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Full text of "Theory and calculation of alternating current phenomena"

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THEORY AND CALCULATION 



ALTERNATING CURRENT 
PHENOMENA/ 



BY 

CHARLES PROTEUS.STEINMETZ 



WITH THE ASSISTANCE OF 



ERNST J. BERG 



THIRD EDITION, REVISED AND ENLARGED 



NEW YORK 
ELECTRICAL WORLD AND ENGINEER 

INCORPORATED 
I9OO 



COPYRIGHT, 1900, 



ELECTRICAL WORLD AND ENGINEER. 
(INCORPORATED.) 



TYPOGRAPHY BY C. J. PETERS * SON, BOSTON. 



1C 27 



DEDICATED 

TO THE 

MEMORY OF MY FATHER, 
CARL HEINRICH STEINMETZ. 



PREFACE TO THE THIRD EDITION. 



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 typographical 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 STEINMETZ. 

CAMP MOHAWK, VIELE'S CREEK, 
July, jgoo. 



PREFACE TO FIRST EDITION. 



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-current circuits, and of their 
application to alternating-current 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 115 and Appendix II. ; and even 106 to 115 
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- 



viii PREFACE. 

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, the transformation 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- 



PREFACE. IX 

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. 

CHARLES PROTEUS STEINMETZ. 
January, 1897. 



CONTENTS. 



CHAP. I. Introduction. 

1, p. 1. Fundamental laws of continuous current circuits. 

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

3, p. 3. Electro-magnetism as source of reactance. 

4, p. 5. Capacity as source of reactance. 

5, p. 6. Joule's law and power equation of alternating circuit. 

6, p. 6. Fundamental wave and higher harmonics, alternating 

waves with and without even harmonics. 

7, p. 9. Alternating waves as sine waves. 

CHAP. II. Instantaneous Values and Integral Values. 
8, p. 11. Integral values of wave. 
9, p. 13. Ratio of mean to maximum to effective value of wave. 

CHAP. III. Law of Electro-magnetic Induction. 
11, p. 16. Induced E.M.F. mean value. 
12, p. 17. Induced E.M.F. effective value. 
13, p. 18. Inductance and reactance. 

CHAP. IV. Graphic Representation. 

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

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

16, p. 21. Parallelogram of sine waves, Kirchhoff's laws, and energy 

equation. 

17, p. 23. Non-inductive circuit fed over inductive line, instance. 
18, p. 24. Counter E.M.F. and component of impressed E.M.F. 
19, p. 26. Continued. 
20, p 26. Inductive circuit and circuit with leading current fed over 

inductive line. Alternating-current generator. 

21, p. 28. Polar diagram of alternating-current transformer, instance. 
22, p. 30. Continued. 

CHAP. V. Symbolic Method. 

23, p. 33. Disadvantage of graphic method for numerical calculatioa 

24, p. 34. Trigonometric calculation. 

25, p. 34. Rectangular components of vectors. 

26, p. 36. Introduction of / as distinguishing index. 

27, p. 36. Rotation of vector by 180 and 90. j = V^HT. 



xii CONTENTS. 

CHAP. V. Symbolic Method Continued. 

28, p. 37. Combination of sine waves in symbolic expression. 

29, p. 38. Resistance, reactance, impedance, in symbolic expression. 

30, p. 40. Capacity reactance in symbolic representation. 

31, p. 40. KirchhofF s laws in symbolic representation. 

32, p. 41. Circuit supplied over inductive line, instance. 

CHAP. VI. Topographic Method. 
33, p, 43. Ambiguity of vectors. 
34, p. 44. Instance of a three-phase system. 
35, p. 46. Three-phase generator on balanced load. 
36, p. 47. Cable with distributed capacity and resistance. 
37, p. 49. Transmission line with self-inductive capacity, resistance, 
and leakage. 

CHAP. VII. Admittance, Conductance, Susceptance. 

38, p. 52. Combination of resistances and conductances in series and 

in parallel. 
39, p. 53. Combination of impedances. Admittance, conductance, 

susceptance. 
40, p. 54. Relation between impedance, resistance, reactance, and 

admittance, conductance, susceptance. 
41, p. 56. Dependence of admittance, conductance, susceptance, upon 

resistance and reactance. Combination of impedances and ad- 

mittances. 

CHAP. VIII. Circuits containing Resistance, Inductance, and Ca- 
pacity. 

42, p. 58. Introduction. 
43, p. 58. Resistance in series with circuit. 
44, p. 60. Discussion of instances. 
45, p. 61. Reactance in series with circuit. 
46, p. 64. Discussion of instances. 
47, p. 66. Reactance in series with circuit. 
48, p. 68. Impedance in series with circuit. 
49, p. 69. Continued. 
50, p. 71. Instance. 

51, p. 72. Compensation for lagging currents by shunted condensance. 
52, p. 73. Complete balance by variation of shunted condensance. 
53, p. 75. Partial balance by constant shunted condensance. 
54, p. 76. Constant potential constant current transformation. 
55, p. 79. Constant current constant potential transformation. 
56, p. 81. Efficiency of constant potential constant current trans- 
formation. 

CHAP. IX. Resistance and Reactance of Transmission Lines. 
57, p. 83. Introduction. 
58, p. 84. Non-inductive receiver circuit supplied over inductive line. 



CONTENTS. xiii 

CHAP. IX. Resistance and Reactance of Transmission Lines. Continued. 

59, p. 86. Instance. 

60, p. 87. Maximum power supplied over inductive line. 

61, p. 88. Dependence of output upon the susceptance of the re- 
ceiver circuit. 

62, p. 89. Dependence of output upon the conductance of the re- 
ceiver circuit. 

63, p. 90. Summary. 

64, p. 92. Instance. 

65, p. 93. Condition of maximum efficiency. 

6, p. 96. Control of receiver voltage by shunted susceptance. 

67, p. 97. Compensation for line drop by shunted susceptance. 

68, p. 97. Maximum output and discussion. 

69, p. 98. Instances. 

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

71, p. 102. Summary and instances. 

CHAP. X. Effective Resistance and Reactance. 

72, p. 104. Effective resistance, reactance, conductance, and suscep- 
tance. 

73, p. 105. Sources of energy losses in alternating-current circuits. 

74, p. 106. Magnetic hysteresis. 

75, p. 107. Hysteretic cycles and corresponding current waves. 

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

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

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

79, p. 115. Angle of hysteretic advance of phase. 

80, p. 116. Loss of energy by molecular magnetic friction. 

81, p. 119. Effective conductance, due to magnetic hysteresis. 

82, p. 122. Absolute admittance of ironclad circuits and angle of 
hysteretic advance. 

83, p. 124. Magnetic circuit containing air-gap. 

84, p. 125. Electric constants of circuit containing iron. 

85, p. 127. Conclusion. 

CHAP. XI. Foucault or Eddy Currents. 

86, p. 129. Effective conductance of eddy currents. 

87, p. 130. Advance angle of eddy currents. 

88, p. 131. Loss of power by eddy currents, and coefficient of eddy 

currents. 

89, p. 131. Laminated iron. 
90, p. 133. Iron wire. 

91, p. 135. Comparison of sheet iron and iron wire. 
92, p. 136. Demagnetizing or screening effect of eddy currents. 
93, p. 138. Continued. 
94, p. 138. Large eddy currents. 



CONTENTS. 

CHAP. XI. Foucault or Eddy Currents. Continued. 

95, p. 139. Eddy currents in conductor and unequal current dis- 
tribution. 

96, p. 140. Continued. 
97, p. 142. Mutual inductance. 
98, p. 144. Dielectric and electrostatic phenomena. 
99, p. 145. Dielectric hysteretic admittance, impedance, lag, etc. 
100, p. 147. Electrostatic induction or influence. 
101, p. 149. Energy components and wattless components. 

CHAP. XII. Power, and Double Frequency Quantities in General. 
102, p. 150. Double frequency of power. 
103, p. 151. Symbolic representation of power. 
104, p. 153. Extra-algebraic features thereof. 
105, p. 155. Combination of powers. 
106, p. 156. Torque as double frequency product. 

CHAP. XIII. Distributed Capacity, Inductance, Resistance, and Leak- 
age. 

107, p. 158. Introduction. 

108, p. 159. Magnitude of charging current of transmission lines. 

109, p. 160. Line capacity represented by one condenser shunted 
across middle of line. 

110, p. 161. Line capacity represented by three condensers. 

111, p. 163. Complete investigation of distributed capacity, induc- 
tance, leakage, and resistance. 

112, p. 165. Continued. 

113, p. 166. Continued. 

114, p. 166. Continued. 

115, p. 167. Continued. 

116, p. 169. Continued. 

117, p. 170. Continued. 

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

119, p. 17-2. Instance. 

120, p. 173. Further instance and discussion. 

121, p. 178. Particular cases, open circuit at end of line, line 
grounded at end, infinitely ong conductor, generator feeding 
into closed circuit. 

122, p. 181. Natural period of transmission line. 

123, p. 186. Discussion. 

124, p. 190. Continued. 

125, p. 191. Inductance of uniformly charged line. 

CHAP. XIV. The Alternating-Current Transformer. 

126, p. 193. General. 

127, p. 193. Mutual inductance and self-inductance of transformer. 

128, p. 194. Magnetic circuit of transformer. 



CONTENTS. 



. XV 



CHAP. XIV. The Alternating-Current Transformer Continued. 
129, p. 195. Continued. 
130, p. 196. Polar diagram of transformer. 
131, p. 198. Instance. 
132, p. 202. Diagram for varying load. 
133, p. 203. Instance. 



134, p. 204. Symbolic method, equations. 



135, p. 206. 
136, p. 208. 



Continued. 

Apparent impedance of transformer. 



Transformer 



equivalent to divided circuit. 

137, p. 209. Continued. 

138, p. 212. Transformer on non-inductive load. 

139, p. 214. Constants of transformer on non-inductive load. 

140, p. 217. Numerical instance. 

CHAP. XV. General Alternating-Current Transformer or Frequency 

Converters. 

141, p. 219. Introduction. 

142, p. 220. Magnetic cross-flux or self-induction of transformer. 

143, p. 221. Mutual flux of transformer. 

144, p. 221. Difference of frequency between primary and secondary 

of general alternate-current transformer. 

145, p. 221. Equations of general alternate-current transformer. 

146, p. 227. Power, output, and input, mechanical and electrical. 

147, p. 228. Continued. 

148, p. 229. Speed and output. 

149, p. 231. Numerical instance. 

150, p. 232. Characteristic curves of frequency converter. 

CHAP. XVI. Induction Machines. 

151, p. 237. Slip and secondary frequency. 

152, p. 238. Equations of induction motor. 

153, p. 239. Magnetic flux, admittance, and impedance. 

154, p. 241. E.M.F. 

155, p. 244. Graphic representation. 

156, p. 245. Continued. 

157, p. 246. Torque and power. 

158, p. 248. Power of induction motors. 

159, p. 250. Maximum torque. 

160, p. 252. Continued. 

161, p. 252. Maximum power. 

162, p. 254. Starting torque. 

163, p. 258. Synchronism. 

164, p. 258. Near synchronism. 

165, p. 259. Numerical instance of induction motor. 

166, p. 262. Calculation of induction motor curves. 

167, p. 265. Numerical instance. 



xvi CONTENTS. 

CHAP. XVI. Induction Machines Continued. 

168, p. 265. Induction generator. 

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

170, p. 269. Constant speed, induction generator. 

171, p. 272. Induction generator and synchronous motor. 

172, p. 274. Concatenation or tandem control of induction motors. 

173, p. 276. Calculation of concatenated couple. 

174, p. 280. Numerical instance. 

175, p. 281. Single-phase induction motor. 

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

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

178, p. 286. Condenser in tertiary circuit. 

179, p. 287. Speed curves with condenser. 

180, p. 291. Synchronous induction motor. 

181, p. 293. Hysteresis motor. 

CHAP. XVII. Alternate-Current Generator. 

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

183, p. 300. Self-inductance and synchronous reactance. 

184, p. 302. Equations of alternator. 

185, p. 303. Numerical instance, field characteristic. 

186, p. 307. Dependence of terminal voltage on phase relation. 

187, p. 307. Constant potential regulation. 

188, p. 309. Constant current regulation, maximum output. 

CHAP. XVIII. Synchronizing Alternators. 

189, p. 311. Introduction. 

190, p. 311. Rigid mechanical connection. 

191, p. 311. Uniformity of speed 

192, p. 312. Synchronizing. 

193, p. 313. Running in synchronism. 

194, p. 313. Series operation of alternators. 

195, p. 314. Equations of synchronous running alternators, synchro- 
nizing power. 

196, p. 317. Special case of equal alternators at equal excitation. 

197, p. 320. Numerical instance. 

CHAP. XIX. Synchronous Motor. 

198, p. 321. Graphic method. 

199, p. 323. Continued. 

200, p. 325. Instance. 

201, p. 326. Constant impressed E.M.F. and constant current. 

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

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

204, p. 334. Constant impressed E.M.F. and constant output. 

205, p. 338. Analytical method. Fundamental equations and power, 
characteristic. 



CONTENTS. xvii 

CHAP. XIX. Synchronous Motor Continued. 

206, p. 342. Maximum output. 

207, p. 343. No load. 

208, p. 345. Minimum current. 

209, p. 347. Maximum displacement of phase. 

210, p. 349. Constant counter E.M.F. 

211, p. 349. Numerical instance. 

212, p. 351. Discussion of results. 

CHAP. XX. Commutator Motors. 

213, p. 354. Types of commutator motors. 

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

215, p. 356. Two types of repulsion motors. 

216, p. 358. Definition of repulsion motor. 

217, p. 359. Equations of repulsion motor. 

218, p. 360. Continued. 

219, p. 361. Power of repulsion motor. Instance. 

220, p. 363. Series motor, shunt motor. 

221, p. 366. Equations of series motor. 

222, p. 367. Numerical instance. 

223, p. 368. Shunt motor. 

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

CHAP. XXI. Reaction Machines. 

225, p. 371. General discussion. 

226, p. 372. Energy component of reactance. 

227, p. 372. Hysteretic energy component of reactance. 

228, p. 373. Periodic variation reactance. 

229, p. 375. Distortion of wave-shape. 

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

231, p. 378. Equations of reaction machines. 

232, p. 380. Numerical instance. 

CHAP. XXII. Distortion of Wave-shape, and its Causes. 

233, p. 383. Equivalent sine wave. 

234, p. 383. Cause of distortion. 

235, p. 384. Lack of uniformity and pulsation of magnetic field^ 

S 236, p. 388. Continued. 

237, p. 391. Pulsation of reactance. 

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

239, p. 393. General discussion. 

240, p. 393. Pulsation of resistance arc. 

241, p. 395. Instance. 

242, p. 396. Distortion of wave-shape by arc. 

243. p. 397. Discussion. 



xvili CO TTENTS. 

CHAP. XXIII. Effects of Higher Harmonics. 

244, p. 393. Distortion of wave-shape by triple and quintuple har- 
monics. Some characteristic wave-shapes. 

245, p. 401. Effect of self-induction and capacity on higher harmonics. 

246, p. 402. Resonance due to higher harmonics in transmission lines. 

247, p. 405. Power of complex harmonic waves. 

248, p. 405. Three-phase generator. 

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

250, p. 407. Increase of hysteresis by distortion of wave-shape. 

251, p. 408. Eddy currents. 

252, p. 408. Effect of distorted waves on insulation. 

CHAP. XXIV. Symbolic Representation of General Alternating Wave. 

253, p. 410. Symbolic representation. 

254, p. 412. Effective values. 

255, p. 4l3. Power torque, etc. Circuit factor. 

256, p. 416. Resistance, inductance, and capacity in series. 

257, p. 419. Apparent capacity of condenser. 

258, p. 422. Synchronous motor. 

259, p. 426. Induction motor. 

CHAP. XXV. General Polyphase Systems. 

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

systems. 
261, p. 430. Flow of power. Balanced and unbalanced systems. 

Independent and interlinked systems. Star connection and ring 

connection. 
262, p. 432. Classification of polyphase systems. 

CHAP. XXVI. Symmetrical Polyphase Systems. 

263, p. 434. General equations of symmetrical systems. 
264, p. 435. Particular systems. 

265, p. 436. Resultant M.M.F. of symmetrical system. 
266, p. 439. Particular systems. 

CHAP. XXVII. Balanced and Uunbalanced Polyphase Systems. 

267, p. 440. Flow of power in single-phase system. 

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

system. 

269, p. 442. Balance factor. 

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

271, p. 413. Inverted three phase system. 

272, p. 444. Diagrams of flow of power. 

273, p. 447. Monocyclic and polycyclic systems. 

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

275, p. 448. The same in rectangular coordinates. 

276, p. 450. Main power axes of alternating-current system. 



CONTENTS. XIX 

CHAP. XXVIII. Interlinked Polyphase Systems. 

277, p. 452. Interlinked and independent systems. 

278, p. 452. Star connection and ring connection. Y connection and 
delta connection. 

279, p. 454. Continued. 

280, p. 455. Star potential and ring potential. Star current and ring 
current. Y potential and Y current, delta potential and delta 
current. 

281, p. 455. Equations of interlinked polyphase systems. 

282, p. 457. Continued. 
CHAP. XXIX. Transformation of Polyphase Systems. 

283, p. 460. Constancy of balance factor. 

284, p. 460. Equations of transformation of polyphase systems. 

285, p. 462. Three-phase, quarter-phase transformation. 

286, p. 463. Some of the more common polyphase transformations. 

287, p. 466.f Transformation with change of balance factor. 

CHAP. XXX. Copper Efficiency of Systems. 

288, p. 468. General discussion. 

289, p. 469. Comparison on the basis of equality of minimum dif- 
ference of potential. 

290, p. 474. Comparison on the basis of equality of maximum dif- 
ference of potential. 

291, p. 476. Continued. 

CHAP. XXXI. Three-phase System. 
292, p. 478. General equations. 

293, p. 481. Special cases: balanced system, one branch loaded, 
two branches loaded. 

CHAP. XXXII. Quarter-phase System. 
294, p. 483. General equations. 
295, p. 484. Special cases : balanced system, one branch loaded. 

APPENDIX I. Algebra of Complex Imaginary Quantities. 

296, p. 489. Introduction. 

297, p. 489. Numeration, addition, multiplication, involution. 

298, p. 490. Subtraction, negative number. 

299, p. 491. Division, fraction. 

300, p. 491. Evolution and logarithmation. 

301, p. 492. Imaginary unit, complex imaginary number. 

302, p. 492. Review. 

303, p. 493. Algebraic operations with complex quantities. 

304, p. 494. Continued. 

305, p. 495. Roots of the unit. 

306, p. 495. Rotation. 

307, p. 496. Complex imaginary plane. 



CONTENTS. 



APPENDIX II. Oscillating Currents. 

308, p. 497. Introduction. 

309, p. 498. General equations. 

310, p. 499. Polar coordinates. 

311, p. 500. Loxodromic spiral. 

312, p. 501. Impedance and admittance. 

313, p. 502. Inductance. 

314, p. 502. Capacity. 

315, p. 503. Impedance. 

316, p. 504. Admittance. 

317, p. 505. Conductance and susceptance. 

318, p. 506. Circuits of zero impedance. 

319, p. 506. Continued. 

320, p. 507. Origin of oscillating currents. 

321, p. 508. Oscillating discharge. 

322, p. 509. Oscillating discharge of condensers 

323, p. 510. Oscillating current transformer. 

324, p. 512. Fundamental equations thereof. 



THEORY AND CALCULATION 

OF 

ALTERNATING-CURRENT PHENOMENA. 



CHAPTER I. 

INTRODUCTION. 

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 
current. 

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 z 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 P is the 
power expended in the circuit of E.M.F., e, and current, /. 

4.) Kirchhoff's laws : 

a.} The sum of all the E.M.Fs. in a closed circuit = 0, 
if the E.M.F. consumed by the resistance, ir, is also con- 
sidered as a counter E.M.F., and all the E.M.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 curr'ents which rapidly and periodically change their 



2 ALTERNATING-CURRENT PHENOMENA. 

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 = e ] s, where z, the 
apparent resistance, or 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 
E.M.F. 

Impedance, z, is, in the system of absolute units, of the 
same dimensions as resistance (that is, of the dimension 
LT~ l = velocity), and is expressed in ohms. 

It consists of two components, the resistance, r, and the 

reactance, x, or , 

0= Vr 2 + Ar 2 . 

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 



INTRODUCTION. 3 

circuit, but depends 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. 

ELECTRO MAGNETISM. 

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 4> = the magnetic flux produced by, and inter- 
linked with, the current i (where those lines of magnetic 
force, which are interlinked w-fold, or pass around n turns 
of the conductor, are counted n times), the ratio, $ / z, is 
denoted by L, and called self -inductance, or the coefficient of 
self-induction 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 : 



4 ALTERNATING-CURRENT PHENOMENA. 

If a conductor surrounds with ;/ turns a magnetic cir- 
cuit of reluctance, (R, the current, i, in the conductor repre- 
sents the M.M.F. of ni ampere-turns, and hence produces 
a magnetic flux of //(R lines of magnetic force, sur- 
rounding each n turns of the conductor, and thereby giving 
<1> =: ;/ 2 //(R interlinkages between the magnetic and electric 
circuits. Hence the inductance is L = $/ i = ;/ 2 /(R. 

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 / is the current, and L is the inductance of 
a circuit, the magnetic flux interlinked with a circuit of 
current, z, is Li, and 4 NLi is consequently the average 
rate of cutting ; that is, the number of lines of force cut 
by the conductor per second, where N ' = 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 
ir/2 H- 1, the maximum rate of cutting is 2-n-N, and, conse- 
quently, the maximum value of E.M.F. induced in a cir- 
cuit of maximum current strength, i, and inductance, L, is, 



Since the maximum values of sine waves are proportional 
(by factor V2) to the effective values (square root of mean 
squares), if i = effective value of alternating current, e = 
2 TT NLi is the effective value of E.M.F. of self-inductance, 
and the ratio, e I i 2 TT NL, is the magnetic reactance : 

x m = 2 TT NL. 
Thus, \ir resistance, x m = reactance, z = impedance, 

the E.M.F. consumed by resistance is : e l = ir ; 
the E.M.F. consumed by reactance is : <? 2 = /v /; , : 



INTRODUCTION. 5 

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



e 

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

CAPACITY. 

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 IV, and 2 TT N the 
maximum rate of charge and discharge, sinusoidal waves sup- 
posed, hence, i 2 TT ./VCV the current passing into the con- 
denser, which is in quadrature to the E.M.F., and leading. 



It is then:- 



the "capacity reactance" or " condensance" 

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 is 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 m -** 

The total resistance of a circuit is equal to the sum of 
all the resistances connected in series ; the total reactance 
of a circuit is 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. 



6 AL TERN A TIA'G- CURRENT PHENOMENA, 

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

5. In Joule's law, P = i 2 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. 

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 : 

P = ei cos <. 

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

Kirchhoff's 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 : 

/ = / s in ^ (/ - 4) = /sin 2 TT yV (/ - 4) ; 



INTRO D UC TION. 



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 / T is the frequency or 
number of complete periods per second ; and t\ is the time, 
where the wave is zero, or the epoch of the wave, generally 
called the pliasc* 

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 count the time from the moment 



T\ 



rS 






Fig. 1. Sine Wave, 

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

= / sin 2 TT Nt ; 
or, / = /cos 2 TT Nt. 

Since it is a univalent function of time, that is, can at a 
given instant have one value 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 : 

/ = 7i sin 2 irN(t A) + 7 2 sin 4 TrJV(t - / 2 ) 
+ 7 3 sin 



* " Epoch " is the time where a periodic function reaches a certain value, 
for instance, zero; and "phase" is the angular position, with respect to a 
datum position, of a periodic function at a given time. Both are in alternate- 
current phenomena only different ways of expressing the same thing. 



8 



ALTERNA TING-CURRENT PHENOMENA. 



where f v 7 2 , 7 3 , . . . are the maximum values of the differ- 
ent components of the wave, f v f v / 3 . . . the times, where 
the respective components pass the zero value. 

The first term, 7 X sin lir N (t tj, is called the fun- 
damental wave, or the first harmonic; the further terms are 
called the higher harmonics, or "overtones," in analogy to 
the overtones of sound waves. I n sin 2 mr N (t /) is the 
th harmonic. 

By resolving the sine functions of the time differences, 
/ f p t / 2 . . . , we reduce the general expression of 
the wave to the form : 

A l sin 2 TrNt + A* sin 4 v Nt + A z sin G TT Nt + . . . 
1 cos27rA?-f^ 2 cos47rA?-f ^ 8 cos67ry\7+ . . . 



F/g. 2. Wave without Even Harmonics. 

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 

ty : i = 7i sin 2 TT N(t A) + 7, sin 6 TT N (t / 3 ) 

+ 7 5 sin 10 TT A^(/ / 5 ) + ... 
or, 

/ = ^ sin 2 TT A7 + A z sin 6 TT A7 + A & sin 10 w A? + . . . 
cos 2 TT Nt + ^ 8 cos 6 TrNt + ^ 5 cos 10 vNt + . . . 



INTR OD UC TION. 



9 



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. 
3 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 



Fig. 3. Wave with Even Harmonics. 

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 v 
nomena, we can safely assume the alternating wave as a 
sine wave, without making any serious error ; and it will be 



10 AL TERN A TING-CURRENT PHENOMENA. 

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. 



INSTANTANEOUS AND INTEGRAL VALUES. 



11 



CHAPTER II 

INSTANTANEOUS VALUES AND INTEGRAL VALUES. 

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 



Fig. 4. Alternating Wave. 

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 pttlsating wave. 



12 



ALTERNA TING-CURRENT PHENOMENA. 



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. 5. 

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- 



Fig. 5. Pulsating Wave. 

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 same 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 



INSTANTANEOUS AND INTEGRAL VALUES. 



13 



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 : 




Fig. 8. 

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

Then, while the angle < traverses the arc -n- / 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- 7r/2, or 2 / TT +- 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 -f- 2 / 77-, and since 
the variations of a sine-function are sinusoidal also, we 
have, 

o 

Mean value of sine wave -r- maximum value = -f- 1 

7T 

= .63663. 

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



14 AL TERNA TING-CURRENT PHENOMENA. 

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

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 : 

TJie effective value of an alternating wave, or tJie 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 : 




Fig. 7. 



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

Then, since the sines of any angle </> and its complemen- 
tary angle, 90 <, fulfill the condition, 

sin 2 $ + sin 2 (90 <) = 1, 

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 
root of the mean square, or the effective value of the sine, 
= 1/V2. That is: 



INSTANTANEOUS AND INTEGRAL VALUES. 



15 



The effective value of a sine function bears to its 
mum value the ratio, 
1 

V2 

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



1 = .70711. 



MAX. 


EFF. 


ARITH. MEAN. 


Half 
Period. 


Whole 
Period. 


1 


1 

V2 


2 

7T 





1 


.7071 


.63663 





1.4142 


1 


.90034 





1.5708 


1.1107 


1 






10. Coming now to the general alternating wave, 

/ = Ai sin 27r Nt + A z sin 4-n- Nt + A 3 sin GTT Nt + . . . 
+ BI cos 2-n-Nt + B* cos TrNt + s cos GTT Nt + . . 

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



1= V* W 

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



16 ALTERNATING-CURRENT PHENOMENA, 



CHAPTER III. 

LAW OF ELECTRO-MAGNETIC INDUCTION. 

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 8 = 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 lines 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 n times the increase or decrease, per second, 
of the flux inclosed by the turns, times 10~ 8 . 

If the change of the flux inclosed by the turn, or by n 
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, n 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. 



LAW OF ELECTRO-MAGNETIC INDUCTION. 17 

Hence, if N= number of complete cycles per second, 
or the frequency of the flux 3>, the average E.M.F. induced 
in n turns is, 

&vg , = 4 3> N 10 ~ 8 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 field 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, <I> = flux inclosed 
per turn, and N = frequency, the E.M.F. induced in the 
machine is E = 44>7V10~ 8 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 n is the number of 
turns in series, $ the maximum flux inclosed per turn, and 
JV the frequency, this formula gives, 

avg = 4 4> JVW ~ 8 volts. 
Since the maximum E.M.F. is given by, 

^maz. = ^avg 

we have 

^"max. = 2 7 r<S>7V 7 10- 8 VOltS. 

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



we have 

es . = 

= 4.44 n 4>^10- 8 volts, 

which is the fundamental formula of alternating-current 
induction by sine waves. 



18 AL TERN A TING-CURRENT PHENOMENA, 

13. If, in a circuit of n turns, the magnetic flux, <t>, 
inclosed by the circuit is produced by the current flowing 
in the circuit, the ratio 

flux X number of turns X 10~ 8 



current . 

is called the inductance, L, of the circuit, in henrys. 

The product of the number of turns, n, into the maxi- 
mum flux, <S>, produced by a current of / amperes effective, 
or / V2 amperes maximum, is therefore 

n =Z/V2 10 8 ; 
and consequently the effective E.M.F. of self-inductance is: 



E = V2 

=' 2 TT NLI volts. 

The product, x = 2 vNL, 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 

E = Ix, 

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. 



GRAPHIC REPRESENTA TION, 



19 






CHAPTER IV. 

GRAPHIC REPRESENTATION. 

14. While alternating waves can be, and frequently are, 
represented graphically in rectangular coordinates, with the 
time as abscissae, 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 
vectores. 




Fig. 8. 



Thus the two waves of Figs. 2 and 3 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 



20 



ALTERNA TING-CURRENT PHENOMENA. 



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, 





Fig. 9. B, Fig. 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, 
1= OC, represents the intensity of the wave ; and the am- 
plitude of the diameter, OC, /_& = AOC, is thefl/iase of the 
wave, which, therefore, is represented analytically by the 

function : 

t = /cos (< w), 

where </> = 2 IT / / T is the instantaneous value of the ampli- 
tude corresponding to the instantaneous value, 2, 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, at the amplitude AOB l = ^ = 2 ^ / T 
(Fig. 10), the instantaneous value is OB' ; at the amplitude 
AO 2 = <f> 2 = 27T/ 2 / T, the instantaneous value is ~OJ3", and 
negative, since in opposition to the radius vector OB Z . 

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. 



GRAPHIC REPRESENTATION. 21 

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 by the length and direction of 
a vector, OC, Fig. 10, and its analytical expression would 
then be c = OC cos (<f> w). 

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, 

l> = cos 

= B cos (</> - fy 
is in Fig. 10# represented by 

T) 

vector OB = , of phase 

A OB = G! ; and the wave, 
c= Ccos 




is in Fig. 10# represented by vector OC=j=, of phase 
AOC= -* 

The former is said to lag by angle ^, the latter to lead 
by angle 2 , with regard to the zero position. 

The wave b lags by angle (o^ + 2 ) behind wave c, or c 
leads b by angle (w x + 2 ). 

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

If, for instance, two sine 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 



22 



ALTERNATING-CURRENT PHENOMEA?A. 



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

For at any time, /, represented by angle <f> = 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 : 

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




Fig. 11. 



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

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

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

a.) 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 
included. 

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



GRAPHIC REPRESENTATION. 



23 



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 



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




Fig. 12. 

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, Of. 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., Ir, is required in phase with the current, repre- 
sented by OE r 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. 



24 



ALTERNA TING-CURRENT PHENOMENA. 



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 X . 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, E , has to give the three E.M.Fs., 
E, E r y and E x , hence it is determined as their resultant. 
Combining by the parallelogram law, OE r and OE X , give 
OE Z , the E.M.F. required to overcome the impedance of 
the line, and similarly OE Z and OE give OE , 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 



or, E = VX 2 (/*) 2 - Jr. 

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., 
E , and have derived E by combining E r , E x , and E. 



E'. 



E? 



Fig. 13. 



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



GRAPHIC REPRESENTA TION. 



25 



Ef and a ! combine to form E g) the counter E.M.F. of 
impedance ; and since Eg and E must combine to form 
E, E is found as the side of a parallelogram, OE EE g) 
whose other side, O z ', and diagonal, OE, are given. 

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

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, 




E.V o 



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 Ix, 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, OI, is the current, 

OE r = E.M.F. consumed by resistance, 
OE r ' = counter E.M.F. of resistance, 
OE X = E.M.F. consumed by inductance, 
OE X ' = counter E.M.F. of inductance, 
OE Z = E.M.F. consumed by impedance, 
OE t ' = counter E.M.F. of impedance. 



26 ALTERNATING-CURRENT PHENOMENA. 

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 
mechanics. 

if. 



\f 
X 



Fig. 15. 



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 



A E = E E = V />' 2 /* 2 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 *, 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 w, and 
we choose again as the zero line, the current OI (Fig. 16), 
the E.M.F., OE is ahead of the current by angle . The 



GRAPHIC REPRESENTA TION. 



27 



E.M.F. consumed by the resistance, Ir, is in phase with the 
current, and represented by OE r ; the E.M.F. consumed 
by the reactance, Ix, is 90 ahead of the current, and re- 
presented by OE X . Combining OE, OE r , and OE X , we 
get OE , 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, 

E = V(^ cos w + Ir) 2 -f- (E sin w + Ix) z . 




Fig. 18. 

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, Of, 
by the angle of lead '; and in this case we get, by 
combining OE with OE r , in phase with the current, and 
OE X , 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 



28 



AL TERN A TING- CURRENT PHENOMENA. 



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. 13 to 17, can be considered polar 
diagrams of an alternating-current generator of an E.M.F., 
E 0> a resistance E.M.F., E r = fr, a reactance E.M.F., 
E x = 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 




'E. 



Fig. 17. 

a circuit 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, 



GRAPHIC REPRESEiVTA TIOiV. 



29 



since the induced E.M.F. lags 90 behind the inducing 
flux. Thus the secondary induced E.M.F., JE 1 , will be 
represented by a vector, O l} in Fig. 18, at the phase 
180. The secondary current, f lf lags behind the E.M.F., 
E lt by an angle a> 1} which is determined by the resistance 
and inductance of the secondary circuit ; that is, by the 
load in the secondary circuit, and is represented in the dia- 
gram by the vector, OF l} of phase 180 + Gj. 




Fig. 18. 



Instead of the secondary current, f lt we plot, however, 



the secondary M.M.F., 



where n 1 is the number 
This. 



of secondary turns, and $ l is given in ampere-turns. 
makes us independent of the ratio of transformation. 

From the secondary induced E.M.F., E ly we get the flux 
3>, required to induce this E.M.F., from the equation 



where 

i = secondary induced E.M.F. , in effective volts, 
JV = frequency, in cycles per second, 
;/ 1 = number of secondary turns, 
3> = maximum value of magnetic flux, in webers. 
The derivation of this equation has been given in a 
preceding chapter. 

This magnetic flux, 4>, is represented by a vector, O<b, at 
the phase 90, and to induce it an M.M.F., ff is required, 



30 ALTERNATING-CURRENT PHENOMENA. 

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 effect of hysteresis, neglected at present, is to shift 
OF ahead of O, by an angle a, the angle of hysteretic 
lead. (See Chapter on Hysteresis.) 

This M.M.F., O 7 , is the resultant of the secondary M.M.F., 
JF lf and the primary M.M.F., SF ; or graphically, OF is the 
diagonal of a parallelogram with OF l and OF as sides. OF 1 
and OF being known, we find OF , the primary ampere- 
turns, and therefrom, and the number of primary turns, n , 
the primary current, I = & / n , which corresponds to the 
secondary current, 7 1 . 

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

To overcome the reactance, x = 2 * n L , of the pri- 
mary coil, an E.M.F. E x = I x is required, 90 ahead of 
the current f , and represented by vector, OE X . 

The resultant magnetic flux, 4>, which in the secondary 
coil induces the E.M.F., E I} induces in the primary coil an 
E.M.F. proportional to E by the ratio of turns n / n l} and 
in phase with E l , or, 

77 f "o zr 
, *m2 lf 

1 

which is represented by the vector OE % '. To overcome this 
counter E.M.F., E t ' t a primary E.M.F., E t , is required, equal 
but opposite to E t ', and represented by the vector, OE,. 

The primary impressed E.M.F., E , must thus consist of 
the three components, OE it OE r , and OE X , and is, there- 
fore, their resultant OE , while the difference of phase in 
the primary circuit is found to be <3 = E OF . 

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



GRAPHIC REPRESENTA TION. 



31 



ondary current, 7 L and therefore secondary M.M.F., & v but 
with different conditions of secondary displacement : 

In Fig. 18, the secondary current, /i , lags 60 behind the sec- 
ondary E.M.F., EI. 

In Fig. 19, the secondary current, 7 1} is in phase with the 
secondary E.M.F., E l . 

In Fig. 20, the secondary current, 7 : , leads by 60 the second- 
ary E.M.F., lf 





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 




Fig. 20. 

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 



32 AL TERNA TING-CURRENT PHENOMENA. 

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. 






SYMBOLIC METHOD. 



CHAPTER V. 

SYMBOLIC METHOD. 

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 l 
100 volts, and 1-^ = 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, x-^ = .08 ohms, giving a lag of ^ = 3.6. We have 
also, 

n^ = 30 turns. 

n = 300 turns. 

CFi = 2250 ampere-turns. 

y = 100 ampere-turns. 

E r = 10 volts. 

JS X = 60 volts. 

E { = 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 1 FF ^ and from the combination of 
vectors of the relative magnitudes 1:6: 100. 

Hence the importance of the graphical method consists 



34 



ALTERNA TING-CURRENT PHENOMENA. 



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 




Fig. 21. 

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

ffo = Vfr 2 + S^ 2 + 20^ sin Wi, 

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

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



Fig. 22. 

25. We have seen that the alternating sine wave is 
represented in intensity, as well as phase, by a vector, Of, 
which is determined analytically by two numerical quanti- 
ties the length, Of, or intensity ; and the amplitude, AOf, 
or phase <3, of the wave, /. 

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



S YMB OL1C ME T11OD. 



35 



/ and <3, we can represent it by its rectangular coordinates, 
a and b (Fig. 22), where 

a = fcos u> is the horizontal component, 

b = I sin co 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 




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 a' and b' the components of another 
sine wave, /' (Fig. 23), their resultant sine wave, I , has the 
rectangular components a (a -f- a!}, and b = (b -f- b'}. 

To get from the rectangular components, a and b, of a 
sine wave, its intensity, i, and phase, o>, we may combine a 
and b by the parallelogram, and derive, 



tan 



36 AL TERN A TING-CURRENT PHENOMENA . 

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 components 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, 

I=a 



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

/ = V^ + // 2 , 

and of phase, tan <3 = b / 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 -f- 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 weans reversing' the wave, or rotating it through 180, 
or one-half period. 

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



SYMBOLIC METHOD. 37 

component, b, and the vertical component, a, and is rep- 
resented symbolically by the expression, ja b, 
Multiplying, however, a + jb by/, we get : 



therefore, if we define the heretofore meaningless symbol, 
j, by the condition, 

y 2 = - i, 

we have 

/(*+/*) =ja 1>; 

hence : 

Multiplying the symbolic expression, a -\- jb, of a sine wave 
by j means rotating the wave through 90, or one-quarter pe- 
riod ; tJiat is, retarding the wave through one-quarter period. 



Fig. 24. 

Similarly, 

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

since y' 2 = 1, j = 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 



38 ALTERNATING-CURRENT PHENOMENA. 

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

the phase by 

tan <o = - , 
a 
and 

a = i cos to, 

b = i sin w ; 

hence the wave a +jb can also be expressed by 
/ (cos <i> -\-j sin <3), 

or, by substituting for cos w and sin w their exponential 

expressions, we obtain 

id. 

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 +jb 
and 



combined give the sine wave 

7- (a + 

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

29. If /= i +/z' is a sine wave of alternating current, 
and r is the resistance, the E.M.F. consumed by the re- 
sistance is in phase with the current, and equal to the prod- 
uct of the current and resistance. Or 
rl ' ri -\- jri' . 

If L is the inductance, and x = 2 TT NL the reactance, 
the E.M.F. produced by the reactance, or the counter 



SYMBOLIC METHOD. 39 

E.M.F. of self-induction, is the product of the current 
and reactance, and lags 90 behind the current ; it 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 -f- xi'. 

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, if / = i -\-ji' is the current, the E.M.F. required 
to overcome the impedance, Z = r jx, is 



hence, sincey" 2 = 1 



or, if E = e -\- 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 



T _ 



or, if E = e -\-je' is the impressed E.M.F., and 7 = i ' -\- ji' 
the current flowing in the circuit, its impedance is 

+./>') O'-./*'') '+^*'' . ' ~ ei ' 

' 



40 ALTERNATING-CURRENT PHENOMENA. 

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

the terminals of the condenser is E = - - , 90 behind 
the current ; and may be represented by - - , or jx^ /, 

where x^ = - is the capacity reactance or condensatice 
2 TT NC 

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 x = 2 IT NL = magnetic reactance. 

If C = capacity, x^ = - = capacity reactance, or conden- 
sance ; 

Z = r j (x JCi), is the impedance of the circuit 
Ohm's law is then reestablished as follows : 



, -, . 

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

31. Since the combination of sine waves takes place by 
the addition of their symbolic expressions, Kirchhoff'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. 



SYMBOLIC METHOD. 41 

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, b = 0. 

Resolving the E.M.Fs. and currents in the expression of 
Kirchhoff '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 sum 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, 7, over an inductive line, we can now represent the 
impedance of the line by 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 

E = E + ZI. 

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

E = E -f ir jix ; 

hence, with non-inductive load, or E = e, 
E =(e + ir) -jix, 



+ /r) 2 + (/X) 2 , tan S> = 



42 ALTERNATING-CURRENT PHENOMENA. 

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



*-*) 2 > tan <S 



e + /> 

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



/V) , tan w = 
values which easily permit calculation. 



TOPOGRAPHIC METHOD. 43 



CHAPTER VI. 

TOPOGRAPHIC METHOD. 

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 OI 
(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 OI- (Fig. 26), or by 7 l = i 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 to B is E l = e je' . 



44 



ALTERNA TING-CURRENT PHENOMENA. 



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. 



-*' 

Fig. 28. 

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




Fig. 27 



from the common connection O of the three branch circuits 
to the terminals A 19 A 2 ,A B , be represented by E lt E 2 , 3 . 
Then the difference of potential from A 2 to A is z lf 



since the two E.M.Fs., E l and 



are connected in cir- 



cuit between the terminals A, and A*, in the direction, 



TOPOGRAPHIC METHOD. 45 

A l O A 2 ; that is, the one, E z , in the direction OA 2 , 
from the common connection to terminal, the other, JS 1 , in 
the opposite direction, A^O, from the terminal to common 
connection, and represented by E l . Conversely, the dif- 
ference of potential from A 1 to A z is E l E z . 

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 
phase. 




Fig. 28. 

Thus, for example, in an interlinked three-phase system 
with three E.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- 



46 



AL TERN A TING-CURRENT PHENOMEMA. 



tial at one of the three-phase terminals, by point E r The 
potentials at the two other terminals will then be given by 
the points E z and E& which have the same distance from 
O as E v and are equidistant from E and from each other. 
The difference of potential between any pair of termi- 
nals for instance E^ and E 2 is then the distance E Z E V 
or EE V according to the direction considered. 

35. If now the three branches OE V ~OE Z and "OE W of 
the three-phase system are loaded equally by three currents 
equal in intensity and in difference of phase against their 



THUEE-PHA8E 8V8TEM 
48 LAO 



BALANCED THREE-PHASE SYSTEM 

NON-INDUCTIVE LOAD 
E 




Fig. 29. 



E.M.Fs., these currents are represented in Fig. 29 by the 
vectors 07^ = 07 2 = Of s = I, lagging behind the E.M.Fs. 
by angles E.O^ = E Z OI Z = E Z OI & = Q. 

Let the three-phase circuit be supplied over a line of 
impedance Z = r^ jx\ from a generator of internal im- 
pedance Z = x -jx . 

In phase OE V the E.M.F. consumed by resistance r^ is 
represented by the distance E^EJ = Ir v in phase, that is 
parallel with current OI V The E.M.F. consumed by re- 
actance #! is represented by E^Ej' = Ix v 90 ahead of cur- 



TOPOGRAPHIC METHOD. 



47 



rent OI r The same applies to the other two phases, and 
it thus follows that to produce the E.M.F. triangle E^E^E^ 
at the terminals of the consumer's circuit, the E.M.F. tri- 
angle E^E^E? is required at the generator terminals. 

Repeating the same operation for the internal impedance 
of the generator we get E"E'" = Ir oi and parallel to OI V 
E'"E = Ix oy and 90 ahead of ~OT V and thus as triangle of 
(nominal) induced E.M.Fs. of the generator EEE. 

In Fig. 29, the diagram is shown for 45 lag, in Fig. 30 
for noninductive load, and in Fig. 31 for 45 lead of the 
currents with regard to their E.M.Fs. 



BALANCED THREE 
-PHASE SYSTEM 

45 LEAD 




THREE-PHASE CIRCUIT 

80LA 

TRANSMISSION LINE' 

WITH DISTRIBUTED 

CAPACITY, INDUCTANCB 

RESISTANCE AUD LEAKAQB 

I, 




Fig. 31. 



Fig. 32. 



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 EEE, the 
E.M.Fs. at the receiver's circuit, E v E z , E 9 fall off more 
with lagging, 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. 



48 ALTERNATING-CURRENT PHENOMENA. 

Let in Fig. 33 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 l equidistant from and opposite each other, 
and the two currents issuing from the terminals are rep- 
resented by the points / and I 1 , equidistant from and 
opposite each other, and under angle & with E and E l 
respectively. 

Considering first an element of the line or cable next to 
the receiver circuit. In this an E.M.F. EE l is consumed 
by the resistance of the line element, in phase with the 
current OI, and proportional thereto, and a current // x 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 OI^ and OE l respectively. 

Passing now to the next cable element we have again an 
E.M.F. E 1 E Z proportional to and in phase with the current 
OI^ and a current IJ Z proportional to and 90 ahead of the 
E.M.F. OE V and thus passing from element to element 
along the cable to the generator, we get curves of E.M.Fs. 
e and e 1 , and curves of currents i and i l , which can be called 
the topographical circuit characteristics, and which corre- 
spond to each other, point for point, until the generator 
terminal voltages OE and OE l and the generator currents 
OI and OIJ are reached. 

Again, adding 'E~E r ' = I r and parallel OI and E"E = 
I x and 90 ahead of ~OI M gives the (nominal) induced 
E.M.F. of the generator OE, where Z = 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. 



TOPOGRAPHIC METHOD. 



49 



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

Let, in Fig. 38, O v ~OE y ~OE Z = three-phase E.M.Fs. 
at receiver circuit, equidistant from each other and = E. 

Let OI V Oly Of 3 = three-phase currents in the receiver 
circuit equidistant from each other and = /, and making 
with E the phase angle <3. 

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




sumed by the reactance of the line element, 90 ahead of 
the current OI V and proportional thereto. 

In the same line element we have a current IJ^ in phase 
with the E.M.F. OE V and proportional thereto, representing 
the loss of energy current by leakage, dielectric hysteresis, 
etc., and a current ^V/', 90 ahead of the E.M.F. OE V 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, E, E s , and generator currents //, 
/ 2 , 7 8 , over the topographical characteristics of E.M.F. e v 
e v e s , and of current i v z' 2 , z' 3 , as shown in Fig. 33. 

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



50 



ALTERNATING-CURRENT PHENOMENA. 



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



TRANSMISSION 

WITH DISTRIBUTED 

CAPACITY, INDUCTANCE 

RESISTANCE AND LEAKAGE 
90 LAO 




Fig. 34. 

Only the circuit characteristics of the first phase are 
shown as ^ and z' r As seen, passing from the receiving 
end towards the generator end of the line, potential and 



TRANSMISSION LINE 

WITH DISTRIBUTED CAPACITY, INDUCTANCE 
RESISTANCE AND LEAKAGE 




Fig. 35. 



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



TOPOGRAPHIC METHOD. 51 

37. a. More 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 co-ordinates 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. 



52 ALTERNATING-CURRENT PHENOMENA. 



CHAPTER VII. 

ADMITTANCE, CONDUCTANCE, SUSCEPTANCE. 

38. If in a continuous-current circuit, a number of 
resistances, ?\, r%, r 3 , . . . are connected in series, their 
joint resistance, R, is the sum of the individual resistances 

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 : 

= J_ _l_ JL + J_ + 

/*! /*2 ^3 

Hence, in the latter case it is preferable to introduce, in- 
stead of the term resistance, its reciprocal, or inverse value, 
the term conductance, g = 1 / r. If, then, a number of con- 
ductances, g^, g^, g z , . . . are connected in parallel, their 
joint conductance is the sum of the individual conductances, 
or G = g l + g z + g z + . . . When using the term con- 
ductance, the joint conductance of a number of series- 
connected conductances becomes similarly a complicated 
expression 



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

The joint resistance of a number of series-connected resis- 
tances is equal to the sum of the individual resistances ; the 



ADMITTANCE, CONDUCTANCE, SUSCEPTANCE. 53 

joint conductance of a number of parallel-connected conduc~ 
tances is equal to the sum 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., Ir; 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., 



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, 

-I- 

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; 



54 ALTERNATING-CURRENT PHENOMENA. 

it consists of the component g, which represents the co- 
efficient of current in phase with the E.M.F., or energy 
current, gE t 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 susceptance, of 
the circuit. Hence the conductance, g, is the energy com- 
ponent, and the susceptance, b, the wattless component, 
of the admittance, Y = g -f jb, while the numerical value of 

admittance is 

y = Vr 1 + P ; 

the resistance, r, is the energy component, and the reactance, 
x, the wattless component, of the impedance, Z r jx, 
the numerical value of impedance being 

z = VV' + x\ 

40. As shown, the term 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 x = 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 = QO , or x = oo , since in this case no current 
passes, and either component of the current = 0. 



ADMITTANCE, CONDUCTANCE, SUSCEPTANCE. 55 

2.) If r = 0, since in this 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 x = oo , or r = oo . 

2.) If * = 0. 

From the definition of admittance, Y ' = g + jb t as the 
reciprocal of the impedance, Z = r jx y 

we have Y , or, g -f- jb = 



Z r jx 

or, multiplying numerator and denominator on the right side 

by(r 



hence, since 

(r-jx) (r + = r 2 + x* = z\ 



x r . . x 



, 
and conversely 



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

gr = 

hence, 

an j _ 1 1 ) the absolute value of 



y V^" 2 + b* ' ) impedance ; 
1 1 ) the absolute value of 

admittance. 



56 



AL TERNA TING-CURRENT PHENOMENA. 



41. If, in a circuit, the reactance, *-, is constant, and the 
resistance, r, is varied from r = to r = oo , the susceptance, 
b, decreases from b = 1 / x at r = 0, to # = at r = cc ; 
while the conductance, g at r = 0, increases, reaches 
a maximum for r = x, where g 1 / 2 r is equal to the 
susceptance, or g = b, and then decreases again, reaching 
g = at r = oo . 





































s 


































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RE; 


CT 


NC 


CO 


NST 


ANT 


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OH 


MS 








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s 


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/ 














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MS 


















l.S 



In Fig. 36, for constant reactance ^- = .5 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, 



ADMITTANCE, CONDUCTANCE, SUSCEPTANCE. 57 

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 g = 1 / r to 0, and the susceptance passing from at 
x = to the maximum, b = 1 / 2 r = g =1 / '2 x at x = r, 
and to b = at x = GO . 

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 =Y, 

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-connected 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. 



58 ALTERNATING-CURRENT PHENOMENA. 



CHAPTER VIII. 

CIRCUITS CONTAINING RESISTANCE, INDUCTANCE, AND 
CAPACITY. 

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

E = ZS, or, / = YE, 

and *%E = in a closed circuit, 

S/ = at a distributing point, 

where E, I, Z, Y, 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 alternating-current 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 
E.M.F., 



o = e. +/V, E. = 






RESISTANCE, INDUCTANCE, CAPACITY. 59 

let the receiving circuit of impedance 

Z = r jx, z = Vr 2 + x' 2 , 

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

Z + r = r + r jx\ 
hence the current is 



____ 
" Z + r r+r -jx (r + r ) 2 -f * 2 ' 

and the E.M.F. of the receiving circuit, becomes 
E = IZ = ^ ( r ~ J ^ = ^ 



or, in absolute values we have the following : 
Impressed E.M.F., 

current, 

zr zr 



V(r + ;- ) 2 + x 2 -Vz 2 + 
E.M.F. at terminals of receiver circuit, 



E = E n J >* + * 2 . Eo 



Vs 2 + 2rr + r 2 
difference of phase in receiver circuit, tan w = - ; 

difference of phase in supply circuit, tan o> = 

since in general, 

tan (phase) = ^aginary component ^ 
real component 

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



1= -=3_ 

r+ r. 



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



60 ALTERNATING-CURRENT PHENOMENA. 

b.} If r is negligible compared with x, as in a wattless 
receiver circuit, 

7= E , = . X - 



or, for small values of r , 

/= , ^ = ^ ; 

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 x z , 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.F:, E, 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, s = 1 ohm ; 
hence / = E, and 

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

with values of reactance, x = V^ 2 r 2 , for abscissae, from 
x = + 1.0 to x = 1.0 ohm. 

As shown, / and E are smallest for x = 0, r = 1.0, 
or for the non-inductive receiver circuit, and largest for 
x = 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 x = 0, x = + .8, x = .8, the polar 
diagrams are shown in Figs. 38 to 40. 



RESISTANCE, INDUCTANCE, CAPACITY. 



61 



2.) Reactance in series witJi a circuit. 
45. In a constant potential system of impressed E.M.F., 



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

Z = r jx, z = -\/r 2 -|- x' 2 . 



IMPRESSED E.M.F. CONSTANT, E =IOO 
IMPEDANCE OF RECEIVER CIRCUIT CONSTANT, Z - 1.0 



LINE RESISTANCE CONSTANT n =.2 



3 - -.4 T-5 ' '.6 T.7 r-8 



Fig. 37. Variation of Voltage at Constant Series Resistance with Phase Relation of 
Receiver Circuit. 

Then, the total impedance of the circuit is 
Z -jx = rj(x +# e ). 



Er Er 



Fig. 38. 

and the current is, 
/= 



E 
Fig. 39. 



Z-jx r j(x + x }' 
/hile the difference of potential at the receiver terminals 



rjx 



62 ALTERNATING-CURRENT PHENOMENA. 

Or, in absolute quantities : 
Current, 

/_ Eo EQ 

* ~ 



Vr* -f- (x + x )' 2 V 'z' 1 + 2xx -\- x a 2 
E.M.F. at receiver terminals, 



r / r' + * = J^ 

V r a + (* + *)* V** + 2*.r + *. a 5 



difference of phase in receiver circuit, 

x 

tan <D = - ; 
r 

difference of phase in supply circuit, 



a.} If JT 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 . 

b.} 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 < z, 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 

^z*-\-2xx + x 2 = 2; 
or, x = 2 x. 

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, 
but E = E ,I= E /z. If x < 2 x, it raises, if x > Zv, 
it lowers, the voltage. 

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



RESISTANCE, INDUCTANCE, CAPACITY. 



63 



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 
circuit. 

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




= (!-}- *)'* expanded by the binomial theorem 



= nx 



Therefore, if x is small compared with r : 



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



Fig. 40. 



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. 



64 



AL TERNA TING-CURRENT PHENOMENA. 



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., E , 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., 

VOLTS E OR AMPERES I 



100 


IMPRESSED E.'M.F! CONSTANT, E 
IMPEDANCE OF RECEIVER CIRC.UI 

I. r=l.o x=o 

II. r=.6 X=H-,8 

111. r=.e i=-.8 


=160 
r CONS 

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1 

















'<! 


HM 

s t 


s 

h 'J 












T UCJTAN CE 




-REACT 


ANC 


E - 


-t-CONDENSANCE 



Fig. 41. 

E = 100 volts, and the following conditions of receiver 
circuit z= 1 Qj r = 1>0> x= ( Curve j) 

2=1.0, r= .6,^= .8(CurveII.) 
2= 1.0, r= .6, AT= .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. 



RESISTANCE, INDUCTANCE, CAPACITY. 



65 



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 x = i .8, or, x = x (the 
condition of resonance), and the E.M.F. reaches the value, 
E = 167 volts, or, E = E z] r. This rise of potential by 
series reactance continues up to x = il.6, or, x = %x, 



Fig. 42. 

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

At x = -8, x = =f .8, the total impedance of the circuit 
is r j (x -f x } = r = .6, x + x = 0, and tan S> = ; 
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. 



\ 



Fig. 43. 

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 

E = 100, x = .6, x = . (Fig. 42) E = 85.7 

x = + .8 (Fig. 43) E = 65.7 

(Fig. 44) E = 158.1 



66 



ALTERNA TING-CURRENT PHENOMENA. 



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

x = .2 (Curve I.) 

x = .6 (Curve II.) 

x = .8 (Curve III.) 

x o = 1.0 (Curve IV.) 

Xo = 1.6 (Curve V.) 

x = 3.2 (Curve VI.) 




Fig. 44. 

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

In Figs. 46 and 47, the same curves are plotted as in 
Fig. 45, but in Fig. 46 with the reactance, .*, of the receiver 
circuit as abscissas ; and in Fig. 47 with the resistance, r, of 
the receiver circuit as abscissae. 

As shown, the receiver voltage, E, is always lowest when 
x 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 x is a 
maximum for x = +1.0, x = 1.0, and r = 0, where 



RESISTANCE, INDUCTANCE, CAPACITY. 



L Q. 4 PHASE D FFERENCE IN CONSUMER SIR UIT 




l-90 80 70 bO 50 40 30 20 10 10 20 30 10 50 60 70 bO 90 OEUHE 

fig. 45. Variation of Voltage at Constant Series Reactance with Phase Angle of 
Receiver Circuit. 




Fig. 46. Variation of Voltage at Constant Series Reactance with Reactance of 
Receiver Circuit. 



68 



AL TERN A TING-CURRENT PHENOMENA. 



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

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 



Volts E 
or Amperes I. 
160 
150 
140 
130 
120 
110 
100 
90 
80 
70 




sfl 




Fig. 47. Variation of Voltage at Constant Series Reactance with Resistance of 
Receiver Circuit. 

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 series witJi 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. 



RESISTANCE, INDUCTANCE, CAPACITY. 69 

Hence the impedance of a reactive coil (choking coil) 
may be written thus : 

&Q = r o JXoi ZQ = V f -j- X o , 

where r is in general small compared with x . 
From this, if the impressed E.M.F. is 



E = e +je '> E = Ve 2 + e ' 2 

and the impedance of the consumer circuit is 

we get the current, /= ^- = -. 

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

. . 7 \ 7 " ( r \ *-\ //_!_ \ ' 

^I^o \ r ~T ' o) J \*- ~T *<>/ 

Or, in absolute quantities, 
the current is, 

~\/(r -f- r o y 2 -|- (x -j- ^; ) 2 V^ 2 + z 2 + 2 (rr 

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

E z E z 



V(r + r )' 2 + (x + x o y V^ 2 + Z * + 2 
the difference of phase in receiver circuit is, 

x 

tan oi = - ; 
r 

and the difference of phase in the supply circuit is, 



49. In this case, the maximum drop of potential will not 
take place for either x = 0, as for resistance in series, or 
for r = 0, as for reactance in series, but at an intermediate 
point. The drop of voltage is a maximum ; that is, E is a 
minimum if the denominator of E is a maximum ; or, since. 
z y z , r , x are constant, if rr + xx is a maximum, that is, 
since x = ~Vz 2 r 2 , if rr -f- x ~\/z 2 r 2 is a maximum. 



70 



AL TERN A TING CURRENT-PHEXOMENA. 



A function, f = rr -+- x V^ 2 r 2 is a maximum when 
its differential coefficient equals zero. For, plotting f as 
curve with r as abscissae, at the point where f 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. 

















































































/ 








































/ 








































/ 






































/ 






































^ 






































/ 


/ 
































^^- 


, " 


*^ 










'"^ 










^^~ 


Z^ 




L 


,~- 


' 


_---* 




































/ 






































/ 
























^__ 





~~ ^ 




. 


^ 


,--- 




















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SiL 




















































9- 


<-* 




























I. 


.9 


.8 


T f 


.0 


J 


.4 


.3 


.2 


., 


- 


-.1 - 


-.2 


-.3 - 


-.4 - 


-} ' 


-.fi 


-.? 


-.* 


2J 



Off. 48. 

Thus we have : 

f = rr + * Vs 2 r 2 = maximum or minimum, if 



Differentiating, we get : 




RESISTANCE, INDUCTANCE, CAPACITY. 



71 



That is, the drop of potential is a maximum, if the re- 
actance factor, x I r, of the receiver circuit equals the reac- 
tance factor, * /r , of the series impedance. 




Fig. 49. 



''o 
Fig. 50. 



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

Z = .S-/.4 (Curve I.) 

Z = 1.2 / 1.6 (Curve II.) 
as functions of the reactance, x, of the receiver circuit. 




Fig. 51. 

Figs. 49 to 51 give the polar diagram for E = 100, 
x = .95, x = 0, x = - .95, and Z = .3 -/ .4. 



72 ALTERNATING-CURRENT PHENOMENA. 

4.) Compensation for Lagging Currents by Shunted 
Condensance. 

51. 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 




Fig. 52. 



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

/= " . 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. 



RESISTANCE, INDUCTANCE, CAPACITY. 73 

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, c, entailing but a negligible loss 
of energy. 

Then, if E = impressed E.M.F., 

the current in receiver circuit is, 



the current in condenser circuit is, 

and the total current is 

J x o J c 



or, in absolute terms, I 



'=VfeJ + fe-'/ ; 



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



52. The main current, 7 , is in phase with the impressed 
E.M.F., E , or the lagging current is completely balanced, 
or supplied by, the condensance, if the imaginary term in 
the expression of I disappears ; that is, if 



This gives, expanded : 



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, 



74 ALTERNATING-CURRENT PHENOMENA. 

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

r 2 4- r 2 
Substituting c = ^/ " , 

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



7 = 





r J x o 

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, <:, 
be varied with the resistance, r; that is, with the varying 
load on the receiver circuit. 

In Fig. 53 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 II.), 
current in main circuit (Curve III.), 

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

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



RESISTANCE, INDUCTANCE, CAPACITY. 



75 



IMPRESSED E.M.F. CONSTANT, E = IOOO VOLTS. 
SERIES REACTANCE CONSTANT, X = IOO OHMS. 
VARIABLE RESISTANCE IN RECEIVER CIRCUIT. 
BALANCED BY VARYING THE SHUNTED CONDENSANCE, 

I. CURRENT IN RECEIVER CIRCUIT. 

II. CURRENT IN CONDENSER CIRCUIT. 

III. CURRENT IN MAIN CIRCUIT. 
JV. E.M.F. AT RECEIVER CIRCUIT. 




100 / 



r. OF RECEIVER 



CIRCUIT OHMS 



10 20 30 40 50 60 70 80 90 100 110 120 130 HO 150 160 170 180 190 200 

Fig. 53. Compensation of Lagging Currents in Receiving Circuit by Variable Shunted 
Condensance. 



53. If, however, the condensance is left unchanged, 
c = x 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 leading. 

We get in this case : 




r-jx 



The difference of phase in the main circuit is, 



tan u> = , 





which is = 0. 



76 



ALTERNA TING-CURRENT PHENOMENA. 



when r = or at no load, and increases with increasing 
resistance, as the lead of the current. At the same time, 
the current in the receiver circuit, 7, is approximately con- 
stant for small values of r, and then gradually decreases. 



IMPRESSED E.M.F. CONSTANT, EOIOOO VOLTS. 

SERIES REACTANCE CONSTANT, Xt, -<OO OHMS. 
SHUNTED CONDENSANCE CONSTANT, C= IOO OH 
VARIABLE RESISTANCE. IN RECEIVER CIRCUIT- 
(.CURRENT IN RECEIVER CIRCUIT. 
II. CURRENT IN CONDENSER C RCUIT. 
III. CURRENT IN MA N CIRCUIT. 
IV.E.M.F. AT RECEIVER CIRCUIT. 


MS. 




































































































































voi 


- 


















ii. 
























?00 




" -. 


~~~, 


^^. 


^ 


\. 
















. 







^^ 






-r_-~ 




















-x 


^ 


_^- 


L-* 






















rnn 
















^ 


' 


^ 


^ 























soo 










IV, 


/ 
















** 


""--^ 


^-^ 


















% 






























- -*. 


-^ ~, 




300 






/ 








































/ 










































/ 






RESISTANCE r OF RECEIVER CIRCUIT, OHMS. 










2 












MINI 










JO 20 80 40 50 60 70 80 90 100 110 120 ' 130 140 150 100 170 1 


JO 190 200 OHMS 



Fig. 54. 

In Fig. 54 are shown the values of /, 7 1} 7 , 7f, in Curves 
I., II., III., IV., similarly as in Fig. 50, for E = 1000 volts, 
c = x = 100 ohms, and r as abscissas. 



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, c = x , so as to balance the lagging current at no 



RESISTANCE, INDUCTANCE, CAPACITY. . 



77 



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 ia 
the diagram, Fig. 55, can be used for raising the potential 
E , with increasing load. 




Fig. 55. 



Let 



be the impressed E.M.F. of the generator, or of the mains, 
and let the condensance be x c = x \ then 
Current in receiver circuit, 



r jx 



current in condenser circuit, 

T 

/I = 



X 



Hence, the total current in main line is 






r x x 



78 A L TERN A TING-CURRENT PHENOMENA. 

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

r JXo 
E.M.F. at condenser terminals, 

E.M.F. consumed in main line, 
hence, the E.M.F. at generator is 



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



current in receiver circuit, 
7 



r jx r (x xj jx? ' 

This value of / contains the resistance, r, only as a fac- 
tor to the difference, x x^\ hence, if the reactance, ;r 2 , 
is chosen = x , r cancels altogether, and we find that if 
#2 = * , the current in the receiver circuit is constant, 

/-/A, 

X 

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

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

E<i = <? 2 + J e *'i E z = V^ 2 2 + ^2' 2 = constant ; 

current in receiver circuit, 

/ =j%L, 7 = ^? = constant; 

x x a 

E.M.F. at receiver circuit, 

E = Ir=j E -^-, E ~ ^^, or proportional to load r; 

' 



RESISTANCE, INDUCTANCE, CAPACITY. 79 

E.M.F. at condenser terminals, 



E* 1 +/ - , = ^ 2 V 1 + - , hence > E, 




.V 

current in condenser circuit, 



main current, 



r 



*.(*.+./>) ' 

( proportional to the load, 
T JZI<L f 1 , . , . , 

/o = V ' J r anC ^ ln P nase Wlt " 

X ( E.M.F., E z . 

The power of the receiver circuit is, 



the power of the main circuit, 

f E z = 2 r , hence the same. 
* 2 

55. This arrangement is entirely reversible ; that is, 
if E z = constant, / = constant ; and 
if I = constant, E = constant. 

In the latter case we have, by expressing all the quanti- 
ties by 7 : 
Current in main line, 

I = constant; 
E.M.F. at receiver circuit, 

E = I x 9 = constant ; 
current in receiver circuit, 

/ =f , proportional to the load -; 
current in condenser circuit, 



80 AL TERNA TING-CURRENT PHENOMENA. 

E.M.F. at condenser terminals, 



Impressed E.M.F. at generator terminals, 

x 2 1 

2 = I , 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 c = x , 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 E^ = 1,000, 7=10, we have : x = 100 ; hence 
the constants of the circuit are : 

E* = 1000 volts ; 

7 = 10 amperes ; 

E 10 r, plotted as Curve I., with the resistances, r, as abscissa;; 

E = 1000 1/1 + I Y plotted as Curve II. ; 
' V 100 y 

7 t = 10 i/1 + ( --Y, plotted as Curve III.- 
V ^-^^ J 

7 = .1 r, plotted as Curve IV. 



RESISTANCE, INDUCTANCE, CAPACITY. 



81 



56. 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. 



u 

13 
12 

11 
10 
9 

j 

|.7 

6 
6 
1 
3 
2 
1 





CURRENT IN RECEIVER CIRCUIT CONSTANT, 
IMPRSSED E.M, F.CONSTANT, E 8 =IOOO VOL 
2 REACTANCES OFOTo =IOO OHMS EACH, SH 
THE CONDENSANCE, Z C = IOO OHMS. 
VARIABLE RES STANCE IN RECEIVER CIRCUI 
1 E.M.F. AT RECEIVER C RCUIT. 
1 II E.M-F. AT CONDENSER CIRCUIT. 
Ill CURRENT IN CONDENSER CIRCUIT. 
IV CURRENT IN MAIN LINE 
V CURRENT IN MAIN LINE INCLUDING tC 
VI EFFICIENCY OF TRANSFORMATION, 


1^10 AMPERES 1 

rs. '~ 

UNTED IN THEIR MID: 


LE 


BY 




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ou 

1100 






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SSES 


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1200 




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<V>0 











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son 




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581 


















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() 


HM8 



F/3. 50. Constant-Potential Constant-Current Transformation. 

Let 

ri = 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 2 ; 
power consumed by first inductance, 7 2 r = 300 ; 
power consumed by second inductance, / 2 r = .03 r*. 
Hence, the total loss of energy is 500 + -05 r 2 ; 
output of system, / 2 r = 100 r 

input, 500 + 100 r -\ 

effidenCy ' 500 + 1W M 

It follows that the main current, f , increases slightly 
by the amount necessary to supply the losses of energy 
in the apparatus. 



82 ALTERNATING-CURRENT PHENOMENA. 

This curve of current, I , 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 
range. 



RESISTANCE OF TRANSMISSION LINES. 



CHAPTER IX. 

RESISTANCE AND REACTANCE OF TRANSMISSION LINES. 

57. 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 = g -f jb, of the receiver circuit. 

The conductance, g y 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, b, however, can be changed by 
shunting the circuit 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 



84 ALTERNATING-CURRENT PHENOMENA. 

receiver circuit can be assumed to consist of two branches, 
a conductance, g, 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- 
lation. 

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. Since 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 

Z = r ,jx = impedance of the line ; 

z = Vr 2 + ^ 2 ; 
Y = g -\-jb = admittance of receiver circuit; 

y = VFTT 2 ; 

E = e -f /<?</ = impressed E.M.F. at generator end of line ; 

E = 
E = e +/<?' = E.lVf.F. at receiver end of line ; 



E = 



I = i -\-jio = current in the line ; 

I = Vtf + 4". 
The simplest condition is the non-inductive circuit. 

1.) Non-inductive Receiver Circuit Sripplied over an 

Inductive Line. 

58. In this case, the admittance of the receiver circuit 
is Y = g, since b = 0. 



RESISTANCE OF TRANSMISSION LINES. 85 

We have then 

current, 7 = Eg; 

impressed E.M.F., E = E + Z 7 = E (1 + Z.g). 

Hence 
E.M.F. at receiver circuit, 

= \^Z g~ \-\-gr.-jgxJ 
current, 7 = J A|_ = ^ . 

Hence, in absolute values 
E.M.F. at receiver circuit, E 

current, 7 : 




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



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

output, P = I E = 



As a function of g, and with a given E ot r , and x , this 
power is a maximum, if 



that is 

-l+^-V^+^^^O; 
hence 

conductance of receiver circuit for maximum output, 



Vr 2 + V ^o 
Resistance of receiver circuit, r m = = z ; 



86 AL TERNA TING-CURRENT PHENOMENA. 

and, substituting this in P 

Maximum output, P m = 2 = g 

and 

ratio of E.M.F. at receiver and at generator end of line, 

a m = -=r = 



efficiency, 



That is, the output which can be transmitted over an 
inductive line of resistance, r , and reactance, x , that is, 
of impedance, z , into a non-inductive receiver circuit, is 
a maximum, if the resistance of the receiver circuit equals 
the impedance of the line, r = z 0) and is 



The output is transmitted at the efficiency of 



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

1 



59. 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. 57 are shown, for the constants 

E = 1000 volts, 

Z g = 2.5 6/ ; that is, r, = 2.5 ohms, x 6 ohms, z = 6.5 ohms, 

with the current I as abscissae, the values 



RESISTANCE OF TRANSMISSION LINES. 



87 



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

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

Efficiency of Transmission, (Curve III.). 

The same quantities, E and P, for a non-inductive line of 
resistance, r = 2.5 ohms, x = 0, are shown in Curves IV., 
V., and VI. 






SUPFUED'OVER INDUCTIVE LINE OF IMPEDAN 
AND OVER NON-INDUCTIVE LII^E OF RESISTAr. 

T = 2.5 
CURVE 1. E. M. F. AT RECEIVER CIRCUIT, INDUCTIVE LI 


3E 














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U K 


100 
90 
80 
70 

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50 
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ii V. 11 ii ii ii NON-INDUCTIVE 


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10 20 30 40 50 60 70 80 
Fig. 57. Non-inductive Receiver Circuit Supplied Over Inductive Line. 

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, 
Impressed E.M.F., 



/ = E Y; 
E = E + I Z == E (1 + KZ ). 



88 AL TERNA TING-CURRENT PHENOMENA. 

Hence 
E.M.F. at receiver terminals, 



1 + FZ (1 + r.g + x.S) - J (x.g - r.6)' 
current, 



or, in absolute values 
E.M.F. at receiver circuit, 



V(l + r.f + x,bf + (x.g - r. 
current, 



= E J _ jr 2 + ^ 2 _ . 

V (i + r og + Xo by + ( Xog - r t>y' 



ratio of E.M.Fs. at receiver circuit and at generator circuit, 
E 1 



and the output in the receiver circuit is, 
P=E*g= E?o?g. 

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

At a given conductance, g, of the receiver circuit, its 
output, P = E?a?g, is a maximum, if a 2 is a maximum ; that 
is, when 

/=!=(! + r.g + x.Vf + (x.g - r b? 

is a minimum. 

The condition necessary is 



or, expanding, ,., , N , , N A 

5 '. *. (1 + r og + jf ^) - r ( Xo g - r b} = 0. 

Hence 
Susceptance of receiver circuit, 

t= ~^^) = ~^ = ~ b ' 
or b + b = 0, 



RESISTANCE OF TRANSMISSION LINES. 89 

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, 



E z (g 
maximum output, 

P l = - 



current, 

E Y E (g 



E (g-jb } 



og - x b.} -J(r b 



Io = E V (1 + r og - Xo b ? + (r b + Xo g)*> 
and, expanding, 

r = * 

' 



phase difference in receiver circuit, 

tan = * = - A . 
^ A" 

phase difference in generator circuit, 



62. b.} Dependence of the output upon the conductance 

of the receiver circuit. 

At a given susceptance, ^, of the receiver circuit, its 
output, P Eo<?g, is a maximum, if 



dP dl\\ 

-r = 0, or I - I = 0, 

dg d^P] 

)* + ( Xo g - 



90 ALTERNATING-CURRENT PHENOMENA. 

that is, expanding, 

C 1 + r g -f x b} 2 + ( Xo g r by 2g(r + r*g -f x*g) = ; 
or, expanding, 

Substituting this value in the equation for a, page 88, 
we get - 
ratio of E.M.Fs., 



power 




As a function of the susceptance, b, this power becomes 
a maximum for dP^j db = 0, that is, according to 61, if 
*'--*. 

Substituting this value, we get 

= bt> g = So* y = y<n hence: Y= g-\- jb= g jb \ 

x = - x , r = r , z = z , Z = r Jx = r + jx ; 

substituting this value, we get 



ratio of E.M.Fs., m . 

power, ^ m = i- 2 - ; 

that is, the same as with a continuous-current circuit ; or, 
in other words, the inductance of the line and of the receiver 
circuit can be perfectly balanced in its effect upon the 
output. 

63. As a summary, we thus have : 

The output delivered over an inductive line of impe- 



RESISTANCE OF TRANSMISSION LINES. 91 

dance, Z = r jx , into a non-inductive receiver circuit, is 
a maximum for the resistance, r = z , or conductance, g = 
y , of the receiver circuit, or 



2 (r. + 
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, 

and is 

EO ' y? 

= 2(^+Vo)' 
at the ratio of potentials, 



With a receiver circuit of constant conductance, g, the 
output, as a function of the susceptance, b, is a maximum 
for the susceptance, b = b , and is 



P= 



tffe+JJ?' 

at the ratio of potentials, 

1 

7o ( + go) ' 

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 = r -\-jx , or 
y=jTo~J6o> 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 r , or independent of the 
reactances, but equal to the output of a continuous-current 



92 



AL TERN A TING-CURRENT PHENOMENA. 



circuit of equal line resistance. The ratio of potentials is, in 
this case, a = z o j 2 r oi while in a continuous-current circuit 
it is equal to . The efficiency is equal to 50 per cent. 




.03 .01 .05 .08 ,07 .08 .09 .10 .11 .12 .13 .14 J5 J6 33 

Fig. 58. Variation of the Potential in Line at Different Loads. 

64. As an instance, in Fig. 58 are shown, for the 
constants 

E = 1000 volts, and Z = 2.5 6/; that is, for 

r = 2.5 ohms, x = Gohms, z = 6.5 ohms, 

and with the variable conductances as abscissae, the values 
of the 

output, in Curve I., Curve III., and Curve V. ; 

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

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



RESISTANCE OF TRANSMISSION LINES, 



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

constant susceptance b = .142 

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

constant susceptance b = .142 ; 

Curves VII. and VIII. refer to a non-inductive re- 
ceiver circuit and non-inductive line. 

In Fig. 59, the output is shown as Curve I., and the 
ratio of potentials as Curve II., for the same line constants, 
fora constant conductance, ^- = .0592 ohms, and for variable 
susceptances, b, of the receiver circuit. 













OUTPUT P /NO RATIO OF POTENTIAL a t 
SENDING END OF LINE OF IMPEDANCE. Z 


T RECEIV 1 NG^ND 
=5.5 -3j 












AT 


CON 


TAN 




g= . 0592 

1 OUTPUT 
II RATIO OF POTENTIALS 














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-.3 -.2 -.1 +.1 +.2 +.3 +.4 

Fig. 59. Variation of Potential in Line at Various Loads. 

3.) Maximum Efficiency. 

65. The output, for a given conductance, g, of a receiver 
circuit, is a maximum if b = b . This, however, is gen- 
erally not the condition of maximum efficiency. 



94 ALTERNATING-CURRENT PHENOMENA. 

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^aximum if the cur- 
rent is in phase with the E.M.F. at the generator terminals. 
Hence the condition of maximum output at given loss, or 
of maximum efficiency, is 

tan > = 0. 
The current is 



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

x + Xo = 0. 

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. 

Substituting x = x , we have, 
ratio of E.M.Fs., 



power, 




RESISTANCE OF TRANSMISSION LINES. 



95 



and depending upon the resistance only, and not upon the 
reactance. 

This power is a maximum if g = g , as shown before; 
hence, substituting g = g , r = r , 

E 2 

maximum power at maximum efficiency, P m = 2 , 

at a ratio of potentials, a m - 2 , 

" r o 

or the same result as in 62. 




.01 .03 .03 .01 .05 .06 .07 .08 

Fig. 60. Load Characteristic of Transmission Line. 

In Fig. 60 are shown, for the constants 
E = 1,000 volts, 
Z =2.5 6/; r = 2.5 ohms, x = 6 ohms, z = 6.5 ohms, 



96 ALTERNATING-CURRENT PHENOMENA. 

and with the variable conductances, g, of the receiver circuit 
as abscissae, the 

Output at maximum efficiency, (Curve I.) ; 

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

Efficiency = , (Curve III.). 

r + r 

4.) Control of Receiver Voltage by Shunted Snsceptance. 

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 



If at constant generator potential E , the receiver potential 
E shall be constant, 

a constant ; 
hence, 

# 2 ' 
or, expanding, 



which is the value of the susceptance, b, as a function of 
the receiver conductance, that is, of the load, which is 
required to yield constant potential, aE , at the receiver 
circuit. 

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



b = - 



RESISTANCE OF TRANSMISSION LINES. 



97 



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 aE , is given by the expression 



hence b = o , 

and g = g -\- 



the susceptance of receiver circuit, 
the conductance of receiver circuit; 



- f the output. 



67. If a = 1, that is, if the voltage at the receiver cir- 
cuit equals the generator potential 

P=E*( t y '-g ). 
If a = 1 when g = 0, b = 

when g > 0, b < ; 
if a > 1 when g = 0, or g > 0, b < 0, 

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

when g = - #, + \/f ^ - <V, ^ = ; 
when^> -g + V/f ^ - V, * < 0, 



or, in other words, if a < 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 / da = ; 

expanding : 2 a ( y JL - g\ _ f!f' = ; 

\a J a 

hence, y = 2ag , 

y o 1 Zo 

" = = = 



98 ALTERNATING-CURRENT PHENOMENA. 

the maximum output is determined by 

S == So i = So I 

and is, P = 2- . 

4 r 

From : a = ^ = -^- , 

the line reactance, x , can be found, which delivers a 
maximum output into the receiver circuit at the ratio of 
potentials, a, 
and z = 2 r a, 

for a == 1, 



If, therefore, the line impedance equals 2# times the line 
resistance, the maximum output, P = E* j r , is trans- 
mitted into the receiver circuit at the ratio of potentials, a. 

If z = 2 r , or x = r V3, the maximum output, P = 
2 /4:r , 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* g a 2 , as abscissae, and a constant impressed E.M.F., 
E = 1,000 volts, and a constant line impedance, Z = 
2.5 6/, or, r = 2.5 ohms, x = 6 ohms, z = 6.5 ohms, 
the following values : 



RATIO'OF RECEIVER VOLTAGE TO SENDER VOLTAGE: d =I.O 

LINE IMPEDANCE: Z = a. 5 6; 

ENERGY CURRENT CONSTANT GENERATOR 

TOTAL CURRENT 

CURRENT IN NON-INDUCTIVE RECEIVER CIRCUIT WITHOUT COMPENSATION 





OUTPUT] IN RECEIVER CIPJCUIT, KILOWJATT 
50 60 70 80 

Fig. 61. Variation of Voltage Transmission Lines. 







. 


RATIO OF RECEIVER VOLTAGE TO SENDER VOLTAGE: 
LINE MPEDANCE:Z_ = 2.5. 6J 
\. ENERGY CURRENT CONSTANT GENRATOR POT 
II. REACTIVE CURRENT 
III. TOTAL CURRENT 
IV. POTENTIAL IN NON-INDUCTIVE CIRCUIT WITHOUT C 

~|Tt-MJJ MINI 


a =.7 
:NTIAL E 

OMPENS 


0= I 

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300 






















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DLTS 
1000 

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GOO 
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400 
300 
200 
100 



















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30 W 50 CO 70 80 

Fig. 62. Variation of Voltaqe Transmission Lines. 



100 



AL TERNA TING-CURRENT PHENOMENA. 




RATIO OF RECEIVER VOLTAGE TO SEN DER VOLTAGE: a =1.3 

INE IMPEDANCE: Z =2.5. ej" 



CONSTANT GENERATOR POTENTIAL E =IOOOl 



I. ENERGY CURRENT 

II. "REACTIVE CURRENT 

III. TOTAL CURRENT 

IV. POTENTIAL IN NON-INDUCTIVE RECEIVER CIRCUIT WITHOUT COMPENSATION 




OUTPUT N RECEIVER C RCUIT, KILOWATTS 



30 10 80 60 70 80 90 

Fig. 63. Variation of Voltage Transn\jssion Lines. 

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) ; a = .7 (Fig. 62) ; a = 1.3 (Fig. 63). 

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 time 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. 



RESISTANCE OF TRANSMISSION LINES. 



101 



5.) Maximum Rise of 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, E . 

The condition is that 



a = maxmum or = mnmum : 
a 2 



that is, 



substituting, 



r g + 



(* g - 



and expanding, we get, 

dg = ; gss ~"'' 

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

Substituting this value, we get, 



and by substituting, in 



, 

b + b = ; 

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

*=-*-, 
we have 



102 AL TERNA TING-CURRENT PHENOMENA. 

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. 











\ 


s 


\ 


\ 














VOLT 










^ 


\ 


\ 




























\ 


\ 












1SOO 
1700 
1COO 
1500 
-1400 
















\ 


\ 


























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\ 


\ 








CONSTANT IMPRESSED E. M. F. Eo^lOOO 
" LINE IMPEDANCE Z =2.5- 
1 MAXIMUM OUTPUT BY COMPENSATION 
II MAXIMUM EFFICIENCY BY COMPENSATIC 
III NON-INDUCTiVE RECEIVER C RCU T 
IV NON-INDUCTIVE LINE AND NON-INDUCT 
RECEIVER CIRCUIT 


If 


\ 








IN 


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IVE 










1200 

1100 
1000 














































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GOO 
800 
700 
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200 
100 








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PUT 


PUT 


K.W 




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it 







Fig. 64. Efficiency and Output of Transmission Line. 

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



RESISTANCE OF TRANSMISSION LINES. 103 

abscissae and the receiver voltages as ordinates, curves 
representing 

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

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

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

the condition b = 0, b = 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. 



104 ALTERNATING-CURRENT PHENOMENA. 



CHAPTER X. 

EFFECTIVE RESISTANCE AND REACTANCE. 

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 : 

r __ Power consumed . 

(Current) 2 

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 
motor. 

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

_ Energy component of E.M.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, 

a ._ Energy component of current 

Total E.M.F. 
is called the effective conductance of the circuit. 



EFFECTIVE RESISTANCE AND REACTANCE. 105 

In the same way, the value, 

_ Wattless component of E.M.F. 

Total current 
is the effective reactance, and 

, _ Wattless component of current 
TotafE.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 effective resistance repre- 
sents the total 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, 

a.) Magnetic hysteresis ; 
b.) Dielectric hysteresis. 



106 .ALTERNATING-CURRENT PHENOMENA. 

2.) Primary electric currents, as, 

a.} Leakage or escape of current through the insu- 
lation, brush discharge ; b.) 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 ; b.} 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. must 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 



EFFECTIVE RESISTANCE AND REACTANCE. 



107 



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, n, of the electric circuit, 
the effective counter E.M.F., E, and the frequency, N, 
of the current, the maximum magnetic flux, <j>, is found 
by the formula : 



hence, 



E 10 8 



A maximum flux, <, and magnetic cross-section, S, give 
the maximum magnetic induction, (B = $ / 6". 

If the magnetic induction varies periodically between 
+ (B and (B, the M.M.F. varies between the correspond- 
ing values -f ff and JF, and describes a looped curve, the 
cycle of hysteresis. 

If the ordinates are given in lines of magnetic force, the 
abscissae 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., JF, 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, i, corresponding to a M.M.F., JF; that is, 
magnetic induction (B, and thus induced E.M.F. e, as : 



75. 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., 



108 



AL TERNA TING-CURRENT PHENOMENA. 



SF = 1.8, 2.8, 4.3, 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, 77 = .0033, 
and are given with ampere-turns per cm as abscissae, and 
kilo-lines of magnetic force as ordinates. 



a 



M 



</. 65. Hysteretic Cycle of Sheet Iron. 

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. 



EFFECTIVE RESISTANCE AND REACTANCE. 109 

As shown in 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, 90 



1=2000 



1.6 



N 



^ 






\ 



(Bfeooo 



T2.8 



3 =2.S 



M\ 



\\ 



Figs. 66 and 67. Distortion of Current Waue by Hysteresis. 

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 
maximum point of the wave coincides in time with the max- 



110 



ALTERNA TING-CURRENT PHENOMENA. 



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 (B = 10,000 ; and at still 



(B- 



10000 



4. 



& 



NX 



\L 



. 16000 






20 






\ 






G 



13 






\ 



F/SfS. 88 and 69. Distortion of Current Waue by Hysteresis. 

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 to the same scale. The maximum values of M.M.F., 



EFFECTIVE RESISTANCE A.\D REACTANCE- 111 

corresponding to the maximum values of magnetic induction, 
(B = 2,000, 6,000, 10,000, and 16,000 lines of force per cm 2 , 
'arc & = 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 
J # r =4 7 r/103 r = 1.257 IF. 

76. The distortion 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 T V 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 ? ^ the length of the magnetic 
circuit. These two curves are drawn to the size of the curve 
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 
nnd equal power to the distorted wave, called the equivalent 



112 



ALTERNATING-CURRENT PHENOMENA. 



sine wave, and a wattless JiigJier harmonic, consisting chiefly 
of a term of triple frequency. 

In Figs. 66 to 71 are shown, as /, the equivalent sine' 



\ 



\ 



v 



\ 



Figs. 70 and 71. Distortion of Current Wave by Hysteresis. 

waves and as i, 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- 



EFFECTIVE RESISTANCE AND REACTANCE. 113 

ism by 34, 44, 38, and 15. 5, respectively. In Fig. 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 V2 -f- 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 electro-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 
& = 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. 



114 



AL TERN A TING-CUR RE KT PHENOMENA. 



The same Fig. 72 gives the curve of hysteretic loss, in 
ergs per cm 3 and cycle, as ordinates, and magnetic induc- 
tions as abscissae. 



TT 






























\ 




































/ 
































/ 


/ / 
/ 


18 






























/ 


/ 
/ 


17 
































r 


1 
1 
































/I 




' 






























/ 




/ 

/ 
































/ 




/ 






























/ 




/ 


' 




























/ 




































/ 




1 


/ 




























/ 






I. 


' 


























/ 








// 




























/ 






/ 


1 


























/ 








I 


























/ 








/ 


























/ 








/' 


























/ 






^ 


^ 


























/ 




^" 


^ 


























^^ 


^ 




























; j^ 


* 


x 






























2=EE 


x^^ 
































=1,000 2,000 3,000 1.0CO 5,000 6,000 7,000 8,000 9,000 10,000 11,000 12,000 13,0<W14,000 15, 
Fig. 72. Magnetization and Hysteresis Curve. 


woie.ooo iv.ow 



The electro-dynamometer method of determining the 
magnetic characteristic is preferable for use with alter- 
nating-current apparatus, since it is not affected by the 
phenomenon of magnetic "creeping," which, especially at 



EFFECTIVE RESISTANCE AND REACTANCE. 115 

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 permissible 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 a, which is called 
the angle of Jiysteretic advance of phase. Hence the cur- 
rent lags behind the E.M.F by ^ 90 a, and the power 



is therefore, p=f cog (9QO _ a) = /E sin a 

Thus the exciting current, 7, consists of an energy compo- 
nent, / sin a, called the Jiysteretic or magnetic energy current, 
and a wattless component, / cos a, which is called the mag- 
netizing current. Or, conversely, the E.M.F. consists of an 
energy component, E sin a, the Jiysteretic energy E.M.F., 
and a wattless component, E cos a, the E.M.F. of self- 
induction. 

Denoting the absolute value of the impedance of the 



116 A L TERNA TING-CURRENT PHENOMENA . 

circuit, E 1 1, by s, 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 jx = z sin a jz cos a ; 
and the admittance by, 

Y = g + j b = - sin a -j- j - cos a = y sin a -f- jy cos a. 

z z 

The quantities, z, 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 
friction is, with sufficient exactness, proportional to the 
1.6 th power of magnetic induction <&. Hence it can be ex- 
pressed by the formula : 



where 

IV a = loss of energy per cycle, in ergs or (C.G.S.) units (= 10~ 7 
Joules) per cm 8 , 

(ft = maximum magnetic induction, in lines of force per cm 2 , and 

77 = the coefficient of hysteresis. 

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, 17 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 

* At present, with the improvements in the production and selection of sheet steel far 
alternating apparatus, .0025 can be considered a fair average in selected material (1899). 



EFFECTIVE RESISTANCE AND REACTANCE. 117 

steel). Soft nickel and cobalt have about the same co- 
efficient of hysteresis as gray cast iron ; in magnetite I 
found rj = .023. 

In the curves of Fig. 62 to 69, r, = .0033. 

At the frequency, N, the loss of power in the volume, V, 
is, by this formula, 

P=- t] N F& 1 - 6 10 - ' watts 



where S is the cross-section of the total magnetic flux, <. 

The maximum magnetic flux, <E>, depends upon the 
counter E.M.F. of self-induction, 

E = V2 -IT Nn 4> 10 - 8 , 



V2 TT Nn 

where n = number of turns of the electric circuit. 

Substituting this in the value of the power, P, and 
canceling, we get, 

E 1 -' FIO 5 - 8 E F10 8 



no 5 - 8 Ka no 3 
where ^ = ^ o. R i. oi. fi ..,.. = 58 -n 



T/- 

or, substituting >; = .0033, we have ^4 = 191.4 ^ ; 

o ' /? * 

or, substituting F= SL, where L = length of magnetic circuit, 
n L 10 5 - 8 58 Z 10 3 Z 





and 10 3 191.4 E 



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., E, as abscissae, for L = 6, 
S = 20, N= 100, and n = 100 ; 



118 



AL TERNA TING-CURRENT PHENOMENA. 





































































































































RELATION 


BE 


TW = 


EN 


EA 


NDP 
























F 


OR 


_ 


5,8 


= 20 


N = 


10 


r5 


= 1 


oo 






























































/ 








































/ 








































/ 






K 


































/ 








o 






























/ 


^/ 










Q. 




























x 


^ 






































x 






































X 


^ 






































x 






































x 


x 




































x 


^ 




































X* 


^ 
































. 





^ 
















E.IV 


l.F. 





















Fig. 73. Hysteresis Loss as Function of . M. F. 



BETW 
OR L T 6. S=20, ^ 



= 100.E= 



SO 100 160 200 250 300 

Fig. 74. Hysteresis Loss as Function of Number of Turns. 



EFFECTIVE RESISTANCE AND REACTANCE. 



119 



II I I II I 



RELATION BETWEEN N AND P 
FOR 8=20, L=6, 71 = 100. E = 100. 




Fig. 75. Hysteresis Loss as Function of Cycles. 

in curve 74, with the number of turns as abscissae, for 
Z = 6, S = 20, JV= 100, and E = 100 ; 

in curve 75, with the frequency, JV, or the cross-section, S, 
as abscissae, for L = 6, n = 100, and E = 100. 

As shown, the hysteretic loss is proportional to the 1.6 th 
power of the E.M.F., inversely proportional to the 1.6 th 
power of the number of turns, and inversely proportional to 
the .6 th power of frequency, and of cross-section. 

81. If g = effective conductance, the energy compo- 
nent of a current is / = Eg, and the energy consumed in 
a conductance, g, is P = IE = E z g. 

Since, however : 

P = A , we have A = E 2 g ; 

or 

A 58r)L 10 s 



191.4 



From this we have the following deduction : 



120 



ALTERNA TING-CURRENT PHENOMENA. 



The effective conductance due to magnetic hysteresis is 
proportional to the coefficient of hysteresis, rj, and to the length 
of the magnetic circuit, L, and inversely proportional to the 
Jj! h power of the E.M.F., to the .6 th power of the frequency, 
N, and of the cross-section of tlie magnetic circuit, S, and to 
tlie 1.6 th power of the number of turns, n. 

Hence, the effective hysteretic conductance increases 
with decreasing E.M.F., and decreases with increasing 





















































RELATION 
FOR L=6, 


BE- 


PWEEN 0AND E 
00. S = 20,?l = 1O 
















































































































































































































V 










































\ 












































\ 










































\ 












































^ 


\ 












































> 


^. 


.^^ 






























__9 

a 
















1 -, 




- -. 




^ 


. 


. 




, 
















































































































E 

























Ftg. 76. Hysteresis Conductance as Function of E.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 
error. 

In Figs. 76, 77, and 78, the hysteretic conductance, g, is 
plotted, for L = 6, E = 100, N= 100, 5 = 20 and n = 100, 
respectively, with the conductance, g, as ordinates, and with 



EFFECTIVE RESISTANCE AND REACTANCE. 



1-21 



RELATION BETWEEN Q AND N 
FOR L-6, E = IOO. S = 20, n=IOO 




Fig. 77. Hysteresis Conductance as Function of Cycles, 










































































































R 


LAI 


,0, 


BE 


WE 


EN 


,AS 


D(/ 






























FOP 


L= 


6,E 


= 1( 


50, 




00 


,8= 


2a 














































































































































































































































\ 








































b 




V 








































a 




\ 


























































































\ 














































s 












































\ 


X. 




































E 


- 




T 


-NL 


\. 

M~B~ 


:RO 


, 

F T 





r= 

























200 250 300 350 

Fig. 78. Hysteresis Conductance as Function of Number of Turns. 



122 ALTERNATING-CURRENT PHENOMENA. 

E as abscissae in Curve 76. 
.A^ as abscissas in Curve 77. 
n as abscissas 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 N or 6" by 50 per cent causes a variation in g of 21 
per cent. 

If (R = magnetic reluctance of a circuit, F A = maximum 
M.M.F., I effective current, since /V2 = maximum cur- 
rent, the magnetic flux, 



(R (R 

Substituting this in the equation of the counter E.M.F. of 
self-induction 



we have 



(R 
hence, the absolute admittance of the circuit is 

(RIO 8 = a& 

E ~ 2 TT n*N ~ N ' 

10 8 

where a = , a constant. 

2 TT n 

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 the number of turns, n. 

82. In a circuit containing iron, the reluctance, (R, 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- 



EFFECTIVE RESISTANCE AND REACl^ANCE. 123 

former, (R is the reluctance of the iron circuit. Hence, 
if p. = permeability, since 



and g: A = jr/7 =Z ge = M.M.F., 



and <R, 10L 



magnetic flux, 



substituting this value in the equation of the admittance, 

(R 10 8 Z 10 9 z 

y= -z- nrv> we have 5 ; 



where L W 127Z10 ' 



TJierefore, in an ironclad circuit, the absolute admittance, 
y, is inversely proportional to the frequency, N, to the perme- 
ability, JJL, to the cross-section, S, and to the square of the 
number of turns, n ; and directly proportional to the length 
of the magnetic circuit, L. 



The conductance is 



= 

and the admittance, y = - ; 

yv/u. 

hence, the angle of hysteretic advance is 



or, substituting for A and z (p. 117), 
N A Z10 68 



or, substituting 
J 

we have sin a = 

- 4 ' 



1 24 AL TERN A TING-CURRENT PHENOMENA. 

which is independent of frequency, number of turns, and 
shape and size of the magnetic and electric circuit. 

Therefore, in an ironclad inductance, tJie angle of Jiysteretic 
advance, a, depends upon the magnetic constants, permeability 
and coefficient of hysteresis, and tipon the maximum magnetic 
induction, but is entirely independent of the frequency, of the 
shape 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 Jiysteretic advance. 

The angle of Jiysteretic advance, a, in a closed circuit 
transformer, depends tipon tJie quality of the iron, and upon 
the magnetic density only. 

The sine of tJie angle of Jiysteretic advance equals 4 times 
the product of the permeability and coefficient of hysteresis, 
divided by the .4 th power of tJie 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 

<R = (R { _|_ <R fl ; 

hence the admittance is 



TJierefore, in a circuit containing iron, the admittance ts 
the sum of the admittance due to the iron part of tJie circuit, 
yi = a&i/ N, and of the admittance due to the air part of the 
circuit, y a = a (& a / N, if the iron and the air are in series in 
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 



sm a = 

y 



EFFECTIVE RESISTANCE AND REACTANCE. 125 

and a maximum, gjy t , for the ironclad circuit, but decreases 
with increasing width of the air-gap. The introduction of 
the air-gap of reluctance, (R , decreases sin a in the ratio, 

<Rj 

* + <* ' 

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, /t = 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 1^ to Ig^, 
or about 6 per cent, that is, remains practically constant ; 
while the angle of hysteretic advance varies from sin a = .035 
to sin a = .064. Thus g is negligible compared with b, and 
b is practically equal to j. 

Therefore, in an electric circuit containing iron, but 
forming an open magnetic circuit whose air-gap is not less 
than T ^ 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, 

b = <R a / N, or : x = N/ (R a . 

The angle of hysteretic advance is small, below 4, and the 
hysteretic conductance is, 

- = A 

E A N* ' 

The current wave is practically a sine wave. 

As an instance, in Fig. 71, Curve II., the current curve 
of a circuit is shown, containing an air-gap of only ^ 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 ; 



126 ALTERNATING-CURRENT PHENOMENA. 

then from the equation, 
E = 



where, 

N '= frequency, 

n = number of turns, 



we get the magnetism, <, and by means of the magnetic cross 
section, S, the maximum magnetic induction : ($> = / S. 

From (B, we get, by means of the magnetic characteristic 
of the iron, the M.M.F., = F ampere-turns per cm length, 
where 



if OC = M.M.F. in C.G.S. units. 

Hence, 
if Z, = length of iron circuit, JFj = Z, F = ampere-turns re- 

quired in the iron ; 
if L a = length of air circuit, CFa = - = ampere-turns re- 

quired in the air ; 

hence, CF= JF, -)- $F a = total ampere -turns, maximum value, 
and JF/ V2 = effective value. The exciting current is 



and the absolute admittance, 



If SF, is not negligible as compared with JF a , this admit- 
tance,^, is variable with the E.M.F., E. 

If 

V = volume of iron, 

rj = coefficient of hysteresis, 

the loss of energy by hysteresis due to molecular magnetic 
friction is, 



hence the hysteretic conductance is g = lV/?, and vari- 
able with the E.M.F., E. 






EFFECTIVE RESISTANCE AND REACTANCE. 127 

The angle of hysteretic advance is, 

sin a=g/y; 

the susceptance, b = Vj* 2 g z \ 

the effective resistance, r = g / y*\ 

and the reactance, x = b / y*. 

85. As conclusions, we derive from this chapter the 
following : 

1.) In an alternating-current 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 
E.M.F. differing from sine shape. 

2.) This distortion is excessive only with a closed mag- 
netic circuit transferring no energy into a secondary circuit 
by mutual inductance. 

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 
harmonic, consisting mainly of a term of triple frequency, 
may be neglected except in resonating circuits. 

4.) Below saturation, the distorted curve of current and 
its equivalent sine wave have approximately the same max- 
imum value. 

5.) The angle of hysteretic advance, that is, the phase 
difference between the magnetic flux and equivalent sine 
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, yu., the coefficient of hys- 
teresis, rj, and the maximum magnetic induction, as shown* in 

the equation, 4 

sin a = fi . 

&' 4 

6.) The effect of hysteresis can be represented by an 
admittance, Y g + j b, or an impedance, Z = r j x. 

7.) The hysteretic admittance, or impedance, varies with 
the magnetic induction; that is, with the E.M.F., etc. 



128 ALTERNATING-CURRENT PHENOMENA. 

8.) The hysteretic conductance, , is proportional to the 
coefficient of hysteresis, 17, and to the length of the magnetic- 
circuit, L, inversely proportional to the .4 th power of the 
E.M.F., E, to the .6^ h power of frequency, N, and of the 
cross-section of the magnetic circuit, S, and to the 1.6 th 
power of the number of turns of the electric circuit, ;/, as 
expressed in the equation, 

58 7 Z 10 3 



9.) The absolute value of hysteretic admittance, 



is proportional to the magnetic reluctance : (R = (R, -f (R a , 
and inversely proportional to the frequency, N, and to the 
square of the number of turns, n, as expressed in the 



> _(. + ) 10- 

2-irNn* 

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, N y 
permeability, /*, and square of the number of turns, n, or 
127 L 10 6 



11.) In an open magnetic circuit, the conductance, g t is 
the same as in a closed magnetic circuit of the same iron part. 

12.) In an open magnetic circuit, the admittance, y t is 
practically constant, if the length of the air-gap is at least 
T J C of the length of the magnetic circuit, and saturation be 
not approached. 

13.) In a closed magnetic circuit, conductance, suscep- 
tance, and admittance can be assumed as constant through 
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, g y b,y; r, x, z, can be calculated. 



FOUCAULT OR EDDY CURRENTS. 129 



CHAPTER XI. 

FOUCAULT OR EDDY CURRENTS. 

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 
circuits. 

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, 
(B, the frequency, N, and to the electric conductivity, y, of 
the iron ; hence, can be expressed by 



The power consumed by eddy currents is proportional to 
their square, and inversely proportional to the electric con- 
ductivity, and can be expressed by 

W= 



130 ALTERNATING-CURRENT PHENOMENA. 

or, since, ($>N is proportional to the induced E.M.F., E, in 
the equation 



it follows that, TJie loss of power by eddy currents is propor- 
tional to the square of the E.M.F., and proportional to tlie 
electric conductivity of the iron ; or, 

W=aE*y. 

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 ^M..^., 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, ft ; 
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, 

sin/3 = , 

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, etc., 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 component, g E, of eddy currents, is of 
interest, since the wattless component is identical with the 
wattless component of hysteresis, discussed in a preceding 
chapter. 



FOUCAULT OR EDDY CURRENTS. 



131 



88. To calculate the loss of power by eddy currents 

Let V = volume of iron ; 

(B = maximum magnetic induction ; 
N= frequency; 

y = electric conductivity of iron ; 
= coefficient of eddy currents. 

The loss of energy per cm 3 , in ergs per cycle, is 



hence, the total loss of power by eddy currents is 

W = e y VN* (B 2 10 - 7 watts, 
and the equivalent conductance due to eddy currents is 



o _ W _ IQey/ __ .507ey/ 






> Tf"2 O 2 C^/2 C2 * 






where : 






/ = length of magnetic circuit, 


d 




S section of magnetic circuit, 
n = number of turns of electric circuit. 






The coefficient of eddy currents, e, 
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. 




x i JC 


The two most important cases are : 






(a). Laminated iron. 
(b). Iron wire. 


1 


' 1 


89. (a). Laminated Iron. 
Let, in Fig. 79, 


i 




d = thickness of the iron plates ; 
(B = maximum magnetic induction ; 
JV = frequency ; 
y = electric conductivity of the iron. 


Fi 


1.79. 



132 ALTERNATING-CURRENT PHENOMENA. 

Then, if x is the distance of a zone, d x, from the center 
of the sheet, the conductance of a zone of thickness, */x, 
and of one cm length and width is y^x ; and the magnetic 
flux cut by this zone is (Bx. Hence, the E.M.F. induced in 
this zone is 

8 E = V2 TrN($> x, in C.G.S. units. 

This E.M.F. produces the current : 

///=S J y</x = V2 TrN<$> y x d x, 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 

dP = EdI= 2 7T 2 ^ 2 (B 2 y x Vx, in C.G.S. units or ergs per second, 

and, consequently, the total power consumed in one cm 2 of 
the sheet of thickness, d, is 



= C + * dP = 27rW 2 (B 2 y C 



in C.G.S. units; 



the power consumed per cm 3 of iron is, therefore, 



. 

/ = = - '- , m C.G.S. units or erg-seconds, 
and the energy consumed per cycle and per cm 3 of iron is 



N 6 

The coefficient of eddy currents for laminated iron is, 
therefore, 

c = ^- = 1.645 d\ 



FOUCAULT OR EDDY CURRENTS. 133 

where y is expressed in C.G.S. units. Hence, if y is ex- 
pressed in practical units or 10 ~ 9 C.G.S. units, 

c = 7rVn '- = 1.645 </ 2 10 - 9 . 

Substituting for the conductivity of sheet iron the ap- 

proximate value, 

y = 10 5 , 

we get as the coefficient of eddy currents for laminated iron, 
2 -= 1.645</ 2 10- 9 - 



loss of energy per cm 3 and cycle, 

W= ey^Wfc 2 = - // 2 y^(B 2 10- 9 = 1.645 </ 2 y N<$? 10 ~ 9 ergs 
6 

= 1.645</ 2 7V~(B 2 10- 4 ergs; 
or, W = c y NW 10 - 7 = 1.645 d* N <S? 10 - " joules ; 

loss of power per cm 3 at frequency, N, 

p = NW '= cy^ 2 2 10- 7 = 1.645 </W 2 (B 2 10 ~ n watts; 
total loss of power in volume, V, 

p = vp = 1.645 ^/ 2 ^ 2 (B 2 10- n watts. 

As an example, 

d = 1 mm = .1 cm ; N= 100 ; OS = 5000; V = 1000 cm 8 . 
e = 1,645 X 10-"; 
^F= 4110 ergs 

= .000411 joules; 
/ = .0411 watts; 
P = 41.1 watts. 

90. (6): 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, d x, 
and one cm in length, 
the conductance of this pig. so. 




134 ALTERNATING-CURRENT PHENOMENA. 

zone is, y^/x/2 TT x, and the magnetic flux inclosed by the 
zone is (B x 2 *. 

Hence, the E.M.F. induced in this zone is : 

8 = V2 7r 2 ^(B x 2 , in C.G.S. units, 
and the current produced thereby is, 



, in C.G.S. units. 

The power consumed in this zone is, therefore, 

dP= EdI = 7T 8 y N' 2 (B 2 x 3 d x, in C.G.S. units 

consequently, the total power consumed in one cm length 
of wire is 

8 P = f~ dW = 7T 3 y N' 1 2 f * x a dx 

= ^-y^ 2 &V 4 , in C.G.S. units. 
Since the volume of one cm length of wire is 

/ ,*?, - 'I 

the power consumed in one cm 3 of iron is 

x P 2 

P = -^- = ^ y ^ 2 (BV 2 , in C.G.S. units or erg-seconds, 

and the energy consumed per cycle and cm 3 of iron is 

ergs. 



Therefore, the coefficient of eddy currents for iron wire is 
c = ^^ 2 = .617 </ 2 ; 

or, if y is expressed in practical units, or 10 ~ 9 C.G.S. units, 

c = -^ 
10 



FOUCAULT OR EDDY CURRENTS. 135 

Substituting ^ = ^ 

we get as the coefficient of eddy currents for iron wire, 

e= ^ 2 10~ 9 = .617 </ 2 10- 9 . 
16 

The loss of energy per cm 3 of iron, and per cycle 
becomes 



= .617 d*N? 10~ 4 ergs, 

loss of power per cm 3 , at frequency, N, 

p = Nh = ey^ 2 (B 2 10- 7 = .617 d 2 N*<$? 10 -" watts; 
total loss of power in volume, V, 

P= Vp = .617 FVJV'&'IO- 11 watts. 
As an example, 
d = 1 mm, = .1 cm ; N= 100 ; 2 = 5,000 ; V= 1000 cm 8 . 

e = .617 X 10- 11 , 
W= 1540 ergs = .000154 joules, 
p = .0154 watts, 
P = 15.4 watts, 

hence very much less than in sheet iron of equal thickness. 

91. Comparison of sheet iron and iron wire. 

If 

//! = thickness of lamination of sheet iron, and 
d z = diameter of iron wire, 

the eddy-coefficient of sheet iron being 
T* j 2 10-9 

* T? 

and the eddy coefficient of iron wire 



136 AL TERNA TING-CURRENT PHENOMENA. 

the loss of power is equal in both other things being 
equal if e x = e 2 ; that is, if, 

# = !</!, or 4 = 1.63 ^. 
o 

It follows that the diameter of iron wire can be 1.63 
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. 



Fig. 81. 

92. Demagnetizing, 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 with 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 lag 
90 behind the flux producing them, their resultant with 
the impressed M.M.F., and therefore the magnetism in the 



FOUCAULT OR EDDY CUKREN7*S. 137 

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, sufficient 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 : 



dl = -rrN&jxdx units (C.G.S.) 

= V2 TT N&jxdx 10 - 8 amperes ; 
hence the total current in sheet is 

/= 



amperes. 



Hence, the maximum possible demagnetizing ampere-turns 
acting upon the center of the lamina, are 

A/9 

- 8 = .555 N&jd* 10 - 8 



8 
= .555 ./V(B</ 2 10~ 3 ampere-turns per cm 

Example : d = .1 cm, N= 100, (B = 5,000, 
or / = 2.775 ampere-turns per cm. 



138 ALTERNATING-CURRENT PHENOMENA. 

93. In iron wire of diameter d, the current in a tubular 
zone of dx thickness and x radius is 

dl= TT JV&j'x dxlO-* amperes; 
hence, the total current is 

I = f$4I~?2. v N&j 10- f* xdx 

Jo " Jo 

A/9 



~ * amperes. 
16 

Hence, the maximum possible demagnetizing ampere-turns, 
acting upon the center of the wire, are 



10 - 



16 

= .2775 N(S> d* 10 - 8 ampere-turns per cm. 

For example, if d= .1 cm, N = 100, = 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 
structure. 

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. 



FOUCAULT OR EDDY CURRENTS. 139 

Eddy Currents in Conductor, and Unequal Current 
Distribution. 

95. 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 
known. 

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- 
where. 

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. 



140 ALTERNATING-CURRENT PHENOMENA. 

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. 




Fig. 82. 



The magnetic reluctance of a tubular zone of unit length 
and thickness dx t of radius x, is 



FOUCAULT OR EDDY CURRENTS. 141 

The current inclosed by this zone is I x = zW, and there 
fore, the M.M.F. acting upon this zone is 

$ x = 47r I x / 10 = 4 **/ 10, 

and the magnetic flux in this zone is 

d$> = $x I G(x = 2 Trixdx / 10. 
Hence, the total magnetic flux inside the conductor is 

, 27T . CR . TTiR* I 



From this we get, as the excess of counter E.M.F. at the 
axis of the conductor over that at the surface 

&E = V27r^0> 10 ~ 8 = V27r7W10 - 9 , per unit length, 



and the reactivity, or specific reactance at the center of the 
conductor, becomes k = &E / i = V2 i^NR* 10 ~ 9 . 
Let p = resistivity, or specific resistance, of the material of 
the conductor. 

We have then, k/p = V^TrW^lO- 9 /?; 
and p/ VFT7, 

the ratio of current densities at center and at periphery. 

For example, if, in copper, p = 1.7xlO 6 , and the 
percentage decrease of current density at center shall not 
exceed 5 per cent, that is 

P -H VF+7 2 = .95 - 1, 

we have, = .51xlO-; 

hence .51 x 10- 6 = V^TrW^lO- 9 
or N2? = 36.6 ; 

hence, when N= 125 100 60 25 

= .541 .605 .781 1.21 cm. 
D = 1R= 1.08 1.21 1.56 2.42cm. 
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, 
however, excluded from use at this frequency by the exter- 
nal self-induction, which is several times larger than the. 



142 ALTERNATING-CURRENT PHENOMENA. 

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 p., then ; d$ = pff x / (& x and 
thus ultimately ; k = V2 wW/^10 ~" and ; k / P = V2 ** 
NpR* 10 '// Thus, for instance, for iron wire at 
/> = 10xlO- 6 , ft = 500 it is, permitting 5% difference 
between center and outside of wire; k = 3.2 X 10 ~ 6 and 
NR* = .46, 
hence when, N = 125 100 60 25 

X = .061 .068 .088 .136 cm. 
thus the effect is noticeable even with relatively small iron 

wire. 

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 circuit upon a parallel circuit, may the effect on the 
primary circuit be considered analogously as in the chapter 
on eddy currents, by the introduction of an energy com- 



FOUCAULT OR EDDY CURRENTS. 143 

ponent, representing the loss of power, and a wattless 
component, representing the decrease of self-inductance. 

Let 

x = 2 TT 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 ; 

.#! = 2-jrNL 1 = reactance of secondary circuit ; that is, 
L l = total number of interlinkages with the secondary 
conductor, of the lines of magnetic force produced by unit 
current in that conductor ; 

x m = 2 TT N L m = mutual inductance of circuits ; that is, 
L m = 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 m * < xx^* 

* As coefficient of self-inductance L, L^, the total flux surrounding the conductor 
is here meant. Usually in the discussion of inductive apparatus, especially of trans- 
formers, that part of the magnetic flux is derroted self-inductance of the one circuit 
which surrounds this circuit, but not the other circuit ; that is, which passes between 
both circuits. Hence, the total self-inductance, L, is in this ease equal to the sum of 
the self-inductance, Z,j, and the mutual inductance, L m . 

The object of this distinction is to separate the wattless part, Z 1? of the 
total self-inductance, L, from that part, L m , which represents the transfer of 
E.M.F. into the secondary circuit, since the action of these two components is 
essentially different. 

Thus, in alternating-current transformers it is customary and will be 
done later in this book to denote as the self-inductance, Z, of each circuit 
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, L m , is usu- 
ally very much greater than the self-inductances, //, and Z/, while, if the 
self-inductances, Z and Zj , represent the total flux, their product is larger 
than the square of the mutual inductance, L m ; or 



144 ALTERNATING CURRENT PHENOMENA. 

Let r x = resistance of secondary circuit. Then the im- 
pedance of secondary circuit is 

^i = r v /*! , z l = V/v + xi 2 ; 

E.M.F. induced in the secondary circuit, = jx m f, 
where / = primary current. Hence, the secondary current is 



and the E.M.F. induced in the primary circuit by the secon- 
dary current, 7 l is 



or, expanded, 

Y z r j~. 2 

x m^ JX m 



2 _i_ r 2 r 2 i JT 

T^ ^i "l " * 2 

Hence, the E.M.F. consumed thereby 



effective resistance of mutual inductance ; 



^ = 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- 
inductance. 

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. 



FOUCAULT OR EDDY CURRENTS. 145 

While the laws of 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 th power of the field strength, that is, following 
the same law as the magnetic hysteresis, 

^ = ^(B'- 6 . 

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 W^ = e7V(B 2 . 

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- 



146 AL TERNA TING-CURRENT PHENOMENA. 

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= b', 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, 

Y=g+jb=g-jb', 

where : g = hysteretic conductance, b' = hysteretic suscep- 
tance ; and the dielectric hysteretic impedance of a con- 
denser, . . . 

Z = r jx r +jx c , 

where : r = hysteretic resistance, x c hysteretic condens- 
ance ; and the angle of dielectric hysteretic lag, tan a = b' / g 
= x c / 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 Arno 
as proportional to the 1.6 th 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 th 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. 



FOUCAULT OR EDDY CURRENTS. 147 

To the phenomenon of mutual inductance corresponds, 
in the electrostatic field, the electrostatic induction, or in- 
fluence. 

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 






148 ALTERNATING-CURRENT PHENOMENA 

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 capacity 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 
strength. 



FOUCAULT OR EDDY CURRENTS. 149 

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 
energy component of current, in phase with E.M.F., and = 

E.M.F. X effective conductance, or g ; 
wattless component of current, in quadrature with E.M.F., and = 

E.M.F. X effective susceptance, or b. 

In many cases the exact calculation of the quantities, 
r, x, g, b, 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 
losses. 

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. 



150 ALTERNATING-CURRENT PHENOMENA. 



CHAPTER XII. 

POWER, AND DOUBLE FREQUENCY QUANTITIES 
IN GENERAL. 

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 



The product, 

P = EI= (M l - *"/") +j (W 



POWER, AND DOUBLE FREQUENCY QUANTITIES. 151 

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 El 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 7, and thus cannot be represented by a 
vector in the same diagram with E and /. 

P = E I 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 w in E and / corre- 
sponds to a phase angle 2 w in the power, it is of interest to 
investigate the product E I formed by doubling the phase 
angle. 

Algebraically it is, 

P=EI= (* +>") (V 1 +/z n ) = 



Since j* = - 1, that is 180 rotation for E and /, for the 
double frequency vector, P,j* = + 1, or 360 rotation, and 

j x 1 =j 
1 x >= -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, 

p = [El] = (W 1 + ^V 11 ) +/ (W 1 - A' u ) 

The symbol [E /] here denotes the transfer from the 
frequency of E and / to the double frequency of P. 



152 AL TERNA TING-CURRENT PHENOMEMA. 

The product, P = \E /] consists of two components ; 
the real component, 

JP 1 = [EIJ = (W 1 + e"i n ) 
and the imaginary component, 

JPJ =j 
The component, 

P 1 

is the power of the circuit, = E I cos (E /) 
The component, 
PJ = 



is what may be called the " wattless power," or the power- 
less or quadrature volt-amperes of the circuit, = E /sin 
(El}. 

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, 

= 

jfc == 

P 
tan <f> = -j 

so the double frequency vector product P = [E /] denotes 
more than the mere power, by giving with its two compo- 
nents P 1 = [E I] 1 and PJ = [E /]>, the true energy volt- 
amperes, and the wattless volt-amperes. 

If 

E = 



POWER, AND DOUBLE FREQUENCY QUANTITIES. 153 

then 



and 

P 1 = 




or 



2 2 22 22 22 22 

+PJ =<* ,1 + *" / 



where ^ = total volt amperes of circuit. That is, 

The true power P 1 and the wattless power P$ are the two 
rectangular components of the total apparent power Q of the 
circuit. 

Consequently, 

In symbolic representation as double freqi'ency 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. 

P 1 

=p = cos w = power factor. 

PJ 

= q = sin w = inductance factor 

of the circuit, and the general expression of power is, 



= Q (cos co -\-j sin o>) 

104. The introduction of the double frequency vector 
product P = \E I~\ brings us outside of the limits of alge- 



154 ALTERNATING-CURRENT PHENOMENA. 

bra, however, and the commutative principle of algebra, 
a X b = b X a, does not apply any more, but we have, 

[El] unlike [IE] 
since 



we have 

[EIJ = [IEJ 

[EI]J=-[IE]J 

that is, the imaginary component reverses its sign by the 
interchange of factors. 

The physical meaning is, that if the wattless power 
[E 7p is lagging with regard to E, it is leading with regard 
to/. 

The wattless component of power is absent, or the total 
apparent power is true power, if 



[EI]J = (W 1 - A' 11 ) = 0. 
that is, 



or, 

tan (E) = tan (/), 

that is, E and / are in phase or in opposition. 

The true power is absent, or the total apparent power 
wattless, if 

[El] 1 = (W 1 + M* = 

that is, 

*" _ i 1 

7 ~ ~/ 
or, 

tan E = cot I 

that is, E and / are in quadrature, 



POWER, AND DOUBLE FREQUENCY QUANTITIES. 155 

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, 



We have, 

[, - 7] = [- E, 7] = - 



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 

\EJT\ = -j [, 7] = [E, iy -j \E, 7] 1 
\JE, 7] =j [E, 7] = - [E, Jy +j [E, 7] 1 
\jEjr\ = [, 7] = [E7? +j [E, jy 



105. If 7- = [^/J, 7> 2 = [E 2 / 2 ] . . . P n = [E n l n } 

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 



7^' = TV + T'ijJ. . + TV 

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 
expression. 

The first equation is obviously directly a result from the 
law of conservation of energy. 



156 ALTERNATING-CURRENT PHENOMENA. 

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 Pi, 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 F$\ 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 
way. 

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 space. 

Thus if 

E 1 = +je a - induced E.M.F. in one direction in 
space. 

7 2 = z 1 +j z 11 = secondary current in the quadrature di- 
rection in space, 



POWER, AND DOUBLE FREQUENCY QUANTITIES. 157 

the torque is 



By this equation the torque is given in watts, the mean- 
ing being that T = \E /]>' is the power which would be 
exerted by the torque at synchronous speed, or the torque 
in synchronous watts. 

The torque proper is then 



where 

/ = number of pairs of poles of the motor. 

In the polyphase induction motor, if 7 2 = i l +/z u is 
the secondary current in quadrature position, in space, to 
E.M.F. Ej. 

The current in the same direction in space as E l is 
/! =y7 2 = z 11 +// 1 ; thus the torque can also be ex- 
pressed as 



158 ALTERNATING-CURRENT PHENOMENA. 



CHAPTER XIII. 

DISTRIBUTED CAPACITY, INDUCTANCE, RESISTANCE, AND 
LEAKAGE. 

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. 



iiiimiiiiumiiiT 

TTTTTTTTTT.TTTTTTTTTT 



i 



Fig. 83. Distributed Capacity. 

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- 



DISTRIBUTED CAPACITY. 159 

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 
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 | 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- 
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 
of applicability of the approximate representation of the line 
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 -K/ -=- 4 log e 2 d/ 8 microfarads, 
where 

K = dielectric constant of the surrounding medium = 1 in air ; 

/ = length of conductor = 5 x 10 6 cm. ; 

d = distance of conductors from each other = 50 cm. ; 

8 = diameter of conductor = 1 cm. 

Since C = .3 microfarads, 

the capacity reactance is x 10 6 / 2 TT NC ohms, 



160 ALTERNATING-CURRENT PHENOMENA. 

where N '= frequency; hence, at N = 60 cycles, 

x = 8,900 ohms ; 

and the charging current of the line, at E = 20,000 volts, 
becomes, ^ = E / 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 

2,000 
/ = -^- = 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, 
30,000, 

capacity reactance at 60 cycles, x = 2,970 ohms ; 

charging current, i = 10.1 amperes ; 

line resistance, r = 66 ohms ; 

main current at 10 per cent loss, 7= 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 + j b = admittance of receiving circuit ; 
z = r j x = impedance of line ; 
b e = condenser susceptance of line. 






DISTRIBUTED CAPACITY. 



161 



Denoting, in Fig. 84, 

the E.M.F., viz., current in receiving circuit by , I t 

the E.M.F. at middle of line by ', 

the E.M.F., viz., current at generator by E 0) I \ 



If 



We have, 



Fig. 84. Capacity Shunted across Middle of Line. 



. = I-jb c E' 



E\\ \ (r 



Jb e (r-Jx) ., (r-j x y( 

~~ 



or, expanding, 



[(* - b c } - (rg+ 



-jx) 



I (r-jx)(g+jt)-\} 
2 Jf 



110. ^.) ZW capacity represented by three condensers^ 
in the middle and at the ends of the line. 
Denoting, in Fig. 85, 

the E.M.F. and current in receiving circuit by , 7, 

the E.M.F. at middle of line by ' ', 



162 



ALTERNATING-CURRENT PHENOMENA. 



the current on receiving side of line by /', 
the current on generator side of line by 7", 
the E.M.F., viz., current at generator by , f , 



Iff 



_L I 



85. Distributed Capacity. 



otherwise retaining the same denotations as in A.), 
We have, 
7 = 



2" = 1' - 



As will be seen, the first terms in the expression of E 
and of I are the same in A.) and in B.). 



DISTRIBUTED CAPACITY. 163 

111. 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 
account. 

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.Fs. 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, 



164 ALTERNATING-CURRENT PHENOMENA. 

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 
shows. 

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 
thereto. 

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 
component. 

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 



DISTRIBUTED CAPACITY. 165 

as consumption of a current, whose component in phase 
with the E.M.F. is the dielectric energy current, which 
may be considered as the power component of the capacity 
current. 

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 I, wattless, and due to : 
Self-inductance, and Mutual inductance. 
Currents consumed in phase with the 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 fo'ur constants : 

Effective resistance, r, 
Effective reactance, x, 
Effective conductance, g, 
Effective susceptance, b b c , 



1GG ALTERNATING-CURRENT PHENOMENA. 

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 l =r l J Xl ; 

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., 

Ei = e \ + J e \i and the current : /i = i\ +///, 

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 : 

Leakage current : JEgdx', 
Capacity current : j E b c d x ; 

hence, the total current consumed by the line element, dx, 
is dl= E(g-jb c }d*, or, 

d -t=E(g-jb c \ (1) 

E.M.F. consumed by resistance, Ird*\ 
E.M.F. consumed by reactance, j 



DISTRIBUTED CAPACITY. 107 

hence, the total E.M.F. consumed in the line element, ^/x, is 
dE = I (r j'x) </x, or, 
ffi. -I(f-jx). (2) 

These fundamental differential equations : 

*L-E(g-jt,),\ (1) 



(2) 



are symmetrical with respect to / and E. 
Differentiating these equations : 
d*I dE , 



and substituting (1) and (2) in (3), we get : 

(4) 



(5) 
the differential equations of E and L 

115. 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 : 

(6) 



and are integrated by 

W = tf 6 rx , 

where e is the basis of natural logarithms ; for, differen- 
tiating this, we get, 






168 ALTERNATING-CURRENT PHENOMENA. 

hence, z> 2 = (g j b c ) (r jx) ; 



(7) 



or, v = V (g - Jb e ) (r joe) \ 

hence, the general integral is : 

tr*.e+-Mr (8) 

where a and b are the two constants of integration ; 
Substituting 

r--/0 (9) 

into (7), we have, 

(a -JP)* = (g - jb c ) (r - jx) ; 
or, 



therefore, _ f 

);-' (10) 



Vl/2 6 - e 



/3= Vl/2 
substituting (9) into (8) : 



= a-c ax (cos/3x /sin^Sx) + ^c~ ax (cos/3x +y sin/3x) ; 
/ = (a x + /5>e~ ax ) cos)8x y(ae ax ^- ax ) sin /3x (12) 

which is the general solution of differential equations (4) 
and (5) 

Differentiating (8) gives : 



hence, substituting, (9) : 
(a JP) {(a 



x}. (13) 

Substituting now / for w, and substituting (13) in (1), 
and writing, 



DISTRIBUTED CAPACITY. 



169 



we get, 



/ \( J f ax. _i_ > 


?e-)cosj8x-y(y 
?-)cos/8x-y(y 


- 


* /_> < \ ' 
a 7/5 

sin /2x} ; 

'** 1 K^" i 


S J^c 
sin ySxf ; 



where ^4 and ^ are the constants of integration. 
Transformed, we get, 

/= J Ae a * (cos )8x j sin 0x) + Bf.~ 

a JP ( ' 

(cos /?x +/ sin /8x) > 
1 



^4e ax (cos /8x y sin 



^-. 

(cos /3x +y sin y8x) 

Thus the waves consist of two components, one, with 
factor ^e ax , increasing in amplitude toward the generator, 
the other, with factor ^e- ax , decreasing toward the genera- 
tor. The latter may be considered as a reflected wave. 

At the point x = 0. 



a-j/3 
A-B 



n 

Thus m (cos to j sin G) = - 

and, 

m = amplitude. 

w = angle of reflection. 

These are the general integral equations of the problem. 



116. If 

/! = /! + /// is the current 
{ is the E.M.F. 



at point, x 



(15) 



170 ALTERNATING-CURRENT PHENOMENA. 

by substituting (15) in (14), we get : 
2 A = {(a t\ + ft //) + (ge v + b c ^') 

(16) 




2 B = {(a /! + /? //) - (ge, + /; c ,/)} 
+ /{(//- 0/0 -(^I'-^ 
a and ft being determined by equations (11). 

117. H Z R j X is the impedance of the receiver 
circuit, E = e + j > ' is the E.M.F. at dynamo terminals 
(17), and / = length of line, we get 
at 



hence 




g jb c 
or 



a-; ft 
At X = /, 



E 



sin/?/}. (19) 



Equations (18) and (19) determine the constants A and B, 
which, substituted in (14), give the final integral equations. 

The length, X = 2 TT / ft is a complete wave length (20), 
,vhich means, that in the distance 2 IT / ft 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 in opposite directions, and the E.M.Fs. are opposite. 

118. The difference of phase, w, between current, /, and 
E.M.F., E y at any point, x, of the line, is determined by 



DISTRIBUTED CAPACITY. 171 



the equation, 

Z?(cos+/sin) =y, : \j JsTI 71 

where Z> is a constant. 

Hence, w varies from point to point, oscillating around a 
medium position, w x , which it approaches at infinity. 

This difference of phase, C> x , towards which current and 
E.M.F. tend at infinity, is determined by the expression, 



^(cos . .. , (/ 

or, substituting for E and /their values, and since e~ a * = 0, 
and A e ax (cos ft x j sin ft x), cancels, and 

D (cos tow +/sin oioc) = 2-p- 



hence, tan ^ = ~ a c + ^ (21) 

This angle, Stx, = ; that is, current and E.M.F. come 
more and more in phase with each other, when 

ab c fig ; that is, 

a -T- ft g -r- b c , or, 

2a/3 ! *2^*/ 5 
substituting (10), gives, 



hence, expanding, r -4- ^ = ^ -f- ^ c ; (22) 

that is, tJie ratio of resistance to inductance equals the ratio 
of leakage to capacity. 

This angle, w x , = 45 ; that is, current and E.M.F. differ 
by th period, if a b c + fig = a.g + pb c ; or, 



which gives : rg + x b c = 0. (23) 



172 



ALTERNA TING-CURRENT PHENOMENA. 



That is, two of the four line constants must be zero; cither 
g and x, or g and b c . 

The case where g = = x, 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- 







< 
































"o^ 

+ 30 


'sr 
I 




\ 




























OLT 
.0,000 


20 


i 




\ 




























8()0 { 




1 
1 




\ 








/ 





*\ 



















j 


ja 






u 






i 






\ 






X 





*> 


V 


u 




( 






\ 














^ - 


+'' 








/ 




-20 








\ 




/ 


















/ 






-30 


/ 








"*"> 
















/ 











-40. 
























: 






., 


us 
Kl 


s 


I 
-50 
























/ 








/.o 




7,0 






















/ 








/ 


z;o 


























/ 






/ 






p. 

. 
















, 


j^, 




/ 












u-j 


c 














/ 














'' 






















/ 






.- 


S 














.ooo 


/ 


? ' 


N 






/ 








/ 
















' 000 


/ 




\ 




^~ 


/ 






V 
















100 


0,00, 








X 


/ 




N >, 


.s 










x = ' 








60 


9 000 






/ 
/ 




















g=i 

b c = 


XI 
'X| 


rj-4 







.ooo 


\ 

\ 




/ 






























4,000 


\, 


-"* 






























JO 


J.OOO 







i 






3 


- 




J 
\ 


L 










5 
1 


L 




i 



Fig. 86. 



DISTRIBUTED CAPACITY. 173 

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, E v =e l = 10,000; 
current at receiving end, 65 amperes, with a power factor 
of .385. 

that is, / = t\ + j // = 25 + 60 j ; 

line constants per unit length, 

r = 1, g = 2 X 10- 5 , 

hence, 

a = 4.95 x 10- 3 , ] 
13 = 28.36 x 10 - 3 , j- 



length of line corresponding to 
one complete period of the wave 



x = L = = 221.5 = 

(^ of propagation. 
A = 1.012 - 1.206 y ) 
B = .812 + .794 / j 

These values, substituted, give, 

/= { x (47.3 cos /?x + 27.4 sin fix) e-* 

(22.3 cos ftx + 32.6 sin fix)} 
+ y (e x (27.4 cos ftx 47.3 sin ftx) + - x 

(32.6 cos y3x 22.3 sin /3x)}; 
E = {e ox (6450 cos /3x + 4410 sin j8x) + c- ax 

(3530 cos fix + 4410 sin /?x)} 
+ y (e ox (4410 cos /3x 6450 sin x) e~ ax 
(4410 cos ft x- 3530 sin /3x)}; 

tan 5, = ~ - lj c + PS = _ .073, JJ = - 4.2. 

120. As a further instance are shown the characteristic 
curves of a transmission line of the relative constants, 

r\x\g>.b = % : 32 : 1.25 X 10 ~ 4 : 25 X 10 ~ 4 , and e 
= 25,000, i = 200 at the receiving circuit, for the con- 
ditions, 

a, non-inductive load in the receiving circuit, Fig. 87. 



174 



ALTERNATING-CURRENT PHENOMENA. 



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 



ZJ 

7 

6sa 



7 




rig. 87. 



DISTRIBUTED CAPACITY. 



175 



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 
angles. 



\ 






V 



Fig. 88. 



176 



AL TERNA TING-CURRENT PHENOMENA. 



For such graphical constructions, polar coordinate paper 
and two angles a and 8 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 

current, tan 8 = - = 20 in above instance. 
g 



\ 





Fig. 89. 



DISTRIBUTED CAPACITY. 



177 



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- 



t 



a +120 






ISSION LINE 



V 



Fig. 90. 

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 
diagram. 



178 DISTRIBUTED CAPACITY. 

121. The following are some particular cases : 
A.) Open circuit at end of lines : 
x = : /! = 0. 

hence, 

E = i-r ^{(e ax + e- ax ) cos/3x y(c ax c- ax )sin/3x} ; 



.?.) Line grounded at end: 



A (a/\ -J- /?//) +/ ( a/ / ^ z i) = -? 
-^- T -^{(e ax c- ax ) cos/?x >(e ax + c~ ax ) sin)8x}; 



(T.) Infinitely long conductor : 

Replacing x by x, that is, counting the distance posi- 
tive in the direction of decreasing energy, we have, 

x = oo : 7= 0, E = 0; 
hence 



and 



I = - ^- x (cos/Sx +y s in/3x), 

' 



revolving decay of the electric wave, that is the reflected 
wave does not exist. 

The total impedance of the infinitely long conductor is 



(q-yff) (g+M 



+ b? g* + b* 



ALTERNATING-CURRENT PHENOMENA. 179 

The infinitely long conductor acts like an impedance 

7 _ -K + P ?>c _ fig Q-b c 

f*+v g* + K' 

that is, like a resistance 



combined with a reactance 



We thus get the difference of phase between E.M.F. 
and current, 



which is constant at all points of the line. 
If g = 0, x = 0, we have, 



hence, 

tan to = 1, or, 

= 45 ; 

that is, current and E.M.F. differ by th period. 
D.) Generator feeding into closed circuit : 
Let x = be the center of cable ; then, 

hence : E at x = ; 



which equations are the same as in B, where the line is 
grounded at x = 0. 

E.) Let the length of a line be one-quarter wave length; 



and assume the resistance r and conductance g as negligible 



180 AL TERN A TING-CURRENT PHENOMENA. 

compared with x and b c . 

r=0=g 
These values substituted in (11) give 

a=0. 

(3= V^ 

Let the E.M.F. at the receiving end of the line be 
assumed zero vector 

l = ei = E.M.F. and 

fi i'i + ji\. current at end of line x = 
= E.M.F. and 

S = current at beginning of line 



Substituting in (16) these values of E l and 7 : and also r = 
= g, we have 



From these equations it follows that 



which values, together with the foregoing values of E v I v r, 
g, a, and /8, substituted in (14) reduce these equations to 



j (i\ +jiC) \~r s ^ 



ALTERNATING-CURRENT TRANSFORMER, 181 

Then at x 



Hence also 



" and 7 are both in quadrature ahead of <? x and 7j 
respectively. 

I l = E Q y = constant, if 7f = constant. That is, at 

constant impressed E.M.F. E& the current 7 X in the receiv- 
ing circuit of a line of one-quarter wave length is constant, 
and inversely (constant potential constant current trans- 
formation by inductive line). In this case, the current 7 at 
the beginning of the line is proportional to the load e l at the 
end of the line. 

If X Q = lx = total reactance, 

b = lb c = total susceptance of line, then 

*<A> = 4- 

Instance* = 4, b c = 20 X 10 ~ 5 , E = 10,000 V. Hence 
/ = 55.5, * = 222, b = .0111, 7j = 70.7, 7 = .00707 e. 

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- 
cuit. 



182 ALTERNATING-CURRENT PHENOMENA. 

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, L, 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 
formula - 



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- 
nary 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 neglecting 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. 

Let / = total length of a transmission line, 
r = resistance per unit length, 
x = reactance per unit length = 2 ?r NL. 



DISTRIBUTED CAPACITY. 183 

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 = capacity susceptance per unit length = 2 TT NC 
where C = capacity per unit length. 

x = the distance from the beginning of the line, 

We have then the equations : 
The E.M.F., 



(^ e ax _ ^ e -ax) CQS x _j (4 

g jb I + ^e~ ax ) sin /3x 
the current, 

1 ^ (Ae a * + ^e~ ax ) COS /3x y (^4e 



where, 



,(14.) 



(r 1 + ^c 2 ) + (^r - 

' (11.) 



c = base of the natural logarithms, and A and B integration 
constants. 

Neglecting the line resistance, r = 0, and the conduc- 
tance (leakage, etc.), g=0, gives, 



These values substituted in (14) give, 
J - 



= J -\(A - B} cos ^fbx^ -j (A + H) sin 



/ = -4= J (^ + -ff) cos V^x y (<4 - B) sin 

; J 



184 ALTERNATING-CURRENT PHENOMENA. 

If the discharge takes place at the point : x = 0, that is, 
if the distance is counted from the discharge point to the 
end of the line ; x = /, hence : 

At x = 0, E = 0, 
Atx=/, 7=0. 

Substituting these values in (25) gives, 

For x = 0, 

^-7^ = A = B 

which reduces these equations to, 

E = sin Nbx x 

b \ 

7= -^4^= cos V&t: x 

VA* I 

and at x = 0, 



At x = /, / = 0, thus, substituted in (26), 

cos V^/ = (28.) 

hence : 

V^/^ 2 ** 1 )", 1 = 0,1, 2,... (29.) 

that is, *Jbx I is an odd multiple of ^ And at x = /, 

2t 

O A 



Substituting in (29) the values, 

we have, 

hence, 

^ = M + l (31.) 

4/VCZ 



DISTRIBUTED CAPACITY. 185 

the frequency of the oscillating discharge, 
where k = 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, 

*-I7^z (32 '> 

Since 

C = 1C = total capacity of transmission line, ) 

L = IL = total self-inductance of transmission line, J ^ '' 

we have, 

2, + 1 

-= the frequency of oscillation, (34.) 



or natural period of the line, and 

NI - - the fundamental, 
- 



or lowest natural period of the line. 
From (30), (33), and (34), 

b = 2irNC= 2/ \T ( 36- ) 

and from (29), 

V ^ = (2 ^2 f / )7r - < 37 ') 

These substituted in (26) give, 

f- (38.) 
4/7 (2 + l)7rx 

/= (2TTi)-^ cosL ^H 

The oscillating discharge of a line can thus follow any of 
the forms given by making k 0, 1, 2, 3 . . .in equation 
(38). 

Reduced from symbolic representation to absolute values 



186 ALTERNATING-CURRENT PHENOMENA. 

by multiplying E with cos 2 * Nt and / with sin 2 TT A 7 / and 
omitting j, and substituting A 7 " from equation (34), we have, 

(2+l)7rx 

sin JT - -cos 
2/ 




where ^4 is an integration constant, depending upon the 
initial distribution of voltage, before the discharge, and 
/ = time after discharge. 

123. The fundamental discharge wave is thus, for k = 0, 



47. Lo . . 7TX 7T/ 

-^ A sin 7^ 
C 2/ 



. o . . 7TX 

= \ -^ A sin 7^ cos 

TT V 



4 / - _, 7T X 7T/ 

fi = A cos 7n - sin - 



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 : k 1, gives : 

47. [Ln . . 3v_ 

(41.) 

4/ " " - ' 

/, = o- A cos 



n 7 
27 



DISTRIBUTED CAPACITY. 



187 



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 : x = _, then increases again, in 

o 



Fig. 



H----0 



Fig. 



\ 



\ 



\ 



\1 



Figs. 91-93. 



188 ALTERNATING CURRENT-PHENOMENA. 

the opposite direction, reaches a second but opposite maxi- 

2/ 

mum at two-thirds of the line : x = ^ , and decreases to 

o 

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- 

2/ 3 
creases to zero at two-thirds of the line : x = IT > 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 = 1, are shown in Fig. 92, 
those with k = 2, 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 
nodes). 

In case of a lightning discharge the capacity C 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 L . 

If d = diameter of line conductor, 

D = distance of conductor above ground, 
and / = length of conductor, 

the capacity is, 



1.11 x 10- 6 / ,. 

~ 



the self-inductance, 



The fundamental frequency of oscillation is thus, by 
substituting (42) in (35), 



DISTRIBUTED CAPACITY. 189 

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 6 cm.. 

hence frequency, 

N-i = 4680 2340 1560 1170 937.5 780 585 475 cycles-.. 

As seen, these frequencies are comparatively low, and 
especially with very long lines almost approach alternator 
frequencies. 

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 k = * j 




where a^ a s a y . . . are constants depending upon the 
initial distribution of potential in the transmission line, at 
the moment of discharge, or at / = 0, and calculated there- 
from. 



190 AL TERN A TING-CURRENT PHENOMENA . 

124. 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 harmonics are determined up to the 11 that is, a v 

a& # 5 > a v a 9> a n- 

These six unknown quantities require six equations, which 

/ 2/ 3/ 4/ 5/ 6/ 
are given by assuming E = e for x = g, _,_,_,_,_. 

At / = 0, E = e, equation (44) assumes the form 

4 / HT ( . TTX , . 3 TTX 
e = V ? j i sm 27 + * 3 sm ~27 + ' ' ' ' + * u 

(45.) 

/ 2/ 6/ 

Substituting herein for x the values : - , , . . . 

gives six equations for the determination of a v <7 3 . . . a n . 
These equations solved give, 



E = e (1.26 sin w cos $ + .40 sin 3 w cos 3 <f + .22 sin ^ 
5 w cos 5 <^ + .12 sin 7 o> cos 7 < + .07 sin 9 co 
cos 9 ^ + .02 sin 11 o> cos 11 ^> 

5 

L 

cos 5 <o sin 5 <^ + .12 cos 7 o> sin 7 < + .07 cos 
9 to sin 9 < + .02 cos 11 o> sin 11 </> 



7 = e i/5 (1.26 cos o> sin < + .40 cos 3 w sin 3 < + .22 

V 7 rt 



,(46.) 



where, 

"-57 1 

r< 47 ') 



Instance, . 

Length of line, / = 25 miles = 4 x 10 6 cm. 
Size of wire : No. 000 B. & S. G., thus : d = 1 cm. 
Height above ground : D 18 feet = 550 cm. 
Let e = 25,000 volts = potential of line in the moment of 
-discharge. 



DISTRIBUTED CAPACITY. 191 

We then have, 

E = 31,500 sin w cos <fr + 10,000 sin 3 <o cos 3 < + 5500 sin 

5 o> cos 5 <J> + 3000 sin 7 o> cos 7 < -j- 1750 sin 9 o> cos 

9 < + 500 sin 11 w cos 11 <. 
/= 61.7 cos w sin < + 19.6 cos 3 o> sin 3 < + 10.8 cos 5 sin 

5 < + 5.9 cos 7 CD sin 7 </> + 3.4 cos 9 to sin 9 < + 1.0 

cos 11 <o sin 11 <J>. 

<o= .39 .r 10 - 6 
</> = 1.18/ 10+ 4 

A simple harmonic oscillation as a line discharge would 
require a sinoidal 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 field 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. 

125. 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 



192 ALTERNATING-CURRENT PHENOMENA. 

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 
intensity. 

Hence the electric distribution in 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. 



ALTERNATING-CURRENT TRANSFORMER. 193 



CHAPTER XIV. 

THE ALTERNATING-CURRENT TRANSFORMER. 

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 



194 ALTERNATING-CURRENT PHENOMENA. 

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 flux is determined by the E.M.F. of the 
transformer, by the number of turns, and by the frequency. 
If 

< = maximum magnetic flux, 
N= frequency, 
n = number of turns of the coil ; 

the E.M.F. induced in this coil is 

E= V2 * JVfc * 10 - 8 = 4.44 .Afo* 10 -'volts; 



hence, if the E.M.F., frequency, and number of turns are 
determined, the maximum magnetic flux is 



To produce the magnetism, $, of the transformer, a 
M.M.F. of 5 ampere-turns is required, which is determined 



ALTERNATING-CURRENT TRANSFORMER. 195 

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 ($> = & /S, where S 
= the cross-section of magnetic circuit. 

129. To induce a magnetic density, ($>, a M.M.F. of 3C TO 
ampere-turns maximum is required, or, 3C OT / V2 ampere- 
turns effective, per unit length of the magnetic circuit ; 
hence, for the total magnetic circuit, of length, /, 

/3C 
& = :r- ampere-turns ; 



*V2 
where n = number of turns. 

At no load, or open secondary circuit, this M.M.F., CF, is 
furnished by the exciting current, T 00 , 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., &, 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 
loss. 

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, f 00 , 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 



196 ALTERNATING-CURRENT PHENOMENA. 

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, 7^, leads the wave of mag- 
netism, 3>, 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. = I 00 sin a; and 
the magnetizing current, in phase with the magnetic fluXj 
and therefore in quadrature with the induced E.M.F., and 
so wattless, = I 00 cos a. 

The exciting current, 7 00 , 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 



I 00 sin a 



Induced E.M.F. 



130. Graphically, the polar diagram of M.M.Fs. ot a 
transformer is constructed thus : 




Fig. 94. 



Let, in Fig. 94, O = the magnetic flux in intensity and 
phase (for convenience, as intensities, the effective values 
are used throughout), assuming its phase as the vertical; 



ALTERNATING-CURRENT TRANSFORMER. 197 

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 
QS, leading O<b by the angle &O = a. 

The induced E.M.Fs. have the phase 180, that is, are 
plotted towards the left, and represented by the vectors 
OZT; and OE. 

If, now, ft' = angle of lag in the secondary circuit, due 
to the total (internal and external) secondary reactance, the 
secondary current I I , and hence the secondary M.M.F., 
JF 1= j /L, will lag behind [ by an angle ft 1 , and have the 
phase, 180 + ft', represented by the vector O^ 1 . Con- 
structing a parallelogram of M.M.Fs., with Off as a diag- 
onal and Oif 1 as one side, the other side or O'S is the 
primary M.M.F., in intensity and phase, and hence, dividing 
by the number of primary turns, n , 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 7 r , in phase with f ot and 
represented by the vector OEr 

E.M.F. consumed by reactance is IoX , 90 ahead of I , and 
represented by the vector OEx ; 

E.M.F. consumed by induced E.M.F. is E', equal and oppo- 
site to E'o, and represented by the vector Off. 

Hence, the total primary impressed E.M.F. by combina- 
tion of OEr , OEx , and OE' by means of the parallelo- 
gram of E.M.Fs. is, 

E = ~OE , 

and the difference of phase between the primary impressed 
E.M.F. and the primary current is 

ft = E O5 . 
In the secondary circuit : 

Counter E.M.F. of resistance is 1^ in opposition with I v 
and represented by the vector OJS'r^ ; 



198 



AL TERNA TING-CURRENT PHENOMENA, 



90 behind 7 X , and 



Counter E.M.F. of reactance is 
represented by the vector OE^x^ 

Induced E.M.Fs., E( represented by the vector OE-[. 

Hence, the secondary terminal voltage, by combination 
of OEr^ OEx{ and OE^ by means of the parallelogram of 

E.M.Fs. is -== 

A = MII 

and the difference of phase between the secondary terminal 
voltage and the secondary current is 



As will 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." 



E r , 



Fig. 95. Transformer Diagram with 80 Lag in Secondary Circuit. 

131. In the construction of the transformer diagram, it 
is usually preferable not to plot the secondary 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 = n / n v for the reason that frequently 
primary and secondary E.M.Fs., etc., are of such different 



AL TERA?A TING-CURRENT TRANSFORMER. 



19!) 



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 E.M.Fs. coincide, or OE-^ = OE^. 





Fig. 96. Transformer Diagram with 50 Lag in Secondary Circuit. 

Figs. 96 to 107 give the polar diagram of a transformer 
having the constants 



r = .2 ohms, 
x = .33 ohms, 
f! = .00167 ohms, 
*! = .0025 ohms, 
g = .0100 mhos, 

for the conditions of secondary circuit, 



= .0173 mhos, 
= 100 volts, 
= 60 amperes, 
=10 degrees. ? 



20 lead in Fig. 99. 
50 lead " 100. 
80 lead " 101. 



ft' = 80 lag in Fig. 95. 

50 lag " 96. 

20 lag " 97. 

O, or in phase, " 98. 

As shown with a change of /?/ the other quantities E , I v 
I , 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 



200 ALTERNATING-CURRENT PHENOMENA. 




Fig. 97. Transformer Diagram with 20 Lag in Secondary Circuit 




Fig. 98. Transformer Diagram with Secondary Current in Phase with E.M.F. 




Fig. 99. Transformer Diagram with 20 Lead in Secondary Current. 



ALTERNATING-CURRENT TRANSFORMER. 201 




(To E O 

Fig. 100. Transformer Diagram with 50 Lead in Secondary Circuit. 




Fig. 101. Transformer Diagram with 80 Lead in Secondary Circuit. 




Fig. 102. 



202 



AL TERNA TING-CURRENT PHENOMENA. 



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. 103. 



Fig. 105 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 
circuit. 

a.) By increase of secondary resistance ; b.} 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, J5 lt and thus of E , etc., are straight lines; and in 
b.) and c.}, parts of one and the same circle a.} is shown 



AL TERNA TING-CURRENT TRANSFORMER. 



203 



in full lines, b.} in heavy full lines, and c.} in light full lines. 
This diagram corresponds to constant maximum magnetic 
flux ; that is, to constant secondary induced E.M.F. The 
diagrams representing constant primary impressed E.M.F. 
and constant secondary terminal voltage can be derived 
from the above by proportionality. 






Fig. 104. 



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 ; 
x = .033 ohms ; 
ri = .00008 ohms j 



#! = .00025 ohms ; 
g = .001 ohms ; 
b = .00173 ohms ; 



that is, about one-tenth as large as assumed. Thus the 
changes of the values of E , E lt etc., under the different 
conditions will be very much smaller. 



204 



ALTERNATING-CURRENT PHENOMENA. 



Symbolic Method. 

134. In symbolic representation by complex quantities 
the transformer problem appears as follows : 

The exciting current, 7 00 , of the transformer depends 
upon the primary E.M.F., which dependance can be rep- 
resented by an admittance, the " primary admittance," 
f tne transformer. 




Fig. 105. 

The resistance and reactance of the primary and the 
secondary circuit are represented in the impedance by 

Z =r - jx , and Z l =r l - j x l . 

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. 

Let 

n = number of primary turns in series ; 
#1 = number of secondary turns in series ; 
a = = ratio of turns ; 

Y = g 4- jb = primary admittance 

Exciting current . ~i I 

Primary counter E.M.F. ' 



.VVWvVl 



rw^ww 



ALTERNATING-CURRENT TRANSFORMER. 205 

Z = r j x = primary impedance 7. 

E.M.F. consumed in primary coil by resistance and reactance. ^ -n-f '" ' j**/\. 

Primary current ~ / 

Z = r jx 1 = secondary impedance 

__ E.M.F. consumed in secondary coil by resistance and reactance . 
Secondary current 

where the reactances, x and ^ , 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 = g -\-jb- total admittance of secondary circuit, 

including the internal impedance ; 
E = primary impressed E.M.F. ; 
E ' = E.M.F. consumed by primary counter E.M.F. ; 
i = secondary terminal voltage ; 
EI = secondary induced E.M.F. ; 
I = primary current, total ; 
/oo = primary exciting current ; 
/i = secondary current. 

Since the primary counter E.M.F.,-"', and the second- 
ary induced E.M.F., E^, are proportional by the ratio of 

turns, a , 

E ' = a E{. (1) 

The secondary current is : 

/i = **/, (2) 

consisting of an energy component, gE^, and a reactive 
component, b E^. 

To this secondary current corresponds the component of 
primary current, 

7o = ~a a* 

The primary exciting current is 

I>=Y O E>. (4) 

Hence, the total primary current is : 



206 AL TERNA TING-CURRENT PHENOMENA. 

(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 EI. 

Therefore, the secondary terminal voltage is 



or, substituting (2), we have 

, = ,' {I- Z,Y} (7) 

The E.M.F. consumed in the primary coil by the inter- 
nal impedance is Z I . 

The E.M.F. consumed in the primary coil by the counter 
E.M.F. is E'. 

Therefore, the primary impressed E.M.F. is 

E = E' + Z S , 
or, substituting (6), 



(8) 

\/ 



136. We thus have, 

primary E.M.F., E = - aE{ j 1 + Z Y + ^Z J , (8) 

secondary E.M.F., E^ = E{ { 1 - Z l Y}, (7) 

primary current, I = -{Y+a*Y }, (6) 

secondary current, /i = YE l ' t (2) 

as functions of the secondary induced E.M.F., EJ, as pa- 
rameter. 



ALTERNATING-CURRENT TRANSFORMER. 207 

From the above we derive 

Ratio of transformation of E.M.Fs. : 



. 1-Z.K 

Ratio of transformations of currents : 



(10) 



From this we get, at constant primary impressed 
E.M.F., 

E = constant ; 

secondary induced E.M.F., 



E.M.F. induced per turn, 
E 1 

n -\ \ 7 y \ 

secondary terminal voltage, 



primary current, 

^ 4- Y 
, . E A Y+a*Y _ w ^^ y 



secondary current, 

Y 



At constant secondary terminal voltage, 
-fi 1 ! = const. ; 



208 AL TERNA TING-CURRENT PHENOMENA. 

secondary induced E.M.F., 

F 1 - l 

1-^F' 
E.M.F. induced per turn, 



^1-Z.F' 
primary impressed E.M.F., 



primary current, 

/ 

secondary current, 



136. Some interesting conclusions can be drawn from 
these equations. 

The apparent impedance of the total transformer is 



(14) 



Substituting now, = V, the total secondary admit- 

tance, reduced to the primary circuit by the ratio of turns, 
it is 



Y -\-Y' is the total admittance of a divided circuit with 
the exciting current, of admittance Y , and the secondary 



AL TERN A TING-CURRENT TRANSFORMER. 



209 



current, of admittance Y 1 (reduced to primary), as branches. 
Thus : 



is the impedance of this divided circuit, and 



That is : 



(17) 



The alternate-current transformer, of primary admittance 
Y , total secondary admittance Y, and primary impedance 
Z , is equivalent to, and can be replaced by, a divided circuit 
with the branches of admittance Y , the exciting current, and 
admittance Y' = Y/a 2 , the secondary current, fed over mains 
of the impedance Z , the internal primary impedance. 

This is shown diagrammatically in Fig. 106. 



Yog 

z 





Fig. 106. 



137. Separating now the internal secondary impedance 
from the external secondary impedance, or the impedance of 
the consumer circuit, it is 

4 -.+ *! (18) 



where Z = external secondary impedance, 



(19) 



210 ALTERNATING-CURRENT PHENOMENA. 

Reduced to primary circuit, it is 



= Z/ + Z 7 . (20) 

That is : 

An alternate-current transformer, of primary admittance 
Y , primary impedance Z , secondary impedance Z v 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 circtiit of impedance Z ' = 
a?Z, fed over mains of the impedance Z -\- Z^, where Z^ = 
a 2 Z lt shunted by a circuit of admittance Y , which latter 
circuit branches off at the points a b, between the impe- 
dances Z and Z-. 



Generator I, Transformer I 





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^ 1 , . . . Z-f, closed by external 
secondary circuits of the impedances Z 1 , Z n , . . . Z x , is equiv- 
alent to a divided circuit of x + 1 branches, one branch of 



AL TERN A TING-CURRENT TRANSFORMER. 
Generator Transformer 



211 




Fig. 108. 

admittance Y 0) the exciting current, the other branches of the 
impedances ZJ + Z 7 , ZJ 1 + Z n , . . . 2f + Z x , 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. , over -mains of the impedance 
Z - 

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 Y . 

Thus, double transformation will be represented by dia- 
gram, Fig. 109. 





212 A L TERN A TING- CURRENT PHENOMENA . 

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 , . 
with a non-inductive secondary circuit Z^ = r v we get the 
condition of transformation from constant primary potential 
to constant secondary current, and inversely, as previously 
discussed. 

Non-inductive Secondary Circuit. 

138. In a non-inductive secondary circuit, the external 
secondary impedance is, 



or, reduced to primary circuit, 

Assuming the secondary impedance, reduced to primary 
circuit, as equal to the primary impedance, 

* is> Y ' i r 



Substituting these values in Equations (9), (10), and (13), 
we have 

Ratio of E.M.Fs. : 



(r jx } 
4- r a jx 



ALTERNATING-CURRENT TRANSFORMER. 213 



+ r -jx f r -jx Y| . . . \ . 

R + r jx \ R + r n /# 



or, expanding, and neglecting terms of higher than third 
order, 

jx 

^ 



or, expanded, 

J|= - 1 1 + 2 r ^'^ + (r, -y^)(.% 

Neglecting terms of tertiary order also, 



t 

Ratio of currents : 

^- = - - 

/I ^ 

or, expanded, 

~=-- 

/! a 

Neglecting terms of tertiary order also, 



Total apparent primary admittance : 



R + r jx 
(r -jx } + R (r - 



= {R + 2 (r - y x } - & ( go +jb } -2 R (r - Jx ) 



214 ALTERNATING-CURRENT PHENOMENA. 

or, 

b }- 2 (r -Jx }( 



Neglecting terms of tertiary order also : 
Z t =R 

Angle of lag in primary circuit : 

tan S> = ^ , hence, 
r t 

2^+Rb + 2r b -2 Xogo -2 
tan S> = a 



Neglecting terms of tertiary order also : 
'R 



139. If, now, we represent the external resistance of 
the secondary circuit at full load (reduced to the primary 
circuit) by R , and denote, 



2 r _ _ . Internal resistance of transformer _ percentage 

R ~ External resistance of secondary circuit ~ na ^ resistance, 

2 X _ __ rat j Q Internal reactance of transformer _ percentage 
J ' External resistance of secondary circuit nal reactance 

X*.- h - ratio - percentage hysteresis, 

,, , , . Magnetizing current percentage magnetizing cur- 

KO o= g = -10 Totalsecondarycurrent = rent ^ 

and if d represents the load of the transformer, as fraction 
of full load, we have 



ALTERNATING-CURRENT TRANSFORMER. 215 



and, 



**.-. 

a 

Substituting these values we get, as the equations of the 
transformer on non-inductive load, 
Ratio of E.M.Fs. : 






or, eliminating imaginary quantities, 



H"-"^) 



Ratio of currents : 

+ ( h +> 

d 



2 f 

. ^ 

or, eliminating imaginary quantities, 



1 f 

a \ 



i i h i 



216 ALTERNATING-CURRENT PHENOMENA. 

Total apparent primary impedance : 
Z, = 




or, eliminating imaginary quantities, 




Angle of lag in primary circuit : 



That is, 

An alternate-current transformer, feeding into a non-induc- 
tive secondary circuit, is represented by the constants : 

R = secondary external 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 2 . 



ALTERNATING-CURRENT TRANSFORMER. 



217 



140. As an instance, a transformer of the following 
constants may be given : 



e =1,000; 
a = 10 ; 



= 120; 

p = .02 



q = .06 ; 
h = .02 ; 
g = .04. 



Substituting these values, gives : 
100 



= 

" 



V(i.oou + .02 </) 2 + (.0002 + .06 <ty 



*-^-- 



.1 ii V/Y 1.0014 + Y + ( - 
\\ d J \ d 



. 0002 . 



- -.0004- 



tan w, 



^- 



1.9972 + . 




Fig. 110. Load Diagram of Transformer. 



218 ALTERNATING-CURKENT PHENOMENA. 

In diagram Fig. 110 are shown, for the values from 
d = to d= 1.5, with the secondary current i x 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 w , is .45 at 
open secondary circuit, and is above .99 from 25 amperes, 
upwards, with a maximum of .995 at full load. 



ALTERNATING-CURRENT TRANSFORMER. 219 



CHAPTER XV. 

THE GENERAL ALTERNATING-CURRENT TRANSFORMER OR 
FREQUENCY CONVERTER. 

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 reaction of the secondary current. 



220 ALTERNATING-CURRENT PHENOMENA. 

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 



AL TERN A TING-CURRENT TRA NSFORMER. 221 

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: 



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. 

145. Let, in a general alternating-current transformer : 

* = ratio iS^ frequency, or slip ; 
thus, if 

N '= primary frequency, or frequency of impressed E.M.F., 
s JV = secondary frequency ; 



222 ALTERNATING-CURRENT PHENOMENA. 

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 ; 

s < represents motion above synchronism driven by external 

mechanical power, as will be seen ; 
s = 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). 
Let 

= number of primary turns in series per circuit ; 

/?! = number of secondary turns in series per circuit ; 

a = = ratio of turns ; 
i 

Y ="0 H~./A) = primary exciting admittance per circuit; 

where 

g Q = effective conductance ; 

b = susceptance ; 

Z = r jx = internal primary self-inductive impedance 

per circuit, 
where 

r = effective resistance of primary circuit ; 

jr = reactance of primary circuit ; 

Z u = TI jx v = internal secondary self -inductive impedance 

per circuit at standstill, or for s = 1, 
where 

rj = effective resistance of secondary coil ; 
Xl 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 : 

Z l = r 1 -jsx l . 

Let the secondary circuit be closed by an external re- 
sistance r, and an external reactance, and denote the latter 



ALTERNATING-CURRENT TRANSFORMER, 223 

by x at frequency N, then at frequency s N, or slip s, it 
will be = s x, and thus : 

Z = r jsx = external secondary impedance.* 
Let 

= primary impressed E.M.F. per circuit, 
E ' = E.M.F. consumed by primary counter E.M.F., 
1 = secondary terminal E.M.F., 
EI = secondary induced E.M.F., 
e = E.M.F. induced per turn by the mutual magnetic flux, 

at full frequency JY, 
I Q = primary current, 
^ = primary exciting current, 
7i = secondary current. 

It is then : 

Secondary induced E.M.F. 

EI = sn^e. 

Total secondary impedance 

Z, + Z= (r, + r) 
hence, secondary current 



Secondary terminal voltage 



* This applies to the case where the secondary contains inductive reac- 
tance only ; or, rather, that kind of reactance which is proportional to the fre- 
quency. In a condenser the reactance is inversely proportional to the frequency, 
in a synchronous motor under circumstances independent of the frequency. 
Thus, in general, we have to set, x = x' + x" -\ x"\ where x' is that part of 
the reactance which is proportional to the frequency, x" that part of the reac- 
tance independent of the frequency, and x'" that part of the reactance which 
is inversely proportional t6 the frequency ; and have thus, at slip s, or frequency 
sN, the external secondary reactance sx' + x" -f- . 



224 AL TERNA TING-CURRENT PHENOMENA, 

E.M.F. consumed by primary counter E.M.F. 

'= -<>'; 

hence, primary exciting current : 

7 00 = E ' Y Q = e (g + /<)) 

Component of primary current corresponding to second- 
ary current 7 X : 



hence, total primary current, 

// 
1 




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, 

\ 1 



ALTERNATING-CURRENT TRANSFORMER. 225 



Secondary current, 

II =7 -7- 



Therefrom, we get : 
Ratio of currents, 



Ratio of E.M.Fs., 



Total apparent primary impedance, 



, , . x" . x'" 

where xx-\ --- \- 

s s 2 

in the general secondary circuit as discussed in foot-note, 
page 221. 

Substituting in these equations : 

*-l, 

gives the 

General Equations of the Stationary Alternating-Current 
Transformer : 



z*+z\ z, + z 

'* = -< \ .,;,* +I U- 

* (Zj + Z) 



ALTERNA TING-CURRENT PHENOMENA. 



r n t e 

y i = 



Z, + Z 
/! a 

P f 1 + *f7\^ + Z ' Y * 
^o_ = _ a } a (Z-j + 2} 

& I- Z * 

( Z, + Z 



1+ 2/ / x+^oKo] 

a 2 (Zj + Z) _ I 

l + ^Fo^ + Z) J 



Substituting in the equations of the general alternating- 
current transformer, 

Z = 0, 
gives the 

General Eqtiations of tJie Induction Motor: 



a'r^-jsx^ 
^ = 0. 

1 i ^o +y^o 



7 = _ s f ] -T, . . 

1 (>-! y**o 



r j,^ 

A = 



5 "^ : ~ + ( r o y^o)(^b +/ 

2 ^i JSXi 



Returning now to the general alternating-current trans^ 
former, we have, by substituting 

(ri + r? + ^ 2 (*i + *) 2 = ** f , 
and separating the real and imaginary quantities, 

-- (r (r, + r)+sx 9 ( Xl + x)) 
22 



ALTERNATING-CURRENT TRANSFORMER, 227 




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, 

,= - e\ I + - fr fa + r) 



146. The true power is, in symbolic representation (see 
Chapter XII.) : 



228 ALTERNATING-CURRENT PHENOMENA. 

denoting, 

safe* 

-7F = W 

gives : 

Secondary output of the transformer 



Internal loss in secondary circuit, 

m -2 t s n\ ^\ 2 

-Pi = 'i 2 n = ( } 

V ** / 

Total secondary power, 



** 

Internal loss in primary circuit, 

ri -9 -9o 

^o = V'o = 4 r t <r 

Total electrical output, plus loss, 

2 



Total electrical input of primary, 



Hence, mechanical output of transformer, 

P=P-P* = w(l-s)(r 
E.atio, 



mechanical 



output _ P 1 S _ speed 



total secondary power P -\- P l 

147. Thus, 

In a general alternating transformer of ratio of turns, a, 
and ratio of frequencies, s, neglecting exciting current, it is : 

Electrical input in primary, 
P 



ALTERNATING-CURRENT TRANSFORMER. 229 

Mechanical output, 

P - jg-j)iV(r+rO 
' 



Electrical output of secondary, 



Losses in transformer, 



Of these quantities, P 1 and P l are always positive ; P Q 
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. 

148. At 

s = 0, synchronism, P Q = 0, P = 0, P l = 0. 
At < s < 1, between synchronism and standstill. 

P l , P and PQ are positive ; that is, the apparatus con- 
sumes electrical power P Q in the primary, and produces 
mechanical power P and electrical power P l -j- P^ in the 
secondary, which is partly, P-^, consumed by the internal 
secondary resistance, partly, P l , available at the secondary 
terminals. 

In this case it is : 

Pi + ^i 1 _ J 
P ~l-s> 

that is, of the electrical power consumed in the primary 
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 1 + P^ 1 , and 
mechanical power, P, in the proportion of the slip, or drop 
below synchronism, s, to the speed : 1 s. 



230 ALTERNATING-CURRENT PHENOMENA. 

In this range, the apparatus is a motor. 
At s > 1 ; or, backwards driving, 

P < 0, or negative ; that is, the apparatus requires mechanical 
power for driving. 

It is then : P - A 1 - A 1 < PI ; 

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 
primary. 

At s < 0, beyond synchronism, 

P < ; that is, the apparatus has to be driven by mechanical 

power. 
/o<0; that is, the primary circuit produces electrical power 

from the mechanical input. 

At r+r! + srj. = 0, or, s < ^^ J ; 

r t 

the electrical power produced in the primary becomes less 
than required to cover the losses of power, and /> becomes 
positive again. 
We have thus : 

K-fl 
r \ 

consumes mechanical and primary electric power ; produces 
secondary electric power. 

- r -^ < s < 
?i 

consumes mechanical, and produces electrical power in 
primary and in secondary circuit. 



ALTERNATING-CURRENT TRANSFORMER. 231 

consumes primary electric power, and produces mechanical 
and secondary electrical power. 



consumes mechanical and primary electrical power ; pro- 
duces secondary electrical power. 



T 



GENERAL ALTERNATE CURRENT TRANSFORMER 



A 



648 

Fig 



H 



149. As an instance, in Fig. Ill are plotted, with the 
slip s as abscissae, 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 : 

n,e = 100.0 ; r = A ; 

r 4. i x = .3; 



232 ALTERNATING-CURRENT PHENOMENA. 

hence, p = 16,000 ^ 2 . 

pl , P i _ 8,000 j. 

""" l -i , j" ? 

_ 4,000 s + (5 + J) . 

~ 1 I 2 ' 

p = 20,000 s (1 - j) 

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. 

Let, 

Y = g +j& = primary exciting admittance per circuit 
of the frequency converter. 

Z^ = r t jx^ internal self inductive impedance per 
secondary circuit, at the secondary frequency. 



ALTERNATING-CURRENT TRANSFORMER. 233 

Z^ = r jx^ = internal self inductive impedance per 
primary circuit at the primary frequency. 

a = ratio of secondary to primary turns per circuit. 

b = ratio of number of secondary to number of primary 
circuits. 

c = ratio of secondary to primary frequencies. 

Let, 

e = induced E.M.F. per secondary circuit at secondary 
frequency. 

Z = r jx = external impedance per secondary circuit 
at secondary frequency, that is load on secondary system, 
where x for noninductive lead. 

We then have, 

total secondary impedance, 

Z + Z 1 = (r-^r l )-j(x + x 1 ) 
secondary current, 






where, 

r + r. x + Xl 



(r + 0> 2 + (* + ^) 2 (r +^i) 2 + (* + 

secondary terminal voltage, 

Ei = IiZ = e ^4-T 



e(r jx) (a t 
where, 



primary induced E.M.F. per circuit, 



primary load current per circuit, 

7 1 = abli = abe (a { 
primary exciting current per circuit, 



234 ALTERNATING-CURRENT PHENOMENA. 

thus, total primary current, 

7 = 7 1 + /oo 

= e (fi 



where, 

<. = **+ <.=**+! 

primary terminal voltage : 



where, 

d - re x d -re -x 

ac 

or absolute, 

e = e vX 2 + 4 2 

. = e - 

V^ 2 + 4 

substituting this value of e in the preceding equations, 
gives, as function of the primary impressed E.M.F., e : 
secondary current, 



7 = > absolu 7 = vi 

V4 + 4 2 v ^i 2 + 

secondary terminal voltage, 



primary current, 

, _ 



primary impressed E.M.F. 

^ _ ^0 (4 

" V4 
secondary output, 



gl ^ + 



AL TERNA TING-CURRENT TRANSFORMER. 



235 



primary electrical input, 



i + 



L r: oj </ + </.* 

primary apparent input, voltamperes, 
<2o = 4/o 

Substituting thus different values for the secondary in- 
ternal impedance Z gives the regulation curve of the fre- 
quency converter. 



REGULATION CURVES 

VOLTS CONSTANT! 25 CYCLES 



DARY 62.5 CYCLE QUARTER-PHASE 



TDARY 
20 



CURRENT PER PHASE, AMP. 

3) 



Fig. 112, 

Such a curve, taken from tests of a 20"0 KW frequency 
converter changing from 6300 volts 25 cycles three-phase, 
to 2500 volts 62.5 cycles quarter-phase, is given in Fig. 
112. 



236 AL TERN A TING-CURRENT PHENOMENA. 

From the secondary terminal voltage, 



it follows, absolute, 































































PRIlt 


ARY 






























VOUT8 


AMP. 















=^ 











/ 






J500 
6000 


J3- 
12 






















/ 










11 


















/ 


/ 












in 
















/ 


/ 














9 














x 


















8 










x' 


X 




















7 








X 


COND/ 


BY, 2 


( 

00 VO 


OMPC 

LT8C( 


SUND 
NSTAt 


NGC 
T 82 


. R c V 5 


LE3 Q 


ARTE 




F 


6 





^ 


^^ 








PRIM 


kRY, ! 


5 CYC 


E? Tl 


REE-P 


HASE 








5 
































4 


































o 


1 


TDAR 
2 


r CUR 
} 


1ENT 
| 


PER P 



HASE, 
4 


AMP. 

t 


.', 


I 


| 


) 









Fig. 113. 



Substituting these values in tne above equation gives 
the quantities as functions of the secondary terminal vol- 
tage, that is at constant e l , or the compounding curve. 

The compounding curve of the frequency converter 
above mentioned is given in Fig. 113. 



INDUCTION MOTOR. 237 



CHAPTER XVI. 

INDUCTION MOTOR. 

151. 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 far as in the 
stationary transformer only the transfer of electrical energy 
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 



238 ALTERNATING-CURRENT PHENOMENA. 

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 = frequency of armature or secondary E.M.F., 

and (1 s) N= 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 // of a period, that is, 
1 // 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 / -phase 
system. Analogously, let the armature or secondary circuits 
consist of a symmetrical / r phase system. 

Let 

n = number of primary turns per circuit or phase ; 
a = number of secondary turns per circuit or phase ; 

a = -^ = ratio of total primary turns to total secondary turns 
n \P\ 
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 



INDUCTION MOTOR. 239 

if E{ = secondary E.M.F. per circuit, E l = aE{ 

= secondary E.M.F. per circuit reduced to primary system; 

if // = secondary current per circuit, f l= 

= secondary current per circuit reduced to primary system ; 
if r^ = secondary resistance per circuit, r t = a 2 r{ 

= secondary resistance per circuit reduced to primary system ; 
if x = secondary reactance per circuit, x t = a 2 x\ 

= secondary reactance per circuit reduced to primary system ; 
if / = secondary impedance per circuit, z 1 = a z z\ 

= 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 
system. 

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 = ?*-. 
iA 
153. Let 

$ = total maximum flux of the magnetic field per motor pole, 
We then have 

E V2 77-72 TV^ 10 ~ 8 = effective E.M.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., 



240 ALTERNATING-CURRENT PHENOMENA. 

where 

e= V2irrt7V<I>10- 8 maybe considered as the "Active E.M.F. 
of the motor," or " Counter E.M.F." 

Since the secondary frequency is s N, the secondary in- 
duced E.M.F. (reduced to primary system) is E l = se. 
Let 

I = exciting current, or current passing through the motor, per 
primary circuit, when doing no work (at synchronism), 

and 

K= g -j- j 'b = orimary admittance per circuit = . 

We thus have, 

ge = magnetic energy current, ge* = loss of power oy hysteresis 
(and eddy currents) per primary coil. 

Hence 



= total loss of energy by hysteresis and eddys, 

as calculated according to Chapter X. 
be = magnetizing current, and 
n be = effective M.M.F. per primary circuit; 

hence ^n be = total effective M.M.F. ; 

z 

and 

l^-n^be = total maximum M.M.F., as resultant of the M.M.Fs. 
of the / -phases, combined by the parallelogram of 
M.M.Fs.* 

If (R = reluctance of magnetic circuit per pole, as dis- 
cussed in Chapter X., it is 

A^^ft*. 

* Complete discussion hereof, see Chapter XXV. 



INDUCTION MOTOR. 241 

Thus, from the hysteretic loss, and the reluctance, the 
constants, g and b, and thus the admittance, Fare derived. 

Let r Q = resistance per primary circuit ; 
X Q = reactance per primary circuit ; 
thus, 

^o = r o j X Q = impedance per primary circuit; 

r v = resistance per secondary circuit reduced to pri- 
mary system ; 

x v = reactance per secondary circuit reduced to primary 
system, at full frequency, .A 7 "; 

hence, 

sx! = reactance per secondary circuit at slip s; 
and 

= secondary internal impedance. 



154. We now have, 
Primary induced E.M.F., 

E = -e. 
Secondary induced E.M.F., 

Hence, 
Secondary current, 

*-$ 



Component of primary current, corresponding thereto, 
primary load current, 

7" --/, = 



Primary exciting current, 

/ =eY= e (g+jfy; hence, 



242 ALTERNATING-CURRENT PHENOMENA. 

Total primary current, 



E.M.F. consumed by primary impedance, 



E.M.F. required to overcome the primary induced E.M.F., 

- E = e; 
hence, 

Primary terminal voltage, 
E. = e + E z 



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, 



INDUCTION MOTOR. 243 

To eliminate e, we divide, and get, 

Primary current, at slip s, and impressed E.M.F., ; 

f=^ 



or, 

/= _ j + (>i-yji _ E 

" ( 



Neglecting, in the denominator, the small quantity 
F, it is 

Z, F 



+ r\ 



or, expanded, 

[(j^ + A' ) + r^ -f s^ (r og - 

+/ [J 3 (jfo+^O + r^+JT! (xtg+r^+fx^ (xj>+ xj- 



Hence, displacement of phase between current and 
E.M.F., 
tan , = ^(^o+^ 



Neglecting the exciting current, /< altogether, that is, 
setting Y = 0, 
We have 

7= sE n ^- 



S 

tan <D = 



244 



AL TEKNA TING-CURRENT PHENOMENA. 



155. In graphic representation, the induction motor dia- 
gram appears as follows : 

Denoting the magnetism by the vertical vector O<b in 
Fig. 114, the M.M.F. in ampere-turns per circuit is repre- 
sented by vector OF, leading the magnetism O<& by the 
angle of hysteretic advance a. The E.M.F. induced in the 
secondary is proportional to the slip s, and represented by 
~OE l at the amplitude of 180. Dividing ~OE l by a in the 
proportion of r t -*- sx v and connecting a with the middle b 
of the upper arc of the circle OE V this line intersects the 
lower arc of the circle at the point 7 X r r Thus, OIj\ is the 
E.M.F. consumed by the secondary resistance, and OI^ 
equal and parallel to EJ^ is the E.M.F. consumed by the 
secondary reactance. The angle, E^OI^\ = ^ is the angle 
of secondary lag. 




\ 



The secondary M.M.F. OG l is in the direction of the 
vector OIf v Completing the parallelogram of M.M.Fs. 
with OF as diagonal and OG l 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 
OIr y is in line with OG, the E.M.F. consumed by the pri- 
mary reactance 90 ahead of OG, and represented by OIx v 
and their resultant Ofz is the E.M.F. consumed by the 



INDUCTION MOTOR. 



245 



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 1 . Com- 
bining OE with OIz Q gives the primary terminal voltage 
represented by vector OE y and the angle of primary lag, 
EOG 




Fig. 115. 



156. 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, 



246 ALTERNATING-CURRENT PHENOMENA. 

Let l = s f 

Assume in opposition to O&, a point A, such that 



O A -r- 7 X r x = E v -* /! J.*!, then 

/ir, x .", /ir, x sE r, _, 

= - ^ = constant. 



That is, /^ lies on a half-circle with OA = E' as 
diameter. 

That means G l lies on a half-circle ^ in Fig. 115 with 
OC as diameter. In consequence hereof, G lies on half- 
circle^ with FB equal and parallel to OCas diameter. 

Thus Ir lies on a half -circle with DH as diameter, which 
circle is perspective to the circle FB, and Ix lies on a half- 
circle with IK as diameter, and Iz Q on a half-circle with LN 
as diameter, which circle is derived by the combination of 
the circles Ir and Ix v 

The primary terminal voltage E Q lies thus on a half- 
circle e equal to the half-circle Iz 9 and having to point 
E the same relative position as the half-circle Iz^ has to 
point 0. 

This diagram corresponds to constant intensity of the 
maximum magnetism, O. If the primary impressed volt- 
age E Q 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, 



INDUCTION MOTOR. 247 



If ! = number of turns, 7 t = current, per circuit, with 
/ r armature circuits, the total maximum current polarization, 
or M.M.F. of the armature, is 



Hence the torque per pole, 



If q = the number of poles of the motor, the total torque 
of the motor is, 



The secondary induced E.M.F., E v 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, I v lags behind its E.M.F., 
E v by angle, <a v the space displacement between armature 
current and field magnetism is 



hence sin (4> fj) = cos o^ 

We have, however, 




thus, ! <$ 

substituting these values in the equation of the torque, it is 
T. 



248 ALTERNATING-CURRENT PHENOMENA. 

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, 
P= TV 
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 / ' = f + 7 X the power is consumed, e I = e I + e 7 r 
The power e I is that consumed by the primary hysteresis 
and eddys. The power e 1^ disappears in the primary circuit 
by being transmitted to the secondary system. 

Thus the total power impressed upon the .secondary 
system, per circuit, is 

Pi-tf, 

Of this power a part, 1 f l , is consumed in the secondary 
circuit by resistance. The remainder, 

P' = f l (e- 1 ), 

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 



INDUCTION MOTOR. 249 

Substituting, 



se; 
se 



it is 



hence, since the imaginary part has no meaning as power, 



and the total power of the motor, 

At the linear speed, 
at unit radius the torque is 



In the foregoing, we found 

= e\ 1 + j|? + Z, Y 
or, approximately, 



or, 
expanded, 



or, eliminating imaginary quantities, 



250 ALTERNATING-CURRENT PHENOMENA. 

Substituting this value in the equations of torque and of 
power, they become, 

torque, T = 






Maximum Torque. 

159. The torque of the induction motor is a maximum 
for that value of slip s, where 



qpi r^ Eg s 
or, since T = -. . T , . 

4 7T JV^ (>1 



for, 

ds 



expanded, this gives, 

r 2 
"7 

or, s t = 



Substituting this in the equation of torque, we get the 
value of maximum torque, 



That is, independent of the secondary resistance, r r 
The power corresponding hereto is, by substitution of s t 
in P, 

Pt = ; 



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. 



INDUCTION MOTOR. 251 

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 t , 
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 s t , 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 t , is between 
synchronism and standstill, rather nearer to synchronism. 
Only in motors of very large armature resistance, that is 
low efficiency, s t > 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, s t can be varied by varying the resistance of the 
secondary circuit, or the motor armature. Since the slip 
of the maximum torque point, s t , is directly proportional to 
the armature resistance, r lf 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 



252 ALTERNATING-CURRENT PHENOMENA. 

to be varied during the operation of the motor, and the 
motor started with high armature resistance, and with in- 
creasing speed this armature resistance cut out as far as 
possible. 

160. If *=:1,__ 

it is ^ = Vr 2 + (x l + * ) 2 . 

In this case the motor starts with maximum torque, and 
when overloaded does not drop out of step, but gradually 
slows down more and more, until it comes to rest. 

If, s t >l, 

then ^ > Vr 2 + (^ + * ) 2 . 

In this case, the maximum torque point is reached only 
by driving the motor backwards, as countertorque. 

As seen above, the maximum torque T 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 depends 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 v , where 



INDUCTION MOTOR. 253 






expanded, this gives 

s n - 



substituted in P, we get the maximum power, 



2 {('i + ''o) + (^ + r ) 2 + (^i + *o) 2 } 

This result has a simple physical meaning : (i\ + 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 Vr x + r ) 2 + (x^ + x } z = z is the total 
impedance of the motor. Hence 



is the maximum output of the induction motor, at the slip, 



The same value has been derived in Chapter IX., as the 
maximum power which can be transmitted into a non- 
inductive receiver circuit over a line of resistance r, and 
impedance z, or as the maximum output of a generator, or 
of a stationary transformer. Hence : 

The 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 p is,. 



254 ALTERNATING-CURRENT PHENOMENA. 

This is not the maximum torque ; but the maximum 
torque, T t , takes place at a lower speed, that is, greater slip, 



since, 



-that is, s t > s p . 

It is obvious from these equations, that, to reach as large 
an output as possible, r and z should be as small as possible ; 
that is, the resistances ^ + r , and the impedances, z, 
and thus the reactances, x + x , should be small. Since 
r + r is usually small compared with x^ -f- x it follows, that 
the problem of induction motor design consists in con- 
structing the motor so as to give the minimum possible 
reactances, x^ + x . 

Starting Torque. 

162. In the moment of starting an induction motor, 
the slip is 

hence, starting current, 



Oo - 

or, expanded, with the rejection of the last term in the 
denominator, as insignificant, 



T _io 11 010 ,io 1 . 
- 8 



and, displacement of phase, or angle of lag, 

fi + r ] + *! [Jf x 4- Jf ]) - jf (r ^ - * r t ) 



_ 
1 W 



r ) 



INDUCTION MOTOR. 255 

Neglecting the exciting current, g = = b, these equa- 
tions assume the form, 



or, eliminating imaginary quantities, 



and tan w = 



+ 'o 



That means, that in starting the induction motor without 
additional resistance in the armature circuit, in which case 
^ + x is large compared with t\ + r , and the total impe- 
dance, z, small, the motor takes excessive and greatly 
lagging currents. 

The starting torque is 



T = 






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. 



256 ALTERNATING-CURRENT PHENOMENA. 

Since, necessarily, 

ri<*, 

''<< 



and since the starting current is, approximately, 

7 =f , 
we have, T a < 



would be the theoretical torque developed at 100 per cent 
efficiency and power factor, by E.M.F., E , and current, /, 
at synchronous speed. 

Thus, T 0< T 00 , 

and the ratio between the starting torque T , and the theo- 
retical maximum torque, T^, gives a means to judge the 
perfection of a motor regarding its starting torque. 

This ratio, T / T w , exceeds .9 in the best motors. 

Substituting 7 = E / z in the equation of starting torque, 
it assumes the form, 

7V,. 



Since 4 IT N / q = synchronous speed, it is : 

The 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 



INDUCTION MOTOR. 257 



dr, 
expanded, this gives, 



the same value as derived in the paragraph on "maximum 
torque." 

Thus, adding to the internal armature 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, t\", required to give 
a certain starting torque, if found from the equation of 
starting torque : 

Denoting the internal armature resistance by rj, the total 
armature resistance is ^ = r^ + r". 

and thus, ?A Eg rj + r" 

4 TT N (r^ + r^ + r ) 2 + ( Xl + * ) 2 ' 
hence, 



This gives two values, one above, the other below, the 
maximum torque point. 



258 ALTERNATING-CURRENT PHENOMENA. 

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 T . 

The smaller value of r 1 " 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 rj' allows to start with minimum 
current, but requires cutting out of the resistance after the 
start, to secure speed regulation and efficiency. 

Synchronism. 
163. At synchronism, s = 0, we have, 



or, 



0, T=Q; 



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 Synchronism. 

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. 



INDUCTION MOTOR. 25 

We then have, current, 



or, eliminating imaginary quantities, 



angle of lag, o*i + *o , 

c2 (r_ -I- <r_\ -4- r.2 h r. 

tan w 

T = 

or, inversely, 



A A 



that is, 

Near sychronism, the slip, s, of an induction motor, or 
its drop in speed, is proportional to the armature resistance> 
i\ and to the power, P, or torque, T. 

Example. 

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 
circuit : 

~ = 63.5 ; or E Q = 63.5. 



260 



AL TERN A TING-CURRENT PHENOMENA. 



The constants of the motor are : 

Primary admittance, Y = .1 + .4 j. 
Primary impedance, Z = .03 .09 j. 
Secondary impedance, Z x = .02 .085/. 

In Fig. 116 is shown, with the speed in per cent of 

synchronism, as abscissae, the torque in kilogrammetres, 

as ordinates, in drawn lines, for the values of armature 
resistance : 




116. Speed Characteristics of Induction Motor. 



r t = .02 : short circuit of armature, full speed. 

^ = .045 : .025 ohms additional resistance. 

^ = .18 : .16 ohms additional, maximum starting torque. 

^ = .75 : .73 ohms additional, same starting torque as r t == .045. 

On the same Figure is shown the current per line, in 
dotted lines, with the verticals or torque as abscissae, and 
the horizontals or amperes as ordinates. To the same 
torque always corresponds the same current, no matter 
what the speed be. 



INDUCTION MOTOR. 



261 



On Fig. 117 is shown, with the current input per line as 
abscissae, the torque in kilogrammetres 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 ^ = .02 
or short circuit. 



20 



lase Induotio Motor. 



. 60Cyc 



110V 



Jiagram 



=.03-.09j 
z0=J&B 



\ 



\\ 



\\ 



12 

-1 



Amperes 
150 1 200 



2,50 



300 



Fig. 117. Current Characteristics of Induction Motor. 

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 ?i = .045, for the whole range of speed from 120 per 



262 



ALTERNA TING-CURRENT PHENOMENA. 



cent backwards speed to 220 per cent beyond synchronism, 
showing the two maxima, the motor maximum at s = .25, 
and the generator maximum at s = .25. 

166. As seen in the preceding, the induction motor is 
characterized by the three complex imaginary constants, 

Y = g +jb w the primary exciting admittance, 
Z = r jx , the primary self-inductive impedance, and 
Zi = r jx^ the secondary self-inductive impedance, 




Fig. 1 18. Speed Characteristics of Induction Motor. 

reduced to the primary by the ratio of secondary to pri- 
mary turns. 

From these constants and the impressed E.M.F. c ot the 
motor can be calculated as follows : 

Let, 

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, se 



INDUCTION MOTOR. 263 



Thus the secondary current, 



where, 



l = -5T 



r* + Atf r? + 

The primary exciting current is, 



thus, the total primary current, 

/ = /! + /oo = * (^i + A) 
where, 



The E.M.F. consumed by the primary impedance is, 
^ = /oZ = * (r -> ) (^ 



the primary counter E.M.F. is e, thus the primary impressed 
E.M.F., 

, 
where, 

c\ 
or, absolute, 

^ = 

hence, 



This value substituted gives, 

Secondary current, 

ffi+A 
A = *b T7= 



Primary current, 

~ 

Impressed E.M.F., 



264 ALTERNATING-CURRENT PHENOMENA. 

Thus torque, in synchronous watts (that is, the watts 
output the torque would produce at synchronous speed), 



tf + tf 

hence, the torque in absolute units, 



= = 



N (f* + r 2 2 ) W 
where N= frequency. 

The power output is torque times speed, thus : 



The power input is, 



^l 2 + 

The voltampere input, 



o 2 ( Vi + V,) /o 2 ( Vi - V 8 ) 



hence, 

efficiency, 

J\ _ a, (I - s) 

J? Vi + V 2 

power factor, 



apparent efficiency, 



<2o 

torque efficiency, * 
a. 



./V Vi + V. 

* That 5s the ratio of actual torque to torque which would be profloced, if there were nc 
losses of energy in the motor, at the same power input. 



INDUCTION MOTOR. 265 



apparent torque efficiency,* 

rrt 

~Q ~ V W~+1?YT^ 



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, 
K = .01 + .!/ 

z* = .i -.3> 

Z, = .1 - .3j 

INDUCTION GENERATOR. 

168. In the foregoing, the range of speed from s = 1, 
standstill, to s = 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, s > 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. All this power is 
consumed in the motor, which thus acts as brake. 

For, s < 0, or negative, P and T become negative, and 
the machine becomes an electric generator, converting me- 
chanical into electric energy. 

* That is the ratio of actual torque to torque which would be produced if there were 
neither losses of energy nor phase displacement in the motor, at the same voltampere input. 



266 



ALTERNA TING-CURRENT PHENOMENA. 



The calculation of the induction generator at constant 
frequency, that is, at a speed increasing with the load by the 
negative slip, s lt is the same as that of the induction motor 
except that s l has negative values, and the load curves for 
the machine shown as motor in Fig. 119 are shown in Fig. 
121 for negative slip s { as induction generator. 



CURV 












POWER 
4000 



"> 



Fig. 119. 



Again, a maximum torque point and a maximum output 
point are found, and the torque and power increase from 
zero at synchronism up to a maximum point, and then de- 
crease again, while the current constantly increases. 



INDUCTION MOTOR. 



267 




Fig. 120. 




268 ALTERNATING-CURRENT PHENOMENA. 

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. 



INDUCTION MOTOR. 269 

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, 

Y = g Q + j\ primary exciting admittance, 

Z = r jx Q = primary self-inductive impedance, 

Zi = r^ jXj_ = 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-j) N = frequency gener- 
ated by machine. 

We then have 

Secondary induced E.M.F. 
se 
thus, secondary current, 



r in 
r \ J sx \ 



270 ALTERNATING-CURRENT PHENOMENA. 

where, 



primary exciting current, 

In = EY = e 
thus, total primary current, 

/ = /i + foo 
where, 

^1 = <*\ + b 

primary impedance voltage, 
& = S (r - 

primary induced E.M.F., 



thus, primary terminal voltage, 

= e(l-s) -S (r -j[l- s] x ) = e 
where, 

f i = ! - s ~ r A - (1 - s 
hence, absolute, 

e = e V^ 
and, 



Thus, 

Secondary current, 

T e O ( a i 



Primary current, 

j _ e o (A + A) 

Primary terminal voltage, 

j-. ^0 \^"l 

= T-, 



INDUCTION MOTOR. 
Torque and mechanical power input, 

T P \f n l e ai 
r * ~ \- e ^ ~ 7^+^ 

Electrical output, 



271 



ELECTRICAL OUTPUT P , WATTS 
1000 2000 3COO 4000 fiOOO fiOOO 7000 8000 




Fig. 122. 



Voltampere output, 
G, = < 

Efficiency, 

j 

power factor, 






272 AL TERNA TING-CURRENT PHENOMENA. 

or, 

p,j b* - V, 
= ^- = ^T^ 

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, 

K = .01 + .!/ 



171. As instance may be considered a power trans- 
mission from an induction generator of constants Y , Z , 
Zj, over a line of impedance Z = r jx, into a synchron- 
ous motor of synchronous impedance Z z = r z jx z , operat- 
ing at constant field excitation. 

Let, e = 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, 
E = e 
thus, terminal voltage at synchronous motor terminals, 



where, 

4 = fi ~ r A ~ C 1 - J ) *A 4 = 

Counter E.M.F. of synchronous motor, 

E 2 

' 

where, 

/ = 4 - r& - (1 
or absolute, 



INDUCTION MOTOR. 



since, however, 




Z=.0|4-6j 

ULL F EQUE 
EXCIT/ 
5 VOL' 



OUTPUT OF SYNCHRONOUS, WATTS 
1000 2000 I 8000 4000 5000 



274 ALTERNATING-CURRENT PHENOMENA. 



Thus, 



Current, _ e 2 (1 - j) (^ +y7; 2 ) 

' 



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 speed gradually falls off with increas- 
ing load (in 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. 
e Q = 125 volts (at full frequency) and synchronous impe- 
dance Z 2 = .04 Gj, over a line of negligible impedance 
is shown in Fig. 123. 

CONCATENATION, OR TANDEM CONTROL OF INDUCTION 
MOTORS. 

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 



INDUCTION MOTOR. 275 

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 slip 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, 1 s, the secondary 
frequency of the second motor is 2^1, hence equal to 
zero at s = .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 increasing 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 in the armature of the second motor. 



276 ALTERNATING-CURRENT PHENOMENA. 

Since, with increasing speed, the frequency impressed 
upon the second motor decreases proportionally to the de- 
crease of voltage, when neglecting 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, 

Let 

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, s N. 

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 



INDUCTION MOTOR. 277 

Let 

e = counter E.M.F. induced in the secondary of the sec- 
ond motor, reduced to full frequency. 

Z = r jx Q = primary self-inductive impedance. 

Z^ = i\ jx v = 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, 



where, 

(2s-l)r 1 



i ~ r*+ (2 J -1) 2 ^ 1 2 z ~ r*+ (2s- 

primary exciting current, 

4 = * (g +JI>} 
thus, total primary current, 

7 2 = 7, + 7 = e ( 
where, 



primary induced E.M.F., 

se 
primary impedance voltage, 

ft ( r o >^o) 
thus, primary impressed E.M.F., 

3 = se + 7 2 (r -jsx ) = e (^ 
where, 



First motor, 
secondary current, 



278 ALTERNATING-CURRENT PHENOMENA. 

secondary induced E.M.F., 

9 = 
where, 



primary induced E.M.F., 

EI = - 
where, 

s 
primary exciting current, 

total primary current, 
where, 



primary impedance voltage, 

|(>o ~> 

thus, primary impressed E.M.F., 
= E, + S(r -> 
where, 

^i =/i + ^o5i + *ba 

or, absolute, 

<- 
and, 



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. 

* At s = these terms/i and/s become indefinite, and thus at and very near synchronism 
have to be derived by substituting the complete expressions fory^ andy" 2 . 



INDUCTION MOTOR. 279 

In the second motor, the torque is, 

T 2 = [,/J = ^ 
hence, its power output, 

/,= (!- s) r 2 = (1 - s) <? ai 
The power input is, 



hence, the efficiency, 

PS (1 - s) fa, 



the power factor, 



etc. 

In the first motor, 
the torque is, 



the power output, 

PI = 71 (1 - j) 

= ^ (1 - ,) (/^ -h/A) 

the power input, 

P 1 = 



Thus, the efficiency, 

^ (1 - Q (/A +/A) 



+ ^ 2 ) - (^ + 
the power factor of the whole system, 



280 ALTERNATING-CURRENT PHENOMENA. 

the power factor of the first motor, 



the total efficiency of the system, 



etc. 




f /ff. 724. Concatenation of Induction Motors. Speed Curves. 
Z=.1 .3/ K=.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 slip from s = + 1.5 to s = .7 for a pair 
of induction motors in concatenation, of the constants : 

Z = Z, = .1 - .Bj 



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 



INDUCTION MOTOR. 



281 



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. 













001 


| | 
CATENATION jOF IN 


SUCTION MOTORS. 










L 










j SPEED CURVES 
|z=.| .3,j Y4=.OI 


H-.l 












it 

rag 










RE! 


. IN S 


;COND 


kRY 


' SECO 


NO MC 


TOR. 










| 
































H 
8000 
































6000 






- 



























4000 










\ 






















2000 


1 











""-s. 


\ 












I 











M 










\\ 












\ 








-2000 














\\ 




X 


^ 












-4000 

















/ 


f 














-60C( 
















./ 
















-8000 


































1 





9 


s 


. 


6 


j 


4 


3 


2 


j 












Fig. 125. Concatenation of Induction Motors. Speed Curves. 



SINGLE-PHASE INDUCTION MOTOR. 

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 



282 AL TERN A TING-CURRENT PHENOMENA. 

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 becomes 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. 



INDUCTION MOTOR. 283 

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 
resistance. 

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. 

3. 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 
mains. 

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.* 

See paper on the Single-phase Induction Motor, A.I.E.E. Transactions, 1898. 



284 ALTERNATING-CURRENT PHENOMENA. 

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 correspond 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. 



INDUCTION MOTOR. 285 

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 F,, 1 , 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 
3> at synchronism, and falls off with decreasing speed to 
zero at standstill, if no starting device is used or to 4^ = /< 
at standstill if by a starting device a quadrature magnetic 
flux is impressed upon the motor, and at standstill t = ratio- 
of quadrature or starting magnetic flux to main magnetic 
flux. 

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 : 

Y* = [!-(!-/) j] Fo 1 

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, 

hence = -^ with a three-phase secondary, and = -^ with a 

two-phase secondary with impedance Z 1 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 2 Z l / 3 with a three-phase, 
and equal to Z^ with a two-phase secondary. Thus the 
effective secondary impedance of the single-phase motor 



286 ALTERNATING-CURRENT PHENOMENA. 

changes with the speed and can at the slip s be represented 

by Zf = - -- -^ - in a three-phase motor, and Z{ = - - <p - 1 

in a two-phase motor, with the impedance Z^ per secondary 
circuit. 

In the single-phase motor without starting device, due to 
the falling off of the quadrature flux, the torque at slip s is : 

T = a^ (I - s) 

In a single-phase motor with a starting device which at 
standstill produces a ratio of magnetic fluxes t, the torque at 
standstill is ; 

T Q = /7I 

where 7^ = total torque of the same motor on polyphase 
circuit. 

. Thus denoting the value ~ = v 
&f 

the single-phase motor torque at standstill is : 



and the single-phase motor torque at slip s is : 
T = of [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 



INDUCTION MOTOR. 287 

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 essentially 
increased. 

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 Q = g -\-jb = primary exciting admittance of the 
motor per delta circuit. 

Z = r jx Q = primary self-inductive impedance per 
delta circuit. 

Z^ = i\ jx^ = secondary self-inductive impedance per 
delta circuit reduced to primary. 

Let 

Y s = g s jb 9 = admittance of the condenser connected 
between terminals 1 and 3. 



288 ALTERNATING-CURRENT PHENOMENA. 

If then, as single-phase motor, 

/ = ratio of auxiliary quadrature flux to main flux in 
starting, 

h = ratio of E.M.F. induced in condenser circuit to 

E.M.F. induced in main circuit in starting, 
starting torque 



It is single-phase 

Fo 1 = 1.5 Y = 1.5 ( +/ ) = primary exciting admit- 

tance, 
Y? = 1.5 Y [1 - (1 - s] 

= 1.5 (g +/<)) [1 (1 J ] = secondary exciting 

admittance at slip s. 

Z l = ?^ = 2 fo~^*o) = primary self-inductive impe- 
o o 

dance. 

Z x i = L^ Zi = ^L + ^ ( ri -j sx j = secondary self- 
o o 

inductive impedance. 

Z, 1 = ^ = 2 (r ~ ***> = tertiary self-inductive impe- 
o o 

dance of motor. 
Thus, 

Y 4 = -^r - T- = total admittance of tertiary circuit. 



Since the E.M.F. induced in the tertiary circuit decreases 
from e at synchronism to he at standstill, the effective ter- 
tiary admittance or admittance reduced to an induced E.M.F. 
e is at slip s 

Y? = [!-(!-*) s] Y 4 
Let then, 

e = counter E.M.F. of primary circuit, 
s = slip. 






INDUCTION MOTOR. 289 



We have, 
secondary load current 

3se 



(1 + s) (r, -jsx,) 
secondary exciting current 

secondary condenser current 
thus, total secondary current 
primary exciting current 



thus, total primary current 

/o = 7 1 + /o 1 
= /, + /, + 

= ' (*i + A) 
primary impressed E.M.F. 



thus, main counter E.M.F. 






or, 



and, absolute 



V^ 2 + c* 
hence, primary current 



T_ s lW + % 

J * - e v f * + ^ 



290 ALTERNATING-CURRENT PHENOMENA. 

voltampere input, 

Qo = **! 
power input 



*t Oo O 2 , 2 

6j T '2 

torque at slip .$ 

2^= r 1 [i - (i - v) s] 



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 /, //, v, which have to be 
determined before calculating the motor, is as follows : 

Let <? = single-phase impressed E.M.F., 

Y total stationary admittance of motor per delta cir- 

cuit, 
E z = E.M.F. at condenser terminals in starting. 

In the circuit between the single-phase mains from ter- 
minal 1 over terminal 3 to 2, the admittances Y + Y 8 , and Y, 
are connected in series, and have the respective E.M.Fs. E^ 
and e - E y It is thus, 

Y+ Y s + Y=e -t+ s , 

since with the same current passing through both circuits, 
the impressed E.M.Fs. are inverse proportional to the re- 
spective admittances. 

Thus, 



INDUCTION MOTOR. 291 

and quadrature E.M.F. 

hence 
thus 




Since in the three-phase E.M.F. triangle, the altitude 
corresponding to the quadrature magnetic flux = y= , and 

the quadrature and main fluxes are equal, in the single-phase 
motor the ratio of quadrature to main flux is 

/ = 2 = 1.155 A a 

V3 

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. 



SYNCHRONOUS INDUCTION MOTOR. 

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 rotati .n 



292 ALTERNATING-CURRENT PHENOMENA. 

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 
admittance. 

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 
small amount of power is required at synchronous rotation, 
and continuous current for field excitation is not available. 



INDUCTION MOTOR. 293 

The current induced in the armature 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 
reached. 

THE HYSTERESIS MOTOR. 

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 



294 AL TERN A TING-CURRENT PHENOMENA. 

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= mF& sin a, 

where 

F= resultant M.M.F., 

< = resultant magnetism, 

a = angle of hysteretic advance of phase, 

m = a constant. 

The apparent or voltampere input of the motor is, 
Q = mF. 

Thus the apparent torque efficiency, 

T 

2 = sma, 

and the power of the motor is, 

P = (1 s) T= (1 s) m F<$> sin a, 
where 

s = slip as fraction of synchronism. 

The apparent efficiency is, 

P 

- = (!_*) sin 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. 



INDUCTION MOTOR. 295 

From the equation of torque it follows, however, that at 
constant impressed E.M.F., or current, that inconstant 
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 and 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 77 = coefficient of hysteresis, 

the energy expended by hysteresis in the movable disk, /, is 
per cycle, 

IV, = V^B, 

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, 

P = 



At the slip, s N, that is, the speed (1 s) N, the energy 
xpended by hysteresis in the rotating disk is, however, 



Hence, in the transfer from the stationary to the revolv- 
ing member the magnetic energy, 



has disappeared, and thus reappears as mechanical work, 
and the torque is, 

'-p^iprW' 

that is, independent of the speed. 



296 AL TERNA TING-CURRENT PHENOMENA. 

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 a, or a fraction of the primary hysteretic 
energy. 

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. 



ALTERNATING-CURRENT GENERATOR. 297 



CHAPTER XVII. 

ALTERNATING-CURRENT GENERATOR. 

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 
field. 

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-directional, 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, 



298 AL TERNA TING-CURRENT PHENOMENA. 

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 t in this case, even on open circuit, no 




Fig. 126. 

rotation through a constant magnetic field, but rotation 
through a pulsating field, which makes the E.M.F. wave 
unsymmetrical, 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 between 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 



AL TERN A TING-CURRENT GENERA TOR. 



299 



density at the trailing-pole corner. Since the internal self- 
inductance of the alternator itself causes a certain lag of 
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 




Fig. 127. 



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 




Fig. 128. 



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. 



300 ALTERNA TING-CURRENT PHENOMENA. 

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 flux, 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 flux 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- 



ALTERNATING-CURRENT GENERATOR. 301 

inductance will be more than 90 behind the induced E.M.F., 
and therefore in partial opposition, and will tend to reduce 
the terminal voltage. 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- 
cJironous reactance of the alternator. In the following, we 
shall represent the total reaction of the armature of the 
alternator by 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. 



"302 ALTERNATING-CURRENT PHENOMENA. 

184. Let E = 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 

where 

n = total number of turns in series on the armature, 

JV = frequency, 

M = total magnetic flux per field pole. 

Let x = synchronous reactance, 

r = internal resistance of alternator ; 
then Z r j x = internal impedance. 

If the circuit of the alternator is closed by the external 
impedance, 

Z = r-jx, 
the current is 

E E 



or, /= 

and, terminal voltage, 



or, 




+x- 



ALTERNA TING-CURRENT GENERA TOR. 



303 



or, expanded in a series, 



As shown, the terminal voltage varies with the condi- 
tions of the external circuit. 

185. As an instance, in Figs. 129-134, at constant 
induced E.M.F., 

Eo = 2500 ; 















































. ^ 
























/ 


' 




x \ 

\ 










*- 











/ 


/ 








\ 
\ 
\ 














\ 


\ 


/ 










\ 
i 
















/ 


\ 


***>. 










1 














/ 

/ 




^ 


X^o 








I 
1 












i 










^J 






\ 






4S . 




( 


.1 

'/ 












\ 




\ 
\ 






Si 
&' 














> 


\ 


\ 






n 

2' 


^ 


f 
















\ 




I 






















\ 


1 






/ 




F 


ELD 


CHA 


MCI 


ERIS 


TIC 






\ 


1 
1 

1 












E = 

1 


250( 
R = 


>, Zo-MOj, 
E, xko 








\ 


I 
, 1 




1 
1 
1 






















\ 




1 



























20 10 60 80 100 180 140 160 18P 2 


X) 2 


210 2 






Fig. 129. Field Characteristic of Alternator on Non-inductive Load. 

' + 

and the values of the internal impedance, 

z = r -j Xo = i - ioy. 

With the current / as abscissae, the terminal voltages E 
as ordinates in drawn line, and the kilowatts output, = / 2 r, 
in dotted lines, the kilovolt-amperes output, = / , in dash- 



304 



AL TEKNA TING-CURRENT PHENOMENA. 



dotted lines, we have, for the following conditions of external 

circuit : 

In Fig. 129, non-inductive external circuit, x = 0. 

In Fig. 130, inductive external circuit, of the condition, r / x 
= -f .75, with a power factor, .6. 

In Fig. 131, inductive external circuit, of the condition, r= <>, 
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. 































\ 








E72 


FIE 
500, 


.D CHARA 
Zf MOj. i 


CTERIST(C 

-.75jop60^P.F 








"\ 


\ 




























S 


\ 






























\ 






























\ 




-^ 
























^X 


\ 






*\ 












I* 






/ 






S 


fe 




\ 










II* 

So 




>/ 










X 






\ 










^ 


"i 




x '' 








\ 




\ 










/ 


J 














^ 


\ 


\ 








/ 


















X^N 


\ 








/ 


















^ 


\\ 

\v 






(/_ 






















\ 


^ 




20 40 60 80 1 


K 120 140 1 


H) 180 200 220 glQ 20 


Amp 



Fig. 130. Field Characteristic of Alternator, at 60% Power-factor on Inductive Load. 

Such a curve is called a field 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. 



ALTERNATING-CURRENT GENERATOR. 305 













\ 
















s, 










FIELD CHARACTt 
: =25OO, Z?1-10j, r = 


RISTIC 

o, 90 Lag 








\ 


\ 












1 R = 


0. 














\ 


\ 




























\ 


\ 




























\ 


\ 




















k 

o 








>C" 




-X 


















A 


/ 


S 


\%< 




\ 












o 2" 

X X 




t 








s 


% 




\ 












/ 


/ 










\ 


\ 




\ 










/ 














\ 


\ 


> 








/ 


















\ 


\ 


\ 






/ 




















\ 









/ 






















s, 


\ 





Fig. 131. Field Characteristic of Alternator, on Wattless Inductive Load. 



5 
I 

li'.'U 

1000 


HM 
















^ 


^ 














N s 
















V 


x^ 






















\ 




.'.X'OU 






^ 


X" 




























\ 






X 


? 


































X 








F 


EU 


Ch 


AR 


ACT 


ER 


ST 


c 




















E 


f 2 


50C 


), Z 


1-1 


3j. : 


= -.75 c 


r 6 


3^F 


.F. 












iloo 






























/ 


"" 
































fc y 


^ 










KM 



































/ 


























/ 












/ 


f 


ItilK 


















< 


/ 












/ 


j 






















/ 










,-* 


'"' 


/ 


j 


^ 


400 

" 


lain.. 














^ 


s 






v 


-- 






1 






> 














> 






, 


,.*' 










/ 


/ 


j 




800 










f* 




. 


X 












/ 


/ 




/ 










, 


7 




,,*" 
















/ 


/ 


/ 






m 




/- 




-*''" 






A 


-n pe 


M 








/ y 


/ x . 


**' 










;-r 


*"' 


1 




B 


, 





I 


| 


2 


0^ 


**! 













m 



Fig. 732. Field Characteristic of Alternator, at 60% Power-factor on Condenser Load. 



306 



AL TERNA TING-CURRENT PHENOMENA. 









1 I 1 1 






'/ 












FIE 


LD CHARACTERISTIC 




/ 
/ 


i 
/ 












f 


E -2500, Zo-1-IOj, 
= o. 90Leading Current 




/ 


/ 
















I'R 


= O 








L 


























/ 


/ 


























/ 


/ 
/ 


























7 


/ 


























/ 

r tu 


/ 
























/ 


2 


























/ 


1 






/ 


? 


















/ 


/ 




/ 


s 

/ 


















?/ 

r 

/ 


J 

/ 


^ 


*X 


/ 


/ 




















7 








I* 

11 










^ 

/ 


/ 


^x 




/ 




















// 




/ 




















// 


/ 


/ 






















// 


! 


/ 






















/ 


/ 


I/ 






















/ 


/ 


// 






















/ 


/ 


/ / 


/ 




















/ 




g 


/ 






















^- x 


^ 


x'' 








xlO 


3- A, 


nps. 













fig. 133. Field Characteristic of Alternator, on Wattless Condenser Load. 

With reactive load the curves are more nearly straight 
lines. 

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. 



AL TERN A TING-CURRENT GENERA TOR. 



307 



M 



VK 



4^z 



W 



Fig. 134. Field Characteristic of Alternator. 

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., 

E = 2,500 volts, 
for the currents, 

1= 50, 100, 150, 200, 250 amperes, 

the terminal voltages, E, as ordinates, with the inductance 
factor of the external circuit, 



as abscissas. 

187. If the internal impedance is negligible compared 
with the external impedance, then, approximately, 



w 



308 



AL TERNA TING-CURRENT PHENOMENA, 




' .C .5 .4 .3 .2 .1 -.1 -.2 -.3 -.1 -.5 -.0 -.7 -.8 

Fig. 135. Regulation of Alternator on Various Loads. 

that is, an alternator with small internal resistance and syn- 
chronous reactance tends to regulate for constant terminal 
voltage. 

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, E 



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. 




ALTERNATING-CURRENT GENERATOR. 309 

With a non-inductive external circuit, if the synchronous 
reactance, X Q , of the alternator is very large compared with 
the external resistance, r, 



current /= 

x 



-g. 1 _E, 



approximately, or constant ; or, if the external circuit con- 
tains the reactance, x, 

T=-** 1 - * 



approximately, or constant. 

The terminal voltage of a non-inductive circuit is 



approximately, or proportional to the external resistance. 
In an inductive circuit, 


x 

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 are generally more or less ma- 



310 ALTERNATING-CURRENT PHENOMENA. 

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, .r . 

On non-inductive circuit, if 




theoutputis 



hence, if 
or 



then 

dr 

That is, the power is a maximum, and 



and 



7 = 




V2 So {so + r ) 

Therefore, with an external resistance equal to the inter- 
nal impedance, or, r ^ = VV 2 + x^ , the output of an 
alternator is a maximum, and near this point it regulates 
for constant output ; that is, an mcrease of current causes 
a proportional decrease of terminal voltage, and inversely. 

The field characteristic of the alternator shows this 
effect plainly. 



SYNCHRONIZING ALTERNATORS. 311 



CHAPTER XVIII. 

SYNCHRONIZING ALTERNATORS. 

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. 



312 ALTERNATING-CURRENT PHENOMENA. 

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 difficulty 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 modern 
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 



SYNCHRONIZING ALTERNATORS. 313 

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 1 . 

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, A l} A 2 , 
are connected in series, but interlinked by the two coils 
of a large transformer, T, of which the one is connected 



314 



AL TERNA TING-CURRENT PHENOMENA. 



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 





Fig. 136. 

be without current. In this case, by interchange of power 
through the transformers, the series connection will be 
maintained stable. 

195. In two parallel operating alternators, as shown in 
Fig. 137, let the voltage at the common bus bars be assumed 





Fig. 137. 



as zero line, or real axis of coordinates of the complex 
representation ; and let 



SYNCHRONIZING ALTERNATORS. 315 

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 



Let 

J?i = e-i je\ = # 2 (cos u> 1 j sin >i) = induced E.M.F. of first 

machine ; 
2 = e. 2 _/>/ = a 2 (cos w 2 j sin w 2 ) = induced E.M.F. of sec- 

ond machine ; 

/! = /! -f-//i' = current of first machine ; 
/ 2 = / 2 -j-yY 2 ' = current of second machine ; 
Z^ = T! jxi = internal impedance, and Y v = gi -\- jb l = inter- 

nal admittance, of first machine ; 
Z 2 = r 2 jx z = internal impedance, and K 2 =gz ~\~ jb<i = inter- 

nal admittance, of second machine. 

Then, 



i^! , or ^ je^= (e 
2 Z 2 , or <? 2 jej= (e 
7 2 , or 



This gives the equations 



4* + *"-**; 

or eight equations with nine variables: ^, ^', ^ 2 , ^/, / lf 






316 ALTERNATING-CURRENT PHENOMENA. 

Combining these equations by twos, 

e l r l -f eSxj. = er^ + t\ 2l 2 - 
e*r 9 + ^/^2 = e 
substituted in 

'i + H = 
we have 



and analogously, 

'1^1 ^iVi + 'a *a <?aV a = ' (^ + ^2 + 
dividing, 



b + ^i + ^2 ^i ^ ; i + <?a ^ ^iVi ^a' ^2 ' 
substituting 

g = V COS a C l = tfj COS Wj ^ 2 = ^2 COS d) 2 

^ = z/ sin a ^/ = ^ sin oJj ^ 2 ' = a 2 sin <o 2 
gives 

a\ v\ cos (en aQ + a 2 z> 2 cos (a 2 a 2 ) 
tfj z/! sin (ai w^) -\- a^Vs sin (a 2 a> 2 ) 

as the equation between the phase displacement angles 
and oi 2 in parallel operation. 

The power supplied to the external circuit is 



of which that supplied by the first machine is, 

/i = \ ; 
by the second machine, 

/ 2 = a 

The total electrical work done by both machines is, 

P = P l + P*, 
of. which that done by the first machine is, 

PI = '! h - e,' // ; 
by the second machine, 



SYNCHRONIZING ALTERNATORS. . 317 

The difference of output of the two machines is, 

denoting 

>! -f- 0)2 <QI o> 2 s 

~2~ ~2~ 

A^>/AS 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 
excitation. 



Substituting this in the eight initial equations, these 
assume the form, 



e- = t x t r 
e 2 ' = / 2 .r // r . 
*g=i\ +'a 
eb = i{ + / a ' 

4* + 4" -</ + *"-< 

Combining these equations by twos, 



substituting e l = a cos o^ 

e{ = a sin o^ 
^ 2 = a cos 0)2 
e 2 f = a sin o) 2 , 

we have a (cos w x + cos w a ) = * (2 + r ^ + 
a (sin Si + sin w 2 ) = e (x^g r li) 

expanding and substituting 



8 = 



318 AL TERN A TING-CURRENT PHENOMENA. 



a cos e cos 8 = e ( 1 + 



r Q g -\-Xzb 



a sin e cos 8 = ^^ ^ 



hence 

That is 

-and cos 8 = - 



tan e = ^ ^ = constant. 



+ A 2 = constant; 



-M 1 * 5 



zaiy , /^o^-^o^\ 2 . 

cos 8 



at no-phase displacement between the alternators, or, 
-we have e = ^ . 



V/(' 



+ 



n>^ + -*o^\ 2 , fxn r *b 



From the eight initial equations we get, by combina- 



(''o 2 



subtracted and expanded 



.or, since 

<?! <? 2 = ^ (cos wj cos G 2 ) = 2 tf 2 sin c sin 8 
^/ <?/ = a (sin wx sin w 2 ) = 2 a cos e sin 8 ; 

we have 

2 a s * n 8 - r sin c} 



2 ay sin 8 cos (c -f a), 



where 



tan d = 2- . 

/"o 



SYNCHRONIZING ALTERNATORS. 319 

The difference of output of the two alternators is 

A/ =/! / 2 = e (/i / 2 ) ; 
hence, substituting, 



substituting, 



2ggsin8 {jfrcos - r sin c}; 



, 
H 



XQ r *b 

2 



i 'of, T *<) " \ i 

2 J + V 2 
we have, 

2a 2 sin 8 cos 8 j * ( 1 + r * + x *\ r 



expanding, 

A / = nr 



Hence, the transfer of power between the alternators, 
A p t is a maximum, if 8 = 45 ; or Wj w 2 = 90 ; that is, 
when the alternators are in quadrature. 



320 ALTERNATING-CURRENT PHENOMENA. 

The synchronizing power, A p / A 8, is a maximum if 
8 = ; that is, the alternators are in phase with each other. 

197. As an instance, curves may be plotted 
for, 

a =2500, 



with the angle 8 = U)l a>2 as abscissae, giving 

the value of terminal voltage, e 

the value of current in the external circuit, / = ey ; 

the value of interchange of current between the alternators, 

*i-* 2 ; 
the value of interchange of power between the alternators, A p 

=A-/ 2 ; 

the value of synchronizing power, ^ . 

A o 

For the condition of external circuit, 

g = 0, b = 0, y = 0, 

.05, 0, .05, 

.08, 0, .08, 

.03, + .04, .05, 

.03, - .04, .05. 



SYNCHRONOUS MOTOR. 321 



CHAPTER XIX. 

SYNCHRONOUS MOTOR. 

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., E, be connected as 
synchronous motor with a supply circuit of E.M.F., E Q , 
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 E 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 Q 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. E Q , and motor of induced' 
E.M.F. E l ; or, more general, two alternators of induced 
E.M.Fs., E , E lf connected together into a circuit of total 
impedance, Z. 

Since in this case several E.M.Fs. are acting in circuit 



322 ALTERNATING-CURRENT PHENOMENA. 

with the same current, it is convenient to use the current, 
/, as zero line OI of the polar diagram. Fig. 188. 

If I=i= current, and Z = impedance, r = effective 
resistance, x = effective reactance, and s = Vr 2 -f x 2 = 
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 OE,, ; and the E.M.F. 
consumed by the reactance is E 2 = xi, and 90 ahead of 
the current, hence the E.M.F. consumed by the impedance 
is E = V(,,) 2 + (E 2 f, or = i Vr 2 + x* = is, and ahead of 
the current by the angle 8, where tan 8 = x / r. 

We have now acting in circuit the E.M.Fs., E, E lf E Q ; 
or E l and E are components of E Q ; that is, E Q is the 
diagonal of a parallelogram, with E l and E as sides. 

Since the E.M.Fs. E lf E z , E, are represented in the 
diagram, Fig. 138, by the vectors OE~ lf OE 2 , OE, to get 
the parallelogram of Q , E lt E, we draw arcs of circles 
around with E Q , and around E with E l . Their point of 
intersection gives the impressed E.M.F., OE Q = E Q , and 
completing the parallelogram OE E Q E we get, OE = E , 
the induced E.M.F. of the motor. 



IOE is the difference of phase between current and im- 
pressed E.M.F., or induced E.M.F. of the generator. 

IOEi is the difference of phase between current and in- 

duced E.M.F. of the motor. 

And the power is the current /times the projection of the E.M.F. 
upon the current, or the zero line OI. 

Hence, dropping perpendiculars, E^EJ and E^E^, from 
E Q and E! upon OI, it is 

P = iX OE^ = power supplied by induced E.M.F. of gen- 

erator. 
PI = / X OE^ = electric power transformed in mechanical 

power by the motor. 
P = / x OE l = power consumed in the circuit by effective 

resistance. 



SYNCHRONOUS MOTOR. 



323 



Since the circles drawn with E Q and E around O and K 
respectively intersect twice, two diagrams exist. In gen- 
eral, in one of these diagrams shown in Fig. 138 in drawn 




Fig. 138. 

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, Q is in the same, E^ 
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- 
considered from the point of view of the motor ; that is,. 



324 



ALTERNATING-CURRENT PHENOMENA. 



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 = OE, Figs. 139 to 141, con- 
sists of three components ; the E.M.F. OE E z , consumed 




Fig. 139. 

by the impedance of the motor, the E.M.F. 
consumed by the impedance of the line, and the E.M.F. 
EZ E = E consumed by the impedance of the generator. 
Hence, dividing the opposite side of the parallelogram E 1 E () , 
in the same way, we have : OE l = E 1 = induced E.M.F. of 
the motor, OE Z = 2? a = E.M.F. at motor terminals or at 
end of line, OE 3 = E 3 = E.M.F. at generator terminals, 
or at beginning of line. OE Q = E Q = induced E.M.F. of 
generator. 



SYNCHRONOUS MOTOR. 



325 



The phase relation of the current with the E.M.Fs. lt 
, depends upon the current strength and the E.M.Fs. E l 



and 



200. Figs. 139 to 141 show several such diagrams for 
different values of E lf but the same value of / and E Q . 
The motor diagram being given in drawn line, the genera- 
tor diagram in dotted line. 




Fig. 140. 

As seen, for small values of E 1 the potential drops in 
the alternator and in the line. For the value of E 1 = E 
the potential rises in the generator, drops in the line, and 
rises again in the motor. For larger values of E ly thfe 
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- 
erator. 



326 



ALTERNATING-CURRENT PHENOMENA, 



It is of interest now to investigate how the values of 
these quantities change with a change of the constants. 




Fig. 747. 

201. A. Constant impressed E.M.F. E v , constant current 
strength I = i, variable motor excitation E v (Fig. 142.) 

If the current is constant, = z; OE, the E.M.F. con- 
sumed by the impedance, and therefore point E, are con- 
stant. Since the intensity, but not the phase of E Q is 
constant, E Q lies on a circle e Q with E Q as radius. From 
the parallelogram, OE E Q E l follows, since E 1 E Q parallel 
and = OE, that E l lies on a circle e l congruent to the circle 
e Q , but with E i} the image of E, as center : OE i = OE. 

We can construct now the variation of the diagram with 
the variation of E l ; in the parallelogram OE E Q E 1 , O and 
E are fixed, and E and E l move on the circles <? e l so that 
EQ E^ is parallel to OE. 



SYNCHRONOUS MOTOR. 



327 



The smallest value of E l consistent with current strength 
/ is Olj = E^, 01 = EQ. In this case the power of the 
motor is Olj 1 x /, hence already considerable. Increasing 
E l to 02"^ OSj, etc., the impressed E.M.Fs. move to 02, 03, 
etc., the power is / x 02^, I x 03^, etc., increases first, 




Fig. 142. 

reaches the maximum at the point 3j, 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 E^ increases and becomes = Q at 
4,, 4. At 5 15 5, the power becomes zero, and further on 
negative ; that is, the motor has changed to a dynamo, and 



328 AL TERNA TING-CURRENT PHENOMENA. 

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 1} 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 1 l , 7, the maximum value 
of E lt consistent with the current /, has been reached, and 
passing still further the E.M.F. E l decreases again, while 
the power still increases up to the maximum at S lt 8, and 
then decreases again, but still E l remaining generator, E Q 
motor, until at 11^ 11, the power of E Q becomes zero; that 
is, E Q changes again to a generator, and both machines are 
generators, up to 12 lf 12, where the power of E l is zero, E l 
changes from generator to motor, and we come again to 
the motor side of the diagram, and while E l still decreases, 
the power of the motor increases until l u 1, is reached. 

Hence, there are two regions, for very large E l from 
5 to 6, and for very small E l from 11 to 12, where both 
machines are generators ; otherwise the one is generator, 
the other motor. 

For small values of E l the current is lagging, begins, 
however, at 2 to lead the induced E.M.F. of the motor E lf 
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 I} lj, 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 



SYNCHRONOUS MOTOR. 



329 



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 Q and E l constant, I variable. 

Obviously E Q lies again on the circle e Q with E Q as radius 
and O as center. 




Fig. 143. 



E lies on a straight line e, passing throtigh the origin; 

Since in the parallelogram OE E E v EE Q = E^ we 
derive E Q by laying a line EE Q = E from any point E 
in the circle e Q , and complete the parallelogram. 

All these lines EE Q envelop a certain curve e lt which 



030 ALTERNATING-CURRENT PHENOMENA. 

can be considered as the characteristic curve of this prob- 
lem, just as circle e^ in the former problem. 

These curves are drawn in Figs. 143, 144, 145, for the 
three cases : 1st, E l = E Q ; 2d, E l < E Q ; 3d, 1 > Q . 

In the first case, E l = E Q (Fig. 127), we see that at 




Fig. 144. 



very small current, that is very small OE, the current / 
leads the impressed E.M.F. E Q by an angle E Q Of = W Q . 
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 Q , and producing 
EE Q until it intersects e it in E if we have ^^ E i OE 90, 
E l = E Q , thus : OE 1 = EE Q =OE Q = E Q E t ; that is, EE { = 



SYNCHRONOUS MOTOR. 



331 



2E Q . That means the characteristic curve e l is the enve- 
lope of lines EE iy of constant lengths 2E Q , sliding between 
the legs of the right angle E t OE; hence, it is the sextic 
hypocyloid osculating circle <? , which has the general equa- 
tion, with e, e i as axes of coordinates : 



In the next case, E 1 < E Q (Fig. 144) we see first, that 
the current can never become zero like in the first case, 




V 



Fig. 145. 



EI = EQ, but has a minimum value corresponding to the 
minimum value of OE l : I{ = , and a maximum 

value : //' = . Furthermore, the current can never 

lead the impressed E.M.F. E^, but always lags. The mini- 



332 ALTERNATING-CURRENT PHENOMENA. 

mum lag is at the point H. The locus e v as envelope of the 
lines EEty is a finite sextic curve, shown in Fig. 144. 

If E l < E Q , at small E Q E l , H can be above the zero 
line, and a range of leading current exist between two ranges 
of lagging current. 

In the case E 1 > E Q (Fig. 145) the current cannot equal 
zero either, but begins at a finite value C, corresponding 

to the minimum value of OE Q : // = * -. At this 

value however, the alternator E 1 is still generator and 
changes to a motor, its power passing through zero, at the 
point corresponding to the vertical tangent, onto e lf 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 : 

Minimum: Maximum: 

1st. 7=0, 7=^. 




Since tfie current passing over the line at E l = O, that 
is, when the motor stands still, is 7 = E Q 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 3. 

203. C. EQ = constant, E l varied so that the efficiency is a 
maximum 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 



SYNCHRONOUS MOTOR. 



333 



the efficiency a maximum, when the current is in phase 
with the induced E.M.F. E Q of the generator, we have as 
the locus of E Q the point E Q (Fig. 146), and when E with 
increasing current varies on <?, E must vary on the straight 
line e v parallel to c. 

Hence, at no-load or zero current, E l = E , decreases 
with increasing load, reaches a minimum at OE^ perpen- 
dicular to c lt and then increases again, reaches once more 




Fig. 146. 



E l = E Q at E?, and then increases beyond E . The cur- 
rent is always ahead of the induced E.M.F. E l 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 Ef, where OEf = EfE Q = 
1/2 x ~OE^ and is then = / x "^7/2. Hence, since OEf = 



EJ2,f=E () /2randP 



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. 



-334 ALTERNATING-CURRENT PHENOMENA. 

In this case, 



In general, it is, taken from the diagram, at the condi- 
tion of maximum efficiency : 



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. 
E l here acting in the same way as the condenser capacity 
in Chapter IX. 




204. 



Fig. 147. 



D. E n = constant ; P = constant. 



If the power of a synchronous motor remains constant, 
we have (Fig. 147) / x OE^ = constant, or, since OE 1 



SYNCHRONOUS MOTOR. 



335 



Ir, I = OE 1 / r, and: OE 1 x OE? = O l X E 1 EJ = 
constant. 

Hence we get the diagram for any value of the current 
/, at constant power P lt by making OE 1 = I r, E 1 E 1 = P l j I 
erecting in E Q l a perpendicular, which gives two points of 
intersection with circle e Q , E Q , one leading, the other lagging. 
Hence, at a given impressed E.M.F. E Q , the same power P 



E, 



1250 7 

1100/1580 31/16.7 

1480 32 

1050/1840 2/25 

2120 
2170 



37.5 
40 



45.5 




16.7 



Fig. U8. 

can be transmitted by the same current I with two different 
induced E.M.Fs. E } of the motor; one, OE l = EE Q small, 
corresponding to a lagging current ; and the other, OE l = 
EE Q 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. 



336 



ALTERNATING-CURRENT PHENOMENA. 



E 1 . In one of the cases the current is leading, in the 
Dther lagging. 

In Figs. 148 to 151 are shown diagrams, giving the points 

E = impressed E.M.F., assumed as constant = 1000 volts, 
E = E.M.F. consumed by impedance, 
E' = E.M.F. consumed by resistance. 



EflOOO 

P=6000 

34O< E,<1920 

7< I < 43 



Fig. 149. 



I 

1450 17.3 

1170/1910 10/30 
1040/1930 8/37.5 




10/30 
17.3 



of the motor, E lt is OE lt equal and 
shown in the diagrams, to avoid 



The counter E.M.F. 
parallel EE Q , but not 
complication. 

The four diagrams correspond to the values of power, 
or motor output, 
P = 1,000, 6,000, 



9,000, 



12,000 watts, and give : 
1 < I < 49 Fig. 132. 



P = 1,000 46 < E l < 2,200, 

P = 6,000 340 < , < 1,920, 7 < I < 43 Fig. 133. 

P = 9,000 540 < E l < 1,750, 11.8 < / < 38.2 Fig. 134. 

P = 12,000 920 < E l < 1,320, 20 < I < 30 Fig. 153. 



SYNCHRONOUS MOTOR. 



337 




E, I 

* 1440 21.2 

3 1200/1660 15/30 

1080/1750 13/34.7 



900/1590 11.8/38.2. 



720/1100 13/34.7 
620/820 15/30 
/ 3 540 21.2 



3 1280 24.5 

2 1120/1320 21/28.6 
all l-QQO/1260 30/30 



920/1100 
020 



21/28.6 
24.5 



P=I200O 

920< E,< 1320 

20<l<30 



Fig. 151. 

As seen, the permissible value of counter E.M.F. E v and 
of current /, becomes narrower with increasing output. 



338 ALTERNATING-CURRENT PHENOMENA. 

In the diagrams, different points of E Q are marked with 
1, 2, 3 . . . , when corresponding to leading current, with 
2 1 , 3 1 , . . . , when corresponding to lagging current. 

The values of counter E.M.F. E v and of current 7 are 
noted on the diagrams, opposite to the corresponding points 

*o- 

In this condition it is interesting to plot the current as 

function of the induced E.M.F. E l of the motor, for con- 
stant power /V Such curves are given in Fig. 155 and 
explained in the following on page 345. 

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 z = Vr 2 -j- x 2 = impedance of the circuit of (equivalent) 
resistance r and (equivalent) reactance x = 2 TT NL, containing 
the impressed E.M.F. e * and the counter E.M.F. e t of 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 1 ; hence 

p = *>! cos ft,^), (1) 

thus, 



* If f = E.M.F. at motor terminals, z = internal impedance of the 
motor; if eo= terminal voltage of the generator, z = total impedance of line 
and motor; if t = E.M.F. of generator, that is, E.M.F. induced in generator 
armature by its rotation through the magnetic field, z includes the generator 
impedance also. 



SYNCHRONOUS MOTOR. 339 

The displacement of phase between current i and E.M.F. 
= z i consumed by the impedance z is : 



cos (ie) = - 



sin (/<?) 



x 



(3) 



Since the three E.M.Fs. acting in the closed circuit : 

e = E.M.F. of generator, 

fi = C.E.M.F. of synchronous motor, 

e = zi = E.M.F. consumed by impedance, 

form a triangle, that is, c^ and e are components of ^ , it is 
(Fig. 152) : 

e 1 2 e Z .1 ?. ^2 ,'2 

hence, cos (*,.#) = *- = - - . (5) 

2 e^e '2,zie^ 

since, however, by diagram : 

cos (e l , e) = cos (/, e /', e^) 

= cos (/, e) cos (/, ^i) + sin (t, e) sin (/, ^) (6) 

substitution of (2), (3) and (5) in (6) gives, after some trans- 
position : 

the Fundamental Equation of tJie Synchronous Motor, relat- 
ing impressed E.M.F., <? ; C. E.M.F., ^ ; current z; power, 
/, and resistance, r ; reactance, x ; impedance s. 

This equation shows that, at given impressed E.M.F. e$ f 
and given impedance s = Vr 2 + x*, three variables are left, 
e v i,p, of which two are independent. Hence, at given ^ 
and s, the current i is not determined by the load / only, 
but also by the excitation, and thus the same current i can 
represent widely different loads p, 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 1 . 

The meaning of equation (7) is made more perspicuous 



340 ALTERNATING-CURRENT PHENOMENA. 



by some transformations, which separate e v and i, as func- 
tion of/ and of an angular parameter <. 
Substituting in (7) the new coordinates : 



V2 



V2 



or, 



_ 

V2 



we get 



substituting again, e< f = a 
Izp = b 

r = Z 
hence, x = z Vl e 2 




jr. 753. 



we jret 



a a V2 e b = V(l e 2 ) (2 a 2 2 2 - 
and, squared, 

substituting 

gives, after some transposition, 

v* -f ze/ 2 = (- 1 ~ *") a (a 2 tb\ 



(9) 




) (11) 

0, (12) 

(13) 
(14) 



SYNCHRONOUS MOTOR. 341 

hence ' if 



i* + w* = * (16) 

the equation of a circle with radius R. 

Substituting now backwards, we get, with some trans- 
positions : 

{r* (ef + z*i 2 ) - z* (Vo 2 - 2 r/)} 2 + {r x (e? - z*i 2 )} 2 = 

* 2 .sV(^ 2 -4r/) (17) 

the Fundamental Rquation of the Synchronous Motor in a 
modified form. 

The separation of e and i can be effected by the intro- 
duction of a parameter < by the equations : 

r 3 - (e? z 2 / 2 ) - z 2 (ef 2rp)=xze () V<r a rp cos < 



rx (e? - z 2 / 2 ) =xze Vtf - 4 r/ sin ' l ' 
These equations (18), transposed, give 

N 

+ sin</> 



The parameter <^> has no direct physical meaning, appar- 
ently. 

These equations (19) and (20), by giving the values ef 
e l and i as functions of / and the parameter < enable us 
to construct the Power Characteristics of the Synchronous 
Motor, as the curves relating e v and i, for a given power /, 
by attributing to < all different values. 



342 ALTERNATING-CURRENT PHENOMENA. 

Since the variables v and w in the equation of the circle 
(16) are quadratic functions of e 1 and /', 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 e l and i [equations (19) and 
(20)] contain the square root, W 2 4 rp, it is obvious 
that the maximum value of / corresponds to the moment 
where this square root disappears by passing from real to 
imaginary ; that is, 

tf _ 4 r p = 0, 

r> 

/ = .. (21) 

This is the same value which represents the maximum 
power transmissible by E.M.F., e Q , 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. 

cor(A,0-^--i 

tc z 

hence, 

tan fa, /) = -?, (23) 



SYNCHRONOUS MOTOR. 343 

that is, the angle of internal displacement in the synchron- 
ous motor i equal, but opposite to, the angle of displace- 
ment of line impedance, 

('i, = - (', 0, 

= ~ <X '), (24) 

and consequently, 

(.-0,0=0; (25) 

that is, the current, z, is in phase with the impressed 
E.M.F., * . 

If 2 < 2 r, e l < <? ; that is, motor E.M.F. < generator E.M.F. 

If z = 2 r, e l = e ; that is, motor E.M.F. = generator E.M.F. 

If z > 2 r, <?! > r ; that is, motor E.M.F. > generator E.M.F. 

In either case, the current in the synchronous motor is 
leading. 

207. B. Running Light, p = 0. 

When running light, or for / = 0, we get, by substitut- 
ing in (19) and (20), 



(26) 



Obviously this condition cannot well be fulfilled, since p 
must at least equal the power consumed by friction, etc. ; 
and thus the true no-load curve merely approaches the curve 
/ = 0, being, however, rounded off, where curve (26) gives 
sharp corners. 

Substituting / = into equation (7) gives, after squar- 
ing and transposing, 

e* + e< * 4- 3*,- - 2 ^V - 2 2 2 rV + 2 r a *'V - 2 * 2 *V = 0. (27) 

This quartic equation can be resolved into the product 
of two quadratic equations, 

0. | (28) 

0. j 



344 ALTERNATING-CURRENT PHENOMENA. 

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., e it is ^ = at / = ^2. (29) 
The minimum value of current, z, is / = at e t = e . (30) 
The maximum value of E.M.F., e lt is given by Equation (28)', 

/= e* + 2 2 z 2 -e<?2 xie l = ; 
by the condition, 



hence, 



The maximum value of current, z, is given by equation 
(28) by 

= 0, as 
de l 

(32) 

If, as abscissas, e lt and as ordinates, zi, 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 A 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 D, whose points do not represent any actual 
condition of the alternator circuit, but make e l , i imaginary. 

A and A' and the same B and B' ', are identical condi- 
tions of the alternator circuit, differing merely by a simul- 



SYNCHRONOUS MOTOR. 



345 



\ 



r 



\ 



I 



\ 



4000 3000 \^ 2000 1000 



Volts 1000 2000\/3000 4000 5000 






\ 



/A' 



\ 



\ 



Fig. 154. 

taneous reversal of current and E.M.F. ; that is, differing 
by the time of a half period. 

Each of the spaces A and B 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, t, at given power, /, 
is determined by the absence of a phase displacement at the 
impressed E.M.F. e Q , 



346 AL TERNA TING-CURRENT PHENOMENA. 

This gives from diagram Fig. 153, 

e 1 * = e ( ? + i*z*-2ie r, (33) 

or, transposed, 

This quadratic curve passes through the point of zero 
current and zero power, 

through the point of maximum power (22), 



and through the point of maximum current and zero power, 



e n x 



r 



(35) 



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 34, 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, 

/ = 'J2, (36) 

while at no-load, the current can reach the maximum value, 
/=^, (35) 

the same value as would exist in a non-inductive circuit of 
the same resistance. 

The minimum value at C.E.M.F. e l} at which coincidence 



SYNCHRONOUS MOTOR. 347 

of phase (e Q , -i) = 0, can still be reached, is determined from 
equation (34) by, 



as 

i e - - (37} 

The curve of no-displacement, or of minimum current, is 
shown in Figs. 138 and 139 in dotted lines.* 

209. D. Maximum Displacement of Phase. 

(e % , i} = maximum. 
At a given power/ the input is, 

A =P + i*r = e,i cos (* , *) ; (38) 

hence, 

cosfo, = /+/V . (39) 

At a given power /, this value, as function of the current 
i, is a maximum when 

d_(p + 

di\ 
this gives, 




(40) 
or, 

(41) 

That is, the displacement of phase, lead or lag, is a 
maximum, when the power of the motor equals the power 

* It is interesting to note that the equation (34) is similar to the value, 
<?! = \/(^ 2 r) 2 z' 2 jr 2 , which represents the output transmitted over an 
inductive line of impedance, z = vV 2 + jr 2 into a non-inductive circuit. 

Equation (34) is identical with the equation giving the maximum voltage, 
e , at current, i, which can be produced by shunting the receiving circuit with a 

condenser; that is, the condition of " complete resonance " of the line, z = 

x 
Vr' 2 + x' 2 , with current, . Hence, referring to equation (35), e l = t ~ is 

the maximum resonance voltage of the line, reached when closed by a con- 
denser of reactance, x. 



348 



ALTERNATING-CURRENT PHENOMENA. 



consumed by the resistance ; that is, at the electrical effi- 
ciency of 50 per cent. 

Substituting (40) in equation (7) gives, after squaring 



/ N 



TSOO 8000^ #WU 3000 3uOO 
Fig. 155. 

and transposing, the Ouartic Equation of Maximum Dis- 
placement, 

<> 2 - e*y + **z 2 (s 2 + 8 r 2 ) + 2 j*e* (5 r 2 - 2 2 ) - 2 / V 

( 3 2 + 3 ^ = Oi (42) 

The curve of maximum displacement is shown in dash- 
dotted lines in Figs. 154 and 155. It passes through the 



SYNCHRONOUS MOTOR. 349 

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 
displacement. 

210. E. Constant Counter E.M.F. 

At constant C.E.M.F., e l = constant, 

If 

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 



the current is lagging at no load ; with increasing load the 
lag decreases, the current comes into coincidence of phase 
with e Q , 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 e Q < <?! , 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 
> <? or < * . 

211. F. 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. 



350 ALTERNATING-CURRENT PHENOMENA. 

In this case we have, 

<? = 2500 volts constant at generator terminals; ^| 
r 10 ohms, including line and motor ; /^gs 

x = 20 ohms, including line and motor ; j 

hence z = 22.36 ohms. 

Substituting these values, we get, 

2500 2 - e* - 500 i* - 20 / = 40 V*V -/ 2 (7) 

{^2 + 500 ? 2 - 31.25 X 10 6 + 100 /} 2 + (2 ^ 2 - 1000 / 2 } 2 = 

7.8125 x 10 15 - 5 + 10 9 /. (17) 

e l = 5590 (19) 

V| {(1 3.2 x 10~ 6 /) + (.894 cos <+ .447sin <) Vl-6.4xlO- 6 /}. 
* = 559 (20) 



6.4xlO- 6 /}. 

Maximum output, 

p = 156.25 kilowatts (21) 

at *i = 2,795 volts 

i = 125 amperes 
Running light, 

^ + 500 / a - 6.25 x 10 4 =p 40 /^ = 
^ = 20 / V6.25 X 10 4 100 i* 

At the" minimum value of C.E.M.F. e 1 = is / = 112 (29) 
At the minimum value of current, / = is e l = 2500 (30) 
At the maximum value of C.E.M.F. e v = 5590 is / = 223.5 (31) 
At the maximum value of current i 250 is e l = 5000 (32) 

Curve of zero displacement of phase, 



l = 10 V(250 - O 2 + 4 * a (34) 

= 10 V6.25 x 10 4 500 / + 5 / 2 
Minimum C.E.M.F. point of this curve, 

/ = 50 ^ = 2240 (35) 

Curve of maximum displacement of phase, 

/ = 10 *' 2 (40) 

(6.25 X 10 6 -^ 2 ) 2 + .65 X 10 6 / - 10 10 / 2 = 0. (42) 



SYNCHRONOUS MOTOR. 351 

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. 
p = 50 kilowatts. 
/ = 100 kilowatts. 
p = 150 kilowatts. 
p = 156.25 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 l . 

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. 155 can be con- 
sidered as approximations only, since a number of assump- 



352 



ALTERNA TING-CURRENT PHENOMENA. 



tions are made which are not, or only partly, fulfilled in 
practice. The foremost of these are : 

1. It is assumed that e l can be varied unrestrictedly, 
while in reality the possible increase of e l is limited by 
magnetic saturation. Thus in Fig. 155, at an impressed 
E.M.F., e Q = 2,500 volts, e l rises up to 5,590 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. 



Fig. 156. 

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 e lt and 
thereby bends the curves upwards at a lower value of ^ 
than represented in Fig. 155. 

It must be understood that the motor reactance is not 
a simple quantity, but represents the combined effect of 



SYNCHRONOUS MOTOR. 353 

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 superposi- 
tion of the M.M.F. of the armature current upon the field 
excitation ; that is, it is the " synchronous reactance." 

3. 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 magnetic induction ; that is, with e lf 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., e lt 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 ^ ; 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. 



354 AL TERN A TING-CURRENT PHENOMENA, 



CHAPTER XX. 

COMMUTATOR MOTORS. 

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- 
tators. 

The main subdivisions of commutator motcrs are the 
repulsion motor, the series motor, and the shunt motor. 

REPULSION 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, arid 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 



COMMUTATOR MOTORS. 



355 



give a rotary effort in the opposite direction, as shown 
by the arrows in Fig. 157. 

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 




Fig. 157. 

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. 



356 



AL TERNA TING-CURRENT PHENOMENA. 



215. In the repulsion motor the difficulty due to the 
equal and opposite 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 




Fig. 158. 

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 
commutator. 

Thus the general construction of a repulsion motor is 
as shown in Figs. 158 and 159 diagrammatically as bipolar 



COMMUTATOR MOTORS. 



357 



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 




Fig. 159. 

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 disadvantage that, in the passage of the brush from 
segment to segment, individual armature coils are short- 



358 



AL TERNA TING-CURRENT PHENOMENA. 



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 




Fig. 160. 

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- 



COMMUTATOR MOTORS. 359 

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 

Motors. 

217. Let 

$ = maximum magnetic flux per field pole ; 
e = effective E.M.F. induced thereby in the field turns ; thus, 



where ;/ = number of turns, N= frequency. 

<?10 8 

thus, 4> = -- 

\&-anN 

The instantaneous value of magnetism is 

<f> = <& sin (3 ; 

and the flux interlinked with the armature circuit 
< x = <I> sin /3 sin X ; 

when X 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 n turns, 
(as reduced to primary circuit), is thus, 

e = _ n ^1 10- 8 , = - n 4- sin B sin X lO" 8 , 

at at 

= - n$> sin X cos (3 + sin (3 cos X 10~ 8 . 



If N= frequency in cycles per second, N : = frequency 
of rotation or speed in cycles per second, and k = N^/ N 
speed 



we have 



frequency 



thus, g l = 2-TrnJV {sin X cos /? + k cos X sin B\ 10~ 8 , 
or, since $ = , 

e t = e V2 {sin X cos /3 + k cos X sin fi\. 



360 ALTERNATING-CURRENT PHENOMENA. 

218. Introducing now complex quantities, and counting 
the time from the zero value of rising magnetism, the mag- 
netism is represented by /4>, 
the primary induced E.M.F., E = e, 
the secondary induced E.M.F., 1 = e {sin X +j"k cos X|; 
hence, if 
Z l = r 1 jx 1 = secondary impedance reduced to primary circuit, 

Z = r jx = primary impedance, 

Y = g jb = exciting admittance, 

we have, 

& sin X -f- jk cos A 

secondary current, 7 X = L = - e - _ - , 

primary exciting current, I = eY= e (g +jb}, 
hence, total primary current, 



Primary impressed E.M.F., E = E + IZ\ 
= e 1 + (sinX 



Neglecting in E the last term, as of higher order, 

= e j 1 + sin X +jk cos X ^ ^4^ j ; 
or, eliminating imaginary quantities, 

e V(?i + r sin X -f- kx cos X) 2 + (x^ + x sin X kr cos X) 2 

The power consumed by the component of primary 
counter E.M.F., whose flux is interlinked with the secondary 
e sin X, is, 

f = [e sin X /]' = ^inXfosuiX-^cosX) , 

r \ + x \ 
the power consumed by the secondary resistance is, 

_ 2 _ ** r i ( sin2 x + ^ cos2 x ) 

hence the difference, or the mechanical power developed by 
the motor armature, 



COMMUTATOR MOTORS. 361 



and substituting for e, 

egk cos X (x^ sin X + r^k cos X) 

~ fa + r sin X + kx cos X) 2 + (x l + x sin \ kr cos X) 2 ' 
and the torque in synchronous watts, 

P <? 2 cos X (x 1 sin X + r^k cos X) 

~~ / ~~ (/i + ?" sin A + # cos X) 2 + (x t + x sin X kr cos X) 2 
or T= V27r^lO- 8 [/!<!> sin X 7 X cos A]' = [^/! cos X}> 
_ ^ cos X (x l sin X + r^k cos X) 

r 2 + x 2 
The stationary torque is, k = 0, 

_ ifo 2 ^ sin X cos X 

= (r x + r sin X) 2 + (^ + * sin X) 2 ' 

and neglecting the primary impedance, r = = x, 
_ e^x^ sin X cos X _ (fo 2 ^ sin 2 X 

which is a maximum at X = 45. 
At speed k, neglecting r = = x, 

<? 2 cos X (X sin X + r^k cos X) 
r 2 j-^2 ~' 

which is a maximum for - = 0, which gives, 

cot 2 X = . For k = 0, X = 45 ; for k = oo , X = 0. 

that is, in the repulsion motor, with increasing speed, the 
angle of secondary closed circuit, X, has to be reduced to 
get maximum torque. 

219. At A = 45 we have, 



(r x V2 + r + *) 2 + (^ V2 + x - krf 
and the power, 

p= ^k (x, + r,K)_ 

(r, V2 + r + kx)*+(xi ^2 + x - krf' 



362 ALTERNATING-CURRENT PHENOMENA. 

this is a maximum, at constant X = 45, for = 0, which 

dk 



gives, k = 1 

At X = we have, 
T-- 



fa + kxf + (* t - krf 

that is, T = at k = 0, or, the motor is not self-starting, 
when X = 0. 



P = 



dP 



which is a maximum at constant X = for, - = 0, which 

dk 
gives, 



rx-, xr-. 



MOO 




















-- 
























'..i i'j 
















S 


^ 
























"~ 


m 


-t^> 










, 


/ 






























no 


1 








/ 


/ 






R 


:PL 


LS 


ON 


M( 


5TC 


3R 














;') 

m 









/ 
















V 


OC 


















rt 




/ 


/ 












r = 


.! 




r, ' 


05 
















joa 


> 




/ 














X 


2. 




x. 


1. 
















M 




/ 












p- 


DO 

1.17 


1 


j^ ^ 
<) 


g 


k 
14 


i, 
-,] 


I 














21 K I 


/ 










































UW 




















K 1 


F^ 


d_ 




















/ 




































s 


2. 


I) 





F/fir. 161. Repulsion Motor. 

As an instance is shown, in Fig. 161, the power output 
as ordinates, with the speed k = N^_ / N as abscissae, of a 
repulsion motor of the constants, 

X = 45 e = 100. 
r= .1 r 1= .05 
* = 2.0 * x = 1.0 
giving the power, 

10,000 f .02 + 1.41 k .05 ffj 
~~ .171 + 2 y&) 2 + (3.14 - .1 Kf ' 



COMMUTATOR MOTORS. 



SERIES MOTOR. SHUNT MOTOR. 

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- 




Fig. 162. Series Motor. 

net, 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 



364 ALTERNATING-CURRENT PHENOMENA. 

magnetic circuit has to be laminated to exclude eddy cur- 
rents. 

Let, in a series 'motor, Fig. 146, 

<l> = effective magnetism per pole, 

n = number of field turns per pole in series, 

i = number of armature turns in series between brushes, 

/ = number of poles, 

(R. = magnetic reluctance of field circuit,* 

(R! = magnetic reluctance of armature circuit,! 

4>i = effective magnetic flux produced by armature current 

(cross magnetization) per pole, 
r = resistance of field (effective resistance, including hys- 

teresis), 
rj = resistance of armature (effective resistance, including hys- 

teresis), 

N = frequency of alternations, 
N = 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 



E.M.F. of self-induction of field, 

E' = 
E.M.F. of self-induction of armature, 

^/ = 2 7 r 1 ^V<I> 1 10- 8 , 
E.M.F. consumed by resistance, 

E r = (r + *i) I, 
where 

/ = current passing through motor, in amperes effective. 

Further, it is : 
Field magnetism : $ = n 710 8 / (R 

* That is, the main magnetic circuit of the motor. 

t That is, the magnetic circuit of the cross magnetization, produced by the armature 
reaction. 



COMMUTATOR MOTORS. 365 



Armature magnetism : 
Wj/10 8 

1 = "V"; 

Substituting these values, 



(R 
ptfNI 



E' = 

(R 

E 1 = ^^ n i NI . 
E r = (r + rj) / 
Thus the impressed E.M.F., 



or, since 

i,2 



x = 2 TT N^- = reactance of field ; 
(R 



2-n-jV = reactance of armature 
fti 



and 

/ 



, 



366 AL TERNA TING-CURRENT PHENOMENA. 

221. The power output at armature shaft is, 
J>= El 



\ (R 



(R 



fi- *Ef 

7T 7V^ 



/2 n N x _j_ r _^_ 
The displacement of phase between current and E.M.F. 



tan CD = 



Neglecting, as approximation, the resistances r + r lf it 

1 + |! 
lan W = ? j ^ 

7T / 7V 

^ n 2 



1+^' 

^ 



/ TV 



COMMUTATOR MOTORS. 367 

hence a maximum for, 




3r 

7T 

substituting this in tan w, it is : 

tan o> = 1, or, w = 45. 

222. Instance of such an alternating-current motor, 
^ = 100 A T =60 p = 2. 

r = .03 ri = .12 
x = .9 *! = .5 

n = 10 j = 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-turn armature in a bipolar field of 20 turns, at 
a ratio of 

armature ampere-turns r> A 

field ampere-turns 

It is in this case, 

100 



V(.023 vVi + ,15) 2 + 1.96 

230 ./v; 

(.023 A! + .15) 2 + 1.96 



368 



AL TERNA TING-CURRENT PHENOMENA. 



In Fig. 163 are given, with the speed N v as abscissae, 
the values of current /, power P, and power factor cos o> 
of this motor. 





















SER 


ES 


MO 


FOP 
































Er 


00 
















^ 


Vaf- 


3 












r = 

n= 


03 
.12 


=(, 
x 


_.y 

= .5 
















>->w 














N = 


60 


P= 


2 


0,, 


hi 












TUMI 












_x 


^ 










^~~~ 


^ 


< 


^ 










































2Stt> 










s 








V( 


J23 


] ( ~ 


QjF 


1.9 










gem 








/ 


































/ 
















1Z'_. 


NI 












-il(N) 






/ 












\'(. 


[23 


^, - 


-)- 


' 9 










>->00 




/ 


























* 










/ 






















| 










JIHIII 




/ 














V( 


).n 


^ ^ 


'SI-' 


1.9 




M 


Am 


P- 


1S'K> 


/ 




























u 


SO 




icon 


/ 






















cos 


>___ 





-sr 


_70 




ij 
















\ 


po* 


ev ^ 










70 


H 







' 





______ 






^J 


< 
















GO 


50 




1000 








^ 


>< ~~ 


*-. 


^_ <ii 
















I 


40 


/ 


sew 




X 


^ 














. 


_Jq 


ssa 


t 




40 


30 




GOO, 


x 




























;',o 


I 


>/ 


M 






























g 


% 




01 111 






























HI 


'oN 




no 




20 




30 




40 




50 




00 




Q 




B 






Fig. 163. Series Motor. 

223. The shunt motor with laminated field will not 
operate satisfactorily in an alternating-current circuit. It 
will 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. 



COMMUTATOR MOTORS. 369 

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 least causes vicious sparking when interrupted by the 
motion of the arm'ature. 

To overcome this difficulty various arrangements have 
been proposed, but have not found an application. 



370 ALTERNATING-CURRENT PHENOMENA. 

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 field 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 irJV/2 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. 



REACTION MACHINES. 371 



CHAPTER XXI. 

REACTION MACHINES. 

225. 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. E reduced to 
zero, and sometimes even if the field circuit is reversed 
and the counter E.M.F. E 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. E Q 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 



372 AL TERNA TING-CURRENT PHENOMENA. 

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 be 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 
can 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. 



REACTION MACHINES. 373 

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 -f- 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 = Ji jx, where 
h is what has been called the " effective hysteretic resis- 
tance." 

A similar phenomenon takes place in alternators of vari- 
able reactance, or what is the same, variable magnetic 
reluctance. 

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 wave of the main current. 
That is, in other words, the reluctance and reactance vary 
with twice the frequency of the alternating main current. 
Such a case is shown in Figs.. 164 and 165. The impressed 
E.M.F., and thus at negligible resistance, the counter E.M.F., 
is represented by the sine wave E, thus the magnetism pro- 
duced thereby is a sine wave 4>, 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 <l> is found from 3> and-^r 
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 



374 AL TERNA TING-CURRENT PHENOMENA. 































f, 


^ 










































/' 


\ 






































<p 




/ 


\ 


























^ 










^ 


^ 


> 




^ 




E 










X 




















/ 




/ 




i 




/ 




s 










\ 








/ 




\ 






/ 




/ 




i, 


A 








\ 






s 




V 


\ 












2 


^ 


s~~~ 


\^ 




// 




"\ 




s 


* 


^ 


\ 


^ 






\ 


> 


. 




}, 


^ / 


s 








y 






^ 


/ 








\ ^ 


\ 






\ 


1 







\ 


// 










i 




1 


\ 


\ 








\/ 


\\ 






/ 


\ 






i 


\ 








y 










\\ 








A 


Vs 




* 




\ 






I 




'^^ 




^ 


/ 












V 


_^- 


' 


\ 


\ 


k * s ' 


x^^ 






\ 




I 








/ 














\ 


^ 


s. 






\ 


N 


\ 






y 


I 






/ 


















\ 


\ 


k 






\ 


\ 




/ 




r\ 




S 




















\ 




\ 






N 


^_ 


\ 
































x 


b:S 












\ 












































\ 


I 










































V 


9 

































Fig. 164, Variable Reactance, Reaction Machine. 






Fig. 165. Variable Reactance, Reaction Machine. 



REACTION MACHINES. 



375 



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 
mqtor. In Figs. 166 and 167 are given the two hysteretic 
cycles or looped curves <J>, / 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 



- 



Fig. 166. Hysteretic Loop of Reaction Machine. 



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 



J76 



ALTERNA TING-CURRENT PHENOMENA. 



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 



Fig. 167. Hysteretic Loop of Reaction Machine. 



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., E Q , of the machine, it follows that, if the 
current is in phase or in quadrature with the E.M.F. E Q , 
the reluctance wave is symmetrical to the current wave, 
and the wave of magnetism therefore symmetrical to the 



REACTION MACHINES. 377 

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 will 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., Q . 

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. 



378 ALTERNATING-CURRENT PHENOMENA. 

231. 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 = a + b cos 2 /8. 

Let the inductance, or the coefficient of self-induction, 
be represented by 

L = I + < cos 2 /3 

= /(I + y COS 2 0) 

where y = amplitude of variation of inductance. 

Let 

u> = 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 V2 sin (/? - ). 

The magnetism produced by this current is, 



where n = number of turns. 
Hence, substituted, 



sin (/? - 5) (1 + y cos 2 0), 
or, expanded, 



n 
when neglecting the term of triple frequency, as wattless. 



REACTION MACHINES, 379 

Thus the E.M.F. induced by this magnetism is, 



hence, expanded 

e = - 2 TT 7W7 V2 !7 1 - 2\ cos cos /3 + /I + sn sn 

IV Z J \ 2 

and the effective value of E.M.F., 



l + 2 



= 2 TT NII\\ + - 7 cos 2 a. ^ 
Hence, the apparent power, or the voltamperes 



+ -J 2 y COS 2 u> 

The instantaneous value of power is 



2 sin(/? c(,)f/l |\ cos w cos y3 + 



sin eo sin /3 [. . 
7 

and, expanded 



sin 2 eo cos 2 /3 + sin 2 /3 ( cos 2 w 2 \ 1 
V 2 /J 

Integrated, the effective value of power is 



380 AL TERNA TING-CURRENT PHENOMENA. 

hence, negative, that is, the machine consumes electrical, 
and produces mechanical, power, as synchronous motor, if 
o> > ; that is, with lagging current; positive, that is, the 
machine produces electrical, and consumes mechanical, 
power, as generator, if to > ; that is, with leading current. 
The power factor is 

r j_ P_ _ y sin 2 ai 



hence, a maximum, if, 

d< 

or, expanded, 1 

cos2 = i 

The power, P, is a maximum at given current, /, if 

sin 2 w = 1 ; 
that is, 

to = 45 

at given E.M.F., E, the power is 
p= __ 



hence, a maximum at 
or, expanded, 



1 + 1T 



232. We have thus, at impressed E.M.F., E, and negli- 
gible resistance, if we denote the mean value of reactance, 

x=lTtNl. 
Current 



REACTION MACHINES. 381 

Voltamperes, 

k- 



Power, 

^g 2 y sin 2 



2^fl+^--ycos2 



Power factor, 

,. / 77 T-N y sin 2 to 
f = cos (E, /) = ' 

2 y/l + J^ _ y cos 2 A 
Maximum power at 



*+i 



Maximum power factor at 



to > : synchronous motor, with lagging current, 
w < : generator, with leading current. 

As an instance is shown in Fig. 168, with angle to as 
abscissae, the values of current, power, and power factor, 
for the constants, 

E = 110 

x = 3 

y =.8 

hence, j 41 

Vl.45 cos 2 
- 2017 sin 2w 



P = 



f= cos (E,I) 



1.45 cos 2 w 

.447 sin 2 G> 



As seen from Fig. 152, the power factor / of such a 
machine is very low does not exceed 40 per cent in this 
instance. 



382 



ALTERNA TING-CURRENT PHENOMENA. 




Fig. 188. Reaction Machine. 



DISTORTION OF WAVE-SHAPE. 383 



CHAPTER XXII. 

DISTORTION OF WAVE-SHAPE AND ITS CAUSES. 

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 
wave. 

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 E.M.F. produces 



384 ALTERNATING-CURRENT PHENOMENA. 

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 higher harmonics we 
have : 

Lack of Uniformity and Pulsation of tJie 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 : 

$ cos /?, 

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- 



DISTORTION OF WAVE-SHAPE. 385 

ing from the field pole of the alternator can be represented 
by the general equation, 

4> = A + A, cos /8 + A* cos 2(3 + A z cos 3/8 + . . . 
+ ^ sin + -# 2 sin 2 + .#, sin 3 ft + . . . 

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 , A z , A t . . . B y B equal zero ; 
that is, successive field poles are equal in strength and dis- 
tribution of magnetism, but of opposite 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, 
8 = $ - 3> 180 

= 2 \Ai cos /3 + A a cos 3 (3 + A 6 cos 5 + . . . 
+ BI sin j3 + B z sin 3 ft + jB 6 sin 5 ft + . . . \ 

Even with an unsymmetrical distribution of the magnetic 
flux in the air-gap, the E.M.F. wave induced in a full-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 



386 ALTERNATING-CURRENT PHENOMENA. 

sine shape, to within 3% 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 multitooth 
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 varying 
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, the instanta- 
neous magnetic flux interlinked with the armature con- 
ductors can be expressed by the equation : 

< = $ cos /? [1 + e cos [2 (3 o>] j 
where, = average magnetic flux, 

c = amplitude of pulsation, 
and to = phase of pulsation. 

In a machine with y slots per pole, the instantaneous flux 
interlinked with the armature conductors will be : 

<f> = & cos /8 { 1 + c cos [2 y ft o>] | , 

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 + 6! cos (2 - SO + e, cos (4 - oV,) + . . . f , 
where the term e y 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. 






DISTORTION OF WAVE-SHAPE. 



387 



Nil LoLd 



,"14 .5 y, 



Fig. 169. No-load 



of E.M.F. of Multitooth Three-phaser. 



130 


JMtfl I 


oad 






































120 '' 


= 12 


7.0 


= 


3 ; 










^ 


'-- 


--- 


>s, 




















110 
















j^ 










5 


















100 














/ 














\ 
















90 












j 


7 
















V 














SO 












/ 


















s 














70 










/ 






















s 












60 










/ 






















^ 












50 








/ 


























\, 










10 








// 


























'^ 


, 








30 






/' 






























\ 








20 






/ 
































\ 






10 




// 


































\\ 











'/ 


/- 


"--v^ 










r- 








1 ^.^ 










^ 


V 






10 




f ' 


10 


50 


30 


10 


g 


(50 


70 


SO 


90 


100 


no 


120 


13(1 


140 


150 


100 


170 


ISO 





Fig. 170. Full-Load Waue of E.M.F. of Multitooth Three-phaser. 



388 ALTERNATING-CURRENT PHENOMENA. 

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 ft {1 + e COS [2 y ft ]}. 

Hence the E.M.F. induced thereby : 

e = n 
dt 

d 

* 

And, expanded : 

e= V27rA^<fc{sin/?+e-^= sin[(2y 1) - J] 



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 (2 y 1) and 
(2 7+1). 

236. If y = 1 it is : 
e = V2 TT Nn <i> (sin /3 + 1 sin (0 ) + ^ sin (3 /? - )} ; 

that is : In a unitooth single-phaser a pronounced triple 
harmonic may be expected, but no pronounced higher 
harmonics. 

Fig. 171 shows the wave of E.M.F. of the main coil of 
a monocyclic alternator at no load, represented by : 

e = E (sin (3 .242 sin ( 3 /3 6.3) .046 sin (5/3- 2.6) 
+ .068 sin (7 3.3) .027 sin (9 ft 10.0) .018 sin 
(11 /3 - 6.6) + .029 sin (13 ft - 8.2)}; 

hence giving a pronounced triple harmonic only, as expected. 
If y = 2, it is : 

e = V2 TT Nn 4> j sin + ^ sin (3 ft - J) + |f sin (5 ft - Si) 



DISTORTION OF WAVE-SHAPE. 



389 



the no-load wave of a unitooth quarter-phase machine, hav- 
ing pronounced triple and quintuple harmonics. 
If 7 = 3, it is : 



in/3+ sin(5j8 fi) + sin (7 ft - S>) I . 



That is : In a unitooth three-phaser, a pronounced quin- 
tuple and septuple harmonic may be expected, but no pro- 
nounced triple harmonic. 



Fig. 155. No-load Wave of E.M.F. of Unitooth Monocyclic Alternator. 



Fig. 156 shows the wave of E.M.F. of a unitooth three- 
phaser at no load, represented by : 

e = E (sin /3 .12 sin (3 2.3) .23 sin (5 (3 1.5) + .134 sin 
(7 ft _ 6.2) - .002 sin (9 /3 + 27.7) - .046 sin (11 /? 
5.5) +.031 sin (13)8-61.5)}. 

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- 
tance. 



390 ALTERNATING-CURRENT PHENOMENA. 



V 



100 



50 60 70 80 90 1 00 



30 140 150 160 170 180 



Fig. 172. No-load Wave of E.M.F. of Unitooth Three-phase Alternator. 



In general, if the pulsation of the magnetic inductance 
is denoted by the general expression : 

l + ^"c Y cos(2 yj 8-a Y ), 
1 

the instantaneous magnetic flux is : 

00 



= $ cos 13 



e y cos (2 y ff - 



cos((2y+l) 



hence, the E.M.F. 



2 ; sm (P 



DISTORTION OF WAVE-SHAPE. 391 

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 
cycle. 

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 B, 



392 ALTERNATING-CURRENT PHENOMENA. 

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 



or, more general, 

X = 

the wave of magnetism is 



X = x 1 + yr ^ cos (2 y ft- & 

l 



hence the wave of induced E.M.F. 






= *sin/3 + sin ()8 - fflO + 




[e, sin ((2 y + 1) 
sin ((2y+ l)/8 -,+!)]} ; 

that is, the pulsation of reactance of frequency, 2y, intro- 
duces two higher harmonics of the order (2y 1), and 

(2y + l\ 

If ^T=^l 



, =*{sin0 + |sinG8-a) + .|l sin (3/J-o,)^ 

Since the pulsation of reactance due to magnetic satu- 
ration and hysteresis is essentially of the frequency, 21V, 



DISTORTION OF WAVE-SHAPE. 393 

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 
\ipon the angle, o>, 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 = Potential difference^between electrodes j g high 

for small currents, low for large currents. Thus in an 
alternating arc the apparent resistance will vary during 



304 ALTERNATING-CURRENT PHENOMENA. 

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- 
rent. 

Let the effective value of current passing through the 
arc be represented by /. 

Then the instantaneous value of current, assuming the 
current wave as sine wave, is represented by 

/ = 7V2sin/3; 

and the apparent resistance of the arc, in first approxima- 
tion, by 

R = r (1 + e cos 2 j8) ; 

thus the potential difference at the arc is 

e = iR = /V2Vsin/3(l -f e cos 2/3) 



Hence the effective value of potential difference, 



and the apparent resistance of the arc, 



r.-f-ry/t-. + f 

The instantaneous power consumed in the arc is, 



Hence the effective power, 



DISTORTION OF WAVE-SHAPE. 395 

The apparent power, or volt amperes consumed by the 
arc, is, 



thus the power factor of the arc, 




that is, less than unity. 

241. 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- 
ment, 

cos w =/ 

of an angle, w, whose sign is indefinite. 

As an instance are shown, in Fig. 173 for the constants, 

1= 12 

r= 3 

=.9 

the resistance, 

R = 3 {I + .9 cos 2 /3) ; 

the current, 

* = 17 sin /3 ; 

tha potential difference, 

e = 28 (sin ft + .82 sin 3 ). 
In this case the effective E.M.F. is 
=25.5; 



396 ALTERNATING-CURRENT PHENOMENA. 

the apparent resistance, 



the power, 



the apparent power, 



the power factor, 



r = 2.13 ; 
P = 244 ; 

El =307; 
/ = .796. 




Fig. 173. Periodically Varying Resistance. 

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 



DISTORTION OF WAVE-SHAPE. 



397 



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.* 



1 13 13 1 15 



ONE PAIR CARBONS 

EG U LATE D BY HAND 

A. C. dynamo e. m. f 

' " " current*. 

" " " watts. 



7 18 19 20 S 



Fig. 174. Electric Arc. 

243. The value of e, 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 f of the arc near 
unity. With hard carbons and an unsteady arc, e rises 
greatly, higher harmonics appear in the pulsation of resis- 
tance, and the power factor f 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, 1890, p- 
376. Tobey and Walbridge, on the Stanley Alternate Arc Dynamo. 



398 



A L TERN A TING-CURRENT PHENOMENA . 



CHAPTER XXIII. 

EFFECTS OF HIGHER HARMONICS. 

244. To elucidate the variation in the shape of alternat- 
ing waves caused by various harmonics, in Figs. 175 and 




Fig. 175. Effect of Triple Harmonic. 



176 are shown the wave-forms produced by the superposi- 
tion of the triple and the quintuple harmonic upon the 
fundamental sine wave. 






EFFECTS OF HIGHER HARMONICS. 399 

In Fig. 175 is shown the fundamental sine wave and 
the complex waves produced by the superposition of a triple 
harmonic of 30 per cent the amplitude of the fundamental, 
under the relative phase displacements of 0, 45, 90, 135, 
and 180, represented by the equations : 

sin ft 

sin ft .3 sin 3 ft 

sin ft- .3 sin (3/3-45) 

sin ft .3 sin (3 ft 90) 

s'm ft - .3 sin (3 ft - 135) 

sin ft .3 sin (3/3 180). 

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-top 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 : 

sin ft 

sin ft .2 sin 5 ft 
sin/3- .2 sin (5,8-45) 
sin/3- .2 sin (5/3-90) 
smft- .2 sin (5/3- 135) 
sin/3- .2 sin (5/8- 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 



400 



AL TERN A TING- CURRENT PHENOMENA. 



decreases, and the wave gradually changes to a triple- 
peaked wave with one main peak, and a sharp zero. 

As seen, with the triple harmonic, flat-top or double- 
peak coincides with sharp zero, while the quintuple har- 
monic flat-top or double-peak coincides with flat zero. 




Distortion of Wave Shapa 
by Quintuple Harmonfc 
Sin./S-.2sin.(5/?-S5j/ 







J 



\J 




Fig. 176. Effect of Quintuple Harmonic. 



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 harmonic 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. 



EFFECTS OF HIGHER HARMONICS. 



401 







Sin/?-.225 sinf3/?-180) , 
""-.05 sin/5/3-180) 



Sin./?- 15 sm.(3/?-180). 




Sin./?-. 15' sin 3/?-.1Q sir 
(5/J-180) 




f/jjr. 777. So/ne Characteristic Wave Shapes. 

Flat top with flat zero : 

sin /3 .15 sin 3 /3 .10 sin 5 0. 
Flat top with sharp zero : 

sin - .225 sin (3 /3 - 180) - .05 sin (5 /3 - 180). 
Double peak, with sharp zero : 

sin (3 - .15 sin (30- 180) - .10 sin 5 /?. 
Sharp peak with sharp zero : 

sin {3 .15 sin 3 .10 sin (5 (3 180). 

245. 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 



402 ALTERNATING-CURRENT PHENOMENA. 

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 difference 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-inductive part of the circuit more 
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 transformers at full load = 5% 

with the fundamental wave. 
The resistance drop in the line at full load = 10%. 



EFFECTS OF HIGHER HARMONICS. 403 

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, 

Ix = .15 , 

-IK E 

or, x = .15 , 

thus the reactance corresponding to the frequency (2/ 1) 
N of the higher harmonic is : 

x(2k- 1) =.15(2- 1) . 
The capacity current at fundamental frequency is : 



hence, at the frequency : (2 k 1) N: 

/ = .2(2-l)/Z, 
if: 

e' = E.M.F. of the (2 k l) th harmonic at the condenser, 

e = E.M.F. of the (2 k l) th harmonic at the generator terminals. 

The E.M.F. at the condenser is : 

e' = V* 2 i a r 2 + ix (2k V) 



404 AL TERNA TING-CURRENT PHENOMENA. 

hence, substituted : 



' l .059856 (2 k I) 2 + .0009 (2 k I) 4 

the rise of voltage by inductance and capacity. 
Substituting : 

k= 1 2 3 4 56 

or, 2 - 1 = 1 3 5 7 9 11 

it is, a = 1.03 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 
reduced. 

The maximum possible rise will take place for : 

= 0, or, 2,- 1 = 5.77 



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 

fundamental, 
the quintuple harmonic must be less than 26.5 per cent of the 

fundamental, 
the septuple harmonic must be less than 46 per cent of the 

fundamental. 

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 



EFFECTS OF HIGHER HARMONICS. 405 

than 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. 

247. 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. 
In 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 = $, the number of 
turns in series per circuit = n, the frequency = N, the 
E.M.F. between any two collector rings is: 

E= V2~7T^2;z<S>10- 8 . 

since 2 armature turns simultaneously interlink with the 
magnetic flux 3>. 

The E.M.F. per armature circuit is : 



hence the E.M.F. between collector rings, as resultant of 
two E.M.Fs. e displaced by 60 from each other, is : 



406 ALTERNATING-CURRENT PHENOMENA. 

while the same E.M.F. was found by direct calculation 
from number of turns, magnetic flux, and frequency to be 
equal to 2e; that is the two values found for the same 
E.M.F. have the proportion V3 : 2 = 1 : 1.154. 




Fig. 178. Three-phase Star-connected Alternator. 

This discrepancy is due to the existence of more pro- 
nounced higher harmonics in the wave e than in the wave 
E = e X V3, 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 diagrammatically in Fig. 
178 is only e x V3, 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. 



EFFECTS OF HIGHER HARMONICS. 407 

It follows herefrom that the distorted E.M.F. wave 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 efficiently. 

250. From another side the same problem can be 
approached. 

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 



408 ALTERNATING-CURRENT PHENOMENA. 

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 
transformer. 

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- 
netism. 

252. 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 



EFFECTS OF HIGHER HARMONICS. 409 

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 
ari 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. 
of very short duration. 

In general, 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. 



410 ALTERNATING-CURRENT PHENOMENA. 



CHAPTER XXIV. 

SYMBOLIC REPRESENTATION OF GENERAL 
ALTERNATING WAVES. 

253. The vector representation, 

A = a 1 +y<z u = a (cos a -\-j sin d) 
of the alternating wave, 

A a cos (< a) 

applies to the sine wave only. 

The general alternating wave, however, contains an in- 
finite series of terms, of odd frequencies, 

A = A l cos (< #1) 4- A z cos (3 < # 3 ) + A & cos (5 < # 5 ) -f 

thus cannot be directly represented by one complex vector 
quantity. 

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. 



REPRESENTATION OF ALTERNATING WAVES. 411 

This can be represented symbolically by combining in 
one formula symbolic representations of different frequen- 
cies, thus, 

00 

A = .-i (a* +j n */) 

i 
where, 

and the index of the/ M merely denotes that the/s of differ- 

entindices n, 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, 



if, 



is the impedance of the fundamental harmonic, where 

x m 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 c is that part of the reactance which is inversely pro- 

portional to the frequency (capacity, etc.). 

The impedance for the nth harmonic is, 



r Jnn x m 



This term can be considered as the general symbolic 
expression of the impedance of a circuit of general wave 
shape. 



412 ALTERNATING-CURRENT PHENOMENA. 

Ohm's law, in symbolic expression, assumes for the 
general alternating wave the form, 

/-Jo, 



E = IZ or, 



Z = or, 



Z = r - n 



The symbols of multiplication and division of the terms 
E, /, ^f, thus represent not algebraic operation, but multi- 
plication and division of corresponding terms of E, T, Z, 
that is, terms of the same index , or, in algebraic multipli- 
cation and division of the series E, /, all compound terms, 
that is terms containing two different w's, vanish. 

254. The effective value of the general wave : 
a = AI cos (< ,) + A s cos (3 < a 8 ) +^ 5 cos (5 <f> # 6 ) +. . 

is the square root of the sum of mean squares of individual 
harmonics, 

A= V i { A? + A 8 2 + A? + . . . | 

Since, as discussed above, the compound terms, of two 
different indices , vanish, the absolute value of the general 
alternating wave, 



REPRESENTATION OF ALTERNATING WAVES. 413 

is thus, 

A 



which offers an easy means of reduction from symbolic to 
absolute values. 

Thus, the absolute value of the E.M.F. 



s, 



the absolute value of the current, 



is, 



255. The double frequency power (torque, etc.) equa- 
tion of the general alternating wave has the same symbolic 
expression as with the sine wave : 



= P l +JPJ 



1 

where, 



41-4 ALTERNATING-CURRENT PHENOMENA. 

The j n enters under the summation sign of the " watt- 
less power " 1$, so that the wattless powers of the different 
harmonics cannot be algebraically added. 

i Thus, 

The total " true power" of a general alternating current 
circuit is the algebraic sum of the powers of the individual 
harmonics. 

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 
is, 



The power factor of the circuit is, 



The term "inductance factor," however, has no mean- 
ing any more, since the wattless powers of the different 
harmonics are not directly comparable. 

The quantity, 



,...._ ... wattless power 

has no physical significance, and is not = 

total apparent power 



REPRESENTATION OF ALTERNATING WAVES. 4] > 

The term, /#. 

El 

= 2/ n ~ 1 7 



where, 



consists of a series of inductance factors q n of the individual 
harmonics. 

As a rule, if <f = 2^-1 ^ n 2 , 



for the general alternating wave, that is q differs from 

fo =vr^7 2 

The complex quantity, 



Q El ~ El 



1 

takes in the circuit of the general alternating wave the 
same position as power factor and inductance factor with 
the sine wave. 

p 

17= -~ may be called the " circuit factor " 

It consists of a real term /, the power factor, and a 
series of imaginary terms j n q n , the inductance factors of 
the individual harmonics. 



416 ALTERNATING-CURRENT PHENOMENA. 

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, 

7 ( *C N 

Z = *./ \nx m -- 



that is, containing resistance r, inductive reactance x m and 
capacity reactance x c in series. 

Let 

e? = 720 ef = 540 

V = 283 4" = - 283 

e = - 104 * 6 " = 138 

or, 

^ = 900 tan e^ = .75 

*, = 400 tan o) 3 = - 1 

^ 5 = 173 tan w 5 = - 1.33 

It is thus in symbolic expression, 

Zj = 10 + 80/; *! = 80.6 

Z 3 = 10 z z = 10 

Z B = 10 - 32/; 2 5 = 33.5 

and, E.M.F., 

^ = (720 + 540/0 + (283 - 283y;) + (- 104 + 138/ 5 ) 

or absolute, 

E = 1000 



REPRESENTATION OF ALTERNATING WAVES. 417 



and current, 

_ _ 720 + 540/ t 283 - 283/ 8 - 104 + 138./; 
Z~~ 10 + 80/i " 10 10-32y 5 



= (7.76 - 8.04/i) + (28.3 - 28.3/ 8 ) + (- 4.86 - 1.73 A) 

or, absolute, 

7=41.85 

of which is of fundamental frequency, l l = 11.15 
" " " " triple " I 3 = 40 

quintuple " I 5 = 5.17 

The total apparent power of the circuit is, 

Q = 7=41,850 
The true power of the circuit is : 

/ = [7i 7] 1 = 1240 + 16,000 + 270 

= 17,510 
the wattless power, 

j PJ =/ [7i 7]J = 10,000^ - 850/ 6 
thus, the total power, 

P= 17,510 + 10,000/; - 850y 5 

That is, the wattless power of the first harmonic is 
leading, that of the third harmonic zero, and that of the fifth 
harmonic lagging. 

17,510 = I 2 r, as obvious. 
The circuit factor is, 



Q El 

= .418 + .239 j\ - .0203/5 

or, absolute, 



u = V.418 2 + .239 2 + .0203 2 
= .482 



The power factor is, 

p = .418 



418 ALTERNATING-CURRENT PHENOMENA. 

The inductance factor of the first harmonic is : q l = .239, 
that of the third harmonic ft = 0, and of the fifth harmonic 
ft = - -0203. 

Considering the waves as replaced by their equivalent 
sine waves, from the sine wave formula, 

f + qf = 1 
the inductance factor would be, 

ft = -914 
and the phase angle, 

tan a, = ^= '-^=2.8 = 65.4 

p .41o 

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 
E.M.F. 

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 symbolism of the complex harmonic 
wave. 



REPRESENTATION OF ALTERNATING WAVES. 419 

257. 2d instance: A condenser of capacity C = 20 
m.f. is connected into the circuit of a 60-cycle alternator 
giving a wave of the form, 

e = E (cos < - .10 cos 3 < - .08 cos 5 <f> + .06 cos 7 <) 
or, in symbolic expression, 

= e(! 1 - .10, - .08 5 + .06 7 ) 
The synchronous impedance of the alternator is, 
ZQ = r j n nx = .3 5 nj n 

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, 
10 6 



or, in symbolic expression, 

Let 

Z^ =.r j n nv = impedance inserted in series with the 
condenser. 

The total impedance of the circuit is then, 

n 
The current in the circuit is, 



(.3 + r) - j (x - 132) (.3 + r) -j 3 (3 x - 29) 

^8 ^6 -j 

(.3 + r) -j, (5x- 1.4) (.3 + r) -j\(7x + 16.1)J 



420 ALTERNATING-CURRENT PHENOMENA. 

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



; 
Jn V 

4.4 j s 



(.3 + r) -A (x - 132) (.3 + r) - j z (3 * - 29) 

__ 2.iiy 5 1.13;; -i 

(.3 + r) -j 6 (5x- 1.4) ^ (.3 + r) -/ 7 (7 x + 16.1) J 
thus the apparent capacity reactance of the condenser is, 



and the apparent capacity, 

10 6 



^.) ^r = : Resistance r in series with the condenser. 
Reduced to absolute values, it is, 

1 .01 .0064 .0036 



17424 19.4 



(.8+r) a + 17424 (.3 +r) 2 + 841 (.3 + r) 2 + 1.96 (.3 -f r) 2 +2 

(.) r = : Inductive reactance x in series with the 
condenser. Reduced to absolute values, it is, 

1 .01 .0064 __ .0036 

1.42 "*". 



1.4)2 .09+(7;r-f 16. 



132)2 . 

From g are derived the values of apparent capacity, 



c= 



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, 



REPRESENTATION OF ALTERNATING WAVES. 421 

and gradually decreases with increasing series resistance, to 
C= 27.5 m.f. = 1.375 times the true capacity at r= 13.2 
ohms, or T V the true capacity reactance, with r = 132 ohms, 
or with an additional resistance equal to the capacity reac- 
tance, C = 20.5 m.f. or only 2.5% in excess of the true 
capacity C , and at r = oo , C = 20,3 m.f. or 1.5% in excess 
of the true capacity. 

With reactances, but no additional resistance r in series, 
the apparent capacity C rises from 4.2 times the true 
capacity at x = 0, to a maximum of 5,03 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 35 % in excess of the true capacity, rises again 
to 60.2 m.f., or 3.01 times the true capacity at x = 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 x = 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. 



422 



ALTERNATING-CURRENT PHENOMENA. 



258. d instance : An alternating current generator 

of the wave, 

E. = 2000 [l t + .12, - .23 B - .13,] 

and of synchronous impedance, 

Z = .3-5*/; 
feeds over a line of impedance, 







































































































































































































































































































































C4PJ 


CITV 


Co = 


= 20 


mf i 


CM 


CL'IT 


OF 


r,E\ 


HAT 


R 








1 
















8 


= E I O-J--I.L .y a -t- uc / OF 

Zo^S-S), n WITH RESIS 


fASC 


DANCE 

k r(I) 








! 


















c 


R RE 


ACT 


NCE 


*^ 


I) 1 


SE 


!ES 










C: 













































100 


/\ 











































90 


J 


















i 
























^ft 


I 


k 
















5 
























rn 


I 


\ 


\ 














i 
























H 








^ 










/ 




\ 






















.w 




\ 






\ 








/ 






X 




















10 




REE 


X 

STAC 


ii 

^=^~ 

CE r 


= 


;=" 


^ 


^ 


= 

REA( 


- 
TAN! 1 






X 

- 


^S 


^ 


* , 




; 





= 


^= 


_ 

=3<F 

10 






I ; 




, 






i !'o ! 


1 


2 1 


1 


1 


1 


1 


- 1 


-t 1 


r, 2, 


1 






a synchronous motor of the wave, 

EI = 2250 [(cos oj +/i sin ) + .24 (cos 3 w -(-y's sin 3 o>)] 
and of synchronous impedance, 

Z 2 = .3 - C /; 

The total impedance of the system is then, 
Z = Z Q + Z l + Z 2 
= 2.6-15/ n 



REPRESENTATION OF ALTERNATING WAVES. 423 

thus the current, 



_ 2000 - 2250 cos o> - 2250/\ sin o> 240 - 540 cos 3a> - 540/; sin 3a> 
2.6 - 15/i 2.6 - 45y 8 

460 260 

~~ 2.6 - 75 j\ 2.6 - 105 jj 

= 

where, 

aj 1 = 22.5 - 25.2 cos co + 146 sin a> 

ag 1 = .306 - .69 cos 3 to + 11.9 sin 3 

a, 1 = - .213 

7 i = - .061 

V 1 = 130 - 146 cos w - 25.2 sin a> 

^ 8 = 5.3 - 11.9 cos 3 o> - .69 sin 3 o> 

a* = - 6.12 

a 7 u = - 2.48 

or, absolute, 

1st harmonic, 

3d harmonic, 

5th harmonic, 

a 6 = 6.12 
7th harmonic, 

7 = 2.48 




/= V 
while the total current of higher harmonics is, 



424 ALTERNATING-CURRENT PHENOMENA. 

The true input of the synchronous motor is, 



= ( 2250 a cos o> + 2250 a? sin o> ) + ( 540 a? cos 3o> + 540 a s n sin 3o>) 

= /V + /'s 1 
^ = 2250 (a? cos <o + af sin o>) 




. 780. Synchronous Motor, 



REPRESENTATION OF ALTERNATING WAVES. 425 

is the power of the fundamental wave, 

P = 540 (a,, 1 cos 3 w + a s 11 sin 3 o>) 

the power of the third harmonic. 

The 5th and 7th harmonics do not give any power, 
since they are not contained in the synchronous motor 
wave. Substituting now different numerical values for u> 
the phase angle between generator E.M.F. and synchronous 
motor counter E.M.F., corresponding values of the currents 
/ 7 , and the powers P\ P*, /Y 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 P\ 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 I , Curve IV., 
the difference between the total current and the current of 
fundamental frequency, / a lt in percentage of the total 
current /, and V the power of the third harmonic, Pj, in 
percentage of the total power P 1 . 

Curves III., IV. and V. correspond to the positive or 
synchronous motor part of the power curve P\ 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 
negative. 

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. 



426 ALTERNATING-CURRENT PHENOMENA. 

259. kth Instance: In a small three-phase induction 
motor, the constants per delta circuit are 

Primary admittance Y= .002 + .03/ 

Self-inductive impedance Z Q = Z l = .6 2.4/ 

and a sine wave of E.M.F. e = 110 volts is impressed upon 
the motor. 

The power output P, current input 7 S , 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 
susceptance 



In this case, let I s = 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 

/i = l s - 4.28/ 

energy current 
and heref rom the power factor = - ; , given in 

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 

E, = e, (l x + .12. - .23 5 - .134,) 

to give the same output, the fundamental wave must be the 
same : e = 110 volts, when assuming the higher harmonics 
in the motor as wattless, that is 

= 110, + 13.2, - 25.3 B - 14.7, 

= *o + <? 
where l = 13.2, - 25.3 B - 14.7 T 

= component of impressed E.M.F. of higher frequency- 






REPRESENTATION^ Of ALTERNATING WAVES. 427 

The effective value is : 

E Q = 114.5 volts. 

The condenser admittance for the general alternating 
wave is 

Y c = -.039/; 

Since the frequency of rotation of the motor is very 
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 

Z 1 = - nj n (X Q + XJ 

= - 4.8 nj n 

The exciting admittance of the motor, for these higher 
harmonics, is, by neglecting the conductance, 



n 
and the higher harmonics of counter E.M.F. 



Thus we have, 



Current input in the condenser, 

fc = E, Y c 

= - 4.28/i - 1.54/3 + 4.93/ 5 + 4.02/ 7 

High frequency component of motor impedance current, 

| = .92/3 - 1.06y 5 - .44/ 7 
High frequency component of motor exciting current, 



= .07/3 - -08/ 5 - . 



428 



AL TERN A TING-CURRENT PHEA'OAIENA. 



thus, total high frequency component of motor current, 

/o 1 = |f + & y 1 

= .99y 3 - 1.14,; - .47/ 7 
and total current, 

without condenser, 

4 = 4 + 4 1 

= I s + .99/ 3 - 1.14,; - .47/ 7 

with condenser, 



= 4 - 4.28,i - . 
and herefrom the power factor. 



3.79,; + 3.55/ 7 




T PER PHASE 



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 sine wave of E.M.F , of 1 10 volts, and with condenser. 

III. With distorted wave of E.M.F., of 114.6 volts, and no condenser. 

IV. With distorted wave of E.M.F., of 114.5 volts, and with condenser. 



REPRESENTATION OF ALTERNATING WAVES. 429 



f 

.01 
.02 
.035 
.05 
.07 
.10 
.13 
.15 


P 


160 
320 
500 
660 
810 
885 
900 
890 


I, 
.24+ 3.10/ 
1.73+ 3.16/ 
3.32+ 3.47> 
5.16+ 4.28/ 
6.95+ 5.4/ 
8.77+ 7.3; 
10.1 + 9.85/ 
10.45 + 11.45/ 
10.75 + 12.9/ 


It 
3.1 
3.6 
4.8 
6.7 
8.8 
11.4 
14.1 
15.5 
16.8 


7.8 
48 
69 
77 
79 
77 
71.5 
67.5 
64 


f 

1.2 
2.1 
3.4 
5.2 
7.0 
9.3 
11.5 
12.7 
13.8 


> 
P 
20 
84 
97.2 
100 
98.7 
94.5 
87 
82 
78 


i 

3.5 
3.9 
5.1 
6.9 
8.9 
11.5 
14.2 
15.6 
16.9 


1 \ 

6.6 
43 
64 
72.5 
76 
73.5 
68 
64.5 
61 


/ 
I 

5.2 
5.5 
6.1 
7.2 
8.6 
10.6 
12.6 
13.7 
14.7 


i 

4. 
81 
64 

(18 
7T 
80 
7T 
73: 
7Q/ 



The curves II. and IV. with condenser are plotted in 
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 using 
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 power 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. 



430 ALTERNATING-CURRENT PHENOMENA. 



CHAPTER XXV. 

GENERAL POLYPHASE SYSTEMS. 

260. A polyphase system is an 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 
independently. 

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 n equal E.M.Fs. 
displaced from each other by 1 / n of a period, the system 
is called a symmetrical system, otherwise an unsymmetrical 
system. 

Thus the three-phase system, consisting of three equal 
E.M.Fs. displaced by one-third of a period, is a symmetrical 
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 



GENERAL POLYPHASE SYSTEMS, 431 

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. 




Fig. 181. 





Fig. 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 diagrammatically in Figs. 181 
and 182, is an interlinked system. 



432 



ALTERNATING-CURRENT PHENOMENA. 



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. 183, 




Fig. 183. 

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. 



-E 



r 



Fig. 184. 

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- 



GENERAL POLYPHASE SYSTEMS. 433 

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. 



434 ALTERNATING-CURRENT PHENOMENA. 



CHAPTER XXVI. 

SYMMETRICAL POLYPHASE SYSTEMS. 

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 w-phase system is a system of n 
E.M.Fs. of equal intensity, differing from each other in 
phase by 1 / n of a period : 

*i = E sin (3 ; 
e 2 =sm((3-^L\', 



e n = E sin ( ft - L V* ~ - 
\ 

The next E.M.F. is again : 

^ = E sin (ft 2 TT) = E sin ft. 

In the polar diagram the n E.M.Fs. of the symmetrical 
0-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 2ir/n, is represented by multiplication with : 



the E.M.Fs. of the symmetrical polyphase system are: 



SYMMETRICAL POLYPHASE SYSTEMS. 435 



/ 9 T- ? -rr 

E( cos + / sin = 
' 



n 

f 2 (n 1) TT . . . 2 ( 1) 
^ f cos -i - L -- \-j sm ^ - ^ 

' V 

The next E.M.F. is again : 

E ( cos 2 -n- +j sin 2 TT) = . e" = .. 
Hence, it is 

27T . . 27T n /? 

e = cos - - -f J sm - = V 1. 
;z 

Or in other words : 

In a symmetrical -phase system any E.M.F. of the 
system is expressed by : 

e'-Ej 

where : e = -y/1. 

264. Substituting now for n different values, we get 
the different symmetrical polyphase systems, represented by 

*E\ 

, n/T 2 7T . . 2 7T 

where, e = vl = cos -- \-j sin . 

n n 

1.) = 1 e = 1 c'^ = ., 
the ordinary single-phase system. 

2.) = 2 e = - 1 J = and - . 

Since ^ is the return of E, n = 2 gives again the 
single-phase system. 



3 
-1-/V3 



436 ALTERNATING-CURRENT PHENOMENA. 

The three E.M.Fs. of the three-phase system are : 



-i-yV3 



Consequently the three-phase system is the lowest sym- 
metrical polyphase system. 

4.) n = 4, c = cos +/ sin =/, 2 = 1, e 3 = - /. 
4 4 

The four E.M.Fs. of the four-phase system are: 

* = , J, -E, -JE. 
They are in pairs opposite to each other : 
E and E j E 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- 
phase system proposed for the same purpose. 

265. A characteristic feature of the symmetrical - 
phase system is that under certain conditions it can pro- 
duce a M.M.F. of constant intensity. 

If 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. 

Let, 
n' = number of turns of each magnetizing coil. 



SYMMETRICAL POLYPHASE SYSTEMS. 437 

E= effective value of impressed E.M.F. 
/ = effective value of current. 

Hence, 
& =n'f= 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 2 *'/ is : 



The two rectangular space components of this M.M.F. are ; 



and 



Hence the M.M.F. of this coil can be expressed by the 
symbolic formula : 



fi 

n \ n 

Thus the total or resultant M.M.F. of the n coils dis- 
placed under the n equal angles is : 



or, expanded : 



n 



438 ALTERNATING-CURRENT PHENOMENA. 

It is, however : 



cos' 2 + / sin cos = ( 1 + cos +/ sin ] 

n n n V w w / 

\ / 

sin 2=1 cos ?Z+ysin2== ^Yl - cos i^'-ysin 4 ^' 

z y 

_ ^ /I _ ,2A X 



2(1-^ 

and, since: 

5t< 2< = 0, 

it is, /= nn ' f ^ (- sin ft _ y cos ft), 

or, 



the symbolic expression of the M.M.F. produced by the 
circuits of the symmetrical -phase system, when exciting 
n equal magnetizing coils displaced in space under equal 
angles. 

The absolute value of this M.M.F. is : 

nn' I n"S n < 5 



V2 V2 2 

Hence constant and equal w/V2 times the effective 
M.M.F. of each coil or /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 ft is : 

tan w = cot /8 ; hence w = /? ^ 

That is, the M.M.F. produced by a symmetrical -phase 
system revolves with constant intensity : 



SYMMETRICAL POLYPHASE SYSTEMS. 439 

F= 

V2 5 

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 w-phase system. 

This is a characteristic feature of the symmetrical poly- 
phase system. 

266. In the three-phase system, n = 3, F= 1.5 < 5 max 
where $ max is the maximum M.M.F. of each of the magne- 
tizing coils. 

In a symmetrical quarter-phase system, n = 4, F = 2 
^ tnax , where $ maje is the maximum M.M.F. of each of the 
four 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 ^ max 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. 



440 ALTERNATING-CURRENT PHENOMENA. 



CHAPTER XXVII. 

BALANCED AND UNBALANCED POLYPHASE SYSTEMS. 

267. If an alternating E.M.F. : 

e = E V2 sin (3, 
produces a current : 

* = 7V2sin (/? a), 

where u> is the angle of lag, the power is : 

p = ei = 2 Ssin ft sin (ft S) 

= S(cos a cos (2 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 is 

constant : 

/-** 

If the angle of lag = it is : 

p = P (1 cos 2 0) ; 

hence the flow of power varies between zero and 2 P t where 
P is the average flow of energy or the effective power of 
the circuit. 



BALANCED POLYPHASE SYSTEMS. 441 

If the current lags or leads the E.M.F. by angle the 
power varies between 

and 



cos u> 

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 = 90, it is : 

O rt , 

" p > 



that is, the effective power : P = 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 

*D e z> *s> = instantaneous values of E.M.F. ; 
h) *2, t'a, = instantaneous values of current pro- 
duced thereby ; 

the total flow of power in the system is : 

p = g lt \ -f <?2/2 -j- e,j, -f . . . . 
The average flow of power is : 

P = i /i cos >! -(- E<i /2 cos w 2 -f- . . . . 

The polyphase system is called a balanced system, if the 
flow of energy : 

/ = e\i\ + <V 2 + W, +.'.'.;.. 

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. 



442 ALTERNATING-CURRENT PHENOMENA. 

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. All the symmetrical systems from the three-phase 
system upward are balanced systems. Many unsymmetrical 
systems are balanced systems also. 

1.) Three-phase system : 
Let 

^ = E V2 sin ft, and t \ = I V2 sin (ft w) ; 

e z = E V2 sin (ft - 120), / 2 = / V2 sin (0 - - 120) ; 

e z = E V2 sin (ft - 240), / 3 = / V2 sin (ft - & - 240) ; 

be the E.M.Fs. of a three-phase system, and the currents 
produced thereby. 

Then the total flow of power is : 

/ = 2 .57 (sin {3 sin (ft fi) + sin ((3 120) sin (ft & 120) 

+ sin (ft 240) sin ($ <* 240)) 
= 3 .7 cos w = T 5 , or constant. 

Hence the symmetrical three-phase system is a balanced 
system. 

2.) Quarter-phase system : 

Let l = ^2s\nft, t\ = I \/2 sin (ft - 5) ; 

e 2 = E V2 cos ft, 4 = 7 V2 cos (ft - ) ; 



BALANCED POLYPHASE SYSTEMS. 443 

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 I(sm J3 sin (/? 5) + cos ft cos (0 >)) 
= 2 Scos w = P, or constant. 

Hence the quarter-phase system is an unsymmetrical bal- 
anced system. 

3.) The symmetrical -phase system, with equal load 
and equal phase displacement in all n branches, is a bal- 
anced system. For, let : 

e ( = E V2 sin ( ft - "\ = E.M.F. ; 
V / 

/ 2 IT A 

*',- = 7V2 sin O S = current 

V V 

the instantaneous flow of power is : 



l V 7 \ 

EI \ yr cos a -57-035^2 /?-- 



or p = n E I cos w = T 7 , or 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- 

* Also called "polyphase monocyclic system," since the E.M.F. triangle is similar to 
that usual in the single-phase monocyclic system. 



444 ALTERNATING-CURRENT PHENOMENA. 

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 branch 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 .333. 

272. In Figs. 185 to 192 are shown the E.M.Fs. as 
e and currents as i in drawn lines, and the power as / in 
dotted lines, for : 




Fig. 185. Single-phase System on Non-inductive Load. 

Balance Factor, 0. 



BALANCED POLYPHASE SYSTEMS. 445 




Fig. 186. Single-phase System on Inductiue Load of 60 Lag. 

Balance Factor, - .333. 






Fig. 187. Quarter-phase System on Non-inductiui Load. 

Balance Factor, + 1. 





Fig. 183. Quarter-phase System on Inductiue Lozd of 60 Lag. 
Balance Factor, + 1. 



446 ALTERNATING-CURRENT PHENOMENA. 





Fig. 189. Three-phase System on Non-induct'we Load. 

Balance Factor, + 1. 




Fig. 190. Three-phase System on Inductive Load of 60 Lag. 

Balance Factor, + 1. 



Fig. 191. Inverted Three-phase System 
on Non-inductive Load. 




Balance Factor, + .333 



BALANCED POLYPHASE SYSTEMS. 



447 




Fig. 174. Inverted Three-phase System on 

Inductive Load of 60 Lag. 

Balance Factor, 0. 

273. The flow of power in an alternating-current 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 
period 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. 



448 ALTERNATING-CURRENT PHENOMENA. 

Since the flow of power in any single-phase branch of 
the alternating-current system can be represented by a sine 
wave of double frequency : 



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 : 

/ = />(! + sin (2 - a.)) 

This is a wave of double frequency also, with c as ampli- 
tude of fluctuation of power. 

This is the equation of 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 o> o /2, so as to make the symmetry 
axes of the power characteristic the coordinate axes. 




hence, sin (2 ft - S> ) = 2 sin ^ - ^ ) cos (/? - ^ j = 
substituted, 

^M' + ^j. 

or, expanded : 

P 2 (x 2 + /* + 2 e A:^) 2 = 0, 



the sextic equation of the power characteristic. 
Introducing : 

a = (! + )/'= maximum value of power, 
b = (1 c) P'= minimum value of power; 



BALANCED POLYPHASE SYSTEMS. 449 

it is **?> 



a + b 
hence, substituted, and expanded : 

(*+/) - \{a (x + j) 2 + b (x -X>T> = 

the equation of the power characteristic, with the main 
power axes a and b, and the balance factor: b I a. 
It is thus : 

Single-phase non-inductive circuit : / = /> (1 + sin 2 <), 
b = 0, a = 2P 



Single-phase circuit, 60 lag : / = P (1 + 2 sin 2 <), 

i*.~ + " 



Single-phase circuit, 90 lag :/ = ^ /sin 2 <, b = E I, 

a = + El 



2 /, &/a= -1. 
Three-phase non-inductive circuit : p = P, ^ = 1, a = 

x^+y* P 2 = 0: circle. & / a = + 1. 
Three-phase circuit, 60 lag : / = P, 6 = 1, a = 1 

a? +/- 7> a = : circle. /= + !. 
Quarter-phase non-inductive circuit :p = P,b = ]-) a = 

x * _|_ y _ ^2 = o . circlei ^ / ^ = _|_ i. 

Quarter-phase circuit, 60 lag : p = P, b = 1, tf = 1 



450 ALTERNATING-CURRENT PHENOMENA. 

Inverted three-phase non-inductive circuit : 



Inverted three-phase circuit 60 lag :/ = f (1 -\- sin 2 <), 
b = 0, a = 2 P 

(y? + /)3 _ />2 ( x _|_ y y = 0< fil a = Q f 

a and <5 are called the main power axes of the alternating- 
current system, and the ratio b [a is the balance factor of 
the system. 



Figs. 193 and 104. Power Characteristic of Single-phase System, at 60 and Lag. 

276. As seen, the flow of power of an alternating-cur- 
rent system is completely characterized by its two main 
power axes a and b. 

The power characteristics in polar coordinates, corre- 



BALANCED POLYPHASE SYSTEM. 



451 



spending to the Figs. 185, 186, 191, and 192 are shown in 
Figs. 193, 194, 195, and 196. 




Figs. 195 and 196. Power Characteristic of Inverted Three-phase System, at and 
60 Lag. 

The balanced quarter-phase and three-phase systems give 
as polar characteristics concentric circles. 



452 ALTERNATING-CURRENT PHENOMENA. 



CHAPTER XXVIII. 

INTERLINKED POLYPHASE SYSTEMS. 

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. 



INTERLINKED POLYPHASE SYSTEMS. 



453 



1st. The star connection, represented diagrammatically 
in Fig. 197. In this connection the n circuits excited by 
currents differing from each other by 1 / n 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- 
lated. 

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. 




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 corners 
or points of connection of adjacent circuits connected to 
the n 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 



454 ALTERNATING-CURRENT PHENOMENA. 




Fig. 198. 



in three-phase systems Y- connected and delta-connected 
apparatus. 

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 n 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 n 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. 



INTERLINKED POLYPHASE SYSTEMS. 455 

280. In the same way as we speak of star connection 
and ring connection of 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- 
tion. 

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 potential 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-connected circuits, the 
E.M.F. per circuit = E, and the common connection or 
neutral point is denoted by zero, the potentials of the n 
terminals are : 



or in general : t* JS, 

at the z' th terminal, where : 

* = 0, 1, 2 ....- 1, e = cos +j sin = -\/l. 



456 ALTERNATING-CURRENT PHENOMENA. 

Hence the E.M.F. in the circuit from the z th to the * 
terminal is : 

E ki = ** E ^E = (c* e') E. 

The E.M.F. between adjacent terminals i and i + 1 is : 

( e .+i -J)E = e* (e - 1) E. 

In a generator with ring-connected circuits, the E.M.F. 

per circuit : 

c l E 

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., IE. 

RingE-M.F., c'Xc-1)^. 

E.M.F. between terminal / and terminal k, (c* e') E. 

In a ring-connected generator with the E.M.F. E per 
circuit, it is : 

Star E.M.F., ^ E. 
e 1 ' 

Ring E.M.F., C E. 

E.M.F. between terminals * and k, e ~ e * E. 

1 ' 

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 E.M.F. and ring 
current. 

In the generator of a symmetrical polyphase system, if : 
c'' E are the E.M.Fs. between the n terminals and the 
neutral point, or star E.M.Fs., 






INTERLINKED POLYPHASE SYSTEMS. 457 

If = the currents issuing from terminal i over a line of 
the impedance Z { (including generator impedance in star 
connection), we have : 

Potential at end of line i : 



Difference of potential between terminals k and i : 



where /,. is the star current of the system, Z t the star im- 
pedance. 

The ring potential at the end of the line between ter- 
minals i and k is E ik , and it is : 

E ile = E ti . 

If now I ik denotes the current passing from terminal i to 
terminal k, and Z ik impedance of the circuit between ter- 
minal i and terminal k, where : 

fit = ~ /*,, 
Zt* = Z ti , 

it is E ik = Z it I ik . 

If I io denotes the current passing from terminal i to a 
ground or neutral point, and Z io is the impedance of this 
circuit between terminal i and neutral point, it is : 

E io = *- ZiSi = Z io l io . 

282. We have thus, by Ohm's law and Kirchhoff 's law : 

If *' E is the E.M.F. per circuit of the generator, be- 
tween the terminal i and the neutral point of the generator, 
or the star E.M.F. 

/,- = the current issuing from the terminal i of the gen- 
erator, or the star current. 

Z t = the impedance of the line connected to a terminal 
i of the generator, including generator impedance. 

E L = the E.M.F. at the end of line connected to a ter- 
minal i of the generator. 



458 ALTERNATING-CURRENT PHENOMENA. 

E ik = the difference of potential between the ends of 
the lines i and k. 

I ik = the current passing from line i to line k. 

Z ik = the impedance of the circuit between lines i and k. 

I io , I ioo . . . . = the current passing from line i to neu- 
tral points 0, 00, .... 

Z io , Z ioo . . . . = the impedance of the circuits between 
line i and neutral points 0, 00, .... 

It is then : 



Z io = Z oi , etc. 
2.) E t =JE-Z i I i . 

3.) Ei = Zi fi = Z ioo fj 00 = . . . . 

4.) E ik = E t '- E { = (t* - e') E - (Z k l k - ZJ^). 

5.) E ik = Z ik I ik . 



7.) If the neutral point of the generator does not exist, 
as in ring connection, or is insulated from the other neutral 
points : 



IE/,, =0; 

n 

5E/ ioo = 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: 



INTERLINKED POLYPHASE SYSTEMS. 459 

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 



P = ^f e 1 ' E Ii cos $i at the generator 

1 

f = "^i ^* E ik I ik cos <f> it in the receiving circuits. 



4GO ALTERNATING-CURRENT PHENOMENA. 



CHAPTER XXIX. 

TRANSFORMATION OF POLYPHASE SYSTEMS. 

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 of 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 lt E 2 , E z . . . . be the E.M.Fs. of the 
primary system which shall be transformed into 

E{, 2 ', s ' . . . . the E.M.Fs. of the secondary 
system. 

Choosing two magnetic fluxes, < and <, of different 



TRANSFORMATION OF POLYPHASE SYSTEMS, 461 

phases, as magnetic circuits of the two transformers, which 
induce the E.M.Fs., e and ?, per turn, by the law of paral- 
lelogram the E.M.Fs., E lf E^, . . . . can be dissolved into 
two components, E l and E lt E^ and E z , .... of the phases* 
"e and J. 
Then, - 

E!, 2 , ' are the counter E.M.Fs. which have to be- 
induced in the primary circuits of the first transformer;. 

E v E 2 , .... the counter E.M.F.'s which have to be in- 
duced in the primary circuits of the second transformer.. 

hence 

EI 1 7, 2 1 J . . . . are the numbers of turns of the primary 
coils of the first transformer. 

Analogously 

EI /T 2 IT . . . . 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 
J and FJ the E.M.Fs. of the secondary system, E-^> E^, .... 

are produced from components, E-f and E^, E and EJ, 
.... in phase with ~e and J, and give as numbers of second 
ary turns, 

i l / J, 2 l /? in the first transformer ; 
EI 1 7, EZ / F, .... in the second transformer. 

That means each of the two transformers m 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 in 
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 



4j^ ALTERNATING-CURRENT PHENOMENA. 

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 

E sin ft, E sin (ft 120), E sin (ft 240), 
in an unsymmetrical balanced quarter-phase system : 

E' sin ft, E' sin (ft 90). 
Let the magnetic flux of the two transformers be 

(/> cos and </> cos (ft 90). 

Then the E.M.Fs. induced per turn in the transformers 
e sin ft and e sin (ft 90) ; 

hence, in the primary circuit the first phase, E sin ft, will 
give, in the first transformer, E/e primary turns; in the 
second transformer, primary turns. 

The second phase, E sin (ft 120), will give, in the 
first transformer, E / 2 e primary turns; in the second 

E x ~\/3 

transformer, primary turns. 

2 e 

The third phase, E sin (ft 240), will give, in the first 
transformer, E /le primary turns; in the second trans- 
former, primary turns. 

2 e 

In the secondary circuit the first phase E' sin ft will give 
in the first transformer: E' / e secondary turns; in the 
second transformer : secondary turns. 

The second phase : E' sin (ft 90) will give in the first 
transformer : secondary turns ; in the second transformer, 
E' I e secondary turns. 

Or, if : 

E = 5,000 E' = 100, e = 10. 



TRANSFORMATION OF POLYPHASE SYSTEMS. 463 



PRIMARY. 
1st. 2d. 



SECONDARY. 
3d. 1st. 2d. Phase. 



first transformer 
second transformer 



+ 500 




- 250 - 250 

4- 433 - 433 



10 





10 turns. 



That means : 

Any balanced polyphase system *.jm 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. 799. 

1. The delta -Y connection of transformers between 
three-phase systems, shown in Fig. 199. One side of the 
transformers is connected in delta, the other in Y. This 
arrangement becomes necessary for feeding four wires 





rwi nnr 
V 



Fig. 200. 



three-phase secondary distributions. The Y 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. 



464 



ALTERNA TING-CURRENT PHENOMENA. 



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. 




Fig. 201. 

3. The main and teaser, or T connection of trans- 
formers between three-phase systems, as shown in Fig. 201. 

V3 
One of the two transformers is wound for ~-~- times the 

voltage of the other (the altitude of the equilateral triangle), 
and connected with one of its ends to the center of the 




Fig. 202. 

other transformer. From the point inside of the teaser 
transformer, a neutral wire can be brought out in this con- 
nection. 

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. 



TRANSFORMATION OF POLYPHASE SYSTEMS. 465 



5. 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 




Fig. 203. 

independent (or interlinked) coils, the three-phase side two 

Vs 

coils with the ratio of turns 1 -=- ^ connected in T. 

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 




Fig 204. 



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. 



466 



AL TERN A TING-CURRENT PHENOMENA. 



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 




Fig. 205. 

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 






\ 

\ 


/ 

/ 


y 

/ \ 


y 


2 ' v ' 




Fig. 208. 



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 



TRANSPORMATION OF POLYPHASE SYSTEMS. 467 

the secondary. The most efficient storing device of electric 
energy is mechanical momentum in revolving 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. 



468 ALTERNATING-CURRENT PHENOMENA. 



CHAPTER XXX. 

EFFICIENCY OF SYSTEMS. 

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. 
used. 

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 



EFFICIENCY OF SYSTEMS. 409 

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 potential 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 
phase. 

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 the 
minimum difference of potential, in low potential lighting 
circuits : 



4TO ALTERNATING-CURRENT PHENOMENA. 

In the single-phase alternating-current circuit, if e 
E.M.F., i = current, r resistance per line, the total power 
is = ei, the loss of power 2z'V. 

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 e, that is, the dis- 
tribution takes place at twice the potential, or only -' the 
copper is needed to transmit the same power at the 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, T % 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 T V f 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, 
/ ? = 10.93 per cent of the copper. 

Coming now to the three-phase system with the poten- 
tial e between the lines as delta potential, if i = the current 
per line or Y current, the current from line to line or delta 
current = ^ / VB ; and since three branches are used, the 
total power is 3 e i\ / V3 == e z' x V3. 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 z'i = i / V3 where i single-phase current, r = 
single-phase resistance per line, at equal power and loss; 
hence if 1\ = resistance of each of the three wires, the loss 
per wire is i? r t = i z r t /.3, and the total loss is z 2 1\, while in 
the single-phase system it is 2 t*r. Hence, to get the same 
loss, it must be : r v = 2 r, that is, each of the three three- 
phase lines has twice the resistance that is, half the cop- 
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. 



EFFICIENCY OF SYSTEMS. 471 

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 = e X V3, since the Y potential = e, and 
the potential of the system is raised thereby from e to 
e V3 ; 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 \ more 
copper, or \ = 33.3 per cent of the copper of the single- 
phase system ; making the neutral of half cross-section, 
requires \ more, or ^ = 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 
by 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 -fo = 31.25 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 e per branch, denoting the current in 
the outside wires by z' 2 , the current in the central wire is 
* a V2 ; and if the same current density is chosen for all 



472 ALTERNATING-CURRENT PHENOMENA. 

three wires, as the condition of maximum efficiency, and 
the resistance of each outside wire denoted by r z , the re- 
sistance of the central wire = r 2 /V2, and the loss of power 
per outside wire is z' 2 2 r 2 , in the central wire 2 z' 2 2 r 2 / V2 
= z' 2 2 r 2 V2 ; hence the total loss of power is 2 z' 2 2 r 2 + z' 2 2 r 2 
V2 = z' 2 2 r 2 (2 -f V2). The power transmitted per branch 
is z' 2 ^, hence the total power 2 z' 2 e. To transmit the same 
power as by a single-phase system of power, e z, it must 

be z 2 = z'/2; hence the loss, * 2; a( 2 + ^ . Since this 
loss shall be the same as the loss 2z' 2 r in the single- 
phase system, it must be 2 r = - r 2 , or r 2 = ~ . . 

2 -}- V 2 

4- V^ 
Therefore each of the outside wires must be times 

o 

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 + V2) (2 + V5) V2 . 9 3 + 2V2 

^~ ~T~ ~T~~ 

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 
/ 8 in the three lines of equal resistance r 3 , gives the out- 
put 2^z' 3 , that is, compared with the single-phase system, 
/ 8 = z'/2. The loss in the three lines is 3 z' 3 2 r 3 = | z 2 r s . 
Hence, to give the same loss 2 z' 2 r as the single-phase sys- 
tem, it must be r s = f r, that is, each of the three wires 
must have f 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. 



EFFICIENCY OF SYSTEMS. 



473 



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, 31.25 
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 half- 
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. 



474 AL TERNA TING-CURRENT PHENOMENA. 

290. Comparison on the 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 single-phase systems. 

In a quarter-phase system with common return, the 
potential between the outside wire is V2 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 system the potential 
per branch will be ej V2, hence the current z' 4 = t/ V2, if i 
equals the current of the single-phase system of equal 
power, and t\ V2 = i will be the current in the central 
wire. 

Hence, if r = resistance per outside wire, r / V2 = 
resistance of central wire, and the total loss in the sys- 
tem is : 



, (2 + V2) = 



EFFICIENCY OF SYSTEMS. 475 

Since in the single-phase system, the loss = 2 i 2 r, it is : 



2 + ~v / 2 
That is, each of the outside wires has to contain - - 

4 
times as much copper as each of the single-phase wires. 

2 x V2 /- 
The central wires have to contain - - V 2 times as 

^ (^ -4- ~v/2^ 
much copper ; hence the total system contains 



2 +V2 
T - V2 times as much copper as each of the single- 

3 + 2 ~\/2 

phase wires ; that is, - times the copper of the 

4 

single-phase system. 
Or, in other words, 
A quarter-phase system with common return requires 

3 + 2 A/2 

== 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 A/2 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 A/2, 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 



476 ALTERNATING-CURRENT PHENOMENA. 

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 

[Continuous-current system, 50.0] 

3 WIRES : Three-phase system, 75.0 

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. ^/V3, 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. <?/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 ( V3 / e) 2 : (2 / ef = 3 : 4 ; 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 



EFFICIENCY OF SYSTEMS. 477 

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, but produces V2 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 copper 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. 



478 ALTERNATING-CURRENT PHENOMENA. 



CHAPTER XXXI. 

THREE-PHASE SYSTEM. 

292. With 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, 

Let: 

E E.M.F. between branches 1 and 2 of a three-phaser. 
Then: 

E = E.M.F. between 2 and 3, 
(*= E.M.F. between 3 and 1, 



where, e = ^1= ~ - 

Let 

ZD Z 2 , Z s = impedances of the lines issuing from genera- 

tor terminals 1, 2, 3, 
and Y l} Y 2 , Y s = admittances of the consumer circuits con- 

nected between lines 2 and 3, 3 and 1, 1 and 2. 
Jf then, 

ID It, / 8 , are the currents issuing from the generator termi- 
nals into the lines, it is, 

/I + /2 + /3 = 0. (1) 



THREE-PHASE SYSTEM. 479 

If //, 7 2 ', 7/ = currents flowing through the admittances Y 1 , 
F 2 , F 3 , from 2 to 3, 3 to 1, 1 to 2, it is, 

/! = /,'-/,', or, / 1 + / 2 '_/ 3 ' = Ol 
>,->/-/.', or, / 2 + / 3 '-7/ = o[ (2) 

> 3 = //->/, or, / 3 + > 1 / -// = OJ 

These three equations (2) added, give (1) as dependent 
equation. 

At the ends of the lines 1, 2, 3, it is : 



(3) 
Il + ztI t ) 

the differences of potential, and 

ti 

(4) 



the currents in the receiver circuits. 

These nine equations (2), (3), (4), determine the nine 
quantities : f lt 7 2 , / 3 , //, 7 a ', 7 3 ', ^', Ti^ & 

Equations (4) substituted in (2) give : 



(5) 



These equations (5) substituted in (3), and transposed, 
give, 

since l = c E 

E z = E \ as E.M.Fs. at the generator terminals. 



480 AL TERNA TING-CURRENT PHENOMENA. 

as three linear equations with the three quantities 2T/, 

Substituting the abbreviations : 

a I \7 7 I I/" 7 \ I/" 7 ~\7 7 i 

~T * 1^2 ~T *1^3)> -tZ^S) *8^'2 I 

7 V 7 /1_1_V7_1_V7 N >/ 

^zt y 2-^D V* 1 ~r -^s^i T *^V / 



A 



c, F 2 Z 3 , F 3 Z 2 

a , - (1 + ^^3 + 

, Y,Z lt -(1 + F 3 Z 1 +F 3 Z 2 ) 

- (1 + Y,Z 2 + FiZ,), c, F 3 Z 2 
F.Z3, c 2 , Y t Z, 
Y.Z,, 1, - (1 + F 3 Z X + F 3 Z 2 ) 

(i + ^iz. + yiz,), F 2 z 3 , 



A = / F I Z S , - (i 

F a Z 2 , F 2 Z X , 



it is: 



D 



7 2 = i 



__ F 2 Z> 2 - 






hence, 



(8) 



(9) 



(10) 



(11) 



THREE-PHASE SYSTEM. 

293. SPECIAL CASES. 

A. Balanced System 

Y, = F 2 = F 8 = F 
Z, = Z 2 = Z 3 = Z. 

Substituting this in (6), and transposing : 



481 



c E 



s = 



EI = 




3FZ 



1 + 3FZ 



1 + 3YZ 
EY 



1 + 3KZJ 



3FZ 



3FZ 



3 YZ 



(12) 



The equations of the symmetrical balanced three-phase 
system. 

B. One circuit loaded, two unloaded: 

F! = F 2 = 0, F 8 = F 
Zj = Z 2 = Z 3 = Z. 

Substituted in equations (6) : 

= ( unloaded branches. 
E E 3 '(l + 2 FZ) = 0, loaded branch. 



hence : 
r./ 

, 



2KZ 



2FZ 



1 + 2 FZ 



unloaded ; 



loaded ; 



all three 

KM.F.'s 

unequal, and (13) 
of unequal 
phase angles. 



482 



AL TERNA TING-CURRENT PHENOMENA. 




(13) 



(13) 



C. Two circuits loaded, one tinloaded. 

F! = F 2 = F, F 8 = 0, 
Z t = Z 2 = Z 3 = Z. 

Substituting this in equations (6), it is : 

e E E{ (1 + 2 FZ) + ./ FZ = 0) 
E El (1 + 2 FZ) + E{ FZ = J 

E s ' + (,,' + ^2') FZ = unloaded branch, 
or, since : 

E E z '' E Z 'Y2 :'= 0, 

E 1 = ? 

\ + FZ 

thus: 



1 + 4 FZ + 3 F 2 Z 2 



1 + 4 FZ + 3 F 2 Z 2 
E 

I+'FZ 



loaded branches. 



unloaded branch. 



(14) 



As seen, with unsymmetrical distribution of load, all 
three branches become more or less unequal, and the phase 
displacement between them unequal also. 



QUARTER-PHASE SYSTEM. 483 



CHAPTER XXXII. 

QUARTER-PHASE SYSTEM. 

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 0. 

It is then : 

EI = E = E.M.F. between and 1 in the generator. 
E z =jE = E.M.F. between and 2 in the generator. 

Let: 

./i and 7 2 = currents in 1 and in 2, 
7 = current in 0, 

Z-L and Z z = impedances of lines 1 and 2, 
Z = impedance of line 0. 

Y l and Y 2 = admittances of circuits to 1, and to 2, 
// and //= currents in circuits to 1, and to 2, 
Eia.-ndE 2 '= potential differences at circuit to 1, and 
to 2. 

it is then, 7, -f 7 8 + 7 = ) v 

or, I =-(/; + 7 2 ) j 

that is, 7 is common return of 7 : and 7 2 . 

Further, we have, 



El =JE - 7 2 Z + Z =jE - 7 2 (Z 2 + Z ) - A 

and 

A = K, E{ 

(3) 



484 AL TERNA TING-CURRENT PHENOMENA. 

Substituting (3) in (2) ; and expanding : 
*/ - * _ l + F 2 Z 2 + F 2 Z (l-y) _ 

'. ( 4 ) 

2 /1_l_VX_l_V7'W'l_l_V7_l_V5^ V V '7 2 

\*- i * 1^0 "T" *\**\)\~ i * i ** T * i^ij *i *J ^o 

Hence, the two E.M.Fs. at the end of the line are un- 
equal in magnitude, and not in quadrature any more. 

295. SPECIAL CASES : 

A. Balanced System. 

Z = Z / V2 ; 
F, = F 2 = F 

Substituting these values in (4), gives : 

i + 1 + V2-y rz 

' 1 + V2 (1 + V2) FZ + (1 + V2) F 2 Z ; 



_ E 1 + (1.707 - .707/) FZ 
1 + 3.414 FZ + 2.414 F 2 Z 2 



(5) 



V2 
~ J 1 + V2 (1 + V2) FZ + (1 + V2) F 2 Z 2 

_ . ^ 1 + (1.707 + .707.;) FZ 
' 1 + 3.414 FZ + 2.414 F 2 Z 2 

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 E.M.Fs. at 
the end of the line are neither equal in magnitude, nor in 
quadrature with each other. 



QUARTER-PHASE SYSTEM. 
B. One branch loaded, one unloaded. 



485 



a.) 
b.) 



Substituting these values in (4), gives : 

i + V2 y 



b.} 



l + FZ 



a.) 1 = E 



V2 



1 + V2 
V2 
j 



2.414 + 



1.414 
YZ 



*+'*f 

= / ^l4-1.707FZ 



1+ ^1^ 



1 + 1.707 FZ 

-t I ^/O 

1 + F2 



V2 



, 



FZ 



1 + 



+ 



V2 



2.414 + 



1.414 
FZ 



(6) 



486 AL TERNA TING-CURRENT PHENOMENA. 

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 
negligible. 

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 regard 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. 



APPENDICES. 



APPENDIX I. 



ALGEBRA OF COMPLEX IMAGINARY 
QUANTITIES. 

INTRODUCTION. 

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 : 



b integers 
introduces the operation of addition, as multiple counting : 

a + b = c. 
It is, a + b = b + a, 



490 APPENDIX 7. 

that is, the terms of addition, or addenda, are interchange- 
able. 

Multiple addition of the same terms : 

a -+- a -\- a -+- . . . + a = c 

b equal numbers 
introduces the operation of multiplication : 

a x b = c. 
It is, a X b = b X a, 

that is, the terms of multiplication, or factors, are inter- 
changeable. 

Multiple multiplication of the same factors : 

aX aX aX . . X a = c 

b equal numbers 
introduces the operation of involution : 



Since a b is not equal to #", 

the terms of involution are not interchangeable. 

298. The reverse operation of addition introduces the 
operation of subtraction : 

If a + 6 = f, 

it is c b = a. 

This operation cannot be carried out in the system of 
absolute numbers, if : 

b> c. 

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: 

_ = (_ 1) X , 
.where (- 1) 

is the negative unit. 

Thereby the system of numbers is subdivided in the 



COMPLEX IMAGINARY QUANTITIES. 491 

positive and negative 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 : 

c - (- b) = c + b, 
thus: (-l)X (-!) = !; 

that is, the negative unit is defined by, (I) 2 = 1. 

299. The reverse operation of multiplication introduces 
the operation of division : 

If a X b = c, then - = a. 

b 

In the system of integral numbers this operation can 
only be carried out, if b 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 infraction: 



:. 

where - is the integer fraction, and is defined by : 



T- x b = 1. 



300. The reverse operation of involution introduces two 
new operations, since in the involution : 



the quantities a and b are not reversible. 

Thus V^ = <z, the evolution, 

= b, the logarithmation. 



The operation of evolution of terms c, which are not 
complete powers, makes a further expansion of the system 



492 APPENDIX I. 

of numbers necessary, by the introduction of the irrational 
number (endless decimal fraction), as for instance : 

V2 = 1.414213. 

301. The operation of evolution of negative quantities 
c with even exponents b, as for instance 

2/ - 

makes a further expansion of the system of numbers neces- 
sary, by the introduction of the imaginary unit. 

-V^l 
Thus -x/^ = -v/^T x #*. 

where : V 1 is denoted by/. 

Thus, the imaginary unity is defined by : 

f = _ 1. 

By addition and subtraction of real and imaginary units, 
compound numbers are derived of the form : 



which are denoted as complex imaginary mimbers. 

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, 



COMPLEX IMAGINARY QUANTITIES. 493 

Real and imaginary numbers and complex imaginary 
numbers. 

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- 
tion. 

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 or 
negative, rational or irrational. 

ALGEBRAIC OPERATIONS WITH COMPLEX IMAGINARY 
QUANTITIES. 

303. Definition of imaginary unit: 

f 2 = - 1. 
Complex imaginary number: 



Substituting : 

a = r cos (3 
b = r sin (3, 
it is A = r (cos /3 -f / sin /?), 

where r = a 2 -- \ 



a 

r = vector, 
/3 = amplitude of complex imaginary number A. 

Substituting : 

e Jft 4- c-JP 

H 



cos 



sin/? = 




494 APPENDIX I. 

it is A = reJP, 

where c = lim (l + -}"= yJT _ 1 
= V n) o~lx2 X 3x 

is the basis of the natural logarithms. 
Conjugate numbers : 

a -\- j b = r (cos ft -\- j sin ft) = reJ'P 
and a jb = r (cos [/?]+> sin [ /?]) 
it is 



Associate numbers: 
a + jb = r (cos ft +/ sin /3) = 



and b + 


ja = r ( cos 1 ^ (3\ -f j sin \ 7 - 


it is 


(a+jb)(b+ja)=j(a*+P) 


If 


a+jb = a' +jb', 


it is 


a = a f 


If 


a +J/= ; 


it is 


a = 0, 



304. Addition and Subtraction : 



Multiplication : 

(a +jb) (a' +jb') = (aa 1 - b b') +j(ab' + b a') 
or r (cos ^3 + / sin ft) X r' (cos /? + / sin ftf) = r r' (cos [ -p 

^]+ysin[/3 + ^]); 
or re J* X r'^'0 7 = rr'ef& + M. 

Division : 

Expansion of complex imaginary fraction, for rationaliza- 
tion of denominator or numerator, by multiplication with 
the conjugate quantity : 



COMPLEX IMAGINARY QUANTITIES. 495*" 

a+jb = (a+jb}(a' -jb'} = (aa r + bb'} +j (b a' - ab'} 
-jb'} , *" + *" 



(a! -f j b'} (a jb} (a a' + b b'} +j(ab' b a') ' 



or, _ r ^_p ^ _ ^ . 

r' 



or> 

r 



involution : 

(a +jbY = {r (cos 



evolution : 



-v/^- (cos /8 + y sin 



305. Roots of the Unit : 
=+l, -1; 



</I=+i, -i, +y, -y ; 

' +i+y +i-y -i +y 



V2 V2 V2 

-i-y 

V2 ' 



306. Rotation : 

In the complex imaginary plane, 
multiplication with 

9 * 2-n- 

VI = cos +y sin = e 



means rotation, in positive direction, by 1 / n of a revolution, 



496 APPENDIX I. 

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 plane : 

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 B r O B 
as imaginary axis. Thus all 

the positive 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. 



APPENDIX II. 



OSCILLATING CURRENTS. 

INTRODUCTION. 

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 

497 



498 



APPENDIX II. 



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 N = 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 



V 


^ ! 








































I"**' 








































\ 




^ 


-. 


































\ 








/ 


S 


r~~ 


-- 


__ 






















1 


> \ 


180 




/ 




3W 


\ 




MO 




^ 


^- 1 


raT 


X 








TWO 


J 


jW8Q 




\ 






/ 








\ 




. 








___ 




^. 




^-i 






\ 






/ 






_^ 





- >T=- 
























Vy 




/.\ 


-' 


-~ 
































t 





















en 


atin 
. 


g E 
135 
cc 


M.F 












X 


" [ 


















E 


=5 




^ 
stf> 














1 




















4afs 


2 













a periodic function, and a function decreasing in geometric 
proportion with the time. The latter is the exponential 
function A f ~ gt . 

309. Thus, the general expression of the oscillating 
current is 

/= ^/-0'COS (2-rrNt S), 

since A'-** = A' A-'* = U~ bt . 

Where e = basis of natural logarithms, the current may 
be expressed 

7= i(.~ bt cos (2-n-JVf ) = ze- a * cos (<#> - ), 

where <#> = %-nNt; that is, the period is represented by a 
complete revolution. 



OSCILLATING CURRENTS. 



499 



In the same way an oscillating electromotive force will 
be represented by 

E = etr a * cos O 5). 

Such an oscillating electromotive force for the values 
e = 5, a = .1435 or - 2 = .4, = 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, 




Fig. 208. 



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.L oscillat- 
ing waves of which one is shown separately, with the same 
constants as Figs. 207 and 208, in Fig. 209. Its character- 



500 



APPENDIX II. 



istic feature is : The angle which any concentric circle 
makes with the curve y ee~ a<t >, is 



tan 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. 209. 



Fig. 210. 



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 j/ = e~ a<t> represent the expo-' 
nential spiral ; 

let z = e cos (< a) 

represent the sine wave ; 
and let E = ef.-** cos (< w) 

represent the oscillating wave. 

We have then 



tan y3 = 



Ed* 
_ sin (< w) a cos 

COS (< oi) 
= {tan (<^> ) + a} ; 



to) 



OSCILLATING CURRENTS. 501 

that is, while the slope of the sine wave, z = e cos (< w), 
is represented by 

tan y = tan (< w), 
the slope of the exponential spiral y = ei' * is 

tan a = a = constant. 
That of the oscillating wave E = *?e~ a * cos (< to) is 

tan /3 = {tan (< w) + a} . 

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 

= ec-**" cos ($ 5). 

In the instance represented by Figs. 181 and 182> we 
have A = .4, a = .1435, a = 8.2. 



Impedance and Admittance. 

312. In complex imaginary quantities, the alternating 
wave * = e cos (* - ffl) 

is represented by the symbol 

E = e (cos w -\-j sin w) = <? x -\-je z . 

By an extension of the meaning of this symbolic ex- 
pression, the oscillating wave E = ee~ a<t> cos (<f> w) can 
be expressed by the symbol 

E = e (cos w -\-j sin w) dec a = (e -\-j'e^) dec a, 
where a = tan a is the exponential decrement, a the angular 
decrement, e~ 27ra the numerical decrement. 



502 APPENDIX II. 

Inductance. 

313. Let r = resistance, L = inductance, and x = 
2 IT N L = reactance. 

In a circuit excited by the oscillating current, 

/= /-* cos (< w) = /(cos to +y sin w) dec a = 

(*i -\-J*z) dec a, 
where /i = / cos w, / 2 = / sin >, a = tan a. 

We have then, 

The electromotive force consumed by the resistance r of 
the circuit ^ 



The electromotive force consumed by the inductance L 
of the circuit, 

E f **L~*iNI&t = *. 

dt d<$> d<$> 

Hence E x = xif.~ a ^> (sin (<J> fy -\- a cos (< w)} 

xi(.~ a ^ . ,. , N 

= sin (^> w -f- a). 

COS a 

Thus, in symbolic expression, 



x = - ^{ sin (w a) +/ cos (w a)} dec a 

COS a 

= x i (a -f y ) (cos w + 7 sin a>) dec a ; 
that is, E x = x I (a +/') dec a . 

Hence the apparent reactance of the oscillating current 
circuit is, in symbolic expression, 

X = x (a +y') dec a. 

Hence it contains an energy component ax, and the 
impedance is 

Z = (r X) dec a = {r x (a +/')} dec a = (r ax jx) dec a. 

Capacity. 

314. Let r = resistance, C = capacity, and x c = 1 /2-n-JVC 
= capacity reactance. In a circuit excited by the oscillating 



OSCILLATING CURRENTS. 503 

current /, the electromotive force consumed by the capacity 
Cis 



or, by substitution, 

E x = x I * e~ a * cos (< 

{sin (< w) a COS (< oi 



2 



(1 + 2 ) COS a 

hence, in symbolic expression, 



sin (</> u> a) ; 



= 2 ( + /) (cos w +y sin w) dec a ; 

hence, 



that is, the apparent capacity reactance of the oscillating 
circuit is, in symbolic expression, 



dec 



315. We have then: 

In an oscillating current circuit of resistance r, induc- 
tive reactance x, and capacity reactance x c , with an expo- 
nential decrement a, the apparent impedance, in symbolic 
expression, is : 



*' 



1 +a 2 / V 1 +** 

= r a jx a ; 



504 APPENDIX 77. 

and, absolute, 



Admittance. 
316. Let / = / e -a* cos ^_ ) ==current< 

Then from the preceding discussion, the electromotive force 
consumed by resistance r, inductive reactance x, and capa- 
city reactance x c , is 



cos $ r ax a * e sin (< 



= iz a (.~ a ^ cos (< w + 8), 
where tan 8 = i_^ , 



a 
r ax . - Xf 



substituting & + 8 for G, and ^ = /^ a we have 



cos <> 



I = e~ a * cos (<#> w 8) 



,1 \ cos 8 / i ~\ i sin 8 . / , 
= e e. a<p \ cos (9 to ) -j sin (9 



hence in complex quantities, 

E = e (cos u> -\-j sin oi) dec a, 
+ sin 



OSCILLATING CURRENTS. 505 

or, substituting, 



r ax 
I =E 



I- dec a. 



317. Thus in complex quantities, for oscillating cur- 
rents, we have : conductance, 



susceptance, 



admittance, in absolute values, 

/ o i To 1 



in symbolic expression, 



Y=g+J 



1 + a 2 / \ 1 + a 2 ' 

Since the impedance is 

Z = ir ax 
we have 



506 APPENDIX II. 

that is, the same relations as in the complex quantities in 
alternating-current circuits, except that in the present case 
all the constants r a , x a , z a , g, z, 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 c , 
the impedance was represented in symbolic expression by 



-jx a = 



! + 

or numerically by 



Thus the inductive reactance x, as well as the capacity 
reactance x c , do not represent wattless electromotive forces 
as in an alternating-current circuit, but introduce energy 
components of negative sign 

a 

ax - - x : 
1 + a 2 

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 : 



In this case we have 



r - ax 






; x -- ^ = 0, 



- c 
1 + a 2 1 + fl a 



OSCILLATING CURRENTS. 507 



substituting in this equation 

x = 2 TT NL x c = 
and expanding, we have 
a 




That is, 

" If in an oscillating-current circuit, the decrement 
1 



and the frequency N = r/4iraL, 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/4:7raL, and the decrement 

1 



That is, the oscillating currents are the phenomena by 
which an electric circuit of disturbed equilibrium returns to 
equilibrium. 

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 



508 APPENDIX II. 

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 

Z = 0, that is, 

~ ~ / .1 r 

2aL 2Z~ ~ 1 ' 




If r = 0, that is, in a circuit without resistance, we have 
a = 0, Af = 1 / 2 TT VZT ; that is, the currents are alter- 
nating with no decrement, and the frequency is that of 
resonance. 

If 4 H r 2 C - 1 < 0, that is, r > 2 V2T/T, 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 V/, / C. In the case r = 2 VZ, / C we 
have = oo , ./V = ; that is, the current dies out without 
oscillation. 

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 denned by the equation : Z = dec a. 

Since 

/ = (/V+y/a) dec a, E r = Ir dec a, 

E x = -x I (a +/) dec a, E xc = _^L_/(- a +/) dec a, 

we have r -a X --^ Xc = ^ 

I + a? 



hence, by substitution, 

E xc = x /( a +/) dec a. 



OSCILLATING CURRENTS. 50 ' 

The two constants, t\ and z' 2 , of the discharge, are deter- 
mined by the initial conditions, that is, the electromotive 
force and the current at the time t = 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 t or < = 0. 
A.t this moment the current is zero ; that is, 

7=// 2 , / 1== 0. 
Since E xe = */( a +/) dec a = e at <f> = 0, 

we have x / 2 Vl + a 2 = e or / 2 = = . 

x V 1 + a 2 
Substituting this, we have, 

I j e dec a, E r =je r dec a, 

x Vl + a 2 x Vl + a z 

E x = e (1 -ja) dec a, ^ c = e (1 +/ ) dec a, 

Vl + 8 Vl + a 2 

the equations of the oscillating discharge of a condense 
of initial voltage e. 

Since x = 2 * N L, 

1 



we have 

x = 



hence, by substitution, 

l 

dec a, 



.510 APPENDIX II. 

E - ef \fC 

-f^r-, rr~ \/ ~r~ 




47TZ 



the final equations of the oscillating discharge, in symbolic 
expression. 

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, 
i\= r{ -\- TJ", where 1\ = external, 1\' = internal, resistance. 
The total inductance L l = Z/ -f /,/', where Z/ = external, 
Zj" = internal, inductance ; total capacity, C v Then the 
total admittance of the secondary circuit is 

) dec a = 



where x l = 2irJVL l = inductive reactance: x cl = \l1-jrNC ' = 
capacity reactance. Let r Q = effecive hysteretic resistance, 
Z = inductance ; hence, x^ = Z-n-N L Q = reactance ; hence, 



admittance 



of the primary exciting circuit of the transformer ; that is, 
the admittance of the primary circuit at open secondary 
circuit. 

As discussed elsewhere, a transformer can be considered 
as consisting of the secondary circuit supplied by the im- 
pressed electromotive force over leads, whose impedance is 



OSCILLATING CURRENTS. 511 

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 ; 

hence ' x = 2 TT NL = inductive reactance, 

x c = 1 / 2 TT N C = capacity reactance of the total primary 
circuit, including the primary coil of the transformer. If 
EI = EI 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 v = E^ Y l 
dec a = secondary current. 

Hence, // = / 7 X dec a = pEJ Y l dec a = primary load 
current, or component of primary current corresponding to 

secondary current. Also, 7 = - 2j/ F dec a = primary 

/ ' 
exciting current ; hence, the total primary current is 

/= // + 7 = -'{Fo +/ 2 Y,} dec a. 

E' 
E' = -^-i- dec a = induced primary electromotive force. 

/ 
Hence the total primary electromotive force is 

E = (' + /Z) dec a = L (1 + Z F +/ 2 Z Y,} dec a. 
P 

In an oscillating discharge the total primary electro- 
motive force E = ; that is, 



or, the substitution 

a 



1 + 



(r - ax ) -. 



. . 



512 APPENDIX II. 



Substituting in this equation, ^r=2 it N C, x c = ~L/'2 
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 
calculated. 

324. If the exciting current of the transformer is neg- 
ligible, that is, if Y Q = 0, the equation becomes essentially 
simplified, 



I a \ . I x \ 

(r a x x c 1 j ( x I 

1+/2 v 1 + * 8 i v Ljt^l =0 ; 



that is, 



or, combined, 

(r, -2a Xl ) +/ 2 (r-2 ax) = 0, 



Substituting for x lt x, x el , x ei we have 




+/ a Z) 



i+/V / 4(A+/ 
+/ 2 Z) V (n +/V) 2 (Ci 

!} dec a, 



7 =pEi YI dec a, 
/! = ^/ F! dec a, 

the equations of the oscillating-current transformer, with 
E{ as parameter. 



INDEX. 



PAGE 

Addition 494. 49 8 

Admittance, conductance, suscep- 

tance, Chap. vn. ... 52 

definition 53 

parallel connection ... 57 
primary exciting, of trans- 
former 204 

of induction motor . . . 240 
Advance of phase, hysteretic . .115 
Algebra of complex imaginary 

Quantities, App. I. . . . 489 
Alternating current generator, 

Chap, xvii 297 

transformer, xiv 193 

motor, commutator, Chap. 

xx 354 

motor, synchronous, Chap. 

xix 321 

Alternating wave, definition . . 11 
general ..."... . 7 
Alternators, Chap. xvii. . . . 297 
parallel operation, Chap. 

xvin 311 

series operation 313 

synchronizing, Chap. xvin. . 311 
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 395 

Arithmetic mean value, or average 

value of alternating wave 11 
Armature reaction of alternators 

and synchronous motors . 297 
51 



Armature reaction of alternators, 
as affecting parallel opera- 
tion 313 

self-induction of alternators 

and synchronous motors . 300 
slots, number of, affecting 
wave shape 384 

Associate numbers 494 

Asynchronous, see induction . . 

Average value, or mean value of 

alternating wave .... 11 

Balance, complete, of lagging 
currents by shunted con- 

densance 74 

Balanced and unbalanced poly- 
phase systems, Chap. 

xxvii 440 

Balanced polyphase system . . 431 
quarter-phase system . . . 484 
three-phase system . . . 481 
Balance factor of polyphase sys- 
tem 441 

of lagging currents by shun- 
ted condensance ... 75 
Biphase, see quarter-phase . . 

Cables, as distributed capacity . 158 
with resistance and capacity 
topographic circuit charac- 
teristic 47 

Calculation of magnetic circuit 

containing iron . . . 125 
of constant frequency induc- 
tion generator .... 269 
of frequency converter . . 232 
of induction motor . . . 262 
of single-phase induction mo- 
tor . . 287 



514 



INDEX. 



Calculation of transmission lines, 

Chap, ix 83 

Capacity and inductance, dis- 
tributed, Chap. xin. . . 158 
as source of reactance . . 6 
in shunt, compensating for 

lagging currents .... 72 
intensifying higher harmon- 
ics 402 

see condenser and conden- 

sance. 
Chain connection of induction 

motors, or concatenation . 274 
Characteristic circuit of cable 
with resistance and capa- 
city 48 

circuit of transmission line 
with resistance, inductance, 
capacity, and leakage . . 49 
curves of transmission lines . 172 
field of alternator .... 304 
power of polyphase systems 447 
Circuit characteristic of cable 
with resistance and capa- 
city 48 

characteristic of transmission 
line with resistance, induc- 
tance, capacity and leakage 49 
factor of distorted wave . . 415 
with series impedance . . 68 
with series reactance ... 61 
with series resistance ... 58 
Circuits containing resistance, in- 
ductance, and capacity, 

Chap, vin 58 

Coefficient of hysteresis . . 116 
Combination of alternating sine 
waves by parallelogram or 
polygon of vectors ... 21 
of double frequency vectors, 

as power 163 

of sine waves by rectangular 

components 35 

of sine waves in symbolic 

representation .... 38 
Commutator motor, Chap. xx. 354 



Compensation for lagging cur- 
rents by shunted conden- 

sance 72 

Complete diagram of transmis- 
sion line in space . . .192 
Complex imaginary number . . 492 
imaginary quantities, algebra 

of, App. i 489 

imaginary quantities, as sym- 
bolic representation of al- 
ternating waves .... 37 

quantity Chap, v 33 

Compounding curve of frequency 

converter 232 

Concatenated couple of 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 ... 6 
see capacity and condenser 
Condensers, distortion of wave 

shape by 393 

see capacity and condensance 
with distorted wave . . . 419 
with single-phase induction 

motor 286 

Conductance, effective, definition 104 
in alternating current cir- 
cuits, definition .... 54 
in continuous current cir- 
cuits 52 

of receiver circuit, affecting 

output of inductive line . 89 
parallel connection ... 52 
see resistance 

Conjugate numbers 494 

Constant current constant po- 
tential transformation . . 76 
current, constant potential 
transformation by trans- 
mission line 181 

potential, constant current 
transformation . . 76 



INDEX. 



515 



Constant potential, constant cur- 
rent transformation by 
transmission line .... 181 

rotating M M.F 436 

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 

susceptance 96 

Converter of frequency, Chap. 

xv 219 

Counter E.M.F. constant in syn- 
chronous motor .... 349 

of impedance 25 

of inductance 25 

of resistance .25 

of self-induction ..... 24 
Counting or numeration . . . 489 
Cross-flux, magnetic, of trans- 
former 193 

of transformer, use for con- 
stant power or constant 
current regulation . . . 194 
Current, minimum, in synchro- 
nous motor 345 

waves, alternating, distorted 

by hysteresis 109 

Cycle, or complete period ... 10 

Decrement of oscillating wave . 501 
Delta connection of three-phase 

system 453 

current in three phase system 455 
potential of three-phase sys- 
tem 455 

Y connection of three-phase 

transformation .... 463 
Demagnetizing effect of armature 
reaction of alternators and 
synchronous motors . . 298 
effect of eddy currents . . 136 
Dielectric and electrostatic phe- 
nomena . . 144 



Dielectic and electrostatic hyste- 
resis 145 

Diphase, see quarter-phase. 
Discharge, oscillating .... 508 
Displacement angle of repulsion 

motor 361 

of phase, maximum, in syn- 
chronous motor .... 347 
Distorted wave, circuit factor . 415 
wave, decreasing hysteresis 

loss 407 

wave, increasing hysteresis 

loss 407 

wave of condenser .... 419 
wave of synchronous motor . 422 
wave, some different shapes . 401 
wave, symbolic representa- 
tion, Chap. xxiv. . . . 410 
wave, in induction motor . . 426 
Distortion of alternating wave . 9 
of wave shape and eddy cur- 
rents 408 

of wave shape, and insulation 

strength 409 

of wave shape and its causes, 

Chap, xxn 383 

of wave shape by hysteresis . 109 
of wave shape, effect of, 

Chap, xxin 398 

of wave shape, increasing ef- 
fective value 405 

Distributed capacity, inductance, 
resistance, and leakage, 

Chap, xni 158 

Distribution efficiency of systems. 468 
Divided circuit, equivalent to 

transformer 209 

Division 491,494 

Double delta connection of three- 
phase six-phase transfor- 
mation 465 

frequency quantities, as pow- 
er, Chap, xii 150 

frequency values of distorted 
wave, symbolic representa- 
tion . . 413 



516 



INDEX. 



Double peaked wave 399 

saw-tooth wave 399 

T connection of three-phase 

six-phase transforma- 
tion 466 

Y connection of three-phase 

six-phase transforma- 
tion 466 

.Eddy currents, unaffected by 

wave-shape distortion . . 408 
demagnetizing or screening 

effect 136 

in conductor, and unequal 

current distribution . . . 139 
Eddy or Foucault currents, Chap. 

xi 129 

Effective reactance and suscep- 

tance, definition .... 105 
resistance and conductance, 

definition 104 

resistance and reactance, 

Chap, x 104 

to maximum value .... 14 
value of alternating wave . 11 
value of alternating wave, 

definition 14 

value of general alternating 

wave 15 

Effects of higher harmonics, 

Chap, xxin 398 

Efficiency, maximum, of induc- 
tive line 93 

Efficiency of systems, Chap. xxx. 468 
Electro-magnetic induction, law 

of, Chap. Ill 16 

.Electrostatic and dielectric phe- 
nomena 144 

hysteresis 145 

Energy component of self-induc- 
tion 372 

flow of, in polyphase system, 441 
Epoch of alternating wave ... 7 
Equations, fundamental, of alter- 
nating current transformer, 

208, 225 



Eauations, fundamental, of gen- 
eral alternating current 
transformer, or frequency 

converter 224 

of induction motor . . 226, 242 
of synchronous motor . . . 339 
of transmission line . . . 169 
Equations, general, of apparatus, 
see equations,fundamental. 
Equivalence of transformer with 

divided circuit 209 

Equivalent sine wave of distorted 

wave in 

Evolution 491, 495 

Exciting admittance of induction 

motor 240 

admittance of transformer . 204 
current of magnetic circuit, 

distorted by hysteresis. .111 
current of transformer . . 195 

Field characteristic of alternator . 304 
First harmonic, or fundamental, 

of general alternating wave, 8 
Five-wire single-phase system, dis- 
tribution efficiency . . . 470 

Flat-top wave 399 

Flow of power in polyphase sys- 
tem 441 

Foucault or Eddy currents, Ch. xi. 129 
Four-phase, see quarter-phase. 

Fraction 491 

Free oscillations of circuit . . . 508 
Frequency converter, Chap. xv. . 219 
converter, calculation . . .232 
converter, fundamental equa- 
tions 224 

of alternating wave ... 7 
ratio of general alternating 
current transformer or fre- 
quency converter . . . 221 
Friction, molecular magnetic . . 106 
Fundamental equations, see equa- 
tions, fundamental, 
frequency of transmission 
line discharge . . . .186 



INDEX. 



517 



Fundamental equations, or first 
harmonic of general alter- 
nating wave 8 

General alternating current trans- 
former, or frequency con- 
verter, Chap. xv. ... 219 

alternating wave . . . . 7, 8 

alternating wave, symbolic 
representation,Chap.xxiv. 410 

equations, see equations, fun- 
damental. 

polyphase systems, Chap. 

xxv 430 

Generator action of concatenated 

couple 280 

of reaction machine . . .377 

alternating current, Chap. 
xvn 297 

synchronous, operating with- 
out field excitation . . . 371 

induction 265 

induction, calculation for con- 
stant frequency .... 269 

reaction, Chap. xxi. . . .371 

vector diagram 28 

Graphical construction of circuit 

characteristic . . . . 48, 49 
Graphic representation, Chap. iv. 19 

limits of method .... 33 

see polar diagram. 

Harmonics, higher, effects of, 

Chap, xxin 398 

higher, resonance rise in 

transmission lines . . . 402 
of general alternating wave . 8 
Hedgehog transformer .... 195 
Hemisymmetrical polyphase sys- 
tem 439 

Henry, definition of 18 

Hexaphase, see six-phase. 

Hysteresis, Chap, x 104 

advance of phase . . . .115 
as energy component of self- 
induction 372 



Hysteresis, coefficient . . . .116 

cycle or loop 107 

dielectric, or electrostatic . 145 
energy current of transformer 196 
loss, effected by wave shape, 407 
loss in alternating field . .114 

magnetic 106 

motor 293 

of magnetic circuit, calcula- 
tion 125 

or magnetic energy current . 115 

Imaginary number 492 

quantities, complex, algebra 

of, App. 1 489 

Impedance 2 

in series with circuit ... 68 
in symbolic representation . 39 
primary and secondary, of 

transformer 205 

see, admittance. 

series connection .... 57 

total apparent, of transformer 208 

Independent polyphase system . 431 

Inductance 4 

definition of 18 

factors of distorted wave . . 415 

mutual ' ... 142 

Induction, electro-magnetic, law of 16 

electrostatic 147 

generator 265 

generator, calculation for 

constant frequency . . 269 
generator, driving synchron- 
ous motor 272 

motor, Chap, xvi 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 



518 



INDEX. 



Induction motor, synchronous . 291 
motor torque, as double fre- 
quency vector . . . .156 
motor with distorted wave . 426 
Inductive devices for starting sin- 
gle-phase induction motor 283 
line, effect of conductance of 
receiver circuit on trans- 
mitted power 89 

line, effect of susceptance of 
receiver circuit on trans- 
mitted power 88 

line, in symbolic representa- 
tion 41 

line, maximum efficiency of 

transmitted power ... 93 
line, maximum power sup- 
plied over 87 

line, maximum rise of poten- 
tial by shunted suseeptance 101 
line, phase control by shunted 

susceptance 96 

line, supplying non-inductive 

receiver circuit .... 84 
Influence, electrostatic . . . .147 
Instantaneous values and inte- 
gral values, Chap. n. . . 11 
value of alternating wave . 1 1 
Insulation strength with distorted 

wave 409 

Integral values of alternating 

wave 11 

Intensity of sine wave .... 20 
Interlinked polyphase systems, 

Chap, xxvin 452 

polyphase system .... 431 
Internal impedance of trans- 
former 205 

Introduction, Chap. 1 1 

Inverted three-phase system . . 434 
three-phase system, balance 

factor 443, 446 

three-phase system, distribu- 
tion efficiency 472 

Involution 490,495 

Iron, laminated, eddy currents . 131 



Iron wire, eddy currents . . . 133 
wire, unequal current distri- 
bution in alternating cir- 
cuit 142 

Irrational number 492 

f, as imaginary unit .... 37 
introduction of, as distin- 
guishing index .... 36 
Joules's law of alternating cur- 
rents 6 

law of continuous currents . 1 

Kirchhoff's laws in symbolic 

representation .... 40 
laws of alternating current 

circuits 58 

laws of alternating sine waves 

in graphic representation . 22 
laws of continuous current 

circuits 1 

Lagging currents, compensation 
for, by shunted conden- 

sance 72 

Lag of alternating wave ... 21 
of alternator current, effect 
on armature reaction and 

self-induction 298 

Laminated iron, eddy currents . 131 
Law of electro-magnetic induc- 
tion, Chap, in 16 

L connection of three-phase, quar- 
ter-phase transformation . 465 
connection of three-phase 

transformation .... 464 
Lead of alternating wave ... 21 
of alternator current, effect 
on armature reaction and 
self-induction . . . . 298 
Leakage current, see Exciting 

current. 

of electric current .... 148 
Lightning discharges from trans- 
mission lines, frequencies 

181, 188 



INDEX. 



519 



Line, inductive, vector diagram . 23 
with distributed capacity and 

inductance 158 

with resistance, inductance, 
capacity, and leakage, 
topographic circuit charac- 
teristic 49 

Logarithmation 491 

Long-distance lines, as distributed 

capacity, and inductance 158 

Loxodromic spiral 500 

Magnetic circuit containing iron, 

calculation 125 

hysteresis 106 

or hysteretic energy current . 116 

Magnetizing current 115 

current of transformer . . 196 
effect of armature reaction 
in alternators and synchro- 
nous motors 298 

Main and teazer connection of 

three-phase transformation 464 
Maximum output of synchronous 

motor 342 

power 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 . . . . 11 
Mechanical power of frequency 

converter 227 

Minimum current in synchronous 

motor 345 

M. M. F. of armature reaction 

of alternator . . 297 



M. M. F. rotating, of constant 

intensity 436 

's acting upon alternator ar- 
mature 297 

Molecular magnetic friction . . 106 
Monocyclic connection of three- 
phase-inverted three-phase 
transformation .... 464 
devices for starting single- 
phase induction motors . 283 

systems 447 

Monophase, see Single-phase. 
Motor, action of reaction ma- 
chine 377 

alternating series .... 363 
alternating shunt .... 368 
commutator, Chap. xx. . . 354 

hysteresis 293 

induction, Chap. xvi. . . . 237 
reaction, Chap. xxi. . . .371 

repulsion 354 

single-phase induction . . 281 
synchronous, Chap, xix . . 321 
synchronous, driven by in- 
duction generator . . . 272 
synchronous induction . . 291 
Multiple frequency of transmis- 
sion line discharge . . . 185 

Multiplication 490,494 

Mutual inductance 142 

inductance of transformer 
circuits 194 

Natural period of transmission 

line 181 

Negative number 490 

Nominal induced E.M.F. of alter- 
nator 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 . . . 489 



520 



INDEX. 



Ohms law in symbolic represen- 
tation 40 

of alternating currents . . 2 

of continuous currents . . 1 

Oscillating currents, App. n. . . 497 

discharge 508 

Oscillation frequency of transmis- 
sion line 181 

Output, see Power. 

Overtones, or higher harmonics 

of general alternating wave 8 

Parallel connection of conduc- 
tances 52 

Parallelogram law of alternating 

sine waves 21 

of double-frequency vectors, 

as power 153 

Parallel operation of alternators, 

Chap, xviir 311 

Peaked wave 399 

Period, natural, of transm. line 181 
of alternating wave ... 7 
Phase angle of transmission line 171 
control, maximum rise of po- 
tential by 101 

control of inductive line by 

shunted susceptance . . 96 
control of transmission line, 

Chap, ix 83 

difference of 7 

displacement, maximum, in 

synchronous motor . . . 347 
of alternating wave ... 7 

of sine wave 20 

splitting devices for starting 
single-phase induction mo- 
tors 283 

Plane, complex imaginary . . . 496 
Polar coordinate of alternating 

waves 19 

diagram of induction motor 244 
diagram of transformer . . 196 
diagram of transmission line 191 
diagrams, see Graphic repre- 
sentation. 



PAGE 

Polarization as capacity . . 6 

distortion of wave shape by . 393 

Polycyclic systems 447 

Polygon of alternating sine waves 22 
Polyphase system, balanced . . 431 
systems, balanced and unbal- 
anced, Chap. xxvn. . . 440 
. systems, efficiency of trans- 
mission, Chap. xxx. . . 468 
systems, flow of power . . 441 
systems, general, Chap. xxv. 430 
systems, hemisymmetrical . 439 
systems, interlinked, Chap. 

xxvin 452 

systems, symmetrical, Chap. 

xxvi 435 

systems, symmetrical . . . 430 
systems, symmetrical, pro- 
ducing constant revolving 

M.M.F 436 

systems, transformation of, 

Chap, xxix 460 

systems, unbalanced . . . 431 
systems, unsymmetrical . . 430 
Power and double frequency 
quantities in general, Chap. 

XII 150 

characteristic of polyphase 

systems 447 

characteristic of synchronous 

motor 341 

equation of alternating cur- 
rents 6 

equation of alternating sine 
waves in graphic represen- 
tation 23 

equation of continuous cur- 
rents 1 

factor of arc 395 

factor of distorted wave . . 414 
factor of reaction machine . 381 
flow of, in polyphase system 441 
flow of, in transmission line 177 
maximum, of inductive line 
with non-inductive receiver 
circuit . . 86 



INDEX. 



521 



PAGE 

Power, maximum of synchronous 

motor 432 

maximum supplied over in- 
ductive line 87 

of complex harmonic wave . 405 
of distorted wave .... 413 
of frequency converter . . 227 
of general polyphase system 459 
of induction motor .... 246 
of repulsion motor .... 360 
parallelogram of, in symbolic 

representation .... 153 
real and wattless, in symbol- 
ic representation . . . 151 
Primary exciting admittance of 

induction motor .... 240 
exciting admittance of trans- 
former 204 

impedance of transformer . 205 
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- 
monics . 391 

of resistance, causing higher 
harmonics 393 

Quadriphase, see Quarter-phase. 
Quarter-phase, five-wire system, 

distribution efficiency . . 471 
system, Chap. xxxn. . . . 483 

system 43^ 

system, balance factor . 442, 445 
system, distribution efficiency 471 
system, symmetry .... 436 
system, transmission effi- 
ciency 474 

three-phase transformation . 465 

unitooth wave 388 

Quintuple harmonic, distortion of 

wave by 400 

Ratio of frequencies in general 
alternating current trans- 
former . . 221 



Ratio of frequencies of transfor- 
mation of transformer . . 207 

Reactance 2 

definition 18 

effective, definition . . . 105 
in series with circuit . . . 61 
in symbolic representation . 39 
periodically varying . . . 373 
pulsation in alternator caus- 
ing higher harmonies . . 391 

sources of 8 

synchronous, of alternator . 301 
see Susceptance. . 

Reaction machines, Chap. xxi. . 371 
machine, power-factor . . 381 
armature, of alternator . . 297 
Rectangular coordinates of alter- 
nating vectors .... 34 
diagram of transmission 

line 191 

Reflected wave of transmission 

line 169 

Reflexion angle of transmission 

line 169 

Regulation curve of frequency 

converter 232 

of alternator for constant 

current . 309 

of alternator for constant 

power 310 

of alternator for constant 

terminal voltage . . . 308 
Reluctance, periodically varying . 373 
pulsation of, causing higher 
harmonics of E.M.F. . . 384 

Repulsion motor 354 

motor, displacement angle . . . 361 

motor, power 360 

motor, starting torque . . 361 

motor, torque 360 

Resistance and reactance of 
transmission Lines, 

Chap, ix 83 

effective, definition . . . 104 
effective, of alternating cur- 
rent circuit . 2 



522 



INDEX. 



22 



153 



Resistance and reactance in alter- 
nating current circuits ... 2 
in series with circuit ... 58 
of induction motor secon- 
dary, affecting starting 

torque 254 

pulsation, causing higher har- 
monics 393 

series connection .... 52 
see Conductance. 
Resonance rise by series induc- 
tance, with leading cur- 
rent 65 

rise in transmission lines with 

higher harmonics . . . 402 
Resolution of alternating sine 
waves by the parallelo- 
gram or polygon of vec- 
tors 

of double frequency vectors, 

as power 

of sine waves by rectangular 

components 35 

of sine waves in symbolic 

representation .... 38 
Reversal of alternating vector by 

multiplication with 1 . 36 
Revolving magnetic field . . . 436 
M. M. F. of constant inten- 
sity 436 

Ring connection of interlinked 

polyphase system . . . 453 
current of interlinked poly- 
phase system 455 

potential of interlinked poly- 
phase system 455 

Rise of voltage by inductance, 

with leading current . . 62 
of voltage by inductance in 
synchronous motor circuit 65 

Roots of the unit 495 

Rotating magnetic field .... 436 
M.M.F. of constant intensity 436 

Rotation 495 

by 90, by multiplication 
with j 37 



Saturation, magnetic, effect on 
exciting current wave . . 

Sawtooth wave 

Screening effect of eddy currents 
Screw diagram of transmission 

line 

Secondary impedance of trans- 
former . 



Self-excitation of alternator and 
synchronous motor by ar- 
mature reaction .... 

Self-inductance 

E.M.F. of 

of transformer 

of transformer for constant 
power or constant current 

regulation 

Self-induction, energy component 

of 

of alternator armature . . 

reducing higher harmonics . 

Series connection of impedances 

of resistances 

impedance in circuit . . . 
motor, alternating .... 
operation of alternators . . 
reactance in circuit . . . 
resistance in circuit 
Shunt motor, alternating 
Sine wave 



circle as polar characteristic 

equivalent, of distorted wave, 
definition 

representation by complex 

quantity 

Single-phase induction motor 

induction motor, calculation 

induction motor, starting de- 
vices 

induction motor, with con- 
denser in tertiary circuit . 

system, balance factor . . 

system, distribution efficiency 

system, transmission effi- 
ciency 

unitooth wave . 



113 
399 
136 

192 
205 



3 

18 

193 



104 



402 
63 
52 
68 

363 

313 
61 
58 

368 

6 

20 

111 

37 
281 
287 

283 

287 
444 
470 

474 



INDEX. 



523 



Six-phase system 434 

three-phase transformation . 465 
Slip of frequency converter or 
general alternating current 

transformer 221 

of induction motor . . . 238 
Slots of alternator armature, af- 
fecting wave shape . . . 384 
Space diagram of transmission 

line 192 

Star connection of interlinked 

polyphase system . . . 453 
current of interlinked poly- 
phase system 455 

potential of interlinked poly- 
phase system 455 

Starting of single-phase induction 

motor 283 

torque of induction motor . 254 
torque of repulsion motor . 361 
Stray field, see Cross flux. 

Subtraction 490, 494 

Suppression of higher harmonics 

by self-induction . . . 402 
Susceptance, definition . . . '. fii 
effective, definition .... 105 
of receiver circuit with in- 
ductive line 88 

shunted, controlling receiver 

circuit 96 

see Reactance. 

Symbolic method, Chap. v. . . 33 
method of transformer . . 204 
representation of general 
alternating waves, Chap. 

xxiV; 410 

Symbolism of double frequency 

vectors 151 

Symmetrical n-phase system . . 435 
polyphase system, Chap. 

xxvi 435 

polyphase systems .... 430 
polyphase system, producing 

constant revolving M.M.F. 436 
Synchronism, at or near induc- 
tion motor . . 258 



Synchronizing alternators, Chap. 

xvm 311 

power of alternators in par- 
allel operation . . . .317 
Synchronous induction motor . 291 

motor, also see Alternator. 

motor, Chap xix 321 

motor, action of reaction ma- 
chine 377 

motor, analytic investiga- 
tion 338 

motor and generator in single 
unit transmission . . . 324 

motor, constant counter 
E.M.F . .349 

motor, constant generator 
and motor E.M.F. . . .329 

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 

motor, minimum current at 
given power 345 

motor, operating without 
field excitation . . . .371 

motor, phase relation of cur- 
rent 325 

motor, polar characteristic . 341 

motor, running light . . . 343 

motor, with distorted wave . 422 

reactance of alternator and 
synchronous motor . . . 301 



524 



INDEX, 



Tandem control of induction 

motors 274 

control of induction motors, 

calculation 276 

T-connection of three-phase, quar- 
ter-phase transformation . 465 
connection of three-phase 

transformation .... 464 
Tertiary circuit with condenser, 
in single-phase induction 

motor 287 

Tetraphase, see Quarter-phase. 
Three-phase,four-wire system, dis- 
tribution efficiency . . . 471 
quarter-phase transformation 465 
six-phase transformation . . 465 
system, Chap. xxxi. . . . 478 

system 433 

system, balance-factor 442, 446 
system, distribution effi- 
ciency 470 

system, equal load on phases, 

topographic method . . 46 
system, interlinked .... 44 
system, symmetry .... 436 
system, transmission effi- 
ciency 474 

unitooth wave 389 

Three-wire, quarter-phase system 483 
single-phase system, distribu- 
tion efficiency 470 

Time constant of circuit ... 3 
Topographic construction of 
transmission line charac- 
teristic - 176 

method, Chap, vi 43 

Torque, as double frequency vec- 
tor 156 

of distorted wave .... 413 
of induction motor . . . 246 
of repulsion motor .... 360 
Transformation of polyphase 

systems, Chap. xxix. . . 460 
ratio of transformer . . . 207 
Transformer, alternating current, 

Chap, xiv 193 



Transformer, equivalent to di- 
vided circuit 209 

fundamental equations 208, 225 
General alternating current, 
or frequency converter, 

Chap, xv 219 

oscillating current .... 510 

polar diagram 196 

symbolic method .... 204 

vector diagram 28 

Transmission efficiency of sys- 
tems, Chap. xxx. . . . 468 
lines, as distributed capacity 

and inductance .... 158 
line, complete space diagram 192 
line, fundamental equations . 169 
line, natural period of . . 181 
lines, resistance 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 .... 34 
method, limits of .... 34 
Triphase, see Three-phase. 
Triple harmonic, distortion of 

wave by 398 

Two-phase, see Quarter-phase. 

Unbalanced polyphase system . 431 

quarter-phase system . . . 485 
three-phase system . . . .481 
Unequal current distribution, 
eddy currents in conduc- 
tor 139 

Uniphase, see Single-phase. 

Unit, imaginary 494 

Unitooth alternator waves . . . 388 
alternator waves, decrease of 

hysteresis loss .... 408 
alternator waves, increase of 

power 405 

Unsymmetrical polyphase system 430 



INDEX. 



525 



Vector, as representation of alter- 
nating wave 21 

of double frequency, in sym- 
bolic representation . .151 

Volt, definition 16 

Wattless power 151 

power of distorted wave . . 413 

Wave length of transmission line 170 
shape distortion and its 

causes, Chap. xxn. . . .383 
shape distortion by hyster- 
esis . .... 109 



Wire, iron, eddy currents . . . 133 

Y-connection of three-phase sys- 
tem 453 

current of three-phase sys- 
tem 455 

delta connection of three- 
phase transformation . . 463 

potential of three-phase sys- 
tem 455 

Zero impedance, circuits of . . 506 



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