 17, pp. 373-421. Jan. 1911. 70A. C. V. DRYSDALE " Propagation of Magnetic Waves in an Iron Bar." The Electrician. April 28, 1911. 71. J. A. FLEMING "The Propagation of Electrical Currents in Telephone and Telegraph Conductors." Constable & Co., London. May 1911. 72. F. BREISIG " Ueberdie Energieverstellung in Fernsprech- kreisen." ElcJctrotechnische ZeitscJirift. Heft 23, pp. 558-561 : Heft 24, pp. 590-593. June 1911. 73. A. E. KENXELLY " Tables of Hyperbolic Functions in Reference to Long Alternating-Current Transmission Lines." Proc. Am. Inst. EL Engrs., pp. 2481-2492. Dec. 1911. -T INDEX ABHENRY, 212 Absohin, 206 Actual attenuation-factors of telephone lines, 116 Addition of complex numbers, 52 Algebraic definition of angles, 3 Alternating continued fractions, 225 Analysis of complex numbers into components, 53 Angle subtended by a terminal load, 23 Angle subtended by a uniform line, 14 Angular point on a line, 21 Aperiodic currents, 202 Apparent surge-resistance of artificial line, 235 Application of hyperbolic functions to wire telephony, 112 Architrave, defined, 29 Argument of a complex quantity, de- nned, 51, 53 Arms of a T denned, 28 Artificial line, apparent surge-resist- ance of, 235 Artificial line, sending- end resistance of, 232 Artificial line tests of signalling speeds, 195 Artificial lines, 38, 231 Artificial lines for telephony, 142 Assumptions, fundamental, of tele- phonic theory, 113 Attenuation-coefficient, 114 Attenuation -constant, 13 Attenuation-constant, complex, 70 Attenuation-constant, vector, 70 Attenuation-constants for twisted pair telephone cables, 120 Attenuation-constants of telephone lines, 116 Attenuation-constants, particular values of, 124 Attenuation-factor, 114 Attenuation-factors observed on tele- phone cables, 134 Auxiliary angle of terminally loaded continued fraction, 227 Ayrtou & Whitehead, 267 Becker, G. F., 17 Becker & Van Orstrand, 271 Bedell & Crehore, 266 Best resistance for receiving instru- ment, 182, 192, 245 Blondel, A., 267 Blondel & Le Roy, 272 Breisig, F., 273 Building up potential and current dis- tribution, 69 Campbell, G. A., 45, 148, 268 Campbell's formula, 240 Canso Rockport Cable, signals over, 200 Casual loads, 42 Casual loads, intermediate, 174 Casual loads on composite lines, 174 Casual loads, terminal, 174 Characteristic constants of uniform lines, 18 Characteristics of loop-lines and wire- lines, 18 Characteristics with respect to unit of length, 20 Circuits, distortionless, 115 Circular angles, generation of, 1 Circular angles in terms of sector area, 4 Circular functions of complex angles, 57 Circular radians, defined, 3 Cohen, B. S., vi, 271, 272 Cohen & Shepherd, 270 Complex angles, trigonometrical func- tions of, 56 Complex attenuation-constant, 70 Complex quantities, 49 Composite lines, 159 Composite lines of two sections, 166 Composite lines, two-section formulas, 172 275 T 2 276 INDEX Composite lines of n sections, 173 Constant, attenuation, 13 Constant continued fractions, 225 Constants, characteristic of uniform lines, 18 Constants, secondary, of lines, 18 Continued fractions, Appendix F, 41, 225 Continued fractions, alternating, 225 Continued fractions, constant, 225 Continued fractions, terminally loaded, 227 Correcting factors of n or T, 31 Crehore & Squier, 268 Damping coefficient in condenser cir- cuit, 203 Definition of attenuation-constant, 13, 70 Definition of surge-resistance, 14, 126 DiPirro, G., 172, 271 Dissipative resistance in condenser circuit, 202 Dissymetrical T or n, 160 Dissymmetrical n or T equivalent to single line, 173 Distortionless circuits, 115 Distortion-ratio, 115 Division of complex numbers, 56 Dot frequency, 188 Dot signalling, 188 Drysdale, C. V., 212, 270, 271. 273 Duddell, W., vi Effect, Ferranti-, 82, 102 Effect of leakage on loaded lines, 154 E.m.f., harmonic components of, 100 Equiangular spiral, 204 Equivalent circuit of telephone cable lines, 141 Equivalent circuits of uniform lines, 28 Equivalent n, defined, 29 Equivalent T, defined, 29 Equivalents, Transmission, 176 Errors, lumpiness, 36 Faccioli, G., 110 Ferranti, S. Z. de, 102 Ferranti-effect, 82, 102 Fleming, J. A., vi, 17,76, 269, 273 Fourier series in cable signalling, 198 Functions, hyperbolic, 7 Fundamental assumptions of telepho- nic theory, 113 Funicular polygon, 238 Gati Bela, 122, 201, 270, 271, 272 General equivalent circuit formulas, 37 Generation of circular angles, 1 Generation of hyperbolic angles, 2 Glazebrook, R. T., 102 Godfrey, C., 148, 267 Graphic method of combining harmonic maximum voltage or current into resultant, 100 Gudermannian angle, 205, 206 Harmonic components of e.m.f., 100 Hayes, H. V., 134, 269 Heaviside, 0., vi, 84, 266 Helmholtz, Yon, 114 Herne, Robert, vi Herzog & Feldman, 269 Hyperbolic angles, generation of, 2 Hyperbolic angles in terms of sector area, 4 Hyperbolic angular velocity, 204 Hyperbolic functions, 7 Hyperbolic functions, inverse, 206 Hyperbolic functions, tables of, 17 Hyperbolic functions of complex angles, 59 Hyperbolic functions of complex angles, tables of, 66 Hyperbolic radians, defined, 3 Hyperbolic reactance, 204 Imaginary components of complex numbers, 53 Impedance loads, 42, 174 Indefinitely long telephone lines, 129 Influence of increasing linear induct- ance on telephony, 143 Influence of impedance load on re- ceiving-end impedance, 174 Influence of loading on normal attenu- ation, 154 Influence of loading on the surge- impedance, 155 Initial disturbances, 78 Initial sending-end impedance, 73 Intermediate casual loads, 174 Intermediate loads, 42 Inverse hyperbolic functions, 206 Irregular loads, 42 ./-operator, defined, 53 Jewett, F. B., vi Judd, Walter, vi Kapp, G., 102 Kelvin, vi, 184, 266 Kelvin, law of silent interval; 184 Kempe, H. R., 266 Leak loading, 46 Leak loads, 174 INDEX 277 Leblanc, M., 268 Ligowski, W., 17, 266 Limit to reduction in telephonic attenuation with increasing in- ductance, 145 Limiting signalling speeds, 183 Limiting telephonic range, 137 Line-angle, relation to length and frequency, 96 Linear capacitance of parallel wires, 211 Lines, artificial, 38 Lines, composite, 159 Lines of uniform resistance and leak- ance, 11 Lines, power-transmission, 86 Lines, single, 159 List of symbols, 253 Loaded composite lines, 174 Loaded lines, effect of leakage on, 154 Loaded telephone lines, 148 Loaded telephone lines, terminal transformers in, 157 Loads, casual, 42 Loads, impedance, 42 Loads, intermediate, 42 Loads, irregular, 42 Loads, series, 42 Loads, terminal, 42 Lodge & Davies, 270 Long line defined, 72 Loop-lines and wire-lines, character- istics of, 18 Lumpiness errors, 36 Macfarlane, A., 206, 267 McMahon, J., 66, 267 Merger T or IT, 160 Miller, W. E., 272 Modulus of a complex quantity de- fined, 51, 53 Multiplication of complex numbers, 55 Nominal II, 31 Nominal T, 31 Normal attenuation, 14, 26 Normal attenuation, influence of load- ing on, 154 Normal attenuation-factors of tele- phone lines, 116 O'Meara, W. A. J., 273 Operator/, defined, 53 Oscillatory currents, 202 n-conductor, 88 Parallel eccentric cylinders, 208 Fender & Osborne, 272 Perfectly insulated lines, 10 Perry, J., vi, 272 Pillars of a n, defined, 29 Plane -vectors, 52 Power-transmission lines, 86 Power vector, defined, 90 Powers of complex numbers, 56 Principal formulas for deriving hyper- bolic functions of complex angles, 65 Pupin, M. L, 148, 268 Quantities, complex, 49 Radians, circular, defined, 3 Radians, hyperbolic, defined, 3 Ratio, Distortion-, 115 Real components of complex numbers, 53 Receiving-end impedance of telephone circuit, 138 Receiving instrument, best resistance of, 245 Reciprocal of complex numbers, 55 Regularly loaded composite lines, 174 Regularly loaded lines, 42 Relation of line-angle to length and frequency, 96 Resistance, Surge-, 13 Reversal frequency, 188 Reversal signalling, 188 Reversion of equivalent n, 36 Reversion of equivalent T, 34 Roessler, G., 270 Roots of complex numbers, 56 Russell, A., 212, 269, 271 Rusch, F., 272 Secondary constants of lines, 18 Semi-amplitude range of telephone lines, 119 Semi- imaginary quantities, 124 Sending and resistance of artificial line/232 Sending-end impedance, 73 Series loads, 42 Short line, defined, 72 Short lines, 22 Signalling speeds on long lines, 183 Silent interval in cable signalling, 184 Single lines, 159 Staff of a T, defined, 29 Standard telephone frequency, 135 Statfarad, 212 Steady state signalling, 182 Steady state tests, 180 Steinmetz, C. P., vi, 126, 266 Strehlke, A. B., 266 Submarine cable impedances, 186 Subtraction of complex numbers, 55 Summation of waves, 82 278 INDEX Surge-admittance defined, 29 Surge-impedance, influence of loading on, 155 Surge-irnpedance of aerial telephone lines, 127 Surge-impedance of cable telephone lines, 128 Surge-impedance of telephone lines, 125 Surge-resistance, 13, 126 Surge-resistance, apparent, of artificial line, 235 Symbols, list of, 253 Symmetrical T or n, 160 T-conductor, 88 Tables of hyperbolic functions, 17 Tables of hyperbolic functions of com- plex angles, 66 Telegraph lines, steady state tests, 180 Telephone cable lines, equivalent cir- cuit of, 141 Telephone circuit, receiving-end im- pedance of, 138 Telephone lines, attenuation-constants of, 316 Telephone lines, hyperbolic angles of, 115 Telephone lines, indefinitely long, 129 Telephone lines, loaded, 148 Telephone lines, normal and actual attenuation-factors of, 116 Telephone lines, surge-impedances of, 125 Telephonic theory, fundamental as- sumptions of, 113 Terminal casual loads, 174 Terminal load, angle subtended by, 23 Terminal load equal to surge-resist- ance, 26 Terminal load greater than surge-re- sistance, 25 Terminal load less than surge-resist- ance, 24 Terminal loads, 42 Terminal reflections, 77 Terminal transformers on loaded tele- phone lines, 157 Terminally loaded continued fractions, 227 Thomas, P. B., 88 Thompson, S. P., vi Thomson, J. J., 212, 266 Transformation between circular and hyperbolic formulas, 213 Transmission equivalents, 176 Transmission, velocity of, 73 Traverse-time, 78 Trigonometrical functions of complex angles, 56 Trigonometrical properties of an angular point on a line, 21 Trigonometrical properties of simple uniform lines, 15 Two-section formulas for composite lines, 172 Ultra-periodic current, 202 Van Orstrand, C. E., 17 Vector attenuation-constant, 70 Vector power, defined, 90 Velocity of transmission, 73 Very short lines, 21 Walmsley, R. M., vi Wave-length, 76 Waves, summation of, 82 Wire telegraphy, application to, 179 Wire telephony, application of hyper- bolic functions to, 112 Printed for the UNIVERSITY OF LONDON PKESS, LTD., by RICHARD CLAY & SONS, LTD., London and Btingay. UNIVERSITY OF CALIFORNIA LIBRARY BERKELEY Return to desfc>from which borrowed. This book is DUE on the last date stamped below. FEB 23 1948 REC'D LD 7 $5 -7PM RECEIVED i 7 . * r >8 -3 p| DEPT. LD 21-100m-9,'47(A5702sl6)476 fC (96 V 7X /i'3 UNIVERSITY OF CALIFORNIA UBRARY