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In preparing the third edition, great improvements have 
been made, and a considerable part of the work entirely re- 
written, with the addition of much new material. A number 
of new chapters have been added, as those on vector rep- 
resentation of double frequency quantities as power and 
torque, and on symbolic representation of general alternating 
waves. Many chapters have been more or less completely 
rewritten and enlarged, as those on the topographical 
method, on distributed capacity and inductance, on fre- 
quency converters and induction machines, etc., and the 
size of the volume thereby greatly increased. 

The denotations have been carried through systematically, 
by distinguishing between complex vectors and absolute 
values throughout the text ; and the typt^raphical errors 
which had passed into the first and second editions, have 
been eliminated with the utmost care. 

To those gentlemen who so materially assisted me by 
drawing my attention to errors in the previous editions, I 
herewith extend my best thanks, and shall be obliged for 
any further assistance in this direction. Great credit is 
due to the publishers, who have gone to very considerable 
expense in bringing out the third edition in its present form, 
and carrying out all my requests regarding changes and 
additions. Many thanks are due to Mr. Townsend Wolcott 
for his valuable and able assistance in preparing and editing 
the third edition. 

Charles Proteus Steinhetz. 

Camp Mohawk, Visle's Ckebe, 
Jtdy, igoo. 

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The following volume is intended as an exposition of 
the methods which I have found useful in the theoretical 
investigation and calculation of the manifold phenomena 
taking place in alternating<urrent circuits, and of their 
application to alternating<urrent apparatus. 

While the book is not intended as first instruction for 
a beginner, but presupposes some knowledge of electrical 
engineering, I have endeavored to make it as elementary 
as possible, and have therefore only used common algebra 
and trigonometry, practically excluding calculus, except in 
§§ 106 to 1 15 and Appendix II. ; and even §§ 106 to 1 15 
have been paralleled by the elementary approximation of 
the same phenomenon in §§ 102 to 105. 

All the methods used in the book have been introduced 
and explicitly discussed, with instances of their application, 
the first part of the book being devoted to this. In the in- 
vestigation of alternating-current phenomena and apparatus, 
one method only has usually been employed, though the 
other available methods are sufficiently explained to show 
their application. 

A considerable part of the book is necessarily devoted 
to the application of complex imaginary quantities, as the 
method which I found most useful in dealing with alternat- 
ing-current phenomena ; and in this regard the book may be 
considered as an expansion and extension of my paper on 
the application of complex imaginary quantities to electri- 
cal engineering, read before the International Electrical Con- 

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gress at Chicago, 1893. The complex imaginary quantity 
is gradually introduced, with full explanations, the algebraic 
operations with complex quantities being discussed in Ap- 
pendix I., so as not to require from the reader any previous 
knowledge of the algebra of the complex imaginary plane. 

While those phenomena which are characteristic to poly- 
phase systems, as the resultant action of the phases, the 
effects of unbalancing, thfi^tiansformation of polyphase sys- 
tems, etc., have been discussed separately in the last chap- 
ters, many of the investigations in the previous parts of the 
book apply to polyphase systems as well as single-phase 
circuits, as the chapters on induction motors, generators, 
synchronous motors, etc. 

A part of the book is original investigation, either pub- 
lished here for the first time, or collected from previous 
publications and more fully explained. Other parts have 
been published before by other investigators, either in the 
same, or more frequently in a different form. 

I have, however, omitted altogether literary references, 
for the reason that incomplete references would be worse 
than none, while complete references would entail the ex- 
penditure of much more time than is at my disposal, with- 
out offering sufficient compensation ; since I believe that 
the reader who wants information on some phenomenon or 
apparatus is more interested in the information than in 
knowing who first investigated the phenomenon. 

Special attention has been given to supply a complete 
and extensive index for easy reference, and to render the 
book as free from errors as possible. Nevertheless, it prob- 
ably contains some errors, typographical and otherwise ; 
and I will be obliged to any reader who on discovering an 
error or an apparent error will notify me. 

I take pleasure here in expressing my thanks to Messrs. 
W. D. Weaver, A. E. Kennelly, and Townsend Wol- 
coTT, for the interest they have taken in the book while in 
the course of publication, as well as for the valuable assist- 

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ance given by them In correcting and standardizing the no- 
tation to conform with the international system, and numer- 
ous valuable suggestions regarding desirable improvements. 
Thanks are due also to the publishers, who have spared 
no effort or expense to make the book as creditable as pos- 
sible mechanically. 


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Chap. I. Introdiictioii. — 

S 1, p. 1. Fondamenta] ]am of continnoiiB current drcnits. 

t 2, p. 2. Impedance, reactance, effective resistance. 

S 3, p. 8. Elect ro-magnetisni as source of reactance. 

i 4, p. 5. Capadtj as source of reactance. 

f 5, p. 8. Joule's law and power equation of alteinating circuit. 

S 6, p. S. Fundamoital wave and higher harmonics, alternating 

waves with and without even hannonics. 
i 7, p. 9. Alternating wares aa sine waves. 

Chap. II. lutAntAneoni VAlties and Integral Valtiea. — 
i 8, p. 11. Integral values of wave. 
J 9, p. 13. Ratio of mean to maximum to effective valne of ware. 

Chap. III. Law of Blectro-nuiEiietlc Indnctlon. — 
{ 11, p. 16. Induced E.M.F. mean value. 
S 12, p. 17. Induced E.M.F. effective value. 
S 13, p. 18. Inductance and reactance. 

Chap. IV. Graphic fiepieMnUtton. — 

S 14, p. 19. Polar characteristic of alternating wave. 

5 15, p. 20. Polar characteristic of sine wave. 

S 16, p. 21. Parallelogram of une waves, KirchhofTs laws, and erieigjr 

S 17, p. 2S. Non-inductive circuit fed over inductive line, instance. 

5 18, p. 2i. Counter E.M.F. and component of impressed E.M.F. 

{ 19, p. 26. Continued. 

S 2(^ p 26. Inductive drcoil and circuit with leading carrenl fed over 

inductive line. Altemating-current generator. 

9 21, p. 28. Polar diagram of alteniating.curTent transformer, instance. 

S 22, p. SO. Continued. 

Chap. V. SrmboUc Httliod.— 

S 23, p. 33. IKsadvantage of graphic method for numerical calculation. 

S 24, p. 34. Trigonometric calculation. 

i 29, p. 84. Rectangular components of vectors. 

S 86, p. 3S. Introduction of / as <^stinguishing index. 

t 27, p. 36. Rotation of vector by 180° and 80'. / = V=T. 

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Chap. V. Symbolle Method — Continuid.— 

j 28, p. 87. ComtriiuUion of sine naves in symbolic expression. 

% 29, p. 38. Redslance, reactance, impedance, in symtxilic expreamon. 

S 30, p. 40. Capacity reactance in symbolic represenlUion. 

9 31, p. 40. KirchhoS's laws in aymlxilic representation. 

I 32, p. 41. Circuit supplied <tver. inductive line, iv 

Chap. VI. Topographic Method.— 
S 33, p, 43. Ambiguity of vectors. 
9 34, p. 44. Instance of a three-phase system. 
5 3S, p. 46. Three-phase generalor on balanced load. 
5 36, p. 4T. Cable witb distributed capacity and it 
§ 37, p. 49. Transmis^on line with self-inductive capacity, resistance, 
and leakage. 

Chap.- VII. AdmitUnce, Condnctonce, Susceptonce. — 

3 38, p. 62. Combination of resistances and conductances in series and 

{ 39, p. ()3. ComUnation of Impedances, Admittance, conductance, 

S40, p. 54. ' Relation between impedance, re^stance, reactance, and 

admittance, conductance, auaceplance. 
} 41. p. 66. Dependence of admittance, conductance, susceplance, upon 
ComtMnation of impedances and ad- 

Chap. Vllt. Circuits contAinins Seslttuice, Inductance, and Ca- 
pacity. — 
g 42, p. 58. Introdaction. 
g 43, p. 68. Rebalance in series with circuit. 
% 44, p. 60. Discussion of insiances. 
% 4S, p. 01. Reactance in series with circuit. 
I 46, p, 04. Discussion of Instances. 
% 47, p. 86. Re^ance in series with circuit. 
{ 48. p. 68. Impedance in series irilh circuit. 
S 49, p. OS. Continued. 
§50, p. 71. Instance. 

S SI, p. 73. Compensatltin for lagging currents by shunted conaensonce. 
S S2, p. 73. Complete balance l>y variation of shunted condensance. 
I 53, p. 75. Partial balance by constant shunted condensance. 
{ 54, p. 76. Constant potential — constant current tiansformatkon. 
% 55. p. 79. Constant current — constant potential transformation. 
S 56, p. 81. Efficiency of constant potential — constant cnnent trai» 

Chap. IX. Keaiataoce and Keactance of TiansmiMion Line*. — 
{ S7, p. 8S. Introduction. 
S 56, p. 84. Non-inductive receiver circuit supplied over inductive line. 

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Chap. IX. Besiatuce and Seactance of Tranamlaaloii Llnea. — Cfntiimed. 
{ 59, p. 80. Instance. 

% 60, p. 87. Maximum power supplied over inductive line. 
S 61, p. 88. Dependence of Output upon the suaceptatice , of the re- 

{ 63, p. 89. Dependence of output upon the conductance of the t«- 

cdvei circuit. 

% 63, p. 00. Summary. 

S 64, p. 92. Instance. 

S 65, p. C3. Condition of majiimam efficiencj'. 

S €6, p. 06. Control of receiver voltage by shunted susceptance. 

j 67, p. 67. Compensation for line drop by shunted susceptance. 

g 68, p. 117. Maximum output and discus^on. 

5 69, p. 98. Instances. 

J 70, p. 101. Maxium rise of potential in receiver circuit 

J 7X, p. 102. Summary and instances. 

Chap. X.. BfiectiTe SMistonce and ScacUnce. — 

S 72, p, IIM, Effective resistance, reactance, conductance, and susc^ 

§ 73, p. 106. Sources of energy losses in alternating-current drcuita. 

5 74, p. IOC. Magnetic hysteresis. 

% 75, p. lOT. Hysteretic cycles and cortespontUng current waves. 

S 76, p. 111. Action of air-gap and of induced current on hysteretic 

distort iotL 

S 77, p. 111. Equivalent sine wave and wattless higher harmonic. 

5 78, p. 113. True and apparent magnetic characteristic. 

5 79, p. 115. Angle of hysleietic advance of phase. 

5 80, p. 116. I^ss of energy by molecular magnetic friction, 

5 81, p. 1 Ifl, Kffeclive conductance, due to magnetic hysteresis. 

S 83, p. 12:!. Absolute admittance of ironclad circuits and angle of 

hysteretic advance. 

. S 03, p. 124. Magnetic circuit containing air-gap, 

% 84, p. \'2->. Electric constants of circuit containing iron. 

g 83, p, 127, Conclusion, 

Chap, XI. Foucatilt or Eddy Currents,— 

§ S6, p. 120, Effective conductance of eddy currents, 
§ 87, p, 13'l, Advance angle of eddy currents, 

5 88, p, 131, I«ss of power by eddy currents, and coefficient of eddy 

589, p, 1-tt, Laminated iron, 

590, p, m. Iron wire, 

S 91, p, 13,'i. . Cornparison of sheet iron and iron wire. 

g 92, p. 136. Demagnetinng or screening effect of eddy curienls. 

A93,p. J3B, Continued.- 

9 94, p. 138. I^rgeeddyci 

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Crap, XL Foneaatt or tAAj Cmn^». — C<mtiHtied.— 

% 95, p. 139. Eddj cnirenta in conductor and nneqiul cnirent dii- 

9 M, p. 140. Conttmwd. 

S 97, p. 143, Mutual inductance. 

9 96. p. 144. Didectiic uid dectroslatic phenomena. 

§ 99. p. 146. Dielectric hysteretic admittance, impedance, ^»g, etc 

1 100, p. 147. Electrostatic indnclian or influence. 

} 101, p. 14S. Energ]' components and wattless components. 

Chap. XIL Powar, and Double Fr*qveiicy QoaBtttlM In Oenanl. 
f 102, p. 160. Double frequency of power. 
S 103. p. 161. Sjimbolic representation of power. 
S 104, p. 153. E«ia.algebtaic featniea thereol 
J 105, p. 166. Cconbinadon of powen. 
S 106, p. 166. Torque as double frequency product. 

Chap. XIII. Diatrlbnted Capacttj, IndocUiiw, KeaiiUnce, and Leak, 

9 107, p. 168. IntrodnctioiL 

9 lOB, p. 15Q. Haiputude of charing current of tianamUsion lines. 

9 109, p. IdO. Line capacity represented by one condenser shunted 
across middle of line. 

9 110, p. 191. Line capadty represented by three condenMrs. 

S 111, p. 1B3. Complete investigation of distributed capacity, induc- 
tance, leakage, and resistance. 

9 112. p. 166. Continued. 

9 113, p. ISe. Continued- 

9 114. p. IW. Continued. 

9 115, p. 187. Continued. 

9 116, p. 169, Continued. 

9 117, p. 170. Continued. 

9 118, p. 170. Difference of phase at any point of line. 

9 119. p. 173. Imtance. 

9 120. p. 17S. Further Instance aitd discussion. 

9 121. p. 17S. Particular cases, open circuit at end of Une, line 
grounded at end, infinitdy ong conductor, generator feeding 
into closed ciicnit. 

S 122, p. 181. Nattual peiiod of transmission tine. 

9 123, p. ISO. Discuaaon. 

9 124, p. 100. Continaed. 

9 12S, p. 191. Inductance of unifoimly charged line. 

Chap. XIV. The AttanaMn'OnMt TnaaHumm. — 
9 136, p. 109. General 

9 127, p. lOS. Mutual inductance and sdfinductaDCe of tranatomtm'. 
9 128, p. 104. Magnetic dicoit of nvaaftHwer. 

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Chap. XIV. Tb* Altnastlni-Cwmnt Znuudtaraur— Cfimiitttiai.r-~ 

S 129, p. 196. Continued. 

1 130, p. 190. P<iv diagmn of irKoSoaaet. 

i 131, p. 198. JwtsDCe. 

% 133, p. 803. DUgnM for Ta:i7in( kMd. 

._ {UXp.W8. Inataooe. 

S 134, p. 204 Sjubedic meihod, equtiotn. 

S 13S, p. SOe. CoMinoed. 

1 136, p. 308. AppareiU iinpedaBce of tnnrfoimeT. Tnuufonner 

eqiuvalent to dividad circnit. 

{ 137, p. 200. Condnnod. 

S 138, p. 212. TianaforaMr on noaiaductiTe load. 

S 139, p. 214. Constants of tiamf onner co noMndnctive load. 

fi 140, p. 217. Nomeiio] inatance. 

Chap. XV. Goiwnil AlUniAting-CiuTeiit Tmuformer or EYequencf 


1 141, p. 219. Introdnctian. 

S 143, p. 220. Magnetic croes-iuz or scU-ondiKtion of tfaiiafomei, 

S 143, p. 231. Mutual flux of tiaufonner. 

} 144, p. 221. DifEerence of freqnenc; between pitmuf twd accondaiy 

of general altematecnmnt tnnsfonner. 

{ 145. p. 221. Equations of genetal altenuUe.caiTent t»a*fcifm*r. 

j 146, p. 227. Power, ontpnt, Biid Input, meduuncal aad elertaca], 

§ 147, p. 22a Continued. 

S 148, p. 329. Speed and ontpnt. 

S 149, p. 2S1. Nmneiical instance. 

S ISO, p. 232. Chajacteristic ctwea of frequency comretter. 

Chap. XVI. IndncUon MacUiiei.— 

t ISl, p. 237. Slip and secondary frequency. 

J 153, p. 238. Equations of inductlan motor. 

S 153, p. 230. Magnetic flnx, admittance, and impedance. 

§154, p. Ml. E,M.F. 

S 155, p. 244. Graphic representation. 

S 156, p. 246. Continued. 

% 157, p. 246. Torque and power. 

5 158, p. 246. Power of indnctian miotots. 

S 199, p. 260. Maximum torque. 

g 160, p. 262. Continued. 

1 161, p. 263. Maximum power. 

\ 162, p. 254. Starting torqne. 

S 163, p. 268. SynchroruHn. 

{ 164, p. 268. Near syadironiam. 

{ 165, pL 269. Nianerica] hidance of induction motor. 

J 166, p. 203. Calculation of induction Jnotor ewes. 

S 167, p. 266. Numerical inotance. 

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Chai*. XVI. Indnnfm Machtaee — Connnuin/. — 

S 16S, p. 266. Induction generator. 

3 169, p. 268. Power factor of induction generator. 

% 170, p. 269. Constant speed, induction g«ierator. 

S 171, p. 272. Induction generator and synchronous tnotoi. 

S 172, p. 274. Concatenation or tandem control (rf induction n 

\ 173, p. l!7e. Calculation of concatenated couple. 

{ 174, p. -Xm. Numerical instance. 

g 17S, p. 2S1. Sing^e.phase induaion motor. 

§ 176, p, 283. Starting devices of single-phase motor. 

§ 177, p. 284. Polyphase motor on single-phase circuit 

9 178, p. 286. Condenser in tertiary circuit. 

\ 179, p. 287. Speed curves with condenser. 

S 180, p. 291. Synchronous induction motor. 

S 181, p. 2ea Hystere^ motor. 

Chap. XVII. Altenute-Coire&t Geoenitoi.— 

S 182, p. 297. Magnetic reaction of lag and lead. 

j 183, p. 800, Self-inductance and synchi 

S 184, p. 302. Equations of alternator. 

S 185. p. 303. Numerical instance, field chi 

S 186, p. 307. Dependence of terminal voltage 

% 167, p. 307. Constant potential regulation. 

% 188, p. 309. Constant current regulati 

Chap. XVIII. SynchroDiiiiiK Alternators.— 

S 189, p. 311. 
§ 190, p. 811. 
S 191, p. 311. 
§ 192, p. 312. Syi 
5193, p. Sl.l Kur 
5 194. p. 313. Seri 
§ 193, p. 314, Equ 

Rigid mechanical 
Uniformity of speed 

ling in synchronism 
s operation of alter 
itions of synchrono 

running alternators, synchro 

S 196. p. 317. Special case of equal alternators at equal 

§ 197, p, .320. Numerical instance. 

Chap. XIX. STiicliroDoaa Motor.— 

S 198, p. 321, Graphic method. 

5 199. p. 333. Continued. 

S 200. p. .12.>. Instance. 

5 201. p, 320. Constant impressed E.M.F, and constant rurrent. 

% 202. p. .329. Constant impressed and counter E.M.F. 

{ 203, p. 332, Constant impressed E.M.F. and maximum ifliciency. 

j! 204, p. 334. Constant impre.ssed E.M.F. and constant output, 

{ 20S,p, •338. .^nalytica1 method. Fundamental equalionit and power. 

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Chap. XIX, SjnchnuMiu TSaXioz — Coniiautd. — 

J 206, p. 342. Majtimum output. 

§ 207, p. 343. No load 

§ 208. p. 345. Minimum current. 

S 209, p. iMT. Maximum displacement of pluue. 

% 210. p MB. ConstanI counter E.M.F. 

% 211. p. S49. Numerical instance. 

S 212, p. 351. Discussion of results. 

Chap. XX. Commntator Motors.— 

% 213, p. 354. Types of commutator motors. 

I 214, p. 354. Repulsion motor as induction motor. 

S 315, p. 356. Two types of repulsion motors. 

g 216, p. 35B. Definition of repulsion motor. 

§ 217, p. 351). Equations of repul^on motor. 

§ 218. p. 360. Continued. 

% 319, p. 301. Power of repulsion motor. Instance; 

S 220, p. 303. Series motor, shunt moloi. 

§ 221, p. 366. Equations of series motor. 

§ 222, p. 3CT. Numerical instance. 

{ 233, p. 368. Shunt motor. 

g 224, p. 370. Power factor of series motor. 

Chap. XXI. Besction MachineB.— 

S 335, p. 371. General discussion. 

§ 226, p. 372. Energy component of reactance^ 

% 237. p. 372. Hysteretic energy component of reactance. 

§ 226. p. 373, Periodic variation reactance. 

§ 229, p. 375. Distortion of waveshape. 

§ 230, p. 377. Unsymmetrical distortion of wave-shape. 

§ 231. p. 37S. Equations of reaction machines. 

j 232, p. 330. Numerical instance. 

Chap. XXII. Diatortion of Wave-ahape, and Its CausM. — 

g 233. p. 383. Equivalent »ne wave. 

S 234, p. 383. Cause of distortion. 

S 233, p. 384. Lack of uniformity and pulsation of magnetic field. 

I 236. p. 388. Contintied. 

S 237, p 381. Pulsation of reactance. 

S 238, p. 391. Pulsation of reactance in reaction machine. 

3 239, p. 393. General discussion. 

S 240, p. 393. Pulsation of resistance arc 

{341, p. 396. Instance. 

3 242, p. 390. Distortion of wave-shape by arc 

$ 243, p. 397. Discussion. 

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Chap. XXIIL BffMta of Highn Haimonica.— 

5 344, p. 883. Distortion of by triple and quintuple har- 
monics. Some characleiiBtic waveshapes. 
f 345, p. 101. Effect of self-induction and capacity on higher haimonics. 
% 246, p. 402. Resonajice due to higher hartnonics in transmission lines. 
S 347, p. 406. Power of complex haimonic waves, 
§ 34S, p. 405. Three-phase generator. 

% 249, p. 407. Decrease of hysteresis by distortion of wave-shape^ 

S 2SO, p. 407. Increase of hysteresis by distortion of wave-shape. 

9 251, p. 40a Eddy conents. 

\ 393, p. 406. Effect of distorted waves on insulation. 

Chap. XXIV. SrmboUcR«preseDtatlon of General AlteniatlnEWBTe. — 

S 253, p. 410. Symbolic representation. 

I 294, p. 412. Effective valnes. 

% 299, p. 413. Power torque, etc Circiut factor. 

S 296, p. 413. Resistance, inductance, and capacity in senes. 

5 257, p. 419. Apparent capacity of condenser. 

\ 298, p. 432. Synchronous motoi, 

\ 299, p. 426. Induction motor. 

Chaf. XXV. General Polyphaae Syatenu.— 

S 260, p. 430. Definition of systems, symmetrical and unsymmetrical 

9 361, p. 430. Flow of power. Balanced and unbalanced systems. 
Independent and interlinked systems. Star connection and ting 

{ 262, p. 432. Classification of polyphase systems. 
Chap. XXVI. Syinmetilcal Poljpliaae Syatenu.— 

9 263, p. 434. General equations of symmetrical systems. 

9 264, p. 43fi. Particular systems 

\ 265, p. 430. Resultant M.M.F. of symmetrica] splem. 

9 366, p. 439. Particular systems. 

Chap. XXVII. Balanced and Unnbalanced Polypliaae Systema. — 

§ 267, p. 440. Plow of power in sing^e^phase system. 

§ 268, p. 441. Flow of power in polyphase systems, balance factor of 

% 269, p. 442. Balance factor. 

9 270, p. 442. Three-phase system, quarter-phase sysltan. 

9 271, p. 44.1 Inverted threephase system. 

9 273, p. 444. Diagrams of flow of power. 

9 273, p, 447. Monocyclic and polycyclic systems. 

9 274, p. 447. Power characteristic of alternating-current system. 

§ 279, p. 448. The same in rectangular coordinates- 

S 276, p. 450. Main power axes of alteniating.carrent syatem. 

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C»AP. XXVIIL Interlinked Poljpliue Systems. — 
% 277, p. 462. Interlinked and independent systems. 
§ 278, p. 452. Star connection and ring connection, V connection and 

delta connection. 
% 279, p. 4M. Continued. 
% 380, p. 4&6. Star potential and ting potential. Star current and ring 

current. Y potential and Y cunreru, delta potential and delta 

3 261, p. 456. Equations of interiinked polyphase systems. 
% 282, p. 457. Continued 
Cha?. XXIX. Transformation of Polypliase Syatema.— 
I 283, p. 460. Constancy of balance factor. 
% 284, p. 4d0. Equations of transfonnation of polyphase systems. 
S 285, p. 402. Three.phase, qnaiter-phase transfonnation. 
S 286, p. 403. Some of the more common polyphase transformations. 
S 387, p. 460j Transformation with change of balance factor. 
Chap. XXX. Copper Effldency of Systems.— 
S 288, p. 468. Genera] discus^on. 
§ 289, p. 400. Comparison on the basis of equality of r 

ference of potential. 
§ 390, p. 474. Comparison on the basis of equality of n 

ference of potential 
S 391, p. 470. CoDtinued. 
Chap. XXXI. Tbree-phaae STitem. — 
j 292, p. 478. General equations. 
$293, p. 481. Spedal cases: balanced system, one branch loaded, 

tno branches loaded. 

Chap. XXXII. Qnartei-phase System.— 
g 294, p. 483. Genera] equations. 
S 295, p. 484. Special cases : balanced system, one branch loaded. 

Appendix I. Algebra of Complex Imactnary Qiuntltiea. — 

3 296, p. 489. Introduction. 

g 297, p. 460. Numeration, addition, moltiidication, Involtition. 

S 298, p. 400. Subtiadion, negative nnmber. 

S 299, p. 491. DiviBon, friction. 

S 300, p. 401. Evolution and logaHthmation. 

3 301. p. 492, Imaginaiy unit, complex imaginary number. 

% 303, p. 402. Review, 

{ 303, p. 498. Algebraic opeiations with complex quantities. 

§ 304, p. 4M. Continued, 

S 305, p, 496. Roots of the unit 

3 306, p, 406. Rotation. 

S 307, p, 406. Complex imaginary jdane. 

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Appendix II. Oscillating Carrenta. — 

§ 308, p. 487. 


§ 309. p. 498. 

General equations. 

§ 310, p. 499. 

Polar coordinates. 

§ 311, p. 500. 

Loxodtomic spiral. 

S 312. p. 501. 

Impedance and admit Unce. 

§ 313, p. 602. 


£ 314. p. 502. 


§ 313, p. 503. 


§ 316, p. 604. 

S 317, p. 606. 

Conductance and ausceptance. 

§ 3ie, p. 606. 

Circuits of tero impedance. 

£ 319, p, 506, 


§ 320, p. 607. 

Origin of oscillating currents. 

§ 321, p. 608. 

OsdDating dischaige. 

S 322. p. 509. 
1 323, p. 510. 
S 324, p. 612. 

Oscillating discharge of condensers 





1. In the practical applications of electrical energy, we 
meet with two different classes of phenomena, due respec- 
tively to the continuous current and to the alternating 

The continuous-current phenomena have been brought 
within the realm of exact analytical calculation by a few 
fundamental laws : — 

1.) Ohm's law: i = e j r, where r, the resistance, is a 
constant of the circuit. 

2.) Joule's law: P= i^r, where P is the rate at which 
energy is expended by the current, i, in the resistance, r. 

3.) The power equation : P^ = ei, where /*, is the 
power expended in the circuit of E.M.F., e, and current, ;'. 

4.) KirchhofTs laws : 

a) The sum of all the E.M.Fs. in a closed circuit = 0, 
if the E.M.F. consumed by the resistance, I'r, is also con- 
sidered as a counter E.M.E,, and all the EM.Fs. are taken 
in their proper direction. 

b.) The sum of all the currents flowing towards a dis- 
tributing point = 0. 

In alternating-current circuits, that is, in circuits con- 
veying currents which rapidly and periodically change their 

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direction, these laws cease to hold. Energy is expended, 
not only in the conductor through its ohmic resistance, but 
also outside of it ; energy is stored up and returned, so 
that large currents may flow, impressed by high E,M.Fs., 
without representing any considerable amount of expended 
energy, but merely a surging to and fro of energy ; the 
ohmic resistance ceases to be the determining factor of 
current strength ; currents may divide into components, 
each of which is larger than the undivided current, etc. 

2. In place of the above-mentioned fundamental laws of 
continuous currents, we find in alternating-current circuits 
the following : 

Ohm's law assumes the form, i = ejs, where s, the 
apparent resistance, o r impedance, is no longer a constant 
of the circuit, but depends upon the frequency of the cur- 
rents ; and in circuits containing iron, etc., also upon the 

Impeda nce, z, is, in the system of absolute units, of the 
same dimensions as resistance (that is, of the dimension 
Z,/"""' = velocity), and is expressed in ohms. 

It consists of two components, the resistance, r, and the 
reactance, x, or — _. 

The resistance, r, in circuits where energy is expended 
only in heating the conductor, is the same as the ohmic 
resistance of continuous-current circuits. In circuits, how- 
ever, where energy is also expended outside of the con- 
ductor by magnetic hysteresis, mutual inductance, dielectric 
hysteresis, etc., r is larger than the true ohmic resistance 
of the conductor, since it refers to the total expenditure of 
energy. It may be called then the effective resistance. It 
is no longer a constant of the circuit. 

The reactance, x, does not represent the expenditure of 
power, as does the effective resistance, r, but merely the 
surging to and fro of energy. It is not a constant of the 

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circuit, but depend s upon the frequency, and frequently, 
as in circuits containing iron, or in electrolytic conductors, 
upon the E.M.F. also. Hence, while the effective resist- 
ance, r, refers to the energy component of E.M.F., or the 
E.M.F. in phase with the current, the reactance, x, refers 
to the wattless component of E.M.F., or the E.M.F. in 
quadrature with the current. 

3. The principal sources of reactance are electro-mag- 
netism and capacity. 


An electric current, i, flowing through a circuit, produces 
a magnetic flux surrounding the conductor in lines of 
magnetic force (or more correctly, lines of magnetic induc- 
tion), of closed, circular, or other form, which alternate 
with the alternations of the current, and thereby induce 
an E.M.F. in the conductor. Since the magnetic flux is 
in phase with the current, and the induced E.M.F. 90°, or 
a quarter period, behind the flux, this E.M.F. of self-induc- 
tance lags 90'", or a quarter period, behind the current ; that 
is, is in quadrature therewith, and therefore wattless. 

If now * =. the magnetic flux produced by, and inter- 
linked with, the current i (where those lines of magnetic 
force, which are interlinked «-foId, or pass around n turns 
of the conductor, are counted n times), the ratio,*/;, is 
denoted by L, and called self -inductance, or the coefficient of 
self-in duction of the circuit. It is numerically equal, in 
absolute units, to the interlinkages of the circuit with the 
magnetic flux produced by unit current, and is, in the 
system of absolute units, of the dimension of length. In- 
stead of the self-inductance, L, sometimes its ratio with 
the ohmic resistance, r, is used, and is called the Time- 
Constant of the circuit : 



If a conductor surrounds with n turns a magnetic cir- 
cuit of reluctance, {R, the current, *', in the conductor repre- 
sents the M.M.F. of ni ampere-turas, and hence produces 
a magnetic fJux of «//(R lines of magnetic force, sur- 
roimding each « turns of the conductor, and thereby giving 
^ = /i*i/(B interlinkages between the magnetic and electric 
circuits. Hence the inductance is L =^ji= n^fS{. 

The fundamental law of electro-magnetic induction is, 
that the E.M.F. induced in a conductor by a varying mag- 
netic field is the rate of cutting of the conductor through 
the magnetic field. 

Hence, if i is the current, and Z is the inductance of 
a circuit, the magnetic flux interlinked with a circuit of 
current, i, is Li, and 4 A'Li is consequently the average 
rate of cutting; that is, the number of lines of force cut 
by the conductor per second, where JV = frequency, or 
number of complete periods (double reversals) of the cur- 
rent per second. 

Since the maximum rate of cutting bears to the average 
rate the same ratio as the quadrant to the radius of a 
circle (a sinusoidal variation supposed), that is the ratio 
v/2 -t-l, the maximum rate of cutting is 2»-A^, and, conse- 
quently, the maximum value of E.M.F. induced in a cir- 
cuit of maximum current strength, /, and inductance, L, is. 

Since the maximum values of sine waves are proportional 
(by factor v'2) to the effective values (square root of mean 
squares), if i = effective value of alternating current, e = 
2rJVLi is the effective value of E.M.F. of self-inductance, 
and the ratio, eji= 2 uNL, is the magnetic reactance: 

*„ = 2 r NL. 

Thus, if r = resistance, x„ = reactance, e = impedance,-* 

the E.M.F. consumed by resistance is : f , = />,- 
the E.M.F. consumed by reactance is : c, = ix,„ -. 

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and, since both E.M.Fs. are in quadrature to each other, 
the total E.M.F. is — 

that is, the impedance, s, takes in alternating-current cir- 
cuits the place of the resistance, r, in continuous-current 


4. If upon a condenser of capacity, C, an E,M.F., e, is 
impressed, the condenser receives the electrostatic charge, Ce. 

If the E.M.F., e, alternates with the frequency, N, the 
average rate of charge and discharge is 4 7\^ and 2i-A''the 
maximum rate of charge and discharge, sinusoidal waves sup- 
posed, hence, »' = 2 ■■ NCe the current passing into the con- 
denser, which is in quadrature to the E.M.F., and leading. 

It is then:— '•-'-)- 27nC' 
the '^capacity reactance" ot " condemattce" 

Polarization in electrolytic conductors acts to a certain 
extent like capacity. 

The capacity reactance is inversely proportional to the 
frequency, and represents the leading out-of-phase wave ; 
the magnetic reactance b directly proportional to the 
frequency, and represents the lagging out-of-phase wave. 
Hence both are of opposite sign with regard to each other, 
and the total reactance of the circuit is their difference, 
X = x„— x^. 

The total resistance of a circuit is equal to the sum of 
all the resistances connected in series ; the total reactance 
of a circuit ts equal to the algebraic sum of all the reac- 
tances connected in series ; the total impedance of a circuit, 
however, is not equal to the sum of all the individual 
impedances, but in general less, and is the resultant of the 
total resistance and the total reactance. Hence it is not 
permissible directly to add impedances, as it is with resist- 
ances or reactances. 

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A further discussion of these quantities will be found in 
the later chapters. 

6. In Joule's law, P = i^r, r is not the true ohmic 
resistance any more, but the " effective resistance ; " that 
is, the ratio of the energy component of E.M.F. to the cur- 
rent. Since in alternating-current circuits, besides by the 
ohmic resistance of the conductor, energy is expended, 
partly outside, partly even inside, of the conductor, by 
magnetic hysteresis, mutual inductance, dielectric hystere- 
sis, etc., the effective resistance, r, is in general larger than 
the true resistance of the conductor, sometimes many times 
larger, as in transformers at open secondary circuit, and is 
not a constant of the circuit any more. It is more fully 
discussed in Chapter VII. ySu jpTa) 

In alternating-current circuits, the power equation con- 
tains a third term, which, in sine waves, is the cosine of 
the difference of phase between E.M.F. and current : — 

/'„ == rt cos *. 

Consequently, even if e and i are both large, P^ may be 
very small, if cos <^ is small, that is, ^ near 90°. 

Kirchhoffs laws become meaningless in their original 
form, since these laws consider the E.M.Fs. and currents 
as directional quantities, counted positive in the one, nega- 
tive in the opposite direction, while the alternating current 
has no definite direction of its own. 

6. The alternating waves may have widely different 
shapes ; some of the more frequent ones are shown in 
a later chapter. 

The simplest form, however, is the sine wave, shown in 
Fig. 1, or, at least, a wave very near sine shape, which 
may be represented analytically by : — 

i =rj sm^ (t - f,) = / sia2^JV (t ~ e,); 

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where / is the maximum value of the wave, or its ampli- 
tude ; T is the time of one complete cyclic repetition, or 
the period of the wave, or N = 1 j T is the frequency or 
number of complete periods per second ; and ^, is the time, 
where the wave is zero, or the epoch of the wave, generally 
called the phase* 

Obviously, "phase" or "epoch" attains a practical 
meaning only when several waves of different phases are 
considered, as "difference of phase." When dealing with 
one wave only, we may coimt the time from the moment 

^^v /^ \ 

J. L\ / N 

t ^ \ ^ \ 

/"^ E 3E 5 

-.^ \ ^^ \ 

7 ^^7 ^^ 

fig. I. Sia Htew. 

where the wave is zero, or from the moment of its maxi- 
mum, and then represent it by : — 

»■ = / sin 2 )r A7 ; 
or, I = /cos 2 s- M. 

Since it is a univalent function of time, that is, can at a 
given instant have one vaKie only, by Fourier's theorem, 
any alternating wave, no matter what its shape may be, 
can be represented by a series of sine functions of different 
frequencies and different phases, in the form : — 

( = 7isin2irA^(/ — /,) + /, sin 4 n-iV (/-/,) 
+ /, sin 6 irJV{t — /,)+.,, 

* " Epoch " is the lime where a periodic lunclion reaches a certain valae, 
lor instance, lero ; and "phase" is the angular position, wilh respect to a 
1 periodic function at a given time. Both are in allemale- 
only different ways of expressing the same thing. 

datum posit: 



where /j, I^, /g, . . . are the maximum values of the differ- 
ent components of the wave, /j, t^, /g . . . the times, where 
the respective components pass the zero value. 

The first term, /j sin 2 ir N {t — /j), is called the fun- 
damental wave, or Mti^ first harmonic; the further terms are 
called the higher harmonics, or "overtones," in analogy to 
the overtones of sound waves. /„ sin ^ nv N {t — /■„) is the 
«'" harmonic. 

By resolving the sine functions of the time differences, 
i ~ tp I — t^ . . . , we reduce the general expression of 
the wave to the form : 

i=j4i sin2«-JW + v4, sin 4 n-iVi' + vj, sin6TrA?-J- . . . 
+ £iCos2rM + £tcosi^M + JftCosGirAt + . . . 



fig. 2. IHwt alUait fam Harmwilcx 

The two half-waves of each period, the positive wave 
and the negative wave (counting in a definite direction in 
the circuit), are almost always identical. Hence the even 
higher harmonics, which cause a difference in the shape of 
the two half-waves, disappear, and only the odd harmonics 
exist, except in very special cases. 

Hence the general alternating-current wave is expressed 


/=./, sin2^.A^(/ — /,) -I-/, sine X..Y (/ — /,) 
-I- 7i sin 10 •ir.V(/ -/,)-!- . . . 

uNt -\- Ai sin 6 TrNt -\- A^ i 
^Nt-\-Bi cos &-!!Nl-\- B^i. 

TtlYt-\- . . 

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Such a wave is shown in Fig. 2, while Fig. 3 shows a 

wave whose half-waves are different. Figs, 2 and 3 repre- 
sent the secondary currents of a Ruhmkorff coil, whose 
secondary coil is closed by a high external resistance : Fig. 
8 is the coil operated in the usual way, by make and break 
of the primary battery current ; Fig. 2 is the coil fed with 
reversed currents by a commutator from a battery. 

7. Self-inductance, or electro-magnetic momentum, which 
is always present in alternating-current circuits, — to a 
large extent in generators, transformers, etc., — tends to 

^ ^^ 

^-^ ^^^ 

^^ F ^' 

'i i t 

- t 

1 1 1 1 ' 1 i ' ^ 1 ' 1 1 1 1 1 1 Li 

suppress the higher harmonics of a complex harmonic wave 
more than the fundamental harmonic, since the self-induc- 
tive reactance is proportional to the frequency, and is thus 
greater with the higher harmonics, and thereby causes a 
general tendency towards simple sine shape, which has the 
effect, that, in general, the alternating currents in our light 
and power circuits are sufficiently near sine waves to make 
the assumption of sine shape permissible. 

Hence, in the calculation of alternating-current phe- 
nomena, we can safely assume the alternating wave as a 
sine wave, without making any serious error ; and it will be 

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sufficient to keep the distortion from sine shape in mind as 
a possible disturbing factor, which generally, however, is in 
practice negligible — perhaps with the only exception of 
low-resistance circuits containing large magnetic reactance, 
and large condensance in series with each other, so as to 
produce resonance effects of these higher harmonics. 

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8. In a periodically varying function, as an alternating 
current, we have to distinguish between the instantaneous 
value, which varies constantly as function of the time, and 
the integral value, which characterizes the wave as a whole. 

As such integral value, almost exclusively the effective 

FIf. 4, AtUritatlng I«b«: 

value is used, that is, the square root of the mean squares ; 
and wherever the intensity of an electric wave is mentioned 
without further reference, the effective value is understood. 

The maximum value of the wave is of practical interest 
only in few cases, and may, besides, be different for the two 
half-waves, as in Fig. 3. 

As arithmetic mean, or average value, of a wave as in 
Figs. 4 and 5, the arithmetical average of all the instan- 
taneous values during one complete period is understood. 

This arithmetic mean is either = 0, as in Fig. 4, or it 
differs from 0, as in Fig. 5. In the first case, the wave 
is called an alternating wave, in the latter a. pulsating wave. 

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Thus, an alternating wave is a wave whose positive 
values give the same sum total as the negative values ; that 
is, whose two half-waves have in rectangular coordinates 
the same area, as shown in Fig. 4. 

A pulsating wave is a wave in which one of the half- 
waves preponderates, as in Fig. 6, 

By electromagnetic induction, pulsating waves are pro- 
duced only by commutating and unipolar machines (or by 
the superposition of alternating upon direct currents, etc.). 

All inductive apparatus without commutation give ex- 
clusively alternating waves, because, no matter what con- 



ditions may exist in the circuit, any line of magnetic force, 
which during a complete period is cut by the circuit, and 
thereby induces an E.M.F., must during the same period 
be cut again in the opposite direction, and thereby induce 
the sftme total amount of E.M.F. (Obviously, this does 
not apply to circuits consisting of different parts movable 
with regard to each other, as in unipolar machines.) 

In the following we shall almost exclusively consider the 
alternating wave, that is the wave whose true arithmetic 
mean value = 0. 

Frequently, by mean value of an alternating wave, the 
average of one half-wave only is denoted, or rather the 

Digitized .yGOO^IC 



average of all instantaneous values without regard to their 
sign. This mean value is of no practical importance, and 
is, besides, in many cases indefinite. 

9. In a sine wave, the relation of the mean to the maxi- 
mum value is found in the following way: — ■ 

Let, in Fig, 6, AOB represent a quadrant of a circle 
with radius 1. 

Then, while the angle + traverses the arc «■ / 2 from A to 
B, the sine varies from to OB = 1. Hence the average 
variation of the sine bears to that of the corresponding arc 
the ratio 1 -j- ir/ 2, or 2 / «■ -s- 1. The maximum variation 
of the sine takes place about its zero value, where the sine 
is equal to the arc. Hence the maximum variation of the 
sine is equal to the variation of the corresponding arc, and 
consequently the maximum variation of the sine bears to- 
its average variation the same ratio as the average variation 
of the arc to that of the sine ; that is, 1 -j- 2 / t, and since 
the variations of a sine-function are sinusoidal also, we 

Mean value of sine wave -i- maximum value = i- 1 

= .63663. 

The quantities, " current," " E.M.F.," " magnetism," etc., 
are in reality mathematical fictions only, as the components- 

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of the entities, "energy," "power," etc. ; that is, they have 
no independent existence, but appear only as squares or 

Consequently, the only integral value of an alternating 
wave which is of practical importance, as directly connected 
with the mechanical system of units, is that value which 
represents the same power or effect as the periodical wave. 
This is called the effective value. Its square is equal to the 
mean square of the periodic function, that is : — 

The effective value of an alternating wave, or tlie value 
representing the same effect as the periodically varying wave, 
is the square root of the mean square. 

In a sine wave, its relation to the maximum value is 
found in the following way : 

Let, in Fig. 7, AOB represent a quadrant of a circle 
with radius 1. 

Then, since the sines of any angle ^ and its complemen- 
iary angle, 90°— <^, fulfill the condition, — 
sin" ^ + sin' (90 — ^) = J, 
-the sines in the quadrant, AOB, can be grouped into pairs, 
so that the sum of the squares of any pair = 1 ; or, in other 
words, the mean square of the sine =1/2, and the square 
TOot of the mean square, or the effective value of the sine, 
- 1 / V2. That is : 

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The effective vaiue of a sine function bears to its maxi- 
mum- value the ratio, — 

— + 1 = .707-11. 


Hence, we have for the sine curve the following rela- 



AllTK. MUH. 

H^[ WholB 














10. Coming now to the general alternating wave, 
i = Ax sin 2b- Nt + A^ sin 4«- A? + -^, sin fer A? + . . . 

we find, by squaring this expression and canceling all the 
products which give as mean square, the effective value, — 

The mean value does not give a simple expression, and 
is of no general interest. 

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ZAW or BLBcrBO-KAOHimc nrouonov. 

11. If an electric conductor moves relatively to a mag- 
netic field, an E.M.F. is induced in the conductor which is 
proportional to the intensity of the magnetic field, to the 
length of the conductor, and to the speed of its motion 
perpendicular to the magnetic field and the direction of the 
conductor ; or, in other words, proportional to the number 
of lines of magnetic force cut per second by the conductor. 

As a practical unit of E.M.F., the volt is defined as the 
E.M.F. induced in a conductor, which cuts 10' = 100,000,000 
lines of magnetic force per second. 

If the conductor is closed upon itself, the induced E.M.F. 
produces a current. 

A closed conductor may be called a turn or a convolution. 
In such a turn, the number of hnes of magnetic force cut 
per second is the increase or decrease of the number of 
lines inclosed by the turn, or n times as large with n turns. 

Hence the E.M.F. in volts induced in n turns, or con- 
volutions, is » times the increase or decrease, per second, 
of the flux inclosed by the turns, times 10~*. 

If the change of the flux inclosed by the turn, or by » 
turns, does not take place uniformly, the product of the 
number of turns, times change of flux per second, gives 
the average E.M.F. 

If the magnetic flux, 4>, alternates relatively to a number 
of turns, « — that is, when the turns either revolve through 
the flux, or the flux passes in and out of the turns, the total 
flux is cut four times during each complete period or cycle, 
twice passing into, and twice out of, the turns. 

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Hence, if N= number of complete cycles per second, 
or the frequency of the flux *, the average E,M.F. induced 
in n turns is, 

E„t, = 4 « * ^ 10 - • volts. 

This is the fundamental equation of electrical engineer- 
ing, and applies to continuous-current, as well as to alter- 
nating-current, apparatus. 

12. In continuous-current machines and in many alter- 
nators, the turns revolve through a constant magnetic 
field ; in other alternators and in induction, motors, the mag- 
netic held revolves ; in transformers, the field alternates 
with respect to the stationary turns. 

Thus, in the continuous-current machine, if n = num- 
ber of turns in series from brush to brush, 4> = flux inclosed 
per turn, and N = frequency, the E.M.F. induced in the 
machine '\% E = 4«* A'^IO"' volts, independent of the num- 
ber of poles, of series or multiple connection of the arma- 
ture, whether of the ring, drum, or other type. 

In an alternator or transformer, if « is the number of 
turns in series, 4 the maximum flux inclosed per turn, and 
N the frequency, this formula gives, 

£.^ = 4«*^10-» volts. 
Since the maximum E.M.F. is given by, — 

we have 

^p»x. = 2 «-«*^10-' volts. 

And since the effective E.M.F. is given by, — 

we have 

= 4.44B*^10-»voIts, 
which is the fundamental formula of alternating-current 
induction by sine waves. 

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■13. If, in a circuit of n turns, the magnetic flux, *, 
inclosed by the circuit is produced by the current flowing 
in the circuit, the ratio — 

flux X number of turns X 10~* 

is called the inducta nce, L, of the circuit, in henrys. 

The product of the number of turns, «, into the maxi- 
mum flux, *, produced by a current of / amperes effective, 
or/V2 amperes maximum, is therefore — 

and consequently the effective E.M.F. of self-inductance is: 

• =2tA'Z/ volts, .^^'ff-^ 
The product, j^ •= 2 irNL, is of the dimension of resistance, 
and is called the reactance of the circuit ; and the E.M.F. 
of self-inductance of the circuit, or the reactance voltage, is 

£ = /x, 
and lags 90* behind the current, since the current is in 
phase with the magnetic flux produced by the current, 
and the E.M.F. lags 90° behind the magnetic flux. The 
E.M,F. lags 90° behind the magnetic flux, as it is propor- 
tional to the change in flux ; thus it is zero when the mag- 
netism is at its maximum value, and a maximum when the 
flux passes through zero, where it changes quickest. 

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14. While alternating waves can be, and frequently are, 

represented graphically in rectangular coordinates, with the 
time as abscissas, and the instantaneous values of the wave 
as ordinates, the best insight with regard to the mutual 
relation of different alternate waves is given by their repre- 
sentation in polar coordinates, with the time as an angle or 
the amplitude, — one complete period being represented by 
one revolution, — and the instantaneous values as radii 

Thus the two waves of Figs. 2 and $ are represented in 
polar coordinates in Figs. 8 and 9 as closed characteristic 
curves, which, by their intersection with the radius vector, 
give the instantaneous value of the wave, corresponding to 
the time represented by the amplitude of the radius vector. 

These instantaneous values are positive if in the direction 
of the radius vector, and negative if in opposition. Hence 
the two half-waves in Fig. 2 are represented by the same 

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polar characteristic curve, which is traversed by the point of 
intersection of the radius vector twice per period, — once 
in the direction of the vector, giving the positive half-wave, 

F(q. S. B, FIf. 10. 

and once in opposition to the vector, giving the negative 
half-wave. In Figs. 3 and 9, where the two half-waves are 
different, they give different polar characteristics. 

15. The sine wave. Fig. 1, is represented in polar 
coordinates by one circle, as shown in Fig. 10. The 
diameter of the characteristic curve of the sine wave, 
/= OC, represents the intensity of the wave ; and the am- 
plitude of the diameter, OC, Z <" = AOC, is the phase of the 
wave, which, therefore, is represented analytically by the 
function ; — ■ r /_, -^ 

t = /cos (^ — 11)), 

where <^=!2a-//7"is the instantaneous value of the ampli- 
Xude corresponding to the instantaneous value, i, of the wave. 

The instantaneous values are cut out on the movable ra- 
dius vector by its intersection with the characteristic circle. 
Thus, for instance, 3t the amplitude AOB^ =.^^=%T,t^f T 
(Fig. 10), the instantaneous value is OB' ; at the amplitude 
AOB^ = ^=%Trt^l T, the instantaneous value is OB", and 
negative, since in opposition to the radius vector OB^. 

The characteristic circle of the alternating sine wave is 
determined by the length of its diameter — the intensity 
of the wave ; and by the amplitude of the diameter — the 
phase of the wave. 

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Hence, wherever the integral value of the wave is con- 
sidered alone, and not the instantaneous values, the charac- 
teristic circle may be omitted altogether, and the wave 
represented in intensity and in phase by the diameter of 
the characteristic circle. 

Thus, in polar coordinates, the alternate wave is repre- 
sented in intensity and phase hy the length and direction of 
a vector, OC, Fig. 10, and its analytical expression would 
then be c = OC cos (^ — «). 

Instead of the maximum value of the wave, the effective 
value, or square root of mean square, may be used as the 
vector, which is more convenient ; and the maximum value 
is then V2 times the vector OC, so that the instantaneous 
values, when taken from the diagram, have to be increased 
by the factor V2. 

Thus the wave, 

^ = .ff cos2fl-.A^(/-i't) 

= .ffc09C*-S^ 

is in Fig. lOu represented by 

vector OB '= —r^, of phase 

AOB = "i ; and the wave, 
= Ccos (^+fi^ 

, of phase 
AOC=-S^. "^ 

The former is said to lag by angle ili, the latter to lead 
by angle w^ with regard to the zero position. 

The wave b lags by angle (a, -t- i^ behind wave c, or c 
leads b by angle (5, + ij. 

16, To combine different sine waves, their graphical rep- 
resentations, or vectors, are combined by the parallelogram 

law. " 

If, for instance, two sme waves, OB and OC (Fig. 11), 
are superposed, — as, for instance, two E.M.F's. acting in 
the same circuit, — their resultant wave is represented by 

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OD, the diagonal of a parallelogram with OB and OC as 

For at any time, t, represented by angle ^ = AOX, the 
instantaneous values of the three waves, OB, OC, OD, are 
their projections upon OX, and the sum of the projections 
of OB and OC is equal to the projection of OD ; that is, the 
instantaneous values of the wave OD are equal to the sum 
of the instantaneous values of waves OB and OC. 

From the foregoing considerations we have the con- 
clusions : 

Tke sine wave is represented graphically in polar coordi- 
nates by a vector, which by its length, OC, denotes the in- 

tensity, and by its amplitude, AOC, the phase, of the sine 

Sine waves are combined or resolved graphically, in polar 
coordinates, by the law of parallelogram or tJte polygon of 
sine waves. 

Kirchhoff's laws now assume, for alternating sine waves, 
the form : — 

iz.) The resultant of all the E.M.Fs, in a closed circuit, 
as found by the parallelogram of sine waves, is zero if 
the counter E.M.Fs. of resistance and of reactance are 

b.) The resultant of all the currents flowing towards a 

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distributing point, as found by the parallelogram of sine 
waves, is zero. 

The energy equation expressed graphically is as follows : 
The power of an alternating-current circuit is repre- 
sented in polar coordinates by the product of the current , 
/, into the projection of,the E.M.F., E, upon the current, or 
by the E.M.F., E, into the projection of the current, /, upon 
the E.M.F., or by IE cos {J,B). 

17. Suppose, as an instance, that over a line having the 
resistance, r, and the reactance, x = 2xNL, — where N = 
frequency and L = inductance, — a current of / amperes 
be sent into a non-inductive circuit at an E.M.F. of £ 

volts. What will be the E.M.F. required at the generator 
end of the line ? 

In the polar diagram. Fig. 12, let the phase of the cur- 
rent be assumed as the initial or zero line, 01. Since the 
receiving circuit is non-inductive, the current is in phase 
with its E.M.F. Hence the E.M.F., E, at the end of the 
line, impressed upon the receiving circuit, is represented by 
a vector, OE. To overcome the resistance, r, of the line, 
an E.M.F,, fr, is required in phase with the current, repre- 
sented by OE^ in the diagram. The self-inductance of the 
line induces an E.M.F. which is proportional to the current 
/ and reactance x, and lags a quarter of a period, or 90°, 
behind the current. To overcome this counter E.M.F. 

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of self-induction, an E.M.F. of the value Ix is required, 
in phase 90° ahead of the current, hence represented by 
vector OE^. Thus resistance consumes E.M,F. in phase, 
and reactance an E.M.F. 90° ahead of the current. The 
E.M.F. of the generator, £"„, has to give the three E.lM.Fs., 
E, Ej,t and E^ hence it is determined as their resultant. 
Combining by the parallelogram law, OE^ and OE^, give 
OE,, the E.M.F. required to overcome the impedance of 
the line, and similarly OE, and OE give 6E„ the E.M.F. 
required at the generator side of the line, to yield the 
E.M.F. E at the receiving end of the line. Algebraically, 
we get from Fig. 12 — 

In this instance we have considered the E.M.F. con- 
sumed by the resistance (in phase with the current) and 
the E.M.F, consumed by the reactance (90" ahead of the 
current) as parts, or components, of the impressed E.M.F., 
Eg, and have derived E, by combining E,, E^ and E. 


li 6 

RS' ra- ' 

18. We may, however, introduce the eiTect of the induc- 
tance directly as an E.M.F., EJ , the counter E.M.F. of 
self-induction = /r, and lagging 90° behind the current ; and 
the E.M.F. consumed by the resistance as a counter E.M.F., 
EJ = Ir, but in opposition to the current, as is done in Fig. 
13; and combine the three E.M.Fs. £,, £/, E^, to form a 
resultant E.M.F., E, which is left at the end of the line- 

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Er and E^ combine to form E^, the counter E.M.F, of 
impedance ; and since E^ and E, must combine to form 
E, E^ is found as the side of a parallelogram, OE^E^, 
whose other side, OE,', and diagonal, OE, are given. 

Or we may say (Fig. 14), that to overcome the counter 
E.M,F. of impedance, OE^, of the line, the component, OE,, 
of the impressed E.M.F. is required which, with the other 
component OE, must give the impressed E.M.F., 0E„. 

As shown, we can represent the E.M.Fs. produced in a 
circuit in two ways — either as counter E.M.Fs., which com- 
bine with the impressed E.M.F., or as parts, or components, 

fig. 14. 

of the impressed E.M.F., in the latter case being of opposite 
phase. According to the nature of the problem, either the 
one or the other way may be preferable. 

As an example, the E.M.F. consumed by the resistance 
is Ir, and in phase with the current ; the counter E.M.F. 
of resistance is in opposition to the current. The E.M.F. 
consumed by the reactance is fx, and 90" ahead of the cur- 
rent, while the counter E.M.F. of reactance is 90° behind 
the current ; so that, if, in Fig. 15, 01, is the current, — 

OMf = E.M.F. consumed by resistance, 
0£/ = counter E.M'.F. of resistance, 
OE i = E.M.F. consumed by inductance, 
OEx' = counter E.M.F. of inductance, 
OE, = RM.F. consumed by impedance, 
OE, ' = counter E.M.F. of impedance. 

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Obviously, these counter E.M.Fs. are different from, for 
instance, the counter E.M.F. of a synchronous motor, in so 
far as they have no independent existence, but exist only 
through, and as long as, the current flows. In this respect 
they are analogous to the opposing force of friction in 



19. Coming back to the equation found for the E.M.F. 
at the generator end of the line, — 

we find, as the drop of potential in the line- 

Ci.E~E,~E~ y/{E + /rf+{Jxf ~ E. 

This is different from, and less than, the E.M.F. of 
impedance — 

Hence it is wrong to calculate the drop of potential in a 
circuit by multiplying the current by the impedance ; and the 
drop of potential in the line depends, with a given current 
fed over the line into a non-inductive circuit, not only upon 
the constants of the line, r and x, but also upon the E.M.F., 
E, at end of line, as can readily be seen from the diagrams. 

20. If the receiver circuit is inductive, that is, if the 
current, /, lags behind the E.M.F., E, by an angle i, and 
we choose again as the zero line, the current 0/ {Fig, 16), 
the E.M.F,, OE is ahead of the current by angle £. The 

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E.M.F. consumed by the resistance, Ir, is in phase with the 
current, and represented by OE^; the E.M.F. consumed 
by the reactance, Ix, is 90° ahead of the current, and re- 
presented by OEr Combining OE, OE^, and OEj., we 
get 0E„, the E,M.F. required at the generator end of the 
line. Comparing Fig, 16 with Fig. 13, we see that in 
the former OE^ is larger ; or conversely, if E^ is the same, 
E will be less with an inductive load. In other words, 
the drop of potential in an inductive line is greater, if the 
receiving circuit is inductive, than if it is non-inductive. 
From Fig. 16, — 

Ea = V(£ cos S> + //-)' + {£ sin S, + /x)\ 

Fig. le. 

If, however, the current in the receiving circuit is 
leading, as is the case when feeding condensers or syn- 
chronous motors whose counter E.M.F. is larger than the 
impressed E.M.F., then the E.M.F. will be represented, in 
Fig, 17, by a vector, OE, lagging behind the current, 0/, 
by the angle of lead i'; and in this case we get, by 
combining OE with OE,., in phase with the current, and 
OEj., 90° ahead of the current, the generator E.M,F., OE^, 
which in this case is not only less than in Fig. 16 and in 
Fig. 13, but may be even less than E ; that is, the poten- 
tial rises in the line. In other words, in a circuit with 
leading current, the self-induction of the line raises the 
potential, so that the drop of potential is less than with 

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a non-inductive load, or may even be negative, and the 
voltage at the generator lower than at the other end of 
the line. 

These diagrams. Figs. Id to 17, can be considered polar 
dif^rams of an alternating-current generator of an E.M.F., 
Eot a resistance E.M.F., E^ = Ir, a reactance E.M.F., 
E^ ==s fx, and a difference of potential, E, at the alternator 
terminals ; and we see, in this case, that with an inductive 
load the potential difference at the alternator terminals will 
be lower than with a non-inductive load, and that with a 
non-inductive load it will be lower than when feeding into 

a cbcuit with leading current, as, for instance, a synchro- 
nous motor circuit under the circumstances stated above. 

21. As a further example, we may consider the dia- 
gram of an alternating-current transformer, feeding through 
its secondary circuit an inductive load. 

For simplicity, we may neglect here the magnetic 
hysteresis, the effect of which will be fully treated in a 
separate chapter on this subject. 

Let the time be counted from the moment when the 
magnetic flux is zero. The phase of the flux, that is, the 
amplitude of its maximum value, is 90° in this case, and, 
consequently, the phase of the induced E.M.F,, is 180°, 

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since tbe induced E.M.F. lags 90° behind the inducing 
flux. Thus the secondary induced E.M.F., E^, will be 
represented by a vector, OE-^, in Fig. 18, at the phase 
180°. The secondary current, /j, lags behind the E.M.F., 
El, by an angle £j, which is determined by the resistance 
^nd inductance of the secondary circuit; that is, by the 
load in the secondary circuit, and is represented in the dia- 
^am by the vector, OFi, of phase 180 + «,. 

Instead of the secondary current, /,, we plot, however, 
the secondary M.M.F., EFj = Wj/j. where n^ is the number 
of secondary turns, and £Fj is given in ampere-turns. This 
makes us independent of the ratio of transformation. 

From the secondary induced E.M.F., E-^^, we get the flux, 
*, required to induce this E.M.F., from the equation — 

£, = V2xn,.A'*10-'; 
where — 

E\ = secondary induced E.M.F., in effective volts, 
JV = frequency, in cycles per second, 
«i = number of secondary turns, 
<t = maximum value of magnetic flux, in webers. 
The derivation of this equation has been given in a 
preceding chapter, {S^^ (»<*-j*- /4) 

This magnetic flux, *, is represented by a vector, 09, at 
the phase 90°, and to induce it an M.M.F., ff is required. 

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which is determined by the magnetic characteristic of the 
iron, and the section and length of the magnetic circuit of 
the transformer ; it is in phase with the flux ♦, and repre- 
sented by the vector OF, in effective ampere-turns. 

The eflfect of hysteresis, neglected at present, is to shift 
OF ahead of <?*, by an angle a, the angle of hysteretic 
lead. (See Chapter on Hysteresis,) 

This M.M.F., ff, is the resultant of the secondary M.M.F., 
ffp and the primary M.M.F,, ff^; or graphically, OF is the 
diagonal of a parallelogram with OF^ and OF^ as sides. OF^ 
and OF being known, we find OF^, the primary ampere- 
turns, and therefrom, and the number of primary turns, n^, 
the primary current, /, = fF,/«j, which corresponds to the 
secondary current, /|. 

To overcome the resistance, r„, of the primary coil, an 
E.M.F., E^ = I^r^, is required, in phase with the current, 
/,, and represented by the vector, OE^. 

To overcome the reactance, x^ = ^itn^L,, of the pri- 
mary coil, an E.M.F. E^ = /„a-„ is required, 90° ahead of 
the current /,, and represented by vector, OE^. 

The resultant magnetic flux, *, which in the secondary 
coil induces the E.M.F., E^, induces in the primary coil an 
E.M.F. proportional to E^ by the ratio of turns n^j ti^, and 
in phase with E^ , or, — 

which is represented by the vector (?£",'. To overcome this 
counter E.M.F., E,', a primary E.M.F., E, , is required, equal 
but opposite to E^', and represented by the vector, OE,. 

The primary impressed E.M.F., £"„, must thus consist of 
the three components, OE,, OE^, and OE^, and is, there- 
fore, their resultant 0E„ while the difference of phase in 
the primary circuit is found to be % = EfiF^. 

22. Thus, in Figs 18 to 20, the diagram of a trans- 
former is drawn for the same secondary E.M.F., .f,, sec- 

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ondary current, I^ and therefore secondary M,M.F., (F„ but 
with different conditions of secondary displacement : — 

la Fig. 18, the secondary current, /i, lags 60* behind the sec- 
ondary KM.F., £,. 
In Fig. 19, the secondary current, /i, is in phase with the 

secondary E.M.F., .£,. 
In Fig. 20, the secondary current, /, , leads by 60° the second- 
»ry E.M.F., £,. 

These diagrams show that lag in the secondary circuit in- 
creases and lead decreases, the primary current and primary 
E,M.F. required to produce in the secondary circuit the 
same E.M.F, and current ; or conversely, at a given primary 





r\g. 10. 

impressed E.M.F., E^, the secondary E.M.F., E.^, will be 
smaller with an inductive, and larger with a condenser 
(leading current) load, than with a non-inductive load. 

At the same time we see that a difference, of phase 
existing in the secondary circuit of a transformer reappears 

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in the primary circuit, somewhat decreased if leading, and 
slightly increased if lagging. Later we shall see that 
hysteresis reduces the displacement in the primary circuit, 
so that, with an excessive lag in the secondary circuit, the 
lag in the primary circuit may be less than in the secondary. 
A conclusion from the foregoing is that the transformer 
is not suitable for producing currents of displaced phase ; 
since primary and secondary current are, except at very 
light loads, very nearly in phase, or rather, in opposition, 
to each other. 

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23. The graphical method of representing alternating- 
current phenomena by polar coordinates of time affords the 
best means for deriving a clear insight into the mutual rela- 
tion of the different alternating sine waves entering into the 
problem. For numerical calculation, however, the graphical 
method is generally not well suited, owing to the widely 
different magnitudes of the alternating sine waves repre- 
sented in the same diagram, which make an exact diagram- 
matic determination impossible. For instance, in the trans- 
former diagrams {cf. Figs. 18-20), the different magnitudes 
will have numerical values in practice, somewhat like E-^ — 
100 volts, and /[ ^ 75 amperes, for a non-inductive secon- 
dary load, as of incandescent lamps. Thus the only reac- 
tance of the secondary circuit is that of the secondary coil, 
or, jTj = .08 ohms, giving a lag of <3, = 8.6°. We have 

ff, = 30 turns. 

«, = 300 turns. 

ffi = 2250 ampere-turns. 

SF = 100 ampere-turns. 

Er = 10 volts. 

Ex= 60 volts. 
Ei = 1000 volts. 

The corresponding diagram is shown in Fig. 21. Obvi- 
ously, no exact numerical values can be taken from a par- 
allelogram as flat as OF^FF^, and from the combination of 
vectors of the relative magnitudes 1 : 6 :100, 

Hence the importance of the graphical method consists 

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not SO much in its usefulness for practical calculation, as to 
aid in the simple understanding of the phenomena involved. 

24. Sometimes we can calculate the numerical values 
trigonometrically by means of the diagram. Usually, how- 
ever, this becomes too complicated, as will be seen by trying 

ffj. 21. 

to calculate, from the above transformer diagram, the ratio 
of transformation. The primary M.M.F. is given by the 

equation : — 

JF, = Vff' + JF," + 251, sin Q),, 

an expression not well suited as a starting-point for further 

A method is therefore desirable which combines the 
exactness of analytical calculation with the clearness of 
the graphical representation. 

26. We have seen that the alternating sine wave is 
represented in intensity, as well as phase, by a vector, 01, 
which is determined analytically by two numerical quanti- 
ties — the length, 01, or intensity ; and the amplitude, AOI, 
or phase ui, of the wave, /. 

Instead of denoting the vector which represents the 
sine wave in the polar diagram by the polar coordinates. 

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/and <u, we can represent it by its rectangular coordinates, 
a and b (Fig. 22), where — 

a = /cosia is the horizontal component, 

^ == /sin (0 is the vertical component of the sine wave. 

This representation of the sine wave by its rectangular 
components is very convenient, in so far as it avoids the 
use of trigonometric functions in the combination or reso- 
lution of sine waves. 

Since the rectangular components a and b are the hori- 
zontal and the vertical projections of the vector represent- 
ing the sine wave, and the projection of the diagonal of a 
parallelogram is equal to the sum of the projections of its 
sides, the combination of sine waves by the parallelogram 

Pg. 23. 

law is reduced to the addition, or subtraction, of their 
rectangular components. That is. 

Sine waves are combined, or resolved, by adding, or 
subtracting, their rectangular components. 

For instance, if a and b are the rectangular components 
of a sine wave, /, and d and b' the components of another 
sine wave, /' (Fig. 23), their resultant sine wave, /,, has the 
rectangular components «„ = {.i -f d), and b^ = (b-\- b'). 

To get from the rectangular components, a and b, of a 
sine wave, its intensity, «, and phase, S, we may combine a 
and b by the parallelogram, and derive, — 

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Hence we can analytically operate with sine waves, as 
with forces in mechanics, by resolving them into their 
rectangular components. 

26. To distinguish, however, the horizontal and the ver- 
tical comjMjnents of sine waves, so as not to be confused in 
lengthier calculation, we may mark, for instance, the vertical 
components, by a distinguishing index, or the addition of 
an otherwise meaningless symbol, as the letter/, and thus 
represent the sine wave by the expression, — 
/= a +jb, 

which now has the meaning, that a is the horizontal and V 
the vertical component of the sine wave /; and that both 
components are to be combined in the resultant wave of 

intensity, — _ 

and of phase, tan S = b j a. 

Similarly, a —jb, means a sine wave with a as horizon- 
tal, and — b AS vertical, components, etc. 

Obviously, the plus sign in the symbol, a -^jb, does not 
imply simple addition, since it connects heterogeneous quan- 
tities — horizontal and vertical components — but implies 
combination by the parallelogram law. 

For the present,/ is nothing but a distinguishing index, 
and otherwise free for definition except that it is not an 
ordinary number. 

27. A wave of equal intensity, and differing in phase 
from the wave a + jb by 180°, or one-half period, is repre- 
sented in polar coordinates by a vector of opposite direction, 
and denoted by the symbolic expression, — a — jb. Or — 

Multiplying the symbolic expression, a +jb, of a sine wave 
by —1 means reversing tfie wave, or rotating it through 18Cf, 
or one-ltalf period. 

A wave of equal intensity, but lagging 90°, or one- 
quarter period, behind a + jb, has (Fig. 24) the horizontal 

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component, — b, and the vertical component, a, and is rep- 
resented symbolically by the expression, ^Vi — b. 
Multiplying, however, a + jb hyj, we get : — 

therefore, if we define the heretofore meaningless symbol, 
/', by the condition, — 

we have — 
hence : — 

Multiplying the sjimbolic expression, a ■\-jb, of a sine wave 
by j means rotating the -wave through 90°, or one-quarter pe- 
riod; that is, retarding the wave through one-quarter period 

Similarly, — 

Multiplying by —j means advancing the wave through 
one-quarter period. 

since y = — 1, / = V— 1 ; 

that is, — 

j is the imaginary unit, and the sine wave is represented 
by a complex imaginary quantity, a -\- jb. 

As the imaginary unit j has no numerical meaning in 
the system of ordinary numbers, this definition of/ = V — 1 
does not contradict its original introduction as a distinguish- 
ing index. For a more exact definition of this complex 
imaginary quantity, reference may be made to the text books 
of mathematics. 

28. In the polar diagram of time, the sine wave is 
represented in intensity as well as phase by one complex 
quantity — , ., 

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where a is the horizontal and b the vertical component of 
the wave ; the intensity is given by — 

the phase by — 

tanu = i, o 


hence the wave a •\-jb can also be expressed by — 

or, by substituting for cos w and sin u their exponential 
expressions, we obtain — 

Since we have seen that sine waves may be combined 
or resolved by adding or subtracting their rectangular com- 
ponents, consequently : — 

Sine waves may be combined or resolved by adding or 
subtracting their complex algebraic expressions. 

For instance, the sine waves, — 

a ->rjb 

«' +Jb', 

combined give the sine wave — 

It will thus be seen that the combination of sine waves 
is reduced to the elementary algebra of complex quantities. 

I 29. If /= i+ji' is a sine wave of alternating current, 

I and r is the resistance, the E.M.F. consumed by the re- 
I sistance is in phase with the current, and equal to the prod- 
uct of the current and resistance. Or — 

i ./-..■+/..'. 

i If X is the inductance, and x = 2 irNL the reactance, 

; the E.M.F. produced by the reactance, or the counter 

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E.M.F. of self-induction, is the product of the current 
and reactance, and lags 90° behind the current ; h is, 
therefore, represented by the expression — 


The E.M.F, required to overcome the reactance is con- 
sequently 90° ahead of the current (or, as usually expressed, 
the current lags 90° behind the E,M.F.), and represented 
by the expression — 

— jxl = — jxi + X?. 

Hence, the E.M.F. required to overcome the resistance, 
r, and the reactance, x, is — 


that is — 

Z = r —jx is the expression of the impedance of the cir- 
cuit, in complex quantities. 

Hence, \l I = i ■\-ji' is the current, the E.M.F. required 
to overcome the impedance, Z = r — Jx, is — 

hence, since ^'^ = — 1 

^_(W + »")+> ("'-»'■)! 
or, if ^ = * ■\-je' is the impressed E.M.F., and Z = r —jx 
the impedance, the current flowing through the circuit is : — 

or, multiplying numerator and denominator by (r +jx) to 
eliminate the imaginary from the denominator, we have — 

j^ (^+J'f^(r-\-jx) ^ er-^x yr+« . 

H + ^r* ^4-^^-' t^^^ ' 

or, ii £ = e +je' is the impressed E.M.F., and / = i -yfi' 
the current flowing in the circuit, its impedance is — 

E , + ,■/ (,+j/)(i-/i') „+y/- y/ - .,- 
/ .+>,' .' + ,'■ i' + i"^-' i' + f 



30. If C is the capacity of a condenser in series in 
a circuit of current / = * +/»', the E.M,F. impressed upon 

of the condenser. 

Capacity reactance is of opposite sign to magnetic re- 
actance ; both may be combined in the name reactance. 

We therefore have the conclusion that 

If r = resistance and L = inductance, 

then « = 2 -wNL = magnetic reactance. 

If C = capacity, x^ = 
sance ; 

Z = r —J (* — ATi), is the impedance of the circuit 

Ohm's law is then reestablished as follows : 

E^Z/, /=|-, ^ = |. 

The more general form gives not only the intensity of 
the wave, but also its phase, as expressed in complex 

31 . Since the combination of sine waves takes place by 
the addition of their symbolic expressions, KirchhofE's laws 
are now reestablished in their original form : — 

a.) The sum of all the E,M.Fs. acting in a closed cir- 
cuit equals zero, if they are expressed by complex quanti- 
ties, and if the resistance and reactance E.M.Fs. are also 
considered as counter E.M.Fs. 

b.) The sum of all the currents flowing towards a dis- 
tributing point is zero, if the currents are expressed as 
complex quantities. 

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If a complex quantity equals zero, the real part as well 
as the imaginary part must be zero individually, thus if 

a -^jb = 0, a = 0, * = 0. 

Resolving the E.M.Fs. and currents in the expression of 
Kirchhofi's law, we find : — 

a.) The sum of the components, in any direction, of all 
the E.M.Fs, in a closed circuit, equals zero, if the resis- 
tance and reactance are considered as counter E.M.Fs. 

b.) The sura of the components, in any direction, of all 
the currents flowing to a distributing point, equals zero. 

Joule's Law and the energy equation do not give a 
simple expression in complex quantities, since the effect or 
power is a quantity of double the frequency of the current 
or E.M.F. wave, and therefore requires for its representa- 
tion as a vector, a transition from single to double fre- 
quency, as will be shown in chapter XII. 

In what follows, complex vector quantities will always 
be denoted by dotted capitals when not written out in full ; 
absolute quantities and real quantities by undotted letters. 

32. Referring to the instance given in the fourth 
chapter, of a circuit supplied with an E.M.F., E, and a cur- 
rent, /, over an inductive line, we can now represent the 
impedance of the line hy Z = r —jx, where r = resistance, 
X = reactance of the line, and have thus as the E.M.F. 
at the beginning of the line, or at the generator, the 
expression — 

£, = £ + ZI. 

Assuming now again the current as the zero line, that 
is, / = t, we have in general — 

£c = -E + ir —jix ; 
hence, with non-inductive load, ox E = e, 
e, =^{e^irf+(ixY, tana„ = -^. 

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In a circuit witli lagging current, tliat is, with leading 
E.M.F., E = e —Je', and 

£.- <-y + (-•-/«) . • 

-(• + ir)-J(/+ix), 
or '.-V(<+.>)'+ (/ + »)■, tanl.-^ig. 

In a circuit with leading current, that is, with lagging 
E.M.F., E = e+je', and 

-(' + '>)+/(<'-'»), 
,. _ V(. + ;r)< + (^-,i)', tan S,. ~ -~ i 

values which easily permit calculation. 

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33. In the representation of alternating sine waves by 
vectors in a polar diagram, a certain ambiguity exists, in so 
far as one and the same quantity — an E.M.F., for in- 
stance — can be represented by two vectors of opposite 
direction, according as to whether the E.M.F. is considered 
as a part of the impressed E.M.F., or as a counter E.M.F. 
This is analogous to the distinction between action and 
reaction in mechanics. 

Further, it is obvious that if in the circuit of a gener- 
ator, G (Fig. 25), the current flowing from terminal A over 
resistance R to terminal B, is represented by a vector 01 
(Fig. 26), or by /= i -^ji', the same current can be con- 
sidered as flowing in the opposite direction, from terminal 
B to terminal A in opposite phase, and therefore represented 
by a vector 0I^ (Fig. 26), or by /j = — * —ji'. 

Or, if the difference of potential from terminal B to 
terminal A is denoted by the E = e ■\-je' , the difference 
of potential from A \o B is E-^ = — e —je'. 

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Hence, in dealing with alternating-current sine waves, 
it is necessary to consider them in their proper direction 
with regard to the circuit. Especially in more complicated 
circuits, as interlinked polyphase systems, careful attention 
has to be paid to this point 

34. Let, for instance, in Fig. 27, an interlinked three- 
phase system be represented diagrammatically, as consist- 
ing of three £.M.Fs., of equal intensity, differing in phase 
by one-third of a period. Let the E.M.Fs. in the direction 

«(. 27. 

from the common connection O of the three branch circuits 
to the terminals y4,, A^, A^, be represented by Ey, £",, E^. 
Then the difference of potential from A^ to Ay is E^—Eyt 
since the two E.M.Fs., E-^^ and £",, are connected in cir- 
cuit between the terminals A^ and A^, in the direction, 

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A^ — — A^\ that is, the one, E^, in the direction OA^, 
from the common connection to terminal, the other, ^p in 
the opposite direction, A^O, from the terminal to common 
connection, and represented by — £", , Conversely, the dif- 
ference of potential from ^i to A^ is £"j — E^. 

It is then convenient to go still a step farther, and 
drop, in the diagrammatic representation, the vector line 
altogether ; that is, denote the sine wave by a point only, 
the end of the corresponding vector. 

Looking at this from a different point of view, it means 
that we choose one point of the system — for instance, the 
common connection O — as a zero point, or point of zero 
potential, and represent the potentials of all the other points 
of the circuit by points in the diagram, such that their dis- 
tances from the zero point gives the intensity; their ampli- 
tude the phase of the difference of potential of the respective 
point with regard to the zero point ; and their distance and 
amplitude with regard to other points of the diagram, their 
difference of potential from these points in intensity and 

Thus, for example, in an interlinked three-phase system 
with three £.M.Fs. of equal intensity, and differing in phase 
by one-third of a period, we may choose the common con- 
nection of the star-connected generator as the zero point, 
and represent, in Fig. 28, one of the E.M.Fs., or the poten- 

cy GoOglc 



tial at one of the three-phase terminals, by point E^. The 
potentials at the two other terminals will then be given by 
the points E^ and E^ which have the same distance from 
O i& E^ and are equidistant from E^ and from each other. 
The difference of potential between any pair of termi- 
nals — for instance E^ and £", — is then the distance Efi^n 
or E^E^ according to the direction considered. 

35. If now the three branches OE^, OE, and OE^ of 
the three-phase system are loaded equally by three currents 
equal in intensity and in difference of phase against their 

Hf. 29. Fig. to. 

E.M.Fs., these currents are represented in Fig. 29 by the 
vectors ^Ti = Of, = 01, = /, lagging behind the E.M.Fs. 
by angles E^O/^ = E,OI, = EjOJ, = a. 

Let the three-phase circuit be supplied over a line of 
impedance Z, = r^ —jx-^ from a generator of internal im- 
pedance Zo = Xg — JX^. 

In phase OEy the E.M.F. consumed by resistance r, is 
represented by the distance E^El = Ir^ in phase, that is 
parallel with current ~0I^. The E.M.F. consumed by re- 
actance x^ is represented by E^^El' =■ Ixy, 90° ahead of cur- 

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rent (?/,. The same ^plies to the other two phases, and 
it thus follows that to produce the E.M.F. triangle E^fi^^ 
at the tenninals of the consumer's circuit, the E.M.F. tri- 
angle E"E^'Ei' is required at the generator terminals. 

Repeating the same operation for the internal impedance 
of the generator we get £"£"' = Ir„, and parallel to 01^ 
E^'E" = Ix„ and 90° ahead of 01^ and thus as triangle of 
(nominal) induced E.M.Fs. of the generator Ei'E^E^. 

In Fig. 29, the diagram is shown for 46° lag, in Fig. 80 
for noninductive load, and in Fig. 81 for 46° lead of the 
currents with regard to their E.M.Fs. 

As seen, the induced generator E.M.F. and thus the 
generator excitation with lagging current must be higher, 
with leading current lower, than at non-inductive load, or 
conversely with the same generator excitation, that is the 
same induced generator E.M.F. triangle E^E°E^, the 
E.M.Fs. at the receiver's circuit, E^, E^ E^ fall off more 
with l^ging, less with leading current, than with non- 
inductive load. 

36. As further instance may be considered the case of 
a single phase alternating current circuit supplied over a 
cable containing resistance and distributed capacity. 

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Let in Fig. 3S the potential midway between the two 
terminals be assumed as zero point 0. The two terminal 
voltages at the receiver circuit are then represented by the 
points E and E} equidistant from and opjMsite each other, 
and the two currents issuing from the terminals are rep- 
resented by the points / and /', equidistant from and 
opposite each other, and under angle s with E and ^ 

Considering first an element of the line or cable next to 
the receiver circuit. In this an E.M.F. EE^ is consumed 
by the resistance of the line element, in phase with the 
current O!, and proportional thereto, and a current //, con- 
sumed by the capacity, as charging current of the line 
element, 90° ahead in phase of the E.M.F. OE and propor- 
tional thereto, so that at the generator end of this cable 
element current and E.M.F. are ^^ and OE^ respectively. 

Passing now to the next cable element we have again an 
E.M.F. £,£■, proportional to and in phase with the current 
<?/, and a current //, proportional to and 90° ahead of the 
E.M.F, OE^ and thus passing from element to element 
along the cable to the generator, we get curves of E.M.Fs. 
e and f^, and curves of currents i and i>, which can be called 
the topographical circuit characteristics, and which corre- 
spond to each other, point for point, until the generator 
terminal voltages 0E„ and OE^ and the generator currents 
Olg and OI^ are reached. 

Again, adding E^E" = I^^ and parallel 01^ and E"E'' = 
7^0 and 90" ahead of '0T„, gives the (nominal) induced 
E.M.F. of the generator OE", where Zg = r^ —jx„ = inter- 
nal impedance of the generator. 

In Fig. 33 is shown the circuit characteristics for 60° 
lag, of a cable containing only resistance and capacity. 

Obviously by graphical construction the circuit character- 
istics appear more or less as broken lines, due to the neces- 
sity of using finite line elements, while in reality when 
calculated by the differential method they are smooth curves. 

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37. As further instance may be considered a three-phase 
circuit supplied over a long distance transmission line of 
distributed Cfqiacity, self-induction, resistance, and leakage. 

Let, in Fig. siz^OZ,, ~0E^ 'OE^ = three-phase E.M.Fs. 
at receiver circuit, equidistant from each other and = £. 

Let O/i, 0/„ 01^ = three-phase currents in the receiver 
circuit equidistant from each other and = /, and making 
with £ the phase angle m. 

Considering again as in § 35 the transmission line ele- 
ment by element, we have in every element an E.M.F. 
£i£ ,' consumed by the resistance in phase with the current 
O/^ and proportional thereto, and an E.M.F. £,', £/' con- 

sumed by the reactance of the line element, 90° ahead of 
the current C/„ and proportional thereto. 

In the same l ine e lement we have a current Z,/,' in phase 
with the E.M.F. OE^, and proportional thereto, representing 
the loss of energy current by leakage, dielectric hysteresis, 
etc., and a current /i'/,", 90° ahead of the E.M.F. 0£^, and 
proportional thereto, the charging current of the line ele- 
ment as condenser, and in this manner passing along the 
line, element by element, we ultimately reach the generator 
terminal voltages £°, £^, E^, and generator currents /j", 
/,*, /,*, over the topographical characteristics of E.M.F. e^ 
e^ e^ and of current (j, i^ i„ as shown in- Fig. 33. 

The circuit characteristics of current i and of E.M.F. e 

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correspond to each other, point for point, the one giving the 
current and the other the E.M.F. in the line element. 

Fig. a*. 
Only the circuit characteristics of the first phase are 
shown as e^ and i^. As seen, passing from the receiving 
end towards the generator end of the line, potential and 

ciurent alternately rise and fall, while their phase angle 
changes periodically between lag and lead. 

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37. a. Mor« markedly this is shown in Fig. 34, the topo- 
graphic circuit characteristic of one of the lines with 90° 
lag in the receiver circuit. Corresponding points of the 
two characteristics e and i are marked by corresponding 
figures to 16, representing equidistant points of the line. 
The values of E.M.F., current and their difference of phase 
are plotted in Fig. 35 in rectangular coordinates with the 
distance as abscissae, counting from the receiving circuit 
towards the generator. As seen from Fig. 35, E.M.F. and 
current periodically but alternately rise and fall, a maximum 
of one approximately coinciding with a minimum of the 
other and with a point of zero phase displacement. 

The phase angle between current and E.M.F. changes 
from 90" lag to 72° lead, 44= lag, 34° lead, etc.. gradually 
decreasing in the amplitude of its variation. 

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38. If in a continuous-current circuit, a number of 
resistances, *-i, /j, r^, . . . are connected in series, their 
joint resistance, R, is the sum of the individual resistances 
^ = r, + r, + r, + . . . 

If, however, a number of resistances are connected in 
multiple or in parallel, their joint resistance, R, cannot 
be expressed in a simple form, but is represented by the 
expression : — , 


Hence, in the latter case it is preferable to introduce, in- 
stead of the term resistance, its reciprocal, or inverse value, 
the term conductance, ^ = 1 j r. If, then, a number of con- 

I ductances, g^, g^, g^, . . . are connected in parallel, their 
joint conductance is the sum of the individual conductances, 
OT G = g^+ g^-^ g^+ . . . When using the term con- 

j ductance, the joint conductance of a number of series- 
connected conductances becomes similarly a complicated 

I expression — 

G= ^ . 

. gx St g* 

Hence the term resistance is preferable in case of series 
connection, and the use of the reciprocal term conductance 
in parallel connections; therefore, £a,fiS 

[| The Joint resistance of a number of series-connected resis- 

tances is equal to the sum of the individual resistances ; tiie 

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joint conductance of a number of parallel-connccled conduc- 
tances is equal to the sunt of the individual conductances. 

39. In alternating-current circuits, instead of the term 
resistance we have the term impedance, Z =^ r —jx, with its 
two components, the resistance, r, and the reactance, x, in the 
formula of Ohm's law, E — IZ. The resistance, r, gives 
the component of E.M,F. in phase with the current, or the 
energy component of the E.M.F., Jr; the reactance, x, 
gives the component of the E.M.F. in quadrature with the 
current, or the wattless component of E.M.F., Ix\ both 
combined give the total E.M.F., — 

/e-./Vr' + ar'. 
Since E.M,Fs. are combined by adding their complex ex- 
pressions, we have : 

The joint impedance of a number of series-connected impe- 
dances is the sum of the individual impedances, when expressed 
in complex quantities. 

In graphical representation impedances have not to be 
added, but are combined in their proper phase by the law 
of parallelogram in the same manner as the E.M.Fs. corre- 
sponding to them. 

The term impedance becomes inconvenient, however, 
when dealing with parallel-connected circuits ; or, in other 
words, when several currents are produced by the same 
E.M.F., such as in cases where Ohm's law is expressed in 
the form, 

It is preferable, then, to introduce the reciprocal of 
impedance, which may be called the admittance of the 
circuit, or 

As the reciprocal of the complex quantity, Z = r —jx, the 
admittance is a complex quantity also, or Y = g+jb ; 

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it consists of the component g, which represents the co- 
efficient of current in phase with the E.M.F., or energy 
current, gE, in the equation of Ohm's law, — 

and the component b, which represents the coefficient of 
current in quadrature with the E.M.F., or wattless com- 
ponent of current, bE. 

g is called the conductance, and b the suscejitance, of 
the circuit. Hence the conductance, g, is the energy com- 
ponent, and the susceptance, b, the wattless component, 
of the admittance, Y= g -\-jby while the numerical value of 

admittance is — ^ 

y = V^^ + b^ ; 

the resistance, r, is the energy component, and the reactance, 
X, the wattless component, of the impedance, Z™ t — jx, 
the numerical value of impedance being — 

40. As shown, the terra admittance implies resolving 
the current into two components, in phase and in quadra- 
ture with- the E.M,F., or the energy current and the watt- 
less current ; while the term impedance implies resolving 
the E.M.F. into two components, in phase and in quad- 
rature with the current, or the energy E.M.F. and the 
wattless E.M.F. 

It must be understood, however, that the conductance 
is not the reciprocal of the resistance, but depends upon 
the resistance as well as upon the reactance. Only when the 
reactance .r = 0, or in continuous-current circuits, is the 
conductance the reciprocal of resistance. 

Again, only in circuits with zero resistance (r == 0) is 
the susceptance the reciprocal of reactance ; otherwise, the 
susceptance depends upon reactance and upon resistance. 

The conductance is zero for two values of the resistance : — 

1.) If r = 00 , or ;r = 00 , since in this case no current 
passes, and either component of the current ^ 0. 

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2.) If r = 0, since in tliis case the current which passes 
through the circuit is in quadrature with the E,M,F., and 
thus has no energy component. 

Similarly, the susceptance, b, is zero for two values of 
the reactance : — 

1.) If jr -« 00 , or r = oo . 

2.) If ^ - 0. 

From the definition of admittance, Y = g-^jb, as the 
reciprocal of the impedance, Z = r —jx, 

we have K =■ — , or, r +jb ■■ ; 

Z r~jx 

or, multiplying numerator and denominator on the right side 
by (r +jx) 


hence, smce 

ir-jx) (r + A) = r" + ;t* - ««. 


^"r^ + **"" »»' 

-f+b* f 

^ + ^* f 
By these equations, the conductance and susceptance can 
be calculated from resistance and reactance, and conversely. 
Multiplying the equations for^ and r, we get : — 

hence, »V = ('^ +^ (^ + i*) =1 ; 

and t ■= 1 = 1 \ the absolute value of 

y ■Vg*-\' b' ' I impedance ; 

1 1 ) the absolute value of 

v'r' 4- x* ' ) admittance. 

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41. If, in a circuit, the reactance, x, is constant, and the 
resistance, r, is varied from r = to r = oo , the susceptance, 
b, decreases from ^ = l/xat ri=iO, to ^ = at r=oo; 
while the conductance, ^ = at r = 0, increases, reaches 
a maximum for r = ^, where ^ = 1 / 2 r is equal to the 
susceptance, or ^ = ^, and then decreases again, reaching 
r=0 at r=«>. 




In Fig. S6, for constant reactance x = .h ohm, the vari- 
ation of the conductance, g, and of the susceptance, b, are 
shown as functions of the varying resistance, r. As shown, 
the absolute value of admittance, susceptance, and conduc- 
tance are plotted in full lines, and in dotted line the abso- 
lute value of impedance, 

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Obviously, if the resistance, r, is constant, and the reac- 
tance, X, is varied, the values of conductance and susceptance 
are merely exchanged, the conductance decreasing steadily 
from ^ = 1 / r to 0, and the susceptance passing from at 
jr = to the maximum, b = \ l^r = g =\ / 1x atx = r, 
and to ^ = at ^ = CO , 

The resistance, r, and the reactance, x, vary as functions 
of the conductance, g, and the susceptance, b, in the same 
manner as g and b vary as functions of r and x. 

The sign in the complex expression of admittance is 
always opposite to that of impedance; this is obvious, since 
if the current lags behind the E.M.F., the E.M.F. leads the 
current, and conversely. 

We can thus express Ohm's law in the two forms — 

E = IZ, 

I = 'ey, 
and therefore — 

The joint impedance of a number of series-connected im- 
pedances is equal to the sum of the individual impedances ; 
the joint admittance of a number of parallel-coAnected admit- 
tances, if expressed in complex quantities, is equal to the sum 
of the individual admittances. In diagrammatic represen- 
tation, combination by the parallelogram law takes the place 
of addition of the complex quantities. 

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42. Having, in the foregoing, reestablished Ohm's law 
and Kirchho^'s laws as being also the fundamental laws 
of alternating-current circuits, when expressed in their com- 
plex form, 17 7 r r trr- 

'^ E = ZI, or, / = YE, 

and S£ = in a closed circuit, 

%! = at a distributing point, 
where E, J, Z, V, are the expressions of E.M.F., current, 
impedance, and admittance in complex quantities, — these 
values representing not only the intensity, but also the phase, 
of the alternating wave, — we can now — by application of 
these laws, and in the same manner as with continuous- 
current circuits, keeping in mind, however, that E, I, Z, Y, 
are complex quantities — calculate a]ternating<urrent cir- 
cuits and networks of circuits containing resistance, induc- 
tance, and capacity in any combination, without meeting 
with greater difficulties than when dealing with continuous- 
current circuits. 

It is obviously not possible to discuss with any com- 
pleteness all the infinite varieties of combinations of resis- 
tance, inductance, and capacity which can be imagined, and 
which may exist, in a system or network of circuits ; there- 
fore only some of the more common or more interesting 
combinations will here be considered. 

1.) Resistance in series with a circuit. 

43. In a constant-potential system with impressed 

■ff» = '» +>/, ■£. = V^T^*. 

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let the receiving circuit of impedance 

Z = r -jx, z = -^T^+xS 

be connected in series with a resistance, r, . 
The total impedance of the circuit is then 

hence the current is 

/= ^' ^ -g. = £o(r + r,+jx) , 

■ Z + r, r+r,~jx (r + r,)* + x' ' 

and the E.M.F, of the receiving circuit, becomes 

A _ VZ _ -^•t'— » _ :*• ('■('■+ '■. )+^->.»l 
• ■ r + r.-J:, (^ + ,.)■ + »• 

,' + 2rr, + r,' ' 
Or, in absolute values we have the following : — 
Impressed E.M.F., 

£,-^'.' + '."-, 

E.M.F. at teminals of receiver circuit, 

r^rJ ^ + ^ '- -g.» ■ 

V (r + r,)» + ^ Vj' + 2rr, + r,* ' 
difference of phase in receiver circuit, tan £ = -; 

difference of phase in supply circuit, tan a^ = 

since in general, 

, , . imai^naiy component 
' real component 

a.) If X is negligible with respect to r, as in a non-induc- 
tive receiving circuit, 

^^ r-^-rl ^ = -^'7+7/ 

and the current and E.M.F. at receiver terminals decrease 
steadily with increasing r, . 

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b) If r is negligible compared with ^, as in £ 
receiver circuit, 

or, for small values of 

that is, the current and E.M.F, at receiver terminals remain 
approximately constant for small values of r^ , and then de- 
crease with increasing rapidity. 

44. In the general equations, x appears in the expres- 
sions for / and E only as j^, so that / and E assume the 
same value when x is negative, as when x is positive ; or, in 
other words, series resistance acts upon a circuit with leading 
current, or in a condenser circuit, in the same way as upon a 
circuit with lagging current, or an inductive circuit. 

For a given impedance, z, of the receiver circuit, the cur- 
rent /, and E.M.P"., £, are smaller, as r is larger; that is, 
the less the difference of phase in the receiver circuit. 

As an instance, in Fig. 37 is shown the E.M.F., E, at 
the receiver circuit, for E^ = const. = 100 volts, 2 = 1 ohm ; 
hence / = E, and — 

a.) r, = .2 ohm (Curve I.) 
i.) r. = .8 ohm (Curve II.) 

with values of reactance, x ^ V«' — r*, for abscissae, from 
X = + 1.0 to .r -= — 1.0 ohm. 

As shown, / and E are smallest for x = G, r = 1.0, 
or for the non-inductive receiver circuit, and largest for 
.r = J; 1.0, r= 0, or for the wattless circuit, in which latter 
a series resistance causes but a very small drop of potential. 

Hence the control of a circuit by series resistance de- 
pends upon the difference of phase in the circuit. 

For r^ = .8, and ;c = 0, x= + .8, x- ~ .8, the polar 
diagrams are shown in Figs. 38 to 40. 

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2.) Reactance in series with a circuit. 
45. In a constant potential system of impressed E.M.F., 
■?.=■;.+ A', £. = V<-,' +<■.'" , 

let a reactance, x^ , be connected in series in a receiver cir- 
cuit of impedance 













;^ r. = .9 







— 1 











— 1 

















Fig. 87. Variation of V 

t Httlitanit uillli Mua Halation ^ 

Then, the total impedance of the circuit is 


and the current is, 

while the difference of potential at the receiver terminals 


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Or, in absolute quantities : 

Vr ^ + (j: + J,/ Vz ' + a :«:, + «,' ' 
KM.F. at receiver terminals, 

difference of phase in receiver circuit, 

tan 5 = - ; 


difference of phase in supply circuit, 

a) If X is small compared with r, that is, if the receiver 
circuit is non-inductive, / and E change very little for small 
values of x, ; but if x is large, that is, if the receiver circuit 
is of large reactance, / and E change much with a change 
of x^ 

6.) If X is negative, that is, if the receiver circuit con- 
tains condensers, synchronous motors, or other apparatus 
which produce leading currents — above a certain value of 
X the denominator in the expression of E, becomes < 3, or 
E > E^\ that is, the reactance, x„ , raises the potential. 

c.) E = E,, or the insertion of a series inductance, x,, 
does not affect the potential difference at the receiver ter- 
minals, if 

or, x,= ~2x. 

That is, if the reactance which is connected in series in 
the circuit is of opposite sign, but twice as large as the 
reactance of the receiver circuit, the voltage is not affected, 
h\xtE = E„l=EJs. If:ro<- 2 .r, it raises, if ;r„ > - 2 «-, 
it lowers, the voltage. 

We see, then, that a reactance inserted in series in 
an alternating-current circuit will lower the voltage at the 

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receiver terminals only when of the same sign as the reac- 
tance of the receiver circuit ; when of opposite sign, it will 
lower the voltage if larger, raise the voltage if less, than 
twice the numerical value of the reactance of the receiver 

d.) If 4r = 0, that is, if the receiver circuit is non- 
inductive, the E.M.F. at receiver terminals is : 

E = -^'^ = -gp 


= {14. *)-* expanded by the binomial theorem 

Therefore, if x^ is small compared with r: — 
E,-E l/it.\' 

£. i{ry 

That is, the percentage drop of potential by the insertion 
of reactance in series in a non-inductive circuit is, for small 


values of reactance, independent of the sign, but propor- 
tional to the square of the reactance, or the same whether 
it be inductance or condensance reactance. 

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46. As an instance, in Fig. 41 the changes of current, 
/, and of E.M.F. at receiver terminals, E, at constant im- 
pressed E.M.F., £", , are shown for various conditions of a 
receiver circuit and amounts of reactance inserted in series. 

Fig. 41 gives for various values of reactance, x^ (if posi- 
tive, inductance — if negative, condensance), the E.M.Fs., 
E, at receiver terminals, for constant impressed E.M.F,, 

^- *• 



l..»!ec(N«V*N+,El=,bo ' 







1. r-I.O X-O 






III. r-.e X — .i 























































"^ -^ 



























*h :i 

£"o = 100 volts, and the following conditions of receiver 
circuit :— ^^jQ^ r=l.O, x= (Curve I.) 
t=l.Q, r= .6, x= .8(Curven.) 
i= 1.0, r= .6, *= - .8 (Curve III.) 

As seen, curve I is symmetrical, and with increasing x, 
the voltage E remains first almost constant, and then drops 
off with increasing rapidity. 

In the inductive circuit series inductance, or, in a con- 
denser circuit series condensance, causes the voltage to drop 
off very much faster than in a non-inductive circuit. 

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Series inductance in a condenser circuit, and series con- 
densance in an inductive circuit, cause a rise of potential. 
This rise is a maximum for jt, => ± .8, or, x^= — x (the 
condition of resonance), and the E.M.F. reaches the value, 
E = 167 volts, or, E = E,sjr. This rise of potential by 
series reactance continues up to ;r, = ± 1.6, or, x,=s — 2x, 

fig. 43. 

where E = 100 volts again ; and for x, > 1.6 the voltage 
drops again. 

At .r, = i .8, :r = T 8, the total impedance of the circuit 
is r —J (x + x^) = r = .6, x + x^ = 0, and tan w, = ; 
that is, the current and E.M.F. in the supply circuit are 
in phase with each other, or the circuit is in electrical 
resonance. ^ , i— A 


fig. 4a. 

Since a synchronous motor in the condition of efficient 
working acts as a condensance, we get the remarkable result 
that, in synchronous motor circuits, choking coils, or reactive 
coils, can be used for raising the voltage. 

In Figs. 42 to 44, the polar diagrams are shown for the 
conditions — 

£, = 100, x, = .Q, x=^ (Fig. 42) £ = 85.7 

x= +.8 (Fig, 43) £ = 65.7 

x= -.8 (Fig. U)E = 158.1 

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47. In Fig. 45 the dependence of the potential, E, upon 
the difference of phase, u, in the receiver circuit is shown 
for the constant impressed E.M.F., E„ == 100 ; for the con- 
stant receiver impedance, ^ = 1.0 (but of various phase 
differences »), and for various series reactances, as follows : 

r«^£ ^-ft*^>v<ilw*>|>(y x,= .2 (Curve I.) '*****'■" ^ (ai«U^ 

x,=- .6 (Curve 11.) 

X, = .8 (Curve III.) 

X, = 1.0 (Curve IV.) 

X, = 1.6 (Curve V.) 

X, = 3.2 (Curve VI.) 

fig. 44. 

Since s = 1.0, the current, /, in all these diagrams has 
the same value as £. 

In Figs. 46 and 47, the same cun-es are plotted as in 
Fig. 45, but in Fig. 46 with the reactance, x, of the receiver 
circuit as abscissae ; and in Fig. 47 with the resistance, r, of 
the receiver circuit as abscissae. 

As shown, the receiver voltage, E, is always lowest when 
Xf and X are of the same sign, and highest when they are 
of opposite sign. 

The rise of voltage due to the balance of x^ and .r is a 
maximum for x^ = + 1.0, x •= — 1.0, and r = 0, where 

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■ E E. 




























































































.E.| 1 1 

Fig. 4S. rarMlm oi 






































— ' 















. V 

















01 R 












^ 4ft VarlcUtm of </eliii9* at Catatant Serlaa Rvtetaiun wM Heaetanct of 

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£ = QO ; that is, absolute resonance takes place. Obvi- 
ously, this condition cannot be completely reached in 

It is interesting to note, from Fig. 47, that the largest 
part of the drop of potential due to inductance, and rise to 
condensance — or conversely — takes place between r= 1.0 
and r = .9 ; or, in other words, a circuit having a power 















































































^ ■ 












. 1 










factor cos £ = .9, gives a drop several times larger than a 
non-inductive circuit, and hence must be considered as 
an inductive circuit. 

3.) Impedance in scries ;vith a circuit. 
48. By the use of reactance for controlling electric 
circuits, a certain amount of resistance is also introduced, 
due to the ohmic resistance of the conductor and the hys- 
teretic loss, which, as will be seen hereafter, can be repre- 
sented as an effective resistance. 

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Hence the impedance of a reactive coil {choking coil) 
may be written thus ; — 

Zo = fo — Aot 'o = Vr-o' + *(,', 

where t„ is in general small compared with Xf^ 
From this, if the impressed E.M.F. is 

-fo = 'o + A', E^ = -JiT+e? 

and the impedance of the consumer circuit is 
Z = r-jx s=-\/?~+^ 

we get the current, / = — — - — = ° 

and the E.M.F. at receiver terminals, 

• -Z + Z. \r + r,)-j(x + X,) 

Or, in absolute quantities, 

the current is, 

j-^ E, ^ E, . 

^/ir + r,y + ix + x.r ^/z* + zj'+2(rr. + xx0 ' 
the E.M.F. at receiver terminals is, 

^^ £,i ^ E^ . 

" V(T+0* + (* + *.)* V^'+V + 2(r^ + ^:..) ' 
the difference of phase in receiver circuit is. 

and the difference of phase in the supply circuit is, 

tan5 = ^-i-^. 
r + ro . 

49. In this case, the maximum drop of potential will not 
take place for either x = 0, as for resistance in series, or 
for y = 0, as for reactance in series, but at an intermediate 
point. The drop of voltage is a maximum ; that is, £' is a 
minimum if the denominator of £ is a maximum ; or, since 
*. -^0 . ''.. ^a are constant, if rr„ + xx^ is a maximum, that is, 
since x = V^ — r^, if rr, + x^ V^* — r* is a maximum. 

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A function, f = rr^ -\- x^ Vi^ — r^ is a maximum when 
its differential coefficient equals zero. For, plotting / as 
curve with r as abscissas, at the point where / is a maxi- 
mum or a minimum, this curve is for a short distance 
horizontal, hence the tangens-function of its tangent equals 
zero. The tangens-function of the tangent of a curve, how- 
ever, is the ratio of the change of ordinates to the change 
of abscissae, or is the differential coefficient of the func- 
tion represented by the curve. 


I t 

- 7 

' ^^-^ 

zt ^- fti - -T^ 

, J. 

X- — 

..(■ j-j.j.j.j_j..» •» J- ■.!■.>.« 

Pg. 48. 

Thus we have : — 

y = rr, + «, Vs' — '^ = maximum or minimum, if 

Differentiating, we get ; — 

or, expanded, — 

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That is, the drop of potential is a maximum, if the re- 
actance factor, xjr, of the receiver circuit equals the reac- 
tance factor, x^jr,, of the series impedance. 


50. As an example, Fig. 48 shows the E.M.F., E, 
at the receiver terminals, at a constant impressed E.M.P., 
E^ = 100, a constant impedance of the receiver circuit, 
e = 1.0, and constant series impedances, 

Z,= .3-/.4 (Curve I.) 

Z. = 1.2 — yi.6 (Curve II.) 
as functions of the reactance, x, of the receiver circuit 


Figs. 49 to 51 give the polar diagram for E^ = 100, 
= .95, X = 0, ^ = - .95, and Z^ = .3 ~j .4. 

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4.) Compensation for Lagging Currents by Shunted 

61 . We have seen in the latter paragraphs, that in a 
constant potential alternating-current system, the voltage 
at the terminals of a receiver circuit can be varied by the 
use of a variable reactance in series to the circuit, without 
loss of energy except the unavoidable loss due to the 
resistance and hysteresis of the reactance ; and that, if 
the series reactance is very large compared with the resis- 
tance of the receiver circuit, the current in the receiver 
circuit becomes more or less independent of the resis- 
tance, — that is, of the power consumed in the receiver 

circuit, which in this case approaches the conditions of a 
constant alternating-current circuit, whose current is. 

I^ °— , or approximately, /^— °. 

This potential control, however, causes the current taken 
from the mains to lag greatly behind the E.M,F., and 
thereby requires a much larger current than corresponds 
to the power consumed in the receiver circuit. 

Since a condenser draws from the mains a leading cur- 
rent, a condenser shunted across such a circuit with lagging 
current will compensate for the lag, the leading and the 
lagging current combining to form a resultant current more 
or less in phase with the E.M.F., and therefore propor- 
tional to the power expended. 

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In a circuit shown diagrammatically in Fig. 52, let the 
non-inductive receiver circuit of resistance, r, be connected 
in series with the inductance, x^ , and the whole shunted by 
a condenser of condensance, £, entailing but a n^ligible loss 
of energy. 

Then, if E, = impressed E.M.F.,— 

the current in receiver circuit is, 



the current in condenser circuit is, 

JC c 

and the total current is 


or, in absolute terms, /, = 

while the E-M.F. at receiver terminals is, 

E-Ir = E, J'. , E = -^°'" . 

62. The main current, /,, is in phase with the impressed 
E.M.F., Eg, or the lagging current is completely balanced, 
or supplied by, the condensance, if the imaginary term in 
the expression of /, disappears ; ttiat is, if 

This gives, expanded : c = "*" ° 

Hence the capacity required to compensate for the 
lagging current produced by the insertion of inductance 
in series to a non-inductive circuit depends upon the resis- 
tance and the inductance of the circuit, x^ being constant. 

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with increasing resistance, r, the condensance has to be 
increased, or the capacity decreased, to keep the balance. 

Substituting c = ^ • , 

we get, as the equations of the inductive circuit balanced 
by condensance : — e ^^ 

_ B. 

^. ('•+>.) 

V^ + X,' 

, .-.i-A^J5 


r' + x- ' 

f' + 'J 






Vr-'+ X.' 

1 AVUi— .-(J^ 

and for the power expended in the receiver circuit : 

that is, the main current is proportional to the expenditure 
of power. 

For r = we have c =- x„, or the condition of balance. 

Complete balance of the lagging component of current 
by shunted capacity thus requires that the condensance, c, 
be varied with the resistance, r; that is, with the varying 
load on the receiver circuit. 

In Fig. 63 are shown, for a constant impressed E.M.F., 
E, = 1000 volts, and a constant series reactance, x^ = 100 
ohms, values for the balanced circuit of, 

current in receiver circuit (Curve I.), 
current in condenser circuit (Curve 1 1.), 
current in main circuit (Curve III.), 

E.M.F. at receiver terminals (Curve IV.), 

with the resistance, r, of the receiver circuit as abscissfe. 

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63. If, however, the condensance is left unchanged, 
^ =s .To at the no-load value, the circuit is balanced for r = 0, 
but will be overbalanced for r> 0, and the main current 
will become leadmg. 

We get in this case : — 

• r-i', Vr^T*? 

■ ■ • ».(*.-i-A) ^.V-^ + j 

The difference of phase in the main circuit is,- 




when r = or at no load, and increases with increasing 
resistance, S*tne lead of the current. At tfie same time, 
the current in the receiver circuit, /, is approximately con- 
stant for small values of r, and then gradually decreases. 







""tidod'dem "'too" 






- = 










1 m 









~ ~~ 










In Fig, 54 are shown the values of /, /], /,, E, in Curves 
I., II., III., IV., similarly as in Fig. 50, for £^ = 1000 volts, 
c = X = 100 ohms, and ras abscissae. 

5.) Constant Potential — Constant Current Transformation. 

54. In a constant potential circuit containing a large 
and constant reactance, x^, and a varying resistance, r, the 
current is approximately constant, and only gradually drops 
off with increasing resistance, r, — that is, with increasing 
load, — but the current lags greatly behind the E.M.F. This 
lagging current in the receiver circuit can be supplied by a 
shunted condensance. Leaving, however, the condensance 
constant, f = x,, so as to balance the lagging current at no 

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load, that is, at r= 0, it will overbalance with increasing 
load, that is, with increasing r, and thus the main current 
will become leading, while the receiver current decreases 
if the impressed E.M.F., E^, is kept constant. Hence, to 
keep the current in the receiver circuit entirely constant, the 
impressed E,M.F., E^, has to be increased with increasing 
resistance, r\ that is, with increasing lead of the main cur- 
rent. Since, as explained before, in a circuit with leading 
current, a series inductance raises the potential, to maintain 
the current in the receiver circuit constant under all loads, 
an inductance, x^ , inserted in the main circuit, as shown in 
the diagram, Fig. 55, can be used for raising the potential, 
E„ with increasing load. 

be the impressed E.M.F. of the generator, or of the mains, 
and let the condensanoe be .r, = jr,; then — 
Current in receiver circuit, 

current in condenser circuit, 

/, = -^. . *<(.^|J.«*-ff.« 

Hence, the total current in main line is 



and the E.M.F. at receiver terminals, 

• ■ r -jxo 
E.M.F. at condenser terminals, 

E.M.F. consumed in main line, 

E' >/.,.-- f--" , 

hence, the E.M.F. at generator is 

Et—E. + E'—E. i 1 T-Jii — : ! . 

■ I '.(r-J'.) S ' 

or, ^E.'' ('•-'•> -''''>■ 

and conversely the E.M.F. at condenser terminals, 

■' r(x.-x^)-jx*' 
current in receiver circuit, 

r-jx. r(x,~xt)-jx,* 
This value of / contains the resistance, r, only as a fac- 
tor to the difference, x^ — x^; hence, if the reactance, Xj , 
is chosen ^^ x^, r cancels altogether, and we find that if 
x^ = Xg, the current in the receiver circuit is constant, 


and is independent of the resistance, r ; that is, of the load. 

Thus, by substituting x^ = x„ we have. 
Impressed E.M.F. at generator, 

Et " tt •Yjft, Ef^ V^s* + ii ' = constant; 

current in receiver circuit, 

/ =/■ — , / = ^ = constant ; 

E.M.F. at receiver circuit, 

E ■=•//■— y— 2—, ,£ >-—^, or proportional to load r; 

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E.M.F. at condenser tenninals, 

-•?.(l+/j^Y £.-i,y/l+(jY,lieiice> A; 

current in condenser circuit, 

f, = _y4fe^, /, _|.y/l + ^£y, 

mam current, 

i T- ( proportional to the load, 

_ -^1^ 7 _ ^t*" ) J ■ (, ■.1, 

= — J- . A =1 ~—^ , < r, and in phase with 

*' "' (em.f., £,. 

The power of the receiver circuit is, 

the power of the main circuit, 

/^Et = —^ , hence the same. 
55. This arrangement is entirely reversible ; that is, 
if Et = constant, / = constant ; and 
if /, = constant, E = constant. 
In the latter case we have, by expressing all the quanti^ 
ties by /^ : — 
Current in main ]in 

KM.F. at receiver ci 

£ ■— /,x, = constant ; 
current in receiver circuit, 

/ =/.-^, proportional to the load - 
current in condenser circuit, 




E.M.F. at condenser terminals, 

Impressed E.M.F. at generator terminals, 

Et = — /, , or proportional to the load - . 

From the above we have the following deduction : 

Connecting two reactances of equal value, x,, in series 
to a non-inductive receiver circuit of variable resistance, r, 
and shunting across the circuit from midway between the 
inductances by a capacity of condensance, x^ = ■*'b> trans- 
forms a constant potential main circuit into a constant cur- 
rent receiver circuit, and, inversely, transforms a constant 
current main circuit into a constant potential receiver cir- 
cuit. This combination of inductance and capacity acts as 
a transformer, and converts from constant potential to con- 
stant current and inversely, without introducing a displace- 
ment of phase between current and E.M.F. 

It is interesting to note here that a short circuit in the 
receiver circuit acts like a break in the supply circuit, and a 
break in the receiver circuit acts like a short circuit in the 
supply circuit 

As an instance, in Fig. 56 are plotted the numerical 
values of a transformation from constant potential of 1,000 
volts to constant current of 10 amperes. 

Since £", = 1,000, / = 10, we have : x, = 100 ; hence 
the constants of the circuit are : — 

£t = 1000 volts ; 
/ =10 amperes; 
£ e= 10r,plottedasCurveI.,withtheresistances,r,asabscissa:; 

o^' + im} 

£, = 1000 \i- + {:—], plotted as Curve II. ; 

A «= 10 yn- (~f, plotted as Curve III.- 
/, » .1 r, plotted as Curve IV. 

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66. In practice, the power consumed in the main circuit 
will be larger than the power delivered to the receiver cir- 
cuit, due to the unavoidable losses of power in the induc- 
tances and condensances. 














i^-slE^Sif "''""" 


















































Ftq. 5ft OoiMtaii(-ft(*R(fii/ — CMtfoitt-OirrFSNt lrKntimtt,tivi, 

Let — 
^1 = 2 ohms = effective resistance of condensance ; 
r, = 3 ohms = effective resistance of each of the inductances. 
We then have : — 

Power consumed in condensance, I* r^ = 200 + .02 r* ; 
power consumed by first inductance, /* r, e= 300 ; 
power consumed by second inductance, /,' r. = .03 r*. — 
Hence, the total loss of energy is 600 + ,05 r* ; 
output of system, /* r = 100 r 

input, 500 + 100 /■ + -05 r* 


100 f- 

600 + 100r + .06r» — 

It follows that the main current, /g, increases slightly 
by the amount necessary to supply the losses of energy 
in the apparatiis. 

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This curve of current, /,, including losses in transforma- 
tion, is shown in dotted lines as Curve V. in Fig. 56 ; and 
the efficiency is shown in broken line, as Curve VI. As 
shown, the efficiency is practically constant within a Wide 

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67. In alternating-current circuits, E.M.F, is consumed 
in the feeders of distributing networks, and in the lines of 
long-distance transmissions, not only by the resistance, but 
also by the reactance, of the line. The E.M.F. consumed by 
the resistance is in phase, while the E.M.F. consumed by the 
reactance is in quadrature, with the current. Hence their 
influence upon the E.M.F. at the receiver circuit depends 
upon the difference of phase between the current and the 
E.M.F. in that circuit. As discussed before, the drop of 
potential due to the resistance is a maximum when the 
receiver current is in phase, a minimum when it is in 
quadrature, with the E.M.F. The change of potential due 
to line reactance is small if the current is in phase with 
the E.M.F., while a drop of potential is produced with a 
lagging, and a rise of potential with a leading, current in 
the receiver circuit. . 

Thus the change of potential due to a line of given re- 
sistance and inductance depends upon the phase difference 
in the receiver circuit, and can be varied and controlled 
by varying this phase difference; that is, by varying the 
admittance, Y=^ff-\-jl!, of the receiver circuit. 

The conductance,^, of the receiver circuit depends upon 
the consumption of power, — that is, upon the load on the 
circuit, — and thus cannot be varied for the purpose of reg- 
ulation. Its susceptance, l>, however, can be changed by 
shunting the cu-cuit with a reactance, and will be increased 
by a shunted inductance, and decreased by a shunted con- 
densance. Hence, for the purpose of investigation, the 





receiver circuit can be assumed to consist of two branches, 
a conductance, /■, — the non-inductive part of the circuit, — 
shunted by a susceptance, b, which can be varied without 
expenditure of energy. The two components of current 
can thus be considered separately, the energy component as 
determined by the load on the circuit, and the wattless 
component, which can be varied for the purpose of regu- 

Obviously, in the same way, the E.M.F. at the receiver 
circuit may be considered as consisting of two components, 
the energy component, in phase with the current, and 
the wattless component, in quadrature with the current. 
This will correspond to the case of a reactance connected 
in series to the non-inductive part of the circuit. Smce the 
effect of either resolution into components is the same so 
far as the line is concerned, we need not make any assump- 
tion as to whether the wattless part of the receiver circuit 
is in shunt, or in series, to the energy part. 

Let — 
Zg = r, ~jx, = impedance of the line ; 

y ^ g +jl> = admittance of receiver circuit ; 

J, = V^*-l-^»i 
£^ = tg -\-jeJ= impressedE.M.F. at generator end of line; 

E =e -^je' = KM. F. at receiver end of line; 

E = V*" +*'*! 
Ig = ig -^jif ™ CTurent in the line ; 

/„ =: V^' + to'*. 
The simplest condition is the non-inductive circuit. 

1.) Non-inductive Receiver Circuit Supplied over an 

Inductive Line. 
58. In this case, the admittance of the receiver circuit 
is y = g, since * = 0. 

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w T. .T. 1=1^ Ye E-^r 

We have then — -r • - - 

current, It = Eg; 

impressed E.M.F., ^, - i + Z./. = ^ (1 + Z,g). {mi- £= tU-4-L^ 

Hence- • • • ■ ^T JfJ 

E.M.F. at receiver circuit, 

1 + Z^ ■S~'\-gr,-jgxJ 
current, y,= _Ai_ = _ :^.£_: 

1 + -^.^ 1 + ^^^ — y^-^. 

Hence, in absolute values — 
EIM.F. at receiver circuit. 



The ratio of E.M.Fs. at receiver circuit and at genera- 
tor, or supply circuit, is — 

^^E_^ 1 . 

and the power delivered in the non-inductive receiver cir- 
cuit, or 
output, P = 


As a function of g; and with a given £,, r^, and jir^, this 
power is a maximum, if — 

that is — 

-~t+gW+gW = Oi 
hence — 

conductance of receiver circuit for maximum output 




^ r J' ^ >• 


and, substituting this in /" — 

Maximum output, -/« = :; ^ — r = " ■ 

2(r, + ^) 2 ir. + Vr,« + *.>}' 
and — 
ratio of E.M.F. at receiver and at generator end of line, 


efficiency, ' = — =- ^ — . 

That is, the output which can be transmitted over an 
inductive line of resistance, r^, and reactance, x,, — that is, 
of impedance, s^, — into a non-inductive receiver circuit, is 
a maximum, if the resistance of the receiver circuit equals 
the impedance of the line, r = s^, and is — 

p. Si—. 

2 ('. + ».) 
The output is transmitted at the efficiency of 

and with a ratio of E.M.Fs. of 

69. We see from this, that the maximum output which 
can be delivered over an inductive line is less than the 
output delivered over a non-inductive line of the same 
resistance — that is, which can be delivered by continuous 
currents with the same generator potential. 

In Fig. 67 are shown, for the constants 
^. = 1000 volts, 

Z, = 2,5 — 6/;thati8, r, = 2.5 ohms, :>:.=« 6ohms,«a = 6.6 ohms, 
with the current /, as abscissae, the values — 1 * ' 

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E.M.F. at Receiver Circuit, E, (Curve I.) ; 

Output of Transmission, P, (Curve II.) ; 

Efficiency of Transmis^on, (Curve III.). 

The same quantities, B and P, for a non-inductive line of 
resistance, r^ = 2.6 ohms, x, = 0, are showm in Curves IV., 
C,,,,^, v., and VI. 


•S 111 lU 
















.. S: "'^,'"" ^ ^"*™1'^."m™« 


































































ffo- 97. ffafl-/iitfHctfBa /ttetlmr Circuit Supplltd Omr liHhietkit Uat. 

2.) Maximum Power Supplied over an Inductive Line. 

60, If the receiver circuit contains the susceptance, b, 
in addition to the conductance, g, its admittance can be 
written thus: — 

Then — 
current, J,= EY-, 

Impressed E.M.F., E^~'e^- /,Z, = -ff (1 + YZ.). 

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Hence — 
E.M.F. at receiver terminals. 

■ 1 + rZ. i). -^ r.g ^ x,b) - j ix.g ~ r,by 


/ _ E,y EJ^->rm . 

or, in absolute values — 
KM.F. at receiver circuit, 

E- , "■ — . 

V(l + ra+'c.if+if.S-r.if 


V (1 + r.g-^.x.b)-' + {x.g -' 
ratio of E.M.Fs. at receiver circuit and at generator circuit, 

■fi. V(i + r,g + X, by + {x,g - rj)' 
and the output in the receiver circuit is, -, 

61. a.) Dependence of the output upon the susceptance of 
the receiver circuit. 

At a given conductance, g, of the receiver circuit, its 
output, /' = £,'a''^, is a maximum, if a' is amaximum; that 
is, when — ^ 

/=! = (! + r,g + + {x.g - 
is a minimum. 

The condition necessary is — 

db ' 

or, expanding, ^_ ^ ^ ^_^ ^ ^_ ^^ _ ^_ ^^_^ _ ^^^ _ „_ 

Hence — 
Susceptance of receiver drcuit, 

^* + x* ».* 

or i + *, = 0, 

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that is, if the sum of the susceptances of line and of receiver 
circuit equals zero. 

Substituting this value, we get — 

ratio of E.M.Fs. at maximum output, 

maximum output, l • — 


J., ■e.f i.(g^-ji.) 

■• 1+Z.y l + (.r,-jx,)(,s-ji.) 


(1 + ^«• - ». ».) - J (r. K + '.s) ' 




'.g-'.i.)'+ {'■.i. + '.sT' 

and, expanding, 

phase difference in receiver circuit, 

phase difference in generator circuit, 

tan u, = — ■ — - = -SA^ ZS-i . 

62. ^.) Dependence of the output upon the conductance 
of Ou receiver circuit. 

At a given susceptance, b, of the receiver circuit, its 
output, P = E*^g, is a maximum, if — 



■</ g + >•.?+».<)' + (■>.? - '-.')' \ 



that is, expanding, — 

or, expanding, — 

ip + io)* = f-g>*; g= V^o' + (* + *.)'. 
Substituting this value in the equation for a, page i 
we get — 

ratio of E.M.FS., 

^ 1 


?(/ + /.) 2 U. + Vf.'+(^ + *,)•} 

^ j_^ I As a function of the susceptance, b, this power becomes 

v~|^ a maximum for p^/ W JL ^ ^that is, according to § 61, if — 

Substituting this value, we get — 

* = — *D.f =/..^=A. hence; 1'=^ + y^ — fo -y^,; 
x^ — Xn, r^ Kf, z ^ tf, Z =i r — Jx = r, -^ jx,; 

substituting this value, we get — 

ratio of E.M.Fs., «■ =" r^ ^ tt^ i 

power, />„ = ^ ; 

that is, the same as with a continuous<urrent circuit ; or, 
in other words, the inductance of the line and of the receiver 
circuit can be perfectly balanced in its efifect upon the 

63. As_a summary, we thus have : 

The output delivered over an inductive line of impe- 

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dance, Z, = r, —j'x,, into a non-inductive receiver circuit, is 
a maximum for the resistance, r = e,, or conductance, g = 
y,, of the receiver circuit, or — 

p. "' , 

at the ratio of potentials. 

With a receiver circuit of constant susceptance, b, the out- 
put, as a function of the conductance, g, is a maximum for 
the conductance, — 

i--V^,* + (* + *.)', 

and is 

at the ratio of potentials, 

V2^U + f.) 
With a receiver circuit of constant conductance, g, the 
output, as a function of the susceptance, ^, is a maximum 
for the susceptance, b = ~ bo, and is 

p Si£—. 


at the ratio of potentials, 

>-^- i:^<"M] 

The maximum output which can be delivered over an in- 
ductive line, as a function of the admittance or impedance 
of the receiver circuit, takes place when Z = fo+J^ai °^ 
Y = ga~jK', that is, when the resistance or conductance 
of receiver circuit and line are equal, the reactance or sus- 
ceptance of the receiver circuit and line are equal but of 
opposite sign, and is, P = E^ / 4 /", ■ o"" independent of the 
reactances, but equal to the output of a continuous-current 

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circuit of equal line resistance. The ratio of potentials is, in 
this case, a '^ s^fir^, while in a continuous-cUTTcnt circuit 
it is equal to J. The efficiency is equal to 50 per cent. 

64. As an instance, in Fig. 68 are shown, for the 
' constants — 

E, = 1000 volts, and Z, = 2.6 — 6/; that is, for 

r, = 2.5 ohms, Xg = Oohms, x, = 6.6 ohms, 

and with the variable conductances as abscissae, the values 

of the — 

output, in Curve I., Curve lU., and Curve V. j 

ratio of potentials, in Curve II., Curve IV., and Curve VI.; 

Curves I. and II. refer to a non-inductive receiver 
circuit ; 

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12 S*^.,^ 







'um iwi Am ''818 iix3sn asr SToan 'aTssri 

'xuoA iKJH JO ANVdiioa somunsNi 3jn ifninii 3h 



ri'"*%tffci<imil lipj r-'rl-inrr The ratio of potentials is, in 

I ^_i^ -~^^-^- 

_ E.'^^_ ~ „ ^i ^3 ^^CA-y^J^ je. 

S-r---'"^ at ,7iT~ — -?4 ^-_' "^^-^ 

,. p _ R.; 32. E^ S J) 



Curves III. and IV. refer to a receiver circuit of ] 

constant susceptance i = .142 - I n^j 

Curves V. and VI. refer to a receiver circuit of | 

constant susceptance ^ = — .142 ; 

Curves VII, and VIII. refer to a non-inductive re- I 

ceiver circuit and non-inductive line. j 

In Fig. 59, the output is shown as Curve I., and the 

ratio of potentials as Curve II., for the same line constants, 

for a constant conductance, g = .0592 ohms, and for variable 

susceptances, b, of the receiver circuit. 

/"\ '.-iSS^ 

7"'i " "''°' 


^ ^ 

i I 

1 t 



^ V 

^2 i - 

4 \ 


A % 

A s^ 

,^t ^^^ 

1 ^ ^-^ 

4 ^\ ^-_ 

•v^ ~ 

BU CI T*i cetiF lo ivi ro HOI it 

S=^ '.3' -\ '-. fl' -t-.l -t-.i-' 4-3 4-.4 ' 


Flf. 69. Variation of Polaiitlal In Unt at Varlot 

3.) Maximum. Efficiency. 
65. The output, for a given conductance, g, of a receiver 
circuit, is a maximum if i = — *,. This, however, is gen- 
erally not the condition of maximum efficiency. 

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The loss of energy in the line is constant if the current 
is constant ; the output of the generator for a given cur- 
rent and given generator E.M.F. is a maximum if the cur- 
rent is in phase with the E.M.F. at the generator terminals. 
Hence the co nditi on of maximum output at given loss, or 
of maximum efficiency, is — 

= 0. 


The current is — 


The current /,, is in phase with the E.M.F., Eg, if its 
quadrature component — that is, the imaginary term — dis- 
appears, or 

This, therefore, is the condition of maximum efficiency, 

Hence, the condition of maximum efficiency is, that the 
reactance of the receiver circuit shall be equal, but of oppo- 
site sign, to the reactance of the line, ^r- t Q "uf^ J, 

Substituting x = — x„ we have, I &«tb«^ ,**«-'f'"*^ 

iofKM.Fs., \f»- 

-E„ (/- + .„) (r + r„) ■ 


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and depending upon the resistance only, and not upon the 

This power is a maximum it £" = £",, as shown before; 
hence, substituting g = g,, r =• r,, 

maximum power at maximum efficiency, /^ =■ -j^- , 

at a ratio of potentials, a„ = —2— , 

or the same result as in § 62. 

Fig. to. lootf Oliaraiittriitic of Tnuimlaalon lint. 

In Fig. 60 are shown, for the constants — 
J-, = 1,000 volts, 
Z, = 2.5 — 6/; r, = 2.5 ohms, .r„ = 6 ohms, 2^ = 6.5 ohms, 

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and with the variable conductances,^, of the receiver circuit 
as abscissx, the — 

Output at maximum efficiency, (Curve I.) ; 

Volts at receiving end of line, (Curve II.) ; 

Efficiency = — - — , (Curve III.). 

4.) Control of Receiver Voltage by Shunted Susceptance. 

66. By varying the susceptance of the receiver circuit, 
the potential at the receiver terminals is varied greatly. 
Therefore, since the susceptance of the receiver circuit can 
be varied at will, it is possible, at a constant generator 
E.M.F., to adjust the receiver susceptance so as to keep 
the potential constant at the receiver end of the line, or to 
vary it in any desired manner, and independently of the 
generator potential, within certain limits. 

The ratio of E.M.Fs. is — 


» V(l + r,g -^ x,i>f + (x,g - r,bf 
If at constant generator potential £",, the receiver potential 
E shall be constant, 

a = constant ; 

or, expanding, 


h + \^{^)-U+^.)', 

which is the value of the susceptance, d, as a function of 
the receiver conductance, — that is, of the load, — which is 
required to yield constant potential, aE^, at the receiver 

For increasing g, that is, for increasing load, a point is 
reached, where, in the expression — 

d= -fi^ + 


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the term under the root becomes imaginary, and it thus 
becomes impossible to maintain a constant potential, aE^. 
Therefore, the maximum output which can be transmitted 
at potential a£'(„ is given by the expression — 


hence ^ = — Joi the susceptance of receiver circuit, 

KoA g = — Sb + "^ > the conductance of receiver circuit ; 

= a" eA— ~ g^ , the output 

67. If a = 1, that is, if the voltage at the receiver cir- 
cuit equals the generator potential — 
^ =J'c-goi 

If a =\ when g=0, i = 
when f > 0, 3 < ; 
if a >\ whenf = 0, or ^> 0, * < 0, 

that is, condensance ; 
if a < 1 when ^ = 0, b>0, 

when g=:-g„ + v/f— Y- *"'' *"°3 



whenf > -go + yJ{^\~h\ *<0. 

or, in other words, if u < 1, the phase difference in the main 
line must change from lag to lead with increasing load. 

68. The value of a giving the maximum possible output 
in a receiver circuit, is determined by dP j da = 0; 

expanding: 2a(^ — ge\ ^ = j 

hence, y, = 2 "go, 



the maximum output is determined by — 

g = - g<, ■\- ^ ■^ g<,S 

and IS, P= —^ . 


From : « = JIs. = Js_ 

the line reactance, x^, can be found, which delivers a 
maximum output into the receiver circuit at the ratio of 
potentials, a, 

and s„=i 2r,«, 

for a = 1, 

X, = r, V4 a' - 


If, therefore, the 
resistance, the maxi 
mitted into the recei 

ine impedance equals "2, a times the Hne 

imum output, P= E^j^r^, is trans- 

iver circuit at the ratio of potentials, a. 

It Sg = 2r^, or x^ = r^ V3, the maximum output, P = 

E^jAr^, can be supplied to the receiver circuit, without 

change of potential at the receiver terminals. 

Obviously, in an analogous manner, the law of variation 
of the susceptance of the receiver circuit can be found which 
is required to increase the receiver voltage proportionally to 
the load ; or, still more generally, — to cause any desired 
variation of the potential at the receiver circuit indepen- 
dently of any variation of the generator potential, as, for in- 
stance, to keep the potential of a receiver circuit constant, 
even if the generator potential fluctuates widely, 

69. In Figs. 61, 62, and 63, are shown, with the output, 
P = E}gc?, as abscissa;, and a constant impressed E.M.F., 
E^ = 1,000 volts, and a constant Hne impedance, Z^ = 
2.5 — Qj, or, r, = 2.5 ohms, .r„ = 6 ohms, s = 6.5 ohms, 
the following values : 

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r. T-^' 

"Mi -^='ff^''' 

£- V3» -*<- 7*r 



TIO'OF {.loilVli vol 






1 1 



' s;i'.rH"x..«.uot.v, »o»v» ««<«T w.T««T oo-««i*T.<,r~ 



































































•— ■ 
















n,. a 


















■— ■ 





















' ' 




I, I 


































TJ«WsvUa<^.<,L...' J. 1_L 

lei^^f "cSS''"'"' C^NSTABT O.NIm™ POT.HTUL s'.-K^OO 




























































^{|. 03. KorftKrcHT 0/ Voltagt Trtuitmlaalon Untt. 

Energy component of current, gE, (Curve I.) ; 

Reactive, or wattless component of current, bE, (Curve II,) ; 
Total current, yE, (Curve III.); 

for the following conditions : 

a = 1.0 (Fig. 61) ; = .7 (Fig. 6: 

= 1.3 (Fig. ( 

For the non-inductive receiver circuit (in dotted lines), 
the curve of E.M.F., E, and of the current, I = gE, are 
added in the three diagrams for comparison, as Curves IV. 
and V. 

As shown, the output can be increased greatly, and the 
potential at the same tiine maintained constant, by the judi- 
cious use of shunted reactance, so that a much larger out- 
put can be transmitted over the line at no drop, or even at 
a rise, of potential. 

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6.) Maximum Rise ^/ Potential at Receiver Circuit: 

70. Since, under certain circumstances, the potential at 

the receiver circuit may be higher than at the generator, 

it is of interest to determine what is the maximum value of 

potential, E, that can be produced at the receiver circuit 

with a given generator potential. Eg. 

The condition is that 

a — maximum or -^ = minimum ; 

that is, 

iig ' di 


^ = (1 + '■of + *. ^y + (^,/ - 'V l>)\ 

and expanding, we get, 

— a value which is impossible, since neither r„ nor g can be 
negative. The next possible value is g = 0, — a wattless 

Substituting this value, we get, 

and by substituting, in 

db ' z* " f 

i + j5„ = 0i - - -^C ^^ 

that is, the sum of the susceptances « 0, or the condition 
of resonance is present. 


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The current in this case is, 


or the same as if the line resistance were short-circuited 
without any inductance. 

This is the condition of perfect resonance, with current 
and E.M.F. in phase. 

X \ 

«s' "sN 

» ^^ 

~ 3-^ 

" it 5 

Z X ^t 

Z -:"'i-T3USia'zK!?j ^ 

Z 1 

^ ^ te sSftiF aN«^ -- ' i 

^>-^ ~%_ 

^ \*' 

W ' z ^ 

|r / 


Z /I ^J^ 

.- 4^ ^i^ 

» y^a^ 


Pa. 04. CfidtiKg onrf Output af 

71. As summary to this chapter, in Fig. 64 are plotted, 
for a constant generator E.M.F., E^ = 1000 volts, and a 
hne impedance, Z, = 2.5 — 6j, or, r, = 2.5 ohms, .r, = 6 
ohms, £, = 6.5 ohms ; and with the receiver output as 

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abscissae and the receiver voltages as ordinates, curves 
representing — 

the condition of maximum output, (Curve I.) j 

the condition of maximum efficiency, (Curve II.) ; 

the condition i — 0, or a non-inductive receiver cir- 
cuit, (Curve III.); 

the condition ^ = 0, ^„ = 0, or a non-inductive line and non- 
inductive receiver circuit. 

In conclusion, it may be remarked here that of the 
sources of susceptance, or reactance, 
a choking coil or reactive coil corresponds to an inductance ; 
a condenser corresponds to a condensance ; 

a polarization cell corresponds to a condensance ■ 

a synchronizing alternator (motor or generator) corresponds to 

an inductance or a condensance, at will ; 
an induction motor or generator corresponds to an inductance. 

The choking coil and the polarization cell are specially 
suited for series reactance, and the condenser and syn- 
chronizer for shunted susceptance. 




"""''^ -^'v 

. B-. 

.. n 

. JSL 


r.- ... '<">» 

T =..«'»°"^- 



, W.wl 

b-t-'O .-. K--J 

«,., .-p^ i"'--y- 

"•■"J <^<*i-'%Y 





72. The resistance of an electric circuit is determined : — 

1.) By direct comparison with a known resistance (Wheat- 
stone bridge method, etc.). 

This method gives what may be called the true ohmic 
resistance of the circuit. 

2.) By the ratio : 

Volts consumed in circuit 
Amperes in circuit 

In an alternating-current circuit, this method gives, not 
the resistance of the circuit, but the impedance, 

3.) By the ratio : 

Power consumed , 
(Current)* ' 
where, however, the " power " does not include the work 
done by the circuit, and the counter E.M.Fs. representing 
it, as, for instance, in the case of the counter E.M.F. of a 

In alternating-current circuits, this value of resistance is 
the energy coefficient of the E.M.F., 

_ Energy component of EM.F. 
Total current 
It is called the effective resistance of the circuit, since it 
represents the effect, or power, expended by the circuit. ■ 
The energy coefficient of current, 

__ Energy component of current 
^ Total E.M.F. ' 

is called the effective conductance of the circuit 

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In the same way, the value, 

Wattless component of RM.F. 
Total current 
is the effective reactance, and 

, Wattless component of current 
" Total E.M.F. 

is the effective susceptance of the circuit. 

While the true ohmic resistance represents the expendi- 
ture of energy as heat inside of the electric conductor by a 
current of uniform density, the e ffective resistance repre- 
sents the to tal expenditure of energy . 

Since, in an alternating-current circuit in general, energy 
is expended not only in the conductor, but also outside of 
it, through hysteresis, secondary currents, etc., the effective 
resistance frequently differs from the true ohmic resistance 
in such way as to represent a larger expenditure of energy. 

In dealing with alternating-current circuits, it is necessary, 
therefore, to substitute everywhere the values "effective re- 
sistance," "effective reactance," "effective conductance," 
and " effective susceptance," to make the calculation appli- 
cable to general alternating-current circuits, such as induc- 
tances, containing iron, etc. 

While the true ohmic resistance is a constant of the 
circuit, depending only upon the temperature, but not upon 
the E,M.F., etc., the effective resistance and effective re- 
actance are, in general, not constants, but depend upon 
the E.M,F., current, etc. This dependence is the cause 
of most of the difficulties met in dealing analytically with 
alternating-current circuits containing iron. 

73. The foremost sources of energy loss in alternating- 
current circuits, outside of the true ohmic resistance loss, 
are as follows : 

1.) Molecular friction, as, 

«.) Magnetic hysteresis; 
b.) Dielectric hysteresis. 

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2.) Primary electric currents, as, 

a.) Leakage or escape of current through the insu- 
lation, brush discharge ; d.) Eddy currents in 
the conductor or unequal current distribution. 
3.) Secondary or induced currents, as, 

a.) Eddy or Foucault currents in surrounding mag- 
netic materials ; i.) Eddy or Foucault currents 
in surrounding conducting materials ; c ,) Sec- 
ondary currents of mutual inductance in neigh- 
boring circuits, 
4.) Induced electric charges, electrostatic influence. 
While all these losses can be included in the terms effec- 
tive resistance, etc., only the magnetic hysteresis and the 
eddy currents in the iron will form the subject of what fol- 
lows, since they are the most frequent and important sources 
of energy loss. 

Magnetic Hysteresis. 
74. In an alternating-current circuit surrounded by iron 
or other magnetic material, energy is expended outside of 
the conductor in the iron, by a kind of molecular friction, 
which, when the energy is supplied electrically, appears as 
magnetic hysteresis, and is caused by the cyclic reversals of 
magnetic flux in the iron in the alternating magnetic field. 

To examine this phenomenon, first a circuit may be con- 
sidered, of very high inductance, but negligible true ohmic 
resistance ; that is, a circuit entirely surrounded by iron, as, 
for instance, the primary circuit of an alternating-current 
transformer with open secondary circuit. 

The wave of current produces in the iron an alternating 
magnetic flux which induces in the electric circuit an E.M.F., 
— the counter E.M.F. of self-induction. If the ohmic re- 
sistance is negligible, that is, practically no E.M.F. con- 
sumed by the resistance, all the impressed E.M.F. be 
consumed by the counter E.M.F. of self-induction, that is, 
the counter E.M.F. equals the impressed E.M.F. ; hence, if 

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the impressed E.M.F. is a sine wave, the counter E.M.F., 
and, therefore, the magnetic flux which induces the counter 
E.M.F. must follow a sine wave also. The alternating wave 
of current is not a sine wave in this case, but is distorted 
by hysteresis. It is possible, however, to plot the current 
wave in this case from the hysteretic cycle of magnetic flux. 
From the number of turns, «, of the electric circuit, 
the effective counter E.M.F., E, and the frequency, N, 
of the current, the maximum magnetic flux, 4, is found 
by the formula : 

£= V2»-«A'*10-'; P/^ 

hence, ^ ^ .glO' 

A maximum flux, *, and magnetic cross-section, S, give 
the maximum magnetic induction, & = * / S. 

If the magnetic induction varies periodically between 
+ (B and — &, the M.M.F. varies between the correspond- 
ing values + SF and — iF, and describes a looped curve, the 
cycle of hysteresis. 

If the ordinates are given in lines of magnetic force, the 
abscissas in tens of ampere-turns, then the area of the loop 
equals the energy consumed by hysteresis in ergs per cycle. 

From the hysteretic loop the instantaneous value of 
M.M.F. is found, corresponding to an instantaneous value 
of magnetic flux, that is, of induced E.M.F. ; and from the 
M.M.F., if, in ampere-turns per unit length of magnetic cir- 
cuit, the length, /, of the magnetic circuit, and the number of 
turns, «, of the electric circuit, are found the instantaneous 
values of current, (', corresponding to a M.M.F., SF; that is, 
magnetic induction <B, and thus induced E.M.F. e, as: 
. J/ 

76. In Fig. 65, four magnetic cycles are plotted, with 
maximum values of magnetic inductions, (B = 2,000, 6,000, 
10,000, and 16,000, and corresponding maximum M.M.Fs., 

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ff = 1.8, 2.8, 4.S, 20.0. They show the well-known hys- 
teretic loop, which becomes pointed when magnetic satu- 
ration is approached. 

These magnetic cycles correspond to average good sheet 
iron or sheet steel, having a hysteretic coefficient, ij = .0033, 
and are given with ampere-turns per cm as abscissae, and 
kilo-lines of magnetic force as ordinates. 


1 1 














f^ i 




\ / 




, 1 



^ 1 




















11 X 







n^ SA Hgittntlc Ctel* vf W**t rrsn 

In Figs. 66, 67, 68, and 69, the curve of magnetic in- 
duction as derived from the induced E.M.F. is a sine wave. 
For the different values of magnetic induction of this sine 
curve, the corresponding values of M.M.F., hence of current, 
are taken from Fig. 65, and plotted, giving thus the exciting 
current required to produce the sine wave of magnetism; 
that is, the wave of current which a sine wave of impressed 
E.M.F. will send through the circuit. 

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As shown ill Figs. 66, 67, 68, and 69, these waves of 
alternating current are not sine waves, but are distorted by 
the superposition of higher harmonics, and are complex 
harmonic waves. They reach their maximum value at the 
same time with the maximum of magnetism, that is, dO'^ 

rigt. ee ana er. eutwtlon of Camnt Want in Hyaltrtila. 

ahead of the maximum induced E.M.F., and hence about 
90" behind the maximum impressed E.M.F., but pass the 
zero line considerably ahead of the zero value of magnet- 
ism, or 42°, 52°, 50°, and 41°, respectively. 

The general character of these current waves is, that the 
1 point of the wave coincides in time with the max- 

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imum point of the sine wave of magnetism ; but the current 
wave is bulged out greatly at the rising, and hollowed in at 
the decreasing, side. With increasing magnetization, the 
maximum of the current wave becomes more pointed, as 
shown by the curve of Fig. 68, for <S> = 10,000 ; and at still 











■ 4. 



















































































ng*. W uiicf B8. DiMiortKHi ef Carmi Wcae hv HgHtrula. 

higher saturation a peak is formed at the maximum point, 
as in the curve of Fig. 69, for (B = 16,000, This is the case 
when the curve of magnetization reaches within the range of 
magnetic saturation, since in the proximity of saturation the 
current near the maximum point of magnetization has to 
rise abnormally to cause even a small increase of magneti- 
zation. The four curves. Figs. 66, 67, 68, and 69, are not 
drawn t.) th;} sani^ seal*;. The maximum values of M.M.F., 

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corresponding to the maximum values of magnetic induction, 
<& = 2,000, 6,000, 10,000, and 16,000 lines of force per cm', 
arc S7 = 1.8, 2.8, 4.3, and 20.0 ampere-turns per cm. In 
the different diagrams these are represented in the ratio of 
8 : 6 : 4 : 1, in order to bring the current curves to approxi- 
mately the same height. The M.M.F., in C.G.S. units, is 
^=4x/10SF = 1.267 !F. 

76. The dbtortion of the wave of magnetizing current 
is as large as shown here only in an iron-closed magnetic 
circuit expending energy by hysteresis only, as in an iron- 
clad transformer on open secondary circuit. As soon as the 
circuit expends energy in any other way, as in resistance, or 
by mutual inductance, or if an air-gap is introduced in the 
magnetic circuit, the distortion of the current wave rapidly 
decreases and practically disappears, and the current becomes 
more sinusoidal. That is, while the distorting component 
remains the same, the sinusoidal component' of the current 
greatly increases, and obscures the distortion. For example, 
in Figs. 70 and 71, two waves are shown, corresponding in 
magnetization to the curve of Fig. 67, as the one most 
distorted. The curve in Fig. 70 is the current wave of a 
transformer at -^ load. At higher loads the distortion is 
correspondingly still less, except where the magnetic flux of 
self-induction, that is, flux passing between primary and sec- 
ondary, and increasing proportionally to the load, is so large 
as to reach saturation, in which case a distortion appears 
again and increases with increasing load. The curve of Fig. 
71 is the exciting current of a magnetic circuit containing 
an air-gap whose length equals ^Jo the length of the magnetic 
circuit. These two curves are drawn to ^ the size of the cur\'e 
in Fig. 67. As shown, both curves are practically sine waves. 
The sine curves of magnetic flux are shown dotted as ^. 

77. The distorted wave of current can be resolved into 
two components : A true sine wave of equal effective intensity 

/'mi ryal fowcr to the distorted wave, called the equivaletit 

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sine wave, and a wattless higher harmonic, consisting chiefly 
of a term of triple frequency. 

In Figs. 66 to 71 are shown, as /, the equivalent sine 
























































1 — ■ 








— ' 

















Fign. 70 anil 71. Dlttariltm of Barn 

t WOM it H^lrttlt. 

waves and as (, the difference between the equivalent sine 
wave and the real distorted wave, which consists of wattless 
complex higher harmonics. The equivalent sine wave of 
M.M.F. or of current, in Figs. 66 to 69, leads the magnet- 

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ism by 34°, 44°, 38", and 15°.5, respectively. In P'ig. 71 
the equivalent sine wave almost coincides with the distorted 
curve, and leads the magnetism by only 9°, 

It is interesting to note, that even in the greatly dis- 
torted curves of Figs. 66 to 68, the maximum value of the 
equivalent sine wave is nearly the same as the maximum 
value of the original distorted wave of M.M.F., so long as 
magnetic saturation is not approached, being 1.8, 2.9, and 
4.2, respectively, against 1.8, 2.8, and 4.3, the maximum 
values of the distorted curve. Since, by the definition, the 
effective value of the equivalent sine wave is the same as 
that of the distorted wave, it follows, that this distorted 
wave of exciting current shares with the sine wave the 
feature, that the maximum value and the effective value 
have the ratio of V'2 -*• 1. Hence, below saturation, the 
maximum value of the distorted curve can be calculated 
from the effective value — which is given by the reading 
of an e]ectro.dynamometer . — by using the same ratio that 
applies to a true sine wave, and the magnetic characteris- 
tic can thus be determined by means of alternating cur- 
rents, with sufficient exactness, by the electro-dynamometer 
method, in the range below saturation. 

78. In Fig. 72 is shown the true magnetic character- 
istic of a sample of good average sheet iron, as found by 
the method of slow reversals with the magnetometer ; for 
comparison there is shown in dotted lines the same char- 
acteristic, as determined with alternating currents by the 
electro-dynamometer, with ampere-turns per cm as ordi- 
nates, and magnetic inductions as abscissas. As repre- 
sented, the two curves practically coincide up to a value of 
(B = 13,000 ; that is, up to the highest inductions practicable 
in alternating-current apparatus. For higher saturations, 
the curves rapidly diverge, and the electro-dynamometer 
curve shows comparatively small M.M.Fs. producing appar- 
ently very high magnetizations. 

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The same Fig. 72 gives the curve of hysteretic loss, in 
ergs per cm* and cycle, as ordinates, and magnetic induc- 
tions as abscissx. 















































y ^ 




«0 T/ 




fjg. 72. 

The electro-dynamometer method of determining the 
magnetic characteristic is preferable for use with alter- 
nating-current apparatiTs, Eince it is not affected by the 
phenomenon of marjnetic "creeping," which, especially at 

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low densities, may in the magnetometer tests bring the mag- 
netism very much higher, or the M.M.F. lower, than found 
in practice in alternating-current apparatus. 

So far as current strength and energy consumption are 
concerned, the distorted wave can be replaced by the equi- 
valent sine wave, and the higher harmonics neglected. 

All the measurements of alternating currents, with the 
single exception of instantaneous readings, yield the equiv- 
alent sine wave only, and suppress the higher harmonic ; 
since all measuring instruments give either the mean square 
of the current wave, or the mean product of instantaneous 
values of current and E.M.F., which, by definition, are the 
same in the equivalent sine wave as in the distorted wave. 

Hence, in all practical applications, it is perrnissible to 
neglect the higher harmonic altogether, and replace the dis- 
torted wave by its equivalent sine wave, keeping in mind, 
however, the existence of a higher harmonic as a possible 
disturbing factor which may become noticeable in those cases 
where the frequency of the higher harmonic is near the fre- 
quency of resonance of the circuit, that is, in circuits con- 
taining capacity besides the inductance. 

79. The equivalent sine wave of exciting current leads 
the sine wave of magnetism by an angle o, which is called 
the angle of hysteretic advance of phase. Hence the cur- 
rent lags behind the E.M.F by ^ 90° — a, and the power 
is therefore, y. = /^ eos (90^ - a) = /£ sin «. 

Thus the exciting current, /, consists of an energy compo- 
nent, / sin a, called the hysteretic or magnetic energy current, 
and a wattless component, / cos o, which is called the mag- 
netising current. Or, conversely, the E.M.F. consists of an 
energy component, E sin a, the hysteretic energy E.M.F., 
and a wattless component, E cos o, the E,M.F. of self- 

Denoting the absolute value of the impedance of the 

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circuit, Ell, by :;, — where s is determined by the mag- 
netic characteristic of the iron, and the shape of the 
magnetic and electric circuits, — the impedance is repre- 
sented, in phase and intensity, by the symbolic expression, 

Z = r-yj:=(asino-y3C0Sa;)E^ 
and the admittance by, 

Y = g -\-jb = - sin a -|-y-COSai=_l'sino -|-y^cosa. 

The quantities, s, r, x, and y, g, b, are, however, not 
•constants as in the case of the circuit without iron, but 
■depend upon the intensity of magnetization, (B, — that is, 
upon the E.M.F. This dependence complicates the investi- 
gation of circuits containing iron. 

In a circuit entirely inclosed by iron, a is quite consider- 
able, ranging from 30° to 50° for values below saturation. 
Hence, even with negligible true ohmic resistance, no great 
lag can be produced in ironclad alternating-current circuits. 

80. The loss of energy by hysteresis due to molecular 
3tL'-vx'>^~*f^ friction is, with sufficient exactness, proportional to the 
T'o-L-''*^ A l.S"' power of magnetic induction (B. Hence it can be ex- 
S-v,»,_",— pressed by the formula: 

where — 

Wa = loss of energy per cycle, in ergs or (C.G.S.) units (= 10~ ' 
Joules) per cm', 

<B = maximum magnetic induction, in lines of force per cm'^, and 

17 ^ the coefficient of liyslerish. 

This I found to vary in iron from .00124 to .0055. As a 
fair mean, .0033 • can be accepted for good average annealed 
sheet iron or sheet steel. In gray cast iron, j; averages 
,013 ; it varies from .0032 to .028 in cast steel, according 
to the chemical or physical constitution ; and reaches values 
as high as .08 in hardened steel (tungsten and manganese 

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Steel). Soft nickel and cobalt have about the same co- 
efficient of hysteresis as gray cast iron ; in magnetite I 
found 7 = .023. 

In the curves of Fig. 62 to 69, 7 = .0033. 
At the frequency, N, the loss of power in the volume, V, 
is, by this formula, — 

/•= 7 Afros'-' 10-^ watts (f*- W^.-^B'*} 
= ,A'F^±Y''lO-» watts, ^/*»a^ ^^^,*4- 

where S is the cross-section of the total magnetic flux, *. 

The maximum magnetic flux, *, depends upon the 
counter E.M.F. of self-induction. 

or * = — ^— — , 

V2 TT Nn 

where « = number of turns of the electric circuit. 

Substituting this in the value of the power, P, and 
canceling, we get, — 

E** t^lQ'-' ' -„ £'■• flO" 

D AE '■' . . no '■* CO ^10' 

" ^- -N^- "'■"' -^ - " 2..^..^...„... - '^■'jTi^.' 

or, substituting ij = .0033, we have A = 191.4 ; 

or, substituting F= ^Z, where L = length of magnetic circuit, 

.f^ .jZlO'-* ^ 58 "J-^: 10* ^1911 X . 

2.»^..^.<i«i.. ^-Ofti-' ■ ^•«„i-«' 

and J, _ 58 „ g '-* Z 10* _ 191.4 .g'-'Z 

In Figs. 73, 74, and 75, is shown a curve of hysteretic 
loss, with the loss of power as ordinates, and 

in curve 73, with the E.M.F., M, as abscissae, for Z = 6, 
S = 2(i,N= 100, and « = 100 ; 

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fTg. 7S. HyH*mli lOM at AmrtrDii of E. K. f. 























■ N 









' » 



F/4. 74, HiiMrali Lata la Functtm of Hambtr of Tun 

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1 1 1 1 1 1 1 1 1 1 







T ^ 


















Ftj. 75, Hgatmilt Lota at function of Oyaloa. 

in curve 74, with the number of turns as abscissae, for 
Z = 6,S=20, N= 100, and £ = 100 ; 

in curve 76, with the frequency, N, or the cross-section, S, 
as abscissae, for Z = 6, « = 100, and £ = 100. 

As shown, the hysteretic loss is proportional to the 1.6* 
power of the E.M.F., inversely pro[>ortional to the 1,6* 
power of the number of turns, and inversely proportional to 
the .6* power of frequency, and of cross-section. 

81. If ^ = effective conductance, the energy comfK)- 
nent of a current is / = Eg, and the energy consumed in 
a conductance, g, is P = IE = E^g. 

Since, however : 



, we have A-- 


- = S?g; 

iL^ ^191.4 — 

From this we have the following deduction ; 

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The effective conductance due to magnetic hysteresis is 
proportional to the coefficient of hysteresis, ij, and to the letigtk 
of the magnetic circuit, L, and inversely proportional to the 
^'^ power of the E.M.F., to the .6'* power of the frequency, 
N, and of the cross-section of tlte magnetic circuit, S, and to 
t/te 1.6'^ power of the number of turns, n. 

Hence, the effective hysteretic conductance increases 
with decreasing E.M.F., and decreases with increasing 

: 1 1 1 1 1 1 1 1 1 1 


« ^ 

; 5 

» i^ 

u "^^ 


" 1 "- — — _ 





Hlittimli CdnduetaRC* oa Fynetltm of C.M.f. 

E.M.F. ; it varies, however, much slower than the E.M,F, 
so that, if the hysteretic conductance represents only a part 
of the total energy consumption, it can, within a limited 
range of variation — as, for instance, in constant potential 
transformers — be assumed as constant without serious 

In Figs. 76, 77, and 78, the hysteretic conductance, g, is 
plotted, for Z; = 6, £ = 100, N = 100, 5 = 20 and « = 100, 
respectively, with the conductance, g, as ordinates, and with 

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1 1 1 1 1 1 

















ffg^ 77. Hi/rleretlM Conductane* as Funttloa of Ci/tlta. 

































FuHCilm af Haiabtr (>f Tktm 



E as abscissae in Curve 76, 
iVas abscissffi in Curve 77. 
n as abscissEe in Curve 78. 

As shown, a variation in the E.M,F. of 50 per cent 
causes a variation in g of only 14 per cent, while a varia- 
tion in iV or 5 by 50 per cent causes a variation in g of 21 
per cent. 

If (R = magnetic reluctance of a circuit, SF^ = maximum 
M.M.F., / = effective current, since / V2 = maximum cur- 
rent, the magnetic flux, 

6t (SI 

Substituting this in the equation of the counter E.M.F. of 

we have E = 


hence, the absolute admittance of the circuit is 

■^ ^ ^ E 2^«W N' '^* 

where a = - — ^ , a constant. 

Therefore, the absolute admittance, y, of a circuit of neg- 
ligible resistance is proportional to the magnetic reluctance, (R, 
and inversely proportional to the frequency, N, and to the 
square of t/te number of turns, n. 

82, In a circuit containing iron, the reluctance, fll, varies 
with the magnetization; that is, with the E.M.F. Hence 
the admittance of such a circuit is not a constant, but is 
also variable. 

In an ironclad electric circuit, — that is, a circuit whose 
magnetic field exists entirely within iron, such as the mag- 
netic circuit of a well-designed alternating-current trans- 

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former, — (R is the reluctance of the iron circuit. Hence, 
if ;* = permeability, since — 

and JF, = LF = 

•fr = S(& = fi SX = magnetic flux, 

substituting this value in the equation of the admittance, 
(RIO* . ZIO* t 

h = ZIP* ^ 127Z 10* 

Therefore, in an ironclad circuit, the absolute admittance, 
y, is inversely proportional to the frequency, N, to the perme- 
ability, ^, to the cross-section, S, and to tke square of the 
number of turns, n ; and directly proportional to the length 
of tltc magnetic circuit, L. 

The conductance is g = — ; — - ; 

and the admittance, v = — ■ ; 

hence, the angle of_hysteretic advance is 

Sina = il = -^tt^; ">MS- 

or, substituting for A and s (p. 117), 

N-* 7 ZIO" 8»'««5 
sin "i^^ "FT T ru cTm t m» > 

E* 10»» 
or, substituting 

we have sino = —^-^, 

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which is independent of frequency, number of turns, and 
shape and size of the magnetic and electric circuit. 

Therefore, in an ironclad inductance, the angle of ky Heretic 
advance, a., depends upon the magnetic constants, permeability 
and coefficient of hysteresis, and upon the maximum magnetic 
induction, but is entirely independent of the frequency, of the 
sltape and other conditions of the magnetic and electric circuit ; 
and, therefore, all ironclad magnetic circuits constructed of the 
same quality of iron and using the same magnetic density, 
give the same angle of hysteretic adz'ance. 

The angle of hysteretic advance, a, in a closed circuit 
transformer, depends upon the quality of the iron, and upon 
the magnetic density only. 

The sine of the angle of hysteretic advance equals 4- I'mes 
the product of the permeability and coefficient of hysteresis, 
divided by the .^'* power of the magnetic density. 

83. If the magnetic circuit is not entirely ironclad, 
and the magnetic structure contains air-gaps, the total re- 
luctance is the sum of the iron reluctance and of the air 
reluctance, or 

CR = (Hi -I- CR, ; 

hence the admittance is 

.y^ VpTT'=~(^< + <R-)■ 
7'^fr^r^?, in a circuit containing iron, the admittance ts 
the sum of the admittatue due to the iron part of the circuit, 
yi = a<Ri/ N, and of the admittance due to the air part of the 
circuit, ya — ^'^a/ ^' if ^^^ "''"' "^ l^^ '"'' '^''^ "• ^^*i^^ "* 
the magnetic circuit. 

The conductance, g, represents the loss of energy in 
the iron, and, since air has no magnetic hysteresis, is not 
changed by the introduction of an air-gap. Hence the 
angle of hysteretic advance of phase is 

sin a = -^ = — £ — = -^ i — , 

y yi-\-y^ 7* («< + «« 

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and a maximum, gjy, , for the ironclad circuit, but decreases 
with increasing width of the air-gap. The introduction of 
the air-gap of reluctance, ifla, decreases sin a in the ratio, 

S^ + l^a 

In the range of practical application, from (B = 2,000 to 
(B = 12,000, the permeability of iron varies between 900 
and 2,000 approximately, while sin a in an ironclad circuit 
varies in this range from .51 to .69. In air, ^ = 1. 

If, consequently, one per cent of the length of the iron 
consists of an air-gap, the total reluctance only varies through 
the above range of densities in the proportion of IJ to \^g, 
or about 6 per cent, that is, remains practically constant ; 
while the angle of hysteretic advance varies from sin a = .035 
to sin o = .064. Thus g is negligible compared with b, and 
b is practically equal to y. 

Therefore, in an electric circuit containing iron, but 
forming an open magnetic circuit whose air-gap is not less 
than jjjf the length of the iron, the susceptance is practi- 
cally constant and equal to the admittance, so long as 
saturation is not yet approached, or, 

The angle of hysteretic advance is small, below 4", and the 
hysteretic conductance is, 

g= -^ . 
. * £-*T^ ^'^"'^ 
The current wave is practicallv,ar^ne wave. 

As an instance, in Fig. 71, Curve II., the current curve 
of a circuit is shown, containing an air-gap of only fjg of 
the length of the iron, giving a current wave much resem- 
bling the sine shape, with an hysteretic advance of 9°. 

84. To determine the electric constants of a circuit 
containing iron, we shall proceed in the following way : 
Let — 

E = counter E.M.F. of self-induction ; 

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then from the equation, 

£= V2,r«A'*10-», 

N = frequency, 

n = number of turns, 

we get the magnetism, *, and by means of the magnetic cross 
section, 5, the maximum magnetic induction : (B = * / i". 

From (B, we get, by means of the magnetic characteristic 
of the iron, the M.M.F., — /? ampere-turns per cm length, 

if 3C = M.M.F. in C.G.S. units. 

if Z( =" length of iron circuit, ^, = L^ F — ampere-turns re- 
quired in the iron ; 

if Za = length of air circuit, ffa = — -~ — = ampere-turns re- 
quired in the air ; 

hence, ?= jFj + ff, = total ampere-turns, maximum value, 

and !F / V2 = effective value. The exciting current is 

and the absolute admittance. 

If £Fj is not negligible as compared with ff„ , this admit- 
tance, ^, is variable with the E.M.F,, E. 

If — 

V = volume of iron, 

ij = coefficient of hysteresis, 
the loss of energy by hysteresis due to molecular magnetic 
friction is, 

hence the hysteretic conductance is ^ — Wj J?, and vari- 
able with the E.M.F., E. 

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The angle of hysteredc advance is, — 

the susceptance, b = V^ — ^; 

the efEective resistance, r = g/^\ 

and the reactance, x = b / ^. 

85. As conclusions, we derive from this chapter the \ 
following : — | 

P/af 1) ^n ^ri altemating-curxent circuit surrounded by iron, , 

' the current produced by a sine wave of E.M.F. is not a true 

sine wave, but is distorted by hysteresis, and inversely, a [ 

sine wave of current requires waves of magnetism and I 

E.M.F. differing from sine shape. 1 

Ph« ^'^ ^^'^ distortion is excessive only with a closed mag- ; 

netic circuit transferring no energy into a secondary circuit r 
by mutual inductance. i 

PiU S7; 3.) The distorted wave of current can be replaced by • 
the equivalent sine wave — that is a sine wave of equal effec- ! 
tive intensity and equal power — and the superposed higher I 
harmonic, consisting mainly of a term of triple frequency, 
may be neglected except in resonating circuits. 

>i(5 4.) Below saturation, the distorted curve of current and 

^di* its equivalent sine wave have approximately the same max- 
imum valye, ' 

>«f ^■) ^''^ angle of hysteretic advance, — that is, the phase 

difference between the magnetic flux and equivalent sine j 
wave of M.M.F., — is a maximum for the closed magnetic 
circuit, and depends there only upon the magnetic constants 
of the iron, upon the permeability, /i, the coefficient of hys- 
teresis, ^ and the maximum magnetic induction, as shown in 
the equation, ^ 


6.) The effect of hysteresis can be represented by an 
admittance, Y = g -^jb, or an impedance, Z = r —jx. 

7.) The hysteretic admittance, or impedance, varies with 
the magnetic induction; that is, with the E.M.F., etc. 

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8.) The hysteretic conductance, g, is proportional to the 
coefficient of hysteresis, ij, and to the length of the magnetic 
circuit, L, inversely proportional to the .4''' power of the 
E.M.F., E, to the .6"" power of frequency, iV, and of the 
cross-section of the magnetic circuit, S, and to the l.e* 
power of the number of turns of the electric circuit, «, as 
expressed in the equation, 

^ 58 t; Z IP* 

9.) The absolute value of hysteretic admittance, — 

y = Vs' + i", 

is proportional to the magnetic reluctance: (R = (B, + CRj, 

and inversely proportional to the frequency, JV, and to the 

square of the number of turns, n, as expressed in the 


^ ((«. + (R^) 10' 



yi = ' 

10.) In an ironclad circuit, the absolute value of admit- 
tance is proportional to the length of the magnetic circuit, 
and inversely proportional to cross-section, S, frequency, J^, 
permeability, it, and square of the number of turns, n, or 
127 Z 10' 

11.) In an open magnetic circuit, the conductance, g, is 
the same as in a closed magnetic circuit of the same iron part. 

12.) In an open magnetic circuit, the admittance,/, is 
practically constant, if the length of the air-gap is at least 
yjo of the length of the magnetic circuit, and saturation be 
not approached. 

13.) In a closed magnetic circuit, conductance, suscep)- 
V\i^ tance, and admittance can be assumed as constant through 

?iw4»>- a limited range only. 

14.) From the shape and the dimensions of the circuits, 
and the magnetic constants of the iron, all the electric con- 
stants, £■, b,y; y, x, z, can be calculated. 

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86, While magnetic hysteresis or molecular friction is 
a magnetic phenomenon, eddy currents are rather an elec- 
trical phenomenon. When iron passes through a magnetic 
field, a loss of energy is caused by hysteresis, which loss, 
however, does not react magnetically upon the field. When 
cutting an electric conductor, the magnetic field induces a 
current therein. The M.M.F. of this current reacts upon 
and affects the magnetic field, more or less ; consequently, 
an alternating magnetic field cannot penetrate deeply into a 
solid conductor, but a. kind of screening effect is produced, 
which makes solid masses of iron .unsuitable for alternating 
fields, and necessitates the use of laminated iron or iron 
wire as the carrier of magnetic flux. 

Eddy currents are true electric currents, though flowing 
in minute circuits ; and they follow all the laws of electric 

Their E.M.F, is proportional to the intensity of magneti- 
zation, (B, and to the frequency, N. 

Eddy currents are thus proportional to the magnetization, 
CB, the frequency, N, and to the electric conductivity, y, of 
the iron ; hence, can be expressed by 
i= ^-i<S>N. 

The power consumed by eddy currents is proportional to 
their square, and inversely proportional to the electric con- 
ductivity, and can be expressed by 


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or, since, (]iA''is proportional to the induced E.M.F., E, in 
the equation 

it follows that, The loss of power by eddy currents is propor- 
tional to the square of the E.M.F., and proportional to the 
electric conductivity of the iron ; or, 
Hence, that component of the effective conductance 
which is due to eddy currents, is 

that is, The equivalent conductance due to eddy currents in 
the iron is a constant of the magnetic circuit ; it is indepen- 
dent of EM.F., frequency, etc., but proportional to the electric 
conductivity of the iron, y. 

87. Eddy currents, like magnetic hysteresis, cause an 
advance of phase of the current by an angle of advance, p ; 
but, unlike hysteresis, eddy currents in general do not dis- 
tort the current wave. 

The angle of advance of phase due to eddy currents is. 

where y = absolute admittance of the circuit, g = eddy 
current conductance. 

While the equivalent conductance, g, due to eddy cur- 
rents, is a constant of the circuit, and independent of 
E.M.F., frequency, eta, the loss of power by eddy currents 
is proportional to the square of the E.M.F. of self-induction, 
and therefore proportional to the square of the frequency 
and to the square of the magnetization. 

Only the energy componenti^f, of eddy currents, is of 
interest, since the wattless component is identical with the 
wattless component of hysteresis, discussed in a preceding 

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88. To calculate the loss of power by eddy currents — 
Let f — volume of iron ; 

(B = maximum magnetic induction ; 
N s= frequency; 

y = electric conductivity of iron ; 
( = coefficient of eddy currents. 

The loss of energy per cm', in ergs per cycle, is 
w= tyN&\ 
hence, the total loss of power by eddy currents is 

Jf = €yri\'*(B' 10-' watts, 
and the equivalent conductance due to eddy currents is 

W 10. y/ 


' 1? i^s<f 



/ = length of magnetic circuit, 
S = section of magne^c circuit, 
n = number of turns of electric circuit. 

The coefficient of eddy currents, «, 
depends merely upon the shape of the 
constituent parts of the magnetic cir- 
cuit ; that is, whether of iron plates 
or wire, and the thickness of plates or 
the diameter of wire, etc. 

The two most important cases are : 
(a). Laminated iron. 
(*). Iron wire. 

89. {a). Laminated Iron. 
Let, in Fig. 79, 

d = thickness of the iron plates ; 
(B '^ maximum magnetic induction ; 
N = frequency ; 
y ^ electric conductivity of the iron. 



— L — 



' 1 



Then, if x is the distance of a zone, d X, from the center 
of the sheet, the conductance of a zone of thickness, dx, 
and of one cm length and width is ydx ; and the magnetic 
flux cut by this zone is tBx. Hence, the E,M.F. induced in 
this zone is 

«£ = V^fl-iVflSx, in C.G.S. units. — ■ ffSo 

This E,M.F, produces the current ; 

dI=hMyd%= ^■aN<S,y%dx, in C.G.S. units, 

provided the thickness of the plate is negligible as compared 
with the length, in order that the current may be assumed 
as flowing parallel to the sheet, and in opposite directions 
on opposite sides of the sheet. 

The power consumed by the induced current in this 
zone, dx, is 

dJ' = %£dl= 2-:^N*<S? y kVx, in C.G.S. units or ergs per second, 

and, consequently, the total power consumed in one cm* of 
the sheet of thickness, d, is 

, in C.G.S. units ; 

the power consumed per cm* of iron is, therefore, 

and the energy consumed per cycle and per cm* of iron is 

^=N^^ ^'^'- 

The coefficient of eddy currents for laminated iron is, 

. = -^ = 1.645 rf», 

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where y is expressed in C.G.S. units. Hence, if y is ex' 
pressed in practical units or 10 ~* C.G.S. units, 
^_ >'rfMO-» _ 1,645 ^s 10-'. 

Substituting for the conductivity of sheet iron the ap- 
proximate value, 

we get as the coefficient of eddy currents for laminated iron, 

. = -</» 10-«= 1.645 rf' 10-»- 

loss of energy per cm' and cycle, 

W =« y .A^04» = ^' «"« Y ■A"®" 10 -• = 1.645 rf» y A'fB* 10 -• ergs 

= 1.645 rf'vVtB*10-*ergs; 
or, W = ty N(^\(i-'' = 1.645 rf« .A' CB» 10"" joules; 

loss of power per cm' at frequency, N, 

/ = A'ir=€y.A^' CBMO-' = 1.645 rfWfi" 10-'^ watts; 
total loss of power in volume, V, 

P= Vp= 1.645 Vii*N*W\0--'^ watts. 

As an example, 

d =1 mm = 

« = 1,645 X 10-"; 

fF= 4110 ergs 
= .000411 joules; 

p = .0411 watts; 

/•= 41.1 watts. 

90. {b). Iron Wire. 

Let, in Fig. 80, d = 
diameter of a piece of 
iron wire ; then if x is 
the radius of a circular 
zone of thickness, (j?x, 
and one cm in length, 
the conductance of this 

; N= 100 ;« = 6000; V = 1000 cm*. 

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zone is, y(/x/2 jrx, and the magnetic flux inclosed by the 
zone is (Bx'b-. 

Hence, the E.M.F. induced in this zone is : 

IE = V2 ir»^tB x', in C.G.S. units, 

and the current produced thereby is, 

= '^:LllyJV<S,xdx, in C.G.S. units. 
The power consumed in this zone is, therefore, 
dJ'= Z£d/=-B*yJV'&?x*ax, in C.G.S. units 
consequently, the total power consumed i n on e cm length 
of wire is 

= ^y.A^(B'rf*, in C.G.S. units. 
Since the volume of one cm length of wire is 

= £li 

4 ' 
the power consumed in one cm' of iron is 

and the energy consumed per cycle and cm' of iron is 
»' = ;^=~y^*'«^ergs. 
Therefore, the coefficient of eddy currents for iron wire is 

or, if y is expressed in practical units, or 10~' C.G.S. units, 
. 'r^ain-» fii7 v> in-t 

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Substituting ^ jn* 

we get as the coefficient of eddy currents for iron wire, 

. = ^-/»10-» = .617</»10-» 

The loss of energy per cm' of iron, and per cycle 

= .617rfWffl''10-'eigs, 
-€yA^Oa*10-' = . 617 rf'jVCB* 10-" joules; 
loss of power per cm*, at frequency, N, 

P= ^>i = iy^«(B'10-T= .617 rf'7V^»aa» 10-" watts; 
total loss of power in volume, V, 

P^Vf^ .617 Vd*N^<S? 10-" watts. 
As an example, 
rf=linm, =.lcm; A'= 100; «" = 6,000; f= 1000 cm*. 


t = .617 X 10-", 
W= 1540 ei^s = .000154 joules, 
/ = .0154 watts,. 
J* = 16.4 watts, 
hence veiy much less than in sheet iron of equal thickness. 

91- Comparison of sheet imn and iron wire. 

dt, = thickness of lamination of sheet iron, and 
d^ » diameter of iron wire, 

the eddy-coefficient of sheet iron being 

«, = ^</,'10-*, -^.-- T»lM 

and the eddy coefficient of iron wire 



the loss of power is equal in both — other things being 
equal — if (^ = c, ; that is, if, 

^» = |rfj« or di = 1.63di. 

It follows that the diameter of iron wire can be 1.68 
times, or, roughly, 1| as large as the thickness of laminated 
iron, to give the same loss of energy through eddy currents, 
as shown in Fig. 81. 

92, Demagnetising, or screening effect of eddy currents. 

The formulas derived for the coefficient of eddy cur- 
rents in laminated iron and in iron wire, hold only when 
the eddy currents are small enough to neglect their mag- 
netizing force. Otherwise the phenomenon becomes more 
complicated; the magnetic flux in the interior of the lam- 
ina, or the wire, is not in phase wjth the flux at the sur- 
face, but lags behind it. The magnetic flux at the surface 
is due to the impressed M.M.F., while the flux in the inte- 
rior is due to the resultant of the impressed M.M.F. and to 
the M.M.F. of eddy currents ; since the eddy currents kg 
90° behind the flux producing them, their resultant with 
the impressed M.M.F., and therefore the magnetism in the 

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interior, is made lagging. Thus, progressing from the sur- 
face towards the interior, the magnetic flux gradually lags 
more and more in phase, and at the same time decreases 
in intensity. While the complete analytical solution of this 
phenomenon is beyond the scope of this book, a determina- 
tion of the magnitude of this demagnetization, or screening 
effect, suflicient to determine whether it is negligible, or 
whether the subdivision of the iron has to be increased 
to make it negligible, can be made by calculating the maxi- 
mum magnetizing effect, which cannot be exceeded by the 
eddys, . 

Assuming the magnetic density as uniform over the 
whole cross-section, and therefore all the eddy currents in 
phase with each other, their total M.M.F. represents the 
maximum possible value, since by the phase difference and 
the lesser magnetic density in the center the resultant 
M.M.F. is reduced. 

In laminated iron of thickness d, the current in a zone 
of thickness, dx at distance x from center of sheet, is : 

rf/= -^-rN&jxdx units (CG.S.) 
= V2»-A'(Byjca'j:10-'amperes; 
hence the total current in sheet is 

I=jjd/= -^ w JV(R J 10 -'jjxdx 


= -~^ iV«><^' 10 -' amperes. 

Hence, the maximum possible demagnetizing ampere-turns 
acting upon the center of the lamina, are 

= .555 J^<S>d^ 10 ~ * ampere-turns per cm. 
Example : d = X zm, N= 100, ffl = 5,000, 
or / = 2.775 ampere-turns per cm. 

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93. In iron wire of diameter </, the current in a tubular 
zone of dx thickness and x radius is 

hence, the total current is 


Hence, the maximum possible demagnetizing ampere-turns, 
• acting upon the center of the wire, are 

I=-l-ZJ^JVS./d^lO-' = .2775/^<S,jd*iO-* 
= .2775 N<& rf» 10 - • ampere-turns per cm. 
For example, if<^=.l cm, A^=100, ffi = 5,000, then 
/= 1,338 ampere-turns per cm; that is, half as much as in 
a lamina of the thickness d. 

94. Besides the eddy, or Foucault, currents proper, which 
flow as parasitic circuits in the interior of the iron lamina 
or wire, under certain circumstances eddy currents also 
flow in larger orbits from lamina to lamina through the 
whole magnetic structure. Obviously a calculation of these 
eddy currents is possible only in a particular structure. 
They are mostly surface currents, due to short circuits 
existing between the laminae at the surface of the magnetic 

Furthermore, eddy currents are induced outside of the 
magnetic iron circuit proper, by the magnetic stray field 
cutting electric conductors in the neighborhood, especially 
when drawn towards them by iron masses behind, in elec- 
tric conductors passing through the iron of an alternating 
field, etc. All these phenomena can be calculated only in 
particular cases, and are of less interest, since they can 
and should be avoided. 

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Eddy Currents in Conductor, and Unequal Current 

©6. If the electric conductor has a considerable size, the 
alternating magnetic field, in cutting the conductor, may 
set up differences of potential between the different parts 
thereof, thus giving rise to local or eddy currents in the 
copper. This phenomenon can obviously be studied only 
with reference to a particular case, where the shape of the 
conductor and the distribution of the magnetic field are 

Only in the case where the magnetic field is produced 
by the current flowing in the conductor can a general solu- 
tion be given. The alternating current in the conductor 
produces a magnetic field, not only outside of the conductor, 
but inside of it also ; and the lines of magnetic force which 
close themselves inside of the conductor induce E.M.Fs. 
in their interior only. Thus the counter E.M.F. of self- 
inductance is largest at the axis of the conductor, and least 
at its surface ; consequently, the current density at the 
surface will be larger than at the axis, or, in extreme cases, 
the current may not penetrate at all to the center, or a 
reversed current flow there. Hence it follows that only the 
exterior part of the conductor may be used for the conduc- 
tion of the current, thereby causing an increase of the 
ohmic resistance due to unequal current distribution. 

The general solution of this problem for round conduc- 
tors leads to complicated equations, and can be found else- 

In practice, this phenomenon is observed only with very 
high frequency currents, as lightning discharges ; in power 
distribution circuits it has to be avoided by either keeping 
the frequency sufficiently low, or having a shape of con- 
ductor such that unequal current distribution does not 
take place, as by using a tubular or a flat conductor, or 
several conductors in parallel. 

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96. It will, therefore, be sufficient to determine the 
largest size of round conductor, or the highest frequency, 
where this phenomenon is still negligible. 

In the interior of the conductor, the current density 
is not only less than at the surface, but the current lags 
behind the current at the surface, due to the increased 
effect of self-inductance. This lag of the current causes the 
magnetic fluxes in the conductor to be out of phase with 
each other, making their resultant less than their sum, while 
the lesser current density in the center reduces the total 
flux inside of the conductor. Thus, by assuming, as a basis 
for calculation, a uniform current density and no difference 
of phase between the currents in the different layers of the 
conductor, the unequal distribution is found larger than it 
is in reality. Hence this assumption brings us on the safe 
side, and at the same time simplifies the calculation greatly. 

Let Fig, 82 represent a cross-section of a conductor of 
radius R, and a uniform current density, 

where /= total current in conductor. 

The magnetic reluctance of a tubular zone of unit length 
and thickness dx, of radius x, is 

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The current inclosed by this zone is /, = iA-, and there 
fore, the M.M.F. acting upon this zone is 

£F, = 4 ,r /, / 10 = 4 »•«* / 10, 
and the magnetic flux in this zone is 

rf* = JFjr / (JU = 2 «■«(& / 10. 
Hence, the total magnetic flux inside the conductor is 

♦=J„''*=lo'Jo"'^ = To- = Io- 
From this we get, as the excess of counter E.M.F. at the 
axis of the conductor over that at the surface — 

AA= V2irA'*10-''= VSiJV/lO-*, per unit length, 
= VSir'iVWlO-'; 

and the reactivity, or specific reactance at the center of the 
conductor, becomes k = li-E / i = V2 w'NR} 10 -*. 
Let p = resistivity, or specific resistance, of the material of 
the conductor. 

We have then, A / p = V2 ■^NH' 10" /p; 
and p/ VF+7, 

the mt io of_current densities at center and at periphery. 

For example, if, in copper, p = 1,7 x 10-*, and the 
percentage decrease of current density at center shall not 
exceed 5 per cent, that is — 

p + V^ + p' = .95 -f- 1, 
we have, i = .51 X lO^*; 

hence .51 x 10"'= "-^jr-A^^ lO"* 
or NJ?' = Z6.6; 

hence, when JV = ' 125 100 60 25 

Ji = .541 .605 .781 1.21 cm. 
Z)=2v?= 1.08 1.21 1.56 2.42 cm. 

Hence, even at a frequency of 125 cycles, the effect of | 
unequal current distribution is still negligible at one cm [ 
diameter of the conductor. Conductors of this size are, i 
however, excluded from use at this frequency by the exter- 
nal self-induction, which is several times larger than the | 

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resistance. We thus see that unequal current distribution 
is usually negligible in practice. The above calculation was 
made under the assumption that the conductor consists of 
unmagnetic material. If this is not the case, but the con- 
■ductor of iron of permeability ft, then; ^* = ^, / (11^ and 
thus ultimately; > = V2ir*AJtj?*10-' and; */p = V2»» 
Np-B^ 10— */p. Thus, for instance, for iron wire at 
p = 10x10-', ;( = 500 it is, permitting 5% difference 
Iwtween center and outside of wire ; k = 3.2 x 10-* and 
NR^ = .46, 
hence when, N= 126 100 60 25 

R = .061 .068 .088 .136 cm. 
thus the effect is noticeable even with relatively small iron 

Mutual Inductance. 

97, When an alternating magnetic field of force includes 
a secondary electric conductor, it induces therein an E.M.F, 
which produces a current, and thereby consumes energy if 
the circuit of the secondary conductor is closed, 

A particular case of such induced secondary currents 
are the eddy or Foucault currents previously discussed. 

Another important case is the induction of secondary 
E.M.Fs. in neighboring circuits ; that is, the interference of 
circuits running parallel with each other. 

In general, it is preferable to consider this phenomenon 
of mutual inductance as not merely producing an energy 
component and a wattless component of E.M.F. in the 
primary conductor, but to consider explicitly both the sec- 
ondary and the primary circuit, as will be done in the 
chapter on the alternating-current transformer. 

Only in cases where the energy transferred into the 
secondary circuit constitutes a small part of the total pri- 
mary energy, as in the discussion of the disturbance caused 
by one cu-cuit upon a parallel circuit, may the effect on the 
primary circuit be considered analogously as in the chapter 
«n eddy currents, by the introduction of an energy com- 

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ponent, representing the loss of power, and a wattless 
component, representing the decrease of self-inductance. 

Let — 

4^ = 2 «■ N L => reactance of main circuit ; that is, L = 
total number of interlinkages with the main conductor, of 
the lines of magnetic force produced by unit current in 
that conductor ; 

x^ = 2rNLi = reactance of secondary circuit ; that is, 
ij = total number of interlinkages with the secondary 
conductor, of the lines of magnetic force produced by unit 
current in that conductor ; 

x^ = 2 X JVL„ = mutual inductance of circuits ; that is, 
i„ = total number of interlinkages with the secondary 
conductor, of the lines of magnetic force produced by unit 
current in the main conductor, or total number of inter- 
linkages with the main conductor of the lines of magnetic 
force produced by unit current in the secondary conductor. 
Obviously : x„* < xx^.* 

■ As coefficient U xlf-fnilucUnce L, Ly, the total fliu lutTonDdliig the conductoi 
ii here meant. UiuallT Id the dlKuulon ol inductive ippuatui, capccbllT ol tiani- 
fonnen, that part of the magnetic flui is denoted lell-inducuuice of the one circuit 
which tutroundi thii circuit, but not the other circuit; that it, which passes between 
both clicutti. Hence, the total xlf-indactance, L, is In Ihii cue equal to the sum of 
tiie self-inductance, L^, aod the mutual inductance, L^,. 

The object of this distinclion is 10 separate the waltlesi part, Z,, of the 
total self 'inductance, L, from itiat part, Lmt which represents the transfer of 
E.M.F. into the secondary circuit, since the action of these two components is 
essentially different. 

Thus, in allernHting-curienI Iransfonners it is CDStoinary — and will be 
done later in this book — to denote as the self4Dductance, L, ol each citcuit 
only that part of the magnetic flux produced by the circuit which passes 
between both circuits, and thus acts in " choking " only, but not in transform- 
ing; while the flux surrounding both circuits is called mutual inductance, or 
useful magnetic flux. 

With this denotation, in transformers the mutual inductance, Zm, is usu- 
ally very much greater than the self-inducunces, /.', and /-i', while, if the 
self .inductances, L and /,, , represent the total flux, th«r product is larger 
than the square of the mutual inductance, /-■■ ; or 

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Let r, *i resistance of secondary circuit. Then the im- 
pedance of secondary circuit is 

E.M.F. induced in the secondary circuit, E^^ =jx^I, 
where/™ primary current. Hence, the secondary current is 

zi n —At" ' 

and the E.M.F. induced in the primary circuit by the secon- 
dary current, J^ is 

or, expanded, 

Hence, the E.M.F. consumed thereby 

= (.r~Jx)l. 

X = , " \ = effective reactance of mutual inductance. 

The susceptance of mutual inductance is negative, or of 
opposite sign from the reactance of self-inductance. Or, 

Mutual inductance consumes energy and decreases the self- 

Dielectric and Electrostatic Phenomena. 
98. While magnetic hysteresis and eddy currents can 
be considered as the energy component of inductance, con- 
densance has an energy component also, namely, dielectric 
hysteresis. In an alternating magnetic field, energy is con- 
sumed in hysteresis due to molecular friction, and similarly, 
energy is also consumed in an alternating electrostatic field 
in the dielectric medium, in what is called electrostatic or 
dielectric hysteresis. 

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While the laws o£ the loss of energy by magnetic hys- 
teresis are fairly well understood, and the magnitude of the 
effect known, the phenomenon of dielectric hysteresis is 
still almost entirely unknown as concerns its laws and the 
magnitude of the effect. 

It is quite probable that the loss of power in the dielec- 
tric in an alternating electrostatic field consists of two dis- 
tinctly different components, of which the one is directly 
proportional to the frequency, — analogous to magnetic 
hysteresis, and thus a constant loss of energy per cycle, 
independent of the frequency; while the other component 
is proportional to the square of the frequency, — analogous 
to the loss of power by eddy currents in the iron, and thus 
a loss of energy per cycle proportional to the frequency. 

The existence of a loss of power in the dielectric, pro- 
portional to the square of the frequency, I observed some 
time ago in paraffined paper in a high electrostatic field and 
at high frequency, by the electro-dynamometer method, 
and other observers under similar conditions have found 
the same result, 

Arno of Turin found at low frequencies and low field 
strength in a larger number of dielectrics, a loss of energy 
per cycle independent of the frequency, but proportional to 
the 1.6"' power of the field strength, — that is, following 
the same law as the magnetic hysteresis, 

This loss, probably true dielectric static hysteresis, was 
observed under conditions such that a loss proportional to 
the square of density and frequency must be small, while at 
high densities and frequencies, as in condensers, the true 
dielectric hysteresis may be entirely obscured by a viscous 
loss, represented by Wf^=: t.N(S,\ 

99. If the loss of power by electrostatic hysteresis is 
proportional to the square of the frequency and of the field 
intensity, — as it probably nearly is under the working con- 

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ditions of alternating-current condensers, — then it is pro- 
portional to the square of the E.M.F., that is, the effective 
conductance, g, due to dielectric hysteresis is a constant ; 
and, since the condenser susceptance, — b— V, is a constant 
also, — unlike the magnetic inductance, — the ratio of con- 
ductance and susceptance, that is, the angle of difference 
of phase due to dielectric hysteresis, is a constant. This I 
found proved by experiment. This would mean that the 
dielectric hysteretic admittance of a condenser, 

where : g = hysteretic conductance, b' = hysteretic suscep- 
tance ; and the dielectric hysteretic impedance of a con- 
denser, . 

Z = r —jx = r +jx„ 

where : r = hysteretic resistance, x^ = hysteretic condens- 
ance; and the angle of dielectric hysteretic lag, tano = b' /g 
= Xc/ r, are constants of the circuit, independent of E.M.F, 
and frequency. The E.M.F. is obviously inversely propor- 
tional to the frequency. 

The true static dielectric hysteresis, observed by Amo 
as proportional to the 1.6'" power of the density, will enter 
the admittance and the impedance as a term variable and 
dependent upon E.M.F. and frequency, in the same manner 
as discussed in the chapter on magnetic hysteresis. 

To the magnetic hysteresis corresponds, in the electro- 
static field, the static component of dielectric hysteresis, 
following, probably, the same law of 1.6"' power. 

To the eddy currents in the iron corresponds, in the 
electrostatic field, the viscous component of dielectric hys- 
teresis, following the square law. 

As a rule however, these hysteresis losses in the alter- 
nating electrostatic field of a condenser are very much 
smaller than the losses in an alternating magnetic field, so 
that while the latter exert a very marked effect on the de- 
sign of apparatus, representing frequently the largest of all 
the losses of energy, the dielectric losses are so small as to 
be very difficult to observe. 

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To the phenomenon of mutual inductance corresponds, 
in the electrostatic field, the electrostatic induction, or in- 

100. The alternating electrostatic field of force of an 
electric circuit induces, in conductors within the field of 
force, electrostatic charges by what is called electrostatic 
influence. These charges are proportional to the field 
strength ; that is, to the E.M.F. in the main circuit. 

If a flow of current is produced by the induced charges, 
energy is consumed proportional to the square of the charge ; 
that is, to the square of the E.M.F. 

These induced charges, reacting upon the main conduc- 
tor, influence therein charges of equal but opposite phase, 
and hence lagging behind the main E.M.F. by the angle 
of lag between induced charge and inducing field. They 
require the expenditure of a charging current in the main 
conductor in quadrature with the induced charge thereon ; 
that is, nearly in quadrature with the E.M.F., and hence 
consisting of an energy component in phase with the 
E.M.F. — representing the power consumed by electrostatic 
influence — and a wattless component, which increases the 
capacity of the conductor, or, in other words, reduces its 
capacity reactance, or condensance. 

Thus, the electrostatic influence introduces an effective 
conductance, g, and an effective susceptance, b, — of the 
same sign with condenser susceptance, — into the equations 
of the electric circuit. 

While theoretically g and b should be constants of the 
circuit, frequently they are very far from such, due to 
disruptive phenomena beginning to appear at high electro- 
static stresses. 

Even the capacity condensance changes at very high 
potentials ; escape of electricity into the air and over the 
surfaces of the supporting insulators by brush discharge or 
electrostatic glow takes place. As far as this electrostatic 

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corona reaches, the space is in electric connection with the 
conductor, and thus the capacity of the circuit is deter- 
mined, not by the surface of the metallic conductor, but 
by the exterior surface of the electrostatic glow surround- 
ing the conductor. This means that with increasing po- 
tential, the cap)acity increases as soon as the electrostatic 
corona appears ; hence, the condensance decreases, and at 
the same time an energy component appears, representing 
the loss of power in the corona. 

This phenomenon thus shows some analogy with the de- 
crease of magnetic inductance due to saturation. 

At moderate potentials, the condensance due to capacity 
can be considered as a constant, consisting of a wattless 
component, the condensance proper, and an energy com- 
ponent, the dielectric hysteresis. 

The condensance of a polarization cell, however, begins 
to decrease at very low potentials, as soon as the counter 
E.M.F. of chemical dissociation is approached. 

The condensance of a synchronizing alternator is of 
the nature of a variable quantity ; that is, the effective 
reactance changes gradually, according to the relation of 
impressed and of counter E.M.F,, from inductance over 
zero to condensance. 

Besides the phenomena discussed in the foregoing as 
terms of the energy components and the wattless compo- 
nents of current and of E.M.F., the electric leakage is 
to be considered as a further energy component ; that is, 
the direct escape of current from conductor to return con- 
ductor through the surrounding medium, due to imperfect 
insulating qualities. This leakage current represents an 
effective conductance, g, theoretically independent of the 
E.M.F., but in reality frequently increasing greatly with the 
E.M.F., owing to the decrease of the insulating strength of 
the medium upon approaching the limits of its disruptive 

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101. In the foregoing, the phenomena causing loss of 
energy in an alternating-current circuit have been dis- 
cussed ; and it has been shown that the mutual relation 
between current and E.M.F, can be expressed by two of 
the four constants : 

Energy component of E.M.F., in phase with current, and = 

current X effective resistance, or r; 
wattless component of E.M.F., in quadrature with current, and = 

current X effective reactance, or x ; 
eneigy component of current, in phase with E.M.F^ and = 

FM.F. X effective conductance, or^; 
wattless component of current, in quadrature with E.M.F., and =i 

E.M.F. X effective susceptance, 01 b. 

In many cases the exact calculation of the quantities, 
'■. ■''f g^ ^1 is not possible in the present state of the art. 

In general, r, x, g, b, are not constants of the circuit, but 
depend — besides upon the frequency — more or less upon 
E,M.F., current, etc. Thus, in each particular case it be- 
comes necessary to discuss the variation of r, x, g, b, or to 
determine whether, and through what range, they can be 
assumed as constant. 

In what follows, the quantities r, x, g, b, will always be 
considered as the coefficients of the energy and wattless 
components of current and E.M.F., — that is, as the effec- 
tive quantities, — so that the results are directly applicable 
to the general electric circuit containing iron and dielectric 

Introducing now, in Chapters VII. to IX., instead of 
"ohmic resistance," the term "effective resistance," etc., 
as discussed in the preceding chapter, the results apply 
also — within the range discussed in the preceding chapter 
— to circuits containing iron and other materials producing 
energy losses outside of the electric conductor. 

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102, Graphically alternating currents and E.M,F's 
are represented by vectors, of which the length represents 
the intensity, the direction the phase of the alternating 
wave. The vectors generally issue from the center of 
■ co-ordinates. 

In the topographical method, however, which is more 
convenient for complex networks, as interlinked polyphase 
circuits, the alternating wave is represented by the straight 
line between two points, these points representing the abso- 
lute values of potential (with regard to any reference point 
chosen as co-ordinate center) and their connection the dif- 
ference of potential in phase and intensity. 

Algebraically these vectors are represented by complex 
quantities. The impedance, admittance, etc, of the circuit 
is a complex quantity also, in symbolic denotation. 

Thus current, E.M.F., impedance, and admittance are 
related by multiplication and division of complex quantities 
similar as current, E.M.F., resistance, and conductance are 
related by Ohms law in direct current circuits. 

In direct current circuits, power is the product of cur- 
rent into E.M.F. In alternating current circuits, if 

E = ^ +y 
The product, 

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is not the power; that is, multiplication and division, which 
are correct in the inter-relation of current, E-M.F., impe- 
dance, do not give a correct result in the inter-relation of 
E.M.F., current, power. The reason is, that £"/ are vec- 
tors of the same frequency, and Z a constant numerical 
factor which thus does not change the frequency. 

The power P, however, is of double frequency compared 
with E and /, that is, makes a complete wave for every 
half wave of E or /, and thus cannot be represented by a 
vector in the same diagram with E and /, 

P^ = E / is a quantity of the same frequency with E 
and /, and thus cannot represent the power. 

103. Since the power is a quantity of double frequency 
of E and /, and thus a phase angle a in E and / corre- 
sponds to a phase angle 2 u in the power, it is of interest to 
investigate the product E I formed by doubling the phase 

Algebraically it is, 

Since _;■'= — 1, that is 180° rotation for E and /, for the 
double frequency vector, P,j* = + 1, or 360* rotation, and 

1 xy=! -j 

That is, multiplication with / reverses the sign, since it 
denotes a rotation by 180° for the power, corresponding to 
a rotation of 90° for E and / 
Hence, substituting these values, we have, 

j'=\En= (^,-t + «",■") +y (^,-1 _ fi/u) 

The symbol \E /] here denotes the transfer from the 
frequency of E and / to the double frequency of P. 

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The product, P =\E I\ consists of two components ; 
the real component, 

/« = \Er^ = (^,1 + «",-") 

and the imaginary component. 
The component, 

i" = [£/]» = {H^ + ^i^-) 

is the power of the circuit, = E I cos {E I) 
The component. 

Pi = [E/y = (<",-> - ^(") 

is what may be called the " wattless power," or the power- 
less or quadrature volt-amperes of the circuit, = E I sin 

The real component will be distinguished by the index 
1, the imaginary or wattless component by the index/ 

By introducing this symbolism, the power of an alternat- 
ing circuit can be represented in the same way as in the 
direct current circuit, as the symbolic product of current 
and E.M.F. 

Just as the symbolic expression of current and E.M.F, 
as complex quantity does not only give the mere intensity, 
but also the phase, 

so the double frequency vector product /' = [£/] denotes 
more than the mere power, by giving with its two compo- 
nents /" = [E /]' and Pf = [E /]■*, the true energy volt- 
amperes, and the wattless volt-amperes. . 


£ = ^ +j^ 

■/=.■> -t-yV" 

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pi = [£/y = (^i^ - ("i") 

/»' + j^f = /i* + <"V + ^V + <"V 

where (2 " total volt amperes of circuit. That is, 

T/ie true power /" and the wattless power Pi are the two 
rectangular components of tlie total apparent power Q of the 


In symbolic representation as double freqvency vector pro- 
ducts, powers can be combined and resolved by the parallelo- 
gram of vectors just as currents and E.M.F's in graphical 
or symbolic representation. 

The graphical methods of treatment of alternating cur- 
rent phenomena are here extended to include double fre- 
quency quantities as power, torque, etc. 

--■=: q ~ sin ui = inductance factor 

of the circuit, and the general expression of power is, 


" =G(cos,«+ysin«,) 

104. The introduction of the double frequency vector 
product P = \E !'\ brings us outside of the limits of alge- 

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bra, however, and the commutative principle of algebra, 
a X i = 6 X a, does not apply any more, but we have, 

[£/] unlike [/B] 

[£/]=. [£/]'+/[£/y 

we have 

that is, the imaginary component reverses its sign by the 
interchange of factors. 

The physical meaning is, that if the wattless power 
[E /y is lagging with regard to £, it is leading with regard 

The wattless component of power is absent, or the total 
apparent power is true power, if 






t»n(£) = 



that is, £ and / are irt phase or in opposition. 

The true power is absent, or the total apparent power 
wattless, if 

[£/]' = (<J|i + *";" = 

that is. 

tan -£■ = — cot / 
that is, E and / are in quadrature, 

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The wattless power is lagging (with regard to E or lead- 
ing with regard to /) if, 


and leading if, 


The true power is negative, that is, power returns, if, 

[-£/]" <0 

We have, 

[£,-/].[-£,/] = -[£/: 

that is, when representing the power of a circuit or a part of 
a circuit, current and E.M,F, must be considered in their 
proper relative phases, but their phase relation with the re- 
maining part of the circuit is immaterial. 
We have further 

\E,jr\ = -j \_E, /] = [£, /y -j \B, /]' 
\jE, /] =j \E,'r\ = - \_B, ly +/ \_E, ij 

\JE, jf] = [£,/] = {£/]' + J [£, I]J 

105. If 7>, = [EJ^l f^ = [^j/J . . . i'„ = [£,/,] 

are the symbolic expressions of the power of the different 
parts of a circuit or network of circuits, the total power of 
the whole circuit or network of circuits is 

J'=A + ^t-\-- ■ 

■ ■+P, 

>' - >,' + F,' + ■ 

■ • • + A 

JV~PJ + P^- ■ ■ 

■ + P.' 

In other words, the total power in symbolic expression 
(true as well as wattless) of a circuit or system is the sum 
of the powers of its individual components in symbolic 

The first equation is obviously directly a result from the 
law of conservation of energy. 

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One result derived herefrom is for instance : 
If in a generator supplying power to a system the cur- 
rent is out of phase with the E.M.F. so as to give the watt- 
less power /y, the current can be brought into phase with 
the generator E.M.F., or the load on the generator made 
non-inductive by inserting anywhere in the circuit an appa- 
ratus producing the wattless power — Pi; that is, compen- 
sation for wattless currents in a system takes place regardless 
of the location of the compensating device. 

Obviously between the compensating device and the 
source of wattless currents to be compensated for, wattless 
currents will flow, and for this reason it may be advisable 
to bring the compensator as near as possible to the circuit 
to be compensated. 

106. Like power, torque in alternating apparatus is a 
double frequency vector product also, of magnetism and 
M.M.F. or current, and thus can be treated in the same 

In an induction motor, for instance, the torque is the 
product of the magnetic flux in one direction into the com- 
ponent of secondary induced current in phase with the 
magnetic flux in time, but in quadrature position therewith 
in space, times the number of turns of this current, or since 
the induced E.M.F. is in quadrature and proportional to 
the magnetic flux and the number of turns, the torque 
of the induction motor is the product of the induced E.M.F. 
into the component of secondary current in quadrature 
therewith in time and space, or the product of the induced 
current into the component of induced E.M.F. in quadra- 
ture therewith in time and s'pace. 

Thus if 

£^ = ^ -t-y-^' = induced E.M.F. in one direction in 

/, = /' -l-y »'" = secondary current in the quadrature di- 
rection in space. 

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the torque is 

T=\Eiy = i^i'~^P^. 

By this equation the torque is given in watts, the mean- 
ing being that T =\E ly is the power which would be 
exerted by the torque at synchronous speed, or the torque 
in synchronous watts. 

The torque proper is then 



p = number of pairs of poles of the motor. 

In the polyphase induction motor, if /, = (' +yi" is 
the secondary current in quadrature position, in space, to 
E.M.F. E,. 

The current in the same direction in space. as £", is 

/, =y7, = — j" +yi' ; thus the torque can also be ex- 

r=[fi/j' = ^»'-^(-^ 

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107. As far as capacity has been considered in the 
foregoing chapters, the assumption has been made that the 
condenser or other source of negative reactance is shunted 
across the circuit at a definite point. In many cases, how- 
ever, the capacity is distributed over the whole length of the 
conductor, so that the circuit can be considered as shunted 
by an infinite number of infinitely small condensers infi 
nitely near together, as diagrammatically shown in Fig. 83. 




r\t. ea. UtrtlxUi Capatlt^. 

In this case the intensity as well as phase of the current, 
and consequently of the counter E.M,F. of inductance and 
resistance, vary from point to point ; and it is no longer 
possible to treat the circuit in the usual manner by the 
vector diagram. 

This phenomenon is especially noticeable in long-distance 
lines, in underground cables, and to a certain degree in the 
high-potential coils of alternating-current transformers for 
very high voltage. It has the effect that not only the 
E.M.Fs., but also the currents, at the beginning, end, and 
different points of the conductor, are different in intensity 
and in phase. 

Where the capacity effect of the line is small, it may 
with sufficient approximation be represented by one con- 

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denser of the same capacity as the line, shunted across the 
line. Frequently it makes no difference either, whether 
this condenser is considered as connected across the line at 
the generator end, or at the receiver end, or at the middle. 

The best approximation is to consider the line as shunted |i 
at the generator and at the motor end, by two condensers 
of \ the line capacity each, and in the middle by a con- 
denser of S the line capacity. This approximation, based 
on Simpson's rule, assumes the variation of the electric 
quantities in the line as parabolic. If, however, the capacity 
of the line is considerable, and the condenser current is of ! 
the same magnitude as the main current, such an approxi- I 
mation is not permissible, but each line element has to be ■ 
considered as an infinitely small condenser, and the differ- 
ential equations based thereon integrated. Or the pheno- 
mena occurring in the circuit can be investigated graphically 
by the method given in Chapter VI. § 37, by dividing the 
circuit into a sufficiently large number of sections or line 
elements, and then passing from line element to line element, 
to construct the topographic circuit characteristics, 

108. It is thus desirable to first investigate the limits '%jL».'''^Uffi^-^ 
of applicability of the approximate representation of the line y»4J, 1, 1^3 
by one or by three condensers. ^ 

Assuming, for instance, that the line conductors are of 
1 cm. diameter, and at a distance from each other of 50 cm., 
and that the length of transmission is 50 km,, we get the 
capacity of the transmission line from the formula — 

C= 1.11 X 10 -•«/-(- 4 log, 2<^/ J microfarads, 

(t = dielectric constant of the surrounding medium = 1 in air; 

/ = len^ of conductor = 5 x 10* cm. ; 

d = distance of conductors from each other = 50 cm. ; 

S = diameter of conductor = 1 cm. 
Since C = .8 microfarads, 
the capacity reactance isx = 10*/2TiVC ohms, f^ 

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where N = frequency ; hence, at jV = 60 cycles, 

* = 8,900 ohms ; 
and the charging current of the line, at £ = 20,000 volts, 
becomes, i„ = £/x^2.25 amperes. 

The resistance of 100 km of line of 1 cm diameter is 22 
ohms; therefore, at 10 per cent = 2,000 volts loss in the 
line, the main current transmitted over the line is 

/ = -^ = 91 amperes, 

representing about 1,800 kw. 

In this case, the condenser current thus amounts to less 
than 2^ per cent., and hence can still be represented by the 
approximation of one condenser shunted across the line. 

If the length of transmission is 150 km., and the voltage, 

capacity reactance at 60 cycles, x = 2,970 ohms ; 

charging ciurent, to = 10.1 amperes ; 

line resistance, r = 66 ohms ; 

main current at 10 per cent loss, /= 45.5 amperes. 
The condenser current is thus about 22 per cent of the 
main current, and the approximate calculation of the effect 
of line capacity still fairly accurate. 

At 300 km length of transmission it will, at 10 per cent, 
loss and with the same size of conductor, rise to nearly 90 
per cent, of the main current, thus making a more explicit 
investigation of the phenomena in the line necessary. 

In most cases of practical engineering, however, the ca- 
pacity effect is small enough to be represented by the approx- 
imation of one ; viz., three condensers shunted across the line. 

109. A.) Line capacity represented by one condenser 
shunted across middle of line. 
Let — 

Y = g -\- jb =■ admittance of receiving circuit; 
% = > — /* = impedance of line ; 
bt = condenser susceptance of line. 

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Denoting, in Fig, 84, 

the E.M.F., viz., current in receiving circuit by ,£, I^ 

tile E.M.F. at middle of line by ,£', 

tile E.M.F., viz., current at generator \iy £,,!,; 

. f. 

■ I 2 ^ 2 

2 ' ' i V 

or, expanding, 

£.- a|i + (/--/AT) (i-+>S) - L^(j--jx) 

11(X 5.) i(>w capacity represented by three eondetuers, 
in the vtiddie and at the ends of the line. 
Denoting, in Fig. 85, 

the E.M.F. and current in receiving circuit by E, It 

the E.M.F. at middle of line by E', 



the current on receiving side of line by 1', 
the current on generator side of line by /", 
the E.M.F., viz., current at generator by -fj, /„, 



Otherwise retaining the same denotations as in A.), 
We have, 





-f {*•+>*- 


6 3 * 






- E^U + ji-ji,) - ih^r-j.){ig+3jt - 5/4) 


As will be seen, the first terms in the expression of E, 
and of /, are the same in A.) and in B.). 

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IIL C.) Complete investigation of distributed capacity, 
inductance, leakage, and resistance. 

In some cases, especially in very long circuits, as in 
lines conveying alternating power currents at high potential 
over extremely long distances by overhead conductors or un- 
derground cables, or with very feeble currents at extremely- 
high frequency, such as telephone currents, the consideration 
of the line resistance — which consumes E,M.Fs, in phase 
with the current — and of the line reactance — which con- 
sumes E.M.Fs. in quadrature with the current — is not 
sufficient for the explanation of the phenomena taking place 
in the line, but several other factors have to be taken into 

In long lines, especially at high potentials, the electro- 
static capacity of the line is sufficient to consume noticeable 
currents. The charging current of the line condenser is 
proportional to the difference of potential, and is one-fourth 
period ahead of the E.M.F. Hence, it will either increase 
or decrease the main current, according to the relative phase 
of the main current and the E.M.F, 

As a consequence, the current will change in intensity 
as well as in phase, in the line from point to point ; and the 
E,M.F"s. consumed by the resistance and inductance will 
therefore also change in phase and intensity from point 
to point, being dependent upon the current. 

Since no insulator has an infinite resistance, and as at 
high potentials not only leakage, but even direct escape of 
electricity into the air, takes place by "silent discharge," we 
have to recognize the existence of a current approximately 
proportional and in phase with the E.M.F. of the line. 
This current represents consumption of energy, and is 
therefore analogous to the E.M.F. consumed by resistance, 
while the condenser current and the E.M.F. of inductance 
are wattless. 

Furthermore, the alternate current passing over the line 
induces in all neighboring conductors secondary currents. 

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which react upon the primary current, and thereby intro- 
duce E.M.Fs. of mutual inductance into the primary circuit. 
Mutual inductance is neither in phase nor in quadrature 
with the current, and can therefore be resolved into an 
energy component of mutual inductance in phase with the 
current, which acts as an increase of resistance, and into 
a wattless component in quadrature with the current, which 
decreases the self-inductance. 

This mutual inductance is not always negligible, as, 
for instance, its disturbing influence in telephone circuits 

The alternating potential of the line induces, by electro- 
static influence, electric charges in neighboring conductors 
outside of the circuit, which retain corresponding opposite 
charges on the line wires. This electrostatic influence re- 
quires the expenditure of a current proportional to the 
E.M.F., and consisting of an energy component, in phase 
with the E.M.F., and a wattless component, in quadrature 

The alternating electromagnetic field of force set up by 
the line current produces in some materials a loss of energy 
by magnetic hysteresis, or an expenditure of E.M.F. in 
phase with the current, which acts as an increase of re- 
sistance. This electromagnetic hysteretic loss may take 
place in the conductor proper if iron wires are used, and 
will then be very serious at high frequencies, such as those 
of telephone currents. 

The effect of eddy currents has already been referred 
to under "mutual inductance," of which it is an energy 

The alternating electrostatic field of force expends 
energy in dielectrics by what is called dielectric hysteresis. 
In concentric cables, where the electrostatic gradient in the 
dielectric is comparatively large, the dielectric hysteresis 
may at high potentials consume considerable amounts of 
energy. The dielectric hysteresis appears in the circuit 

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as consumption of a current, whose component in phase 
with the E.M.F. is the dieUchic energy current, whith 
may be considered as the power component of the capacity 

Besides this, there is the increase of ohmic resistance 
due to unequal distribution of current, which, however, is 
usually not large enough to be noticeable. 

112. This gives, as the most general case, and per unit 
length of line : 

E.M.Fs. consumed in phase with the current I, and = rl, 
representing consumption of energy, and due to : 
Resistance, and its increase by unequal current distri- 
tribution ; to the energy component of mutual 
inductance; to induced currents ; to the energy 
component of self-inductance; or to electromag- 
netic hysteresis. 
E.M.Fs. consumed in quadrature -with the current I, and 
= X ly wattless, and due to : 
Self-inductance, and Mutual inductance. 
Currents consumed in phase with tlie E.M.F., E, and 
= gE, representing consumption of energy, and 
due to : 
Leakage through the insulating material, including 
silent discharge ; energy component of electro- 
static influence ; energy component of capacity, or 
of dielectric hysteresis. 
Currents consumed in quadrature to the E.M.F., E, and 
= bE, being wattless, and due to : 
Capacity and Electrostatic influence. 

Hence we get four constants: — 

Effective resistance, r, 
Effective reactance, x. 
Effective conductance, g. 
Effective susceptance, b = — b„ 

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per unit length of line, which represent the coefficients, per 
unit length of line, of 

E-M.F. consumed in phase with current; 
E.M.F, consumed in quadrature with current; 
Current consumed in phase with E.M.F. ; 
Current consumed in quadrature with E.M.F. 

113- This line we may assume now as feeding into a 
receiver circuit of any description, and determine the current 
and E.M.F. at any point of the circuit. 

That is, an E.M.F. and current (differing in phase by any 
desired angle) may be given at the terminals of receiving cir- 
cuit. To be determined are the E.M.F. and current at any 
point of the line ; for instance, at the generator terminals. 
Or, Z^=ri— j'xi ; 

the impedance of receiver circuit, or admittance, 

and E.M.F., E^, at generator terminals are given. Current 
and E.M.F. at any point of circuit to be determined, etc. 

114. Counting now the distance, X, from a point, 0, of 
the line which has the E.M.F., 

-^1 = 'i +y*t'i and the current: /i =* /| +Jii, 
and counting X positive in the direction of rising energy, 
and negative in the direction of decreasing energy, we have 
at any point, x, in the line differential, dx: aJ i • dVt'Civ,^ 

Leakage current : Egdx; ' 

Capacity current : ~ jBbfdt.; 

hence, the total current consumed by the line element, dx, 
'* dI=E{g~jb,)dx, or, 

E.M.F. consumed by resistance, Irdx; 
E.M.F. consumed by reactance, — jlxd%: 

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hence, the total E.M.F, consumed in the line element, dx, is 
iE ~Iir-jx)dyi, or, 

Ih.^^ fundamental differential equations : 

are symmetrical with respect to / and £. 
Differentiating these equations : 

dx' dx ' 


dx.' dx " 
and substituting (1) and (2) in (S), we get : 


= /"U- A) ('•-/■*). I (5) 

the differential equations of E and I. 

116. These differential equations are identical, and con- 
sequently I and E are functions differing by their limiting 
conditions only. 

These equations, (4) and (5), are of the form : 

^=wig-jb:)(r-Jx). (6) 

and are integrated by 

where c is the basis of natural logarithms ; for, differen- 
tiating this, we get. 

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hence, *^-= {g —Jh) ir —J^) \ 

or, » = ± ^ {g-j'l>i)i.'--jx) i 

hence, the general integral is : 

«i IB a« + *" -|- bt~", 
where a and b are the two constants of integratic 

into (7), we have, 



therefore, __^^ 

') + {gr-i,x)} -X 





substituting (9) into (8) : 

= a«"(cos^x — /sin/Sx) + ^<~"(cos/?x +ysin/5x); 
w= {at"+ *<-")cosj8x —/(at" - *»-•■) sin j9x (12) 
which is the general solution of differential equations (4) 
and (5) 

Differentiating (8) gives : 
dw/t/x = p(at" — ^t""); 
hence, substituting, (9) : 

^w/dK = (a -y^) {(«." - *.-") COS ^x -y 

(«." + ^<-")sin/3x}. (13) 

Substituting now / for w, and substituting (13) in (1), 
and writing. 

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sin ^) 


(-^t-*')co3^x ~j(A^* — Bi-"-) 

where A and £ are the constants of integration. 
Transformed, we get, 

« + B. 


(cos ^ +_;' sin ^) J 


Ix) - J<-" 

(cos/3x+ysin j9x) ] 


Thus the waves consist of two components, one, with 
factor ^c", increasing in amplitude toward the generator, 
the other, with factor Bt—", decreasing toward the genera- 
tor. The latter may be considered as a reflected wave. 

At the point x = 0. 

^ a-j0 


Thus m (cos £ — / sin ") = S ■* -£- 

nd, ^ 

« = amplitude. 
Si = angle o( reflection. 

These are the general integral equations of the problem. 

U6. If — 

/i = »\ + ji{ is the current 1 
.ffj = <■, -f je{ is the R M.F. j 


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by substituting (15) in (14), we get ; 

2 A- {{p. ;, + ^ /,') + (y^, + b, »,')) \ 

+ / ((« >■' - /8 ;,) + U'l - ■».'.)). I 

2 J- ((./, + /3;,')-U ', + «,',')) I 

+ /{("'.'-^'',) - U'^'-i,',)),! 

a and ^ being determined by equations (11). 


117. li Z :=■ R ~jX is the impedance of the receiver 
circuit, E, = e^ -\- j e^ is the E.M.F. at dynamo terminaJs 
(17), and / = length of line, we get 
at X = 0, 


At X = /, 

/ A + £ g-Jb.' 

■i-i - ■, g-ib, 

A + JI .->/»■ 



1 -£.-■') COS f/-j\A,- 

' + £,-') 

Equations (18) and (19) determine the constants A and B, 
which, substituted in (14), give the final integral equations. 

The length, x„ = 2»/^ is a complete wave length (20), 
#hich means, that in the distance 2v/ p the phases of the 
components of current and E.M.F. repeat, and that in half 
this distance, they are just opposite. 

Hence the remarkable condition exists that, in a very 
long line, at different points the currents at the same time 
flow ia opposite directions, and the E.M.Fs. are opposite, 

118. The difference of phase, £, between current, f, and 
E.M.F. , £, at any point, x, of the line, is determined by 

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the equation, 

ZJ(cos5+ysma) = :|, 

where /? is a constant. 

Hence, w varies from point to point, oscillating around a 
medium position, wa, which it approaches at infinity. 

This difference of phase, 5^, towards which current and 
E.M.F, tend at infinity, is determined by the expression, 

/?(cos5, +ysin5x) = \M. \ 

or, substituting for £'and /their values, and since*-" =0, 
and^«" {C0S|8x —/sin j9X), cancels, and 


This angle, u<c, = ; that is, current and E.M.F. come 
more and more in phase with each other, when 
ad, — |8^ = 0; that is, 
o.^^ = g^b,. or, 

2a^ 2f*, ' 

substituting (10), gives, 

g r — b ^x _ g^ — b* . 

gx-\-b^r 2gb^ 

hence, expanding, r -k- x = g -^ b,; (22) 

that is, the ratio of resistance to inductance equals the ratio 

of leakage to capacity. 

This angle, u>x, = 45° ; that is, current and E.M.F. differ 
by Jth period, if — a ^^ + ^S = <^£' + fi^d o^ 
a _ K+g . 

which gives : rg+ xb^ = <i. (23) 

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That is, two of the four line constants must be zero; either 
g and X, or g and b^. 

The case where ^ = = j:, that is a line having only 
resistance and distributed capacity, but no self-induction is 
approximately realized in concentric or multiple conductor 
cables, and in these the phase angle tends towards 45° lead 
for infinite length. 

119. As an instance, in Fig. 86 a line diagram is shown, 
with the distances from the receiver end as abscissae. 
The diagram represents one and one-half complete waves, 
and gives total effective current, total E.M.F., and differ- 





\ ■ 




* -i* 







. i 



„ 1 














■a. iJK 






•« ... 



M. LM 


m. - 









M M 

■ ■ 






• f 



ence of phase between both as function of the distance from 
receiver circuit ; under the conditions, 

E.M.F. at receiving end, 10,000 volts ; hence, £, =f, = 10,000 ; 
current at receiving end, 6S amperes, with a power factor 
of .385. 
that is, /= i,+/ii' = 25 + 60/; 

line constants per unit length, 

^ = 2 X 10-', 1 
*, = 20x 10-'; i 

length of line corresponding to 
one complete period of the wave 
of propagation. 

These values, substituted, give, 

/= {t«" (47.3 cos |8x + 27.4 sin ;3x) - %-" 
(22.3 cos ;3x + 32.6 sin ^x)} 
+y {«" (27.4 cos /3x - 47.3 sin ^x) + «-«» 
(32.6 cos /3x - 22.3 sin ^x)}; 
M= {." (6450 co5^x + 4410 sin ^x) + €-« 
(3S30 cos j8x + 4410 sin j9x)} 
+/■ {«" (4410 cos ;3x ~ 6450 sin ^x) - «-" 
(4410 cos j8k - 3530 sin ;3x)}; 

tan il. = ~~ °-''' + ^^ = - .073, <3- = - 4.2°. 

If + ^l>c 

120, As a further instance are shown the characteristic 
curves of a transmission line of the relative constants, 

r:x:g:b = ^ : 82 : 1.25 X 10 "• : 25 X 10-', and <■ 
= 25,000, i = 200 at the receiving circuit, for the con- 

a, non-inductive load in the receiving circuit, Fig. 87. 



4.95 X 10-", ■ 
28.36 X 10-*, 


_ ?Jr_ 221.5- 

A = 

1.012 - 1.206/; 
i .812+ .794/ 




b, wattless receiving circuit of 90° lag, Fig. 88. 

c, wattless receiving circuit of 90° lead, Fig. 89. 
These curves are determined graphically by constructing 

the topographic circuit characteristics in polar coordinates 
as explained in Chapter VI., paragraphs 36 and 37, and de- 
riving corresponding values of current, potential difference 
and phase angle therefrom. 

As seen from these diagrams, for wattless receiving cir- 
cuit, current and E.M.F. oscillate in intensity inversely to 





























^ J 













:. . 





each other, with an amplitude of oscillation gradually de- 
creasing when passing from the receiving circuit towards 
the generator, while the phase angle between current and 
E.M.F. oscillates between lag and lead with decreasing am- 
plitude. Approximately maxima and minima of current co- 
incide with minima and maxima of E.M.F. and zero phase 


























ve I 

















S / 
















For such graphical constructions, polar coordinate paper 
and two angles a. and S are desirable, the angle a being the 

angle between current and change of E.M.F., tan a = - = 4, 

and the angle 8 the angle between E.M.F. and change of 



^ ^"^ 

^ t V 

Z ^ Y-3!:/^ ^ 

3 / ^ 

— i ^. Z 

1.7 ^--^ 



.,...,.. „.,^ 2 

^ s 

-/ 2^ 

7 J^ 

^-C ^ ^ 

^-^^ 7 ^^ ^ 

^ ?^^/ 

^--# ^/ ^^ 

< 2i. 1 

^ y -'^^ 




With non-inductive load, Fig. 87, these oscillations of 
intensity have almost disappeared, and only traces of them 
are noticeable in the fluctuations of the phase angle and the 
relative values of current and E.M.F. along the line. 

Towards the generator end of the line, that is towards 
rising power, the curves can be extended indefinitely, ap- 
proaching more and more the conditions of non-inductive 
circuit, towards decreasing power, however, all curves ulti- 
mately reach the conditions of a wattless receiving circuit, 
as Figs, 88 and 89, at the point where the total energy in- 

y^- ^ 

y"-^ 1 \ ,.,. i.l_,.,. 

'Z^'^ 31™ 

-s"" U — 

^^ -=-t /^ '^ ^ 

% -I -> ^ 2^ 

\ .^_7 ^ = ^-" 

\^^\ 3-1 l~% i 

N^^ jT ^ t-^*^ 

S^L %^'^^4-i , 

\ Z%^\7 S^ ^:^7^ 



put into the line has been consumed therein, and at this 
point the two curves for lead and for lag join each other as 
shown in Fig. 90, the one being a prolongation of the other, 
and the flow of power in the line reverses. Thus in Fig. 90 
power flows from both sides of the line towards the point of 
zero power marked by 0, where current and E.M.F. are in 
quadrature with each other, the current being leading with 
regard to the flow of power from the left, and lagging with 
regard to the flow of power from the right side of the 

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121. The following are some particular cases; 
A.) Open circuit at end of lines : 
x = 0:7, = 0. 
A = (£e^ + b,e{) +j\gei'~ b,e^ ^-B; 

E= — L_^.^{(." + «-")cos^x -/(«"-«-") sin /3x};] 
g—JOt I 

/ ^ <*{(«" - «-") cosy5 X -/■(<" + «-") sin ^x}. I 

0-—JP } 

B.) Line grounded at end: 

x-0: ^, = 0. 

A = (ail + i9'.') +)(«'.' - P'\) = -? 

^=-A_^{(."-«-")cos^x-/(.- + .-")sin^x};| 

/ ^^{(«" + «— ) cos ;3x ->(.-■ -«-")sin^x}. 

C.) Infinitely long conductor:^ a JjW 
Replacing x by — x, that is, counting the distance posi- 
tive in the direction of decreasing energy, we have, 

x = x : 7=0, .ff-0; 

and J, 1 

revolving decay of the electric wave, that is the reflected 
wave does not exist. 

The total impedance of the infinitely long conductor is 

g' + i'c' g* + h* 

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The infinitely long conductor acts like an impedance 

g' + «,' 

^ g' + i.' 

that is, 

like a resistance 


' + «."' 

combined with a reactance 


We thus get the difference of phase between E.M.F. 
and current, 

tana = ^=^^-°^S 

which is constant at all points of the line. 
If ^ = 0, jf = 0, we have, 


tan ui = 1, or, 
5 = 46" ; 

that is, current and E.M.F. differ by Jth period. 
ZJ.) Generator feeding into closed circuit : 
Let X = be the center of cable ; then, 

£^= — £_,; hence: £ = at x = 0; 

i- = /— J 

which equations are the same as in B, where the line is 
grounded at x = 0. 

M.) Let the length of a line be one-quarter wave length, 

and assume the resistance r and conductance g as negligible 

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compared with x and bp 

r = 0=g 
These values substituted in (11) give 

i8= V^ 

Let the E.M.F, at the receiving end of the line be 
assumed zero vector 

Ey=ey = KM.F. and 

Iy = iy +ji{ = current at end of line x = 
E^ = KM.F. and 

I^ = current at beginning of line 

Substituting in (16) these values of £■, and I^ and also r = 
= g, we have 

From these equations it follows that 

A-~B= -Jteti 

which values, together with the foregoing values of Ej, I„ r, 
g, a, and A substituted in (14) reduce these equations to 

■^ = (ti + Jh') <^^ Vi^x -Jeiy— sin V^ x 

^ = «, cos V^ X —J (ii +jii') Y J- sin V^ x 

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Then at x = — ^ 
Hence also 

£", and /, are both in quadrature ahead of ^^ and /, 

A = -^0 V " = constant, !££■„ = constant. That is, at 

constant impressed E.M.F, £", the current /, in the receiv- 
ing circuit of a line of one-quarter wave length is constant, 
and inversely (constant potential — constant current trans- 
fonnation by inductive line). In this case, the current /, at 
the beginning of the line is proportional to the load e^ at the 
end of the line. 

If x^ = he = total reactance, 

bg = Ibe = total susceptance of line, then 

Instance;r =- 4, ^„ = 20 X 10 -», E^ = 10,000 K Hence 
/=55.6, :r, = 222, *,= .0111, /, = 70.7, /o=-. 00707 tf. 

122. An interesting application of this method is the 
determination of the natural period of a transmission line ; 
that is the frequency at which such a line discharges an 
accumulated charge of atmospheric electricity (lightning), 
or oscillates at a sudden change of load, as a break of cir- 

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The discharge of a condenser through a circuit contain- 
ing self-induction and resistance is oscillating (provided that 
the resistance does not exceed a certain critical value de- 
pending upon the capacity and the self-induction). That is, 
the discharge current alternates with constantly decreasing 
intensity. The frequency of this oscillating discharge de- 
pends upon the capacity, C, and the self-induction, Z, of the 
circuit, and to a much lesser extent upon the resistance, so 
that if the resistance of the circuit is not excessive the fre- 
quency of oscillation can, by neglecting the resistance, be 
expressed with fair, or even close, approximation by the 



IT Vex' 

An electric transmission line represents a capacity as well 
as a self-induction ; and thus when charged to a certain 
potential, for instance, by atmospheric electricity, as by in- 
duction from a thunder-cloud passing over or near the line, 
the transmission line discharges by an oscillating current. 

Such a transmission line differs, however, from an ordi- 
naiy condenser, in that with the former the capacity and 
the self-induction are distributed along the circuit. 

In determining the frequency of the oscillating discharge 
of such a transmission line, a sufficiently close approximation 
is obtained by neglectmg the resistance of the line, which, 
at the relatively high frequency of oscillating discharges, 
is small compared with the reactance. This assumption 
means that the dying out of the discharge current through 
the influence of the resistance of the circuit is neglected,. 
and the current assumed as an alternating current of ap- 
proximately the same frequency and the same intensity as 
the initial waves of the oscillating discharge current. By 
this means the problem is essentially simplified. 

LiCt / = total length of a transmission line, 
r = resistance per unit length, 
X = reactance per unit length = 2 w JVZ. 

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where L = coefficient of self-induction or inductance per unit 
length ; 
g = conductance from line to return (leakage and dis- 
charge into the air) per unit length ; 
b = capaci^ susceptance per unit length = 2 ir JVC 
where C = capacity per unit length. 

X = the distance from the beginning of the line, 

We have then the equations : 
The E.M.F., 

»_ 1 J(^<«-^«— ') cos /9x-/ (A»«l -) 

■ g-Jb I ■ + J?«-") sin /3x ' S 

the current, Ml^-) 

1 ( (-4«« + j€-«) cos j9x -j (^.« I 

■ ~ o.-j^\ ■-^.— ')sin^x ■ ( J 



c = base of the natural logarithms, and A and B integration 

Neglecting the line resistance, r = 0, and the conduc- 
tance leakage, etc.), g=% gives. 

These values substituted in (14) give, 
B = ^-\{A- ff) COS. -^x - j {A •'r B) %\Vi -^brA 

. ■ ■ K2«-) 

I = J^ \{A-¥B)&^ 'Jbx'ii -J iA-S)sin -JWn \ 

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If the discharge takes place at the point : j; = 0, that is, 
If the distance is counted from the discharge point to the 
end of the line ; x = /, hence : 

At X = 0, ^ =: 0, 

Atx = /, 7=0. 

Substituting these values in (25) gives. 
For ;r = 0, 

A-B=^ A=B 

which reduces these equations to, 



■ V5 


= 0, 



-/, / = 

0, thus, substituted in 
cos ■4M=(i 

I (2S), 



?"t'>',* 0.1. 

2, . . 


I (27.) 



that is, VAt / is an odd multiple of ^ ■ And at x = ^ 

S, = ±^- (30.) 

Substituting in (29) the values, 
b = 2irNC 
x = %-wNL 
we have, 


2* + 1 
N= ^"Zl (81.) 

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the frequency of the oscillating discharge, 
where A = 0, 1, 2. . . . 

That is, the oscillating discharge of a transmission line 
of distributed capacity does not occur at one definite fre- 
quency (as that of a condenser), but the line can discharge 
at any one of an infinite number of frequencies, which are 
the odd multiples of the fundamental discharge frequency, 


} (33.) 


N^ = 


Co = IC= total capacity of transmission line, 

Zo = IL = total self-inductance of transmission line, 

we have, 



the frequency of oscillation, 

or natural period of the line, and 

or lowest natural period of the line. 
From (30), (88), and (34), 

and from (29), 

These substituted in ( 

8) give, 
A cos 

2i + l),, 

(2* + l)irx 




r-(2i + l) 

. (2i + l),'. 

The oscillating discharge of a hne can thus follow any of 
the forms given by making A = 0, 1, 2, 3 ... in equation 

Reduced from symbolic representation to absolute values 

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by multiplying E with cos 2 x A7 and / with sin 2 » A'i" and 
omittingy, and substituting N from equation (34), we have, 

^ ^'•"^o (39.) 

" (2^+1)t x . (2,6+1) IT ' 

(2i+l)x 2 /«" 2 vcTo 

where ^ is an integration constant, depending upon the 
initial distribution of voltage, before the discharge, and 
t = time after discharge. 

123. The fundamental discharge wave is thus, for ^ = 0, 

/5 _...,_ _ 

With this wave the current is a maximum at the begin- 
ning of the line : x = 0, and gradually decreases to zero at 
the end of the line : x == /, 

The voltage is zero at the beginning of the line, and 
rises to a maximum at the end of the line. 

Thus the relative intensities of current and potential 
along the line are as represented by Fig. 91, where the cur- 
is shown as /, the potential as E. 

The next higher discharge frequency, for : i = 1, gives : 





cos - 










Here the current is again a maximum at the beginning 
of the line : x = 0, and gradually decreases, but reaches 

zero at one-third of the line : >< = «• then increases again, in 









































































the opposite direction, reaches a second but opposite maxi- 
mum at two-thirds of the line : x = -5- , and decreases to 

zero at the end of the line. There is thus a nodal point of 
current at one-third of the line. 

The E,M.F. is zero at the beginning of the line : x = 0, 

rises to a maximum at one-third of the line : x = ^ , de- 

11 ^ 
creases to zero at two-thirds of the line : k = -^ , and rises 

again to a second but opposite maximum at the end of the 
line: x = /. The E.M.F. thus has a nodal point at two- 
thirds of the line. 

The discharge waves : k = \, are shown in Fig, 92, 
those with k = % with two nodal points, in Fig, 93. 

Thus k is the number of nodal points or zero points of 
current and of E.M.F. existing in the line (not counting 
zero points at the ends of the line, which of course are not 

In case of a lightning discharge the capacity Cg is the 
capacity of the line against ground, and thus has no direct 
relation to the capacity of the line conductor against its 
return. The same applies to the inductance Lg. 
\id= diameter of line conductor, 
D = distance of conductor above ground, 
and / = length of conductor, 
the capacity is. 

the self-inductance, 


The fundamental frequency of oscillation is thus, by 
substituting (42) in (35), 

4V5S ' ' 



That is, the frequency of oscillation of a line discharging 
to ground is independent of the size of line wire and its 
distance from the ground, and merely depends upon the 
length / of the line, being inversely proportional thereto. 
We thus get the numerical values, 
Length of line 

10 20 30 40 50 60 80 100 miles. 
= 1.6 3.2 4.8 6.4 . 8 9.6 12.8 16 X 10* cm. 

hence frequency, 

.V, = 4680 2340 1560 1170 937.5 780 685 475 cycles. 

As seen, these frequencies are comparatively low, and 
especially with very long lines almost approach alternator 

The higher harmonics of the oscillation are the odd 
multiples of these frequencies. 

Obviously all these waves of different frequencies repre- 
sented in equation (39) can occur simultaneously in the 
oscillating discharge of a transmission line, and in general 
the oscillating discharge of a transmission line is thus of 
the form, 

(by substituting: a* = 2k\^\ ] 

\ (44-) 

where a, a, a^ . . . are constants depending upon the 
initial distribution of potential in the transmission line, at 
the moment of discharge, or at / = 0, and calculated there- 

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V2A. As an instance the following discharge equation 
of a line charged to a uniform potential e over its entire 
length, and then discharging at x = 0, has been calculated. 

The harmonica are determined up to the 11 — that is, «,, 
a„ a„ a^ a^ «„. 

These six unknown quantities require six equations, which 

, . „ , I 21 Zl ^l &l Ql 

are given by assuming E~eiQxx = ^,-^,^,-^,-^,-^. 

At i = 0, £ = a, equation (44) assumes the form 


11 ^x ) 

nin + «t sm-i 

Substituting herein for x the values '■ ^$ - 

gives six equations for the determination of a,, u, . 
These equations solved give, 

£ = € (1.26 sin u cos i^ + .40 sin 3 u cos 3 ^ + .22 sin 
6 01 cos 5 ^ + .12 sin 7 m cos 7 ^ + .07 sin 9 u 
cos 9 ♦ + .02 sin 11 « cos 11 *) 

/=*t/^(1.26cos «o sin + .40 cos 3 <o sin 3 + +.22 

cos 6 u sin 5 ^ + .12 cos 7 u sin 7 ^ + .07 cos 
9 « sin 9 ^ + .02 cos 11 o> sin 11 <^) 







Length of line, 1=25 miles = 4 x 10* cm. 
Size of wire : No. 000 B. & S. G., thus : rf = 1 cm. 
Height above ground : Z> = 18 feet = 650 cm. 
Let e s= 26,000 volts = potential of line in the moment of 

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We then have, 

E = 31,500 sin u cos •^ + 10,000 sin 3 u cos 3 <^ + 5S00 sin 

5 u> cos 5 ^ + 3000 sin 7 u cos 7 ^ + 1760 sin 9 u cos 

9 ^ + 500 sin 11 w cos 11 ^. 
1= 61.7 cos 01 sin <^ + 19.6 cos 3 <u sin 3 ^ + 10.8 cos S u sin 

5 ^ + 5.9 cos 7 u sin 7 ^ + 3.4 cos 9 <u sin 9 ^ + 1.0 

cos 11 w sin 11 ^. 

o.=> .39a:10-« 
+ = 1.18/ 10+*. 

A simple harmonic oscillation as a line discharge would 
require a sinotdal distribution of potential on the trans- 
mission line at the instant of discharge, which is not proba- 
ble, so that probably all lightning discharges of transmission 
lines or oscillations produced by sudden changes of circuit 
conditions are complex waves of many harmonics, which in 
their relative magnitude depend upon the initial charge and 
its distribution — that is, in the case of the lightning dis- 
charge, upon the atmospheric electrostatic iield of force. 

The fundamental frequency of the oscillating discharge 
of a transmission line is relatively low, and of not much 
higher magnitude than frequencies in commercial use in 
alternating current circuits. Obviously, the more nearly 
sinusoidal the distribution of potential before the discharge, 
the more the low harmonics predominate, while a very un- 
equal distribution of potential, that is a very rapid change 
along the line, as caused for instance by a sudden short 
circuit rupturing itself instantly, causes the higher harmo- 
nics to predominate, which as a rule are more liable to cause 
excessive rises of voltage by resonance. 

126. As has been shown, the electric distribution in a 
transmission line containing distributed capacity, self-induc- 
tion, etc., can be represented either by a polar diagram 
with the phase as amplitude, and the intensity as radius 
vector, as in Fig. 34, or by a rectangular diagram with the 

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distance as abscissae, and the intensity as ordinate, as in 
Fig. 35 and in the preceding paragraphs. 

In the former case, the consecutive points of the circuit 
characteristic refer to consecutive points along the trans- 
mission line, and thus to give a complete representation of 
the phenomenon, should not be plotted in one plane but in 
front of each other by their distance along the transmission 
line. That is, if 0, 1, 2, etc., are the polar vectors in Fig. 
34, corresponding to equi-distant points of the transmission 
line, 1 should be in a plane vertically in front of the plane 
of 0, 2 by the same distance in front of 1, etc. 

In Fig. 35 the consecutive points of the circuit charac- 
teristic represent vectors of different phase, and thus should 
be rotated out of the plane around the zero axis by the 
angles of phase difference, and then give a length view of 
the same space diagram, of which Fig. 34 gives a view along 
the axis. 

Thus, the electric distribution in a transmission line can 
be represented completely only by a space diagram, and as 
complete circuit characteristic we get for each of the lines 
a screw shaped space curve, of which the distance along the 
axis of the screw represents the distance along the transmis- 
sion line, and the distance of each point from the axis rep- 
resents by its direction the phase, and by its length the 

Hence the electric distribution tn a transmission line 
leads to a space problem of which Figs. 34 and 35 are par- 
tial views. The single-phase line is represented by a double 
screw, the three-phase line by a triple screw, and the quarter- 
phase four-wire line by a quadruple screw. In the symbolic 
expression of the electric distribution in the transmission 
line, the real .part of the symbolic equation represents a pro- 
jection on a plane passing through the axis of the screw, 
and the imaginary part a projection on a plane perpendicular 
to the first, and also passing through the axis of the screw. 

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126. The simplest alternating-current apparatus is the 
transformer. It consists of a magnetic circuit interlinked 
with two electric circuits, a primary and a secondary. The 
primary circuit is excited by an impressed E.M.F., while in 
the secondary circuit an E.M.F. is induced. Thus, in the 
primary circuit power is consumed, and in the secondary 
a corresponding amount of power is produced. 

Since the same magnetic circuit is interlinked with both 
electric circuits, the E.M.F. induced per turn must be the 
same in the secondary as in the primary circuit ; hence, 
the primary induced E.M.F, being approximately equal to 
the impressed E,M,F,, the E.M.Fs, at primary and at sec- 
ondary terminals have approximately the ratio of their 
respective turns. Since the power produced in the second- 
ary is approximately the same as that consumed in the 
primary, the primary and secondary currents are approxi- 
mately in inverse ratio to the turns. 

127. Besides the magnetic flux interlinked with both 
electric circuits — which flux, in a closed magnetic circuit 
transformer, has a circuit of low reluctance — a magnetic 
cross-flux passes between the primary and secondary coils, 
surrounding one coil only, without being interlinked with 
the other. This magnetic cross-flux is proportional to the 
current flowing in the electric circuit, or rather, the ampere- 
turns or M.M.F, increase with the increasing load on the 
transformer, and constitute what is called the self-induc- 
tance of the transformer ; while the flux surrounding both 

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coils may be considered as mutual inductance. This cross- 
flux of self-induction does not induce E.M.F, in the second- 
ary circuit, and is thus, in general, objectionable, by causing 
a drop of voltage and a decrease of output. It is this 
cross-flux, however, or flux of self-inductance, which is uti- 
lized in special transformers, to secure automatic regulation, 
for constant power, or for constant current, and in this 
case is exaggerated by separating primary and secondary 
coils. In the constant potential transformer however, the 
primary and secondary coils are brought as near together as 
possible, or even interspersed, to reduce the cross-flux. 

As will be seen by the self-inductance of a circuit, not 
the total flux produced by, and interlinked with, the circuit 
is understood, but only that (usually small) part of the flux 
which surrounds one circuit without interlinking with the 
other circuit. 

128. The alternating magnetic flux of the magnetic 
circuit surrounding both electric circuits is produced by 
the combined magnetizing action of the primary and of the 
secondary current. 

This magnetic ilux is determined by the E.M.F. of the 
transformer, by the number of turns, and by the frequency. 

4 = maximum magnetic flux, 
N= frequency, 
n •= number of turns of the coil ; 

the E.M.F. induced in this coil is 

E = V2».M»* 10-' = 4.44 Mt^ 10-* volts; 

hence, if the E.M.F., frequency, and number of turns are 
determined, the maximum magnetic flux is 

To produce the magnetism, 4, of the transformer, a 
M.M.F. of ff ampere-turns is required, which is determined 

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by the shape and the magnetic characteristic of the iron, in 
the manner discussed in Chapter X. 

For instance, in the closed magnet circuit transformer, 
the maximum magnetic induction is lB = */5, where 5 
= the cross-section of magnetic circuit. 

129. To induce a magnetic density, (B, a M.M.F. of 3C,. 
ampere-turns maximum is required, or, JC„ / V2 ampere- 
turns effective, per unit length of the magnetic circuit ; 
hence, for the total magnetic circuit, of length, /, 

_ /3C„ 

" V2 

or , SF /K_ „ 

/ = — = ^- amps, eft., 

« «V2 

where « = number of turns. 

At no load, or open secondary circuit, this M.M.F., ff, is 
furnished by the exciting current, /„, improperly called the 
leakage current, of the transformer ; that is, that small 
amount of primary current which passes through the trans- 
former at open secondary* circuit. 

In a transformer with open magnetic circuit, such as 
the "hedgehog" transformer, the M.M.F., JF, is the sum 
of the M.M.F. consumed in the iron and in the air part of 
the magnetic circuit (see Chapter X.), 

The energy of the exciting current is the energy con- 
sumed by hysteresis and eddy currents and the small ohmic 

The exciting current is not a sine wave, but is, at least 
in the closed magnetic circuit transformer, greatly distorted 
by hysteresis, though less so in the open magnetic circuit 
transformer. It can, however, be represented by an equiv- 
alent sine wave, /„, of equal intensity and equal power with 
the distorted wave, and a wattless higher harmonic, mainly 
of triple frequency. 

Since the higher harmonic is small compared with the 

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total exciting current, and the exciting current is only a 
small part of the total primary current, the higher harmonic 
can, for most practical cases, be neglected, and the exciting 
current represented by the equivalent sine wave. 

This equivalent sine wave, /„, leads the wave of mag- 
netism, *, by an angle, a, the angle of hysteretic advance of 
phase, and consists of two components, — the hysteretic 
energy current, in quadrature with the magnetic flux, and 
therefore in phase with the induced E.M.F. = /„ sin a ; and 
the magnetizing current, in phase with the magnetic flux, 
and therefore in quadrature with the induced E.M,F,, and 
so wattless, = /„ cos o. 

The exciting current, /„, is determined from the shape 
and magnetic characteristic of the iron, and number of 
turns ; the hysteretic energy current is — 

Power consumed in the iron 


Induced KM.F. 

130- Graphically, the polar diagram of M.M.Fs. of j 
transformer is constructed thus : 

Let, in Fig. 94, t?* = the magnetic flux in intensity and 
phase (for convenience, as intensities, the effective values 
are used throughout), assuming its phase as the vertical; 

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that is, counting the time from the moment where the 
rising magnetism passes its zero value. 

Then the resultant M.M.F, is represented by the vector 
O'S, leading O* by the angle ffO* = «. 

The induced £.M.Fs. have the phase 180°, that is, are 
plotted towards the left, and represented by the vectors 
Ve;, and OEi. 

If, now, ^' = angle of lag in the secondary circuit, due 
to the total (internal and external) secondary reactance, the 
secondary current /j, and hence the secondary M.M.F., 
{Fj= «, /j, will lag behind £^ by an angle fi', and have the 
phase, 180° + j8', represented bj' the vector 0^^. Con- 
structing a parallelogram of M.M.Fs., with (?£F as a diag- 
onal and OxF^ as one side, the other side or Oif^ is the 
primary M.M.F., in intensity and phase, and hence, dividing 
by the number of primary turns, «„ , the primary current is 

To complete the diagram of E.M,Fs., we have now, — 
In the primary circuit : 

E.M.F. consumed by resistance is f^o< in phase with ^, and 
represented by the vector 0£r„ ; 

E.M.F. consumed by reactance is f^Xo, 90° ahead of fg, and 
represented by the vector 0£xo ; 

E.M.F. consumed by induced E.M.F. is £", equal and oppo- 
site to £"0, and represented by the vector OS". 

Hence, the total primary impressed E.M.F. by combina- 
tion of OBfa, OExg, and OE' by means of the parallelo- 
gram of E.M.Fs. is, 

Eo = 'OE„, 

and the difference of phase between the primary impressed 
E.M.F. and the primary current is 

ft = E„m„. 
In the secondary circuit : 

Counter E.M.F. of resist ance is ^^t in opposition with I^, 
and represented by the vector OEr^ ; 

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Counter KM.F, of reactance is /,:e„'90° behind /j. and 
represented by the vector OE^x(. 

Induced E.M.Fs., £( represented by the vector 0E(. 

Hence, the secondary terminal voltage, by combination 
of OEr(, OExl and OEl by means of the parallelogram of E, = ^E,, 

ajid the difference of phase between the secondary terminal 
voltage and the secondary current is 

As wUI be seen in the primary circuit the " components 
of impressed E.M.F. required to overcome the counter 
E.M.Fs.'* were used for convenience, and in the secondary 
circuit the "counter E.M.Fs." 

131. In the construction of the transformer diagram, it 
is usually preferable not to plot the seconc^ary quantities, 
current and E.M.F., direct, but to reduce them to corre- 
spondence with the primary circuit by multiplying by the 
ratio of turns, a = «„/«„ for the reason that frequently 
primary and secondary E.M.Fs., etc., are of such different 

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magnitude as not to be easily represented on the same 
scale; or the primary circuit may be reduced to the sec- 
ondary in the same way. In either case, the vectors repre- 
senting the two induced EXM.Fs. coincide, or OE{ = OE^. 

Fig. H. TraiufOrmtr Diagram altli SO' Lag In Steomiarii Circuit 

Figs. 96 to 107 give the polar diagram of a transformer 
having the constants — 

r, = .2 ohms, h = 0173 mhos, 

X, = .33 ohms, £i = 100 volts, 

r, = .00167 ohms, f\ =60 amperes, 

xt = .0025 ohms, a =10 degrees. 
g, = .0100 mhos, 

for the conditions of secondary circuit, 

ft' = 80" lag in Fig. 96. ft' = 20° lead in Fig. 99. 

60" lag " 96. SOMead " 100. 

20° lag " 97. 80" lead " JOl, 

O, or in phase, " 98. 

As shown with a change of ft' the other quantities E„, /„ 
To, etc., change in intensity and direction. The loci de- 
scribed by them are circles, and are shown in Fig. 102, 
with the point corresponding to non-inductive load marked. 
The part of the locus corresponding to a lagging secondary 

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Fig. 99. TnmifariHtr Ohgnun with 2<r Itail In Bmondarn Cur 



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current is shown in thick full lines; and the part correspond- 
ing to leading current in thin full lines. 

132. This diagram represents the condition of constant 
secondary induced E.M.F., £■,', that is, corresponding to a 
constant maximum magnetic flux. 

By changing all the quantities proportionally from the 
diagram of Fig. 102, the diagrams for the constant primary 
impressed E.M.F. {Fig. 103), and for constant secondary 
terminal voltage (Fig. 104), are derived. In these cases, 
the locus gives curves of higher order. 

Fig. 106 gives the locus of the various quantities when 
the load is changed from full load, /j = 60 amperes in a 
non-inductive secondary external circuit to no load or open 

a.) By increase of secondary resistance ; *.) by increase 
of secondary inductive reactance ; c.) by increase of sec- 
ondary capacity reactance. 

As shown in a.), the locus of the secondary terminal vol- 
tage, £*!, and thus of E^, etc, are straight lines; and in 
^.) and <:.), parts of one and the same circle a.) is shown 

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in full lines, ^.) in heavy full lines, and f.) in light full lines. 
This diagram corresponds to constant maximum magnetic 
flux; that is, to constant secondary induced RM.F. The 
diagrams representing constant primary impressed E.M.F. 
and constant secondary terminal voltage can be derived 
from the above by proportionality. 

flf. tot. 

133. It must be understood, however, that for the pur- 
pose of making the diagrams plainer, by bringing the dif- 
ferent values to somewhat nearer the same magnitude, the 
constants chosen for these diagrams represent, not the mag- 
nitudes found in actual transformers, but refer to greatly 
exaggerated internal losses. 

In practice, about the following magnitudes would be 
found : 

r, =,01 ohms; 

*i = .00026 ohms 

x„ = .033 ohms j 

f„ = .001 ohms 

n = .00008 ohms ; 

h = .00173 ohms 

that is, about one-tenth as large as assumed. Thus the 
changes of the values of E^, fj, etc., under the different 
conditions will be very much smaller. 

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Symbolic Method. 
134. In symbolic representation by complex quantities 
the transformer problem appears as follows : 

The exciting current, J^, of the transformer depends 
upon the primary E.M.F,, which dependance can be rep- 
resented by an admittance, the " primary admittance," 
^o = ^«+y^o> of t**^ transformer. 

Fit. fOS. 

The resistance and reactance of the primary and the 
secondary circuit are represented in the impedance by 
Zo=rg~jXt, and Zi=ri—jxx. 
Within the limited range of variation of the magnetic 
density in a constant potential transformer, admittance and 
impedance can usually, and with sufficient exactness, be 
considered as constant. 
«B = number of primary turns in series; 
Wi = number of secon<Ury turns in series; 
a. = -^ = ratio of turns; 

i^ =fo + J6e = primary admittance 

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= fe —jxa = primary impedance 

= secondary impedance 

where the reactances, x^ and x-^, refer to the true self-induc- 
tance only, or to the cross-flux passing between primary and 
secondary coils ; that is, interlinked with one coil only. 
Let also 

y =f+_/J = total admittance of secondary circuit, 
including the internal impedance ; 

£a ^ primary impressed E.M.F. ; 

£ ' = E.M.F. consumed by primary counter E.M.F. ; 

Ex = secondary terminal voltage ; 

E( = secondary induced KM. P.; 

/o = primary current, total ; 

f„ = primary exciting current ; 

/, = secondary current. 
Since the primary counter E.M.F., E', and the second- 
ary induced E.M.F., E^, are proportional by the ratio of 

turns, a, 

E' =-«.£(. (1) 

The secondary current is ; 

7, = YEl, (2) 

consisting of an energy component, gE^, and a reactive 
component, b E^. 

To this secondary current corresponds the component of 
primary current, ., ., 

/; =^i^=i-|-. (3) 

The primary exciting current is — 

/«=!;£'. (4) 

Hence, the total primary current is : 

I. = /„'+ /» (P) 

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or, /. =^'{l'+«'n} (6) 

The E.M.F, consumed in the secondary coil by the 
internal impedance is Z.J^ . 

The E,M.F. induced in the secondary coil by the mag- 
netic flux is £(. 

Therefore, the secondary terminal voltage is 
£, = E( _ Z,/, , 
or, substituting (2), we have 

^1 = £/{!- Z,r} (7) 

The E.M.F. consumed in the primary coil by the inter- 
nal impedance is Z^I^ 

The RM.F. consumed in the primary coil by the counter 
E.M.F. is£'. 

Therefore, the primary impressed E.M.F. is 

or, substituting (6), 


136. We thus have, 

primary E.M.F., E^= - aE{ j 1 + Z„K. + ^ j , (8) 

secondary E.M.F., E^ = E({\~ Z,Y}, (7) 

primary current, fg = — - — !- { F + a'l'o}, (6) 

secondary current, /( = VE,', (2) 
as functions of the secondary induced E.M.F., £^, as pa- 

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From the above we derive 

Ratio of transformation of E.M.Fs. : 

Ratio of transformations of currents : 



From this we get, at constant primary impressed 

E, = constant ; 

secondary induced E.M.F., 

E.M.F. induced per turn, 

secondary terminal voltage, 

p i, ^-^-y 



' l+Z.K. + 
primary current, 

/. „ -E. Y+a-Y. 

"'i + z.y.+is^ 

secondary current, 





At constant secondary terminal voltage, 
£i = const ; 




secondary induced E.M.F., 
Ex - 

E.M.F. induced per turn, 
iS = 

primary impressed E.M.F., 

E, =- 
primary current, 

h = - 
secondary current, 







i + z.y. + ^ 



y+ a'y. 




1- z,y' 

136. Some interesting conclusions can be drawn from 
these equations. 

The apparent impedance of the total transformer is 

i + z,n + 


^ + z>(k + Pj 




Substituting now, — = V, the total secondary admit- 
tance, reduced to the primary circuit by the ratio of turns, 
it U 

— + Z.. 


1^+ Y' is the total admittance of a divided circuit with 
the exciting current, of admittance Y,, and the secondary 

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current, of admittance Y' (reduced to primary), as branches. 
Thus : 

z: (16) 

is the impedance of this divided circuit, and 

Z, =Z/ + Z,. (17) 

That is : 

TAe alternate-current transformer, of primary admittance 
Yg, total secondary admittance Y, and primary impedance 
Zg , is equivalent to, and can be replaced by, a divided circuit 
with the branches of admittance Yg, the exciting current, and 
admittance Y' = Yjct', the secondary current, fed over mains 
of the impedance Z„ the internal primary impedance. 

This is shown diagrammatically in Fig. 106. 

137. Separating now the internal secondary impedance 
from the external secondary impedance, or the impedance of 
the consumer circuit, it is 

y-2, + 2; (18) 

where Z = external secondary impedance. 



Reduced to primary circuit, it is 

That is : 

= 7,' + Z'. 


An alternate-current transformer, of primary admittance 
Y,, primary impedance 2",, secondary impedance Z-^, and 
ratio of turns a, can, when the secondary circuit is closed by 
an impedance Z (the impedance of the receiver circuit), be 
replaced, and is equivalent to a circuit of impedance 2" = 
a^Z, fed over mains of the impedance Z,-\- Z{, where Z{ = 
a^Z^, shunted by a circuit of admittance V^, which latter 
circuit branches off at the points a — b, between the in^e- 
dances Z,, and Z-^. 

fig. 107. 

This is represented diagrammatically in Fig, 107, 

It is obvious therefore, that if the transformer contains 
several independent secondary circuits they are to be con- 
sidered as branched off at the points a, i, in diagram 
Fig. 107, as shown in diagram Fig. 108. 

It therefore follows : 

An alternate-current transformer, of x secondary coils, of 
the internal impedances Z^', Z^", . . . Z^, closed by external 
secondary circuits of the impedances Z', Z", . . . Z', is equiv- 
alent to a divided circuit of x + 1 branches, one branch of 

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admittance Y^, the exciting current, the other branches of the 
impedances Z^ + Z', Z^" + Z", . . . Z^ -Y Z', the latter 
impedances being reduced to the primary circuit by the ratio 
of turns, and the whole divided circuit being fed by the 
primary impressed E.M.F. E^, over vtains of the impedance 

Consequently, transformation of a circuit merely changes 
all the quantities proportionally, introduces in the mains the 
impedance Z^ + Z^, and a branch circuit between Z^ and 
Z(, of admittance K„. 

Thus, double transformation will be represented by dia- 
gram, Fig. 109. 

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With this the discussion of the alternate-current trans- 
former ends, by becoming identical with that of a divided 
circuit containing resistances and reactances. 

Such circuits have explicitly been discussed in Chapter 
VIII., and the results derived there are now directly appli- 
cable to the transformer, giving the variation and the con- 
trol of secondary terminal voltage, resonance phenomena, etc. 

Thus, for instance, if Z^ = Z^, and the transformer con- 
tains an additional secondary coil, constantly closed by a 
condenser reactance of such size that this auxiliary circuit, 
together with the exciting circuit, gives the reactance — x^, i 
with a non-inductive secondary circuit Z, = r^, we get the ' 
condition of transformation from constant primary potential 
to constant secondary current, and inversely, as previously 

Non-indtictive Secondary Circuit. 
138. In a non-inductive secondary circuit, the external 
secondary impedance is, 

or, reduced to primary circuit, 

ceZ = a*.ff, = R 
Assuming the secondary impedance, reduced to primary 
circuit, as equal to the primary impedance, 

a^Z^ = Z^ = r^-~jx„, 
it is, 

Y \ 1 

a» a\ZJrZ^ R + r,-jx^' 
Substituting these values in Equations (9), (10), and (18), 
we have 

Ratio of E.M.FS. : 



■-'■i'+ ^;./_%. +(--->-)c^-+>'-)} 

+ J+ r, -}., + \r + r, -}«.)■'■• 

or, expanding, and neglecting teims of tiigiier than third 

Ji + r.-jx,^ \X + r,-/x.)^ 
{'■.-jx.)U. +/'■•)}: 

or, expanded, 

Neglecting terms of tertiary order also. 
Ratio of currents : 

^ - - J {1 + (f. +>«.)(■«+'•. - >»i)> i 

or, expanded. 

Neglecting terms of tertiary order also. 
Total apparent primary admittance : 

-[X + ir.-jx.)+Jt(r.-jx.)ie.+/i.)}il-U.+ji.) 

^Jt + r.-Ji.)+^g. + Jl■.')•(.« + r.-/i,r+ ■ ■ .} 
- {Jf + 2 (^ - y«.) - Jf (^. + /«.) - 2 jt (r. - jx.) 




Neglecting terms of tertiary order also : 

Angle of lag in primary circuit : 
tan So = *' , hence, 

2^^Rb„ + 2 r,3, - 2:r,f, -2/^g„b„ 

tan a„ ^- . 

1 + =^ - ^^» - 2 r,f, - 2 *,*. + JPg,'+X^H* 

Neglecting terms of tertiary order also : 
2f + JC*, 

1 + 2^-^^. 

139. If, now, we represent the external resistance of 
the secondary circuit at full load (reduced to the primary 
circuit) by Ji,, and denote, 

inttnul rcKiuoa of traoilormt r percentage inter- 

«mii ™i.una oi Kcond^ dnuu " ^i resistance, 

i iiernti n,.cm.c of i.^ii .fonner _ perccutage intcr- 
r«ma] rc>i>UD« oi Kcond^j ciraiii nal rcactancc. 

J„^„= h = rat 

- = percentage hysteresis, 
_ percentage magnetizing cur- 

and if d represents the load of the transformer, as fractio 
of full load, we have 



and, ^= prf, 

Substituting these values we get, as the equations of the 
transformer on non-inductive load. 
Ratio of E.M.FS. : 

i-o{l + i(p-/.l)) 
or, eliminating imaginaty quantities, 

i«V(l + ./p)' + ■/"<!• 

-"■{l + ^p}. 
Ratio of currents : 

/;_ Ml I (h+yg) I (P-yq)(h+>g) ) 

7, or ^ ^ 2 ) 


or, eliminating imaginary quantities, 



Total apparent primary impedance : 


or, eliminating imaginary quantities. 

Angle of lag in primary circuit : 

l + rfp-^-ph-qg + ^ii 

That is, ^ 

An alternate-current transformer, feeding into a non-induC' 
live secondary circuit, is represented by the constants: 

Rb = secondary ezteinal resistance at full load; 

p = percentage resistance ; 

q ^ percentage reactance \ 

h = percentage hysteresis ; 

g = percentage magnetizing current ; 

d = secondary percentage load. 
All these qualities being considered as reduced to the primary 
circuit by the square of the ratio of turns, a*. 

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140. As an instance, a transformer of the following 
constants may be given ; 

t, -1,000; R,— 120; q - .06 ; 

o - 10; p -.02- h-.02; 

g = .04. 
Substituting these values, gives : 

V(O)014 + .02 dy + (.0002 + .06 ,/)» 
_ e^da' e,d . 
^ ~ '~«r 1.2 ' 
(, _ .1 1,y/^1.0014 + :51Y+ ^:5t _ jm2\ 

.06 ,/+:«_. 0004 -:2«M. 

1.9972 +.02 d+ —■ - Si- 










J it 

-DSL 10 
















It nn 







ft ai 







I "> 













WSi (/ft fanrf maqtam ef Transfwinar. 

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In diagram Fig. 110 are shown, for the values from 
«/= to (/= 1.5, with the secondary current ij as abscis- 
sae, the values : 

secondary terminal voltage, in volts, 

secondary drop of voltage, in per cent, 

primary current, in amps, 

excess of primary current over proportionality with 

secondary, in per cent, 
primary angle of lag. 

The power-factor of the transformer, cos u,, is .45 at 
open secondary circuit, and is above .99 from 25 amperes, 
upwards, with a maximum of .995 at full load. 

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141- The simplest alternating-current apparatus is the 
alternating-current transformer. It consists of a magnetic 
circuit, Interlinked with two electric circuits or sets of 
electric circuits. The one, the primary circuit, is excited 
by an impressed E.M.F., while in the other, the secondary 
circuit, an E.M.F. is induced. Thus, in the primary circuit, 
power is consumed, in the secondary circuit a correspond- 
ing amount of power produced ; or in other words, power 
is transferred through space, from primary to secondary 
circuit. This transfer of power finds its mechanical equiv- 
alent in a repulsive thrust acting between primary and 
secondary. Thus, if the secondary coil is not held rigidly 
as in the stationary transformer, it will be repelled and 
move away from the primary. This mechanical effect is 
made use of in the induction motor, which represents a 
transformer whose secondary is mounted movably with re- 
gard to the primary in such a way that, while set in rota- 
tion, it still remains in the primary field of force. The 
condition that the secondary circuit, while revolving with 
regard to the primary, does not leave the primary field of 
magnetic force, requires that this field is not undirectional, 
but that an active field exists in every direction. One way 
of producing such a magnetic field is by exciting different 
primary circuits angularly displaced in space with each 
other by currents of different phase. Another way is to 
excite the primary field in one direction only, and get the 
cross magnetization, or the angularly displaced magnetic 
field, by the leaction of the secondary current. 

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We see, consequently, that the stationary transformer 
and the induction motor are merely different applications 
of the same apparatus, comprising a magnetic circuit in- 
terlinked with two electric circuits. Such an apparatus 
can properly be called a "general alternating-current trans- 
former" The equations of the stationary transformer and 
those of the induction motor are merely specializations of 
the general alternating-current transformer equations. 

Quantitatively the main differences between induction 
motor and stationary transformer are those produced by 
the air-gap between primary and secondary, which is re- 
quired to give the secondary mechanical movability. This 
air-gap greatly increases the magnetizing current over that 
in the closed magnetic circuit transformer, and requires 
an ironclad construction of primary and secondary to keep 
the magnetizing current within reasonable limits. An iron- 
clad construction again greatly increases the self-induction 
of primary and secondary circuit. Thus the induction 
motor is a transformer of large magnetizing current and 
large self-induction ; that is, comparatively large primary 
exciting susceptance and large reactance. 

The general alternating-current transformer transforms 
between electrical and mechanical power, and changes not 
only E.M.Fs. and currents, but frequencies also, and may 
therefore be called a "frequency converter." Obviously, 
it also may change the number of phases. 

142- Besides the magnetic flux interlinked with both 
primary and secondary electric circuit, a magnetic cross- 
flux passes in the transformer between primary and second- 
ary, surrounding one coil only, without being interlinked 
with the other. This magnetic cross-flux is proportional to 
the current flowing in the electric circuit, and constitutes 
what is called the self-induction of the transformer. As 
seen, as self-induction of a transformer circuit, not the total 
flux produced by and interlinked with this circuit is under- 
stood, but only that — usually small — part of the flux 

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which surrounds the one circuit without interlinking with 
the other, and is thus produced by the M.M.F. of one 
circuit only. 

143. The mutual magnetic flux of the transformer is 
produced by the resultant M.M.F. of both electric circuits. 
It is determined by the counter E.M.F., the- number of 
turns, and the frequency of the electric circuit, by the 
equation: £_V5,^,,10-, 

Where E = effective E.M.F. 

JV^ frequency. 

n = number of turns. 

* = maximum magnetic flux. 
The M.M.F, producing this flux, or the resultant M.M.F. 
of primary and secondary circuit, is determined by shape 
and magnetic characteristic of the material composing the 
magnetic circuit, and by the magnetic induction. At open 
secondary circuit, this M.M.F. is the M.M.F. of the primary 
current, which in this case is called the exciting current, 
and consists of an energy component, the magnetic energy 
current, and a reactive component, the magnetizing current. 

144- In the general alternating-current transformer, 
where the secondary is movable with regard to the primary, 
the rate of cutting of the secondary electric circuit with the 
mutual magnetic flux is different from that of the primary. 
Thus, the frequencies of both circuits are different, and the 
induced E.M.Fs. are not proportional to the number of 
turns as in the stationary transformer, but to the product 
of number of turns into frequency. 

146. Let, in a general alternating-current transformer: 

s = ratio """''"^ frequency, or " slip " ; 
thus, if 

jV= primary frequency, or frequency of impressed E.M.F., 
sJV= secondary frequency ; 

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and the E.M.F. induced per secondary turn by the mutual 
flux has to the E.M.F. induced per primary turn the ratio s, 

■S ^ represents synchronous motion of the secondary ; 

^ < represents motion above synchronism — driven by external 

mechanical power, as will be seen ; 
,f =: 1 represents standstill ; 
s> 1 represents backward motion of the secondary 

that is, motion against the mechanical force acting between 
primary and secondary {thus representing driving by ex- 
ternal mechanical power). 

«o = number of primary turns in series per circuit ; 

ni ^ number of secondary turns in series per circuit; 

a = -5 = ratio of turns ; 

y^ =fi+yii, = primary exciting admittance per circuit; 

gf = effective conductance ; 

^0 = susceptance ; 

Zo = To — Jxq ^ internal primary self-inductive impedance 
per circuit, 

rg = effective resistance of primary circuit; 

jTo = reactance of primary circuit ; 

Z„ = r, — Ai = internal secondary self-inductive impedance 
per circuit at standstill, or for j = 1, 

Ti = effective resistance of secondary coil ; 

*i = reactance of secondary coil at standstill, or full fre- 
quency, s ^ 1. 

Since the reactance is proportional to the frequency, at 

the slip s, or the secondary frequency s N, the secondary 

impedance is: 

Zi = ri—jsx^. 

Let the secondary circuit be closed by an external re- 
sistance r, and an external reactance, and denote the latter 

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by X at frequency N, then at frequency sN, ox slip s, it 
will be ™ sx, and thus : 

Z — r— jix = external secondary impedance.* 
E,t = primary impressed E.M.F. per circuit, 
E ' = E.M.F. consumed by primary counter KM.F., 
£i — secondary terminal E.M.F., 
£^' ~ secondary induced E.M.F., 
i = E.M.F. induced per turn by the mutual magnetic flux, 

at full frequency ^, 
^ = primary current, 
Ji( = primary exciting current, 
Ii = secondary current. 

It is then : 

Secondary induced E.M.F. 

E{ = sn^e. 
Total secondary impedance 

Zi + Z=(r, + r)- js (x, + x) j 
hence, secondary current 

f El' Bftte . 

Secondary terminal voltage 


\ in + r)- js (xi + x) ( (r, + r) - js (a;, + x) 

■ Tbis applies lo the case where the secondary contains inductive reac' 
tance only ; or, rather, thai kind of reactance which is proportional to the fre- 
quency. In a condenser (he reactance is inversely proportional to the frequent^, 
in a synchronous motor under circumstances independent of the frequency. 
Thus, in general, we have lo set, j = r' + j" 4 jr'", where j:' is ihal pari of 
the reactance which is proporiional to the frequency, jt" that part of the reac- 
tance independent o( the frequency, and x"" that pert of the reactance which 
b inversely proportional to Ihe Irequency ; and have thus, at slip i, 01 Irequency 
iN, the external secondary r< 



E.M.F. consumed by primary counter E.M.F. 
E' = —n„ts 
hence, primary exciting current : 

Component of primary current corresponding to second- 
ary current /, : i' = —Jx 

hence, total primary current, 

Primary impressed E.M.F., 

' + 

We get thus, as the 
Equations of the General Alternating-Current Transformer: 

Of ratio of turns, a ; and ratio of frequencies, s ; with the 
E.M,F, induced per turn at full frequency, e, as parameter, 

the values : 

Primary impressed E.M.F., 
Secondary terminal voltage. 

Primary current, 

! «'C'-. + '-)->»('i + «) 



Secondary current, 

{x = 

(n + '■)-/■' (*! + *) 

Therefrom, we get : 
Ratio of currents, 
? = - M 1 + y (f <. + y V [(n + r) -js{x^ + :r)] j 

Katio of E.M.PS., 

1 + ^ 





Total appuent primary impedance, 

1 + 

s «i_/,. 

+ ('■. 


•>■('-. + '■)+>•' (».+«) 

i + yto+>*.)[('-i + 




*=-*'+— 4 

in the general secondary circuit as discussed in foot-note, 
page 221. 

Substituting in these equations: 

j = l, 
gives the 

General Equations of the Stationary Alternating-Current 

z, + z] 




t*(Z, + Z) 

1 + ^ 1- z^ kI 

Substituting in the equations of the general alternating- 
current transformer, 

Z = 0, 
gives the 

General Equations of the Induction Motor: 

E^ = -n^e [ 

Il, = — J «o tf 

' + 7.?: 


, a.+ji. 

fl*(r, -/j^i) ' J 

^.-—('■l -/'».) 

i + -=-(n->'»0(*,+/A) 

Returning now to the general alternating-current trans- 
former, we have, by substituting 

(Ti + '•)' + '* (*i + ■>)' - «.", 

and separating the real and imaginaiy quantities, 

£. ,^,, I [l + -i- (r, (r, + r) + <*.(», + j:)) 

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+ frof. + «.*.)"]+ y fj;^, (» ^ («.+«)-*, (n + >•)) 

Neglecting the exciting current, or rather considering 
it as a separate and independent shunt circuit outside of 
the transformer, as can approximately be done, and assum- 
ing the primary impedance reduced to the secondary circuit 
as equal to the secondary impedance. 

Substituting this in the equations of the general trans- 
former, we get, 

+ "^ ["•.('. + «)-».('■. + '■)] !■ 

^" =- It? '<"+'■> +^'<'-+ '"■ 

146. The true power is, in symbolic representation (see 
Chapter XII.) : 




gives : 

Secondary output of the transformer 

Internal loss in secondary circuit, 

Total secondary power, 

P, + F^ _ (^)V + '•.)-'■"('•+ rO- 
Internal loss in primary circuit, 

/V = ^j'^o = 'i'''i«* = (~^) 'i = ■''>"' 
Total electrical output, plus loss, 
P' = P^ + Pi-^Pi = ( £^y (r + 2 rO - JH- (r + 2 rO. 
Total electrical input of primary. 

Hence, mechanical output of transformer, 

mechinL ca l outpnt " \ ^ S tpecd 

taul KGODdaiy pomr s i si ^ ^Lp * 

147. Thus, 

In a general alternating transformer of ratio of turns, a, 
and ratio of frequencies, f, neglecting exciting current, it is ; 

Electrical input in primary, 

• C. + '■)' + <• ('I + *)■ 


Mechanical output, 

(n+r)' + <■(». + «)■■ 

Electrical output of secondary, 

Losses in transformer, 

/i' + /",' 


Cr, + r)« + ^(x, + *)» 

Of these quantities, /" and /*, are always positive ; /"^ 
and P can be positive or negative, according to the value 
of s. Thus the apparatus can either produce mechanical 
power, acting as a motor, or consume mechanical power; 
and it can either consume electrical power or produce 
electrical power, as a generator. 

14a At 

J = 0, synchronism, P^^d, P=0, Pi = 0. 
At < f < 1, between synchronism and standstill 

/*! , P and Pf, are positive ; that is, the apparatus con- 
sumes electrical power /•(, in the primary, and produces 
mechanical power P and electrical power /*, + P^ in the 
secondary, which is partly, P^, consumed by the internal 
secondary resistance, partly, Pi, available at the secondary 

In this case it is : 

Py, + -Pt' ^ J_ 
P 1-t' 

that is, of the electrical power consumed in the primarj- 
circuit, P^, a part P^ is consumed by the internal pri- 
mary resistance, the remainder transmitted to the secon- 
dary, and divides between electrical power, P^ + P^\ and 
mechanical power, P, in the proportion of the slip, or drop 
below synchronism, s, to the speed : 1 — s. 

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In this range, the apparatus is a motor. 

At f > 1 ; or, backwards driving, 

/* < 0, or negative ; that is, the apparatus requires mechanical 
power for driving. 

It is then : P^ - /"o' - i^' < ^i ; 

that is : the secondary electrical power is produced partly 
by the primary electrical power, partly by the mechanical 
power, and the apparatus acts simultaneously as trans- 
former and as alternating-current generator, with the sec- 
ondary as armature. 

The ratio of mechanical input to electrical input is the 
ratio of speed to synchronism. 

In this case, the secondary frequency is higher than the 

At J < 0, beyond synchronism, 

P <0\ that is, the apparatus has to be driven by mechanical 

/o< 0; that is, the primary circuit produces electrical power 
from the mechanical input. 

At r + n + ^n = 0, or, s<- ^1±-^' ; 

the electrical power produced in the primary becomes less 
than required to cover the losses of power, and Pq becomes 
positive again. 
We have thus : 

consumes mechanical and primary electric power ; produces 
secondary electric power. 

consumes mechanical, and produces electrical power in 
primary and in secondary circuit. 

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consumes primary electric power, and produces mechanical 
and secondary electrical power, 

consumes mechanical and primary electrical power ; pro- 
duces secondary electrical power. 

149. As an instance, in Fig. Ill are plotted, with the 
slip s as abscissfe, the values of : 

Secondary electrical output as Curve I. ; 
Total internal loss as Curve II. ; 

Mechanical output as Curve III. ; 

Primary electrical input as Curve IV. ; 
for the values : 

«,^= 100.0; r = .4; 

n = -1; X ~ .3: 

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8,000 i". 
1 + '"' 

4,000 , + (5 + j) 

1 + .- 

20,000 »(!-<) 


P^ + y,' = 

1 + ^ 

150. Since the most common practical application of 
the general alternating current transformer is that of fre- 
quency converter, that is to change from one frequency to 
another, either with or without change of the number of 
phases, the following characteristic curves of this apparatus 
are of great interest. 

1. The regulation curve ; that is, the change of second- 
ary terminal voltage as function of the load at constant im- 
pressed primary voltage. 

2. The compounding curve ; that is, the change of pri- 
mary impressed voltage required to maintain constant sec- 
ondary terminal voltage. 

In this case the impressed frequency and the speed are 
constant, and consequently the secondary frequency. Gen- 
erally the frequency converter is used to change from a low 
frequency, as 25 cycles, to a higher frequency, as 62,5 
cycles, and is then driven backward, that is, against its 
torque, by mechanical power. Mostly a synchronous motor 
is employed, connected to the primary mains, which by 
over-excitation compensates also for the lagging current of 
the frequency converter. 


Y^= ga +/^o — primary exciting admittance per circuit 
of the frequency converter. 

Z^ = rj —jx^= internal self inductive impedance per 
secondary circuit, at the secondary frequency. 

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Z^ = r^ —jx^ = internal self inductive impedance per 
primary circuit at the primary frequency. 

a = ratio of secondary to primary turns per circuit. 

d = ratio of number of secondary to number of primary 

c = ratio of secondary to primary frequencies. 


^ = induced E.M.F. per secondary circuit at secondary 

Z =r —jx — external impedance per secondary circuit 
at secondary frequency, that is load on secondary system, 
where jr = for noninductive lead. 

We then have, 

total secondary impedance, 

Z + Z, = (r+r^-j{x + x;) 

secondary current, 


' C'- + n)' + (' + ^0' ' ('• +rd' + C* + *0* 

secondary terminal voltage, 

= e{r -Jx) (a, -k-jai) = e (h +Jb^ 

*i = ("»! + ^<*i) ^t = (ra^ — *■«!) 

primary induced E.M.F. per circuit, 

.£■' = — 

primary load current per circuit, 

primary exciting current per circuit. 



thus, total primary current, 

/o = /» + 7„ 


primary terminal voltage : 

-^ = v. - ^«w 

substituting this value of e in the preceding equations, 
gives, as function of the primary impressed E.M.F., e : 
secondary current, 

secondary terminal voltage, 

Vai* 4- d} 
primary current, 

primary impressed E.M.F. 

secondary output, 

•4' + ^" 


primary electrical input, 

P. = IEJ.\' 


primary apparent input, voltamperes, 

Substituting thus diiferent values for the secondary in- 
ternal impedance Z gives the regulation curve of the fre- 
quency converter. 









































fig. 112. 

Such a curve, taken from tests of a 200 KW frequency 
fonverter changing from 6300 volts 25 cycles three-phase, 
to 2500 volts 62.5 cycles quarter-phase, is given in Fig. 

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From the secondary terminal voltage, 
it follows, absolute, 














NQ 1 


— ■ 









a p 




Substituting these values in tne above equation gives 
the quantities as functions of the secondary terminal vol- 
tage, that is at constant ^,, or the compounding curve. 

The compounding curve of the frequency converter 
above mentioned is given in Fig, H3, 

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161. A specialization of the general alternating-current 
transformer is the induction motor. It differs from the 
stationary alternating-current transformer, which is also a 
specialization of the general transformer, in so ^ as in the 
stationary transformer only the transfer of electrical enei;gy 
from primary to secondary is used, but not the mechanical 
force acting between the two, and therefore primary and 
secondary coils are held rigidly in position with regard to 
each other. In the induction motor, only the mechanical 
force between primary and secondary is used, but not the 
transfer of electrical energy, and thus the secondary circuits 
closed upon themselves. Transformer and induction motor 
thus are the two limiting cases of the general alternating- 
current transformer. Hence the induction motor consists 
of a magnetic circuit interlinked with two electric circuits or 
sets of circuits, the primary and the secondary circuit, which 
are movable with regard to each other. In general a num- 
ber of primary and a number of secondary circuits are used, 
angularly displaced around the periphery of the motor, and 
containing E.M.Fs. displaced in phase by the same angle. 
This multi-circuit arrangement has the object always to 
retain secondary circuits in inductive relation to primary 
circuits and vice versa, in spite of their relative motion. 

The result of the relative motion between primary and 
secondary is, that the E.M.Fs. induced in the secondary or 
the motor armature are not of the same frequency as the 
E.M.Fs. impressed upon the primary, but of a frequency 
which is the difference between the impressed frequency 

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and the frequency of rotation, or equal to the " slip," that is, 
the difference between synchronism and speed (in cycles). 
Hence, if 

N = frequency of main or primary E,M.F., 

and s = percentage slip ; 

sJV =s frequency of armature or secondary KM.F., 

and (1 — j) JV= frequency of rotation of armature. 

In its reaction upon the primary circuit, however, the 
armature current is of the same frequency as the primary 
current, since it is carried around mechanically, with a fre- 
quency equal to the difference between its own frequency 
and that of the primary. Or rather, since the reaction of 
the secondary on the primary must be of primary frequency 
— whatever the speed of rotation — the secondary frequency 
is always such as to give at the existing speed of rotation a 
reaction of primary frequency. 

152. Let the primary system consist of /„ equal circuits, 
displaced angulary in space by 1 /p^ of a period, that is, 
1 /fi^ of the width of two poles, and excited by/,, E.M.Fs. 
displaced in phase by 1 //, of a period ; that is, in other 
words, let the field circuits consist of a symmetrical p^-phaise 
system. Analogously, let the armature or secondary circuits 
consist of a symmetrical /,-phase system. 

Mo = number of primary turns per circuit or phase ; 
ni = number of secondary turns per circuit or phase ; 
a = —- ss ratio of total primary turns to total secondary turns 
or ratio of transformation. 

Since the number of secondary circuits and number of 
turns of the secondary circuits, in the induction motor — as 
in the stationary transformer — is entirely unessential, it is 
preferable to reduce all secondary quantities to the primary 
system, by the ratio of transformation, a ; thus 

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if El = secondary E.M.F. per circuit, £i = a£i' 

= secondary RM.F. per circuit reduce to primary system; 

if /,' = secondary current per circuit, ^ = — 

= secondary current per circuit reduced to primary system j 
if r^ = secondary resistance per circuit, r^ = a' r{ 

=s secondary resistance per circuit reduced to primary system ; 
if .T,' jss secondary reactance per circuit, jc, e= a'jc', 

= s'.condary reactance per circuit reduced to primary system ; 
if 2,' = secondary impedance per circuit, a, =^/, 

= secondary impedance per circuit reduced to primary system ; 

that is, the number of secondary circuits and of turns per 
secondary circuit is assumed the same as in the primary 

In the following discussion, as secondary quantities, the 
values reduced to the primary system shall be exclusively 
used, so that, to derive the true secondary values, these 
quantities have to be reduced backwards again by the factor 

a = — 

4> = total maximum flux of the magnetic field per motor pole, 
We then have 

£= ■^■rtiN'bW-* =effective KM.F. induced by the mag- 
netic field per primary circuit. 

Counting the time from the moment where the rising 

magnetic flux of mutual induction * (flux interlinked with 

both electric circuits, primary and secondary) passes through 

zero, in complex quantities, the magnetic flux is denoted by 

* =!_/■*, 

and the primary induced E.M.F., 

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tf— '^w«^* 10-* may be considered as the "Active E.M.F. 
ol the motor," or « Counter E.M.F." 

Since the secondaiy frequency is sN, the secondary in- 
duced E.M.F. (reduced to primary system) is ^i = — se. 

Z„ = exciting current, or current passing through the motor, per 

primary circuit, when doing no work (at synchronism), 

Y = g +J6 = orimaiy admittance per circuit = — . 
We thus have, 

ge = magnetic energy current, g^ = loss of power oy hysteresis 
(and eddy currents) per primary coil. 

Ptg^ = total loss of energy by hysteresis and eddys, 
as calculated according to Chapter X. 
^« = magnetizing current, and 
fi^ie = effective M.M.F, per primary circuit; 

hence "Tj/i^^e = total effective M.M.F.; 

^«,^« = total maximum M.M.F., as resultant of the M.M.Fs. 
^ of the /o-phases, combined by the parallelogram of 


If (R = reluctance of magnetic circuit per pole, as dis- 
cussed in Chapter X., it is 

• CompIeK diKiuaLon hereof, lee Chapter XXV. 

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Thus, from the hysteretic loss, and the reluctance, the 
constants, g and b, and thus the admittance Kare derived. 

Let r^ = resistance per primary drcuit ; 
x^ = reactance per primary drcuit ; 

Z^ = ro —j'x^ = impedance per primary circuit; 
rj = resistance per secondary circuit reduced to pri- 
mary system ; 
Xi SS5 reactance per secondary circuit reduced to primary 
system, at full frequency, JV; 

sxj = reactance per secondary circuit at slip * j 

Zi = Ti —jsxi = secondary internal impedance. 

164. We now have, 
Primary induced E.M.F., 

Secondary induced RM.F,, 

Secondary current, 

Component of primary current, corresponding thereto, or 
primary load current, 

Primary exciting current, 

/„ =ffr=((f+y*); hence. 

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Total primary current, 

E.H.F. consumed by primaiy impedance, 

- ' (.r, -/X) j * + (/ +/•») j i 

E.M.F. required to overcome the primary induced E.M.F., 
-E =e; 

Primaiy terminal voltage, 

We get thus, in an induction motor, at slip s and active 
E.M.F. e, 

Primary terminal voltage, 

Primary current, 

or, in complex expression, 
Primary terminal voltage, 

Primary current, 



To eliminate e, we divide, and get, 

Frimary current, at slip s, and impressed E.M.F., E^\ 


,_ ' + (ri-j"^){g+J^ , 

Neglecting, in the denominator, tlie small quantity 
^iZ,r, itis 

Zi + sZa 

(s + r,H-,x,i) +J (r, i - ,x,g) 


or, expanded, 

[(.r, + J'r.) + r.'f + >/■, (/ii- - «^) + ^«, (»^ + :.,<•+ V)] + 
r . +jV(.x,Jrx,)+r-!,+sr,(x^+rJ,)+fx,(,xJ.+x,i-r^)-\ ^ 

Hence, displacement of phase between current and 

tan u, =: 

'' {x, + J.) + rfi + sr, (irf + >■/) + ^x, (x^ + x,i- r„f ) 
(^, + A,) + r,-g +<r, (r^- V) + fx, ix^+ x,g + r^) ' 

Neglecting the exciting current, /(, altogether, that is, 
setting K= 0, 
We have 

^ W + r^i.) +/'»('. + ■»■) 
'\r, + ,r,)' + f(x, + xy 


■ '(». + »■) 




166. In graphic representation, the induction tnotor dia- 
gram appears as follows : — 

Denoting the magnetism by the vertical vector 0* in 
Fig. 114, the M.M.F. in ampere-turns per circuit is repre- 
sented by vector OF, leading the magnetism tJ* by the 
angle of hysteretic advance a. The E.M.F. induced in the 
secondary is proportional to the slip J, and represented by 
OE^ at the amplitude of 180°. Dividing OE^ by a in the 
proportion of r, -i- sx^, and connecting a with the middle b 
of the upper arc of the circle OE^, this line intersects the 
lower arc of the circle at the point /, r,. Thus, Ol^r^ is the 
E.M.F. consumed by the secondary resistance, and (3/^, 
equal and parallel to EJ^^ is the E.M.F. consumed by the 
secondary reactance. The angle, Efil^r^ = w, is the angle 
of secondary lag. 

The secondary M.M.F. OG^ is in the direction of the 
vector (?/,r,. Completing the parallelogram of M.M-Fs. 
with C^as diagonal and OC, as one side, gives the primary 
M.M.F. OG as other side. The primary current and the 
E.M.F. consumed by the primary resistance, represented by 
Olr^ is in line with OG, the E.M.F. consumed by the pri- 
mary reactance 90° ahead of OG, and represented by OIx„ 
and their resultant OIc^ is the E.M.F. consumed by the 

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primary impedance. The E.M.F. induced in the primary 
circuit is OE', and the E.M.F. required to overcome this 
counter E.M.F. is OE equal and opposite to OE'. Com- 
bining OE with OIs^ gives the primary terminal voltage 
represented by vector OE^ and the angle of primary lag, 
EfiG = *v 

166. Thus far the diagram is essentially the same as 
the diagram of the stationary alternating-current trans- 
former. Regarding dependence upon the slip of the motor, 
the locus of the different quantities -for different values of 
the slip s is determined thus, 

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Let E^ = sS 

Assume in opposition to 04, a point A, such that 
0-4-»-/,r, = £i + 7iJj:„ then 
OA = -i-4 = -^ = -i E = constant. 

That is, 7,^1 lies on a half-circle with OA =—£' as 
diameter. ' 

That means £7, lies on a half-circle ^, in Fig. 115 with 
^C as diameter. In consequence hereof, G^ lies on half- 
circle ^, with FB equal and parallel to OC as diameter. 

Thus /r, lies on a half -circle with DH as diameter, which 
circle is perspective to the circle F£, and /x^ lies on a half- 
circle with /A' as diameter, and /s^ on a half-circle with L/^ 
as diameter, which drcle is derived by the combination of 
the circles Ir^ and /x„ 

The primary terminal voltage E^ lies thus on a half- 
circle e^ equal to the half-circle /z„ and having to point 
£ the same relative position as the half-circle /s^ has to 
point 0. 

This diagram corresponds to constant intensity of the 
maximum magnetism, O*. If the primary impressed volt- 
age £^ is kept constant, the circle e^ of the primary im- 
pressed voltage changes to an arc with O as center, and all 
the corresponding points of the other circles have to be 
reduced in accordance herewith, thus giving as locus of the 
other quantities curves of higher order which most con- 
veniently are constructed point for point by reduction from 
the circle of the loci in Fig. 115. 

Torque and Power. 

157. The torque developed per pole by an electric motor 
equals the product of effective magnetism, */ V2, times ef- 
fective armature M.M.F., F / V2, times the sine of the 
angle between both. 

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7- = -^sin(*^. 

If », = number of turns, /; = current, per circuit, with 
/i-armature circuits, the total maximum current polarization, 
or M.M.F. of the armature, is 

' V2 
Hence the torque per pole. 

If y = the number of poles of the motor, the total torque 
of the motor is, 

2 V2 ^ '' 

The secondary induced E,M.F., £",, lags 90° behind the 
inducing magnetism, hence reaches a maximum displaced in 
space by 90° from the position of maximum magnetization. 
Thus, if the secondary current, /j, lags behind its E.M.F., 
E^, by angle, \, the space displacement between armature 
current and field magnetism is 

hence sin (* /J = cos 5, 

We have, however, 

cos W| 

/, = 

tf= ^/§1^«J*A'10-^ 
substituting tliese values in the equation of the torque, it i 



or, in practical (C.G.S.) units. 

is the Torque of the Induction Motor. 

At the slip s, the frequency N, and the number of poles 
q, the linear speed at unit radius is 

hence the output of the motor, 
or, substituted, 

is the Power of the Induction Motor. 

158. We can arrive at the same results in a different 
way : 

By the counter E.M.F. e of the primary circuit with 
current I = I^-^ I^ the power is consumed, e I = el^-V el^. 
The power e !„ is that consumed by the primary hysteresis 
and eddys. The power ^/(disappears in the primary circuit 
by being transmitted to the secondary system. 

Thus the total power impressed upon the secondary 
system, per circuit, is 

/", = */i. 
Of this power a part, ^,/„ is consumed in the secondary 
circuit by resistance. The remainder, 

disappears as electrical power altogether ; hence, by the law 
of conservation of energy, must reappear as some other 
form of energy, in this case as mechanical power, or as the 
output of the motor (including friction). 

Thus the mechanical output per motor circuit is 

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7, = 

<■>(!-.) (r.+y,»,) . 

hence, since the imaginary part has no meaning as power, 

•^ - r; + fx; ' 

and the total power of the motor, 


'•." + '■«■■ ■ 

At the linear speed. 


at unit radius the torque is 

7, fAn^^ 

4ir^(r,« + J*^*)- 

In the forgoing, we found 

M. = ,\i + >§ + z.y\ 

or, approximately, 



or, eliminating imaginary quantities, 



Substituting this value in the equations of torque and of 
power, they become, 
tonjue, . T IJiTl^- 

power, /"= 

Afi E},{ \-,) ■ 

Maximum Torque. 
169. The torque of the induction motor is a maximum 
for that value of slip s, where 

expanded, this gives, 

-^' + r,« + (x,+V = 0, 

Substituting this in the equation of torque, we get the 
value of maximum torque, 

J. ^ ?A-g»' 

That is, independent of the secondary resistance, r^ 
The power corresponding hereto is, by substitution of j, 

/"/ = 

A- gflMW+C- ^i + V ■ 

2 V^TTC*! + V ! Vv + (*, + x;f + r„j ' 

This power is not the maximum output of the motor, 
but already below the maximum output. The maximum 
output is found at a lesser slip, or higher speed, while at 
the maximum torque point the output is already on the 
decrease, due to the decrease of speed. 

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With increasing slip, or decreasing speed, the torque of 
the induction motor increases ; or inversely, with increasing 
load, the speed of the motor decreases, and thereby the 
torque increases, so as to carry the load down to the slip s,, 
corresponding to the maximum torque. At this point of 
load and slip the torque begins to decrease again ; that is, 
as soon as with increasing load, and thus increasing slip, 
the motor passes the maximum torque point j^ it "falls out 
of step," and comes to a standstill. 

Inversely, the torque of the motor, when starting from 
rest, will increase with increasing speed, until the maximum 
torque point is reached. From there towards synchronism 
the torque decreases again. 

In consequence hereof, the part of the torque-speed 
curve below the maximum torque point is in general un- 
stable, and can be observed only by loading the motor 
with an apparatus, whose countertorque increases with the 
speed faster than the torque of the induction motor. 

In general, the maximum torque point, s^ is between 
synchronism and standstill, rather nearer to synchronism. 
Only in motors of very large armature resistance, that is 
low efficiency, j, > 1, that is, the maximum torque falls 
below standstill, and the torque constantly increases from 
synchronism down to standstill. 

It is evident that the position of the maximum torque 
point, J, can be varied by varying the resistance of the 
secondary circuit, or the motor armature. Since the slip 
of the maximum torque point, s^, is directly proportional to 
the armature resistance, r^, it follows that very constant 
speed and high efficiency will bring the maximum torque 
point near synchronism, and give small starting torque, 
while good starting torque means a maximum torque point 
at low speed ; that is, a motor with poor speed regulation 
and low efficiency. 

Thus, to combine high efficiency and close speed regula- 
tion with large starting torque, the armature resistance has 

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to be varied during the operation of the motor, and the 
motor started with high annature resistance, and with in- 
creasing ^)eed this armature resistance cut out as far as 

160. If J, = l, 

it is r, = VV + (j;, + x^. 

In this case the motor starts with maximtmi torque, and 
when overloaded does not drop out of step, but gradually 
slows down more and more, until it comes to rest. 

If, H>\, 

then n> Vro'+(x,+ 3^)». 

In this case, the maximum torque point is reached only 
by driving the motor backwards, as countertorque. 

As seen above, the maximum torque T„ is entirely in- 
dependent of the armature resistance, and likewise is the 
current corresponding thereto, independent of the armature 
resistance. Only the speed of the motor dep»ids upon the 
armature resistance. 

Hence the insertion of resistance into the motor arma- 
ture does not change the maximum torque, and the current 
corresponding thereto, but merely lowers the speed at which 
the maximum torque is reached. 

The effect of resistance inserted into the induction motor 
is merely to consume the E.M.F., which otherwise would 
find its mechanical equivalent in an increased speed, analo- 
gous as resistance in the armature circuit of a continuous- 
current shunt motor. 

Further discussion on the effect of armature resistance 
is found under *' Starting Torque." 

Maximum Power, 
161. The power of an induction motor is a maximum 
for that slip, s., where 

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„.,o„... . (r.+,r.)- + <■{;., + », 


J ((r, +.r.)> + ^(«, + »,y| 

Js\ <(!-') ) 

expanded, this gives 


substituted in P, we get the maximum power, 

2 J (r, + r.) + V(ri + r«)» + {x^ + x^f j 

This result has a simple physical meaning ; (r, 4- r„) = r 
is the total resistance of the motor, primary plus secondary 
(the latter reduced to the primary), {x^ + x^ is the total 
reactance, and thus Vfj + r^)' + (;c, + x^ = ^ is the total 
impedance of the motor. Hence 

p - t,B; 

^'-^ir + .r 
is the maximum output of the induction motor, at the slip, 

^'' ^ r^ + s' 

The same value has been derived in Chapter IX., as the 
maximum power which can be transmitted into a non- 
mductive receiver circuit over a line of resistance r, and 
impedance s, or as the maximum output of a generator, or 
of a stationary transformer. Hence ; 

T/te maximum output of an induction motor is expressed 
by the same formula as the maximum output of a generator, 
or of a stationary transformer, or the maximum output which 
can be transmitted over an inductive line into a non-inductive 
receiver circuit. 

The torque corresponding to the maximum output P^ i^ 
' ^■,rNz{r+zy 

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This is not the maximum torque ; but the maximum 
torque, T^ takes place at a lower speed, that is, greater slip. 

■ Vr„' + (*,+«(,)= r, + V(r, + r,) + (ar, + :ro)» ' 
that is, J( > JJH 

It is obvious from these eqiiations, that, to reach as large 
an output as possible, rand z should be as small as possible ; 
that is, the resistances r, + Vg, and the impedances, z, 
and thus the reactances, x^ + x„ should be small. Since 
r^ + 'o is usually small compared with x^ + Xg it follows, that 
the problem of induction motor design consists in con- 
structing the motor so as to give the minimum possible 
reactances, ;r, + x^- 

Starting Torque. 

162, In the moment of starting an induction motor, 
the slip is 

s = \; 

hence, starting current, 

{'-1 - Ai) + ira - >tt) + ('"I - Ai) ('ii - >o) {.s-^jb) ■ "' 

or, expanded, with the rejection of the last term in the 
■denominator, as insignificant, 

and, displacement of phase, or angle of lag, 

. . _ (x,-\-x,)-\-b (r. [r. + rp] + x, [x, + x,]) - g jr^x, - x„r,-) 

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Neglecting the exciting current, g—^ = b, these equa- 
tions assume the form, 

or, eliminating imaginary quantities, 

r - ^ - ^-. 

■^ir^ + r^ + (x^ + ^.)' 


'i + 'V * 

That means, that in starting the induction motor without 
■ additional resistance in the armature circuit, — in which case 
x^ + x„ is large compared with r, + fo, and the total impe- 
dance, s, small, — the motor takes excessive and greatly 
lagging currents. 

The starting torque is 

Ti = 

9 Pi n A* 

irrN z'' 

That is, the starting torque is proportional to the 
armature resistance, and inversely proportional to the square 
of the total impedance of the motor. 

It is obvious thus, that, to secure large starting torque, 
the impedance should be as small, and the armature resis- 
tance as large, as possible. The former condition is the 
condition of large maximum output and good efficiency 
and speed regulation ; the latter condition, however, means 
inefficiency and poor regulation, and thus cannot properly 
be fulfilled by the internal resistance of the motor, but only 
by an additional resistance which is short-circuited while 
the motor is in operation. 

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Since, necessarily, 


we have, 

and since 



current is, approximately. 

we have, 

would be the theoretical torque developed at 100 per cent 
efficiency and power factor, by E.M.F., E„ and current, I, 
at synchronous speed. 

Thus, T,<T^ 

and the ratio between the starting torque T^ and the theo- 
retical maximum torque, 7"a„ gives a means to judge the 
perfection of a motor regarding its starting torque. 

This ratio, 7^/ 7"„, exceeds .9 in the best motors. 

Substituting / = ^'o / .s in the equation of starting torque, 
it assumes the form. 


T — y*^' nr 

Since ^tr N f q = synchronous speed, it is : 

TAe starting torque of the induction motor is equal to the 
resistance loss in the motor armature, divided by the synchro- 
nous speed. 

The armature resistance which gives maximum starting 
torque is 

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expanded, this gives, 

the same value as derived in the paragraph on "maximum 

Thus, adding to the internal annature resistance, r,' in 
starting the additional resistance, 

makes the motor start with maximum torque, while with in- 
creasing speed the torque constantly decreases, and reaches 
zero at synchronism. Under these conditions, the induc- 
tion motor behaves similarly to the continuous-current series 
motor, varying in the speed with the load, the difference 
being, however, that the induction motor approaches a 
definite speed at no load, while with the series motor the 
speed indefinitely increases with decreasing load. 

The additional armature resistance, r", required to give 
a certab starting torque, if found from the equation of 
starting torque : 

Denoting the internal armature resistance by r,', the total 
armature resistance is r, = r/ + r". 

and thus, „ _ gp^ E^ r( + r" 

■'' 4 ^ A^ (r( + r," + r^ + {x^ + x^ ' 


'."= -'.--'•.+ ISi:±V U^S^S -^i^--<'.+'.)'- 

This gives two values, one above, the other below, the 
maximum torque point. 

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Choosing the positive sign of the root, we get a larger 
armature resistance, a small current in starting, but the 
torque constantly decreases with the speed. 

Choosing the negative sign, we get a smaller resistance, 
a large starting current, and with increasing speed the 
torque first increases, reaches a maximum, and then de- 
creases again towards synchronism. 

These two points correspond to the two points of the 
speed-torque curve of the induction motor, in Fig. 116, 
giving the desired torque 7^ 

The smaller value of r," will give fairly good speed regu- 
lation, and thus in small motors, where the comparatively 
large starting current is no objection, the permanent arma- 
ture resistance may be chosen to represent this value. 

The larger value of r," allows to start with minimum 
current, but requires cutting out of the resistance after the 
start, to secure speed regulation and efficiency. 

163, At synchronism, i =0, we have. 

that is, power and torque are zero. Hence, the induction 
motor can never reach complete synchronism, but must 
slip sufficiently to give the torque consumed by friction. 

Running near Synchrenism. 

164. When running near synchronism, at a slip s above 
the maximum output point, where s is small, from .02 to 
.05 at full load, the equations can be simplified by neglect- 
ing terms with s, as of higher order. 

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We then have, current, 

tj . " 

or, eliminating imaginary quantities, 

= V(^ 

+ y +/f£.; 

angle of lag, 

"B = 

''(*, + ».> 


" «r, + , 






or, inversely. 


■ = 

that is. 

Near sychronism, the slip, s, of an induction motor, or 
its drop in speed, is proportional to the armature resistance, 
r^ and to the power, P, or torque, T. 


165. As an instance are shown, in Fig. 116, character- 
istic curves of a 20 horse-power three-phase induction motor, 
of 900 revolutions synchronous speed, 8 poles, frequency 
of 60 cycles. 

The impressed E.M.F. is 110 volts between lines, and 
the motor star connected, hence the E.M.F. impressed per 


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The constants of the motor are : 

Primary admittance, Y = 
Primary impedance, Z = 
Secondary impedance, Z^ = 

.1 +.4 
: .03 - .09 / 
.02 - .086/ 

In Fig. 116 is shown, with the speal in per cent of 

synchronism, as abscissae, the torque in kilogrammetres, 

as ordinates, in drawn lines, for the values of armature 
resistance : 































































— - 





























Flg.Ve. 8p—il OharacfrlttlBi of Inauatlon H 

ri = .02 : short circuit of annature, full speed. 

^1 = .045 : .025 ohms additional resistance. 

r, = .18 : ,16 ohms additional, maximum starting torque. 

r^ =s ,75 : .73 ohms additional, same starting torque as r, = .046. 

On the same Figure is shown the current per line, in 
dotted lines, with the verticals or torque as abscissas, and 
the horizontals or amperes as ordinates. To the same 
torque always corresponds the same current, no matter 
what the speed be. 

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On Fig. 117 is shown, with the current input per line as 
abscissae, the torque in kUogrammetres and the output in 
horse-power as ordinates in drawn lines, and the speed and 
the magnetism, in per cent of their synchronous values, as 
ordinates in dotted lines, for the armature resistance r, = ,02 
or short circuit. 













.. t 








































































fit. 117, 

rt Cliaraettrlitia of luduetian IMar, 

In Fig. 118 is shown, with the speed, in per cent of 
synchronism, as abscissae, the torque in drawn line, and 
the output in dotted line, for the value of armature resist- 
ance ^1= .045, for the whole range of speed from 120 per 

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cent backwards speed to 220 per cent beyond synchronism, 
showing the two maxima, the motor maximum at ^ = .25, 
and the generator maximum at j = — .25. 

166. As seen in the preceding, the induction motor is 
characterized by the three complex imaginary constants, 
y<i = Sa +/^a the primary exciting admittance, 
Zo = To —Am the primary self-inductive impedance, and 
Zy = r, —j'xi, the secondary self -inductive impedance, 












..».,. 'oo 
































































£ \ 



j \ 


Fig. f 18. SpMrf Chanuttrlitlei of Maetlwi Matw. 

reduced to the primary by the ratio of secondary to pri- 
mary turns. 

From these constants and the impressed E.M.F. Cg, the 
motor can be calculated as follows : 


e = counter E,M.F. of motor, that is E.M.F. induced in 
the primary by the mutual magnetic flux. 

At the slip s the E.M.F. induced in the secondary cir. 
cuit is, j^ 

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Thus the secondaiy current, 


The primary exciting current is, 

thus, the total primary current, 

/, = /. + /„ = tf (^ +^»^ 

*1 = «1 + ^0. 6t= "% + *• 

The E.M.F. consumed by the primary impedance is, 

f = A^o = « (To -A) (^ + A) 
the primary counter E.M.F. is ^, thus the primary impressed 


<■! = 

1 + rA + * 

or, absolute, 

<0 = 

e Vf,» + r,» 



A' + .,' 

This value substituted gives. 

Secondary current 

A = 

Primary current, 



Impressed E.M.F., 


.„ ^'+^^ 

V^,' • 



Thus torque, in synchronous watts (that is, the watts 
output the torque would produce at synchronous speed), 

~ 'i* + (k* 
hence, the torque in absolute units, 

T f,'<7i 

where JV= frequency. 

The power output is torque times speed, thus : 

The power input is, 

'.■+',- •' 


The voltampere input, 


power factor, 

apparent efEciency, 

P, _ «, (1 - rt 

torque efEciency, * 
T a, 

^.' ~ Vi + Vi 

■ Itut !• the nliD ol Ktiul torqu ta lorqaa which mDU b* pRMocnl, U Oun m 


apparent torque efficiency,* 

167, Most instructive in showing the behavior of an 
induction motor are the load curves and the speed curves. 

The load curves are curves giving, with the power out- 
put as abscissae, the current imput, speed, torque, power 
factor, efficiency, and apparent efficiency, as ordinates. 

The speed curves give, with the speed as abscissae, the 
torque, current input, power factor, torque efficiency, and 
apparent torque efficiency, as ordinates. 

The load curves characterize the motor especially at its 
normal running speeds near synchronism, the speed curves 
over the whole range of speed. 

In Fig. 119 are shown the load curves, and in Fig. 120 

the speed curves of a motor of the constants, 

n = .01 + .1/ 

Z^ = X -.3/ 

2, = .l -.3/ 


168. In the foregoing, the range of speed from s=l, 
standstUl, to j = 0, synchronism, has been discussed. In 
this range the motor does mechanical work. 

It consumes mechanical power, that is, acts as generator 
or as brake outside of this range. 

For, J > 1, backwards driving, P becomes negative, 
representing consumption of power, while T remains posi- 
tive ; hence, since the direction of rotation has changed, 
represents consumption of power also. AU this power is 
consumed in the motor, which thus acts as brake. 

For, J<0, or negative, P and 7" become negative, and 
the machine becomes an electric generator, converting me- 
chanical into electric energy. 

• That is Ihe rillo of uluil torque to torque which mold be produced if there wtrt 

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The calculation of the induction generator at constant 
frequency, that is, at a speed increasing with the load by the 
negative slip, j,, is the same as that of the induction motor 
except that j, has negative values, and the load curves for 
the machine shown as motor in Fig. 119 are shown in Fig. 
121 for negative slip j, as induction generator. 











































































Fit. "A 

Again, a maximum torque point and a maximum output 
point are found, and the torque and power increase from 
2ero at synchronism up to a maximum px>int, and then de- 
crease again, while the current constantly increases. 

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% eS^irves . 

'^^ '"^•^' 'X 


¥ -» z^'^^lk 

. ====^::>g^::j;:?ss 

„.„e^^^=^^^x 5: »- 

j=s-2--'- ,,sS^ Vj- • " 

» a! ? A •"" 

15 " \ "" 

^ 3S 

"7 ^Sm 


^ 1U 


-fi Y =0 +, i 

_ ; -»■" 

110 ^ 

_-,s .» 


2'tB y, 

"2 -». 

~^c.icy J ' 


.— ■'" " • ' " 

rr-^a-^ «« « 

al &' 

-^^ ^s ^ 

/ -■• 

^ ^- 

-"^ »,. 

^^ ^^ - 

-' ' » 

^ ^'^ 


' ^^^ 


"! T 

ion M 




i ^. jre."^ 




169. The induction generator differs essentially from 
the ordinary synchronous alternator in so far as the induc- 
tion generator has a definite power factor, while the syn- 
chronous alternator has not. That is, in the synchronous 
alternator the phase relation between current and terminal 
voltage entirely depends upon the condition of the external 
circuit. The induction generator, however, can operate 
only if the phase relation of current and E.M.F,, that is, the 
power factor required by the external circuit, exactly coin- 
cides with the internal power factor of the induction gen- 
erator. This requires that the power factor either of the 
external circuit or of the induction generator varies with 
the voltage, so as to permit the generator and the external 
circuit to adjust themselves to equality of power factor. ■ 

Beyond magnetic saturation the power factor decreases ; 
that is, the lead of current increases in the induction ma- 
chine. Thus, when connected to an external circuit of con- 
stant power factor the induction generator will either not 
generate at all, if its power factor is lower than that of the 
external circuit, or, if its power factor is higher than that of 
the external circuit, the voltage will rise until by magnetic 
saturation in the induction generator its power factor has 
fallen to equality with that of the external circuit. This, 
however, requires magnetic saturation in the induction gen- 
erator, which is objectionable, due to excessive hysteresis 
losses in the alternating field. 

To operate below saturation, — that is, at constant inter- 
nal power factor, — the induction generator requires an exter- 
nal circuit with leading current, whose power factor varies 
with the voltage, as a circuit containing synchronous motors 
or synchronous converters. In such a circuit, the voltage 
of the induction generator remains just as much below the 
counter E.M.F. of the synchronous motor as necessary to 
give the required leading exciting current of the induction 
generator, and the synchronous motor can thus to a certain 
extent be called the exciter of the induction generator. 

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When operating self-exciting, that is shunt-wound, con- 
verters from the induction generator, below saturation of 
both the converter and the induction generator, the condi- 
tions are unstable also, and the voltage of one of the two 
machines must rise beyond saturation of its magnetic field. 

When operating in parallel with synchronous alternat- 
ing generators, the induction generator obviously takes its 
leading exciting current from the synchronous alternator, 
which thus carries a lagging wattless current. 

170. To generate constant frequency, the speed of the 
induction generator must increase with the load. Inversely, 
when driven at constant speed, with increasing load on the 
induction generator, the frequency of the current generated 
thereby decreases. Thus, when calculating the character- 
istic curves of the constant speed induction generator, due 
regard has to be taken of the decrease of frequency with 
increase of load, or what may be called the slip of fre- 
quency, s. 

Let in an induction generator, 

Jo = ^0 -t- jl'r, = primary exciting admittance, 

Zf, = r^ —j'xa = primary self-inductive impedance, 

Z, = ^1 —jxy sa secondary self-inductive impedance, 

reduced to primary, all these quantities being reduced to 
the frequency of synchronism with the speed of the ma- 
chine, N. 

Let e = induced E.M.F., reduced to full frequency. 

s = slip of frequency, thus ; (1-f) N= frequency gener- 
ated by machine. 

We then have 

Secondary induced E.M.F. 

'I +M) 




primary exciting current, 

thus, total primary current, 

/. = -f. + /» = <^(^.+A) 

*i = "i + fi, ^, = </» + *,) 

primary impedance voltage, 

primary induced E.M.F., 

thus, primary terminal voltage, 

^. = ^ (1 - .) - 7. (r, -y [1 - .] X,) = e (f. +A) 

r, = 1 - J - r^. - (1 - .) V, f, = (1 - /) M - r-A 

hence, absolute, 



Secondary current, 

Primary current, 

Primary terminal voltage. 




Torque and mechanical power input, 

Electrical output, 

P, - i".' +/y/ - [EM - (-«./.]■ +/■[«•.]' 
- ;t^ j (V, + V,) +y(v, - v.) j 




















— i 


































Voltampere output, 


power factor. 

Pi Vi + », + >^ 




t,n - - :^ - V- - ^'^ 

^ /i' hc^ + V. 

In Fig. 122 is plotted the load characteristic of a con- 
stant speed induction generator, at constant terminal vol- 
tage e^ = 110, and the constants, 

n = .01 + .V 

Zo = .l -.Zj 

Z, = .1 -.3/ 
171. As instance may be considered a power trans- 
mission from an induction generator of constants Y^, Z^, 
Z^, over a line of impedance Z = r —jx, into a synchron- 
ous motor of synchronous impedance Z, = r, —jx^, operat- 
ing at constant field excitation. 

Let, ^0 = counter E.M.F, or nominal induced E.M.F. of 
synchronous motor at full frequency; that is, frequency of 
synchronism with the speed of the induction generator. 
By the preceding paragraph the primary current of the 
induction generator was, 

primary terminal voltage, 

^o = *(^,+>.) 

thus, terminal voltage at synchronous motor terminals, 


d-, = <-, - r^b^ - (1 - j) V» <^ = (1 - j) x^b^ - r,^ 

Counter E.M.F. of synchronous motor, 

£, = E;-Ur,-j{l-iixi 

= 'CA+>0 

/, = rf. - rA - (1 - ^) V, /. = (1 - J) Vi - Vi 

or absolute, ^___ 



we have, 



















— ' 






















































J " 











































Cunent, ^ _ t, (1 - ») (i, +/<J 

Terminal voltage at induction generator. 

Terminal voltage at synchronous motor, 

and herefrom in the usual way the efficiencies, power fac- 
tor, etc. are derived. 

When operated from an induction generator, a syn- 
chronous motor gives a load characteristic very similar to 
that of an induction motor operated from a synchronous 
generator, but in the former case the current is leading, in 
the latter lagging. 

In either case, the Sf)eed gradually falls off with increas- 
ing load ^n the synchronous motor, due to the falling off 
of the frequency of the induction generator), up to a maxi- 
mum output point, where the motor drops out of step and 
comes to standstill. 

Such a load characteristic of the induction generator in 
Fig. 121, feeding a synchronous motor of counter E.M.F. 
^0=125 volts (at full frequency) and synchronous impe- 
dance Z^ = .04 — 6_/i over a line of negligible impedance 
is shown in Fig. 123. 

172. - If of two induction motors the secondary of the 
first motor is connected to the primary of the second motor, 
the second machine operates as motor with the E.M.F. and 
frequency impressed upon it by the secondary of the first 
machine, which acts as general alternating-current trans- 
former, converting a part of the primary impressed power 

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into secondary electrical power for the supply of the second 
machine, and a part into mechanical work. 

The frequency of the secondary E.M.F, of the first motor, 
and thus the frequency impressed upon the second motor, is 
the frequency of shp below complete synchronism, s. The 
frequency of the secondary induced E.M.F, of the second 
motor is the difference between its impressed frequency, 
s, and its speed ; thus, if both motors are connected together 
mechanically to turn at the same speed, \ — s, the secondary 
frequency of the second motor is 2 j — 1, hence equal to 
zero at f = .5. That is, the second motor reaches its syn- 
chronism at half speed. At this speed its torque becomes 
equal to zero, the energy current flowing into it, and conse- 
quently the energy component of the secondary current of 
the first motor, and thus the torque of the first motor be- 
comes equal to zero also, when neglecting the hysteresis 
energy current of the second motor. That is, a system of 
concatenated motors with short-circuited secondary of the 
second motor approaches half synchronism, in the same 
manner as the ordinary induction motor approaches syn- 
chronism. With mcreasing load, its slip below half syn- 
chronism increases. 

More generally, any pair of induction motors connected 
in concatenation divide the speed so that the sum of their 
two respective speeds approaches synchronism at no load ; 
or, still more generally, any number of concatenated motors 
run at such speeds that the sum of the -speeds approaches 
synchronism at no load. 

With mechanical connection between the two motors, 
concatenation thus offers a means to operate a pair of 
induction motors at full efficiency at half speed in tandem, 
as well as at full speed in parallel, and thus gives the same 
advantage as the series-parallel control of the continuous- 
current motor. 

In starting, a concatenated system is controlled by re- 
sistance m the armature of the second motor. 

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Since, with increasing speed, the frequency impressed 
upon the second motor decreases proportionally to the de- 
crease of voltage, when Delecting internal losses in the 
first motor, the magnetic density of the second motor re- 
mains practically constant, and thus its torque the same as 
when operated at full voltage and full frequency under the 
same conditions. 

At half synchronism the torque of the concatenated 
couple becomes zero, and above half synchronism the sec- 
ond motor runs beyond its impressed frequency ; that is, 
becomes generator. In this case, due to the reversal of 
current in the secondary of the first motor, its torque 
becomes negative also, that is the concatenated couple 
becomes induction generator above half synchronism. At 
about two-thirds synchronism, with low resistance armature, 
the torque of the couple becomes zero again, and once more 
positive between about two-thirds synchronism and full syn- 
chronism, and negative once more beyond full synchronism. 
With high resistance in the secondary of the second motor, 
the second range of positive torque, below full synchronism, 
disappears, more or less. 

173. The calculation of a concatenated couple of in- 
duction motors is as follows. 


N = frequency of main circuit, 

s = slip of the first motor from synchronism, 

the frequency induced in the secondary of the first motor 
and thus impressed upon the primary of the second motor 
is, J A^ 

The^peed of the first motor is (1 — s) N, thus the slip 
of the second motor, or the frequency induced in its sec- 
ondary, is 

sN - {\ - s) N = {2s - \) n: 

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/ = counter E.M.F. induced in the secondary of the sec- 
ond motor, reduced to full frequency. 

Z^ = r^ — jXc = primary self-inductive impedance. 

Z^^ = r, —jx^^ = secondary self-inductance impedance. 

Y = g ■¥jb = primary exciting admittance of each mo- 
tor, all reduced to full frequency and to the primary by the 
ratio of turns. 

We then have. 

Second motor, 
secondary induced E.M.F,, 

secondary current, 


■'■ = 







". = - 


- D'V ^ 


primary exciting current. 


thus, total 


■ current. 

/,-/! + /. = ' 

(«, + A) 


*, = J, 

+ t 

«,-^ + * 

primary induced E.M.F., 

primary impedance voltage, 

thus, primary impressed E.M.F., 

£, = «+/,( r, - jsx,) = ^ (^. -HA) 

c, = J + r^i + jjr/, f, = f^, — sx^. 

First motor, 
secondary current. 

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secondary induced E.M.F., 

f , = f , + 7, (r. -/Mi) = e W -^jd^) 

i^, = f, + r,^, + jjT,^, ^ = r, + r,i, — f;»^i, 

primary induced E.M.F.-, 

primary exciting current, 

total primary current, 

7= 7, + 7. = ^ (^1 +^f,) 

^1 = *i +£A - */, ^, = ^ +«/;+ */i 

primary impedance voltage, 

thus, primary impressed E.M.F., 

E^ = E,-^I (r, -j\) = € (A, +/:*,) 

^1 =/i + 'afi + ^ttf* ^1 =/a + 'off. - *b^i 

or, absolute, 

?o = ^ V V + V 

Substituting now this value of ^ in the preceding gives 
the values of the currents and E.M.F.'s in the different 
circuits of the motor series. 

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In the second motor, the torque is, 
T, = KJ - <■«, 
hence, its power output, 

/I =(l-.)r,= (l -.)<>«, 
The power input is, 

= '■[('.+ A.) («,+A)] 
hence, the efficiency, 

P, (l-,)<-», _ (l-<)^ 

the power factor, 

■P.' _ [-g-q _ 'A + 'A 

a ^^. V(v + <-,') w + «,") 


In the first motor, 
the torque is, 

T, - lS,/,f .<■[(/, +y,) (4, + A)? 
= <*(/A+/A) 
the power output, 

/I = r, (1 - ») 

-^•(l-') W".+/A) 
the power input, 

A - [£,/] = <■[(*, +.?»,) ta +*,)] 
- [EJf +jlEjy 

Thus, the efficiency, 

_ g - ') (/A +/A) 

[f ,/]' - [f .4]' (*.*■. + 'Srf,)-('A + <A) 
the power factor of the whole system. 



the power factor of the first motor, 

£^-M,/, V(/t,- + hi) {;,- + if) - -J(t{ + 1!) («,■ + «,•) 
the total efKciency of the system, 

■P. + P , - (' - ') (/A +/A + »J 

;?^£«0,.i.. rr. 1 

X A " we oli* lOlilil 

X ^ "^ *'"' CIRVES 

X ISp- -' "^ '- / \ ■'■" " 1-^1 1 

T « '2° — -- -" / ^ 

_X / '-■ " V_ 

_X i 'A . 

_J„ k U = * -''' 

-«L - / i 

X f T 

» "" — r- - '■' '"■ V 

\L \ /.'^ 

■ f_ '-^-i 

» ji / 

t firduMiwi Katart. Spnd Ountt. 
Z = .l— .3/ J'=.01 + .l/ 

174. As instance are given in Fig. 124, the curves of 
total torque, of torque of the second motor, and of current, 
for the range of shp from j = + 1.5 to j = — .7 for a pair 
of induction motors in concatenation, of the constants : 

Z^ = Z, = .\- ZJ 
K=.01 + .l/ 

As seen, there are two ranges of positive torque for the 
whole system, one below half synchronism, and one from 
about two-thirds to full synchronism, and two ranges of 

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negative torque, or generator action of the motor, from half 
to two-third synchronism, and above full synchronism. 

With higher resistance in the secondary of the second 
motor, the second range of positive torque of the system 
disappears more or less, and the torque curves become as 
shown in Fig. 125. 





.P.^. c« 

^ 1 


























Fff. 12S. Comatnaatn of li 
Z = .l—.3j 

I MatorM. ap*»d Caratt, 
= .01+.1> 


175, The magnetic circuit of the induction motor at or 
near synchronism consists of two magnetic fluxes super- 
imposed upon each other in quadrature, in time, and in 
position. In the polyphase motor these fluxes are produced 
by E.M.Fs. displaced in phase. In the monocyclic motor 
one of the fluxes is due to the primary energy circuit, the 
other to the primary exciting circuit. In the single-phase 

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motor the one flux is produced by the primary circuit, the 
other by the currents induced in the secondary or armature, 
which are carried into quadrature position by the rotation 
of the armature. In consequence thereof, while in all these 
motors the magnetic distribution is the same at or near syn- 
chronism, and can be represented by a rotating field of 
uniform intensity and uniform velocity, it remains such in 
polyphase and monocyclic motors ; but in the single-phase 
motor, with increasing slip, — that is, decreasing speed, — 
the quadrature field decreases, since the induced armature 
currents are not carried to complete quadrature position ; 
and thus only a component available for producing the 
quadrature flux. Hence, approximately, the quadrature flux 
of a single-phase motor can be considered as proportional to 
its speed ; that is, it is zero at standstill. 

Since the torque of the motor is proportional to the 
product of secondary current times magnetic flux in quad- 
rature, it follows that the torque of the single-phase motor 
is equal to that of the same motor under the same condition 
of operation on a polyphase circuit, multiplied with the • 
speed ; hence equal to zero at standstill. 

Thus, while single-phase induction motors are quite sat- 
isfactory at or near synchronism, their torque decreases 
proportionally to the speed, and beccanes zero at standstill. 
That is, they are not self-starting, but some starting device 
has to be used. 

Such a starting device may either be mechanical or elec- 
trical. All the electrical starting devices essentially consist 
in impressing upon the motor at standstill a magnetic quad- 
rature flux. This may be produced either by some outside 
E.M.F., as in the monocyclic starting device, or by displa- 
cing the circuits of two or more primary coils from each 
other, either by mutual induction between the coils, — that 
is, by using one as secondary to the other, — or by impe- 
dances of different inductance factors connected with the 
different primary coils. 

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176. The starting-devices of the single-phase induc- 
tion motor by producing a quadrature magnetic flux can be 
subdivided into three classes : 

1. Phase-Splitting Devices. Two or more primary 
circuits are used, displaced in position from each other, and 
either in series or in shunt with each other, or in any other 
way related, as by transformation. The impedances of 
these circuits are made different from each other as much 
as possible, to produce a phase displacement between them. 
This can be done either by inserting external impedances 
into the circuits, as a condenser and a reactive coil, or by 
making the internal impedances of the motor circuits differ- 
ent, as by making one coil of high and the other of low 

2. Inductive Devices, The different primary circuits 
of the motor are inductively related to each other in such a 
way as to produce a phase displacement between them. 
The inductive relation can be outside of the motor or inside, 
by having the one coil induced by the other ; and in this 
latter case the current in the induced coil may be made 
leading, accelerating coil, or lagging, shading coil. 

8. Monocyclic Devices. External to the motor an 
essentially wattless E.M.F. is produced in quadrature with 
the main E,M.F. and impressed upon the motor, either 
directly or after combination with the single-phase main 
E.M.F. Such wattless quadrature E.M.F. can be produced 
by the common connection of two impedances of different 
power factor, as an inductance and a resistance, or an in- 
ductance and a condensance connected in series across the 

The investigation of these starting-devices offers a very 
instructive application of the symbolic method of investiga- 
tion of alternating-current phenomena, and a study thereof 
is thus recommended to the reader,* 

■ S« paper on the Single-phuc Inducllaa Motor, A.I.E.E. Tr*DUcIunu, iBgS. 

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177. As a rule, no special motors are built for single- 
phase operation, but polyphase motors used in single-phase 
circuits, since for starting the polyphase primary winding is 
required, the single primary coil motor obviously not allow- 
ing the application of phase-displacing devices for produ- 
cing the starting quadrature flux. 

Since at or near synchronism, at the same impressed 
E.M.F. — that is, the same magnetic density — the total 
voltamperes excitation of the single-phase induction motor 
must be the same as of the same motor on polyphase circuit, 
it follows that by operating a quarter-phase motor from 
single-phase circuit on one primary coil, its primary excit- 
ing admittance is doubled. Operating a three-phase motor 
single-phase on one circuit its primary exciting admittance 
is trebled. The self-inductive primary impedance is the 
same single-phase as polyphase, but the secondary impe- 
dance reduced to the primary is lowered, since in single- 
phase operation all secondary circuits corresfMnd to the 
one primary circuit used. Thus the secondary impedance 
in a quarter-phase motor running single-phase is reduced to 
one-half, in a three-phase motor running single-phase re- 
duced to one-third. In consequence thereof the slip of 
speed in a single-phase induction motor is usually less than 
in a polyphase motor ; but the exciting current is consider- 
ably greater, and thus the power factor and the efficiency 
are lower. 

The preceding considerations obviously apply only when 
running so near synchronism that the magnetic field of the 
single-phase motor can be assumed as uniform, that is the 
cross magnetizing flux produced by the armature as equal 
to the main magnetic flux. 

When investigating the action of the single-phase motor 
at lower speeds and at standstill, the falling off of the mag- 
netic quadrature flux produced by the armature current, the 
change of secondary impedance, and where a starting device 
is used the effect of the magnetic field produced by the 
starting device, have to be considered. 

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The exciting current of the single-phase motor consists 
of the primary exciting current or current producing the 
main magnetic flux, and represented by a constant admit- 
tance Y^, the primary exciting admittance of the motor, and 
the secondary exciting current, that is that component of 
primary current corresponding to the secondary current 
which gives the excitation for the quadrature magnetic flux. 
This latter magnetic flux is equal to the main magnetic flux 
*, at synchronism, and falls off with decreasing speed to 
zero at standstill, if no starting device is used or to *, = t^^ 
at standstill if by a starting device a quadrature magnetic 
flux is impressed upon the motor, and at standstill / = ratio- 
of quadrature or starting magnetic flux to main magnetic 

Thus the secondary exciting current can be represented 
by an admittance Y^ which changes from equality with the 
primary exciting admittance Y^ at synchronism, to Y^ = 0, 
respectively to y,' = t Y^ at standstill. Assuming thus that 
the starting device is such that its action is not impaired by 
the change of speed, at slip s the secondary exciting admit- 
tance can be represented by : 

r,' = [1 - (1 - /) ,] n' 

The secondary impedance of the motor at synchronism 
is the joint impedance of all the secondary circuits, since all 
secondary circuits correspond to the same primary circuit, 

Z Z 

hence = -^ with a three-phase secondary, and = -^ with a 

two-phase secondary with impedance Z^ per circuit. 

At standstill, however, the secondary circuits correspond 
to the primary circuit only with their projection in the direc- 
tion of the primary flux, and thus as resultant only one-half 
of the secondary circuits are effective, so that the secondary 
impedance at standstill is equal to 2Z,/3 with a three-phase, 
and equal to Z, with a two-phase secondary. Thus the 
effective secondary impedance of the single-phase motor 

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changes with the speed and can at the slip s be represented 

^ ,, (1+^)Z,. , , , ,, (H-J-)Z 
oy Z/ = q in a three-phase motor, and Z, = ^ 

in a two-phase motor, with the impedance Z^ per secondary 

In the single-phase motor without starting device, due to 
the falling off of the quadrature flux, the torque at slip s is : 
T = V (1 - J) 

In a single-phase motor with a starting device which at 
standstill produces a ratio of magnetic fluxes /, the torque at 
standstill is ; 

where T, = total torque of the same motor on polyphase 

Thus denoting the value — i = v 

the single-phase motor torque at standstill is : 

and the single-phase motor torque at slip s is : 
r = a,^ [1 - (1 - V) S-] 

178. In the single-phase motor considerably more 
advantage is gained by compensating for the wattless mag- 
netizing component of current by capacity than in the 
polyphase motor, where this wattless current is relatively 
small. The use of shunted capacity, however, has the dis- 
advantage of requiring a wave of impressed E.M,F. very 
close to sine shape; since even with a moderate variation 
from sine shape the wattless charging current of the con- 
denser of higher frequency may lower the power factor 
more than the compensation for the wattless component of 
the fundamental wave raises it, as will be seen in the chap- 
ter on General Alternating Current Waves. 

Thus the most satisfactory application of the condenser 
in the single-phase motor is not in shunt to the primary 

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circuit, but in a tertiary circuit; that is, in a circuit stationary 
with regard to the primary impressed circuit, but induced 
by the revolving secondary circuit. 

In this case the condenser is supplied with an E.M.F. 
transformed twice, from primary to secondary, and from 
secondary to tertiary, through multitooth structures in a 
uniformly revolving field, and thus a very close approxi- 
mation to sine wave produced at the condenser, irrespective 
of the wave shape of primary impressed E.M.F. 

With the condenser connected into a tertiary circuit of 
a single-phase induction motor, the wattless magnetizing 
current of the motor is supplied by the condenser in a 
separate circuit, and the primary coil carries the energy cur- 
rent only, and thus the efficiency of the motor is essentblly 

The tertiary circuit may be at right angles to the pri- 
mary, or under any other angle. Usually it is applied on an 
angle of 60", so as to secure a mutual induction between 
tertiary and primary for starting, which produces in start- 
ing in the condenser a leading current, and gives the quad- 
rature magnetic flux required. 

179. The most convenient way to secure this arrange- 
ment is the use of a three-phase motor which with two of 
its terminals 1-2, is connected to the single-phase mains, 
and with terminals 1 and 3 to a condenser. 

Let Y^ = g^ +J% = primary exciting admittance of the 
motor per delta circuit. 

Z^ ~ r^ ~ jx^ = primary self-inductive impedance per 
delta circuit, 

Z^ = r, —jXi = secondary self-inductive impedance per 
delta circuit reduced to primary. 


J^ =^,—y^, = admittance of the condenser connected 
between tenninals 1 and 3. 

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If then, as single-phase motor, 

/ = ratio of auidliaiy quadrature flux to main flux in 

h = ratio of E.M.F. induced in condenser drcuit to 
E.M.F. induced in main circuit in starting, 
_ starting torque 
a^i? in starting * 

It is single-phase 

1? = 1.6 J^ = 1.5 (^, ■Vjh^ = primary exdting admit- 
r,! = 1,5 rjl - (1 - i] 

= 1.5 (fo -f-y^o) [1 — (1 — /) s] = secondary exciting 
admittance at slip s. 


(1 + s) (1 + i) 

Zf = - — s — - Zi = ^^ — = — - (r, — /sx{) = secondary self- 
inductive impedance. 
2!^ = —^ s= — -^ = tertiary seU-inductive impe- 

dance of motor. 

I4 = -,| ^ = total admittance of tertiary circuit 

Since the E.M.F. induced in the tertiary circuit decreases 
from e at synchronism to /te at standstill, the effective ter- 
tiary admittance or admittance reduced to an induced E.M.F. 
e is at slip s 

y,> = [1 - (I - A) .] y, 

Let then, 

e = counter RM.F. of primary circuit, 
J = slip. 

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We have, 
secondary load current 

secondary exciting current 

/,' = eY} = 1.5 eY^ [1 - (1 - *] 
secondary condenser current 

J^^eYi = eY^\\-<^-K)s\ 
thus, total secondary current 

/» = /, + 7,1 + /, 
primary exciting current 

Ii = eYi = \.^eY^ 
thus, total primary current 

/o = /' + li 

= >, + /; + /i' -I- >? 

= ^ (^ +^»0' 
primary impressed E,M.F, 

thus, main counter E.M.F. 
' ~ "i + A 

and, absolute 

hence, primaiy current 



voltampere input, 

power input 
torque at slip s 

r- y [1 - (1 - !.) <] 

•! + <■' ' 
and, power output 


and herefrom in the usual manner the efficiency, apparent 
efficiency, torque efficiency, apparent torque efficiency, and 
power factor. 

The derivation o*^ the constants t, k, v, which have to be 
determined before calculating the motor, is as follows : 

Let e^ = single-phase impressed E,M.F., 

Y= total stationary admittance of motor per delta cir* 
Ef = KM.F. at condenser tenninab in starting. 

In the circuit between the single-phase mains from ter- 
minal 1 over terminal 3 to 2, the admittances V+V^ and V, 
are connected in series, and have the respective E.M.Fs. £", 
and g^ — E^. It is thus, 

Y+ K, -(- Y=e^-£t-t- Et, 

since with the same current passing through both circuits, 
the impressed E.M.Fs. are inverse proportional to the re- 
spective admittances. 

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and quadrature E.M,F. 



Since in the three-phase E.M.F, triangle, the altitude 

corresponding to the quadrature magnetic flux = ~t^ , and 

the quadrature and main fluxes are equal, in the single-phase 
motor the ratio of quadrature to main flux is 

/ = — ' = 1.155A, 


From /, V is derived as shown in the preceding. 

For further discussion on the Theory and Calculation of 
the Single-phase Induction Motor, see American Institute 
Electrical Engineers Transactions, January, 1900. 


180. The induction motor discussed in the foregoing 
consists of one or a number of primary circuits acting upon 
a movable armature which comprises a number of closed 
secondary circuits displaced from each other in space so as 
to offer a resultant circuit in any direction. In consequence 
thereof the motor can be considered as a transformer, having 
to each primary circuit a corresponding secondary circuit, 
— a secondary coil, moving out of the field of the primary 
coil, being replaced by another secondary coil moving into 
the field. 

In such a motor the torque is zero at synchronism, posi- 
tive below, and negative above, synchronism. 

If, however, the movable armature contains one closed 
circuit only, it offers a closed secondary circuit only in the 
direction of the axis of the armature coil, but no secondary 
circuit at right angles therewith. That is, with the rotation 

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of the armature the secondary circuit, corresponding to a 
primary circuit, varies from short circuit at coincidence of 
the axis of the armature coil with the axis of the primary 
coil, to open circuit in quadrature therewith, with the 
periodicity of the armature speed. That is, the apparent 
admittance of the primary circuit varies periodically from 
open-circuit admittance to the short-circuited transformer 

At synchronism such a motor represents an electric cir- 
cuit of an admittance varying with twice the periodicity of 
the primary frequency, since twice per period the axis of the 
armature coil and that of the primary coil coincide. A vary- 
ing admittance is obviously identical in effect with a varying 
reluctance, which will be discussed in the chapter on reac- 
tion machines. That is, the induction motor with one 
closed armature circuit is, at synchronism, nothing but a 
reaction machine, and consequently gives zero torque at 
synchronism if the maxima and minima of the periodically 
varying admittance coincide with the maximum and zero 
values of the primary circuit, but gives a definite torque if 
they are displaced therefrom. This torque may be positive 
or negative according to the phase displacement between 
admittance and primary circuit ; that is, the lag or lead 
of the maximum admittance with regard to the primary 
maximum. Hence an induction motor with single-armature 
circuit at synchronism acts either as motor or as alternat- 
ing-current generator according to the relative position of 
the armature circuit to the primary circuit. Thus it can be 
called a synchronous induction motor or synchronous in- 
duction generator, since it is an induction machine giving 
torque at synchronism. 

Power factor and apparent efficiency of the synchron- 
ous induction motor as reaction machine are very low. 
Hence it is of practical application only in cases where a 
smalt amount of power is required at synchronous rotation, 
and continuous current for field excitation is not available. 

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The current induced in the annature of the synchronous 
induction motor is of double the frequency impressed upon 
the primary. 

Below and above synchronism the ordinary induction 
motor, or induction generator, torque is superimposed upon 
the synchronous induction machine torque. Since with the 
frequency of slip the relative position of primary and of 
secondary coil changes, the synchronous induction machine 
torque alternates periodically with the frequency of slip. 
That is, upon the constant positive or negative torque be- 
low or above synchronism an alternating torque of the fre- 
quency of slip is superimposed, and thus the resultant 
torque pulsating with a positive mean value below, a nega- 
tive mean value above, synchronism. 

When started from rest, a synchronous induction motor 
will accelerate like an ordinary single-phase induction mo- 
tor, but not only approach synchronism, as the latter does, 
but run up to complete synchronism under load. When 
approaching synchronism it makes definite beats with the 
frequency of slip, which disappear when synchronism is 


181. In a revolving magnetic field, a circular iron disk, 
or iron cylinder of uniform magnetic reluctance in the 
direction of the revolving field, is set in rotation, even if 
subdivided so as to preclude the induction of eddy currents. 
This rotation is due to the effect of hysteresis of the revolv- 
ing disks or cyclinder, and such a motor may thus be called 
a hysteresis motor. 

Let / be the iron disk exposed to a rotating magnetic 
field or resultant M.M.F. The axis of resultant magneti- 
zation in the disk / does not coincide with the axis of the 
rotating field, but lags behind the latter, thus producing a 
couple. That is, the component of magnetism in a direction 
of the rotating disk, /, ahead of the axis of rotating M.M.F., 
is rising, thus below, and in a direction behind the axis 

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of rotating M.M.F. decreasing ; that is, above proportion- 
ality with the M.M.F., in consequence of the lag of magnet- 
ism in the hysteresis loop, and thus the axis of resultant 
magnetism in the iron disk, /, does not coincide with the 
axis of rotating M.M.F., but is shifted backwards by an 
angle, a, which is the angle of hysteretic lead in Chapter 
X., § 79. 

The induced magnetism gives with the resultant M.M.F. 
a mechanical couple, — 

T= m F^ sin a, 

F= resultant M.M.F., 

4> = resultant magnetism, 

a. = angle of hysteretic advance of phase, 

ffi = a constant. 

' The apparent or voltampere input of the motor is, — 

Q = mF*. 

Thus the apparent torque efficiency, — 


^ =, 

and the pov/er of the motor is, — 

/- = (1 - j) r= {1 - j) « /•* sin o, 

s = slip as fraction of synchronism. 

The apparent efficiency is, — 

- = (l-s) sm a. 

Since in a magnetic circuit containing an air gap the 
angle a is extremely small, a few degrees only, it follows 
that the apparent efficiency of the hysteresis motor is ex- 
tremely low, the motor consequently unsuitable for produ- 
cing larger amounts of mechanical work. 

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From the equation of torque it follows, however, that at 
constant impressed E.M.F., or current, — that is, constant 
F, — the torque is constant and independent of the speed ; 
and therefore such a motor arrangement is suitable, and 
occasionally used as alternating-current meter. 

The same result can be reached from a different point 
of view. In such a magnetic system, comprising a mov- 
able iron disk, /, of uniform magnetic reluctance in a 
revolving field, the magnetic reluctance — and thus the dis- 
tribution of magnetism — is obviously independent of the 
speed, and consequently the current ^nd energy expenditure 
of the impressed M.M.F. independent of the speed also. If, 
now, — 

V= volume of iron of the movable part, 
B = magnetic density, 
and i; = coefficient of hysteresis, 

the energy expended by hysteresis in the movable disk, /, is 
per cycle, — 

iV„= V-qB**, 

hence, if N = frequency, the energy supplied by the M.M.F. 
to the rotating iron disk in the hysteretic loop of the 
M.M.F. is, — 

At the slip, sN, that is, the speed (1 — s) N, the energy 
expended by hysteresis in the rotating disk is, howe\'er, — 
I>t = sNV^B^' 

Hence, in the transfer from the stationary to the revolv- 
ing member the magnetic energy, — 

F= F^- r^ = {X - s) NVriB'^* 

has disappeared, and thus reappears as mechanical work, 
and the torque is, — 

that is, independent of the speed. 

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Since, as seen in Chapter X., sin a. is the ratio of the 
energy of the hysteretic loop to the total apparent energy, 
in voltampere, of the magnetic cycle, it follows that the 
apparent efficiency of such a motor can never exceed the 
value (1 — s) sin o, or a fraction of the primary hysteretic 

The primary hysteretic energy of an induction motor, as 
represented by its conductance, g, being a part of the loss 
in the motor, and thus a very small part of its output only, 
it follows that the output of a hysteresis motor is a very 
small fraction only of the output which the same magnetic 
structure could give with secondary short-circuited winding, 
as regular induction motor. 

As secondary effect, however, the rotary effort of the 
magnetic structure as hysteresis motor appears more or less 
in all induction motors, although usually it is so small as to 
be neglected. 

If in the hysteresis motor the rotary iron structure has 
not uniform reluctance in all directions — but is, for in- 
stance, bar-shaped or shuttle-shaped — on the hysteresis 
motor effect is superimposed the effect of varying magnetic 
reluctance, which tends to accelerate the motor to syn- 
chronism, and maintain it therein, as shall be more fully 
investigated under " Reaction Machine " in Chapter XX. 

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182. In the alternating-current generator, E.M.F. is 
induced in the armature conductors by their relative motion 
through a constant or approximately constant magnetic 

When yielding current, two distinctly different M.M.Fs. 
are acting upon the alternator armature — the M.M.F. of 
the field due to the field-exciting spools, and the M.M.F. 
of the armature current. The former is constant, or approx- 
imately so, while the latter is alternating, and in synchro- 
nous motion relatively to the former ; hence, fixed in space 
relative to the field M.M.F., or uni-tlirectional, but pulsating 
in a single-phase alternator. In the polyphase alternator, 
when evenly loaded or balanced, the resultant M.M.F, of 
the armature current is more or less constant. 

The E.M.F, induced in the armature is due to the mag- 
netic flux passing through and interlinked with the arma- 
ture conductors. This flux is produced by the resultant of 
both M.M.Fs., that of the field, and that of the armature. 

On open circuit, the M.M.F. of the armature is zero, and 
the E.M.F. of the armature is due to the M.M.F. of the 
field coils only. In this case the E.M.F. is, in general, a 
maximum at the moment when the armature coil faces the 
position midway between adjacent field coils, as shown in 
Fig. 126, and thus incloses no magnetism. The E.M.F. 
wave in this case is, in general, symmetrical. 

An exception from this statement may take place only 
in those types of alternators where the magnetic reluctance 
of the armature is different in different directions ; thereby, 

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during the synchronous rotation of the armature, a pulsa- 
tion of the magnetic flux passing through it is produced. 
This pulsation of the magnetic flux induces E.M.F. in the 
field spools, and thereby makes the field current pulsating 
also. Thus, we have^ in this case, even on open circuit, no 

rotation through a constant magnetic field, but rotation 
through a pulsating field, which makes the E.M.F, wave 
unsymnietrical, and shifts the maximum point from its the- 
oretical position midway between the field poles. In gen- 
eral this secondary reaction can be neglected, and the field 
M.M.F. be assumed as constant. 

The relative position of the armature M.M.F. with re- 
spect to the field M.M.F. depends upon the phase rela- 
tion existing in the electric circuit. Thus, if there is no 
displacement of phase between current and E.M.F., the 
current reaches its maximum at the same moment as the 
E.M.F. ; or, in the position of the armature shown in Fig. 
126, midway betwe'en the field poles. In this case the arma- 
ture current tends neither to magnetize nor demagnetize the 
field, but merely distorts it ; that is, demagnetizes the trail- 
ing-pole corner, a, and magnetizes the leading-pole corner, 
b. A change of the total flux, and thereby of the resultant 
E.M.F., will take place in this case only when the magnetic 
densities are so near to saturation that the rise of density 
at the leading-pole corner will be less than the decrease of 

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density at the trailing-pole corner. Since the internal self- 
inductance of the alternator itself causes a certain lag ot 
the current behind the induced E.M.F,, this condition of no 
displacement can exist only in a circuit with external nega- 
tive reactance, as capacity, etc. 

If the armature current lags, it reaches the maximum 
later than the E.M.F. ; that is, in a position where the 
armature coil partly faces the following-field pole, as shown 
in diagram in Fig. 127. Since the armature current flows 

in opposite direction to the current in the following-field 
pole {in a generator), the armature in this case will tend to 
demagnetize the field. 

If, however, the armature current leads, — that is, reaches 
its maximum while the armature coil still partly faces the 

preceding-field pole, as shown in diagram Fig, 128, — it tends 
to magnetize this field coil, since the armature current flows 
in the same direction with the exciting current of the pre- 
ceding-field spools. 

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Thus, with a leading current, the armature reaction of 
the alternator strengthens the field, and thereby, at con- 
stant-field excitation, increases the voltage ; with lagging 
current it weakens the field, and thereby decreases the vol- 
tage in a generator. Obviously, the opposite holds for a 
synchronous motor, in which the armature current flows in 
the opposite direction ; and thus a lagging current tends to 
magnetize, a leading current to demagnetize, the field. 

183. The E.M.F. induced in the armature by the re- 
sultant magnetic flux, produced by the resultant M.M.F, of 
the field and of the armature, is not the terminal voltage 
of the machine; the terminal voltage is the resultant of this 
induced E.M.F. and the E.M.F. of self-inductance and the 
E.M.F. representing the energy loss by resistance in the 
alternator armature. That is, in other words, the armature 
current not only opposes or assists the field M.M.F. in cre- 
ating the resultant magnetic fiux, but sends a second mag- 
netic flux in a local circuit through the armature, which 
flux does not pass through the field spools, and is called the 
magnetic fiux of armature self-inductance. 

Thus we have to distinguish in an alternator between 
armature reaction, or the magnetizing action of the arma- 
ture upon the field, and armature self-inductance, or the 
E.M.F. induced in the armature conductors by the current 
flowing therein. This E.M,F. of self-inductance is (if the 
magnetic reluctance, and consequently the reactance, of 
the armature circuit is assumed as constant) in quadrature 
behind the armature current, and will thus combine with 
the induced E.M.F. in the proper phase relation. Obvi- 
ously the E.M.F. of self-inductance and the induced E.M.F. 
do not in reality combine, but their respective magnetic 
fluxes combine in the armature core, where they pass'through 
the same structure. These component E.M.Fs. are there- 
fore mathematical fictions, but their resultant is real. This 
means that, if the armature current lags, the E.M.F. of self- 

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inductance will be more than 90° behind the induced E.M.F., 
and therefore in partial opposition, and will tend to reduce 
the terminal voltaga On the other hand, if the armature 
current leads, the E.M.F. of self-inductance will be less 
than 90° behind the induced E.M.F., or in partial conjunc- 
tion therewith, and increase the terminal voltage. This 
means that the E.M.F. of self -inductance increases the ter- 
minal voltage with a leading, and decreases it with a lagging 
current, or, in other words, acts in the same manner as the 
armature reaction. For this reason both actions can be 
combined in one, and represented by what is called the syn- 
chronous reactance of the alternator. In the following, we 
shall represent the total reaction of the armature of the 
alternator hy the one term, synchronous reactance. While 
this is not exact, as stated above, since the reactance should 
be resolved into the magnetic reaction due to the magnet- 
izing action of the armature current, and the electric reac- 
tion due to the self-induction of the armature current, it is 
in general sufficiently near for practical purposes, and well 
suited to explain the phenomena taking place under the 
various conditions of load. This synchronous reactance, x, 
is frequently not constant, but is pulsating, owing to the 
synchronously varying reluctance of the armature magnetic 
circuit, and the field magnetic circuit ; it may, however, be 
considered in what follows as constant ; that is, the E.M.Fs. 
induced thereby may be represented by their equivalent sine 
waves. A specific discussion of the distortions of the wave 
shape due to the pulsation of the synchronous reactance is 
found in Chapter XX, The synchronous reactance, x, is 
not a true reactance in the ordinary sense of the word, but 
an equivalent or effective reactance. Sometimes the total 
effects taking place in the alternator armature, are repre- 
sented by a magnetic reaction, neglecting the self-inductance 
altogether, or rather replacing it by an increase of the arma- 
ture reaction or armature M.M.F. to such a value as to in- 
clude the self-inductance. This assumption is mostly made 
in the preliminary designs of alternators. 

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184. Let Eo = induced E.M.F. of the alternator, or the 
E.M.F. induced in the armature coils by their rotation 
through the constant magnetic field produced by the cur- 
rent in the field spools, or the open circuit voltage, more 
properly called the "nominal induced E.M.F.," since in 
reality it does not exist, as before stated. 

Then e, = -^/^-wnNMyi-*; 


n = total number of turns in series on the armature, 

JV ■• frequency, 

M s= tot^ magnetic flux per field pole. 

Let Xt = synchronous reactance, 

r-g B> internal resistance of alternator ; 
then Zg = r, ~jXa — internal impedance. 

If the circuit of the alternator is closed by the external 

the current is 

and, terminal voltage. 

■ Z 

('•. + '•)-/(«. + »)' 



, + r)- +(*, + »)■ 

„ ., E.l.r-J^-) 

DiBiiu.d, Google 


■, expanded in a series. 



(r,r + x.x')-(r,x + x.r) . 

As shown, the terminal voltage varies with the condi- 
tions of the external circuit. 

186. As an instance, in Figs. 129-134, at constant 
induced E.M,F., 

E, = 2500 ; 









































R = 










Flq. t29. FItU CHaractirlitle of Alttrnalat on Man-fnduetlai Load. 

and the values of the internal impedance, 
Z, = r^ -jx^ = 1 - 10/ 
With the current / as abscissae, the terminal voltages E 
as ordinates in drawn line, and the kilowatts output, = /' r, 
in dotted lines, the kilovolt-amperes output, = IE, in dash- 

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dotted lines, we have, for the following conditions of external 

circuit ; 

In Fig. 129, non-inductive external circuit, « = 0, 

In Fig. 130, inductive external circuit, of the condition, r / x 
= ■+■ .75, with a power factor, .6. 

In Fig. 131, inductive external circuit, of the condition, r = 0. 
with a power factor, 0. 

In Fig. 132, external circuit with leading current, of the condi- 
tion, r / x= — .75, with a power factor, ,6. 

In Fig. 133, external circuit with leading current, of the condi- 
tion, r = 0, with a power factor, 0. 

In Fig. 134, all the volt-ampere curves are shown together as 
complete ellipses, giving also the negative or synchronous 
motor part of the curves. 













































i— T 







ng. ISO. FMt ChatoBtirlitlc of Mlliniator, at eofi Poav-fnetor on Induetlet lead. 

Such a curve is called a ^e/d characteristic. 

As shown, the E.M.F. curve at non-inductive load is 
nearly horizontal at open circuit, nearly vertical at short 
circuit, and is similar to an arc of an ellipse 

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U=8 00. 





































Id 1 

»> u 

U in Amp 





































































F/^ 132. FltU ChancUrlatlc of Altmator, at BOX Pavtr-fiKlor on Coiultaiv Load 


























































' / 









1 — r 



ii ' 

[ — 

■ — ■ 

np. IS* FM4Ci 

if Alitraatar, air Watllm Candtnmr taut 

With reactive load the curves are more nearly straight 

The voltage drops on inductive, rises on capacity load. 

The output increases from zero at open circuit to a maxi- 
mum, and then decreases again to zero at short circuit. 

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( * 


^U L 



i ^ 




































Fl,. J 

! of AlUmatof. 

186, The dependence of the terminal voltage, E, upon 
the phase relation of the external circuit is shown in Fig. 
135, which gives, at impressed E.M.F,, 

£, = 2,500 volts, 
for the currents, 

/= 50, 100, 150, 200, 250 amperes, 

the terminal voltages, E, as ordinates, with the inductance 
factor of the external circuit. 

as abscissae. 

187. If the internal impedance is negligible compared 
with the external impedance, then, approximately, 

V(r. -I- >)'+(«. + ,)< 

- = -ffo; 



















.= 0^ 





















— ■ 













































— ' 








— ' 


Fig. 135, Rigulatlim of Alttnalar sa Varleia lootfk 

that is, an alternator with small internal resistance and syn- 
chronous reactance tends to regulate for constant terminal 


Every alternator does this near open circuit, especially 
on non-inductive load. 

Even if the synchronous reactance, x^, is not quite neg- 
ligible, this regulation takes place, to a certain extent, on 
non-inductive circuit, since for 

^^0, £- 

and thus the expression of the terminal voltage, E, contains 
the synchronous reactance, x^, only as a term of second 
order in the denominator. 

On inductive circuit, however, x^ appears in the denom- 
inator as a term of first order, and therefore constant poten- 
tial regulation does not take place as well. 

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With a non-inductive external circuit, if the synchronous 
reactance, x^, of the alternator is very large compared with 
the external resistance, r, 

current 1= ^ ^- = ^. 


approximately, or constant ; or, if the external circuit con- 
tains the reactance, x, 


approximately, or constant. 

The terminal voltage of a non-inductive circuit is 

approximately, or proportional to the external resistance. 
In an inductive circuit, 

£ = £ — Vr' + jc* ) 

approximately, or proportional to the external impedance. 

188. That is, on a non-inductive external circuit, an 
alternator with very low synchronous reactance regulates 
for constant terminal voltage, as a constant-potential ma- 
chine ; an alternator with a very high synchronous reac- 
tance regulates for a terminal voltage proportional to the 
external resistance, as a constant-current machine. 

Thus, every alternator acts as a constant -potential ma- 
chine near open circuit, and as a constant-current machine 
near short circuit. Between these conditions, there is a 
range where the alternator regulates approximately as a 
constant power machine, that is current and E.M.F. vary 
in inverse proportion, as between 130 and 200 amperes in 
Fig. 129. 

The modern alternators arc generally more or less ma- 

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chines of the first class ; the old alternators, as built by 
Jablockkoff, Gramme, etc., were machines of the second 
class, used for arc lighting, where constant-current regula- 
tion is an advantage. 

Obviously, large external reactances cause the same reg- 
ulation for constant current independently of the resistance, 
r, as a large internal reactance, x^. 

On non-inductive circuit, if 

/ ^^ , 

V(»- +.-.)■ + V 

the output IS 

p = !£ = ■; --^ , 

(' + »•.)■ + «. 


dP v-^'+V J, 

III- {('•+»•.)■ + *.*}" " 

hence, if 



r - Vr.' + ;.■,■- lii 



That is, the 

power is a maximum, and 

P„ ^•' . 

2 («. + r, 



and J- ^ 

V2 a, (a, -t- r^ " 

Therefore, with an external resistance equal to the inter- 
nal impedance, or, r=^^— ^f^-\-x^, the output of an 
alternator is a maximum, and near this point it regulates 
for constant output ; that is, art increase of current causes 
a proportional decrease of terminal voltage, and inversely. 

The field characteristic of the alternator shows this 
effect plainly. 

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189. All alternators, when brought to synchronism with 
each other, will operate in parallel more or less satisfactorily. 
This is due to the reversibility of the alternating-current 
machine ; that is, its ability to operate as synchronous motor. 
In consequence thereof, if the driving power of one of sev- 
eral parallel-operating generators is withdrawn, this gene- 
rator will keep revolving in synchronism as a synchronous 
motor ; and the power with which it tends to remain in 
synchronism is the maximum power which it can furnish 
as synchronous motor under the conditions of running. 

190. The principal and foremost condition of parallel 
operation of alternators is equality of frequency ; that is, 
the transmission of power from the prime movers to the 
alternators must be such as to allow them to run at the 
same frequency without slippage or excessive strains on 
the belts or transmission devices. 

Rigid mechanical connection of the alternators cannot be 
considered as synchronizing ; since it allows no flexibility or 
phase adjustment between the alternators, but makes them 
essentially one machine. If connected in parallel, a differ- 
ence in the field excitation, and thus the induced E.M.F. of 
the machines, must cause large cross-current ; since it cannot 
be taken care of by phase adjustment of the machines. 

Thus rigid mechanical connection is not desirable for 
parallel operation of alternators. 

191. The second important condition of parallel opera- 
tion is uniformity of speed ; that is, constancy of frequency. 

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If, for instance, two alternators are driven by independent 
single-cylinder engines, and the cranks of the engines hap- 
pen to be crossed, the one engine will pull, while the other 
is near the dead-point, and conversely. Consequently, alter- 
nately the one alternator will tend to speed up and the 
other slow down, then the other speed up and the first 
slow down. This effect, if not taken care of by fly-wheel 
capacity, causes a " hunting " or pumping action ; that is, a 
fluctuation of the lights with the period of the engine revo- 
lution, due to the alternating transfer of the load from one 
engine to the other, which may even become so excessive 
as to throw the machines out of step, especially when by an 
approximate coincidence of the period of engine impulses 
(or a multiple thereof), with the natural period of oscillation 
of the revolving structure, the effect is made cumulative. 
This difliculty as a rule does not exist with turbine or water- 
wheel driving. 

192. In synchronizing alternators, we have to distin- 
guish the phenomena taking place when throwing the ma- 
chines in parallel or out of parallel, and the phenomena 
when running in synchronism. 

When connecting alternators in parallel, they are first 
brought approximately to the same frequency and same 
voltage ; and then, at the moment of approximate equality 
of phase, as shown by a phase-lamp or other device, they 
are thrown in parallel. 

Equality of voltage is much less important with modem 
alternators than equality of frequency, and equality of phase 
is usually of importance only in avoiding an instantaneous 
flickering of the lights on the system. When two alter- 
nators are thrown together, currents pass between the 
machines, which accelerate the one and retard the other 
machine until equal frequency and proper phase relation 
are reached. 

With modern ironclad alternators, this interchange of 
mechanical power is usually, even without very careful 

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adjustment before synchronizing, sufficiently limited net 
to endanger the machines mechanically ; since the cross- 
currents, and thus the interchange of power, are limited 
by self-induction and armature reaction. 

In machines of very low armature reaction, that is, 
machines of "very good constant potential regulation," 
much greater care has to be exerted in the adjustment 
to equality of frequency, voltage, and phase, or the inter- 
change of current may become so large as to destroy the 
machine by the mechanical shock ; and sometimes the 
machines are so sensitive in this respect that it is prefer- 
able not to operate them in parallel. The same applies 
in getting out of step. 

193. When running in synchronism, nearly all types 
of machines will operate satisfactorily ; a medium amount 
of armature reaction is preferable, however, such as is given 
by modern alternators — not too high to reduce the 
synchronizing power too much, nor too low to make the 
machine unsafe in case of accident, such as falling out of 
step, etc. 

If the armature reaction is very low, an accident, — such 
as a short circuit, falling out of step, opening of the field 
circuit, etc., — may destroy the machine. If the armature 
reaction is very high, the driving-power has to be adjusted 
very carefully to constancy ; since the synchronizing power 
of the alternators is too weak to hold them in step, and 
carry them over irregularities of the driving-power. 

194. Series operation of alternators is possible only by 
rigid mechanical connection, or by some means whereby 
the machines, with regard to their synchronizing power, 
act essentially in parallel ; as, for instance, by the arrange- 
ment shown in Fig. 120, where the two alternators, Ay, Aj, 
are connected in series, but interlinked by the two coils 
of a large transformer, T, of which the one is connected 

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across the terminals of one alternator, and the other across 
the terminals of the other alternator in such a way that, 
when operating in series, the coils of the transformer will 

be without current. In this case, by interchange of power 
through the transformers, the series connection will be 
maintained stable. 

196. In two parallel operating alternators, as shown in 
Fig. 1S7, let the voltage at the common bus bars be assumed 

as zero line, or real axis of coordinates of the complex 
representation ; and let — 

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e -> difference of potential at the common bus bars of 

the two alternators, 
Z =r — jx = impedance of external circuit, 
Y = g-\-jb = admittance of external circuit; 

hence, the current in external circuit is 


Ex = Cx —Jcx' = at (cosii — /sinii) = induced E.M.F. of first 
machine ; 

Ei = ft —jtt' = a, (cos wt —/'sin lu,) = induced RM.F. of sec- 
ond machine ; 

/i = J, +jt\' = current of first machine ; 

/j = (j -\-/it = current of second machine ; 

Zi = rx —jxx = internal impedance, and Yx =gi +jii = inter- 
nal admittance, of first machine ; 

Z, ™ r, —jx, = internal impedance, and Y, =gt -^j'it = inter- 
nal admittance, of second machine. 


<■»' + fa' * = «»'; 
Ex =e + IxZi, or e^ — yV^ (« + /i''i + '/jsi) —JiiiXi — tin); 
£i='<r+ >,Z, , or c, —je4=' {e + /,r, + V««) —j ('»*» — 'i''"*) ; 
I =Ii + /,, or eg +Je^ = (r, + ,,) +/ (// + *,'). 

This gives the equations — 

'i =e + 'i'-t + ii'xf, 
e{ = t\xi — ix'ri ; 
tt = iiXt — it'r,; 
eg=H +H\ 
eb = »i' + V ; 

'a' + <;/* = fl)'; 
or eight equations with nine variables: e^, e{, e^, e^, i|, i^', 

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Combining these equations by twos, 

i\r^ + '/■»'i = "■! + i\*\ ; 
■fi^'i + 'I'^i = 'rt + i,s,* ; 
substituted in 

<i + 'i = eg, 
we have 

e\g\ + t\ii. + fi/. +'f«'^ ■= "iSx -^-gi^-g)-, 
and analogously, 

■^A — '\g\ + '>*a — '^gt = ' (^ + ^1 + *) ; 

f' + ^1 + f a ^ ^\g\-\- fjgt + "i' ^1 + g/ ^« ■ 
^ + ^1 + *» t\h-\-'xl>t — <ig\ — ta'gt' 

g = v cos a e^ — Oj COS a, e^ = a^ cos ui 
^ = ir sin a ei = III sin ui i,' = at sin u, 

f + fi*+ jfl _ gi V, c os (g, — a,) + a^Vt COS (g, - w,) 
i + it + ^, «! »! sin (ai — <u,) + a,»a sin (o, — i,) 

as the equation between the phase displacement angles « 
and 5, in parallel operation. 

The power supplied to the external circuit is 

/ = '*g, 
of which that supplied by the first machine is. 

Pi = «i ; 
by the second machine, 

/, =■ «, . 

The total electrical work done by both machines is, 
of which that done by the first machine is, 

/>! = ,?,(, — ^,'/l'; 

by the second machine, 

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The difference of output of the two machines is, 

M| + frl _ "1 — " ■ „ s 

2 2 

A^/A8 may be called the synchronizing power of the- 
machines, or the power which is transferred from one ma- 
chine to the other by a change of the relative phase angle. 

196. Special Case. — Two equal alternators of equal 


ai = ot = a 

Substituting this in the eight initial equations, these 
assume the form, — 

'i =« + /]' +ii'xt 

(t = /, Xa —if rj . 

<;/=■ *i + '1 

ei = /,' + /,' 

e,* + <r,'* = ^j* + <■/»=» a\ 

Combining these equations by twos, 

substituting c^ =■ a cos u. 

et = a sin ui,, 

we have a (cos S^ + cos a,) = f (2 + '^o^ + *♦ *) 
<i (sin <u, + sin Sj) = e (x^ — r,,*) ; 

expanding and substituting — 

g ^ "1 — Si ■ 

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a COS < COS 8 = ^ (l + ^iii^^ 
sin . cos S = e' ^^~TA . 


2 + r,i- + x^b 
That is Si + ut >- constant; 

at no-phase displacement between the alternators, or, 

8 = "■ ~ ^ - ; 

we have ' =■ - ■ ■ ■ " — ■- ■ . 

From the eight initial equations we get, by combina- 
'I'o + 'i'*B = <('■» + *o) + <i (ni* + *0*) 

•fi '■o + 'I'^o = * (''I. + *o) + '. ('■e' + *o*) ; 
subtracted and expanded — 
'i — '» 

( , — /; = ^i> f^i — <•!) + '^ (^i — 'i') . 

e^ — ti = a (cos 5] — cos u,) = — 2 it' sin < sin S 
fi' — tx^a (sin Ml — sin a^ = 2 a cos c sin 8 ; 
e have 

/i — It s= ^ — {*■„ COS < — ^0 sin €} 

= 2 a}„ sin 8 cos (i + a). 

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The difference of output of the two alternators i; 

A/ =/,-/, = *(/■,-,;); 
hence, substituting, 


Xf,g -r„i 

^ 2 

cos t = ■ 

we have, 

,. j ^1 + ^•i±^J+ ( »•■!•-'-•■' )• j 

.«sin2s{.. + £|!.} 
A/- ' -^ J - 

V {1 + ''e^ + Xg6 + V»^ 


Afi = - i ^; 

^» +^■^ + ^^0 + 1' 


Hence, the transfer of power between the alternators, 
A/, is a maximum, if 8 i= 45° ; or wi — £, = 90° ; that is, 
when the alternators are in quadrature. 

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a COS ( cos 

a sin < cos 8 = c'-^ 

8=.(l + ti£±^') 

hence tan t =. „ ^^^ — p" — - = constant. 

2 + rof 4- x^b 

That is Ml + "« = constant ; 

^nd co,8_|y/^rTS|i5y7^iIZEHiJ, 

or, ,~ , ''°°'* . 

at no-phase displacement between the alternators, or, 

we have e ■• — " , 

From the eight initial equations we get, by corabina- 
fi '■» + "f/^i) = « ('•o + ^) + »i (V + ^D*) 

'I'-o + 'I'jfo = ' (''I) + *•) + 'j ('■o' + *ll") i 

subtracted and expanded — 

or, since 

tx — ^, = a (cos »i — cos fl^ = — 2 fl" sin c sin 8 
t( — ei = a (sin £[ — sin 5^ = 2 u cos t sin S ; 

we have 

A - /. _ 2£ji!Li {^^ cos * - r, sin «} 

= 2 dj'o sin 8 cos (t + a), 

tan a =" — . 

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The difference of output of the two alternators is 
A/ — /i —px => e (/[ — *",) ; 
hence, substituting, 






2 j'l 2 


we have, 

^a^sin8cos8|^.(H-'M±^^-^. ^^^-'■^' ^j. 

.■.in28{<. + |} 
2«'cos28|«, + |} 


Hence, the transfer of power between the alternators, 
A /, is a maximum, if 8 = 45° ; or 5, — fij = 90° ; that is, 
when the alternators are in quadrature. 

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The synchronizing power, ^p j ^h, is a maximum if 
& «0 ; that is, the alternators are in phase with each other. 

197. As an instance, curves may be plotted 

a -2500, 

Z, a r, -Jx^ = 1 - lOy; or K, = f, +/ *, = .01 + .1/, 

with the angle h =• - ^ - "' as abscissae, giving 

the value of terminal voltage, e ; 

the value of current in the external circuit, i^^ey, 

the value of interchange of current between the alternators, 

'i — ^: 

the value of interchange of power between the alternators, A/ 
the value of synchronizing power, -— ^ . 
For the condition of external circuit. 















198. In the chapter on synchronizing alternators we 
have seen that when an alternator running in synchronism 
is connected with a system of given E.M.F., the work done 
by the alternator can be either positive or negative. In 
the latter case the alternator consumes electrical, and 
consequently produces mechanical, power; that is, runs 
as a synchronous motor, so that the investigation of the 
synchronous motor is already contained essentially in the 
equations of parallel-running alternators. 

Since in the foregoing we have made use mostly of 
the symbolic method, we may in the following, as an 
instance of the graphical method, treat the action of the 
synchronous motor diagrammatically. 

Let an alternator of the E.M.F., ^j, be connected as 
synchronous motor with a supply circuit of E.M.F., E,^, 
by a circuit of the impedance Z. 

If E^ is the E.M.F. impressed upon the motor termi- 
nals, Z is the impedance of the motor of induced E.M.F., 
E-^. If £"(, is the E.M.F. at the generator terminals, Z is 
the impedance of motor and line, including transformers 
and other intermediate apparatus. If E^ is the induced 
E.M.F. of the generator, Z is the sum of the impedances 
of motor, line, and generator, and thus we have the prob- 
lem, generator of induced E.M.F. £"(,, and motor of induced 
E.M.F. E^\ or, more general, two alternators of induced 
E.M.Fs., E(^, E^, connected together into a circuit of total 
impedance, Z. 

Since in this case several E.M.Fs. are acting in circuit 

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with the same current, it is Convenient to use the current, 
/, as zero line 01 of the polar diagram. Fig. 188. 

If f =c^ i = current, and Z =■ impedance, r = effective 
resistance, x = effective reactance, and s «= Vr' + x^ = 
absolute value of impedance, then the E.M.F. consumed 
by the resistance is E„ = ri, and in phase with the cur- 
rent, hence represented by vector 0E„; and. the E.M.F. 
consumed by the reactance is £, = xi, and 90° ahead of 
the current, hence the E.M.F. consumed by the impedance 
\sE= ^{E.,f + (£a)*, or = ; v7>+^ = is, and ahead of 
the current by the angle S, where tan i = x j r. 

We have now acting in circuit the E.M,Fs., E, E-^, E^; 
or El and E are components of £"(,; that is, E^ is the 
diagonal of a parallelogram, with E^ and E as sides. 

Since the E.M.Fs. ^^i, E^, E, are represented in the 
diagram, Fig. 138, by the vectors OE^, OE^, OE, to get 
the parallelogram of E^, E^, E, we draw arcs of circles 
around with E^ , and around E with Ej . Their point of 
intersection gives the impressed E.M.F., OE^, = E^, and 
completing the parallelogram OE £"(, E-^ we get, £?£, = E^, 
the induced E.M.F, of the motor. 

j^ IOE(, is the difEerence of phase between current and im- 
pressed E.M.F., or induced E.M.F, of the generator. 

f^ lOEx is the difference of phase between current and in- 
duced KM.F. ot the motor. 

And the power is the current /' times the projection of the E.M.F. 
upon the current, or the zero line 01. 

Hence, dropping perpendiculars, E^E^ and EiE^, from 
£■(, and Ey upon 01, it is — 

/'o — 'X OEa^ = power supplied by induced E.M.F. of gen- 

Pi = i X OE} = electric y^wgt transformed in mechanical 
power by the motor. 

P = i X OEi = power consumed in the circuit by effective 

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Obviously P^ = P^-\- P. 

Since the circles drawn with E^ and E-y around O and E 
respectively intersect twice, two diagrams exist. In gen- 
eral, in one of these diagrams shown in Fig. 1S8 in drawn 

lines, current and E.M,F, are in the same direction, repre- 
senting mechanical work done by the machine as motor. 
In the other, shown in dotted lines, current and'E.M.F. are 
in opposite direction, representing mechanical work con- 
sumed by the machine as generator. 

Under certain conditions, however, E^ is in the same, £", 
in opposite direction, with the current ; that is, both ma- 
chines are generators. 

199. It is seen that in these diagrams the E.M.Fs. are 
consideretl from the point of view of the motor ; that is. 

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work done as synchronous motor is considered as positive, 
work done as generator is negative. In the chapter on syn- 
chronizing generators we took the opposite view, from the 
generator side. 

In a single unit-power transmission, that is, one generator 
supplying one synchronous motor over a line, the E.M.F. 
consumed by the impedance, E = OB, Figs. 139 to 141, con- 
sists of three components ; the E.M.F. 0£^ = £*,, consumed 

by the impedance of the motor, the E.M.F. £% Eg = £'j 
consumed by the impedance of the hne, and the E.M.F. 
£^ E = E^ consumed by the impedance of the generator. 
Hence, dividing the opposite side of the parallelogram E^E^, 
in the same way, we have : OE^ = E^= induced E.M.F. of 
the motor, OE^ ^ E^ = E.M.F. at motor terminals or at 
end of line, OEg = E^ = E.M.F. at generator terminals, 
or at beginning of line. OEf, = Eq = induced E.M.F, of 

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The phase relation of the current with the RM.Fs. £■,, 
E^, depends upon the current strength and the E.M.Fs. £■, 
and Et^. 

aOO. Figs. 189 to 141 show several such diagrams for 
different values of E^, but the same value of / and E^. 
The motor diagram being given in drawn line, the genera- 
tor diagram in dotted line. 

fit. 140. 

As seen, for small values of E^ the potential drops in 
the alternator and in the line. For the value of Ey = Eq 
the potential rises in the generator, drops in the line, and 
rises again in the motor. For larger values of E^, the 
potential rises in the alternator as well as in the line, so 
that the highest potential is the induced E.M.F. of the 
motor, the lowest potential the induced E.M.F. of the gen- 

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It is of interest now to investigate liow the values of 
these quantities change with a change of the constants. 

201. A. — Constant impressed E.M.F. E„, constant current 
strength I = i, variable motor excitation £,. (Fig. 142.) 

If the current is constant, = t; OF, the E.M.F. con- 
sumed by the impedance, and therefore point £, are con- 
stant. Since the intensity, but not the phase of Ef, is 
constant, E^ lies on a circle e^ with E^ as radius. From 
the parallelogram, OE £g E^ follows, since E^E^ parallel 
and = OE, that E■^ lies on a circle e^ congruent to the circle 
tf,, but with Ef, the image of E, as center : OEf = OE. 

We can construct now the variation of the diagram with 
the variation of Fj ; in the parallelogram OE E^, E^, O and 
E are fixed, and E^ and E^ move on the circles Co e^ so that 
£■(, £1 is parallel to OE. 

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The smallest value of E-^ consistent with current strength 
/ is Olj = E-^, 01 = £■(,. In this case the power of the 
motor is Olj' x /, hence already considerable. Increasing 
E^ to 02p OSp etc., the impressed E.M.Fs. move to 02, 03, 
etc., the power is / X 02,i, / x 03,', etc., increases first, 

reaches the maximum at the point 3,, 3, the most extreme 
point at the right, with the impressed E.M.F. in phase with 
the current, and then decreases again, while the induced 
E.M.F. of the motor £, increases and becomes = £■„ at 
4[, 4, At 5,, 5, the power becomes zero, and further on 
negative ; that is, the motor has changed to a dynamo, and 

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produces electrical energy, while the impressed E.M.F. E^ 
still furnishes electrical energy, that is, both machines as 
generators feed into the line, until at 6, , 6, the power of the 
impressed E.M.F, E^ becomes zero, and further on power 
begins to flow back ; that is, the motor is changed to a gen- 
erator and the generator to a motor, and we are on the 
generator side of the diagram. At 7i, 7, the maximum value 
of £■,, consistent with the current /, has been reached, and 
passing still further the E.M.F. Ey decreases again, while 
the power still increases up to the maximum at 8,, 8, and 
then decreases again, but still E^ remaining generator, E^ 
motor, until at ll,, 11, the power of E^ becomes zero ; that 
is, £"o changes again to a generator, and both machines are 
generators, up to \%, 12, where the power of E^ is zero, £", 
changes from generator to rhotor, and we come again to 
the motor side of the diagram, and while E^ still decreases, 
the power of the motor increases until 1,, 1, is reached. 

Hence, there are two regions, for very large £, from 
5 to 6, and for very small E^ from 11 to 12, where both 
machines are generators ; otherwise the one is generator, 
the other motor. 

For small values of Ey the current is lagging, begins, 
however, at 2 to lead the induced E.M.F. of the motor E^, 
at 3 the induced E.M.F, of the generator E^. 

It is of interest to note that at the smallest possible 
value of E^, Ij, the power is already considerable. Hence, 
the motor can run under these conditions only at a certain 
load. If this load is thrown off, the motor cannot run with 
the same current, but the current must increase. We have 
here the curious condition that loading the motor reduces, 
unloading increases, the current within the range between 
1 and 12, 

The condition of maximum output is 3, current in phase 
with impressed E.M.F. Since at constant current the loss 
is constant, this is at the same time the condition of max- 
imum efficiency : no displacement of phase of the impressed 

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E.M.F., or self-induction of the circuit compensated by the 
effect of the lead of the motor current. This condition of 
maximum efficiency of a circuit we have found already in 
the Chapter on Inductance and Capacity. 

202. B. E^ and E-^ constant, I variable. 

Obviously Eq lies again on the circle e^ with E^ as radius 
and O as center. 

E lies on a straight line e, passing through the origin. 

Since in the parallelogram OE E^ E-y, EE^^ = E^, we 
derive E^ by laying a line EE;^ = Ey from any point E 
in the circle e^, and complete the parallelogram. 

All these lines EEq envelop a certain curve ei, which 

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can be considered as the characteristic curve of this prob- 
lem, just as circle Cj in the former problem. 

These curves are drawn in Figs. 148, 144, 145, for the 
three cases : 1st, E^ = E^; 2d, E^KE^; 8d, E^>E^. 

In the first case, E^ = E^ (Fig. 127), we see that at 

very small current, that is very small OE, the current / 
leads the impressed E.M.F. E^, by an angle E(,0/ = uiq. 
This lead decreases with increasing current, becomes zero, 
and afterwards for larger current, the current lags. Taking 
now any pair of corresponding points E, E,^, and producing 
EE„ until it intersects e^, in E,, we have ^^ E^ OE = 90°, 
£i = £o , thus : OEi ^ ££„ = OE^ = E^i ; that is, EEi = 

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S^ij. That means the characteristic curve e^ is the enve- 
lope of lines EE^, of constant lengths S^g, sliding between 
the I^s of the right angle £", OE; hence, it is the sextic 
hypocyloid osculating circle e^, which has the general equa- 
tion, with e, tf as axes of coordinates : 

In the next case, £, < E^ (Fig. 144) we see first, that 
the current can never become zero like in the first case. 

E^ = E^, but has a minimum value corresponding to the 

E — E 

minimum value of OE-^ ; /['= — ^ ^—, and a maximum 

£■„ + £■, 
value : II' = ::: . Furthermore, the current can never 

lead the impressed E.M.F, £"q, but always lags. The mini- 

fy GoOglc 


mum lag is at the point H. The locus e^, as envelope of the 
lines EE^, is a finite sextic curve, shown in Fig. 144. 

n Ey< E^, at small E^ — E^, H can be above the zero 
line, and a range of leading current exist between two ranges 
of lagging current. 

In the case .E^ > E,^ (Fig. 145) the current cannot equal 

zero either, but begins at a finite value C,', corresponding 

E, - Ef. 

to the minimum value of OEn : // = -. At this 

value however, the alternator Ey is still generator and 
changes to a motor, its power passing through zero, at the 
point corresponding to the vertical tangent, onto e^, with 
a very large lead of the impressed E.M.F. against the cur- 
rent. At H the lead changes to lag. 

The minimum and maximum value of current in the 
three conditions are given by : 

Since the current passing over the line at £, = O, that 
is, when the motor stands still, is /^ = Ef^j z, we see that 
in such a synchronous motor-plant, when running at syn- 
chronism, the current can rise far beyond the value it has 
at standstill of the motor, to twice this value at 1, some- 
what less at 2, but more at S. 

203. C. E^ = constant, Ei varied so that the efficiency is a 
mctximum for all currents, (Fig. 146.) 

Since we have seen that the output at a given current 
■ strength, that is, a given loss, is a maximum, and therefore 

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the efficiency a maximum, when the current is in phase 
with the induced E.M.F. E^ of the generator, we have as 
the locus of Eq the point E^^ (Fig- 146), and when E with 
increasing current varies on e, £, must vary on the straight 
line ^1 parallel to ^. 

Hence, at no-load or zero current, Ey = E^, decreases 
with increasing load, reaches a minimum at OE^ perpen- 
dicular to e^, and then increases again, reaches once more 

El = E^ at Ei', and then increases beyond E^. The cur- 
rent is always ahead of the induced E.M.F. £, of the motor, 
and by its lead compensates for the self-induction of the 
system, making the total circuit non-inductive. 

The power is a maximum at Ej^, where 0£i* = £i*Eg ^ 
1/2 X OEf,, and is then = / x ^/2. Hence, since 0E\* = 
/r = Eo/2, / = E^/2r and P = E^'/ir, hence = the maxi- 
mum power which, over a non-inductive line of resistance r 
can be transmitted, at 50 per cent, efficiency, into a non- 
inductive circuit. 

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In general, it is, taken from the diagram, at the condi- 
tion of maximum efficiency : 

^1= V(£,-/r)'' +/»*'. 
Comparing these results with those in Chapter IX. on 
Self-induction and Capacity, we see that the condition of 
maximum efficiency of the synchronous motor system is 
the same as in a system containing only inductance and 
capacity, the lead of the current against the induced E.M.F. 
£'1 here acting in the same way as the condenser capacity 
in Chapter IX. 

D. Ea = constant ; P ~ constant. 

If the power of a synchi 
we have {Fig. 147) I X OE 

motor remams constant, 
constant, or, since OE^ = 

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Ir, I = OE^jr, and: OE^ X OE^ = tJA' x "E^ = 

Hence we get the diagram for any value of the current 
/, at constant power /*, , by making OE ^Ir, EKE^ = F^ j I 
erecting in E^ a perpendicular, which gives two points of 
intersection with circle ef,, E^, one leading, the other lagging. 
Hence, at a given impressed E.M.F. Eq, the same power P^ 

FI3. 149. 

can be transmitted by the saine current / with two different 
induced E.M.Fs. E, of the motor; one, OEi^^EE^ small, 
corresponding to a lagging current ; and the other, OE, = 
EEq large, corresponding to a leading current. The former 
is shown in dotted lines, the latter in drawn lines, in the 
diagram, Fig. 147. 

Hence a synchronous motor can work with a given out- 
put, at the same current with two different counter E.M.Fs 

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£■,. In one of the cases the current is leading, in the 
Dther lagging. 

In Figs. 148 to 161 are shown diagrams, giving the points 

Et, = impressed E.M.F., assumed as constant — 1000 volts, 
E = E.M.F. consumed by impedance, 
£' s= E.M.F. consumed by resistance. 

^^s, =. 


X i«» 


/ V 1170/1910 



/ y\ loa/iiso 


^^ , 1 , ; . *oo/ino 



\ N^T^'' W/IW 



\ \)^ «»/» 


'\ X' 310 










ftg. 149. 

The counter E,M.F. of the motor, £■,, is OE^, equal and 
parallel EE^, but not shown in the diagrams, to avoid 

The four diagrams correspond to the values of power, 
or motor output, 
P = 1,000, 6,000, 9,000, 

/» = 1,000 46 < £, < 2,200, 

J' = 6,000 340 < ^, < 1,920, 

/" = 9,000 640 <£,< 1,750, 

J' = 12,000 920 <£i< 1,320, 

12,000 watts, and give : 
1 < 7 < 49 Fig. 132. 

7 < 7 < 43 Fig. 133. 

11.8 </< 38.3 Fig. 134. 

20 < / < 30 Fig. 153. 

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(t 1440 

5. zoeon™ 



,1 doo/iiieo 

113/aBj, , 


V ) 

-%' 7ao/iK» 








m/iioo a/t&s 

As seen, the permissible value of counter E.M.F. E^ and 
f current /, becomes narrower with increasing output. 

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In the diagrams, different points ,of E^ are marked with 
1, 2, 3 . . ., when corresponding td leading current, with 
2^, 3', . . . , when corresponding to lagging current. 

The values of counter E.M.F. ^i and of current / are 
noted on the diagrams, opposite to the corresponding points 

In this condition it is interesting to plot the current as 
function of the induced E.M.F. E^ of the motor, for con- 
stant power P^. Such curves are given in Fig. 165 and 
explained in the following on page 845. 

205. While the graphic method is very convenient to 
get a clear insight into the interdependence of the different 
quantities, for numerical calculation it is preferable to ex- 
press the diagrams analytically* 

For this purpose, 

Let s — V/^ + J^ = impedance of the circuit of (equivalent) 
resistance r and (equivalent) reactance x^2irNL, containing 
the impressed E.M.F. e^* and the counter EM.F. e^ o£ the syn- 
chronous motor; that Is, the E.M.F. induced in the motor arma- 
ture by its rotation through the (resultant) magnetic field. 

Let i = current in the circuit (effective values). 

The mechanical power delivered by the synchronous 
motor (including friction and core loss) is the electric 
power consumed by the C.E.M.F. e^^; hence — 

/ = /^,cos(.„<rO, (1) 

thus, — 



= v/^^-- 


* If fg =• E.M.F. al motor terminals, ( — inlernal impedance of the 
motor; ii fg— terminal voltage of the generator, ■ — total impedance of line 
and motor; if e^— E.M.F. of generator, that is, E.M.F. induced in generator 
armature by its rotation ihrough the magnetic field, ■ includei the generator 
impedance also. 



The displacement of phase between current i and E.M.F. 
e = si consumed by the impedance z is : 

cos (ie) = 

sin (ie) . 

Since the three E.M.Fs. acting in the closed circuit : 

ifi = EM.F. o( generator, 

Ci = C.KM.F. of synchronous motor, 

e ^ si ^ E.M.F, consumed by impedance, 

form a triangle, that is, f, and e are components of e^, it is 
(Fig. 152) : 

^o' = ^,' + ^ + 2 ^ f, cos {c, c), (4) 

hence, cos (^i, e) = " ■■ ' — = -" — ' ■ . , (^ij 

since, however, by diagram : 

cos (.r, , €) = cos (/, ^ - i, e,) 

= cos {/, e) cos (,; .,) + sin (/, .) sin (/, .,) (C) 

substitution of (2), (3) and (5) in (6) gives, after some trans- 
position : 

the Fundamental Equation of the Synchronous Motor, relat- 
ing impressed E.M.F., e^; C.E.M.F., c-^; current /; power, 
/, and resistance, r \ reactance, x ; impedance s. 

This equation shows that, at given impressed E.M.F, Cp, 
and given impedance s = '■Jr^ + ^^i three variables are left, 
e^, i, p, of which two are independent. Hence, at given c,, 
and s, the current * is not determined by the load / only, 
but also by the excitation, and thus the same current i can 
represent widely different loads /, according to the excita- 
tion ; and with the same load, the current i can be varied 
in a wide range, by varying the field excitation e-^. 

The meaning of equation (7) is made more perspicuous 

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by some transformations, which separate e^ and ti as func- 
tion of/ and of an angular parameter ^. 
Substituting in (7) the new coordinates : 




U*'* = ° 

we get 

V--V2-2r/ = 2^y/^l^- 8»/'i 
substituting again, (* = « 



we get 

, a_„V2-f^ = Vi:l- t')(2a'-2/J»-i^, (11) 

and, squared, 

.'.•+(l-.')^-aV? (..—«) + *'<'-■') + 6!=^-0, (12) 


(i.-c>)V2 1 

gives, after some transposition, 

■d'^-m'^ ^ ~P a(a-2ti), (14) 

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'* '^ z^ + «/=> = J« (16) 

the equation of a circle with radius R. 

Substituting now backwards, we get, with some trans- 
positions : 

{r'(<r,' + ««/•) - *• (V - 2 rp)Y + {rx (..• - ^/«)}* = 

x*^e^'{e,*-irp) (17) 

the Fundametital Equation of the Synchronous Motor in a 
modified form. 

The separation of <■, and ( can be effected by the intro- 
duction of a parameter ^ by the equations ; 

These equations (18), transposed, give 





sin + JVi? 






.* + J. 






i(f=— . 



The parameter ^ has no direct physical meaning, appar- 

These equations (19) and (20), by giving the values of 
^1 and i as functions of fi and the parameter 4' enable us 
to ccnstruct the Power Characteristics of the Synchronous 
Motor, as the curves relating c, and i, for a given power /, 
by attributing to ^ all different values. 

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Since the variables v and w in the equation of the circle 
(16) are quadratic functions of e^ and i, the Power Charac- 
teristics of the Synchronous Motor are Quartic Curves. 

They represent the action of the synchronous motor 
under all conditions of load and excitation, as an element 
of power transmission even including the line, etc. 

Before discussing further these Power Characteristics, 
some special conditions may be considered. 

206. A. Maximum Output. 

Since the expression of *■, and i [equations (19) and 
(20)] contain the square root, Vc^^ — 4 rp, it is obvious 
that the maximum value of p corresponds to the moment 
where this square root disappears by passing from real to 
imaginary ; that is, 

i^ - 4 r/ = 0, 

P-fy (21) 

This is the same value which represents the maximum 
power transmissible by E.M.F., e^, over a non-inductive line 
of resistance, r; or, more generally, the maximum power 
which can be transmitted over a line of impedance, 

into any circuit, shunted by a condenser of suitable capacity. 
Substituting (21) in (19) and (20), we get. 

and the displacement of phase in the synchronous motor. 

cos (^1,/) = -^ = -; 

taii(.„/) = -?, (23) 

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that is, the angle of internal displacement in the synchron- 
ous motor is equal, but opposite to, the angle of displace- 
ment of line impedance, 

(<„ .■) - - (,, 0, 

_-(.,r), (24) 

and consequently, 

(<.,0-0; (25) 

that is, the current, i, is in phase with the impressed 
E.M.F., ^,. 

If » < 2 r, .f, < e^; that is, motor E.M.F. < generator E.M.F. 

\iz^2r, e^=tet\ that is, motor E.M.F. =« generator KM. F. 

If«>2r, ^,><ro; that is, motor KM.F. > generator RM,F. 

In either case, the current in the synchronous motor is 

207. B. Running Light, / = 0. 

When running light, or for / = 0, we get, by substitut- 
ing in (19) and (20), 

/ =^Y||l-HjCOS + -^sin*| 

Obviously this condition cannot well be fulfilled, since/ 
must at least equal the power consumed by friction, etc. ; 
and thus the true no-load curve merely approaches the curve 
/ = 0, bemg, however, rounded off, where curve (26) gives 
sharp comers. 

Substituting / = into equation (7) gives, after squar- 
ing and transposing, 
.ri* + V + «''*-2<-iV-2;'i*^o' + 2r'(V-2ji:"»V = 0.(27) 

This quartic equation can be resolved into the product 
of two quadratic equations, 

e* + ,*,-« - ^„« -I- 2 *iV, * 0. > ^28) 

^1* -|- «*»■* — Co* — 2 xity = 0. j 

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which are the equations of two ellipses, the one the image 

of the other, both inclined with their axes. 

The minimum value of C.E.M.F., (u is i-i = at / = ^. (29) 
The minimum value of current, i, is /' «- at rf, =- a, . (30) 
The maximum value of E.M.F., e^, is given by Equation (2S), 
f='\ +«*»' — V±2j:iV, = 0; 

by the condition, 


ai v^/^. i^'i = 0- 

' = eD^. fi = =F'o~ (31) 

The maximum value of current, i, is given by equation 
(28) by 

-^ = 0, as 

i-^ fi= fo-- (32) 

If, as abscissx, «,, and as ordinates, si, are chosen, the 
axis of these ellipses pass through the points of maximum 
power given by equation (22). 

It is obvious thus, that in the V-shaped curves of syn- 
chronous motors running light, the two sides of the curves 
are not straight lines, as usually assumed, but arcs of ellipses, 
the one of concave, the other of convex, curvature. 

These two ellipses are shown in Fig. 154, and divide the 
whole space into six parts — the two parts .fl and A', whose 
areas contain the quartic curves (19) (20) of synchronous 
motor, the two parts B and B', whose areas contain the 
quartic curves of generator, and the interior space C and 
exterior space Z>, whose points do not represent any actual 
condition of the alternator circuit, but make e,, ( imaginary. 

A and A' and the same B and B', are identical condi- 
tions of the alternator circuit, differing merely by a simul- 

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1 , 











































F\3. IS*. 

taneous reversal of current and E,M.F. ; that is, differing 
by the time of a half period. 

Each of the spaces A and £ contains one point of equa- 
tion (22), representing the condition of maximum output 
of generator, viz., synchronous motor. 

208. C. Minimum Current at Given Power. 

The condition of minimum current, (', at given power, /, 
is determined by the absence of a phase displacement at the 
impressed E.M.F. e^, 


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This gives from diagram Fig. 158, 

«i' = t^ + »«^ _ 2 ie^r, (33) 

or, transposed, 

'1 = V(.tf-/r)' + /«;c». (34) 

This quadratic curve passes through the point of zero 
current and zero power, 

,- = 0, e^=e„ 

through the point of maximum power (22), 

and through the point of maximum current and zero power^ 
,■ = % *, =. ^ (36) 

and divides each of the quartic curves or power character- 
istics into two sections, one with leading, the other with 
lagging, current, which sections are separated by the two 
points of equation 84, the one corresponding to minimum, 
the other to maximum, current. 

It is interesting to note that at the latter point the 
current can be many times larger than the current which 
would pass through the motor while at rest, which latter 
current is, 

«■ =. ^ , (36) 

while at no-load, the current can reach the maximum value, 

»■=-", (36) 

the same value as would exist in a non-inductive circuit of 
the same resistance. 

The minimum value at C.E.M.F. e-^, at which coincidence 

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of phase (fg, i) = 0, can still be reached, is determined from 
equation (34) by, 


,_,.r „_,.?. (37) 

The curve of no-displaceraent, or of minimum current, is 
shown in Figs. 138 and 139 in dotted lines.* 

209. D. Maximum Displacement of Phase. 

(«,, /) = maximuni. 
At a given power / the input is, 

A =P + '■''• = '.'' cos (^., i (38) 


cos (.To, = ^i^. (39) 

At a given power/, this value, as function of the current 
i, is a maximum when 

dt \ «o» / 
this gives, 

/ = iV; (40) 



That is, the displacement of phase, lead or lag, is a 
maximum, when the power of the motor equals the power 

* II is interesting lo note that the equation (34) is similar to the value, 
<1 = -/l^t^ — irY — i^x^, which represents the output transmitted over sn 
inductive line ol impedance, i = Vr^ + x^ into a non-inductive circuit. 

Equation (34) is identical with the equation giving the maiimiim vollSEe, 
t\ , at cuirent, i, which can be produced by shunting the receiving circuit with & 
condenser; that is, the condition of " complete resonance " o[ the line, c = 
Vr* + J*, with current, l'. Hence, referring lo equation (35), t^^ e^-" is 
the maximum resonance voltage ol the line, reached when closed \if a con- 
denser of r( 

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consumed by the resistance ; that is, at the electrical effi- 
ciency of 50 per cent. 

Substituting (40) in equation (7) gives, after squaring 

and transposing, the Quartic Equation of Maximum Dis- 

(3'+3r') = 0. (42) 

The curve of maximum displacement is shown in dash- 
dotted lines in Figs. 154 and 155. It passes through the 

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point of zero current — as singular or nodal point — and 
through the point of maximum power, where the maximum 
displacement is zero, and it intersects the curve of zero 

210. E, Constant Counter E.M.F. 

At constant C.E.M.F., e^ = constant. 



the current at no-load is not a minimum, and is lagging. 
With increasing load, the lag decreases, reaches a mini- 
mum, and then increases again, until the motor falls out of 
step, without ever coming into coincidence of phase. 

If «oyl f-<«i<<^., 

the current is lagging at no load ; with increasing load the 
lag decreases, the current comes into coincidence of phase 
with (■(,, then becomes leading, reaches a maximum lead; 
then the lead decreases again, the current comes again into 
coincidence of phase, and becomes lagging, until the motor 
falls out of step. 

If ffl < ^1 , the current is leading at no load, and the 
lead first increases, reaches a maximum, then decreases ; 
and whether the current ever comes into coincidence of 
phase, and then becomes lagging, or whether the motor 
falls out of step while the current is still leading, depends, 
whether the C.E.M.F, at the point of maximum output is 
> ^ijor < <•(,- 

211. P. Numerical Instance, 

Figs, 154 and 155 show the characteristics of a 100- 
kilowatt motor, supplied from a 2500-volt generator over a 
distance of 5 miles, the line consisting of two wires, No. 
2 B. & S.G., 18 inches apart. 

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In this case we have, 

^0 = 2500 volts constant at generator terminals; 
r = 10 ohms, including line and motor ; 
x= 20 ohms, including line and motor ; 
hence < = 22.36 ohms. 

Substituting these values, we get, 

2500' - «,» - 500 (» - 20/ = 40 V*'V-/* (7) 

{V + SOO »■»- 31.25 X 10'+ 100/}*+ {2 V- 1000 »■»}» = 

7.8125 X 10" - 5 + lOV- (17) 

tx = 5590 (19) 

V'i{(l-3.2xl0-^) + (.804cos*+.447sin<^)Vl-6.4xl0-V}. 
*■ = 559 (20) 

Vi {(1-3-2 Xl0-'/)+(.894cos^-.447 sin ^) Vl^ 6.4 XlO-'/7. 

Maximum output, 

/ = 156.25 kilowatts (21) 

at ^, = 2,795 volts J .^^, 

i = 125 amperes ( 
Running light, 

V + 600 »» - 6.25 X 10* ^ 40 »>, = 1 .^o. 

<;, = 20/i V6.25 X 10'- 100/" ( * ^ 

mum value of C.E.M.F. f i = Is / = 112 (29) 
mum value of current, i = is r, = 2500 (30) 
mum value of C.E.M.F. e^ = 5."90 is / = 223.5 (31) 
imum value of current * = 250 is ti = 5000 (32) 

At the mini 
At the mini 
At the max 
At the m.ix 

Curve of zero displacement of phase, 

= 10 V(250 _()* + 4»* (34) 

= 10 v'6:25 X 10* - 500 / + 5 i« 
Minimum C.E.M.F. point of this curve, 

,■ = 50 f, = 2240 (35) 

Curve of maximum displacement of phase, 

/ = 10 /' (40) 

(6.25 X 10»-<r,y + -65 X 10'/' - 10">i' = 0. (42) 

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Fig. 154 gives the two ellipses of zero power, in drawn 
lines, with the curves of zero displacement in dotted, the 
curves of maximum displacement in dash-dotted lines, and 
the points of maximum power as crosses. 

Fig. 155 gives the motor-power characteristics, for, 

/ = 10 kilowatts. 
/ = 50 kilowatts. 
/ = 100 kilowatts. 
/ = 150 kilowatts. 
p = 166.25 kilowatts, 

together with the curves of zero displacement, and of maxi- 
mum displacement. 

212. G. Discussion of Results. 

The characteristic curves of the synchronous motor, as 
shown in Fig. 155, have been observed frequently, with 
their essential features, the V-shaped curve of no load, with 
the point rounded off and the two legs slightly curved, the 
one concave, the other convex; the increased rounding off 
and contraction of the curves with increasing load ; and 
the gradual shifting of the point of minimum current with 
increasing load, first towards lower, then towards higher, 
■ values of C.E.M.F. e^. 

The upper parts of the curves, however, I have never 
been able to observe experimentally, and consider it as 
probable that they correspond to a condition of synchro- 
nous motor-running, which is unstable. The experimental 
observations usually extend about over that part of the 
curves of Fig. 155 which is reproduced in Fig. 156, and in 
trying to extend the curves further to either side, the motor 
is thrown out of synchronism. 

It must be understood, however, that these power char- 
acteristics of the synchronous motor in Fig. 1 55 can be con- 
sidered as approximations only, since a number of assump- 

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tions are made which are not, or only partly, fulfilled in 
practice. The foremost of these are : 

1, It is assumed that e^ can be varied unrestrictedly, 
while in reality the possible increase of ^i is limited by 
magnetic saturation. Thus in Fig. 155, at an impressed 
E.M.F., e^ = 2,500 volts, ^, rises up to 6,690 volts, which 
may or may not be beyond that which can be produced 
by the motor, but certainly is beyond that which can be 
constantly given by the motor. 

























a a 

ff — Ti 

f JA 



. — i 

i 4 

Pg. 169. 

2. The reactance, x, is assumed as constant. While 
the reactance of the line is practically constant, that of the 
motor is not, but varies more or less with the saturation, 
decreasing for higher values. This decrease of x increases 
the current (, corresponding to higher values of ?j, and 
thereby bends the curves upwards at a lower value of ei 
than represented in Fig. 15S. 

It must be understood that the motor reactance is not 
a simple quantity, but represents the combined effect of 

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self-induction, that is, the E.M.F. induced in the armature 
conductor by the current flowing therein and armature 
reaction, or the variation of the C.E.M.F. of the motor 
by the change of the resultant field, due to the super[>osi- 
tion of the M.M.F. of the armature current upon the field 
excitation ; that is, it is the " synchronous reactance." 

8. These curves in Fig. 155 represent the conditions 
of constant electric power of the motor, thus including the 
mechanical and the magnetic friction (core loss). While 
the mechanical friction can be considered as approximately 
constant, the magnetic friction is not, but increases with 
the m^^netic induction ; that is, with ^, , and the same holds 
for the power consumed for field excitation. 

Hence the useful mechanical output of the motor will 
on the same curve, / = const,, be larger at points of lower 
C.E.M.F,, *,, than at points of higher e^; and if the curves 
are plotted for constant useful mechanical output, the whole 
system of curves will be shifted somewhat towards lower 
values of e^ ; hence the points of maximum output of the 
motor correspond to a lower E.M.F. also. 

It is obvious that the true mechanical power-character- 
istics of the synchronous motor can be determined only 
in the case of the particular conditions of the installation 
under consideration. 

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213. Commutator motors — that is, motors in which 
the current enters or leaves the armature over brushes 
through a segmental commutator — have been built of 
various types, but have not found any extensive appli- 
cation, in consequence of the superiority of the induction 
and synchronous motors, due to the absence of commu- 

The main subdivisions of commutator motcrs are the 
repulsion motor, the series motor, and the shcnt motor. 


214. The repulsion motor is an induction motor or 
transformer motor; that is, a motor in which the main 
current enters the primary member or field only, while 
in the secondary member, or armature, a current is in- 
duced, and thus the action is due to the repulsive thrust 
between induced current and inducing magnetism. 

As stated under the heading of induction motors, a 
multiple circuit armature is required for the purpose of 
having always secondary circuits in inductive relation to 
the primary circuit during the rotation. If with a single- 
coil field, these secondary circuits are constantly closed 
upon themselves as in the induction motor, the primary 
circuit will not exert a rotary effect upon the armature 
while at rest, since in half of the armature coils the cur- 
rent is induced so as to give a rotary effort in the one 
direction, and in the other half the current is induced to 

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give a rotary effort in the opposite direction, as shown 
by the arrows in Fig. 167. 

In the induction motor a second magnetic field is used 
to act upon the currents induced by the first, or inducing 
magnetic field, and thereby cause a rotation. That means 
the motor consists of a primary electric circuit, inducing 

in the armature the secondary currents, and a primary 
magnetizing circuit producing the magnetism to act upon 
the secondary currents. 

In the polyphase induction motor both functions of the 
primary circuit are usually combined in the same coils ; that 
is, each primary coil induces secondary currents, and pro- 
duces magnetic flux acting upon secondary currents induced 
by another primary coil. 

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215. In the repulsion motor the difficulty due to the 
equal and oppwsite rotary efforts, caused by the induced 
armature currents when acted upon by the inducing mag- 
netic field, is overcome by having the armature coils closed 
upon themselves, either on short circuit or through resist- 
ance, only in that position where the induced currents give 

a rotary effort in the desired direction, while the armature 
coils are open-circuited in the position where the rotary 
effort of the induced currents would be in opposition to 
the desired rotation. This requires means to open or close 
the circuit of the armature coils and thereby introduces the 

Thus the general construction of a repulsion motor is 
as shown in Figs. 158 and 159 diagrammatically as bipolar 

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motor. The field is a single-phase alternating field F^ the 
armature shown diagrammatically as ring wound A consists 
of a number of coils connected to a segmental commutator 
C, in general in the same way as in continuous-current ma- 
chines. Brushes standing under an angle of about 45° with 
the direction of the magnetic field, short-circuit either a 

part of the armature coils as shown in Fig. 158, or the 
whole armature by a connection from brush to brush as 
shown in Fig, 159. 

The former arrangement has the disadvantage of using a 
part of the armature coils only. The second arrangement 
has the disadvant^e that, in the passage of the brush from 
s^ment to segment, individual armature coils are short- 

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circuited, and thereby give a torque in opposite direction to 
the torque developed by the main induced current flowing 
through the whole armature from brush to brush. 

216. Thus the repulsion motor consists of a primary 
electric circuit, a magnetic circuit interlinked therewith, 
and a secondary circuit closed upon itself and displaced in 


Flo. MO. 

Space by 45" — in a bipolar motor — from the direction of 

the magnetic flux, as shown diagrammatically in Fig. 160. 

This secondary circuit, while set in motion, still remains 
in the same position of 45° displacement, with the magnetic 
flux, or rather, what is theoretically the same, when moving 
out of this position, is replaced by other secondary circuits 
entering this position of 45° displacement. 

For simplicity, in the following all the secondary quan- 

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titles, as E.M.F., current, resistance, reactance, etc., are 
assumed as reduced to the primary circuit by the ratio of 
turns, in the same way as done in the chapter on Induction 

217. Let 
* — maximum magnetic flux per field pole ; 
e = effective E.M.F, induced thereby in the field turns; thus, 
e =5 V2«-iV»*10-'; 

where « = number of turns, N= frequency, 

thus, * = ^ 

The instantaneous value of magnetism is 
^ = * sin /S ; 
and the flux interlinked with the armature circuit 

when A is the angle between the plane of the armature coil 
and the direction of the magnetic flux. (Usually about 45°.) 
The E.M.F. induced in the armature circuit, of « turns, 
(as reduced to primary circuit), is thus, 

fj=_„^10-» =-»*j^siny5sinAlO-', 

If ^= frequency in cycles per second, N^ = frequency 

of rotation or speed in cycles per second, and k= N^/ N 

speed , 

= J — ~ we have 


thus, e^= — 2-irnNt f sin A cos (S + i cos X sin ^j 10-», 
or, since 

■, =e'\!2 i sin A cos fi + k cos X sin ^ j . 

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218. Introducing now complex quantities, and counting 
the time from the zero value of rising magnetism, the mag- 
netism is represented by _/'*, 
the primary induced E.M.F., E — ^ e, 
the secondary induced E.M.F,, E^ = ~ e jsin A,+yitcosXf; 
hence, if 
Z, ^ *'\—Jxi= secondary impedance reduced to primary circuit, 

Z = r —jx =s primary impedance, 

Y = g ~jb = exciting admittance, 

we have, 

, i , -^1 sinX + /icosX 
secondary current, 1. = -^ = — e ~ , 

primary exciting current, I^ = eY= e {g +ji>), 
hence, total primary current. 

Primary impressed E.M.F., E^= ~ £ + !Z; 

= ^ j 1 + (sin A +>i cos X) ""^-"^ + (f +Jb) (r -jx) \ 

( M — /*1 ) 

Neglecting in E^ the last term, as of higher order, 
^„ = f i 1 + sin A, + ik cos \ IZlJl. \ ; 
or, eliminating imaginary quantities, 

t V(r| + r sin \ + ^Jc cos X ) ' + (j r . + :t sin \- kr cos A)* 
'•"' V(n* + x^ 

The power consumed by the component of primary 
counter E.M.F., whose flux is interlinked with the secondary 
e sin A, is, 

_, r ■ 1 n/ ^ sin A (r. sin A — kx, cos A) 
I^ = \e sm X /] = ^ , — j 1 

the power consumed by the secondary resbtance is, 

_ _ <*r,(sin'X + ^cos'A) 

' ~ ' ' ~ r-i + x^ ' 

hence the difference, or the mechanical power developed by 
the motor armature. 

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P= _ (/v_ i>) = ^*^(a:,3mX + r,-6cosA), 

and substituting for e, 

_ e^k cos X {xx sin X + r,^ cos A) 

~ (ri + r sin A + kx cos X)' + {xy^ x sin X — iir cos \)* ' 
and the torque in synchronous watts, 

_ P _ e} cos A (jr, sin A + r^k cos A) 

~ k~ (r, + r sin A + *j: cos A)* + («, + « sin A — kr cos A)' 
or - 7*= %^ ,^10-' [>,* sin A /, cos A]' = [?,/, cos Ay 
_ ^^ cos A (j^t sin A + r^k cos A) 

The stationary torque is, k = % 

_ ^i,*a:i sin X cos X 

" ~ (r, + r sin X)' + (*, + * sin X/ ' 
and neglecting the primary impedance, r = = ;r, 
_ V^i sin X cos X _ V*i sin* X 

which is a maximum at A = 45°. 
At speed k, neglecting r = = jt, 

_ _ V cos A (a:, sin A + r^k cos A) 

cot 2 A = '^- For A = 0, X = 45°; for i = oo , X = 0. 

that is, in the repulsion motor, with increasing speed, the 
angle of secondary closed circuit, A, has to be reduced to 
get maximum torque. 

219. At A = 45° we have. 

T= , 

(r, V2 + r + i*)» + (.(i V2 + * - hf 

and the power. 



gives, k = 1 . 

At A = we have, 

that is, 7" = at A = 0, or, the motor is not self-starting, 
when \ = 0. 

which is a maximum at constant X = for, -^j- = 0, which 




























P = 




















Fig. 181. Ri/mlilBn jWMOr. 

As an instance is shown, in Fig. 161, the power output 
as ordinates, with the speed k == N^/ N as abscisses, of a 
repulsion motor of the constants, 

giving the power, 

10,0 00 i.Q2 + 1.41 k - .05 ^j 
~ (.171 + 2 kf^i^M - .1 kf' ' 

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220. If, in a continuous-current motor, series motor as 
well as shunt motor, the current is reversed, the direction 
of rotation remains the same, since field magnetism and 
armature current have reversed their sign, and their prod- 

uct, the torque, thus maintained the same sign. There- 
fore such a motor, when supplied by an alternating current, 
will operate also, provided that the reversals in field and 
in armature take place simultaneously. In the series motor 
this is necessarily the case, the same current passing through 
field and through armature. 

With an alternating current in the field, obviously the 

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magnetic circuit has to be laminated to exclude eddy cur- 
Let, in a series motor. Fig. 146, 

♦ » effective magnetism per pole, 

« ■! number of field turns per pole in series, 

Ri = number of armature turns in series between brushes, 

/ a Dumber of poles, 

(R = magnetic reluctance of field circuit,* 

(R, = magnetic reluctance of armature circuit.t 

4i = effective magnetic flux produced by armature current 
(cross magnetization) per pole, 

r ~ resistance of field (effective resistance, including hys- 

ri = resistance of armature (effective resistance, including hys- 

N = frequency of alternations, 

a; = speed in cycles per second. 

It is then, 

E.M.F. induced in armature conductors by their rotation 
through the magnetic field (counter E.M.F. of motor). 

E =4«,ir,*10-' 
E.M.F. of self-induction of field, 

.E' =2ir/«A'*10-», 
E.M.F. of self-induction of armature, 

E.M.F. consumed by resistance, 


I — current passing through motor, in amperes effective. 

Further, it is : 
Field magnetism : « = n/10'/(R 

• ThU ii, Iht miis nmKDelic drcukl of tht motor. 

t Tlial It, the magDctk circoit of the aon pupiclizition, produced \fj Ibc >matan 



Armature magnetism : 

' '^\ 

Substituting these values, 

^, _ 2,pn'NI . 
, _ ■J.nfNI 

E, -(r+r,)Z 
Thus the impressed E.M.F., 

E, - V(£ + £r)' + (^' + E,y 

or, smce 

* = 2 IT N^—- = reactance of field ; 

V(^^+'+")''^ *'+"'■ 





'+'■.')' + (* + «,)■ 

DiBiiu.d, Google 


221. The power output at armature shaft is, 
P= EI 

tML^ + , + ..Y +(, + „)• 


- -!- —xE^ 

ir pn N 

/2 ft, N, , , V , , , M 

The displacement of phase between current and E.M.F, 
. £'+ ■£,' 


^■ + '^ 

2 «, ^ , 
■K pn N 

N^lecting, as approximation, the resistances r + r^, it is, 





- + 






ence a maximum for, 

2 A, 


' N 



, pn N 


2 ;,■ 

ibstituting this in tan 

», it 


tanu = 


or, S - t 

222. Instance of such an altemating-curren*- motor, 

£, = 100 iV'=60 / = 2. 

r = .03 r, — .12 

:<: = .9 x^= .^ 

n = 10 »! = 48 

Special provisions were made to keep the armature re- 
actance a minimum, and overcome the distortion of the 
field by the armature M.M.F., by means of a coil closely 
surrounding the armature and excited by a current of equal 
phase but opposite direction with the armature current 
(Eickemeyer). Thereby it was possible to operate a two- 
circuit, 96-tum armature in a bipolar field of 20 turns, at 
a ratio of 

= 2.4. 

field BTnpere-turns 

It is 

in this case. 


V(.023^, + .15)'+1.96 


230 a; 

(.023iV;+.16)' + 1.96 

n S - 

1.4 ... ... .-. 

3 A'l + .15 ' ' V(.023 N^ + .15)* + 1.96' 

Digitized .yGOOgle 


In Fig. 168 are given, with the speed A'l as abscissae, 
the values of current /, power /*, and power factor cos i 
of this motor. 

























































































^ f03. SflttMotar. 

223. The shunt motor with laminated field will not 
operate satisfactorily in an alternating-current circuit. It 
wOl start with good torque, since in starting the current in 
armature, as well . as in field, are greatly lagging, and thus 
approximately in phase with each other. With increasing 
speed, however, the armature current should come more 
into phase with the impressed E.M,F., to represent power. 
Since, however, the field current, and thus the field mag 
netism, lag nearly 90°, the induced E.M.F, of the armature 
rotation will lag nearly 90°, and thus not represent power. 

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Hence, to make a shunt motor work on alternating-cur- 
rent circuits, the magnetism of the field should be approxi- 
mately in phase with the impressed E,M.F., that is, the field 
reactance negligible. Since the self-induction of the field is 
far in excess to its resistance, this requires the insertion of 
negative reactance, or capacity, in the field. 

If the self-induction of the field circuit is balanced by 
capacity, the motor will operate, provided that the armature 
reactance is low, and that in starting sufficient resistance 
is inserted in the armature circuit to keep the armature 
current approximately in phase with the E.M.F, Under 
these conditions the equations of the motor will be similar 
to those of the series motor. 

However, such motors have not been Introduced, due to 
the difficulty of maintaining the balance between capacity 
and self-induction in the field circuit, which depends upon 
the square of the frequency, and thus is disturbed by the 
least change of frequency. 

The main objection to both series and shunt motors is 
the destructive sparking at the commutator due to the in- 
duction of secondary currents in those armature coils which 
pass under the brushes. As seen in Fig. 162, with the 
normal position of brushes midway between the field poles, 
the armature coil which passes under the brush incloses the 
total magnetic flux. Thus, in this moment no E.M.F. is 
induced in the armature coil due to its rotation, but the 
E.M.F. induced by the alternation of the magnetic flux 
has a maximum at this moment, and the coil, when short- 
circuited by the brush, acts as a short-circuited secondary 
to the field coils as primary ; that is, an excessive current 
flows through this armature coil, which either destroys it, 
or at leist causes vicious sparking when interrupted by the 
motion of the armature. 

To overcome this difficulty various arrangements have 
been proposed, but have not found an application. 

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224. Compared with the synchronous motor which has 
practically no lagging currents, and the induction motor 
which reaches very high power factors, the power factor of 
the series motor is low, as seen from Fig. 163, which repre- 
sents about the best possible design of such motors. 

In the alternating-series motor, as well as in the shunt 
motor, no position of an armature coil exists wherein the 
coil is dead ; but in every position E.M.F". is induced in the 
armature coil : in the position parallel with the held flux an 
E.M.F. in phase with the current, in the position at right 
angles with the field flux an E.M.F, in quadrature with the 
current, intermediate E.M.Fs. in intermediate positions. 
At the speed tr N jl the two induced E.M.Fs. in phase and 
in quadrature with the current are equal, and the armature 
coils are the seat of a complete system of symmetrical and 
balanced polyphase E.M.Fs. Thus, by means of stationary 
brushes, from such a commutator polyphase currents could 
be derived. 

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226. In the chapters on Alternating-Current Genera- 
tors and on Induction Motors, the assumption has been 
made that the reactance x of the machine is a constant. 
While this is more or less approximately the case in many 
alternators, in others, especially in machines of large arma- 
ture reaction, the reactance x is variable, and is different in 
the different positions of the armature coils in the magnetic 
circuit. This variation of the reactance causes phenomena 
which do not find their explanation by the theoretical cal- 
culations made under the assumption of constant reactance. 

It is known that synchronous motors of large and 
variable reactance keep in synchronism, and are able to 
do a considerable amount of work, and even carry under 
circumstances full load, if the field-exciting circuit is 
broken, and thereby the counter E.M.F. £"1 reduced to 
zero, and sometimes even if the field circuit is reversed 
and the counter E.M. F. Ey made negative. 

Inversely, under certain conditions of load, the current 
and the E.M.F. of a generator do not disappear if the gene- 
rator field is broken, or even reversed to a small negative 
value, in which latter case the current flows against the 
E.M.F, Eq of the generator. 

Furthermore, a shuttle armature without any winding 
will in an alternating magnetic field revolve when once 
brought up to synchronism, and do considerable work as 
a motor. 

These phenomena are not due to remanent magnetism 
nor to the magnetizing effect of Foucault currents, because 

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they exist also in machines with laminated fields, and exist 
if the alternator is brought up to synchronism by external 
means and the remanent magnetism of the field poles de- 
stroyed beforehand by application of an alternating current. 

226. These phenomena cannot l?e explained under the 
assumption of a constant synchronous reactance; because 
in this case, at no-field excitation, the E.M.F. or counter 
E.M.F. of the machine is zero, and the only E.M.F, exist- 
ing in the alternator is the E,M.F. of self-induction; that 
is, the E.M,F. induced by the alternating current upon 
itself. If, however, the synchronous reactance is constant, 
the counter E.M.F. of self-induction is in quadrature with 
the current and wattless; that is, can neither produce nor 
consume energy. 

In the synchronous motor running without field excita- 
tion, always a large lag of the current behind the impressed 
E.M.F. exists ; and an alternating generator will yield an 
E.M.F, without field excitation, only when closed by an 
external circuit of large negative reactance; that is, a circuit 
in which the current leads the E.M.F., as a condenser, or 
an over-excited synchronous motor, etc. 

Self-excitation of the alternator by armature reaction 
cnn be explained by the fact that the counter E.M.F. of 
.self-induction is not wattless or in quadrature with the cur- 
rent, but contains an energy component ; that is, that the 
reactance is of the form X = h —jx, where x is the wattless 
component of reactance and h the energy component of 
reactance, and h is positive if the reactance consumes 
power, — in which case the counter E.M.F. of self-induc- 
tion lags more than 90° behind the current, — while h is 
negative if the reactance produces power, — in which case 
the counter E.M.F. of self-induction lags less than 90° 
behind the current. 

227. A case of this nature has been discussed already 
in the chapter on Hysteresis, from a different point of view. 

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There the effect of magnetic hysteresis was found to distort 
the current wave in such a way that the equivalent sine 
wave, that is, the sine wave of equal effective strength and 
equal power with the distorted wave, is in advance of the 
wave of magnetism by what is called the angle of hysteretic 
advance of phase a. Since the E.M.F. induced by the 
magnetism, or counter E.M.F. of self-induction, lags 90° 
behind the magnetism, it lags 90 + a behind the current ; 
that is, the self-induction in a circuit containing iron is not 
in quadrature with the current and thereby wattless, but 
lags more than 90° and thereby consumes power, so that 
the reactance has to be represented by X= A —jx, where 
k is what has been called the " effective hysteretic resis- 

A similar phenomenon takes place in alternators of vari- 
able reactance, or what is the same, variable magnetic 

228. Obviously, if the reactance or reluctance is vari- 
able, it will perform a complete cycle during the time the 
armature coil moves from one field pole to the next field 
pole, that is, during one-half \vave of the main current. 
That is, in other word.s, the reluctance and reactance vary 
with twice the frequency of the alternating main current. 
Such a case is shown in Figs. 164 and 16$, The impressed 
E.M.F., and thus at negligible resistance, the counter E.M.F., 
i.? represented by the sine wave E, thus the magnetism pro- 
duced thereby is a sine wave *, 90" ahead of E. The 
reactance is represented by the sine wave x, varying with 
the double frequency of E, and shown in Fig. 164 to reach 
the maximum value during the rise of magnetism, in Fig. 
165 during the decrease of magnetism. The current /re- 
quired to produce the magnetism * is found from ♦ and x 
in combination with the cycle of molecular magnetic friction 
of the material, and the power P is the product IE As 
seen in Fig. 164, the positive part of P is larger than the 

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» r-' 

hB ^'2^" E 

^^ 3 V y^ 5 ' ^^ 

2, f - t^ t^^ ^ 

=-^^^^ i,i;2'^v^7 r,^"^-. 

v?».-i;^ 5-. 5^ ^3 

L Y 2 ' ^^V ^' 

% 2v t / % A 

vLW ^ y^Hrf Vm 1 

'"n St 2 °"^ - 

\ \- y- / \\- 

v^ ^ •- ' ^^ 

^^S"^ ^-s 



F. I«4. Varlabia Reacttua, Seactloa McftJn*. 




J -'^"■* E 

^■^ 4y \ i,-' ^■^ - 

^ 1' ^.5 ^^ 

S ^-^v, * .--g &^ s,'-- 

"^^ iZ^'^ Zt ^"^JsE^^^ ^?, 

t Y "^i - f* ^L NZ 

\ J\. A I \ 1\ 

V ^ 77^ r ^\, nl 

% ^ % ' V I I -, - 

\\i It \A \\i , 

L'^/^.-^ f t^^ 

^ i- -. ^^■' 

= C7 3 , 

' :t ^^ - 

^r^ I - 

Fig. 195. Yarlabit Hmctana, Htaetlen MacHln. 

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negative part ; that is, the machine produces electrical energy 
as generator. In Fig. 165 the negative part of P is larger 
than the positive ; that is, the machine consumes electrical 
energy and produces mechanical energy as synchronous 
motor. In Figs. 166 and 167 are given the two hysteretic 
cycles or looped curves *, / under the two conditions. They 
show that, due to the variation of reactance x, in the first 
case the hysteretic cycle has been overturned so as to 
represent not consumption, but production of electrical 










1 . 










flQ- '00. Hytttrttic Loop of Siaetlon Mmliliit. 

energy, while in the second case the hysteretic cycle has 
been widened, representing not only the electrical energy 
consumed by molecular magnetic friction, but also the me- 
chanical output. 

229. It is evident that the variation of reluctance must 
be symmetrical with regard to the field poles ; that is, that 
the two extreme values of reluctance, maximum and mini- 
mum, will take place at the moment where the armature 

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coil stands in front of the field pole, and at the moment 
where it stands midway between the field poles. 

The effect of this periodic variation of reluctance is a 
distortion of the wave of E.M.F., or of the wave of current, 
or of both. Here again, as before, the distorted wave can 
be replaced by the equivalent sine wave, or sine wave of 
equal effective intensity and equal power. 

The instantaneous value of magnetism produced by the 


-— ^ 


" "5 




/ / 







^z J 





1 ■■ 

---^ . 

flu- 197. Hf$teretlc Uop of ItvKllmt MacUn: 

armature current — which magnetism induces in the arma- 
ture conductor the E.M.F. of self-induction — is propor- 
tional to the instantaneous value of the current, divided 
by the instantaneous value of the reluctance. _ Since the 
extreme values of the reluctance coincide with the sym- 
metrical positions of the armature with regard to the field 
poles, — that is, with zero and maximum value of the in- 
duced E.M.F., £fj, of the machine, — it follows that, if the 
current is in phase or in quadrature with the E.M.F. E^,, 
the reluctance wave is symmetrical to the current wave, 
and the wave of magnetism therefore symmetrical to the 

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current wave also. Hence the equivalent sine wave of 
magnetism is of equal phase with the current wave ; that 
is, the E.M.F. of self-induction lags 90° behind the cur- 
rent, or is wattless. 

Thus at no-phase displacement, and at 90° phase dis- 
placement, a reaction machine can neither produce electri- 
cal power nor mechanical power. 

230. If, however, the current wave differs in phase 
from the wave of E.M.F. by less than 90°, but more than 
zero degrees, it is unsymmetrical with regard to the 
reluctance wave, and the reluctance wOl be higher for ris- 
ing current than for decreasing current, or it will be 
higher for decreasing than for rising current, according 
to the phase relation of current with regard to induced 
E.M.F., £■(. 

In the first case, if the reluctance is higher for rising, 
lower for decreasing, current, the magnetism, which is pro- 
portional to current divided by reluctance, is higher for 
decreasing than for rising current ; that is, its equivalent 
sine wave lags behind the sine wave of current, and the 
E.M.F, or self-induction will lag more than 90" behind the 
current ; that is, it will consume electrical power, and 
thereby deliver mechanical power, and do work as syn- 
chronous motor. 

In the second case, if the reluctance is lower for rising, 
and higher for decreasing, current, the magnetism is higher 
for rising than for decreasing current, or the equivalent sine 
wave of magnetism leads the sine wave of the current, and 
the counter E.M.F. at self-induction lags less than 90° be- 
hind the current ; that is, yields electric power as generator, 
and thereby consumes mechanical power. 

In the first case the reactance will be represented by 
X = h — jx, similar as in the case of hysteresis ; while in 
the second case the reactance will be represented by 
X= -h-jx. 

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23L The influence of the periodical variation of reac- 
tance will obviously depend upon the nature of the variation, 
that is, upon the shape of the reactance curve. Since, 
however, no matter what shape the wave has, it can always 
be dissolved in a series of sine waves of double frequency, 
and its higher harmonics, in first approximation the assump- 
tion can be made that the reactance or the reluctance vary 
with double frequency of the main current ; that is, are 
represented in the form; 

X — >i-|-3cos2/3. 
Let the inductance, or the coefficient of self-induction, 
be represented by — 

Z = /+ ^cos2/S 
= /(I + y cos 2 ^) 

where y = amplitude of variation of inductance. 

£ = angle of lag of zero value of current behind maximum value 
of inductance L. 

It is then, assuming the current as sine wave, or repla- 
cing it by the equivalent sine wave of effective intensity /, 

Current, _ 

i = /V5sin(^-i). 

The magnetism produced by this current is, 

where « = number of turns. 
Hence, substituted, 

* = <£:^sin 03 - £) (1 + ycos2^), 

or, expanded, 

when neglecting the term of triple frequency, as wattless. 

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Thus the E.M.F. induced by this magnetism is, 

hence, expanded — 

e 2'wNlI'^Ul -^^cosaco3;8 + /'l + |^sinwsi^^j 

and the effective value of E.M..F., 


E-2,NuJ(l~r\ cos'O + fl+J^ 


= 2-wNl iJl + ^ - y cos 2 a. 
Hence, the apparent power, or the voltamperes — 
J'^ = IE = 2wNlI*\Jl + ^ — ycos21a 
^ E^ 


The instantaneous value of power is 

= ~4^7V//'sin03-w)JCl-|Vos»co3/9 + 

/l + XNsinwsin^j. 

and, expanded — 

p = - 2ir Nl n Ul +^%m2 m sin» ^ - /l - 1\ 

sin 2 w cos' ^ + sin 2 jS/^ cos 2a» -^^l 

Integrated, the effective value of power is 
/"= — »^//»ysin2w; 

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hence, negative, that is, the machine consumes electrical, 
and produces mechanical, power, as synchronous motor, if 
a > ; that is, with lagging current; positive, that is, the 
machine produces electrical, and consumes mechanical, 
power, as generator, if S > ; that is, with leading current. 
The power factor is 

f _P^^ y sin 2 a 

■^^ 2^i + x'_y^,os2a; 
hence, a maximum, if, , , 

or, expanded, ^ - 

The power, P, is a maximum at given current, /, if 

sin2a = l; 
that is, 

£ = 45° 

at given E.M.F., E, the power is 

p_ A'ysm 


4tJV//1 + J-')-cos2il'\ 
hence, a maximum at . n 


or, expanded, , 

cos25= ^y . 

232, We have thus, at impressed E.M.F., E, and negli- 
gible resistance, if we denote the mean value of reactance, 

Current „ 


5y- — y COS 2 5. 

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„ £^ y sin 2 w 

2x{\-Y ^ — rcos2uj\ 
Power factor, 

- , — r\ 7 sin 3 S 

2^1 + i_ycos2a 
Maximum power at 

cos 2 w = — 3! — . 

Maximum power factor at 

ui > ; synchronous motor, with bgging current, 
» < : generator, with leading current. 

As an instance is shown in Fig, 168, with angle « as 
abscissas, the values of current, power, and power factor, 
for the constants, — 

E= 110 


1.45 — cos 2 Si 

,4-17 sin 2 3 

As seen from Fig. 152, the power factor / of such a 
machine is very low — does not exceed 40 per cent in this 

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' t 












/ 1 










s \ 























— L 




















TOBswe^tss aw watx^hapb ahd ns causxb. 

233. In the preceding chapters we have considered 
the alternating currents and alternating E.M.Fs. as sine 
waves or as replaced by their equivalent sine waves. 

While this is sufficiently exact in most cases, under 
certain circumstances the deviation of the wave from sine 
shape becomes of importance, and with certain distortions 
it may not be possible to replace the distorted wave by an 
equivalent sine wave, since the angle of phase displacement 
of the equivalent sine wave becomes indefinite. Thus it 
becomes desirable to investigate the distortion of the wave, 
its causes and its effects. 

Since, as stated before, any alternating wave can be 
represented by a series of sine functions of odd orders, the 
investigation of distortion of wave-shape resolves itself in 
the investigation of the higher harmonics of the alternating 

In general we have to distinguish between higher har- 
monics of E.M.F. and higher harmonics of current. Both 
depend upon each other in so far as with a sine wave of 
impressed E.M.F. a distorting effect will cause distortion 
of the current wave, while with a sine wave of current 
passing through the circuit, a distorting effect will cause 
higher harmonics of E.M.F. 

234. In a conductor revolving with uniform velocity 
through a uniform and constant magnetic field, a sine wave 
of E.M.F. is induced. In a circuit with constant resistance 
and constant reactance, this sine wave of EM.F. produces 

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a sine wave of current. Thus distortion of the wave-shape 
or higher harmonics may be due to : lack- of uniformity of 
the velocity of the revolving conductor ; lack of uniformity 
or pulsation of the magnetic field ; pulsation of the resis- 
tance ; or pulsation of the reactance. 

The first two cases, lack of uniformity of the rotation or 
of the magnetic field, cause higher harmonics of E.M.F. at 
open circuit. The last, pulsation of resistance and reac- 
tance, causes higher harmonics only with a current flowing 
in the circuit, that is, under load. 

Lack of uniformity of the rotation is of no practical in- 
terest as cause of distortion, since in alternators, due to 
mechanical momentum, the speed is always very nearly 
uniform during the period. 

Thus as causes of higher harmonics remain : 

1st. Lack of uniformity and pulsation of the magnetic 
field, causing a distortion of the induced E,M,F. at open 
circuit as well as under load. 

2d. Pulsation of the reactance, causing higher harmonics 
under load. 

3d. Pulsation of the resistance, causing higher harmonics 
under load also. 

Taking up the different causes of h^her harmonics we 
have ; — 

Lack of Unifortnity and Pulsation of the Magnetic Field. 

235. Since most of the alternating-current generators 
contain definite and sharply defined field poles covering in 
different types different proportions of the pitch, in general 
the magnetic flux interlinked with the armature coil will 
not vary as simply sine wave, of the form: 

but as a complex harmonic function, depending on the shape 
and the pitch of the field poles, and the arrangement of the 
armature conductors. In this case, the magnetic flux issu' 



ing from the field pole of the alternator can be represented 
by the general equation, 

♦ = ^,'+^iCOs^ + ^,cos2/J + ^,cos30 + . . . 
+ J, sin;3 + ^,sm2^ + ^,sm3^ + . . . 

If the reluctance of the armature is uniform in all directions, 
so that the distribution of the magnetic flux at the field-pole 
face does not change by the rotation of the armature, the 
rate of cutting magnetic flux by an armature conductor is *, 
and the E.M.F. induced in the conductor thus equal thereto 
in wave shape. As a rule A„ Aj,A^ . . . B^ B, equal zero ; 
that is, successive field poles are equal in strength and dis- 
tribution of magnetism, but of op[>osite polarity. In some 
types of machines, however, especially induction alternators, 
this is not the case. 

The E.M.F, induced in a full-pitch armature turn — that 
is, armature conductor and return conductor distant from 
former by the pitch of the armature pole (corresponding to 
the distance from field pole center to pole center) is, 
6 = *, - *^ 
= 2 S^, C0S/9+ .^,cos30 + j4,cos50-f-. . . 
-H Bi sin + ^, sin 3^ + 5, sin 6^ + . . . J 

Even with an unsymmetrical distribution of the magnetic 
flux in the air-gap, the E.M.F. wave induced in a fuH-pitch 
armature coil is symmetrical ; the positive and negative half 
waves equal, and correspond to the mean flux distribution 
of adjacent poles. With fractional pitch windings — that 
is, windings whose turns cover less than the armature pole 
pitch — the induced E.M.F. can be unsymmetrical with 
unsymmetrical magnetic field, but as a rule is symmetrical 
also. In unitooth alternators the total induced E.M.F. has 
the same shape as that induced in a single turn. 

With the conductors more or less distributed over the 
surface of the armature, the total induced E.M.F. is the 
resultant of several E.M.Fs. of different phases, and is thus 
more uniformly varying ; that is, more sinusoidal, approaching 

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sine sh^>e, to within 8 % or less, as for instance the curves 
Fig. 169 and Fig, 170 show, which represent the no-load 
and full-load wave of E.M.F. of a three-phase multitcoth 
alternator. The principal term of these harmonics is the 
third harmonic, which consequently appears more or less in 
all alternator waves. As a rule these harmonics can be 
considered together with the harmonics due to the var^'ing 
reluctance of the magnetic circuit. In ironclad alternators 
with few slots and teeth per pole, the passage of slots across 
the field poles causes a pulsation of the magnetic reluc- 
tance, or its reciprocal, the magnetic inductance of the 
circuit. In consequence thereof the magnetism per field 
pole, or at least that part of the magnetism passing through 
the armature, will pulsate with a frequency 2 y if y = num- 
ber of slots per pole. 

Thus, in a machine with one slot per pole, Ihe instanta- 
neous magnetic flux interlinked with the armature con- 
ductors can be expressed by the equation : 

* = *cosj3 )1 -J-tcos[2j8-u.]j 
where, • = average magnetic flux, 

c = amplitude of pulsation, 
and <o = phase of pulsation. 

In a machine with y slots per pole, the instantaneous flux 
interlinked with the armature conductors will be : 

^ = *cos^ |H-«cos [Sy^-ujj, 
if the assumption is made that the pulsation of the magnetic 
flux follows a simple sine law, as first approximation. 

In general the instantaneous magnetic flux interlinked 
with the armature conductors will be : 

^ = * cos ^ } 1 -H <t cos (2 - £,) + «» cos (4 ^ - £,) -H . . . j, 
where the term k, is predominating if y = number of arma- 
ture slots per pole. This general equation includes also the 
effect of lack of uniformity of the magnetic flux. 

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m ^ ^i ^'~'~ ^ 

„ ., 14 .S »- . . ^ . 

/ \ 1 

-.^ \ 


t x 

_f \ 

I ■ V 

^ \ 

1 V 

r ^ 

3 \ 

.. ^ t 

L lE-^ -- --. --- 

„ »;^ 

,„ -. =1 '-0 f,'- 3.J x;-- -">^ 

^ ^ 

/ s 

^ V 

Z 3 

» / \ 

/ ^ 

J % 

-/ V 

7 ^ 

t \ 

j \ 

fit. in. Fall-loaa Wottt ef C.M.f. of mililiooa THif-plmir. 

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In case of a pulsation of the magnetic flux with the 
frequency 2y, due to an existence of y slots per pole in the 
armature, the instantaneous value of magnetism interlinked 
with the armature coil is : 

.^ = *cos^{l + .cos[2y^-a]}. 
Hence the E.M.F. induced thereby : 


'~ " 4t ■ 

= - V5 IT A^«*-^ {cos ,3(1 + < cos [2yj8 -«])}. 
And, expanded : 

* = V2 wNn * {sin fi + t IsLIlL sin [ (^2 y - 1) ^ - 5] 

Hence, the pulsation of the magnetic flux with the 
frequency 2 y, as due to the existence of y slots per pole, 
introduces two harmonics, of the orders (2y — 1) and 


236. If r = 1 it is : 

^=V2T^«»{sin^ + isin(^-a) + |5sin(3^-i)}; 

that is : In a unitooth singlc-phaser a pronounced triple 
harmonic may be expected, but no pronounced higher 

Fig, 171 shows the wave of E.M.F. of the main coil of 
a monocyclic alternator at no load, represented by ; 

^ = A{sin/3- .242 sin ( 3 ^ - CS) - .046 sir (5 jS - 2.6) 
+ .068 sin (7 ^ - 3.3) - .027 sin (9 ^ - 10.0) - .018 sin 
(11 ^ - 6.6) + .029 sin (13 /J - 8.2)}; 

hence giving a pronounced triple harmonic only, as expected. 
If y = 2, it is : 

<r = V5 IT .A^w * I sin /3 + ^^' sin |3 ;3 - 5) + "'Ji sin (6 - £) | 

Digitized .yGOO^Ie 


the no-load wave of a unitooth quarter-phase machine, hav< 
ing pronounced triple and quintuple harmonics. 
If y = 8, it is : 

<f= V§irJV»* j sin/? +|5 sin (5^—5)4- y sin (7^-5) j . 

That is : In a unitooth three-phaser, a pronounced quin- 
tuple and septuple harmonic may be expected, but no pro- 
nounced triple harmonia 

m ,g 

Mi. V 

- I \ 


« " Z S 

. " zl :'^^ : 

? Jl ^^ 

n ^'^ ^fa^tf "^ > 

,.^t -^iSaafi _/i-\., .,^. 

T,, :rTj1Te-:^Es»^i?Si'SJ&>«- 

Flq. 155. Ho-loait Want ef t.M.F. of UnlteoU MmceiKillc Altttnatar. 

Fig. 156 shows the wave of E.M.F. of a unitooth three- 
phaser at no load, represented by ; 

e^E {sin^- .12sin (3^9 — 2.3) - .23 sin (6^9- 1.5) -|-. 134 sin 

(7 - 6.2) - .002 sin (9 j8 + 27.7) - .046 sin (11 ^ — 
B.5) +.031 sin (13)3-61.5)}. 

Thus giving a pronounced quintuple and septuple and 
a lesser triple harmonic, probably due to the deviation of 
the field from uniformity, as explained above, and deviation 
of the pulsation of reluctance from sine shape. In some 
especially favorable cases, harmonics as high as the 23d and 
25th have been observed, caused by pulsation of the reluc- 

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Ho- m- HtyJBia Wmt of f.H.F. of l/nlUoVi Fkiw^taM Alttmator. 

In general, if the pulsation of the magnetic inductance 
is denoted by the general expression: 

l + ;g:scos(2y^-»,), 
the instantaneous magnetic flux is ; 
♦ - tcos/S j 1 + s;-,™ (2y/S - S,) j • 

-t jeos;3 + -5-cos(^-S,)+Er^"s((2»+l) 

^-s,) + ^=o!((2r+l)^-"...)]j 
hence, the E.M.F. 
«- V2»yV/.»jsin^ + |sinO-i,)+^^^^ 

[ssin((2y+l)^-:,,) + .„,sin((2y+l)^-a„,)] j 



Pulsation of Reactance. 

237. The main causes of a pulsation of reactance are : 
magnetic saturation and hysteresis, and synchronous motion. 
Since in an ironclad magnetic circuit the magnetism is not 
proportional to the M.M.F., the wave of magnetism and 
thus the wave of E.M.F. will differ from the wave of cur- 
rent. As far as this distortion is due to the variation of 
permeability, the distortion is symmetrical and the wave 
of induced E.M.F. represents no power. The distortion 
caused by hysteresis, or the lag of the magnetism behind 
the M.M.F., causes an unsymmetrical distortion of the wave 
which makes the wave of induced E.M.F. differ by more 
than 90° from the current wave and thereby represents 
power, — the power consumed by hysteresis. 

In practice both effects are always superimposed ; that 
is, in a ferric inductance, a distortion of wave-shape takes 
place due to the lack of proportionality between magnetism 
and M.M.F. as expressed by the variation in the hysteretic 

This pulsation of reactance gives rise to a distortion 
consisting mainly of a triple harmonic. Such current waves 
distorted by hysteresis, with a sine wave of impressed 
E.M.F., are shown in Figs. 66 to 69, Chapter X„ on Hy- 
steresis. Inversely, if the current is a sine wave, the mag- 
netism and the E.M.F, will differ from sine shape. 

For further discussion of this distortion of wave-shape 
by hysteresis, Chapter X. may be consulted. 

238. Distortion of wave-shape takes place also by the 
pulsation of reactance due to synchronous rotation, as dis- 
cussed in chapter on Reaction Machines. 

In Figs. 148 and 149, at a sine wave of impressed 
E.M.F., the distorted current waves have been constructed. 
Inversely, if a sine wave of current, 

(" = /cos ft 

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passes through a circuit of synchronously varying reac- 
tance ; as for instance, the armature of a unitooth alterna- 
tor or synchronous motor — or, more general, an alternator 
whose armature reluctance is different in different positions 
with regard to the field poles — and the reactance is ex- 
pressed by 

A'-*{l + <cos(2;3-5)}; 

or, more general, 

the wave of magnetism is 
^ 2-,Nn '^ 2,rJVn\ 

hence the wave of induced E.M,F. 

= *{sin^ + |si„(,S 




+ 1) 

#-»,) + S + ls 



that is, the pulsation 

of reactance ol 




duces two higher harmonics of the 

order (2y 



(i!r + l\ 

If Jf-»{l + ,co. 


' 2,Nn\ 

^ + |cos(^- 

l) + icos(3;8- 


<=,{sin^+isin{^-S) + Afsi„(3;3- 


Since the pulsation of reactance due to magnetic satu- 
ration and hysteresis is essentially of the frequency, 2A^ 

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— that is, describes a complete cycle for each half -wave of 
current, — this shows why the distortion of wave-shape by 
hysteresis consists essentially of a triple harmonic. 

The phase displacement between e and i, and thus the 
power consumed or produced in the electric circuit, depend ■ 
upon the angle, w, as discussed before. 

239. In case of a distortion of the wave-shape by 
reactance, the distorted waves can be replaced by their 
equivalent sine waves, and the investigation with suffi- 
cient exactness for most cases be carried out under the 
assumption of sine waves, as done in the preceding chapters. 
Similar phenomena take place in circuits containing 
polarization cells, leaky condensers, or other apparatus 
representing a synchronously varying negative reactance. 
Possibly dielectric hysteresis in condensers causes a dis- 
tortion similar to that due to magnetic hysteresis. 

Pulsation of Resistance. 

240. To a certain extent the investigation of the effect 
of synchronous pulsation of the resistance coincides with 
that of reactance ; since a pulsation of reactance, when 
unsymmetrical with regard to the current wave, introduces 
an energy component which can be represented by an 
"effective resistance." 

Inversely, an unsymmetrical pulsation of the ohmic 
resistance introduces a wattless component, to be denoted 
by "effective reactance," 

A typical case of a synchronously pulsating resistance is 
represented in the alternating arc. 

The apparent resistance of an arc depends upon the 
current passing through the arc ; that is, the apparent 
resistance of the arc = p°»"'1'I diff.n,«c^h.»,«., .kcoj.. jg j^j^j^ 

for small currents, low for large currents. Thus in an 
alternating arc the apparent resistance will vary during 

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every half-wave of current between a maximum value at 
zero current and a minimum value at maximum current, 
thereby describing a complete cycle per half-wave of cur- 

Let the effective value of current passing through the 
arc be represented by /. 

Then the instantaneous value of current, assiuning the 
current wave as sine wave, is represented by 

I = /Vasin/3; 

and the apparent resistance of the arc, in first approxima- 
tion, by 

JP = r(l + «cos2j8); 

thus the potential difference at the arc is 

e = iJi~ 7 V2 r sin j8 (1 + « cos2 j8) 

Hence the effective value of potential difference. 


and the apparent resistance of the a 

The instantaneous power consumed in the arc is, 
/ = ,> = 2r/'j^l-|yinV + |8ini8sin3pj 

Hence the effective power, 




The apparent power, or volt amperes consumed by the 
arc, is, ^__^^_ 

thus the power factor of the arc, 




that is, less than unity. 

341. We find here a case of a circuit in which the 
power factor — that is, the ratio of watts to volt amperes 
— differs from unity without any displacement of phase ; 
that is, while current and E.M.F, are in phase with each 
other, but are distorted, the alternating wave cannot be 
replaced by an equivalent sine wave ; since the assumption 
of equivalent sine wave would introduce a phase displace- 

cos m =f 

o£ an angle, S, whose sign is indefinite. 

As an instance are shown, in Fig, 173 for the constants, 

/= 12 
r= 3 
( = .9 
the resistance, 

.ff = 3{l + .9cos2/3); 

the current, 

(■ =17sini9; 

the potential difference, 

<f = 28 (sin /5 + .82 sin 3 yS). 

In this case the effective E.M.F. is 

B = 25.6 ; 

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the apparent resistance, 

FI9. Va, PtrMlmUg fnryfirg »ml 

As seen, with a sine wave of current the E.M.F, wave 
in an alternating arc will become double-peaked, and rise 
very abruptly near the zero values of current. Inversely, 
with a sine wave of E.M.F. the current wave in an alter- 
nating arc will become peaked, and very flat near the zero 
values of E.M.F. 

242. In reality the distortion is of more complex nature ; 
since the pulsation of resistance in the arc does not follow 

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a simple sine law of double frequency, but varies much 
more abruptly near the zero value of current, making 
thereby the variation of E.M,F. near the zero value of 
current much more abruptly, or, inversely, the variation 
of current more flat. 

A typical wave of potential difference, with a sine wave 
of current passing through the arc, is given in Fig. 174.* 

flf. 114. Elietrit Are. 

243. The value of c, the amplitude of the resistance 
pulsation, largely depends upon the nature of the electrodes 
and the steadiness of the arc, and with soft carbons and a 
steady arc is small, and the power factor / of the arc near 
unity. With hard carbons and an unsteady arc, t rises 
greatly, higher harmonics appear in the pulsation of resis- 
tance, and the power factor / falls, being in extreme cases 
even as low as .6. 

The conclusion to be drawn herefrom is, that photo- 
metric tests of alternating arcs are of little value, if, besides 
current and voltage, the power is not determined also by 
means of electro-dynamometers. 

• From American Institute of Electrical Engineers, Transactions, 1800, p. 
8T9. Tohey and Walbrldge, on the Stanley Alteinate Arc Dynamo. 





244. To elucidate the variation in the shape of alternat- 
ing waves caused by various harmonics, in Figs. 175 and 

176 are shown the wave-forms produced by the superposi- 
tion of the triple and the quintuple harmonic upon the 
fundamental sine wave. 

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In Fig. 175 is shown the fundamental sine wave and 
the complex waves produced by the superposition of a triple 
harmonic of SO per cent the amplitude of the fundamental, 
under the relative phase displacements of 0°, 45°, 90°, 135°, 
and 180°, represented by the equations : 


n ^ - .3 sin 3 y9 
n 0— .3 sin {3 ^ - 90°) 
,n ja-.3sin (3^3-135°) 
-.3sin(3jS- 180°). 

As seen, the effect of the triple harmonic is in the first 
figure to flatten the zero values and point the maximum 
values of the wave, giving what is called a peaked wave. 
With increasing phase displacement of the triple harmonic, 
the flat zero rises and gradually changes to a second peak, 
giving ultimately a flat-lop or even double-peaked wave with 
sharp zero. The intermediate positions represent what is 
called a saw-tooth wave. 

In Fig. 176 are shown the fundamental sine wave and 
the complex waves produced by superposition of a quintuple 
harmonic of 20 per cent the amplitude of the fundamental, 
under the relative phase displacement of 0°, 45°, 90°, 135", 
180°, represented by the equations : 


n (5^-90°) 
in {5 0-135°) 
in {5 -180°). 

The quintuple harmonic causes a flat-topped or even 
double-peaked wave with flat zero. With increasing phase 
displacement, the wave becomes of the type called saw- 
tooth wave also. The flat zero rises and becomes a third 
peak, while of the two former peaks, one rises, the other 

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decreases, and the wave gradually changes to a triple- 
peaked wave with one main peak, and a sharp zero. 

As seen, with the triple hannonic, flat-top or double- 
peak coincides with sharp zero, while the quintuple har- 
monic flat-top or double-peak coincides with flat zero. 

fit- 179. Effttt of Qamtupli Harmonk. 

Sharp peak coincides with flat zero in the triple, with 
sharp zero in the quintuple harmonic With the triple har- 
monic, the saw-tooth shape appearing in case of a phase 
difference between fundamental and hannonic is single, 
while with the quintuple harmonic it is double. 

Thus in general, from simple inspection of the wave 
shape, the existence of these first harmonics can be discov- 
ered. Some characteristic shapes are shown in Fig. 177. 

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Fig. 177, Semt OluatustttliUe 

Flat top with flat zero : 

sin ^ - .15 sin 3 yS - .10 sin 6 jS. 
Flat top with sharp zero : 

sin fi - .225 sin (3 ^8 - 180") - .05 sin (5 ^ - 180"). 
Double peak, with sharp zero : 

sin P - .16 sin (3 )3 - 180") - .10 sin 5 0, 
Sharp peak with sharp zero : 

sin ^— .16 sin 3/3 - .10 sin {5 fi - 180°). 

246. Since the distortion of the wave-shape consists in 
the superposition of higher harmonics, that is, waves of 
higher frequency, the phenomena taking place in a circuit 

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supplied by such a wave will be the combined effect of the 
different waves. 

Thus in a non-inductive circuit, the current and the 
potential difiference across the different parts of the circuit 
are of the same shape as the impressed E.M.F. If self- 
induction is inserted in series to a non-inductive circuit, the 
self-induction consumes more E.M.F. of the higher harmon- 
ics, since the reactance is proportional to the frequency, 
and thus the current and the E.M.F. in the non-inductive 
part of the circuit shows the higher harmonics in a reduced 
amplitude. That is, self-induction in series to a non-induc- 
tive circuit reduces the higher harmonics or smooths out 
the wave to a closer resemblance with sine shape. In- 
versely, capacity in series to a non-inductive circuit con- 
sumes less E.M.F. at higher than at lower frequency, and 
thus makes the higher harmonics of current and of poten- 
tial difference in the non-inducttve part of the circuit mor*" 
pronounced — intensifies the harmonics. 

Self-induction and capacity in series may cause an in- 
crease of voltage due to complete or partial resonance with 
higher harmonics, and a discrepancy between volt-amperes 
and watts, without corresponding phase displacement, as 
will be shown hereafter. 

246. In long-distance transmission over lines of notice- 
able inductance and capacity, rise of voltage due to reso- 
nance may occur with higher harmonics, as waves of higher 
frequency, while the fundamental wave is usually of too low 
a frequency to cause resonance. 

An approximate estimate of the possible rise by reso- 
nance with various harmonics can be obtained by the inves- 
tigation of a numerical instance. Let in a long-distance 
line, fed by step-up transformers at 60 cycles. 

The resistance drop in the transformers at full load = 1^. 
The inductance voltage in the transfonners at full load = 6% 

with the fundamental wave. 
The resistance drop in the line at full load = 10%. 

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The inductance voltage in the line at full load = 20*^ with the 
fundamental wave. 

The capacity or charging current of the line = 20"^ of the full- 
load current / at the frequency of the fundamental. 

The line capacity may approximately be represented by 
a condenser shunted across the middle of the line. The 
E.M.F. at the generator terminals E is assumed as main- 
tained constant. 

The E.M.F. consumed by the resistance of the circuit 
from generator terminals to condenser is 

Ir = .06 £, 

or, r=.06^. 

The reactance E.M.F. between generator terminals and 
condenser is, for the fundamental frequency, 

thus the reactance corresponding to the frequency {2i — 1) 
iVof the higher harmonic is : 

The capacity current at fundamental frequency is : 

/ = .2 i; 

hence, at the frequency: (2i — 1)A'': 



/ = E.M.F- of the (2 i — 1)"* harmonic at the condenser, 

e = E.M.F. of the {2 i — 1)* harmonic at the generator terminals. 

The E.M.F. at the condenser is : — 

y _ V^* - <V' + ■* (2 i - 1) , 



hence, substituted : 

VI - .059856 (2 i - 1)* + .0009 (2 * - 1)* 

the rise of voltage by inductance and capacity. 
Substituting : 

it is, a = l.OS 1.36 3.76 2.18 .70 .38 

That is, the fundamental will be increased at open circuit 
by 3 per cent, the triple harmonic by 36 per cent, the 
quintuple harmonic by 276 per cent, the septuple harmonic 
by 118 per cent, while the still higher harmonics are 

The maximum possible rise will take place for : 

- = 0, or, 2A- 1 = 6.77 

That is, at a frequency : N= 346, and a = 14.4. 

That is, complete resonance will appear at a frequency 
between quintuple and septuple harmonic, and would raise 
the voltage at this particular frequency 14.4 fold. 

If the voltage shall not exceed the impressed voltage by 
more than 100 per cent, even at coincidence of the maximum 
of the harmonic with the maximum of the fundamental, 

the triple harmonic must be less than 70 per cent of the 

the quintuple harmonic must be less than 26.5 per cent of the 

the septuple harmonic must be less than 46 per cent of the 


The voltage will not exceed twice the normal, even at 
a frequency of complete resonance with the higher har- 
monic, if none of the higher harmonics amounts to more 

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ihan 7 per cent, of the fundamental. Herefrom it follows 
that the danger of resonance in high potential lines is in 
general greatly over-estimated, since the conditions assumed 
in this instance are rather more severe than found in prac- 
tice, the capacity current of the line very seldom reaching 
20^^ of the main current. 

347. The power developed by a complex harmonic wave 
in a non-inductive circuit is the sum of the powers of the 
individual harmonics. Thus if upon a sine wave of alter- 
nating E.M.F. higher harmonic waves are superposed, the 
effective E.M.F., and the power produced by this wave in a 
given circuit or with a given effective current, are increased, 
tn consequence hereof alternators and synchronous motors 
of ironclad unitooth construction — that is, machines giving 
waves with pronounced higher harmonics — give with the 
same number of turns on the armature, and the same mag- 
netic flux per field pole at the same frequency, a higher 
output than machines built to produce sine waves. 

248. This explains an apparent paradox : 
If in the three-phase star-connected generator with the 
magnetic field constructed as shown diagrammatically in 
Fig. 162, the magnetic flux per pole =s *, the number of 
turns in series per circuit = «, the frequency = N, the 
E.M.F. between any two collector rings is : 

since 2« armature turns simultaneously interlink with the 
magnetic flux 4. 

The E.M,F. per armature circuit is : 
e = V2iriV»*10-'; 

hence the E.M.F. between collector rings, as resultant of 
two E.M.Fs. e displaced by 60° from each other, is ; 

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while the same E.M.F. was found by direct calculation 
from number of turns, magnetic flux, and frequency to be 
equal to 2 ^ ; that is the two values found for the same 
E.M.F. have the proportion V3 : 2 = 1 : 1.154. 

This discrepancy is due to the existence of more pro- 
nounced higher harmonics in the wave e than in the wave 
E = e X V^ which have been neglected in the formula : 

Hence it follows that, while the E.M.F. between two col- 
lector rings in the machine shown diagrammalically in Fig. 
178 is only e x VS, by massing the same number of turns 
in one slot instead of in two slots, we get the E.M.F. 2 e 
or 15,4 per cent higher E.M.F., that is, larger output. 

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It follo\/s herefrom that the distorted E.M.F. w-ave of 
a unitooth alternator is produced by lesser magnetic flux per 
pole — that is, in general, at a lesser hysteretic loss in the 
armature or at higher efficiency — than the same effective 
E.M.F. would be produced with the same number of arma- 
ture turns if the magnetic disposition were such as to pro- 
duce a sine wave. 

249. Inversely, if su':h a distorted wave of E.M.F. is 
impressed upon a magnetic circuit, as, for instance, a trans- 
former, the wave of magnetism in the primary will repeat 
in shape the wave of magnetism interlinked with the arma- 
ture coils of the alternator, and consequently, with a lesser 
maximum magnetic flux, the same effective counter E.M.F. 
will be produced, that is, the same power converted in the 
transformer. Since the hysteretic loss in the transformer 
depends upon the maximum value of magnetism, it follows 
that the hysteretic loss in a transformer is less with a dis- 
torted wave of a unitooth alternator than with a sine wave. 

Thus with the distorted waves of unitooth machines, 
generators, transformers, and synchronous motors — and 
induction motors in so far as they are transformers — 
operate more eflUciently. 

250. From another side the same problem can be 

If upon a transformer a sine wave of E.M.F. is im- 
pressed, the wave of magnetism will be a sine wave also. 
If now upon the sine wave of E.M.F. higher harmonics, 
as sine waves of triple, quintuple, etc., frequency are 
superposed in such a way that the corresponding higher 
harmonic sine waves of magnetism do not increase the 
maximum value of magnetism, or even lower it by a 
coincidence of their negative maxima with the positive 
maximum of the fundamental, — in this case all the power 
represented by these higher harmonics of E.M.F. will be 

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transformed without an increase of the hysteretic loss, or 
even with a decreased hysteretic loss. 

Obviously, if the maximum of the higher harmonic wave 
of magnetism coincides with the maximum of the funda- 
mental, and thereby makes the wave of magnetism more 
pointed, the hysteretic loss will be increased more than in 
proportion to the increased power transformed, i.e., the 
efficiency of the transformer will be lowered. 

That is : Some distorted waves of E.M.F. are transformed 
at a lesser, some at a larger, hysteretic loss than the sine 
wave, if the same effective E.M.F. is impressed upon the 

The unitooth alternator wave and the first wave in Fig, 
175 belong to the former class; the waves derived from 
continuous-current machines, tapped at two equi-distant 
points of the armature, in general, to the latter class. 

251. Regarding the loss of energy by Foucault or eddy 
currents, this loss is not affected by distortion of wave 
shape, since the E,M,F, of eddy currents, as induced 
E.M.F., is proportional to the secondary E.M,F. ; and 
thus at constant impressed primary E.M.F., the energy 
consumed by eddy currents bears a constant relation to 
the output of the secondary circuit, as obvious, since the 
division of power between the two secondary circuits — 
the eddy current circuit, and the useful or consumer cir- 
cuit — is unaffected by wave-shape or intensity of mag- 

262. In high potential lines, distorted waves whose 
maxima are very high above the effective values, as peaked' 
waves, may be objectionable by increasing the strain on 
the insulation. It is, however, not settled yet beyond 
doubt whether the striking-distance of a rapidly alternat- 
ing potential depends upon the maximum value or upon 

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some value between effective and maximum. Since dis- 
ruptive phenomena do not always take place immediately 
after application of the potential, but the time element plays 
an important part^ it is possible that insulation-strain and 
striking-distance is, in a certain range, dependent upon the 
effective potential, and thus independent of the wave-shape. 

In this respect it is quite likely that different insulating 
materials show a different behavior, and homogeneous solid 
substances, as paraffin, depend in their disruptive strength 
upon the maximum value of the potential difference, while 
heterogeneous materials, as mica, laminated organic sub- 
stances, air, etc., that is substances in which the disruptive 
strength decreases with the time application of the potential 
difference, are less affected by very high peaks of E.M.F. 
o£ very short duration. 

In genera], as conclusions may be derived that the im- 
portance of a proper wave-shape is generally greatly over- 
rated, but that in certain cases sine waves are desirable, 
in other cases certain distorted waves are preferable. 

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253. The vector representation, 

A = ^ +/«" = a (cos a +_/' sin a) 
of the alternating wave, 

A = a^ cos (^ — d) 
applies to the sine wave only. 

The general alternating wave, however, contains an in- 
finite series of terms, of odd frequencies, 

A = A^ cos {* - <7,) ■»- A^ cos (3 - a,) + /4.cos (5 ^ - «()+ 
thus cannot be directly represented by one complex vector 

The replacement of the general wave by its equivalent 
sine wave, as before discussed, that is a sine wave of equal 
effective intensity and equal power, while sufficiently accu- 
rate in many cases, completely fails in other cases, espe- 
cially in circuits containing capacity, or in circuits containing 
periodically (and in synchronism with the wave) varying 
resistance or reactance (as alternating arcs, reaction ma- 
chines, synchronous induction motors, oversaturated mag- 
netic circuits, etc.). 

Since, however, the individual harmonics of the general 
alternating wave are independent of each other, that is, all 
products of different harmonics vanish, each term can be 
represented by a complex symbol, and the equations of the 
general wave then are the resultants of those of the indi- 
vidual harmonics. 



This can be represented symbolically by combining in 
one formula symbolic representations of different frequen- 
cies, thus. 

and the index of they„ merely denotes that the/'s of differ- 
entindices «, while algebraically identical, physically rep- 
resent different frequencies, and thus cannot be combined. 
The general wave of E.M.F. is thus represented by, 

the general wave of current by, 


z, = r — y(*„ + *o + *e) 

is the impedance of the fundamental harmonic, where 

x„ is that part of the reactance which is proportional to 
the frequency (inductance, etc.). 

x„ is that part of the reactance which is independent of 
the frequency {mutual induction, synchronous motion, etc.). 
x^ is that part of the reactance which is inversely pro- 
portional to the frequency (capacity, etc.). 
The impedance for the «th harmonic is, 

Z = .-^.(„«. + .. + ?!) 

This term can be considered as the general symbolic 
expression of the impedance of a circuit of general wave 

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Ohm's law, in symbolic expression, assumes for the 
general alternating wave the form, 

I 1 '■->('««» + «;, + =5 1 

E'-^IZ or, 

^"-i (».' +^w.'') = X"-' [' -•'" ("*» + *• + J)] 

.Z = ior, 



The symbols of multiplication and division of the terms 
E, I, Z, thus represent not algebraic operation, but multi- 
plication and division of corresponding terms of E, I, Z, 
that is, terms of the same index n, or, in algebraic multipli^ 
cation and divbion of the series E, I, all compound terms» 
that is terms containing two different »'s, vanish. 

254. The effective value of the general wave : 
a = .^iCos(^ ~ ai^-\- A^ cos (S ^ — «») +^iCos (5 ^ — a() +.. 

is the square root of the sum of mean squares of individual 

A= V i i ^,' + ^,' + /*.* + ... i 
Since, as discussed above, the compound terms, of two 
different indices n, vanish, the absolute value of the general 
alternating wave, 

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is thus. 

^ = V^.-.'"'+'^°' 

«.' + c 

which offers an easy means of reduction from symbolic to 
absolute values. 

Thus, the absolute value of the E.M.F. 


.E-y/X"— ('»'' + '■"') 

the absolute value of the current, 


/.y/j;.'.-. (4'" + ,,"') 

265. The double frequency power (torque, etc.) equa- 
tion of the general alternating wave has the same symbolic 
expression as with the sine wave : 

p- \Er\ 

- F- +jPI 

= {EiY+j\sr\' 

= S'— ('»v + ^.''i") + 2;'"-v;(<»"4' - 'A") 


p = i£/y = 2'— <'•■'■' + '•"'■"> 

DiBiiu.d, Google 


The /, enters under the summation sign of the " watt- 
less power " pi, so that the wattless powers of the different 
harmonics cannot be algebraically added. 


The total " true power" of a general alternating current 
circuit is the algebraic sum of the powers of the individual 

The total "wattless power" of a general alternating 
current circuit is not the algebraic, but the absolute sum of 
the wattless powers of the individual harmonics. 

Thus, regarding the wattless power as a whole, in the 
general alternating circuit no distinction can be made be- 
tween lead and lag, since some harmonics may be leading, 
others lagging. 

The apparent power, or total volt-amperes, of the circuit 

The power factor of the circuit is, 

The term "inductance factor," however, has no mean- 
ing any more, since the wattless powers of the diflFerent 
harmonics are not directly comparable. 

The quantity, 

wattless power 

has no physical significance, and is not 

total apparent power 

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The term, Pi 


?» = -— ;g^^ 

consists of a series of inductance factors q^ of the individual 

As a rule, if ^= 5)*»-' ?«'. 

for the general alternating wave, that is g differs from 

The complex quantity, 

.~Q£/ EI 

takes in the circuit of the general alternating wave the 
same position as power factor and inductance factor with 
the sine wave. 

U= -= may be called the " arcutt factor." 

It consists of a real term /, the power factor, and a 
series of imaginary terms _/« g^ the inductance factors of 
the individual harmonics. 

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The absolute value of the circuit factor ; 

as a rule, is < 1. 

256. Some applications of this symbolism will explain 
its mechanism and its usefulness more fully. 

\st Instance: Let the E.M.F., 

be impressed upon a circuit of the impedance, 


that is, containing resistance r, inductive reactance x^ and 
capacity reactance x^ in series. 


a' = 720 


= 540 


= -283 


= 138 


•1 = 



»,= - 

- 1 


»,= - 


«, = 400 
«, = 1T3 

It is thus in symbolic expression, 

Z, = 10 + 80^; .,-.806 

Z, - 10 «, - 10 

Z, » 10 - 32/; ., - 33.6 

and, E.M.F., 

E = (720 + MO/a + (283 - 283/;) + (- 104 + 138/;) 

or absolute, 

E = 1000 



and current, 

M _ 720 + 540a , 283 - 283/; - 104 + 138^; 

. z~ 10 + 80/, "*■ 10 ■*" 10 - 32/; 

- (7.76 - 8.04/;) + (28.3 - 28.3/;^ + (- 4.86 - 1.73/;) 

or, absolute, 


of which b of fundamental frequency, I, = 11.15 
" " " " triple " I, = 40 

" " " " quintuple " I, = 5.17 

The total apparent power of the circuit is, 

(3 = £7=41,850 
The true power of the circuit is : 

7"= \E If = 1240 + 16,000 + 270 
= 17,610 
the wattless power, 

/■ Fi =/ \E ly = 10,000 /i - 850/; 
thus, the total power, 

P= 17,510 + 10,000/; - 850/; 

That is, the wattless power of the first harmonic is 
leading, that of the third harmonic zero, and that of the Mth 
harmonic lagging. 

17,510 = I' r, as obvious. 

The circuit factor is, 

= .418 + .239/; - .0203/; 
or, absolute, 

a= V.418»+. 239' + .0203* 
= .482 

The power factor is. 

p = .418 

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The inductance factor of the first hannonic is : ^, = . 239 
that of the third harmonic ^, = 0, and of the fifth harmonic 
ft = - .0203. 

Considering the waves as replaced by their equivalent 
sine waves, from the sine wave formula, 

/• + ?o' = 1 

the inductance factor would be, 

?. = .914 

and the phase 



■^^* 2 8 

a = 65.4' 

giving apparently a very great phase displacement, while in 
reality, of the 41.85 amperes total current, 40 amperes (the 
current of the third harmonic) are in phase with their 

We thus have here a case of a circuit with complex har- 
monic waves which cannot be represented by their equiva- 
lent sine waves. The relative magnitudes of the different 
harmonics in the wave of current and of E.M.F. differ 
essentially, and the circuit has simultaneously a very low 
power factor and a very low inductance factor; that is, a low 
power factor exists without corresponding phase displace- 
ment, the circuit factor being less than one-half. 

Such circuits, for instance, are those including alternat- 
ing arcs, reaction machines, synchronous induction motors, 
reactances with over-saturated magnetic circuit, high poten- 
tial lines in which the maximum difference of potential ex- 
ceeds the voltage at which brush discharges begin, polariza- 
tion cells, and in general electrolytic conductors above the 
dissociation voltage of the electrolyte, etc. Such circuits 
cannot correctly, and in many cases not even approxi- 
mately, be treated by the theory of the equivalent sine 
waves, but require the symboHsm of the complex harmonic 

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257. 2d instance: A condenser of capacity C", = 20 
in.f. is connected into the circuit of a 60-cycle alteriuitor 
giving a wave of the form, 

e = £ (iMs ^ - .10 cos 3 <A - .08 cos C ^ + .06 cos 7 <^) 
or, in symbolic expression, 

£ = * (1, - .10, - .08, + .06,) 
The synchronous impedance of the alternator is, 

Z^ = r^ — j^nxa = .3 — Bnjn 

What is the apparent capacity C of the condenser (as cal- 
culated from its terminal volts and amperes) when connected 
directly with the alternator terminals, and when connected 
thereto through various amounts of resistance and induc- 
tive reactance. 

The capacity reactance of the condenser is, 


Xc = 2 ji/r ~ ^^^ ohms, 

or, in symbolic expression, 

_L - *c 132 . 


Z^ = r — y„ «r = impedance inserted in series with the 

The total impedance of the circuit is then, 

Z = Z^ + Z^ + /;ffl 

The current in the circuit is, 


L(.3 + r)-yXJ'-132) (.3 + r)-AC3»- 


(■3 + »-)-A(5«-1.4) ^ (.3 + r) -/V(7»+ I6.1)J 



and the E.M.F. at the condenser tenninals, 

.r 13^^ - ■'■^ 

U.3 + r)-/i(*-132) (.3 + r)-A(3«-i») 

2Ua 113.>, - 1 

(•3 + r) - A (6 « - 11) (-3 + ')-/;('» + 1« J)J 

thus the apparent capacity reactance of the condenser is, 
and the apparent capacity, 

(a.) * = : Resistance r in series with the condenser. 
Reduced to absolute values, it is, 

1 01 .0064 .0036 

1 <.8 + r) + 174Jj"'"(,3 + >-)'+84l"'"(.3+^)' + 1.98"*'(.a+r)' + a60 
X^ 'in.i4 19-4 4.46 1.26 

(.3+r)'+17424 + <.3+r)» + l)4l'''(.3 + r)a+l.l»'''(.3 + r)' + a69 

(i.) r = : Inductive reactance 4: in series with the 
condenser. Reduced to absolute values, it is, 

1 .01 .0064 .0030 

1 ■00+(j--l«li)' .0t'+(3-i--a9|'"'".0l>+[6j--1.4)'''".0»+(7j-+lfl.l)* 
^'~ 17424 18.4 4.45_ 1.28 


From — ^ are derived the values of apparent capacity, 

c L"L 

and plotted in Fig, 179 for values of r and x respectively 
varying from to 22 ohms. 

As seen, with neither additional resistance nor reactance 
in series to the condenser, the apparent capacity with this 
generator wave is 84 m.f., or 4.2 times the true capacity, 

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and gradually decreases with increasing series resistance, to 
C= 27.5 mi. = 1.375 times the true capacity at r= 18.2 
ohms, or Vff the true capacity reactance, with r = 182 ohms, 
or with an additional resistance equal to the capacity reac- 
tance, C= 20.6 m.f. or only 2.5% in excess of the true 
capacity C^, and at r = oo , C= 20,8 m.f. or 1.5% in excess 
of the true capacity. 

With reactance x, but no additional resistance r in series, 
the apparent capacity C rises from 4.2 times the true 
capacity at :r = 0, to a maximum of 5,08 times the true 
capacity, or C= 100.6 m.f. at x = .28, the condition of res- 
onance of the fifth harmonic, then decreases to a minimum 
of 27 m.f., or 85 % in excess of the true capacity, rises again 
to 60.2 m.f., or 8.01 times the true capacity at ;r=9.67, 
the condition of resonance with the third harmonic, and 
finally decreases, reaching 20 m.f., or the true capacity at 
X = 132, or an inductive reactance equal to the capacity 
reactance, then increases again to 20.2 m.f. at :r = oo . 

This rise and fall of the apparent capacity is within cer- 
tain limits independent of the magnitude of the higher 
harmonics of the generator wave of E.M.F., but merely de- 
pends upon their presence. That is, with such a reactance 
connected in series as to cause resonance with one of the 
higher harmonics, the increase of apparent capacity is ap- 
proximately the same, whatever the value of the harmonic, 
whether it equals 25% of the fundamental or less than 5%, 
provided the resistance in the circuit is negligible. The 
only effect of the amplitude of the higher harmonic is that 
when it is small, a lower resistance makes itself felt by re- 
ducing the increase of apparent capacity below the value it 
would have were the amplitude greater. 

It thus follows that the true capacity of a condenser 
cannot even approximately be determined by measuring 
volts and amperes if there are any higher harmonics present 
in the generator wave, except by inserting a very large re- 
sistance or reactance in series to the condenser. 

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268. Zd instance : An alternating current generator 
of the wave, 

£, = 2000 [li + .12, - .23, - .13,] 
and of synchronous impedance, 

Z, = .3-6«/W 
feeds over a line of impedance, 










■ (I 

) " 













































a synchronous motor of the wave, 

£i = 2250 [(cos <u +j\ sin «.) + .24 (cos 3 » +y; sin 3 »)] 
and of synchronous impedance, 

Z, = .3-n/y; 
The total impedance of the system is then, 
Z = Z„ + Z, + Z, 
= 2.6-15/./, 

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thus the current. 

10 cos M- 2250/; sin <n , 24Q-54Qcos3<tt-540;;sin3« 

2.6 - X5j\ 2.6 - 45y; 


~ 2.6 ~ 76/; 2.6 - 106/V 


a,> =s 22.6 - 26.2 cos u + 146 sin » 
i^> = .306 - .69 cos 3 u + 11.9 sin 3 u 
V = - -213 
tf,» « - .061 

a,'* = 130 - 146 cos o> - 25.2 sin w 
a," = 5.3 — 11.9 cos 3 » — .69 sin 3 » 
</," = - 6.12 
B," =s - 2.48 

or, absolute, 

1st harmonic, 

8d harmonic, 

5th harmonic, 
7th harmonic, 

B, = 6.12 

while the total current of higher harmonics is, 

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The true input of the synchronous motor is, 

= (2250Vcos«+2260«,"sm-) + (540a,'cos3«+S40<7,»8in3») 

= Pi + Pi 

Pi = 2250 (ai cos « + a," sin w) 

" _= » 

' c. ^'' \ 

•E ^/ S - 

•X;^ \ - 

• ^ / ± 3 » 

■i-V ^ - 

.55^ ^ Ei. 

■ /_-— -^^^ "^s. -jX' • 

. ^S;<^ "^4- . 

\ -» 

i^ "v"— ^ 

i\ , „ ,:t'.. ,.,,.. .....,, ^ 



X_ -« 

^^ ^ - 

^ I ^ 

X I ^ 

t,"f"oi"°".,T""«t \ ' j» 

""'.°™M'r3r"' "^^j!.^' J, 

Z,-2. i-li j.^ 

»0. r«a AyRMnu 

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is the power of the fundamental wave, 

Pi = 540 (fl,^ cos 3 » + fl," sin 3 o>) 

the power of the third harmonic. 

The 6th and 7th harmonics do not give any power, 
since they are not contained in the synchronous motor 
wave. Substituting now different numerical values for m 
the phase angle between generator E.M.F. and synchronous 
motor counter E,M.F., corresponding values of the currents 
/ /fl, and the powers /", /*,', /*,' are derived. These are 
plotted in Fig. 180 with the total current /as abcissae. To 
each value of the total current / correspond two values of 
the total power /", a positive value plotted as Curve I. — ■ 
synchronous motor — and a negative value plotted as 
Curve II. — ^alternating current generator — . Curve III. 
gives the total current of higher frequency /„, Curve IV., 
the difference between the total current and the current of 
fundamental frequency, / — a^, in percentage of the total 
current /, and F the power of the third harmonic, /",', in 
percentage of the total power PK 

Curves III., IV. and V, correspond to the positive or 
synchronous motor part of the power curve /". As seen, 
the increase of current due to the higher harmonics is 
small, and entirely disappears at about 180 amperes. The 
power of the third harmonic is positive, that is, adds to the 
work of the synchronous motor up to about 140 amperes, 
or near the maximum output of the motor, and then becomes 

It follows herefrom that higher harmonics in the E.M.F. 
waves of generators and synchronous motors do not repre- 
sent a mere waste of current, but may contribute more or 
less to the output of the motor. Thus at 75 amperes total 
current, the percentage of increase of power due to the 
higher harmonic is equal to the increase of current, or in 
other words the higher harmonics of current do work with 
the same efficiency as the fundamental wave. 

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259. 4/A Instance: In a small three-phase induction 
motor, the constants per delta circuit are 

Primaiy admittance Y= .002 + .03_/' 

Self-inductive impedance ^j = Z, = .6 — 2.4/ 
and a sine vrave of E.M.F. e^ = 110 volts is impressed upon 
the motor. 

The power output P, current input /„ and power factor 
/, as function of the slip s are given in the first columns of 
the following table, calculated in the manner as described In 
the chapter on Induction Motors. 

To improve the power factor of the motor and bring it 
to unity at an output of 500 watts, a condenser capacity is 
required giving 4.28 amperes leading current at 110 volts, 
that is, neglecting the energy loss in the condenser, capacity 

In this case, let /, = current input into the motor per 
delta circuit at slip s, as given in the following table. 

The total current supplied by the circuit with a sine 

wave of impressed E.M.F., is 

r = I,- 4,28/ 

, , , , , energy current 

and herefrom the power factor = — ■, given m 

"^ , , , , total current ° 

the second columns of the table. 

If the impressed E.M.F. is not a sine wave but a wave 
of the shape 

-£■(, = 'o (li + -12, - .23. - .134,) 
to give the same output, the fiuidamental wave must be the 
same: e^^ 110 volts, when assuming the higher harmonics 
in the motor as wattless, that is 

E^ = 110, + 13.2, - 25.3. - 14.7, 
= ^, + E^ 
where E^ = 13.2, - 25.3. - 14.7, 

= component of impressed KM.F. of higher frequency- 

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The effective value is ; 

Ba = 114.6 volts. 

The condenser admittance for the general alternating 
■wave is 

K„ = - .039 nj^ 

Since the frequency of rotation of the motor is veiy 
small compared with the frequency of the higher harmonics, 
as total impedance of the motor for these higher harmonics 
can be assumed the stationary impedance, and by neglecting 
the resistance it is 

= -4.8/,/; 

The exdting admittance of the motor, for these higher 
harmonics, is, by neglecting the conductance, 

and the higher harmonics of counter E.M.F. 

Thus we have, 
Current input in the condenser, 

Ic = Ea K 

= - 4.28/, - 1.54>, + 4.93/; + 4.02/; 

High frequency component of motor impedance current, 

^ = .92/, - 1.06/; - .44/; 
High frequency component of motor exciting current 

= ,07/; - ,08/; - .03y, 

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thus, total high frequency component of motor current, 

and total current, 

without condenser, 

/„=/. + n 

' = /. + !99a - 1.14a - •^'^h 
with condenser, 

/=/, + /,'-/„ 

' = >, - 4.28^' - .66a + 3-79y; + 3.56/; 
and herefrom the power factor. 





K. « 




« •:*• ■ 




_ J! 










^ = 

















- — 







= !? 




_ t i 







' . 


i- :z, ' 


c . 




/ , 












In the following table and in Fig. 181 are given the 
values of current and power factor : — 

I. With sine wave of E.M.F., of 110 volts, and no condenser. 

II. With ^ne wave of E.M.F, of 110 volts, and with condenser. 

III. With distorted wave of E.M.F., of 114.5 volts, and no condenser. 

IV. With distorted wave of KM.F., of 114.6 volts, andirith condenser. 

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.24+ 3.10/ 










1.73+ 8.16/ 










8.32+ 8.47/ 










5.16+ 4.28/ 










6.96+ 5.4/ 










8.77+ 7.3/ 










10.1 + 9,86/ 




















10.76 + 12.9/ 









The curves II. and IV. with condenser are plotted m. 
dotted lines in Fig. 181. As seen, even with such a dis- 
torted wave the current input and power factor of the motor 
are not much changed if no condenser is used. When usin^ 
a condenser in shunt to the motor, however, with such a 
wave of impressed E.M.F. the increase of the total current, 
due to higher frequency currents in the condenser, is greater 
than the decrease, due to the compensation of lagging cur- 
rents, and the power factor is actually lowered by the con- 
denser, over the total range of load up to overloads, and 
especially at light loads. 

Where a compensator or transformer is used for feeding 
the condenser, due to the internal self-induction of the com- 
pensator, the higher harmonics of current are still more 
accentuated, that is the power factor still more lowered. 

In the preceding the energy loss in the condenser and 
compensator and that due to the higher harmonics of cur- 
rent in the motor has been neglected. The effect of this 
energy loss is a slight decrease of efficiency and correspond- 
ing increase of j)0wer factor. The power produced by the 
higher harmonics has also been neglected ; it may be posi- 
tive or negative, according to the index of the harmonic, 
and the winding of the motor primary. Thus for instance, 
the effect of the triple harmonic is negative in the quarter- 
phase motor, zero in the three-phase motor, etc., altogether,, 
however, the effect of these harmonics is very small. 

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260- A polyphase system is a.n alternating-current sys- 
tem in which several E.M.Fs. of the same frequency, but 
displaced in phase from each other, produce several currents 
of equal frequency, but displaced phases. 

Thus any polyphase system can be considered as con- 
sisting of a number of single circuits, or branches of the 
polyphase system, which may be more or less interlinked 
with each other. 

In general the investigation of a polyphase system is 
carried out by treating the single-phase branch circuits 

Thus all the discussions on generators, synchronous 
motors, induction motors, etc., in the preceding chapters, 
apply to single-phase systems as well as polyphase systems, 
in the latter case the total power being the sum of the 
powers of the individual or branch circuits. 

If the polyphase system consists of « equal E.M.Fs. 
displaced from each other by 1 / « of a period, the system 
is called a symmetrical system, otherwise an unsymmetrical 

Thus the three-phase system, consisting of three equal 
E.M.Fs. displaced by one-third of a period, is a symmetncal 
system. The quarter-phase system, consisting of two equal 
E.M.Fs. displaced by 90°, or one-quarter of a period, is an 
unsymmetrical system. 

261. The flow of power in a single-phase system is 
pulsating ; that is, the watt curve of the circuit is a sine 

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wave of double frequency, alternating between a maximum 
value and zero, or a negative maximum value. In a poly- 
phase system the watt curves of the different branches of 
the system are pulsating also. Their sum, however, or the 
total flow of power of the system, may be either constant 
or pulsating. In the first case, the system is called a 
balanced system, in the latter case an unbalanced system. 

The three-phase system and the quarter-phase system, 
with equal load on the different branches, are balanced sys- 
tems; with unequal distribution of load between the indi- 
vidual branches both systems become unbalanced systems. 

FI3. 182. 

The different branches of a polyphase system may be 
either independent from each other, that is, without any 
electrical interconnection, or they may be interlinked with 
each other. In the first case, the polyphase system is 
called an independent system, in the latter case an inter- 
linked system. 

The three-phase system with star-connected or ring-con- 
nected generator, as shown diagram matically in Figs, 181 
and 182, is an interlinked system. 

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The four-phase system as derived by connecting four 
equidistant points of a continuous-current armature with 
four collector rings, as shown diagrammatically in Fig, 188, 

is an interlinked system also. The four-wire quarter-phase 
system produced by a generator with two independent 
armature coils, or by two single-phase generators rigidly 
connected with each other in quadrature, is an independent 
system. As interlinked system, it is shown in Fig. 184, as 
star-connected four-phase system. 



262. Thus, polyphase systems can be subdivided into : 
Symmetrical systems and unsymmetrical systems. 
Balanced systems and unbalanced systems. 
Interlinked systems and independent systems. 
The only polyphase systems which have found practical 
application are : 

The three-phase system, consisting of three E-M.Fs, dis- 

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placed by one-third of a period, used exclusively as inter- 
linked system. 

The quarter-phase system, consisting of two E.M,Fs. in 
quadrature, and used with four wires, or with three wires, 
which may be either an interlinked system or an indepen- 
dent system. 

The six-phase system, consisting of two three-phase sys- 
tems in opposition to each other, and derived by transforma- 
tion from a three-phase system, in the alternating supply 
circuit of large synchronous converters. 

The inverted three-phase system, consisting of two 
E.M.F.'s displaced from each other by 60°, and derived 
from two phases of a three-phase system by transformation 
with two transformers, of which the secondary of one is 
reversed with regard to its primary (thus changing the 
phase difference from 120° to 180° - 120° = 60°), finds a 
limited application in low tension distribution. 

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263. If all the E.M.Fs. of a polyphase system are equal 
in intensity, and differ from each other by the same angle 
of difference of phase, the system is called a symmetrical 
polyphase system. 

Hence, a symmetrical »-phase system is a system of » 
E.M.F3. of equal intensity, differing from each other in 
phase by l/w of a period: 

*, =Esmfi\ 

The next E.M.F. is again : 

4r, t= £sm(p -^2^)-* £ sin j8. 

In the polar diagram the « E.M.Fs, of the symmetrical 
ff-phase system are represented by n equal vectors, follow- 
ing each other under equal angles. 

Since in symbolic writing, rotation by l/» of a period, 
or angle 2»/a, is represented by multiplication with: 

the E.M.F8. of the symmetrical polyphase system are: 

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2» , ,,.:_2, 

^f COS 


£ ( cos — i i- Yj sin — i 1— \ =Et* '. 

■ \ " " / " 

The next E.M.F. is again : 

^Ccos2r+/sin2,) = A»»=^. 
Hence, it is 

Or in other words : 
• In a symmetrical «-phase system any E.M,F. of the 
system is expressed by : 

where : j = -v/I. 

264. Substituting now for « different values, we get 
the different symmetrical polyphase systems, represented by 


1.) « = 1 «=.! i*E = £, 
the ordinary single-phase system. 

2.) « = 2 « = - 1 ^E=EiR6-E. 

Since — £■ is the return of ^, « i- 2 gives again the 
single-phase system. 


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The three E.M.Fs. of the three-phase system are : 

■' 2 ■ ' 2 ■ 

Consequently the three-phase system is the lowest sym- 
metrical polyphase system. 

4.) « = 4, . = cos^-f/sm^=/; «*=-!, .»=-/ 
4 4 

The four E.M.Fs. of the four-phase system are : 

€*£=-£, jE, -E, -j'E. 

They are in pairs opposite to each other : 

E and ~E\jE and —jE. 

Hence can be produced by two coils in quadrature with 
■each other, analogous as the two-phase system, or ordinary 
.alternating-current system, can be produced by one coil. 

Thus the symmetrical quarter-phase system is a four- 
phase system. 

Higher systems, than the quarter-phase or four-phase 
system, have not been very extensively used, and are thus 
of less practical interest, A symmetrical six-phase system, 
derived by transformation from a three-phase system, has 
found application in synchronous converters, as offering a 
higher output from these machines,and a symmetrical eight- 
jihase system proposed for the same purpose. 

265. A characteristic feature of the symmetrical bi- 
phase system is that under certain conditions it can pro- 
duce a M.M.F. of constant intensity. 

If n equal magnetizing coils act upon a point under 
equal angular displacements in space, and are excited by the 
n E.M.Fs. of a symmetrical w-phase system, a M.M.F. of 
■constant intensity is produced at this point, whose direction 
revolves synchronously with uniform velocity. 

ri = number of turns of each magnetizing coiL 

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£= effective value of impressed RM.F. 

/ = effective value of current 

■7 = V/si effective M.M.F. of one of the magnetizing coils. 
Then the instantaneous value of the M.M.F. of the coil 
acting in the direction 2vi jn is : 

= «'/V2sin/';3-?iA. 
The two rectangular space components of this M.M.F. are; 
fi =^cos?Ji* 

= ^I-J2 cos ^iisin U - ^-^ 

, ,,, , . 2«-i 

and A = /i sm — 

= «'/V2 sin ^"sin U - ?^A . 

Hence the M.M.F. of this coil can be expressed by the 
symbolic formula : 

Thus the total or resultant M.M.F. of the « coils dis- 
placed under the n equal angles is : 

or, expanded : 

/=V/V2 j sin/35t('cos'^'+ysin?^'cos^^- 
( 1 \ « « « / 

„ Vr/ ■ 2ir» 2i»" , . . ,25r<\ 1 

cos p /i I sin — COS —— +y sin* J . 

1 \ « « « / 1 

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It is, however : 
ff n ft \ 1 « / 

Sin cos l-^sin" — — ^ 1 — cos ^sin — — ) 

and, since: 

P" = '' ^-"''' 

it is, /_ "'^^^ (sin ^ - y COS fi). 

= ^(sin^-ycos«; 

the symbolic expression of the M.M.F. produced by the 
» circuits of the symmetrical n-phase system, when exciting 
» equal magnetizing coils displaced in space under equal 

The absolute value of this M.M.F. is : 

V2 V2 2 • 

Hence constant and equal «/V2 times the effective 
M.M.F. of each coil or n/2 times the maximum M.M.F. 
of each coil. 

The phase of the resultant M.M.F. at the time repre- 
sented by the angle fi is ; 

tan u = — cot fl ; hence fi = ^ — ^ 

That is, the M.M.F. produced by a symmetrical »-phase 
system revolves with constant intensity : 



and constant speed, in synchronism with the frequency of 
the system ; and, if the reluctance of the magnetic circuit 
is constant, the magnetism revolves with constant intensity 
and constant speed also, at the point acted upon symmetri- 
cally by the n M.M.Fs. of the »-phase system. 

This is a characteristic feature of the symmetrical poly- 
phase system, 

266. In the three-phase system, « = 3, F= 1.5 fF„^, 
where SF,„„ is the maximum M.M.F. of each of the magne- 
tizing coils. 

In a symmetrical quarter-phase system, « = 4, F =% 
^maxy where ff„„ is the maximum M.M.F. of each of the 
fout* magnetizing coils, or, if only two coils are used, since 
the four-phase M.M.Fs. are opposite in phase by two, F = 
^max< where ^„,„ is the maximum M.M.F. of each of the 
two magnetizing coils of the quarter-phase system. 

While the quarter-phase system, consisting of two E.M.Fs. 
displaced by one-quarter of a period, is by its nature an 
unsymmetrical system, it shares a number of features — 
as, for instance, the ability of producing a constant result- 
ant M,M.F. — with the symmetrical system, and may be 
considered as one-half of a symmetrical four-phase system. 

Such systems, consisting of one-half of a symmetrical 
system, are called hemisymmetrical systems. 

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267. If an alternating E.M.F. : 
* = £ V2 sin ft 
produces a current : 

where i is the angle of lag, the power is : 

/ = « = 2 ^/sin j8 sin (/3 — a) 

= £/(cosa-cos(2y3-a)), 

and the average value of power : 

Substituting this, the instantaneous value of power is 
found as: 

Hence the power, or the flow of energy, in an ordinary 

single-phase alternating-current circuit is fluctuating, and 

varies with twice the frequency of E.M.F. and current, 

unlike the power of a continuous-current circuit, which b 

constant : 

p = et. 

If the angle of lag w = it is : 

/ = ^(l-cos2/3)i 

hence the flow of power varies between zero and 2/*, where 
P is the average flow of energy or the effective power of 
the circuit. 

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If the current lags or leads the E.M.F. by angle w the 
power varies between 


:os ut y \^ cos i» j 

that is, becomes negative for a certain part of each half- 
wave. That is, for a time during each half-wave, energy 
flows back into the generator, while during the other part 
of the half-wave the generator sends out energy, and the 
difference between both is the effective power of the circuit. 
If 5 = 90°, it is : 

/ = — £/sin2^; 

that is, the effective power : /• = 0, and the energy flows 
to and fro between generator and receiving circuit. 

Under any circumstances, however, the flow of energy in 
the single-phase system is fluctuating at least between zero 
and a maximum value, frequently even reversing. 

268. If in a polyphase system 

'ii'it'it ■ ' • . ^ instantaneous'values of E-M.F. ; 
'it ^1 it, . ■ . ■ = instantaneous values of current pro- 
duced thereby ; 

the total flow of power in the system is : 

/ = 'i'i + '^.'i + -;i'. + - ■ - ■ 

The average flow of power is : 

F = Eil\ cos «i -J- Et /, cos £i -|- . . • . 

The polyphase system is called a balanced system, if the 
flow of energy : 

P = 'I'l + ^ih -f-'i'V + - ■ ■ ■ 
is constant, and it is called an unbalanced system if the 
flow of energy varies periodically, as in the single-phase sys- 
tem ; and the ratio of the minimum value to the maximum 
value of power is called the balance factor of the system. 

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Hence in a single-phase system on non-inductive circuit, 
that is, at no-phase displacement, the balance factor is zero ; 
and it is negative in a single-phase system with lagging or 
leading current, and becomes = — 1, if the phase displace- 
ment is 90° — that is, the'circuit is wattless. 

269. Obviously, in a polyphase system the balance of 
the system is a function of the distribution of load between 
the different branch circuits. 

A balanced system in particular is called a polyphase 
system, whose flow of energy is constant, if all the circuits 
are loaded equally with a load of the same character, that 
is, the same phase displacement. 

270. Ail the symmetrical systems from the three-phase 
system upward are balanced systems. Many unsymmetrical 
systems are balanced systems also. 

1.) Three-phase system : 

t^=-E->/2 sin A and i, = / V5 sin (/3 — «) ; 

<r, = ^ V2 sin (^ - 120), i^ = Iy^2 sin {fi - S> ~ 120) ; 

<r, = £ ■v'2 sin (^ - 240), f, = 7V2 sin (^ - fi - 240) ; 

be the E.M.Fs. of a three-phase system, and the currents 
produced thereby. 

Then the total flow of power is : 

/ = 2 ^/(sin ^ sin ()3 - S) + sin (j8 - 120) sin {fi-S.- 120) 
4- sin (^ - 240) sin (^ - £ - 240)} 
= 3£/cosi ^ P, or constant. 

Hence the symmetrical three-phase system is a balanced 

2.) Quarter-phase system : 

Let €^=^ E-J2 sin A h = / V2 sin (fi-S,)\ 

^j = £ V2 cosft «; = / V2 cos 03 - S) ; 

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be the E.M.Fs. of the quarter-phase system, and the cur- 
rents produced thereby. 

This is an unsymmetrical system, but the instantaneous 
flow of power is ; 

/ = 2 £/(sin ^ sin (yS - £) + cos j8 cos 03 - £)) 
= 2 EIcos £ = /*, or constant. 

Hence the quarter-phase system is an unsymmetrical bal- 
anced system. 

3.) The symmetrica] w-phase system, with equal load 
and equal phase displacement in all n branches, is a bal- 
anced system. For, let : 

^,= £V2sin(^-^'^ = E.M.F.; ■ 

/■( = / V5 sin f j8 — 5 ~ ] = current 

the instantaneous flow of power is : 

j -^ cos a - yr cos /'2 ff - a - i^'\ \ -, 

or p = n EIqos St = P,ot constant. 

271. An unbalanced polyphase system is the so-called 
inverted three-phase system,* derived from two branches of 
a three-phase system by transformation by means of two 
transformers, whose secondaries are connected in opposite 
direction with respect to their primaries. Such a system 
takes an intermediate position between the Edison three- 
wire system and the three-phase system. It shares 'with 
the latter the polyphase feature, and with the Edison three- 

= EI 

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wire system the feature that the potential difference be- 
tween the outside wires is higher than between middle 
wire and outside wire. 

By such a pair of transformers the two primary E-M-Fs. 
of 120° displacement of phase are transformed into two 
secondary E.M.Fs. differing from each other by 60°, Thus 
in the secondary circuit the difference of potential between 
the outside wires is V3 times the difference of potential 
between middle wire and outside wire. At equal load on 
the two branches, the three currents are equal, and differ 
from each other by 120", that is, have the same relative 
proportion as in a three-phase system. If the load on 
one bi^'K^h is maintained constant, while the load of the 
other branch is reduced from equality with that in the 
first branch down to zero, the current in the middle wire 
first decreases, reaches a minimum value of 87 per cent of 
its original value, and then increases again, reaching at no 
load the same value as at full load. 

The balance factor of the inverted three-phase system 
on non-inductive load is .383. 

272. In Figs. 185 to 192 are shown the E.M.Fs. as 
/ and currents as i in drawn lines, and the power as / in 
dotted lines, for : 

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np. lai. Quartar-iiliaMi Byitim on Mon-lailum 

Balance Factor, + 1. 

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Fig. IM. nrM-pJhu* 8g»t*m on /«Aiet/w t-oa^ a/ »0° Lt 

B dance Factor, + 1. 

nibnceFactor. + .333. 



273. The flow of power in an alternating-cuirent system 
is a most important and characteristic feature of the system, 
and by its nature the systems may be classified into : 

Monocyclic systems, or systems with a balance factor zero 
or negative. 

Polycyclic systems, with a positive balance factor. 
Balance factor — 1 corresponds to a wattless circuit, 
balance factor zero to a non-inductive single-phase circuit, 
balance factor + 1 to a balanced polyphase system. 

274. In polar coordinates, the flow of power of an 
alternating-current system is represented by using the in- 
stantaneous flow of power as radius vector, with the angle 
$ corresponding to the time as amplitude, one complete 
f)eriod being represented by one revolution. 

In this way the power of an alternating-current system 
is represented by a closed symmetrical curve, having the 
zero point as quadruple point. In the monocyclic systems 
the zero point is quadruple nodal point ; in the polycyclic 
system quadruple isolated point. 

Thus these curves are sextics. 

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Since the flow of power in any single-phase branch of 
the alternatmg-ciirrent system can be represented by a sine 
wave of double frequency : 

V cosi ; 

the total flow of power of the system as derived by the 
addition of the powers of the branch circuits can be rep- 
resented in the form : 

/ = /'(l + «sin(2^-£,)) 

This is a wave of double frequency also, with < as ampli- 
tude of fluctuation of power. 

This is the equation pf the power characteristics of the 
system in polar coordinates. 

275. To derive the equation in rectangular coordinates 
we introduce a substitution which revolves the system of 
coordinates by an angle m^ 2, so as to make the symmetry 
axes of the power characteristic the coordinate axes. 

henct, sin (2^ - 1.) = 2si,>(^ - |)cos(/S - |) = ^^, 


the sextic equation of the power characteristic. 
Introducing : 

a = {\ •\- i) P = maximum value of power, 
i ^ (1 — <) y ^ minimum value of power ; 

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_ a-h _ 
* fl + *' 

hence, substituted, and expanded : 

(^ + /)' - H« (* + ^)* + n* - ^)'}', = 

the equation of the power characteristic, with the main 
power axes a and b, and the balance factor: bfa. 
It is thus : 

Single-phase non-inductive circuit : / = ^ (1 -I- sin 2 ^), 
^ = 0, a = 2 /» 

(^ +/)■ - i" ix + yy - 0, i/a = 0. 

Single-phase circuit, 60° lag : / = /• (1 -|- 2 sin 2 ^), b^ 
~P, a = -\-ZP 

Single-phase circuit, 90° lag :/ = £/sin 2 ^, b = ~ E I, 
a = +£/ 

(■«"+/)• -4 (£/)•**/, i/a = - 1. 

Three-phase non-inductive circuit : p -^ P, b ^1, a = l 

:(» -f- ^ - 7" = : circle. A/n = +1. 
Three-phase circuit, 60° lag : / = /", i = 1, a = 1 

:c* + _y* _ /•« = ; circle. ^ /« = + 1- 
Quarter-phase non-inductive circuit ■.p = P, A = 1, a = 1 

a^ + _^ _ /■> = : circle. A/« = -|- 1. 
Quarter-phase circuit, 60° lag :/ = /•,. i = 1, a = 1 

^ + /- /** = : circle. bla = -\-l. 

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Inverted three-phase non-inductive circuit : 

Inverted three-phase circuit 60° lag : /=■/*(! + sin 2 ^), 
^ = 0, a = 2P 

a and b are called the main power axes of the alternating- 
current system, and the ratio */« is the balance factor of 
the system. 

. PoKtt Charmttrlttle of Blnglt-ph 

t SiiiUm, at 00° ai 

276. As seen, the flow of power of an alternating-cur- 
rent system is completely characterized by its two main 
power axes a and A, 

The power characteristics in polar coordinates, corre- 

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spending to the Figs. 185, 186, 191, and 192 are shown in 
Figs. 193, 194, 195, and 196. 

The balanced quarter-phase and three-ptiase systems give 
as polar characteristics concentric circles. 

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277. In a polyphase system the different circuits of 
displaced phases, which constitute the system, may either 
be entirely separate and without electrical connection with 
each other, or they may be connected with each other 
electrically, so that a part of the electrical conductors are 
in common to the different phases, and in this case the 
system is called an interlinked polyphase system. 

Thus, for instance, the quarter-phase system will be 
called an independent system if the two E.M.Fs. in quadra- 
ture with each other are produced by two entirely separate 
coils of the same, or different but rigidly connected, arma- 
tures, and are connected to four wires which energize inde- 
pendent circuits in motors or other receiving devices. If 
the quarter-phase system is derived by connecting four 
equidistant points of a closed-circuit drum or ring-wound 
armature to the four collector rings, the system is an inter- 
linked quarter-phase system. 

Similarly in a three-phase system. Since each of the 
three currents which differ from each other by one-third 
of a period is equal to the resultant of the other two cur- 
rents, it can be considered as the return circuit of the other 
two currents, and an interlinked three-phase system thus 
consists of three wires conveying currents differing by one- 
third of a period from each other, so that each of the three 
currents is a common return of the other two, and inversely. 

278. In an interlinked polyphase system two ways exist 
of connecting apparatus into the system. 

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1st. The star connection, represented diagrammatically 
in Fig. 197. In this connection the n circuits excited by 
currents differing from each other by 1 /« of a period, are 
connected with their one end together into a neutral point 
or common connection, which may either be grounded or 
connected with other corresponding neutral points, or insu- 

In a three-phase system this connection is usually called 
a Y connection, from a similarity of its diagrammatical rep- 
resentation with the letter Y, as shown in Fig. 181. 

000 , 



2d. The ring connection, represented diagrammatically 
in Fig. 198, where the n circuits of the apparatus are con- 
nected with each other in closed circuit, and the comers 
or points of connection of adjacent circuits connected to 
the « lines of the polyphase system. In a three-phase 
system this connection is called the delta connection, from 
the similarity of its diagrammatic representation with the 
Greek letter Delta, as shown in Fig. 182. 

In consequence hereof we distinguish between star- 
connected and ring-connected generators, motors, etc., or 

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B. ry 

in three-phase systems Y- connected and delta-connected 

279. Obviously, the polyphase system as a whole does 
not differ, whether star connection or ring connection is 
used in the generators or other apparatus ; and the trans- 
mission line of a symmetrical «-phase system always con- 
sists of M wires carrying current of equal strength, when 
balanced, differing from each other in phase by l/« of a 
period. Since the line wires radiate from the « terminals 
of the generator, the lines can be considered as being in 
star connection. 

The circuits of all the apparatus, generators, motors, 
etc., can either be connected in star connection, that is, 
between one line and a neutral point, or in ring connection, 
that is, between two adjacent lines. 

In general some of the apparatus will be arranged in 
star connection, some in ring connection, as the occasion 
may require. 

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280. In the same way as we speak of star connection 
and ring connection o£ the circuits of the apparatus, the 
term star potential and ring potential, star current and ring 
current, etc., are used, whereby as star potential or in a 
three-phase circuit Y potential, the potential difference be- 
tween one of the lines and the neutral point, that is, a point 
having the same difference of potential against all the lines, 
is understood ; that is, the potential as measured by a volt- 
meter connected into star or Y connection. By ring or 
delta potential is understood the difference of potential 
between adjacent lines, as measured by a voltmeter con- 
nected between adjacent lines, in ring or delta connec- 

In the same way the star or Y current is the current 
flowing from one line to a neutral point ; the ring or delta 
current, the current flowing from one line to the other. 

The current in the transmission line is always the star 
or Y current, and the fxjtential difference between the line 
wires, the ring or delta potential. 

Since the star potential and the ring potential differ 
from each other, apparatus requiring different voltages can 
be connected into the same polyphase mains, by using either 
star or ring connection. 

281 . If in a generator with star<onnected circuits, the 
E.M.F. per circuit = E, and the common connection or 
neutral point is denoted by zero, the potentials of the a 
terminals are : 

E,tE,^E . . . . «--'£; 
or in general : i' E, 

at the ("' terminal, where : 

.'=0,1,2 n~\, « = cos^-|-/sin — =-!/l. 

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Hence the E.M.F. in the circuit from the i^ to the ** 
terminal is : 

The E.M.F. between adjacent terminals i and i + 1 is : 

(."'-. 1) A -.'(.-1)^. 

In a generator with ring-connected circuits, the E.M.F. 
per circuit : ^ 

is the ring E.M.F., and takes the place of 

while the E.M.F. between terminal and neutral point, or 
the star E.M.F., is ; 

Hence in a star-connected generator with the E.M.F. 
E per circuit, it is : 

Star E.M.F., .'£. 

Ring E.M.F., €* (« - 1) E. 

KM.F. between terminal i and terminal k, («* — «*) E. 

In a ring-connected generator with the E.M.F. E per 
circuit, it is : 

RingE.M.F., «'^. 

E.M.F. between terminals i and k, * ~ *. E. 

In a star-connected apparatus, the E.M.F. and the cur- 
rent per circuit have to be the star E.M.F. and the star 
current. In a ring-connected apparatus the E.M.F. and 
current per circuit have to be the ring RM.F. and ring 

In the generator of a symmetrical polyphase system, if : 
«* E are the E.M.Fs. between the n terminals and the 
neutral point, or star E.M.Fs:, 

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/( — the currents issuing from terminal i over a line of 
the impedance Zi (including generator impedance in star 
connection), we have : 

Potential at end of line i : 

t'E - Ztlf. 
Difference of potential between terminals k and i : 

where // is the star current of the system, Z/ the star im- 

The ring potential at the end of the line between ter- 
minals i and k is E^, and it is : 

£it = - £ii- 
If now /^ denotes the current passing from terminal i to 
terminal k, and Z;t impedance of the circuit between ter- 
minal i and terminal k, where : 

/it = - /**. 
Z,i = Z,i, 

it is £,t = Zit fit- 

If /^ denotes the current passing from terminal (' to a 
ground or neutral point, and Z^^ is the impedance of this 
circuit between terminal i and neutral point, it is : 
£io = t'E — ZJi = Zi,Iu' 

282. We have thus, by Ohm's law and Kirchhoff's law : 

I£ t.' E is the E.M.F. per ci'rcuit of the generator, be- 
tween the terminal ( and the neutral point of the generator, 
or the star E.M.F. 

li = the current issuing from the terminal i of the gen- 
erator, or the star current. 

Z,- = the impedance of the line connected to a terminal 
i of the generator, including generator impedance. 

£,■ = the E.M.F. at the end of line connected to a ter- 
minal I of the generator. 

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En = the difference of potential between the ends of 
the lines i and k. 

/,i = the current passing from line i to line k. 

Zg, = the impedance of the circuit between lines t and k. 

/^> /«•■•■■ ™ the current passing from line i to neu- 
tral paints 0, 00 

Z/,, Z^ . . . . = the impedance of the circuits between 
line / and neutral points 0, 00, .... 

It is then : 

1.) Eit= - Eh^ /<*=-/"*(. Z,* = Zn, fu=~/M, 

Zio = 2,i, etc 
2.) Ei =%'E-Zili. 

3.) Ei = Z(J/„ = Z(^ v;„ = . . . . 

4.) £« = £/-£, =-(^ -€*)£- {Zilt~ Zi/f). 
5.) ^** = Zi,/,i.' 

6.) /, ~^I». 

7,) If the neutral point of the generator does not exist, 
as in ring connection, or is insulated from the other neutral 
points : 

^/i« = 0, etc. 
1 ■ 

Where 0, 00, etc., are the different neutral points which 
are insulated from each other. 

If the neutral point of the generator and all the other 
neutral points are grounded or connected with each other, 
it is: 

^/,-5|: (/..+/.»+■•■•) 



If the neutral point of the generator and all other neu- 
tral points are grounded, the system is called a grounded 
system. If the neutral points are not grounded, the sys- 
tem is an insulated polyphase system, and an insulated 
polyphase system with equalizing return, if all the neutral 
points are connected with each other, 

8.) The power of the polyphase system is — 

■^= "^ji t' Eli cos ^i at the generator 

In cos <^* in the receiving circuits. 

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283. In transforming a polyphase system into another 
polyphase system, it is obvious that the primary system 
must have the same flow of power as the secondary system, 
neglecting losses in transformation, and that consequently 
a balanced system will be transformed again in a balanced 
system, and an unbalanced system into an unbalanced sys- 
tem of the same balance factor, since the transformer is an 
apparatus not able to store energy, and thereby to change 
the nature of the flow of power. The energy stored as 
magnetism, amounts in a well-designed transformer only to 
a very small percentage of the total energy. This shows 
the futility of producing symmetrical balanced polyphase 
systems by transformation from the unbalanced single-phase 
system without additional apparatus able to store energy 
efficiently, as revolving machinery. 

Since any E.M.F. can be resolved into, or produced by, 
two components of given directions, the E.M.Fs. of any 
polyphase system can be resolved into components or pro- 
duced from components o£ two given directions. This en- 
ables the transformation of any polyphase system into any 
other polyphase system of the same balance factor by two 
transformers only, 

284. Let £",, E^, £■,.... be the EM.Fs. of the 
primary system which shall be transformed into — 

Ei, £",', £,'.... the E.M.FS. of the secondary 

Choosing two magnetic fluxes, ^ and 0, of different 

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phases, as magnetic circuits of the two transformers, which 
induce the KM.Fs., ?and ", per turn, by the law of paraU 
lelogram the E.M.Fs., E^, E^, .... can be dissolved into 
two components, E^ and E^ , E^ and E^, . . . . of the phases, 
?and 7. 
Then, — 

^1, Eti . ■ ■ ■ are the counter E.M.Fs. which have to be 
induced in the primary circuits of the first transfonner %, 
E^, E^ . . . . the counter KM.F.'s which have to be in- 
duced in the primary circuits of the second transformer, 

^1 /?, £t je . . . . are the numbers of turns of the primary 

coils of the first transformer. 


E\fT E^fT ■ . . . are the number of turns of the primary coils 
in the second transformer. 

In the same manner as the E.M.Fs. of the primary 
system have been resolved into components in phase with 
e and ~. the E.M.Fs. of the secondary system, E^, E^, .... 
are produced from components, E^ and E^, E^ and .^r 
.... in phase with e and e, and give as numbers of second 
ary turns, — 

^i' / e, Ei fe, .... in the first transformer ; 

£i je, Ei 17, ... i in the second transformer. 

That means each of the two transformers w and m con. 
tains in general primary turns of each of the primary 
phases, and secondary turns of each of the secondary 
phases. Loading now the secondary polyphase system ia 
any desired manner, corresponding to the secondary cur- 
rents, primary currents will flow in such a manner that the 
total flow of power in the primary polyphase system is the 

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same as the total flow of power in the secondary system, 
plus the loss of power in the transformers. 

285. As an instance may be considered the transforma- 
tion of the symmetrical balanced three-phase system 

£ sin (8, M sin 08 — 120), £ sin 03 — 240), 
in an unsymmetrical balanced quarter-phase system : 
E' sin A -*" sin (0 — 90). 
Let the magnetic flux of the two transformers be 

^ cos jS and i^ cos (j3 — 90). 
Then the E.M.Fs. induced per turn in the transformers 
^"^ ^sin/3 and ^ sin (^ - 90) ; 

hence, in the primary circuit the first phase, E sin A will 
give, in the first transformer, Eje primary turns; in the 
second transformer, primary turns. 

The second phase, E sin (^ — 120), will give, in the 
first transformer, —Ejle primary turns; in the second 

r X Vt 


The third phase, £"sin (jS - 240), will give, in the first 

transformer, — Ejle primary turns; in the second trans- 

, - £ X V3 . 

former, primary turns. 

In the secondary circuit the first phase E' sin j3 will give 
in the first transformer: E' j e secondary turns; in the 
second transformer : secondary turns. 

The second phase : E' sin (^ — 90) will give in the first 
transformer : secondary turns; in the second transformer, 
E' j e secondary turns. , 

Or, if r 

E = 5,000 E' = 100, e = 10. 

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IK. ftL OIL IM. Sd. Huh. 

first transformer + 600 - 260 - 250 10 

second transformer 0-1-433—433 10 turns. 

That means ; 

Any balanced polyphase system ►jw be transformed by two 
transformers only, without storage of energy, into any other 
balanced polyphase system. 

286. Some of the more common methods of transfor- 
mation between polyphase systems are : 

fig. ISO. 

1. The delta -I' connection of transformers between 
three-phase systems, shown in Fig. 199. One side of the 
transformers is connected in delta, the other in K This 
arrangement becomes necessary for feeding four wires 

three-phase secondary distributions. The V connection of 
the secondary allows to bring out a neutral wire, while the 
delta connection of the primary maintains the balance be- 
tween the phases at unequal distribution of load. 

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2. The L connection of transformers between three-phase 
systems, consisting in using two sides of the triangle only, 
as shown in Fig, 200. This arrangement has the disadvan- 
tage of transforming one phase by two transformers in 
series, hence is less efficient, and is liable to unbalance the 
system by the internal impedance of the transformers. 

Flq. 201. 

S. The main and teaser, or T connection of trans- 
formers between three-phase systems, as shown in Fig. 201. 

voltage of the other (the altitude of the equilateral triangle), 
and connected with one of its ends to the center of the 



other transformer. From the point ^ inside of the teaser 
transformer, a neutral wire can be brought out in this con- 

4. The monocyclic connection, transforming between 
three-phase and inverted three-phase or polyphase mono- 
cycle, by two transformers, the secondary of one being 
reversed regarding its primary, as shown in Fig. 202, 

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6, The L connection for transformation between quar- 
ter-phase and three-phase as described in the instance, para- 
graph 257. 

6. The T connection of transformation between quarter- 
phase and three-phase, as shown in Fig. 203. The quar- 
ter-phase side of the transformers contains two equal and 

OOflOoJ UfiOQ. 

independent (or interlinked) coils, the three-phase side two 

7. The double delta connection of transformation from 
three-phase to six-phase, shown in Fig. 204. Three trans- 
formers, with two secondary coils each, are used, one set of 






' — 

-H 1 


3 2' Is' ' 

secondary coils connected in delta, the other set in delta 
also, but with reversed terminals, so as to give a reversed 
E.M.F, triangle. These E.M.F.'s thus give topographically 
a six-cornered star. 

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8. The double Y connection of transformation from 
three-phase to six-phase, shown in Fig. 205. It is analo- 
gous to (7), the delta connection merely being replaced by 
the Y connection. The neutrals of the two F's may be 
connected together and to an external neutral if desired. 

9. The double T connection of transformation from 

three-phase to six-phase, shown in Fig. 206. Two trans- 
formers are used with two secondary coils which are T con- 
nected, but one with reversed terminals. This method 
allows a secondary neutral also to be brought out. 

287. Transformation with a change of the balance 
factor of the system is possible only by means of apparatus 

7 ViV uuiixJuumL 
/\ I A. msn rmn rBin (W\ 




able to store energy, since the difference of power between 
primary and secondary circuit has to be stored at the time 
when the secondary power is below the primary, and re- 
turned during the time when the primary power is below 

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the secondary. The most efficient storing device of electric 
energy is mechanical momentum in revolvii^ machinery. 
It has, however, the disadvantage of requiring attendance ; 
fairly efficient also are capacities and inductances, but, as a 
rule, have the disadvantage not to give constant potential 

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288. In electric power transmission and distribution, 
wherever the place of consumption of the electric energy 
is distant from the place of production, the conductors 
which transfer the current are a sufficiently large item to 
require consideration, when deciding which system and 
what potential is to be used. 

In general, in transmitting a given amount of power at a 
given loss over a given distance, other things being equal, 
the amount of copper required in the conductors is inversely 
proportional to the square of the potential used. Since 
the total power transmitted is proportional to the product 
of current and E.M.F., at a given power, the current will 
vary inversely proportional to the E.M.F., and therefore, 
since the loss is proportional to the product of current- 
square and resistance, to give the same loss the resistance 
must vary inversely proportional to the square of the cur- 
rent, that is, proportional to the square of the E.M.F. ; and 
:since the amount of copper is inversely proportional to the 
resistance, other things being equal, the amount of copper 
varies inversely proportional to the square of the E.M.F. 

This holds for any system. 

Therefore to compare the different systems, as two-wire 
single-phase, single-phase three-wire, three-phase and quar- 
ter-phase, equality of the potential must be assumed. 

Some systems, however, as for instance, the Edison 
three-wire system, or the inverted three-phase system, have 

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different potentials in the different circuits constituting the 
system, and thus the comparison can be made either — 

1st. On the basis of equality of the maximum potential 
difference in the system ; or 

2d. On the basis of the minimum potential difference 
in the system, or the potential difference per circuit or 
phase of the system. 

In low jwtential circuits, as secondary networks, where 
the potential is not limited by the insulation strain, but by 
the potential of the apparatus connected into the system, 
as incandescent lamps, the proper basis of comparison is 
equality of the potential per branch of the system, or per 

On the other hand, in long distance transmissions where 
the potential is not restricted by any consideration of ap- 
paratus suitable for a certain maximum potential only, but 
where the limitation of potential depends upon the problem 
of insulating the conductors against disruptive discharge, 
the proper comparison is on the basis of equality of the 
maximum difference of potential in the system ; that is, 
equal maximum dielectric strain on the insulation. 

The same consideration holds in moderate potential 
power circuits, in considering the danger to life from live 
wires entering human habitations. 

Thus the comparison of different systems of long-dis- 
tance transmission at high potential or power distribution 
for motors is to be made on the basis of equality of the 
maximum difference of potential existing in the system. 
The comparison of low potential distribution circuits for 
lighting on the basis of equality of the minimum difference 
of potential between any pair of wires connected to the 
receiving apparatus. 

289. 1st. Comparison on the basis of equality of t/te 
minimum difference of potential, in low potential lighting 
^rcuits : 

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In the single-phase alternating-current circuit, if ^ = 
E.M.F., /= current, r= resistance per line, the total power 
is = ei, the loss of power '^i^r. 

Using, however, a three-wire system, the potential be- 
tween outside wires and neutral being given = e, the 
potential between the outside wires is = 2 c, that is, the dis- 
tribution takes place at twice the potential, or only J the 
copper is needed to transmit the same power gt t^e same 
loss, if, as it is theoretically possible, the neutral wire has 
no cross-section. If therefore the neutral wire is made of 
the same cross-section with each of the outside wires, \ of 
the copper of the two-wire system is needed ; if the neutral 
wire is \ the cross-section of each of the outside wires, -ff of 
the copper is needed. Obviously, a single-phase five-wire 
system will be a system of distribution at the potential 4 e, 
and therefore require only tV of the copper of the single- 
phase system in the outside wires ; and if each of the three 
neutral wires is of i the cross-section of the outside wires, 
a'f = 10.93 per cent of the copper. 

Coming now to the three-phase system with the proten- 
tial c between the lines as delta potential, if ( = the current 
per line or Y current, the current from line to line or delta 
current = j^/ VS; and since three branches are used, the 
total power is 3 «■ i, / VS = e i, VS. Hence if the same 
power has to be transmitted by the three-phase system as 
with the single-phase system, the three-phase line current 
must be i^ ~ i / "^ where * = single-phase current, r = 
single-phase resistance per line, at equal power and loss ; 
hence if n = resistance of each of the three wires, the loss 
per wire is I'lVi = i^ri/3, and the total loss is I'ri, while in 
the single-phase system it is 2 i'r. Hence, to get the same 
loss, it must be : r, = 2 r, that is, each of the three three- 
phase lines has twice the resistance — that is, half the copi- 
per of each of the two single-phase lines ; or in other words, 
the three-phase system requires three-fourths of the copper 
of the single-phase system of the same potential. 

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Introducing, however, a fourth or neutral wire into the 
three-phase system, and connecting the lamps between the 
neutral wire and the three outside wires — that is, in Y con- 
nection — the potential between the outside wires or delta 
potential will be = f X V3, since the Y potential = c, and 
the potential of the system is raised thereby from e to 
e VB ; that is, only J as much copper is required in the out- 
side wires as before — that is \ as much copper as in the 
single-phase two-wire system. Making the neutral of the 
same cross-section as the outside wires, requires J more 
copper, or J =1 33.3 per cent of the copper of the single- 
phase system ; making the neutral of half cross-section, 
requires \ more, or i^ = 29.17 per cent of the copper of 
the single-phase system. The system, however, now is a 
four-wire system. 

The independent quarter-phase system with four wires 
is identical in efficiency to the two-wire single-phase sys- 
tem, since it is nothing but two independent single-phase 
systems in quadrature. 

The four-wire quarter-phase system can be used as two 
independent Edison three-wire systems also, deriving there- 
from the same saving by doubling the potential between 
the outside wires, and has in this case the advantage, that 
hy interlinkage, the same neutral wire can be used for both 
phases, and thus one of the neutral wires saved. 

In this case the quarter-phase system with common neu- 
tral of full cross-section requires ■{■^ = 31. 2o per cent, the 
quarter-phase system with common neutral of one-half cross- 
section requires ^^ = 28.125 per cent, of the copper of the 
two-wire single-phase system. 

In this case, however, the system is a five-wire system, 
and as such far inferior to the five-wire single-phase system. 

Coming now to the quarter-phase system with common 
return and potential £ per branch, denoting the current in 
the outside wires by (j, the current in the central wire is 
(3 V2 ; and if the same current density is chosen for all 

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three wires, as the condition of maximum efficiency, and 
the resistance of each outside wire denoted by r^, the re- 
sistance of the central wire = r^j^/^, and the loss of power 
per outside wire is i^r^, in the central wire li^r^jW^ 
= i^ rj V2 ; hence the total loss of power is 2 i^ r^ + i^ «, 
V2 = i^ r^ (2 4- Vl), The power transmitted per branch 
is i^e, hence the total power 2/jf, To transmit the same 
power as by a single-phase system of power, ei, it must 

(3 y /2 4- "v'21 
be i^^ t/2; hence the loss, — ^ a • Since this 

loss shall be the same as the loss 2 I'r in the single- 
phase system, it must be 2 r = ^^- . — '- r,, or rg = . 

^ ^ • 4 " ' 2-1- V2 

as large as each single-phase wire, the central wire V2 
times larger ; hence the copper required for the quarter- 
phase system with common return bears to the copper 
required for the single-phase system the relation : 
2(2 + V5) (2+V5)V2 3-H2v^ ^ 

8^8 ' ' 8 

per cent of the copper of the single-phase system. 

Hence the quarter-phase system with common return 
saves 2 per cent more copper than the three-phase system, 
but is inferior to the single-phase three-wire system. 

The inverted three-phase system, consisting of two 
E.M.Fs. e at 60° displacement, and three equal currents 
I'j in the three lines of equal resistance rj, gives the out- 
put 2etg, that is, compared with the single-phase system, 
ij = // 2. The loss in the three lines is 3 i^ 'i = i »' 'a- 
Hence, to give the same loss 2i* ras the single-phase sys- 
tem, it must be fg = 5 r, that is, each of the three wires 
must have | of the copper cross-section of the wire in the 
two-wire single-phase system ; or in other words, the in- 
verted three-phase system requires ^^ of the copper of the 
two-wire single-phase system. 

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We get thus the result, 

If a given power has to be transmitted at a given loss, 
and a given minimum potential, as for instance 110 volts 
for lighting, the amount of copper necessary is : 

2 Wires : Single-phase system, 100.0 

3 Wires: Edison three-wire single-phase sys- 

tem, neutral full section, 37.5 
Edison three-wire single-phase sys- 
tem, neutral half-section, 81.25 
Inverted three-phase system, 56.25 
Quarter-phase system with common 

return, 72.9 

Three-phase system, 75.0 

4 Wires : Three-phase, with neutral wire full 

section, 33.3 

Three-phase, with neutral wire halt- 
section, 29.17 
Independent quarter-phase system, 100.0 

5 Wires: Edison five-wire, single-phase system, 

full neutral, 15.625 

Edison five-wire, single-phase system, 

half-neutral, 10.93 

Four-wire, quarter- phase, with com- 
mon neutral full section, 31.25 
Four-wire, quarter-phase, with com- 
mon neutral half-section, 28.125 

We see herefrom, that in distribution for lighting — that 
is, with the same minimum potential, and with the same 
number of wires — the single-phase system is superior to 
any polyphase system. 

The continuous-current system is equivalent in this 
comparison to the single-phase alternating-current system 
of the same effective potential, since the comparison is 
made on the basis of effective potential, and the power 
depends upon the effective potential also. 

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290. Comparison on tke Basis of Equality of the Maximum 
Difference of Potential in the System, in Long-Distance 
Transmission, Power Distribution, etc. 

Wherever the potential is so high as to bring the ques- 
tion of the strain on the insulation into consideration, or in 
other cases, to approach the danger limit to life, the proper 
comparison of different systems is on the basis of equality 
of maximum potential in the system. 

Hence in this case, since the maximum potential is 
fixed, nothing is gained by three- or five-wire Edison sys- 
tems. Thus, such systems do not come into consideration. 

The comparison of the three-phase system with the 
single-phase system remains the same, since the three- 
phase system has the same maximum as minimum poten- 
tial ; that is : 

The three-phase system requires three-fourths of the 
copper of the single-phase system to transmit the same 
power at the same loss over the same distance. 

The four-wire quarter-phase system requires the same 
amount of copper as the single-phase system, since it con- 
sists of two singte-phase systems. 

In a quarter-phase system with common return, the 
potential between the outside wire is "^2 times the poten- 
tial per branch, hence to get the same maximum strain on 
the insulation — that is, the same potential e between the 
outside wires as in the single-phase sy.stem — the potential 
per branch will be c j V2, hence the current Z, = i/ V2, if /' 
equals the current of the single-phase system of equal 
power, and i^ V2 = ( will be the current in the central 

Hence, if r^ = resistance per outside wire, r^ / V2 =« 
resistance of central wire, and the total loss in the sys- 
tem is : 

2 ,^... + i!^ _ ,-.V. (2 + V5) - ,-., i^±^ . 


Digitized .yGOOgle 


Since in the single-phase system, the loss = 2 1 ' r, it is : 

* ~ 2 + V2 ■ 

That is, each of the outside wires has to contain — i 

times as much copper as each of the single-phase wires. 

The centra] wires have to contain — V2 times as 

much copper ; hence the total system contains — — ^ 

2 +V2 - . 

H 7 v'2 times as much copper as each of the single- 

single-phase system. 
Or, in other words, 
A quarter-phase system with common return requires 

-^ = 1.457 times as much copper as a single-phase 

system of the same maximum potential, same power, and 
same loss. 

Since the comparison is made on the basis of equal 
maximum potential, and the maximum potential of alter- 
nating system is V2 times that of a continuous-current 
circuit of equal effective potential, the alternating circuit 
of effective potential e compares with the continuous-cur- 
rent circuit of potential e V2, which latter requires only 
half the copper of the alternating system. 

This comparison of the alternating with the continuous- 
current system is not proper however, since the continuous- 
current potential introduces, besides the electrostatic strain, 
an electrolytic strain on the dielectric which does not exist 
in the alternating system, and thus makes the action of the 
continuous-current potential on the insulation more severe 
than that of an equal alternating potential. Besides, self- 
induction having no effect on a steady current, continuous 
current circuits as a rule have a self-induction far in excess 

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of any alternating circuit. During changes of current, as 
make and break, and changes of load, especially rapid 
changes, there are consequently induced in these circuits 
E.M.F.'s far exceeding their normal potentials. At the 
voltages which came under consideration, the continuous 
current is excluded to begin with. 

Thus we get : 

If a given power is to be transmitted at a given loss, 
and a given maximum difference of potential in the system, 
that is, with the same strain on the insulation, the amount 
of copper required is : 

2 Wires : Single-phase system, 100.0 

[Continue us -current system, 50.0] 

3 Wires : Three-phase system, 76.0 

Quarter-phase system, with common return, 145.7 

4 Wires : Independent Quarter-phase system, 100.0 

Hence the quarter-phase system with common return is 
practically excluded from long-distance transmission. 

291 ■ In a different way the same comparative results 
between single-phase, three-phase, and quarter-phase sys- 
tems can be derived by resolving the systems into their 
single-phase branches. 

The three-phase system of E.M.F. e between the lines 
can be considered as consisting of three single-phase cir- 
cuits of E.M.F. el^fZ, and no return. The single-phase 
system of E.M.F. e between lines as consisting of two 
single-phase circuits of E.M.F. f/2 and no return. Thus, 
the relative amount of copper in the two systems being 
inversely proportional to the square of E.M.F., bears the 
relation { VS j ef\{^l ef = Z -A; that is, the three-phase 
system requires 75 per cent of the copper of the single- 
phase system. 

The quarter-phase system with four equal wires requires 
the same copper as the single-phase system, since it consists 

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of two single-phase circuits. Replacing two of the four 
quarter-phase wires by one wire of the same cross-section 
as each of the wires replaced thereby, the current in this 
wire is V2 times as large as in the other wires, hence, the 
loss twice as large — that is, the same as in the two wires 
replaced by this common wire, or the total loss is not 
changed — while 25 per cent of the copper is saved, and 
the system requires only 75 per cent of the copper of the 
single-phase system, hut produces V5 times as high a 
potential between the outside wires. Hence, to give the 
same maximum potential, the E,M.Fs. of the system have 
to be reduced by V2, that is, the amount of coppw doubled, 
and thus the quarter-phase system with common return of 
the same cross-section as the outside wires requires 150 
per cent of the copper of the single-phase system. In this 
case, however, the current density in the middle wire is. 
higher, thus the copper not used most economical, and 
transferring a part of the copper from the outside wire to 
the middle wire, to bring all three wires to the same current 
density, reduces the loss, and thereby reduces the amount 
of copper at a given loss, to 145.7 per cent of that of a 
single-phase system. 

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292. A\^th equal load of the same phase displacement 
in all three branches, the symmetrical three-phase system 
offers no special features over those of three equally loaded 
single-phase systems, and can be treated as such ; since the 
mutual reactions between the three phases balance at equal 
distribution of load, that is, since each phase is acted upon 
by the preceding phase in an equal but opposite manner 
.as by the following phase. 

With unequal distribution of load between the different 
branches, the voltages and phase differences become more or 
less unequal. These unbalancing effects are obviously maxi- 
mum, if some of the phases are fully loaded, others unloaded, 


E 1- E.M.F, between branches 1 and 2 of a three-phaser. 

c £= KM.F. between 2 and 3, 
€*i'=E.M.F. between 3 and 1, 

wTiere, ,^-^= -^+/-^ . 


Zi, Z,, Zx ~ impedances of the lines issuing from genera- 
tor terminals 1, 2, 3, 
and Y^, Yt, Y, = admittances of the consumer circuits con- 
nected between lines 2 and 3, 3 and 1, 1 and 2. 
If then, 

A) A> A, are the currents issuing from the generator termi- 
nals into the lines, it is, 

/. + /.+ /, = 0. (1) 

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If //, /|', Ii= currents flowing through the admittances Y^ 
Yt, y„ from 2 to 3, 3 to 1, 1 to 2, it is, 

I^~It'~I^, or, /, + /,'-// = O' 
'lt~Ii-It', or, >, + >,'-/,'=. 
>,=-■/,'->,', or, It + /i'-/s 

These three equations (2) added, give (1) as dependent 

At the ends of the lines 1, 2, 8, it is : 

Ei = >, - Z, >, + Z, >, \ (3) 

i,' = i', ~Zi>, + Z,>J 

the differences of potential, and 


the currents in the receiver circuits. 

These nine equations (2), (S), (4), determine the nine 
quantities: l^, /j, /g, /j', /j', /,', ^i', E^, E^. 

Equations (4) substituted in (2) give : 

>, = 'e{ Y^ - £,' y, 1- (5) 


These equations (5) substituted in (3), and transposed, 

since E^^tE \ 

Ei = ^E\as E.M.FS. at the generator terminals. 
Et = E ] 

t£-~£i'(l + Y, Z, + Vj Z.) + £/ r, Z, + E,' K, Z, = O] 

^£ - 'e4 (1 +YjZt + J', Z,) + E,' Y^Zy + >/ r, Z. = 1 (6) 

£■ - ^,' (1 + r,z, + r,Zj) + -ki' I'lZ, + ^/y, z, = o] 

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as three linear equations with the three quantities £",', 

¥i. El. 

Substituting tiie abbreviations : 

/' - (1 + y;z, + KiZ,), YiZt, YxZ, i 
yA, -ii + y-z, + y,z,), y,Zi I 
r,z„ y,z„ - (1 + y,z, + y,z,) / 

1 1, YtZt, y^Zt I 

i>, _/,', - (1 + y,z, + y,z,), y,z, / 
/i, y,z„ -(i + y,z, + y,z,) / 

/- (1 + y,z, + y.z,), ,, y,z, i 
y,z„ <", y,z, / 

y,z,; 1, - (1 + y,z, + y,z,) / 

/- (1 + y,z, + y,z,), y,z„ < / 
J'A, - (I + y^z, + y,z,), <•/ 
Kz„ y,z„ 1 / 



e; = ed,i 


e; = ed.i 










_ "i-^A 














+ -«.' + ■?. 



+ 7, +7. 






293, Special Cases. 

A. Balanced System 

Y,= y,= y^^ y 

Zi-Z, — Z, — Z- 
Substituting tliis in (6), and transposing : 



i + ayz 


i + 3yz 

i + syz 
_ f(.-i)jsy 

1 + 3YZ 


Tlie equations of tlie symmetrical balanced three-phase 

B. One circuit loaded, two unloaded; 

y, -r, -0, Y, — y 

Z, = Zt = Z, = z. 

Substituted in equations (6) : 
. B-El + E^yz=« j 

fk-ki+'Ejyz = f> \ 


I unloaded branches. 
: 0, loaded branch. 

.g(.+ (l + 2.)KZ} 

l-)-2F2 I , , . 

.S(^+Tl + 2.-)m ■"■'^"'^ 

1 + 2 yz 

loaded ; 

1 + 2KZ 

all three 
I E-M.F.'s 
w unequal, and (13) 

of unequal 
I phase angles. 

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i; _ f^ \ (13) 


C. Two circuits loaded, one unloaded, 
y, = r, = >^ Yt = % 

Substituting this in equations (6), it is : 

E-jS^jf. (E^ + E^) VZ ^0 unloaded branch, 
or, since : 

E-E/ - £t'yz = o, 
E,' ^- 

1 +yz 


^,_ £,{l + (2+.)yZ} ] -1 

■' l + iYZ + 3V'Z* , ^ j^ ^ 
JF 1/1 J- /9 J- 4\ V7i f loaded branches. 

■' i + ir^ + ar^Z" J I 

£.' = ■ unloaded branch. ' 

■ • 1 + yz 

As seen, with unsymmetrical distribution of load, all 
three branches become more or less unequal, and the phase 
displacement between them unequal also. 

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294. In a three-wire quarter-phase system, or quarter- 
phase system with common return wire of both phases, let 
the two outside terminals and wires be denoted by 1 and 2, 
the middle wire or common return by fl- 
it is then : 

^1 = £ = E.M.F. between and 1 in the generator. 
Et =JE = E.M.F. between and 2 in the generator. 

/, and /, = currents in 1 and in 2, 

/^ = current in 0, 

Zi and Zj = impedances of lines 1 and 2, 

Zt = impedance of line 0. 

Ki and Kj = admittances of circuits to 1, and to 2, 

// and //= currents in circuits to 1, and to 2, 

^/and£i'= potential differences at circuit to 1, and 
to 2. 

it is then, /: + /. + /« = 1 

or, >»=-(/,"+/,) 1 

that is, /o is common return of /] and ^. 

Further, we have, ■ 


= E- 

-/,Z, + /,Z,- 


-/.(2, + 2.)- 




->,Z. + /.Z,-> 


-/,(Z, + Z^- 



y, = j^ Ex 

>,_ Y,'Ei 









Substituting (3) in (2) ; and expanding : 

l+V,Z, + y,Z,(l-j) 

■ (i + y,z,+ y,z,)(i + y,z,+y,z,)- 
.jE i + y,z, + y,z,ii+j ) 



' (1 + >',z,+ KiZiXi + r,z, + 1-,^,) - K.iiiv 

Hence, the two E,M.Fs. at tlie end of the line are un- 
equal in magnitude, and not in quadrature any more. 

2?5. Special Cases : 

A. Balanced System. 


y^-y,^ y. 
Sobstttuting these values in (4), gives : 
1 + V2 

1 + - 


!— M 

1 + V2 (1 + V2) FZ + (1 + V2) y<Z' 
y 1 + (1.707 - .707/) rz 
■ l + 3AUyZ + 2Aliyz' 

S,' ^jE 

1 + V2 (1 + V2) rz + (1 + V2) VZ' 



1 + 3.414 yz + 2.414 y»z' 

Hence, the balanced quarter-phase system with common 
return is unbalanced with regard to voltage and phase rela- 
tion, or in other words, even if in a quarter-phase system with 
common return both branches or phases are loaded equally, 
with a load of the same phase displacement, nevertheless 
the system becomes unbalanced, and the two £.M.Fs. at 
the end of the line are neither equal in magnitude, nor in 
quadrature with each other. 



B. One branch loaded, one unloaded. 

Z, = Z, — Z; 

Z, = ZW2. 

o.) y, — i>, Y,~ V. 

i.) K, = y, Yt = 0. 

Substituting these values in (4), gives : 


1 + rz- 

i + yz 

1 + V2 

1 + V2 + 




1 + KZ1±V2 

■ 1 + 1.707 yz 

i + rz' + ^ 


-E I 

■ 1 + 1.707 yz 

i + yz '^ + ^+Z 

E ^1— 

l + yzl±J^ 


1+ i— 




These two E.M.Fs. are unequal, and not in quadrature 
with each other. 

But the values in case a.) are different from the values 
in case b.). 

That means : 

The two phases of a three-wire quarter-phase system 
are unsymmetrical, and the leading phase 1 reacts upon 
the lagging phase 2 in a different manner than 2 reacts 
upon 1. 

It is thus undesirable to use a three-wire quarter-phase 
system, except in cases where the line impedances Z are 

In all other cases, the four-wire quarter-phase system 
is preferable, which essentially consists of two independent 
single-phase circuits, and is treated as such. 

Obviously, even in such an independent quarter-phase 
system, at unequal distribution of load, unbalancing effects 
may take place. 

If one of the branches or phases is loaded differently 
from the other, the drop of voltage and the shift of the 
phase will be different from that in the other branch ; and 
thus the E.M.Fs. at the end of the lines will be neither 
equal in magnitude, nor in quadrature with each other. 

With both branches however loaded equally, the system 
remains balanced in voltage and phase, just like the three- 
phase system under the same conditions.' 

Thus the four-wire quarter-phase system and the three- 
phase system are balanced with r^ard to voltage and phase 
at equal distribution of load, but are liable to become un- 
balanced at unequal distribution of load ; the three-wire 
quarter-phase system is unbalanced in voltage and phase, 
even at equal distribution of load. 

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296. The system of numbers, of which the science 
of algebra treats, finds its ultimate origin in experience. 
Directly derived from experience, however, are only the 
absolute integral numbers ; fractions, for instance, are not 
directly derived from experience, but are abstractions ex- 
pressing relations between different classes of quantities. 
Thus, for instance, if a quantity is divided in two parts, 
from one quantity two quantities are derived, and denoting 
these latter as halves expresses a relation, namely, that two 
of the new kinds of quantities are derived from, or can be 
combined to one of the old quantities. 

297. Directly derived from experience is the operation 
of counting or of numeration. 

a, a + 1, a + 2, a + 3 . . . . 
Counting by a given number of integers : 

l> integers 
introduces the operation of addition, as multiple counting : 

a + i-c. 
It is, 3 + i = « + «, 


490 APPENDIX r. 

that is, the terms of addition, or addenda, are interchange- 

Multiple addition of the same terms : 

g + g + a+. ■ . + a »c 
b equal numbers 
introduces the operation of multiplication: 

axb = £. 
It is, a'x.b = by.a, 

that is, the tenns of multiplication, or factors, are inter- 

Multiple multiplication of the same factors : 
aX<iXflX...Xfl = f 

b equal numbers 
introduces the operation of involution : 
tf» = /- 
Since o* is not equal to ^, 

the terms of involution are not interchangeablie. 

298. The reverse operation of addition introduces the 
operation of subtraction : 

If a+b'-c, 

it is c~ b ~ a. 

This operation cannot be carried out in the system of 
absolute numbers, if : 


Thus, to make it possible to carry out the operation of 
subtraction under any circumstances, the system of abso- 
lute numbers has to be expanded by the introduction of 
the negative number: 

where (—1) 

is the negative unit. 

Thereby the system of numbers is subdivided in the 

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positive and ne^tive numbers, and the operation of sub- 
traction possible for all values of subtrahend and minuend. 
From the definition of addition as multiple numeration, and 
subtraction as its inverse operation, it follows : 

thus: (-l)x (-1) = !; 

that is, the negative unit is defined by, (— Vf = 1. 

299. The reverse operation of multiplication introduces 
the operation of division : 

If a% b = c, then t = «■ 

In the system of integral numbers this operation can 
only be carried out, if ^ is a factor of c. 

To make it possible to carry out the operation of division 
under any circumstances, the system of integral numbers 
has to be expanded by the introduction of 'i^^ fraction : 

c n\ 

300, The reverse operation of involution introduces two 
new operations, since in the involution : 

the quantities a and b are not reversible. 

Thus Vc = a, the evolution, 

logaf = b, the logaritkmatum. 

The operation of evolution of terms c, which are not 
complete powers, makes a further expansion of the system 



of numbers necessary, by the introduction of the irrational 
number (endless decimal fraction), as for instance ; 
-v^ = 1.414213. 

301. The operation of evolution of negative quantities 
c with even exponents i, as for instance 

makes a further expansion of the system of numbers neces- 
sary, by the introduction of the imaginary unit 

Thus A/— a = -v^— \ X -v^- 

where : V— 1 is denoted byy. 

Thus, the imaginary unity is defined by : 

y» = - 1. 

By addition and subtraction of real and imaginary units, 
compound numbers are derived of the form : 

a +Jb, 
which are denoted as complex imaginary numbers. 

No further system of numbers is introduced by the 
operation of evolution. 

The operation of logarithmation introduces the irrational 
and imaginary and complex imaginary numbers also, but 
no further system of numbers. 

302. Thus, starting from the absolute integral num- 
bers of experience, by the two conditions : 

1st. Possibility of carrying out the algebraic operations 
and their reverse operations under all conditions, 

2d. Permanence of the laws of calculation, 
the expansion of the system of numbers has become neces- 
sary, into 

Positive and negative numbers. 

Integral numbers and fractions, 

Rational and irrational numbers, 

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Real and imaginary numbers and complex imagmary 

Therewith closes the field of algebra, and all the alge- 
braic operations and their reverse operations can be carried 
out irrespective of the values of terms entering the opera- 

Thus within the range of algebra no further extension 
of the system of numbers is necessary or possible, and the 
most general number is 

a -^jb. 

where a and b can be integers or fractions, positive oc 
negative, rational or irrational. 

Algebraic Operations with Complex Imaginary 

303. Definition of imaginary unit: 

J' - - 1- 

Complex imaginary number: 

Substituting : 

A = 

= r COS ^ 



A = 

b = 

-rsin ft 



' = 

V«' + « 




r = vector 

^= amplit 

jde of complex imaginary 

number A. 

Substituting : 


/s = 

," + ,-'t 




€>P — €-'fl 



it is ^ = rtJf, 

where . = lim fl + -l"='!^ '^ 

V ^ nl o-lx2x3x . . .-Kk 

is the basis of the natural logarithms. 
Conjugate numbers : 

a+ji = r {cos P + / sin ff) = re^'fl 
and a -jb = r(cos [- y3] +jsm [- ^]) = rt-/P 
it is (a +y*) {a ~jb) = a" + ^ = r» 

Associate numbers : 
ii+yi = r(cos^+/sin/3) = r«/« 
and b ^ja - r (cos [| - ^] +ysm |j - ^]) = ^<^'f5~'> ; 

it is (<»+/*) (*+/«) =/■("" + ^ =>'-». 

If a+jb = a! +jy, 

it is fl = y 

d = ^. 

If a+yi = Oj 

it is fl = 0, 

304. Addition and Subtraction : 

(a +j6) ± <y +yi') = (fl ± y) -|-y(i J^ ^. 

Multiplication : 

(a +jb) (</ +jir) = {ad - bl/) +J{al^+ bcT) 
or r(cos0+/sin/3) x r' (cos j9' +/ sin ^ = rr' (cos |j3-i- 

^] +/3in Ci8 + ,9^) ; 
or reJ» X r'.fJP' = rr'(r/»+«. 

Division : 

Ejqiansion of complex imaginary fraction, for rationaliza- 
tion of denominator or numerator, by multiplication with 
the conjugate quantity : 

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,.+/« (a-yjf)(,J -JIT) (aa'+<y) +/(<!»'- ay ) 

_ (.+/<)(»-/<) »■+<• 

(^ +/iO (« ->i) (»«■ + *«-)+/(««-- *y) • 

involution : 

= r"(cosMj8+ysin«j8) = r-^/-^. 
evolution : 

^/a+ji = -C^ (cos P -\-J sin ^) = -^reJB. 

305. Roots of tfie Unit : 
^/1=+1, -1; 

-i^^i 1 -i+yV3 -l-yV3 

^ ' 2 2 ' 

<^=+i, -1, +/; -y; 

-Ji = +i, -1, +y, 

+i-y -i+y 

-4i ' V2 ' 

306. Rotation: 
In the complex imaginary plane, 
multiplication with 

Vl=.cos^+y3in^ = /-^' 

means rotation, in positive direction, by l/« of a revolution. 

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multiplication with (— 1) means reversal, or rotation by 180°, 
multiplication with (+y) means positive rotation by 90°, 
multiplication with (— » means negative rotation by 90°, 

307. Complex imaginary platu : 

While the positive and negative numbers can be rep- 
resented by the points of a line, the complex imaginary 
numbers are represented by the points of a plane, with the 
horizontal axis A' O A as real axis, the vertical axis ROB 
as imaginary axis. Thus all 

the po^tive real numbers are represented by the points of half 

axis OA towards the right ; 
the negative real numbers are represented by the points of half 

axis OA' towards the left \ 
the positive imaginary numbers are represented by the points of 

half axis OB upwards ; 
the negative imaginary numbers are represented by the points of 

half axis OB downwards ; 
the complex imaginary numbers are represented by the points 

outside of the coordinate axes. 

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308. An electric current varying periodically between 
constant maximum and minimum values, — that is, in equal 
time intervals repeating the same values, — is called an 
alternating current if the arithmetic mean value equals 
zero ; and is called a pulsating current if the arithmetic 
mean value differs from zero. 

Assuming the wave as a sine curve, or replacing it by 
the equivalent sine wave, the alternating current is charac- 
terized by the period or the time of one complete cyclic 
change, and the amplitude or the maximum value of the 
current. Period and amplitude are constant in the alter- 
nating current. 

A very important class are the currents of constant 
period, but geometrically varying amplitude ; that is, cur- 
rents in which the amplitude of each following wave bears 
to that of the preceding wave a constant ratio. Such 
currents consist of a series of waves of constant length, 
decreasing in amplitude, that is in strength, in constant 
proportion. They are called oscillating currents in analogy 
with mechanical oscillations, — for instance of the pendu- 
lum, — in which the amplitude of the vibration decreases 
in constant proportion. 

Since the amplitude of the oscillating current varies, 
constantly decreasing, the oscillating current differs from 

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the alternating current in so far that it starts at a definite 
time, and gradually dies out, reaching zero value theoreti- 
cally at infinite time, practically in a very short time, short 
even in comparison with the time of one alternating half- 
wave. Characteristic constants of the oscillating current 
are the period T or frequency iV ■= 1/7", the first ampli- 
tude and the ratio of any two successive amplitudes, the 
latter being called the decrement of the wave. The oscil- 
lating current will thus be represented by the product of 

a periodic function, and a function decreasing in geometric 
proportion with the time. The latter is the exponential 
function A-^-''. 

309. Thus, the general expression of the oscillating 
current is 

/= A^-9' cos (2wJVt - S>), 
since ^y-'' = A^A^'' ■= W". 

Where c = basis of natural logarithms, the current may 
be expressed 

/=. (■«-»' cos (2,rJV/- 5) = (.—♦ cos (* - S), 

where ^ = 2 ■■A'/; that is, the period is represented by a 
complete revolution. 

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In the same way an oscillating electromotive force will 
be represented by 

E = et.-'^ cos («^ — a). 

Such an oscillating electromotive force for the values 

e = &, 0. = .1435 or e-*-" = .4, Z, = 0, 

is represented in rectangular coordinates in Fig. 207, and 
in polar coordinates in Fig. 208. As seen from Fig. 207, 
the oscillating wave in rectangular coordinates is tangent 
to the two exponential curves. 

ftq. 209. 

310. In polar coordinates, the oscillating wave is repre- 
sented in Fig. 208 by a spiral curve passing the zero point 
twice per period, and tangent to the exponential spiral, 

The latter is called the envelope of a system o^ oscillat- 
ing waves of which one is shown separately, with the same 
constants as Figs. 207 and 208, in Fig. 209. Its character- 

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istic feature is : The angle which any concentric circle 
makes with the curve/ = ^ «"'*, is 

tan a = -■ ■' •■■ = — a, 

which is, therefore, constant; or, in other words: "The 
envelope of the oscillating current is the exponential spiral, 
which is characterized by a constant angle of intersection 

Fig. 200. flf. 310. 

with all concentric circles or all radii vectores." The oscil- 
lating current wave is the product of the sine wave and the 
exponential or loxodromic spiral 

311. In Fig. 210 let y 
nential spiral ; 

let » = ' cos (0 - a) 

represent the sine wave ; 
and let £ = et-'* cos (* — ») 

represent the oscillating wave. 

We have then 

■♦ represent the expo- 

tan ^ = 

— sin (^ - 

a) — a cos (^ - w) 

cos (^ — <o) 

= - {tan (*-£)+«}; 

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that is, while the slope of the sine wave, z = e cos {4> ~ ^), 
is represented by 

tany tan (^ - £), 

the slope of the exponential spiral _>' — ef* is 

tan 11=— a = constant 
That of the oscillating wave E = et~'* cos (+ — S) is 

tan^=-{taii(*-a) + 4. 
Hence, it is increased over that of the alternating sine 
wave by the constant a. The ratio of the amplitudes of 
two consequent periods is 

A is called the numerical decrement of the oscillating 
wave, a the exponential decrement of the oscillating wave, 
a the angular decrement of the oscillating wave. The 
oscillating wave can be represented by the equation 

In the instance represented by Figs. 181 and 182> we 
have A = A, a = .1435, a = 8.2". 

Impedance and Admittance. 

312. In complex imaginary quantities, the alternating 

wave - _ - .„„ cj. "I 

s = ^ cos (* - «) 

is represented by the symbol 

^ = ^(cos5 +y sinS) = Ci +/>>. 
By an extension of the meaning of this symbolic ex- 
pression, the oscillating wave E=^e*r'* cos (^ — ») can 
be expressed by the symbol 

E = e (cos 5 +/' sin ") dec a = (fi -\-jei) dec a, 
where a = tan a is the exponential decrement, a the angular 
decrement, c"*"' the numerical decrement. 

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313, Let r = resistance, L — inductance, and x « 
2 r NL ■= reactance. 

In a circuit excited by the oscillating current, 

/= i*""* cos (^ — ui) = /(cos w +y sin w) dec a™ 
Oi +/'») dec a, 
where »i = »' cos i, »» ™ / sin «, « •= tan a. 

We have then, 

The electromotive force consumed by the resistance r of 
the circuit £,= r7deca. 

The electromotive force consumed by the inductance L 
of the circuit, 

dt d^ d^ 

Hence £, = — j:/«-''*{sin (^ — «) + a cos (^ — £)} 
= _ ^IIZ!^ sin (^ _ a + „). 

cos o 

Thus, in symbolic expression, 

£^ = £i_{_ sin (£ — a) +ycos (« — a)} deca 

= —xi{a +j) (cos i + y sin a) dec o ; 
that is, £, = -x/(a +» dec a. 

Hence the apparent reactance of the oscillating current 
circuit is, in symbolic expression, 


Hence it contains an energy component ax, and the 
impedance is 

^■'-(r — X)dec n=;{r — j: («+/)} dec o = {r~ax—jx) dec a. 


314. Let r = resistance, C = capacity, and Xc = l/ 2tNC 
= capacity reactance. In a circuit excited by the oscillating 

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current /, the electromotive force consumed by the capacity 
C is 

or, by substitution, 

E^ = x fit-'* cos (* — a> d4i 

« ~^ /«— ♦ {sin (^ - £) _ „ cos (* - a)} 

hence, in symbolic expression, 

_ {_ sin (a + a) +ycos (£ + a)} dec « 

(1 -h 0») COS a 

= '^' , (« +» (cos a +y sin a) dec a 

that is, the apparent capacity reactance of the oscillating 
circuit is, in symbolic expression, 

C = ^-^, (--*+» dec a. 

316. We have then : 

In an oscillating current circuit of resistance r, induc- 
tive reactance jr, and capacity reactance j:^, with an expo- 
nential decrement a, the apparent impedance, in symbolic 
expression, is : 





and, absolute. 

Vt-('+rfp)] + ['-rf^]- 

Then from the preceding discussion, the electromotive force 
consumed by resistance r, inductive reactance x, and capa- 
city reactance x^, is 

£ = />-♦ Jcos(0-£)[r- a*-^-^:r,]- sin (*- a) 

■v/('-rf7.)'+(— -rfr.")' 

substituting a + S for u, and ^ = is, we have 
^ = ^.-*cos(*-S), 
/ = —(-"♦cos (* — £ — 8) 

.-i ( cos S /. -\ ■ sin 8 ■ /, -^ ) 
"♦ } cos (^ — w) + . sin (^ — o>) t ; 

hence in complex quantities, 

E = e (cos 5 +y sin <u) dec a, 

, c-fcosS , -sinS") . 
/ =£-j H^ [-decai 

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or, substituting. 

f r-ax--^—x. 

317. Thus in complex quantities, for oscillating cur- 
rents, we have : conductance, 


^^ ^"fTP ^, 

(* — — 1 +fr — fla:— ■ '^ , x\ 

\ \-\- aV \ 1 + tf' / 

admittance, in absolute values, 

in symbolic expression, 

[r~ax ^ — xA +J (x S — ) 

y.,+y..) _l+fLi±:L4±4. 


Since the impedance is 

^- ('— "-rf^") -•'■('-rf^-)-'----''" 

we have 

J'=^; .»' = — ; i-=-f; * = -§; 

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that is, the same relations as in the complex quantities in 
alternating-current circuits, except that in the present case 
all the constants r„, x^, s^, g, t,y, depend upon the dec- 
rement a. 

Circuits of Zero Impedance. 
318. In an oscillating-current circuit of decrement a, of 
resistance r, inductive reactance x, and capacity reactance x^ 
the impedance was represented in symbolic expression by 

or numerically by 

= v;?T^=v/(r 

1 + . 

Thus the inductive reactance x, as well as the capacity 
reactance x^, do not represent wattless electromotive forces 
as in an alternating-current circuit, but introduce energy 
components of negative sign 

that means, 

" In an oscillating-current circuit, the counter electro- 
motive force of self-induction is not in quadrature behind 
the current, but lags less than 90", or a quarter period ; and 
the charging current of a condenser is less than 90°, or a 
quarter period, ahead of the impressed electromotive force." 

319. In consequence of the existence of negative en- 
ergy components of reactance in an oscillating-current cir- 
cuit, a phenomenon can exist which has no analogy in an 
alternating-current circuit ; that is, under certain conditions 
the total impedance of the oscillating-current circuit can 
equal zero : 7 _ n 

In this case we have 

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substituting in this equation 
and expanding, we have 

If in an oscillating-current circuit, the decrement 

. _ 2oZ 

That is, 

and the frequency N =^ rji-naL, the total impedance of 
the circuit is zero ; that is, the oscillating current, when 
started once, will continue without external energy being 
impressed upon the circuit." 

320. The physical meaning of this is : " If upon an 
electric circuit a certain amount of energy is impressed 
and then the circuit left to itself, the current in the circuit 
will become oscillating, and the oscillations assume the fre- 
quency N = r/4-iraL, and the decrement 

That is, the oscillating currents are the phenomena by 
which an electric circuit of disturbed equilibrium returns to 

This feature shows the origin of the oscillating currents, 
and the means to produce such currents by disturbing 
the equilibrium of the electric circuit ; for instance, by 
the discharge of a condenser, by make and break of the 
circuit, by sudden electrostatic charge, as lightning, etc. 
Obviously, the most important oscillating currents are 

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those flowing in a circuit of zero impedance, representing 
oscillating discharges of the circuit. Lightning strokes 
usually belong to this class. 

Oscillating Discharges. 
321. The condition of an oscillating discharge is 



2 Z V r» C 

If r = 0, that is, in a circuit without resistance, we have 
(1 = 0, //=l/2ir Vi C ; that is, the currents are alter- 
nating with no decrement, and the frequency is that of 

If 4i/r*C-l<0, that is, r>2\/LjC. a and N 
become imaginary ; that is, the discharge ceases to be os- 
cillatory. An electrical discharge assumes an oscillating 
nature only, if r < 2 VZ,/ C. In the case r = 2 Vi/ C we 
have >E 00 , N =^\ that is, the current dies out without 

From the foregoing we have seen that oscillating dis- 
charges, — as for instance the phenomena taking place if 
a condenser charged to a given potential is discharged 
through a given circuit, or if lightning strikes the line 
circuit, — are defined by the equation : Z = dec a. 


/ =. (/, -|-y*i) dec o, Er = Ir dec a, 

-ff, xlia +>) dec a, E„= —^£—/(^-a+j) deco, 

I + a • 

we have r—ax— " r »= 

hence, by substitution, 


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The two constants, i^ and ij, of the discharge, are deter- 
mined by the initial conditions, that is, the electromotive 
force and the current at the time / = 0. 

322. Let a condenser of capacity C be discharged 
through a circuit of resistance r and inductance L. Let 
e = electromotive force at the condenser in the moment 
of closing the circuit, that is, at the time ^ = or ^ = 0, 
A.t this moment the current is zero ; that b, 

Since -^= •*/(— <* +/) dec o = * at ^ = 0, 

we have * /, Vl + a^ = ^ or /» = — ■ ■ ■ • 

Substituting this, we have, 

I = j ' dec a, Er=je *" deco, 

* Vl + d^ a: Vl + a' 

E^ = — (1 - ia) dec a. £„=» ' a+/a^deca. 

Vi + ^» " vrqr? 

the equations of the oscillating discharge of a condense 
of initial voltage e. 
Since * = 2xiVZ, 


2 ^ 2 V r' C 
hence, by substitution, 

i —J'Sl'T deco, Er—jersJ — dec a, 


APPEifDix n. 



N= ■ 




the final equations of the oscillating discharge, in symbolic 

Oscillating Current Transformer. 
323. As an instance of the application of the symbolic 
method of analyzing the phenomena caused by oscillating 
currents, the transformation of such currents may be inves- 
tigated. If an oscillating current is produced in a circuit 
including the primary of a transformer, oscillating currents 
will also flow in the secondary of this transformer. In a 
transformer let the ratio of secondary to primary turns be/. 
Let the secondary be closed by a circuit of total resistance, 
^1= ^1'+ r-l' , where r^= external, r-^' = internal, resistance. 
The total inductance Z-j = £;' ■+■ L^", where Lj = external, 
i," = internal, inductance; total capacity, Cj. Then the 
total admittance of the secondary circuit is 

Vi = (gi +Jh} dec a = L , 

wherejr,= 2irNLi= inductive reactance: x^j = lj1rNC=» 
capacity reactance. Let r^ = effecivc hysteretic resistance, 
Z(, = inductance ; hence, jr, = 2 «■ N L^ = reactance ; hence, 

>» = go -^Jh = :; ^ — = admittance 

of the primary exciting circuit of the transformer ; that is, 
the admittance of the primary circuit at open secondary 

As discussed elsewhere, a transformer can be considered 
as consisting of the secondary circuit supplied by the im- 
presses! electromotive force over leads, whose impedance is 

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equal to the sum of primary and secondary transformer im- 
pedance, and which are shunted by the exciting circuit, out- 
side of the secondary, but inside of the primary impedance. 
Let r = resistance ; L = inductance ; C = capacity ; 

' X = 2w N L = inductive reactance, 

x^ = 1 /2itJ^C = capacity reactance of the total primary 
circuit, including the primary coil of the transformer. If 
£{ = E{ dec a denotes the electromotive force induced in 
the secondary of the transformer by the mutual magnetic 
flux ; that is, by the oscillating magnetism interlinked 
with the primary and secondary coil, we have I^!=E^Yy 
dec a = secondary current. 

Hence, I( = pli dec o ^pE^Y^ dec a = primary load 
current, or component of primary current corresponding to 
secondary current. Also, /q = - E^ Y^ dec a •= primary 
exciting current ; hence, the total primary current is 

£■' = -J- dec a = induced primary electromotive force. 
Hence the total primary electromotive force is 

= f< 

In an oscillating discharge the total primary electro- 
motive force .£■ = ; that is, 

1 + Zro + /»ZF, =0; 
or, the substitution 


(r — ax - 

\ ' ' l + o' I ^ \' l + a'l 


612 APPENDIX 11. 

Substituting in this equation, X'='i-K N C, x^=\l%'' N C, 
etc., we get a complex imaginary equation with the two 
constants a and N. Separating this equation in the real 
and the imaginary parts, we derive two equations, from 
which the two constants a and N of the discharge are 

324. If the exciting current of the transformer is neg- 
ligible, — that is, if J^ = 0, the equation becomes essentially 
simplified, — 

that is, 
(..-.., __£_,..)+^.(,_„«__i_.,) = Oi 


or, combined, — 

(/•, - 2«»,) +/•(»•- 2oi) _ 0, 

',x +/>■»,-(! + »•) (*, +/••«). 
Substituting for :ri , :r, ;r,i , ;r„, we have 

/ 4(z:, + /-Z) 


« (A +fL) 

E- Sill^f'ZYA dec a, 

/ =./^,' Yi dec a, 

/, _ e; r, dec a, 
the equations of the oscillating-current transformer, witli 
Ej as parameter. 



Addition 494. 49S 

Admittance, conductance, auscep- 

tance, Chap. vii. . . , 52 

definiiion 6S 

parallel connection ... 67 
primary exciting, of trans- 
former 204 

of induction motor . . . 240 
Advance of phase, hysterecic , . 115 
Algebra of complex ima^oar; 

Quantities, App, l. . . . 489 
Alternating current generator, 

Chap. XVII 297 

transformer, xiv 103 

XX. 354 

motor, synchronous, Chap. 

XIX 321 

Alternating wave, definition . . 11 

general 7 

Alletnatots, Chap. xvii. . , ,297 
parallel operation. Chap. 

xviii 311 

series operation 813 

synchronizing, Chap, xvitl. . 811 
synchronizing power in paral- 
lel operation 317 

Ambiguity of vectors , , , . 43 
Amplitude of alternating wave . 7 
Angle of brush displacement in 

repulsion motor .... 361 
Apparent total impedance of 

transformer 208 

Arc, distortion of wave shape by 394 

power factor of 396 

Arithmetic mean value, or average 

value of alternating wave 11 
Armature reaction of alternators 

and synchronous motors . 297 

Armature reaction of altematora, 
a« affecting parallel opera- 

self-induction of alternators 

and synchronous mol 
slots, number of, affecting 
wave shape , . , . 
Associate numbers .... 
Asynchronous, see induction . 
Average value, or mean value of 
alternating wave . . 

B&Uoce, complete, of lagging 
currents by shunted c 
densance .... 

Balanced and unbalanced poly- 
phase systems. Chap. 

Balanced polyphase system 
quarter-phase system . 
tbree-phase system 

Balance factor of polyphase 

of lagging cunenis by shun- 
ted condensance . . 
Biphase, see quarter-phase 

Cables, as distributed capadly 
with resistance and capacity 

CaJculation of magnetic c 
containing iron . . 
of constant frequency indue- 

of frequency c 

of induction motor . . 

of single-phase induction ir 

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Calculation of 
Chap, i: 

Capacity a.nd induct suice, dh 
tributed, Chap. xni. . 
as source of reactance 
in shunt, compensating for 

intensifying higher hannon- 

see condenser and conden- 

of induction 

with resistance, inductance, 

capacity, and leaJiage . . 

curves of transmis^n lines ■ 1 

field of allemalor . . . . S 

power of polyphase systems 1- 

Circuit characteristic of cable 

with resistance and c^a- 

characteristic of transmission 
line with resistance, induc- 
tance, capacity and leakage < 
factor of distorted wave , . 4 
with series impedance . . < 
with series reactance . . , < 
with series resistance ... I 
Circuits containing rebalance, in- 
ductance, and capacity, 

Chap, viii I 

Coefficient of hysteresis . . 1] 
Comtnnation of allemating nne 
waves by parallelogram or 
polygon of vectors . . . ' 
of double frequency vectoi^ 

ai power If 

of sine waves by rectangular 

components i 

ot sine waves in symbolic 
representation . . . . i 
IT, Chap. XX. 31 

Compensation for lagging cur- 
rents by shnnled conden- 

sance 72 

Complete diagram of transmis- 
sion line in space . . . 162 
Complex imaginary number . . 492 
imaginary quantities, algebra 

of, App. 1 489 

imaginary quantities, as sym- 
bolic representation of al- 
ternating waves .... 37 

quantity Chap, v 33 

Compounding curve of frequency 

convener 232 

Concatenated couple ot induction 

motors, calculation . . . 276 
Concatenation of induction mo- 
tors 274 

Condensance in shunt, compen- 
sating for lagging currents 72 
in symbolic representation . 40 
or capacity reactance ... b 
see capacity and condenser 
Condensers, distortion of wave 

shape by SOS 

see capacity and condensance 
wiih distorted wave . . . 419 
with »ngle-phase induction 

motor 286 

Conductance, elTeclive, definition 104 
alternating current dr- 
uits, definition , . . . (A 


of receiver ciicuit, affecting 

output of inductive line . 811 
parallel c< 
see re^sti 

Conjugate numbers 494 

Constant current — constant po- 
tential transformation . . 70 
current, constant potential 
transformation by trans- 
mission line IBl 

potential, constant current 
transformation .... 70 


Constant potential, 

rent transformation by 

transmission line .... 

rolaiing M M.F 438 

Constants, characteristic, of in- 
duction motor .... 262 

Continuous current system, distri- 
bution efficiency .... 473 

Control, by change of phase, of 

transmission line, Chap. ix. 83 
of receiver circuit by shunted 
ausceptance 06 

Converter of frequency, Cbap. 

XV 210 

T E.M.F. 

of inductance 25 

ENelectic and electrostatic hyale- 

Diphase, see quarter-phase. 
Discharge, oscillating . . . 
Displacement angle of repuUior 

of phase, j 

Distorted wave, circuit factor . 4 
wave, decreasing hysteresis 

ve, increasing hysteresis 

of self-indue tioi 
Counting o 
Cross-Hun, magnetic, of trans- 

of transfoimer, use for con- 

-wavea, alternating, distorted 

by hysteresis 1 

Cycle, Of complete period . . . 

Decremeat of oscillating wave . ( 
Delta connection of three-phase 

current in tbieephase system 15i 
potential of three-phase sys- 

n of three-phase 
transformation .... 4 
Demagnetizing effect of armature 
reaction of alternators and 
synchronous molors , . 2 
effect of eddy currents . . 1. 
Dielectric and electrostatic phe- 
nomena 1 

wave of condenser .... 4 
wave of synchronous motor. 4 
wave. somedifferent shapes , 4 
wave, symbolic representa- 
tion. Chap. XXIV. ... 4 
wave, in induction motor, . 4 
Distortion of alternating wave . 
of wave shape and eddy cur- 

of waveshape, and insulation 
strength 4 

of wave shape and its caoses. 
Chap, xxii 3 

of wave shape by hysteresis . 1 

of wave shape, effect of. 
Chap. XXIII 3 

of wave shape, increasing ef- 
fective value 4 

Distributed capacity, inductance, 
resistance, and leakage, 

Chap. XHI 1 

Distribution efficiencyof systems. 4 
Divided circuit, equivalent to 
transformer 2 

Double delta connection of three- 
phase — six-phase transfor- 
mation 4 

frequency auanlilies, as pow- 
er, Chap. XII 1 

frequency values of distorted 
wave, symbolic representa- 

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Double peaked wave 3 

$a.i*-iooih wave 3 

T connection of three-phase 

— ux-phaae transfonna- 
tion * 

V connection of three-phase 

— six-phase Inuisfonna- 

Bddy cotrents, unafFeded by 
nave-shape distOTtion . . 4 
demagnetizing or screening 

effect 1 

in conductor, and unequal 
current distiibation . . . 1 
Eddy or Foucauit currents, Chap, 

XI 1 

Effective reactance and suscep- 

tance, definition . ■ . . 1' 
resistance and conductance, 

definition li 

reusiance and reactance, 
Chf^LX. H 

value of alternating vave 
value of alternating wave, 


value of general alternating 

Effects of higher harmonics, 

Chap. Kxiii S 

Efficiency, maximum, of induc- 

Efiiciency of systems. Chap. xxx. 4 
Electro-magnetic induction, law 

of. Chap. Ill 

Electrostatic and dielectric phe- 
nomena 1 

hysteresis 1 

Energy component of self.induc- 

llow of, in polyphase system, 4 
Epoch of alternating wave . . . 
Equations, fundamental, of alter. 
nating current transformer, 

Eouations, fundamental, of gen- 
eral alternating cnirent 
tranafotmer, or frequency 

converter 2' 

of induction motor . , 226, 2' 

of synchronous motor ... 3 

of transmission line . . . 1< 

Equations, general, of apparatus, 

see equal i ons^undamentaL 

Equivalence of iransfonner with 

divided circuit 2' 

Equivalent sine wave of distorted 

wave 1 

Evolution 491, 4' 

Exdting admittance of induction 

motor 2, 

admittance of transformer . 2' 

distorted by hysteresis . . 1 
current of transformer . . 1' 

Field characteristic of alternator . S 

First harmonic, or fundamental, 
of general alternating wave, 
Five.Vfi re single-phase syslem, dis- 
tribution efficiency ... 4 

Flat-top wave 3 

Flow of power in polyphase sys- 
tem 4 

Foucault orEddycurTents,Ch.KL 1 
Four-phase, see quarter-phase. 

Fraction 4 

Free oscillations of circuit ... 6 
Frequency converter, Chap. XV. . 2 
converter, calculation ... 2. 
converter, fundamental equa- 

of alternating wave . . . 
ratio of general alternating 
current transformer or fre- 
quency converter ... 2 
Friction, molecular magnetic . . li 
Fundamental equations, see equa- 
tions, fundamental, 
freqtiency of transmission 
line discharge . . . . 1 

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Fundamental equations, or fiist 
bannonic of general alter- 

General alternating current trans- 
fonner, or frequency con- 
vener, Chap. XV. . . . S 
alternating wave . . . . 'i 

altemaiing wave, symbolic 
represent alion.Chap.KX IV. 410 

equations, see equations, fun- 

polyphase systems. Chap. 

XXV 430 

Generator action of concatenated 

couple 280 

of reaction machine ■ . . 877 

alternating current, Chap. 
XVII 297 

synchronous, operating with- 
out field excitation . . .871 

induction 205 

induction, calculation for con- 
stant frequency .... 26S 

reaction, Chap. xxL . . . 871 

vector diagram 28 

Gr^hical construction of circuit 

characteristic .... 48, 40 
Graphic representation, Chap, iv. It) 

limits of method .... 38 

see polar diagram. 

Hutnonics, higher, effects of. 

Chap, xxm 3M 

higher, resonance rise in 

transmission lines ... 402 
of general alternating wave . 8 
Hedgehog transformer .... 195 
Hemisyramelrical polyphase sys- 
tem 439 

Henry, definition of 18 

Hexaphase, see six-phase. 

Hysieresis, Chap, x 104 

advance of phase . . . .115 
as energy component of self- 
induction 372 

Hysteresis, coefficient .... 110 

cycle or loop 107 

dielectric, or electrostatic . 145 

energy current of transformer 196 

loss, efiected by wave shape, 407 

loss in alternating fidd . . 114 

magnetic 100 

motor 283 

of magnetic circuit, calcula- 
tion 126 

or magnetic energy cnirent . 115 

Imagiiuurf number 492 

quantities, comi^ei, algebra 

of, App. I. 489 

Impedance 2 

in series wilh drcuit ... 98 
in symbolic representation . 89 
primary anTl secondary, of 

transformer 205 

see, admittance. 

series connectbn .... 67 

total apparent, of transformer 208 

Independent polyphase system . 431 

Induaance 4 

definition of 18 

factors of distorted wave. . 416 

mutual 142 

Induction, electro-magnetic, law of 10 

electrostatic 147 

generator 286 

generator, calculation for 

constant frequency . . 269 
generator, driving synchron. 

ous motor 272 

motor, Chap. XVL .... 237 

motor . ' 281 

motor, calculation . , . . 262 
motor, concatenation or tan- 
dem control 274 

motor, fundamental equa. 

tions 226, 242 

motor, graphic representa- 
tion ■ ... 244 

motors in concatenation, cal- 
culation 276 

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IndDCtion motor, aynchTonous • 291 
motor torque, as double fre- 
quency vector .... 156 
motor with distorted wave . 
Induaive devices for starting sin- 
gle-phase inductioti motor 
line, effect of conductance of 

mitted power 

line, effect of susceptance of 

milted power 

line, in symbolic representa- 

Une, maximum efficiency of 

(tsnsmitted power . . . 

line, maximimi power sup. 

line, maximuni nsk of poten- 
tial by shunted susceptance 

line, phase control by shanted 

line, supplying non-inductive 

Influence, electrostatic .... 
Instantaneous values and inte- 
gral values. Chap. 11. . . 
value of alternating wave 
Insulation strength with distorted 

Integral valnes of alternating 

Intensity of nne wave .... 
Interlinked polyphase systems, 

Chap, xxviii 

polyphase system .... 
Interna) impedance <!f trans- 
Introduction, Chap. I 

Inverted three-phase system . . 
three-phase system, balance 

factor 443, 

three-phase system, distribu- 
tion efficiency 

Involution 400, 

Iron. laminale.'J, eddy currents 

Iron wire, eddy cnirents . . . 1 
wire, unequal current dtstri- 
bntioQ in alternating cir- 
cuit 1 

Irrational number 4 

/ a» inutcinaiy unit .... 

introduction of, as dislin. 
guishing index .... 
Joules's law of allemaling cur- 
law of continuous currents . 

EiicUioff'a lAWi In symbolic 

representatioD .... 

laws of alternating current 

laws of alternating sine waves 
in graphic tepreseiitation . 

IfSKKins cnnrenta, comp«iuation 
for, by abtuted ooaAta- 

Lag of alternating wave 

of alternator current, effect 

self-induction 2 

Lanunated iron, eddy cunenls . 1 
Law of elect CO- magnetic induc- 

L connection of three-phase, quar- 
ter phase iransfoimation . 4' 
connection of Ihree-phase 
transformation .... 4 
Lead of alternating wave ... 
of alternator current, e^ect 

self-induction . . . . 2 
Leakage current, see Exciting 

of eleclric current . . . . 1 
Lightrting discharges from trans- 
mission lines, frequencies 


line, inductive, vector Oiagram . 23 
mth dist ribuled capacity and 

indnctance 168 

with resistance, inductance, 
capacity, and leakage, 
topographic circuit charac- 

Logarithmation 401 

Long-distance lines, as distributed 

capacity, and inductance 15S 
Loxodromic spiral 600 

Magnetic circtiit cootainiiic iron, 

calculation 12u 

hysteresis 106 

or hysteretic energy current . 115 

Magnetizing current 116 

current of transformer . . 106 
effect of annature reaction 
in alternators and synchro- 

M^n and teazer connection of 

three-phase iransfoimation 4ft4 
Maximum output of synchronous 

motor 342 

poner of induction motor . 252 
power of synchronous motor 342 
power supplied over induc- 
tive line 87 

rise of potential in inductive 
line, by shunted suscep- 

tance 101 

to effective value .... 14 

to mean value 13 

torque of induction motor . 250 

value of alternating wave . 11 

Mean to maximum value ... 13 

value 12 

value, or average value of 
alternating wave . ... II 
Mechanical power of frequency 

convener 227 

Minimum current in synchronous 

motor 345 

M. M. F, of 

of alternator 

M. M, F. rotating, of i 

intensity 430 

's acting upon alternator ar- 
mature 207 

Molecular magnetic friction . . 100 
Monocyclic connection of three- 
phase-inverted three-phase 
transformation .... 464 
devices for starting single- 
phase induction motors . 28.1 

systems 447 

Monophase, see Single-phase. 
Motor, action of reaction ma- 
chine 377 

alternating series .... 303 
alternating shunt .... 368 
commutator, Chap. XX. , . 364 

hysteresis 203 

induction. Chap. xvi. ... 237 
reaction, Chap. xsi. . . .871 

repulsion 364 

single phase indnction . . 281 
synchronous, Chap. XIX . . 321 
synchronous, driven by in- 
duction generator . . .272 
synchronous induction . . 2S1 
Multiple frequency of transmis- 

uon line discharge . . . 165 

Multiplication 400,401 

Mutual indadance 142 

inductance of transformer 

natural period of transmission 

line 181 

Negative number 400 

Nominal induced E.M.F. of alter. 

nalor 302 

Non-inductive load on trans- 
former 212 

receiver circuit supplied over 
inductive line .... 84 

N'phase system, balance fac- 
tor 443 

phase system, symmetrical . 435 

Numeration or counting . . . 480 


Oluu Uw in BTmboUc TepreMO- 

of alternating currents . . 

of continuoDS currents . . 
Oscillating currents, App. ii. . .4 

discharge G 

OsciOation frequency of 

Output, see Power. 
Overtonsa, or higher hannoTucs 
of general ahematingnave 

Parallel coonection of condnc- 

Paialielogiam law of alternating 

of double-frequency vectors, 

Parallel operation of alternators, 

Chap, xviu 3 

* Peaked wave 3 

Period, natural, of transm. line 1 

of alternating wave . . . 
Phase angle of transmission line 1 
control, maximum rise of po- 
tential by 1 

control of inductive line by 

shunted suscepiance . . 

control of t 

Chap. a.. . 
difference of , 
displacement, t 

synchronous motor . . . & 
of alternating wave . . . 

of ane wave 

splitting devices for starting 
single-phase induction mo- 
tors 2 

Plane, complex imaginary ... 4 
Polar co6idinate of alternating 

iMagram of induction motor 2 
diagram of transformer . . I 
diagram of tiansmisnion line 1 
diagrams, see Giaphic repre- 

Polarization as capacity . . 

distortion of wave shape by . 3i 

Polycyclic systems ii 

Polygon of alternating sine waves i 

Polyphase system, balanced . . 4! 

systems, balanced and unbal' 

anced, Chap, xxvti. , . 4< 
systems, elficiency of trans- 
mission, Chap. XXX. . . 4( 
systEms, flow of power . . i' 
systems, general. Chap. xxv. 41 
systems, hemisymmetrical . 4^ 
systems, interlinked. Chap. 

xxvui 41 

systems, symmetrical, Chap. 

XXVL 41 

systems, symmetrical . . . 4i 
systems, symmetrical, pro- 
ducing constant revolving 

M.M.F. 4: 

systems, transformation of. 

Chap. XXIX. 4 

systems, unbalanced , , . 4 
systems, unsymmelrical . .4 
Power and double frequency 
quantities in geiteial, Chap. 

XII 1. 

characteristic of polyphase 

systems 4 

characteristic of synchronous 

motor S 

equation of alternating cur- 
equation of alternating sine 
waves in graphic represen- 

equalion of continuous cur- 
factor of arc 3 

factor of distorted wave . . 4 
factor of reaction machine . 3 
flow of, in polyphase system 4 
fiow of. in transmission line 1 
maximum, of inductive line 
with non-inductive recover 


Fower, majdmnm of synch ronODH 

motor 432 

maiimiun supplied over in- 
ductive Hne 87 

of complex harmonic wave . 406 
of distoned wave , , . . 413 
of frequency converter . . 227 
of geneial polyphase sjntem 469 
of induction motor .... 248 
of repulsion motor .... 360 
pardUelogntm of, in symbolic 

representation .... 163 
real and wattless, in symbol- 
ic representation . . . 151 
Primarj exciting admittance of 

induction motor .... 240 
ezdting admittance of trans- 

. 204 

lm))edance of transformer . 206 
Pulsating wave, definition ... 11 
Pulsation of magnetic field caus- 
ing higher harmonics of 

E.M.F 384 

of reactance of alternator ar- 
mature causing higher har- 

of resistance, causing higher 
harmonics 393 

Qnadiipbaw, see Quarter-phase. 
Quarter-phase, live-wire system, 

dislribntion efficiency . . 471 
system. Chap, xxxil. ... 48-3 

system 43* 

system, balance factor . 442, 445 
system, distribution effiuency 471 
system, symmetry .... 436 
system, transmission effi. 

ciency 474 

three-phase (ransEormalion . 4S5 

umlooth wave 388 

Quintnplc harmonic, distortion of 

wave by 400 

SatlO of frequencies in general 

alternating current trans- 
former 221 

Ratio of freqnendea of transfor- 
mation of transformer. . 2 


effective, deGnition . . . 1 

in symbolic representation . 
periodically varying ... 3 
pulsation in alternator caus- 
ing higher harmonies . . 3 

a of . 

synchronous, of alternator . 3 

Reaction machines, Chap. xxi. . 3 
machine, power-factor . . 3 
armature, of alternator . . 2 

Rectangular coordinates of alter- 

diagram of transmission 

line 1 

Reflected wave 

Reflexion angle of i 

line 1 

Regulation curve of frequency 

converter 2 

of alternator for constant 

current 3 

of allemaior for constant 

power 3 

of alternator for constant 
terminal voltage ... 3 
Reluctance, periodically varying . 3 
pulsation of, causing higher 
liarmonics of E.M.F. . . 8 

Repulsion motor 3 

motor, displacement angle ... 3 

motor, starting torque . . 3 

motor, (orque 8 

Resistance and reactance of 
transmission lines. 

Chap. IX 

effective, definition . . . 1 
effective, of alternating cm. 

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Resistance a.nd reactance in alter- 
in series with circuit . . . 
of induction motor secon- 
dary, affecting starting 

pulsation, cauung higher har- 

see Conductance. 
Resonance rise by series indac- 
tance, with leading car- 
rise in transmission lines with 
higher harmomcs . . . 
Resolution of alternating ^ne 
waves by the panUlelo- 
grain or polygon of vec- 

of double frequency vectois, 

of Nne waves by rectangular 


of sine waves in symbolic 

representation .... 

Reversal of alternating vector by 

multiplication with ~ 1 . 

Revolving magnetic field . . . 436 

M. M. F, of constant inten- 

aiy 436 

Ring connection of interliniied 

polyphase system . . . 453 
current of interlinked poly- 
phase system 466 

potential of interlinked poly- 
phase system 4G5 

Rise of voltage by inductance, 

viith leading current . . 82 
of voltage by inductance in 

Roots of the unit 495 

Rotating magnetic field. . . . 436 

M.M.F. of constant intensity 436 
Rotation 405 

by 00°, by multiplication 
with ± j 37 

Satttiation, magnetic, effect on 
eliciting current wave . . 1 

Sawtooth wave S 

Screening effect of eddy currents 1 
Screw diagram of transmisuon 

line 1 

Secondaiy impedance of trans^ 

former ! 

Self-excitation of alternator and 
synchronous motor by ar- 

Self-inductance . . . 
E.M.F. of . . . 
of transformer . 
of transformer for 
power or constant 


1, energy component 

Lgher harmonics 
of impedanci 

motor, alternating . . . 
operation of alternators . 

Shunt motor, altemaling . 

Sine wave 

circle as pohir characteristic 
equivalent, of distorted wave, 


representation by complex 

Single-phase induaion motor 
induction motor, calculation 
induction motor, starling de- 

induction motor, with con- 
denser in lertiaiy circuil . 287 
system, balance factor . . 444 
.distribution efficiency 470 


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Six-phase system 434 

three-phase transformation . 405 
Slip of frequency converter or 
genera] alternating current 

transformer 221 

of induction motor . . . 238 
Slots of alternator 

feeling wave shapi 
Space diagram of 

line 192 

Star connection of interlinked 

polyphase system . . . 453 
cutient of interlinked poly- 
phase system 466 

potential of interUnked poly- 
phase system 466 

Startiog of single-phase induction 

torque of induction motor . 254 
torque of repulsion motor . 361 

Stray field, see Cross flux. 

Subtraction 400, 494 

Suppression of higher harmonics 
by self-induction 

Snsceptance, definition 
effective, detinition 

of receiver ci 
ducCive line 
shunted, controlling 

see Reactance. 

Symbolic method. Chap. v. 

method of transformer 

representation of general 

alternating waves. Chap. 

Symbolism of double frequency 

Symmettical n-phase systen 

polypha-iie system. Chap. 

polyphase systems . . 

polyphase system, producing 

constant revolving M.M.F. 

Synchronism, at or near indue- 

Synchronizing alternators, Chap. 

power of alternators in par- 
allel operation . . . .317 
Synchronous induction motor . 291 
motor, also see Alternator. 
motor. Chap xix. .... 321 

chine 377 

motor, analytic investiga- 
tion 338 

motor and generator in single 
unit transmission . . . 324 

E.M.F. 849 

motor, constant generator 
andmoiorE-M.F- . . , 32B 

motor, constant generator 
E.M.F. and constant 
power 334 

motor, constant generator 
E.M.F. and maximum effi- 
ciency 332 

. motor, constant impressed 
-E.M.F, and constant cur- 
rent 326 

motor driven by induction 
generator 272 

motor, fundamental equa- 
tions 339 

motor, graphic representa- 
tion 321 

motor, maximum phase dis- 
placement 347 

motor, maximum output . . 342 

given power 345 

motor, operating without 

field excitation . . , .371 
motor, phase relation of cur- 
motor, polar characteristic . 341 
motor, running light . , . 343 
motor, with distorted wave . 422 
reactance of alternator and 

syachronons motor . . . 301 


Tuid«m control of inducdoii 

rootois 2 

conirol of induction motoie, 

calculation .... 
T-connection of three phase, qnar- 

ter-phase tntnafonnation 
connection of Ihree-phtu 

transformaiion . . . 
Tertiary circuit with condenser, 

in ^ngle-phase induction 

Tetraphase, see Qoarter-phi 
Three-phasejour-wire tyaxec 

tribution etRdency . 
qiujter-phase transfotmaiion 

ni-phase tranaformation 
system. Chap. xxxi. . 

system, balance-factor 
system, distribution 

system, equal load on phases, 

topographic method 
system, interlinked . , 
system, symmBtry . , 
system, tiansnussion 

unitooth mve . . ■ 
Three-wire, quaiter-phase system 
single-phase system, distribu- 
tion efficiency 
Time constant of circ 

transmission line chaiac- 

method. Chap. 
Torque, m double frequency vec- 

of distorted wave . . . 

of induction motor . . 

of repulsion motor . . . 

Transformation of polypbi 

systems. Chap. xxix. . 

ratio of transformer . . 

Transformer, alternating current. 

Chap. XIV 

Transformer, equivalent to di- 
vided circuit 208 

fundamental equations 208, 226 
General alternating current, 
or frequency converter. 

Chap, XV. 219 

osciUaling current .... 610 

polar diagram 196 

symbolic method .... 201 

vector diagram 28 

Transmission efficiency of sys- 
tems. Chap. XXX. ... 468 
lines, as distributed c^odty 

and inductance .... 168 
line, complete space <Uagram 192 
line, fundamental equations , 160 
line, natural period of , . 181 
lines, re^tance and re- 
actance of (Phase Con- 
trol, Chap. IX. .... 83 
line, resonance rise with 

higher harmonics . . . 402 
lines with resistance, induc- 
tance, capacity, topo- 
graphic characteristic . . 49 
Trigonometric method . . . . S4 
method, Kmiis of .... 34 
Triphase, see Threephase. 
Triple harmonic, distortion of 

Two-phase, see Quarter-phase. 

VnbaUnced polypliAM Byatem . 431 
quartei-phase system . . . 48G 
three-phase system .... 461 
Unequal current distribution, 
eddy currents in conduc- 
tor 189 

Uniphase, see Single-phase. 

Unit, imaginary 494 

Urutooth alternator waves . . . 368 
alternator waves, decrease of 

hysteresis loss .... 408 
alternator waves, increase of 

power 405 

Unsymmetrical polyphasesystem 480 


Vector, as representation of alter- 
nating wave 21 

of double frequency, in sym- 
boKc representation . . 161 

Volt, definition 18 

Wittleu power 151 

power of distorted nave . , 413 

Wave length of tranEmisaon line ITO 
shape distortion and its 

causes. Chap. xxii. ... 883 
shape distortion by hyster- 

Wire. iron, eddy currents . . . 1 
T-condection of three-phase sj-s- 
cnrrenl of three-phase ays- 
delta conneciion of three- 
phase transformation . . 4 
potential of three-phase sys- 
tem 4 

Zero ImpedanM, circuits of . . a 

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