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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= izr, where P is the rate at which
energy is expended by the current, i, in the resistance, r.
3.) The power equation : P0 = ei, where P0 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= Vr2 + Ar2.
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 :
xm = 2 TT NL.
Thus, \ir— resistance, xm = reactance, z = impedance,—
the E.M.F. consumed by resistance is : el = 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,
* ' = Xm -•**•
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 = i2r, 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 : —
P0 = ei cos <£.
Consequently, even if e and i are both large, P0 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 : —
/ = / sin ^ (/ - 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) + 72 sin 4 TrJV(t - /2)
+ 73 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 fv 72, 73, . . . are the maximum values of the differ-
ent components of the wave, fv fv /3 . . . the times, where
the respective components pass the zero value.
The first term, 7X 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. In sin 2 mr N (t — /„) is the
«th harmonic.
By resolving the sine functions of the time differences,
/ — fp t — /2 . . . , we reduce the general expression of
the wave to the form :
Al sin 2 TrNt + A* sin 4 vNt + Az sin G TT Nt + . . .
1cos27rA?-f^2cos47rA?-f ^8cos67ry\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)
+ 75 sin 10 TT A^(/ — /5) + ...
or,
/ = ^ sin 2 TT A7 + Az 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 phev
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 0 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, —
sin2 $ + sin2 (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
0
1
.7071
.63663
0
1.4142
1
.90034
0
1.5708
1.1107
1
0
10. Coming now to the general alternating wave,
/ = Ai sin 27r Nt + Az sin 4-n- Nt + A3 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 0 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 108 = 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 = 4«4>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. = 27r»<S>7V710-8VOltS.
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 108;
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 AOBl = ^ = 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 OBZ.
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 (wx + £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 OEr 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 OEX. 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, E0, has to give the three E.M.Fs.,
E, Ery and Ex, hence it is determined as their resultant.
Combining by the parallelogram law, OEr and OEX, give
OEZ, the E.M.F. required to overcome the impedance of
the line, and similarly OEZ and OE give OE0, 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 = VX2 — (/*)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.,
E0, and have derived E0 by combining Er, Ex, and E.
E'.
E? 0
Fig. 13.
18. We may, however, introduce the effect of the induc-
tance directly as an E.M.F., Ex , 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. E0, EJ, Ex , 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 Eg) the counter E.M.F. of
impedance ; and since Eg and E0 must combine to form
E, E0 is found as the side of a parallelogram, OE0EEg)
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, OEZ, of the line, the component, OEZ,
of the impressed E.M.F. is required which, with the other
component OE, must give the impressed E.M.F., OE0.
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, —
OEr = E.M.F. consumed by resistance,
OEr' = counter E.M.F. of resistance,
OEX = E.M.F. consumed by inductance,
OEX' = counter E.M.F. of inductance,
OEZ = E.M.F. consumed by impedance,
OEt ' = 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 OEr; the E.M.F. consumed
by the reactance, Ix, is 90° ahead of the current, and re-
presented by OEX. Combining OE, OEr, and OEX, we
get OE0, the E.M.F. required at the generator end of the
line. Comparing Fig. 16 with Fig. 13, we see that in
the former OE0 is larger ; or conversely, if E0 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, —
E0 = 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 OEr, in phase with the current, and
OEX, 90° ahead of the current, the generator E.M.F., OE~0,
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.,
E0> a resistance E.M.F., Er = fr, a reactance E.M.F.,
Ex = 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., JE1, will be
represented by a vector, O£l} in Fig. 18, at the phase
180°. The secondary current, flf lags behind the E.M.F.,
Elt 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, OFl} of phase 180 + Gj.
Fig. 18.
Instead of the secondary current, flt we plot, however,
the secondary M.M.F.,
where n1 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., Ely 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., O7, is the resultant of the secondary M.M.F.,
JFlf and the primary M.M.F., SF0; or graphically, OF is the
diagonal of a parallelogram with OFl and OF0 as sides. OF1
and OF being known, we find OF0, the primary ampere-
turns, and therefrom, and the number of primary turns, n0,
the primary current, I0 = &0/ n0, which corresponds to the
secondary current, 71.
To overcome the resistance, r0, of the primary coil, an
E.M.F., Er = f0r0, is required, in phase with the current,
J0, and represented by the vector, OEr.
To overcome the reactance, x0 = 2 •*• n0 L0 , of the pri-
mary coil, an E.M.F. Ex = I0x0 is required, 90° ahead of
the current f0, and represented by vector, OEX.
The resultant magnetic flux, 4>, which in the secondary
coil induces the E.M.F., EI} induces in the primary coil an
E.M.F. proportional to E± by the ratio of turns n0/ nl} and
in phase with El , or, —
77 f "o zr
£, *m—2£lf
»1
•which is represented by the vector OE%'. To overcome this
counter E.M.F., Et't a primary E.M.F., Et, is required, equal
but opposite to Et', and represented by the vector, OE,.
The primary impressed E.M.F., E0, must thus consist of
the three components, OEit OEr, and OEX, and is, there-
fore, their resultant OE0, while the difference of phase in
the primary circuit is found to be <30 = E0OF0.
22. Thus, in Figs 18 to 20, the diagram of a trans-
former is drawn for the same secondary E.M.F., Ev sec-
GRAPHIC REPRESENTA TION.
31
ondary current, 7L 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, 71} is in phase with the
secondary E.M.F., El.
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., E0, 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 El —
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.
n0 = 300 turns.
CFi = 2250 ampere-turns.
y = 100 ampere-turns.
Er = 10 volts.
JSX = 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 OF1FF0^ 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 = Vfr2 + 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, I0, has the
rectangular components a0 — (a -f- a!}, and b0 = (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, —
y2 = - 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 —
0 +./>') 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 —
E0 = E + ZI.
Assuming now again the current as the zero line, that
is, / = /, we have in general —
E0 = E -f ir —jix ;
hence, with non-inductive load, or E = e,
E0=(e + ir) -jix,
+ /r)2 + (/X)2, tan S>0 =
42 ALTERNATING-CURRENT PHENOMENA.
In a circuit with lagging current, that is, with leading
E.M.F., E = e -je', and
*-*)2> tan <S0
e + />
In a circuit with leading current, that is, with lagging
E.M.F., E = * +>', and
— /V) , tan w0 =
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 7l = — 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 El = — 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 A19 A2,AB, be represented by Elt E2, £3.
Then the difference of potential from A2 to A± is £z — £lf
since the two E.M.Fs., El and
are connected in cir-
cuit between the terminals A, and A*, in the direction,
TOPOGRAPHIC METHOD. 45
Al — O — A2; that is, the one, Ez, in the direction OA2,
from the common connection to terminal, the other, JS1, in
the opposite direction, A^O, from the terminal to common
connection, and represented by — El. Conversely, the dif-
ference of potential from A1 to Az is El — Ez.
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 Er The
potentials at the two other terminals will then be given by
the points Ez and E& which have the same distance from
O as Ev and are equidistant from E± and from each other.
The difference of potential between any pair of termi-
nals — for instance E^ and E2 — is then the distance EZEV
or E±EV according to the direction considered.
35. If now the three branches OEV ~OEZ and "OEW 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^ = 072 = Ofs = I, lagging behind the E.M.Fs.
by angles E.O^ = EZOIZ = EZOI& = Q.
Let the three-phase circuit be supplied over a line of
impedance Z± = r^ —jx\ from a generator of internal im-
pedance Z0 = x0 -jx0.
In phase OEV the E.M.F. consumed by resistance r^ is
represented by the distance E^EJ = Irv in phase, that is
parallel with current OIV The E.M.F. consumed by re-
actance #! is represented by E^Ej' = Ixv 90° ahead of cur-
TOPOGRAPHIC METHOD.
47
rent OIr 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'" = Iroi and parallel to OIV
E'"E° = Ixoy and 90° ahead of ~OTV and thus as triangle of
(nominal) induced E.M.Fs. of the generator E°E£E°.
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
80°LA»
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 E°E£E°, the
E.M.Fs. at the receiver's circuit, Ev Ez, E9 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 El equidistant from 0 and opposite each other,
and the two currents issuing from the terminals are rep-
resented by the points / and I1, equidistant from 0 and
opposite each other, and under angle & with E and El
respectively.
Considering first an element of the line or cable next to
the receiver circuit. In this an E.M.F. EEl 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 OEl respectively.
Passing now to the next cable element we have again an
E.M.F. E1EZ proportional to and in phase with the current
OI^ and a current IJZ proportional to and 90° ahead of the
E.M.F. OEV and thus passing from element to element
along the cable to the generator, we get curves of E.M.Fs.
e and e1, and curves of currents i and il, which can be called
the topographical circuit characteristics, and which corre-
spond to each other, point for point, until the generator
terminal voltages OE0 and OE0l and the generator currents
OI0 and OIJ are reached.
Again, adding 'E~Er' = I0r0 and parallel OI0 and E"E° =
I0x0 and 90° ahead of ~OIM gives the (nominal) induced
E.M.F. of the generator OE°, where Z0 = r0 — jx0 = 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 ~OEy ~OEZ = three-phase E.M.Fs.
at receiver circuit, equidistant from each other and = E.
Let OIV Oly Of3 = 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 OIV and proportional thereto.
In the same line element we have a current IJ^ in phase
with the E.M.F. OEV 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. OEV 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°, Es°, and generator currents //,
/2°, 78°, over the topographical characteristics of E.M.F. ev
ev es, and of current iv 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 0 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%, r3, . . . 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^, gz, . . . are connected in parallel, their
joint conductance is the sum of the individual conductances,
or G = gl + gz + gz + . . . 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, gEt 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 = Vr1 + 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 + jbt as the
reciprocal of the impedance, Z = r — jxy
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 +» = r2 + 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 = 0 to r = oo , the susceptance,
b, decreases from b = 1 / x at r = 0, to # = 0 at r = cc ;
while the conductance, g — 0 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 = 0 at r = oo .
s
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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 0 at
x = 0 to the maximum, b = 1 / 2 r = g =1 / '2 x at x = r,
and to b = 0 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 = 0 in a closed circuit,
S/ = 0 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 = Vr2 + x'2,
be connected in series with a resistance, r0 .
The total impedance of the circuit is then
Z + r0 = r + r0—jx\
hence the current is
____
•" Z + r0 r+r0 -jx (r + r0)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 + ;-0)2 + x2 -Vz2 +
E.M.F. at terminals of receiver circuit,
E = EnJ >* + *2 . Eo
Vs2 + 2rr0 + r02
difference of phase in receiver circuit, tan w = - ;
difference of phase in supply circuit, tan o>0 =
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 r0 .
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 r0 ,
/=— °, ^ = ^0;
that is, the current and E.M.F. at receiver terminals remain
approximately constant for small values of r0, and then de-
crease with increasing rapidity.
44. In the general equations, x appears in the expres-
sions for / and E only as xz, 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 E0 = const. = 100 volts, s = 1 ohm ;
hence / = E, and —
a.) r0 = .2 ohm (Curve I.)
b.) r0 = .8 ohm (Curve II.)
with values of reactance, x = V^2 — r2, 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 r0 = .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, x0 , be connected in series in a receiver cir-
cuit of impedance
Z = r — jx, z = -\/r2 -|- x'2.
IMPRESSED E.M.F. CONSTANT, E0=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 -jx0 = r—j(x +#e).
Er Er0
Fig. 38.
and the current is,
/=
E
Fig. 39.
Z-jx0 r—j(x + x0}'
/hile the difference of potential at the receiver terminals
r—jx
62 ALTERNATING-CURRENT PHENOMENA.
Or, in absolute quantities : —
Current,
/_ Eo EQ
•* ~
Vr* -f- (x + x0)'2 V 'z'1 + 2xx0 -\- xa2
E.M.F. at receiver terminals,
r / r' + *« = J^
° V ra + (* + *„)* V** + 2*.r0 + *.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 x0 ; but if x is large, that is, if the receiver circuit
is of large reactance, / and E change much with a change
of x0.
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 > E0 ; that is, the reactance, x0 , raises the potential.
c.) E = E0 , or the insertion of a series inductance, x0 ,
does, not affect the potential difference at the receiver ter-
minals, if
^z*-\-2xx0 + x02 = 2;
or, x0 = — 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 = E0,I= E0/z. If x0 < — 2 x, it raises, if x0 > — 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 x0 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., E0, are shown for various conditions of a
receiver circuit and amounts of reactance inserted in series.
Fig. 41 gives for various values of reactance, x0 (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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^
-^
.
n
*<
_- — '
_~. —
---
, —
— -
|
o
.
0
1
0
0 '<!
HM
s t
s
h 'J
TUCJTANCE
-REACT
ANC
E -
-t-CONDENSANCE
Fig. 41.
E0 = 100 volts, and the following conditions of receiver
circuit •— z= 1 Qj r = 1>0> x= 0 (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 x0
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 x0 = i .8, or, x0 = — x (the
condition of resonance), and the E.M.F. reaches the value,
E = 167 volts, or, E = E0z] r. This rise of potential by
series reactance continues up to x0 = il.6, or, x0 = — %x,
Fig. 42.
where E = 100 volts again ; and for x0 > 1.6 the voltage
drops again.
At x0 = ± -8, x = =f .8, the total impedance of the circuit
is r — j (x -f x0} = r = .6, x + x0 = 0, and tan S>0 = 0 ;
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 —
E0 = 100, x0 = .6, x = 0 . (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., E0 = 100 ; for the con-
stant receiver impedance, z = 1.0 (but of various phase
differences to), and for various series reactances, as follows :
x0 = .2 (Curve I.)
x0 = .6 (Curve II.)
x0 = .8 (Curve III.)
xo = 1.0 (Curve IV.)
Xo = 1.6 (Curve V.)
x0 = 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
x0 and x are of the same sign, and highest when they are
of opposite sign.
The rise of voltage due to the balance of x0 and x is a
maximum for x0= +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 0 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 = ro JXoi ZQ = V f0 -j- Xo ,
where r0 is in general small compared with x0.
From this, if the impressed E.M.F. is
E0 = e0 +je0'> E0 = Ve02 + e0'2
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- roy2 -|- (x -j- ^;0)2 V^2 + z02 + 2 (rr0
the E.M.F. at receiver terminals is,
E0z E0z
V(r + r0)'2 + (x + xoy V^2 + Z0* + 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.
zy z0, r0, x0 are constant, if rr0 + xx0 is a maximum, that is,
since x = ~Vz2 — r2, if rr0 -f- x0 ~\/z2 — r2 is a maximum.
70
AL TERN A TING CURRENT-PHEXOMENA.
A function, f = rr0 -+- x0 V^2 — r2 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
,~-—
— '
_---*
/
/
^__
• •
•~~ ^
. •
^ — •
,---
J^-
~~ -
SiL
9-
<-*
I.
.9
.8
Tf
.0
J
.4
.3
.2
.,
-
-.1 -
-.2
-.3 -
-.4 -
-•} '
-.fi
-.?
-.*
2J
Off. 48.
Thus we have : —
f = rr0 + *0 Vs2 — r2 = 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, *0/r0, 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.,
E0 = 100, a constant impedance of the receiver circuit,
s = 1.0, and constant series impedances,
Z0= .S-/.4 (Curve I.)
Z0 = 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 E0 = 100,
x = .95, x = 0, x = - .95, and Z0 = .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, x0 , and the whole shunted by
a condenser of condensance, c, entailing but a negligible loss
of energy.
Then, if E0 = impressed E.M.F.,—
the current in receiver circuit is,
the current in condenser circuit is,
and the total current is
— Jxo Jc
or, in absolute terms, I0
'•=VfeJ+fe-'/;
while the E.M.F. at receiver terminals is,
r
52. The main current, 70, is in phase with the impressed
E.M.F., E0, or the lagging current is completely balanced,
or supplied by, the condensance, if the imaginary term in
the expression of I0 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. x0 being constant,
74 ALTERNATING-CURRENT PHENOMENA.
with increasing resistance, r, the condensance has to be
increased, or the capacity decreased, to keep the balance.
r2 4- r2
Substituting c = ^/ " ,
we get, as the equations of the inductive circuit balanced
by condensance : —
7 =
r — Jxo
and for the power expended in the receiver circuit : —
that is, the main current is proportional to the expenditure
of power.
For r = 0 we have c = x0, 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.,
E0 = 1000 volts, and a constant series reactance, x0 = 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, E0 = IOOO VOLTS.
SERIES REACTANCE CONSTANT, X0= 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 = x0 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>0 = ,
«0
which is = 0.
76
ALTERNA TING-CURRENT PHENOMENA.
when r = 0 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, EO—IOOO 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 /, 71} 70, 7f, in Curves
I., II., III., IV., similarly as in Fig. 50, for E0 = 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, x0, 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 = x0, 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., E0, is kept constant. Hence, to
keep the current in the receiver circuit entirely constant, the
impressed E.M.F., E0, 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
E0, with increasing load.
Fig. 55.
Let —
be the impressed E.M.F. of the generator, or of the mains,
and let the condensance be xc = x0\ then — •
Current in receiver circuit,
r —jx0
current in condenser circuit,
T
/I = —
X0
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 —jx0 r (x0 — xj —jx? '
This value of / contains the resistance, r, only as a fac-
tor to the difference, x0 — x^\ hence, if the reactance, ;r2 ,
is chosen = x0 , r cancels altogether, and we find that if
#2 = *0, the current in the receiver circuit is constant,
/-/A,
X0
and is independent of the resistance, r ; that is, of the load.
Thus, by substituting xz = x0, we have,
Impressed E.M.F. at generator,
E<i = <?2 + Je*'i Ez = V^22 + ^2' 2 = constant ;
current in receiver circuit,
/ =j%L, 7 = ^? = constant;
x0 xa
E.M.F. at receiver circuit,
E = Ir=jE-^-, E ~ ^^, or proportional to load r;
'
RESISTANCE, INDUCTANCE, CAPACITY. 79
E.M.F. at condenser terminals,
E* 1 +/ - , £0= ^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 Pnase Wlt"
° X° ( E.M.F., Ez .
The power of the receiver circuit is,
the power of the main circuit,
f0Ez = 2 r , hence the same.
*02
55. This arrangement is entirely reversible ; that is,
if Ez = constant, / = constant ; and
if I0 = constant, E = constant.
In the latter case we have, by expressing all the quanti-
ties by 70 : —
Current in main line,
I0 = constant;
E.M.F. at receiver circuit,
E = I0x9 = constant ;
current in receiver circuit,
/ =f0 — , 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 = —I0 , or proportional to the load - .
From the above we have the following deduction :
Connecting two reactances of equal value, x0, 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, xc = x0, 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 : x0 = 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;;
E0 = 1000 1/1 + I — Y plotted as Curve II. ;
»' V 100 y
7t = 10 i/1 + ( -£-Y, plotted as Curve III.-
V ^-^^ J
70 = .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,
IMPR£SSED E.M, F.CONSTANT, E8=IOOO VOL
2 REACTANCES OFOTo =IOO OHMS EACH, SH
THE CONDENSANCE, ZC = 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
r.
ou
1100
^
^*-
1300
SSES
,--
^
"
,''"
1200
^-«
-^
„
^
1100
\^
— '
•^
,•'
!'• u
<V>0
—
, '
«-— —
VI
"^
X
-
~
^
-^
-
son
X
--
581
^
''
^xi
?
700
/
x^
X-1
/
^
600/
^
'
^ \
•^
4
^
-''
^
•^
L
--'•
^
^
1
}joo
,x
X
^
IMO
^x
^>
^
100
,.
^
^
=iES
ST*
NCE
—
r c
F R
ECE
VE
H Cl
RCL
IT,
OH
AS
X
1) .
1
.0 1
(1 1
I. V-
1 V
1) 2
» ()
HM8
F/3. 50. Constant-Potential — Constant-Current Transformation.
Let —
ri = 2 ohms = effective resistance of condensance ;
r0 = 3 ohms = effective resistance of each of the inductances.
We then have : —
Power consumed in condensance, I* r± = 200 + .02 r2 ;
power consumed by first inductance, 72 r0 = 300 ;
power consumed by second inductance, /02r0 = .03 r*.
Hence, the total loss of energy is 500 + -05 r2 ;
output of system, /2 r = 100 r
input, 500 + 100 r -\
effidenCy' 500 + 1W M
It follows that the main current, f0, increases slightly
by the amount necessary to supply the losses of energy
in the apparatus.
82 ALTERNATING-CURRENT PHENOMENA.
This curve of current, I0, 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, gy 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—
Z0 = r0 —,jx0 = impedance of the line ;
z0 = Vr02 + ^2;
Y = g -\-jb = admittance of receiver circuit;
y = VFTT2;
E0 = e0 -f /<?</ = impressed E.M.F. at generator end of line ;
E0 =
E = e +/<?' = E.lVf.F. at receiver end of line ;
E =
I0 = i0 -\-jio = current in the line ;
I0 = 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, 70 = Eg;
impressed E.M.F., E0 = E + Z0 70 = E (1 + Z.g).
Hence —
E.M.F. at receiver circuit,
= \^Z0g~ \-\-gr.-jgxJ
current, 70 = JA|_ = ^ .
Hence, in absolute values —
E.M.F. at receiver circuit, E
current, 70 :
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 = I0 E =
As a function of g, and with a given Eot r0, and x0, this
power is a maximum, if —
that is —
-l+^-V^+^^^O;
hence —
conductance of receiver circuit for maximum output,
Vr02 + V ^o
Resistance of receiver circuit, rm = — = z0 ;
86 AL TERNA TING-CURRENT PHENOMENA.
and, substituting this in P —
Maximum output, Pm = 2 = — g —
and —
ratio of E.M.F. at receiver and at generator end of line,
am = -=r =
efficiency,
That is, the output which can be transmitted over an
inductive line of resistance, r0 , and reactance, x0 , — that is,
of impedance, z0 , — into a non-inductive receiver circuit, is
a maximum, if the resistance of the receiver circuit equals
the impedance of the line, r = z0) 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
E0 = 1000 volts,
Zg = 2.5 — 6/ ; that is, r, = 2.5 ohms, x0 — 6 ohms, z0 = 6.5 ohms,
with the current I0 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, r0 = 2.5 ohms, x0 = 0, are shown in Curves IV.,
V., and VI.
SUPFUED'OVER INDUCTIVE LINE OF IMPEDAN
AND OVER NON-INDUCTIVE LII^E OF RESISTAr.
T0 = 2.5
CURVE 1. E. M. F. AT RECEIVER CIRCUIT, INDUCTIVE LI
3E
CE
SE
UK
100
90
80
70
CO
50
40
30
40
10
^^x-
---1
.
ii V. 11 ii ii ii NON-INDUCTIVE »
^
x'.
0
t
VI
"
"
sos
NDL
CTIV
/
•4
"?"
z
5
-S-
o
/
cr:
fc
/
5
0
/
o
^
/
|
co
/
jjj
0
/,
/*
*~^
IIMl'
m
+*
"^
!5^-
//
/
<>
,„,
m
^^
^
^
^^^
**as.
\
gpj
JQJ
/
^^
^^
<^
^~,
f^
\
B3
TOO
/
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>>
/r
5
-~^.
jj^
300
^
Xs-
x
S
x
\
.-»i ) 1
"~ — .
no
/
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40j
wo
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ai-r
.300
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s
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/
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100
1
cu
^RE
NT
N L
!NE
AMF
ERE
s
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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;
E0 = E + I0Z0 == E (1 + KZ0).
88 AL TERNA TING-CURRENT PHENOMENA.
Hence —
E.M.F. at receiver terminals,
1 + FZ0 (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 _ jr2 + ^2 _ .
° V (i + rog + Xoby + (Xog - r0t>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 a2 is a maximum ; that
is, when —
/=!=(! + r.g + x.Vf + (x.g - r0b?
is a minimum.
The condition necessary is —
or, expanding, ,., ,N , ,N A
5'. *. (1 + rog + jf0^) - r0 (Xog - r0b} = 0.
Hence —
Susceptance of receiver circuit,
t=~^^)=~^= ~b°'
or b + b0 = 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,
E0 z0 (g
maximum output,
Pl = -
current,
E0Y E0 (g
E0(g-jb0}
og - x0b.} -J(r0b0
Io = E° V (1 + rog - Xob0? + (r0b0 + Xog)*>
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]
)* + (Xog -
90 ALTERNATING-CURRENT PHENOMENA.
that is, expanding, —
C1 + r0g -f x0 b}2 + (Xog — r0by — 2g(r0 + r*g -f x*g) = 0 ;
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= g0 — jb0\
x = - x0 , r = r0 , z = z0, Z = r — Jx = r0 + jx0 ;
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, Z0 = r0 —jx0 , into a non-inductive receiver circuit, is
a maximum for the resistance, r = z0, or conductance, g =
y0 , 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 = — b0, 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 = r0 -\-jx0, 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 r0 , 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 = zo j 2 roi 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 —
E0 = 1000 volts, and Z0 = 2.5 — 6/; that is, for
r0 = 2.5 ohms, x0 = Gohms, z0 = 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. Z0
T RECEIV1 NG^ND
=5.5 -3j
AT
CON
TAN
g= . 0592
1 OUTPUT
II RATIO OF POTENTIALS —
/
\
/
\
/
\
/
\
/
\\
/
\\
/
I
/
/
Ns
f
\
1
/
\
\
/
\\
/
5
/
\\
/
'/
\
\
/
7
\°
/
/
\
\
/
P
\
*
\
X
-<,
~^_
^
^«
' — -.
SUSCEfA
°T'
iECE
IVE
R C
KCU
IT
-.3 -.2 -.1 0 +.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 = — b0. 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 = 0.
The current is —
The current I0, is in phase with the E.M.F., E0, 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 = — x0, 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 = g0, as shown before;
hence, substituting g = g0, r = r0,
E 2
maximum power at maximum efficiency, Pm = —2— ,
at a ratio of potentials, am — — -2— ,
" ro
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 —
E0 = 1,000 volts,
Z0 =2.5 — 6/; r0 = 2.5 ohms, x0 = 6 ohms, z0 = 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 + r0
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 E0, 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, aE0, 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, aE0.
Therefore, the maximum output which can be transmitted
at potential aE0, is given by the expression —
hence b = — o0 ,
and g = — g0 -\-
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*(ty00'-g0).
If a = 1 when g = 0, b = 0
when g > 0, b < 0 ;
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, ^ = 0 ;
when^> -g0 + 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 = 0 ;
expanding : 2 a (yJL - g\ _ f!f' = 0 ;
\a J a
hence, y0 = 2ag0,
yo 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, x0, can be found, which delivers a
maximum output into the receiver circuit at the ratio of
potentials, a,
and z0 = 2 r0 a,
for a == 1,
If, therefore, the line impedance equals 2# times the line
resistance, the maximum output, P = E* j ± r0, is trans-
mitted into the receiver circuit at the ratio of potentials, a.
If z0 = 2 r0, or x0 = r0 V3, the maximum output, P =
£02/4:r0, 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 a2, as abscissae, and a constant impressed E.M.F.,
E0 = 1,000 volts, and a constant line impedance, Z0 =
2.5 — 6/, or, r0 = 2.5 ohms, x0 = 6 ohms, z = 6.5 ohms,
the following values :
RATIO'OF RECEIVER VOLTAGE TO SENDER VOLTAGE: d =I.O
LINE IMPEDANCE: Z0= 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
~
300
• ' .
DLTS
1000
uoo
too
roo
GOO
M)
400
300
200
100
0
~""~-
\:
~~~~-.
-^.
—
—
*-•,
~— ^.
*
V
nr
\
x
//'
"^
-^
//s
\
\
"*x-
^^
\
x
2
S
/^
A
1
,
s
^-
^
^T
)
^S
^~
^^-*
•*^=:
—
—
—
^
>^
/
^
^y
*~^
-^.
•*fc
x
f
-"
*
^^
•^,
^>
-^
^
—1 —
— -
_____
-. —
=rrT
—-
, — •
01
r?v
T IN
RE
iEIV
x c
RC
IT,
<ILO
.VAT
TS
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: Z0=2.5.— ej"
CONSTANT GENERATOR POTENTIAL E0=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, E0 .
The condition is that
a = maxmum or — = mnmum :
a2
that is,
substituting,
r0g +
(*0g -
and expanding, we get,
dg = °; gss~"£''
— a value which is impossible, since neither r0 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 + b0 = 0 ;
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
\
\
\\
\
\
CONSTANT IMPRESSED E. M. F. Eo^lOOO
" LINE IMPEDANCE Z0=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
\
IVE
1200
1100
1000
l\
>
s 1
GOO
800
700
COO
*
"-^
•^
SEC
JFF
C1EN_
*-.
-*
fa
"N
^
^
/^
5
L
•\
//
tV
i
/
^
\
8
/f
//
300
200
100
A
^
' .,
/\v
^
tc)P
^>
4
/
^ —
*.ovJS
^"\
PUT
PUT
K.W
0 '
i) i
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., E0 = 1000 volts, and a
line impedance, Z0 = 2.5 — 6/, or, r0 = 2.5 ohms, x0 = 6
ohms, z0 = 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, b0 = 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 108
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 cm2,
'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=47r/103r = 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 TV 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 cm3 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
1.6th 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 cm8,
(ft = maximum magnetic induction, in lines of force per cm2, 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, —
E1-' FIO 5-8 E™ F108
no5-8 Ka no3
» 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 103 Z
— —
and 103 191.4 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 LT6. 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 1.6th
power of the E.M.F., inversely proportional to the 1.6th
power of the number of turns, and inversely proportional to
the .6th 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 = Ezg.
Since, however :
P = A , we have A = E2 g ;
or
A 58r)L 10s
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 .6th power of the frequency,
N, and of the cross-section of tlie magnetic circuit, S, and to
tlie 1.6th 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, £FA = 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
(RIO8 = a&
E ~ 2 TT n*N ~ N '
108
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=Zge = M.M.F.,
and <R, 10L
magnetic flux,
substituting this value in the equation of the admittance,
(R 108 Z 109 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),
NA «Z1068
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 .4th 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 { _|_ <Rfl ;
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, ya = 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, gjyt, for the ironclad circuit, but decreases
with increasing width of the air-gap. The introduction of
the air-gap of reluctance, (R0, 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 = <Ra / N, or : x = N/ (Ra.
The angle of hysteretic advance is small, below 4°, and the
hysteretic conductance is,
-= A
EAN* '
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 La = length of air circuit, CFa = — — - — = ampere-turns re-
quired in the air ;
hence, CF= JF, -)- $Fa = 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 JFa, 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 — gz\
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 = — f—i .
&'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 .4th 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 1.6th
power of the number of turns of the electric circuit, ;/, as
expressed in the equation,
58 7 Z 103
9.) The absolute value of hysteretic admittance, —
is proportional to the magnetic reluctance : (R = (R, -f (Ra ,
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, Ny
permeability, /*, and square of the number of turns, n, or
127 L 106
11.) In an open magnetic circuit, the conductance, gt is
the same as in a closed magnetic circuit of the same iron part.
12.) In an open magnetic circuit, the admittance, yt is
practically constant, if the length of the air-gap is at least
TJC 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, gy 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 cm3, in ergs per cycle, is
hence, the total loss of power by eddy currents is
W = e y VN* (B2 10 - 7 watts,
and the equivalent conductance due to eddy currents is
o_ W _ IQey/ __ .507ey/
£> Tf"2 O 2 C^/2 C«2 *
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($> x, in C.G.S. units.
This E.M.F. produces the current :
///=SJ£y</x = V2 TrN<$> 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 7T2^2(B2 y x Vx, in C.G.S. units or ergs per second,
and, consequently, the total power consumed in one cm2 of
the sheet of thickness, d, is
= C+* dP = 27rW2(B2y C
° in C.G.S. units;
the power consumed per cm3 of iron is, therefore,
.
/ = — = - — '- — , m C.G.S. units or erg-seconds,
and the energy consumed per cycle and per cm3 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 = 105,
we get as the coefficient of eddy currents for laminated iron,
2-»= 1.645</210-9-
loss of energy per cm3 and cycle,
W= ey^Wfc2 = - //2y^(B210-9 = 1.645 </2y N<$? 10 ~9 ergs
6
= 1.645</27V~(B210-4ergs;
or, W = c y NW 10 - 7 = 1.645 d* N <S? 10 - " joules ;
loss of power per cm3 at frequency, N,
p = NW '= cy^2«210-7 = 1.645 </W2(B2 10 ~n watts;
total loss of power in volume, V,
p = vp = 1.645 ^/2^2(B210-n watts.
As an example,
d = 1 mm = .1 cm ; N= 100 ; OS = 5000; V = 1000 cm8.
e = 1,645 X 10-";
^F= 4110 ergs
= .000411 joules;
/ = .0411 watts;
P = 41.1 watts.
90. (6): Iron Wire.
Let, in Fig. 80, d =
diameter of a piece of
iron wire ; then if x is
the radius of a circular
zone of thickness, d x,
and one cm in length,
the conductance of this pig. so.
134 ALTERNATING-CURRENT PHENOMENA.
zone is, y^/x/2 TT x, and the magnetic flux inclosed by the
zone is (B x2 *.
Hence, the E.M.F. induced in this zone is :
8£ = V2 7r2^(B x2, in C.G.S. units,
and the current produced thereby is,
, in C.G.S. units.
The power consumed in this zone is, therefore,
dP= §EdI = 7T8 y N'2 (B2 x3 d x, in C.G.S. units
consequently, the total power consumed in one cm length
of wire is
8 P = f~ dW = 7T3 y N'1 ®2 f * xa dx
= ^-y^2&V4, in C.G.S. units.
Since the volume of one cm length of wire is
/ ,*?, - 'I
the power consumed in one cm3 of iron is
x P 2
P = -^- = ^ y ^2(BV2, in C.G.S. units or erg-seconds,
and the energy consumed per cycle and cm3 of iron is
ergs.
Therefore, the coefficient of eddy currents for iron wire is
c = ^^2 = .617 </2;
or, if y is expressed in practical units, or 10 ~9 C.G.S. units,
c = -^
10
FOUCAULT OR EDDY CURRENTS. 135
Substituting ^ = ^
we get as the coefficient of eddy currents for iron wire,
e= — ^210~9 = .617 </210-9.
16
The loss of energy per cm3 of iron, and per cycle
becomes
= .617 d*N®? 10~4 ergs,
loss of power per cm3, at frequency, N,
p = Nh = ey^2(B210-7 = .617 d 2 N*<$? 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 cm8.
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
dz = 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 ex = e2 ; 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</210~3 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. vN&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 dxt of radius x, is
FOUCAULT OR EDDY CURRENTS. 141
The current inclosed by this zone is Ix = zW, and there
fore, the M.M.F. acting upon this zone is
$x = 47r Ix/ 10 = 4 **«»/ 10,
and the magnetic flux in this zone is
d$> = $x I G(x = 2 Trixdx / 10.
Hence, the total magnetic flux inside the conductor is
, 27T . CR . TTiR* I
From this we get, as the excess of counter E.M.F. at the
axis of the conductor over that at the surface —
&E = V27r^0> 10 ~8 = V27r7W10 -9, per unit length,
and the reactivity, or specific reactance at the center of the
conductor, becomes k = &E / i = V2 i^NR* 10 ~9.
Let p = resistivity, or specific resistance, of the material of
the conductor.
We have then, k/p = V^TrW^lO-9/?;
and p/ VFT7,
the ratio of current densities at center and at periphery.
For example, if, in copper, p = 1.7xlO— 6, and the
percentage decrease of current density at center shall not
exceed 5 per cent, that is —
P -H VF+72 = .95 - 1,
we have, £ = .51xlO-«;
hence .51 x 10-6= V^TrW^lO-9
or N2? = 36.6 ;
hence, when N= 125 100 60 25
£ = .541 .605 .781 1.21 cm.
D = 1R= 1.08 1.21 1.56 2.42cm.
Hence, even at a frequency of 125 cycles, the effect of
unequal current distribution is still negligible at one cm
diameter of the conductor. Conductors of this size are,
however, excluded from use at this frequency by the exter-
nal self-induction, which is several times larger than the.
142 ALTERNATING-CURRENT PHENOMENA.
resistance. We thus see that unequal current distribution
is usually negligible in practice. The above calculation was
made under the assumption that the conductor consists of
unmagnetic material. If this is not the case, but the con-
ductor of iron of permeability p., then ; d$ = pffx / (&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-jrNL1 = reactance of secondary circuit ; that is,
Ll = total number of interlinkages with the secondary
conductor, of the lines of magnetic force produced by unit
current in that conductor ;
xm = 2 TT N Lm = mutual inductance of circuits ; that is,
Lm = 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 : xm* < 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, Lm.
The object of this distinction is to separate the wattless part, Z1? of the
total self-inductance, L, from that part, Lm, 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, Lm, 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, Lm ; or
144 ALTERNATING— CURRENT PHENOMENA.
Let rx = resistance of secondary circuit. Then the im-
pedance of secondary circuit is
^i = rv — /*! , zl = V/v + xi2 ;
E.M.F. induced in the secondary circuit, £± = jxmf,
where / = primary current. Hence, the secondary current is
and the E.M.F. induced in the primary circuit by the secon-
dary current, 7l is
or, expanded,
Y zr j~. 2
xm^ JXm
2 _i_ r 2 r2 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 1.6th 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(B2.
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 +jxc,
where : r = hysteretic resistance, xc — hysteretic condens-
ance ; and the angle of dielectric hysteretic lag, tan a = b' / g
= xc / r, are constants of the circuit, independent of E.M.F.
and frequency. The E.M.F. is obviously inversely propor-
tional to the frequency.
The true static dielectric hysteresis, observed by Arno
as proportional to the 1.6th 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 1.6th 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,
P0 = EI= (Ml - *"/") +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 /.
P0 = 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= (* +>") (V1 +/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] = (W1 + ^V11) +/ (W1 - 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,
JP1 = [EIJ = (W1 + e"in)
and the imaginary component,
JPJ =j
The component,
P1
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 P1 = [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
P1 =
or
2 2 22 22 22 22
+PJ =<* ,1 + *" /
where ^ = total volt amperes of circuit. That is,
The true power P1 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.
P1
— =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 = (W1 - 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 = (W1 + M* = 0
that is,
*" _ i1
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 = [E2/2] . . . Pn = [Enln}
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
E1 = £ +jea- — induced E.M.F. in one direction in
space.
72 = z1 +j z11 = 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 72 = il +/zu is
the secondary current in quadrature position, in space, to
E.M.F. Ej.
The current in the same direction in space as El is
/! =y72 = — z11 +//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 loge 2 d/ 8 microfarads,
where
K = dielectric constant of the surrounding medium = 1 in air ;
/ = length of conductor = 5 x 106 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 — 106 / 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, i0 = 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 ;
be = condenser susceptance of line.
DISTRIBUTED CAPACITY.
161
Denoting, in Fig. 84,
the E.M.F., viz., current in receiving circuit by £, It
the E.M.F. at middle of line by £',
the E.M.F., viz., current at generator by E0)I0\
If
We have,
Fig. 84. Capacity Shunted across Middle of Line.
. = I-jbcE'
E\\ \ (r
Jbe(r-Jx) ., (r-jxy(
~~
or, expanding,
[(* - bc} - (rg+
-jx)
I (r-jx)(g+jt)-\}
2 Jf
110. ^.) Z«W 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 £0, f0,
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 E0
and of I0 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 — — bc,
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, Zl=rl— JXl ;
the impedance of receiver circuit, or admittance,
and E.M.F., E0, 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\ + Je\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 bc d x ;
hence, the total current consumed by the line element, dx,
is dl= E(g-jbc}d*, or,
d-t=E(g-jbc\ (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 6rx,
where e is the basis of natural logarithms ; for, differen-
tiating this, we get,
168 ALTERNATING-CURRENT PHENOMENA.
hence, z>2 = (g — j bc) (r — jx) ;
(7)
or, v = ± V (g - Jbe) (r — joe) \
hence, the general integral is :
tr*.«e+«-M«r«« (8)
where a and b are the two constants of integration ;
Substituting
r-«-/0 (9)
into (7), we have,
(a -JP)* = (g - jbc) (r - jx) ;
or,
therefore, _ f
);-' (10)
Vl/2 6 - e
/3= Vl/2
substituting (9) into (8) :
= a-cax (cos/3x — /sin^Sx) + ^c~ax (cos/3x +y sin/3x) ;
«/ = (a£«x + /5>e~ax) cos)8x — y(aeax — ^«-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,
/• \( Jfax. _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 Aea* (cos )8x — j sin 0x) + Bf.~™
a — JP ( '
(cos /?x +/ sin /8x) >
1
^4eax (cos /8x — y sin
^-.
(cos /3x +y sin y8x)
Thus the waves consist of two components, one, with
factor ^eax, 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 //) + (gev + bc ^')
(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, E0 = e0 + j >0' is the E.M.F. at dynamo terminals
(17), and / = length of line, we get
at
hence
g — jbc
or
a-; ft
At X = /,
E0
sin/?/}. (19)
Equations (18) and (19) determine the constants A and B,
which, substituted in (14), give the final integral equations.
The length, X0 = 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., Ey at any point, x, of the line, is determined by
DISTRIBUTED CAPACITY. 171
the equation,
Z?(cos«+/sin£) =y, : \j JsTI71
where Z> is a constant.
Hence, w varies from point to point, oscillating around a
medium position, wx, 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 eax (cos ft x — j sin ft x), cancels, and
D (cos tow +/sin oioc) = — 2-p-
hence, tan ^ = ~a° c + ^ • (21)
This angle, Stx, = 0 ; that is, current and E.M.F. come
more and more in phase with each other, when
abc — fig — 0 ; that is,
a -T- ft — g -r- bc , 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, wx, = 45° ; that is, current and E.M.F. differ
by £th period, if — a bc + fig = a.g + pbc ; or,
which gives : rg + x bc = 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 bc.
The case where g = 0 = 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
bc=
XI
'X|
rj-4
»0
«.ooo
\
\
/
4,000
\,
-"*
JO
J.OOO
0
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, Ev =el = 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
x0 = 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 = {eox (6450 cos /3x + 4410 sin j8x) + c-ax
(3530 cos fix + 4410 sin /?x)}
+ y (eox (4410 cos /3x — 6450 sin £x) — e~ax
(4410 cos ft x- 3530 sin /3x)};
tan 5, = ~ °-ljc + 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 : /! = 0.
hence,
E = i-r— ^{(eax + e-ax) cos/3x — y(cax — c-ax)sin/3x};
.£?.) Line grounded at end:
A — (a/\ -J- /?//) +/ (a// — ^zi) = -?
-^-T-^{(eax — c-ax) cos/?x — >(eax + 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 +ysin/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-bc
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 = 0 be the center of cable ; then,
hence : E — 0 at x = 0 ;
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 bc.
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 = 0
£0 = E.M.F. and
S0 = current at beginning of line
Substituting in (16) these values of El and 7: and also r = 0
= g, we have
From these equations it follows that
which values, together with the foregoing values of Ev Iv 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 70 are both in quadrature ahead of <?x and 7j
respectively.
Il = EQ y — = constant, if 7f0 = constant. That is, at
constant impressed E.M.F. E& the current 7X 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 70 at
the beginning of the line is proportional to the load el at the
end of the line.
If XQ = lx = total reactance,
b0 = lbc = total susceptance of line, then
*<A> = 4-
Instance* = 4, bc = 20 X 10 ~5, E0 = 10,000 V. Hence
/ = 55.5, *0 = 222, b0 = .0111, 7j = 70.7, 70 = .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.,
(^eax _ ^e-ax) CQS £x _j (4€
g — jb I + ^e~ax) sin /3x
the current,
1 ^ (Aea* + ^e~ax) COS /3x — y (^4e
where,
,(14.)
(r1 + ^c2) + (^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^ = 0 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^/ = 0 (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
C0 = 1C = total capacity of transmission line, )
L0 = 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/ \T0 (36-)
and from (29),
V^ = (2^2f/)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 A7/ and
omitting j, and substituting A7" 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^
C0 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 C0 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 L0.
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 106 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: ak = * j
where a^ as ay . . . 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, av
•a& #5> av a9> an-
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 av <73 . . . an.
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
L0
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 7rt
,(46.)
where,
"-57 1
„ r<47')
Instance, .
Length of line, / = 25 miles = 4 x 106 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 3CTO
ampere-turns maximum is required, or, 3COT / 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, T00, 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, f00, 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. = I00 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, = I00 cos a.
The exciting current, 700, 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
I00 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 II , and hence the secondary M.M.F.,
JF1= «j /L, will lag behind £•[ by an angle ft1, 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 Oif1 as one side, the other side or O'S0 is the
primary M.M.F., in intensity and phase, and hence, dividing
by the number of primary turns, n0, 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 70r0, in phase with fot and
represented by the vector OEr0 •
E.M.F. consumed by reactance is IoX0, 90° ahead of I0, and
represented by the vector OEx0 ;
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 OEr0, OEx0, and OE' by means of the parallelo-
gram of E.M.Fs. is,
E0 = ~OE0,
and the difference of phase between the primary impressed
E.M.F. and the primary current is
ft0 = E0O50.
In the secondary circuit :
Counter E.M.F. of resistance is 1^ in opposition with Iv
and represented by the vector OJS'r^ ;
198
AL TERNA TING-CURRENT PHENOMENA,
90° behind 7X, 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 = M»II
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."
Er,
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 = n0/ nv 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 —
r0 = .2 ohms,
x0 = .33 ohms,
f! = .00167 ohms,
*! = .0025 ohms,
g0 = .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 E0, Iv
I0, 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 EO
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, J5lt and thus of E0, 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 :
r0 = .01 ohms ;
x0 = .033 ohms ;
ri = .00008 ohms j
#! = .00025 ohms ;
g0 = .001 ohms ;
b0 = .00173 ohms ;
that is, about one-tenth as large as assumed. Thus the
changes of the values of E0, Elt 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, 700, 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
Z0=r0- jx0, and Zl=rl- j xl .
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
n0 = number of primary turns in series ;
#1 = number of secondary turns in series ;
a = — = ratio of turns ;
Y0 = g0 4- jb0 = primary admittance
Exciting current . ~i I
Primary counter E.M.F. '
.VVWvVl
rw^ww
ALTERNATING-CURRENT TRANSFORMER. 205
Z0 = r0 — j x0 = primary impedance 7. — —
E.M.F. consumed in primary coil by resistance and reactance. ^ -n-f '" ' j**/\.
Primary current ~ /
Z± = r± —jx1= secondary impedance
__ E.M.F. consumed in secondary coil by resistance and reactance .
Secondary current
where the reactances, x0 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 ;
E0 = 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. ;
I0 = 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«>=YOE>. (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 Z0 I0.
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
E0 = E' + Z0S0,
or, substituting (6),
(8)
\°/
136. We thus have,
primary E.M.F., E0 = - aE{ j 1 + Z0 Y0 + ^Z J , (8)
secondary E.M.F., E^ = E{ { 1 - Zl Y}, (7)
primary current, I0 = — — -{Y+a*Y0}, (6)
secondary current, /i = YEl'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.,
E0 = constant ;
secondary induced E.M.F.,
E.M.F. induced per turn,
E 1
n0 -\ \ 7 y \
secondary terminal voltage,
primary current,
^ 4- Y
, . EA Y+a*Y0 _ w ^^ y°
secondary current,
Y
At constant secondary terminal voltage,
-fi1! = const. ;
208 AL TERNA TING-CURRENT PHENOMENA.
secondary induced E.M.F.,
F1 - £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
Y0-\-Y' is the total admittance of a divided circuit with
the exciting current, of admittance Y0, and the secondary
AL TERN A TING-CURRENT TRANSFORMER.
209
current, of admittance Y1 (reduced to primary), as branches.
Thus :
is the impedance of this divided circuit, and
That is :
(17)
The alternate-current transformer, of primary admittance
Y0 , total secondary admittance Y, and primary impedance
Z0 , is equivalent to, and can be replaced by, a divided circuit
with the branches of admittance Y0 , the exciting current, and
admittance Y' = Y/a2, the secondary current, fed over mains
of the impedance Z0, 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/ + Z7. (20)
That is :
An alternate-current transformer, of primary admittance
Y0, primary impedance Z0, secondary impedance Zv 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 Z0-\- Z^, where Z^ =
a2Zlt shunted by a circuit of admittance Y0, 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 Z1, Zn, . . . Zx, 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 Y0) the exciting current, the other branches of the
impedances ZJ + Z7, ZJ1 + Zn, . . . 2f + Zx, 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. £0, over -mains of the impedance
Z0-
Consequently, transformation of a circuit merely changes
all the quantities proportionally, introduces in the mains the
impedance Z0 + Z^, and a branch circuit between Z0 and
Z^, of admittance Y0.
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/ = Z0, 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 — x0, .
with a non-inductive secondary circuit Z^ = rv 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. :
(r0 — jx0}
4- ra—jx0
ALTERNATING-CURRENT TRANSFORMER. 213
+ r0-jx0 f r0-jx0 Y| . . . \ .
R + r0 — jx0 \ R + rn — /#„
or, expanding, and neglecting terms of higher than third
order,
— jx0
^
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 + r0— jx
(r0 -jx0} + R (r0-
= {R + 2 (r0 - y x0} - & (go +jb0} -2 R (r0 - Jx0)
214 ALTERNATING-CURRENT PHENOMENA.
or,
b0}- 2 (r0 -Jx0}(
Neglecting terms of tertiary order also :
Zt=R
Angle of lag in primary circuit :
tan S>0 = ^ , hence,
rt
2^+Rb0 + 2r0b0-2Xogo-2
tan S>0 = 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 R0, and denote,
2 r0 _ _ . Internal resistance of transformer _ percentage
R0 ~ External resistance of secondary circuit ~ na^ resistance,
2 X0 _ __ ratjQ 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 :
R0 = 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, a2.
ALTERNATING-CURRENT TRANSFORMER.
217
140. As an instance, a transformer of the following
constants may be given :
e0 =1,000;
a = 10 ;
£0= 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 = 0 to d= 1.5, with the secondary current ix 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 w0, 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 = 0 represents synchronous motion of the secondary ;
s < 0 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
«0 = number of primary turns in series per circuit ;
/?! = number of secondary turns in series per circuit ;
a = — = ratio of turns ;
«i
Y0 =£"0 H~./A) = primary exciting admittance per circuit;
where
gQ = effective conductance ;
b0 = susceptance ;
Z0 = r0 —jx0 = internal primary self-inductive impedance
per circuit,
where
r0 = effective resistance of primary circuit ;
jr0 = reactance of primary circuit ;
Zu = TI — jxv = 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 :
Zl = r1-jsxl.
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
£0 = 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,
IQ = primary current,
^0 = 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 :
700 = E ' YQ = — «0 e (g0 + /£<))•
Component of primary current corresponding to second-
ary current 7X :
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 x—x-\ --- \- —
s s2
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
'* = -»•< \ .,;,* +IU-
»* (Zj + Z)
ALTERNA TING-CURRENT PHENOMENA.
r nte
yi =
Z, + Z
/! a
P f1 + *f7\^ + Z'Y*
^o_= _ a } a (Z-j + 2}
& I- Z*
( Z, + Z
1+ 2//° x+^oKo]
a2 (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
70 = _ s «0 f ] -T, . .
1 «•(>-!— y**o
r j«,^
A = —
—5 "^ : — ~ + (ro — 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,
-±- (r0 (r, + r)+sx9(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,
£,= - «0 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 = 'i2 n = ( — — }
V ** /
Total secondary power,
**
Internal loss in primary circuit,
r»i -9 -9o
^o = V'o = 4 rt<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, P1 and Pl are always positive ; PQ
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, PQ = 0, P = 0, Pl = 0.
At 0 < s < 1, between synchronism and standstill.
Pl , P and PQ are positive ; that is, the apparatus con-
sumes electrical power PQ in the primary, and produces
mechanical power P and electrical power Pl -j- P^ in the
secondary, which is partly, P-^, consumed by the internal
secondary resistance, partly, Pl , available at the secondary
terminals.
In this case it is :
•Pi + ^i1 _ J
P ~l-s>
that is, of the electrical power consumed in the primary
circuit, P0, a part P^ is consumed by the internal pri-
mary resistance, the remainder transmitted to the secon-
dary, and divides between electrical power, P1 + 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 : P0 - A1 - A1 < 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 < 0 ; 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 ;
rt
the electrical power produced in the primary becomes less
than required to cover the losses of power, and />0 becomes
positive again.
We have thus :
K-£±fl
r\
consumes mechanical and primary electric power ; produces
secondary electric power.
- r-±^ < s < 0
?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, Pi _ 8,000 j«.
0 """ 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,
Y0 = g0 +j&0 = primary exciting admittance per circuit
of the frequency converter.
Z^ = rt —jx^— internal self inductive impedance per
secondary circuit, at the secondary frequency.
ALTERNATING-CURRENT TRANSFORMER. 233
Z^ = r0 — 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 — 0 for noninductive lead.
We then have,
total secondary impedance,
Z + Z1 = (r-^rl)-j(x + x1)
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) (at
where,
primary induced E.M.F. per circuit,
primary load current per circuit,
71 = abli = abe (a{
primary exciting current per circuit,
234 ALTERNATING-CURRENT PHENOMENA.
thus, total primary current,
70 = 71 + /oo
= e (fi
where,
<. = •**+£ <•.=«**+!
primary terminal voltage :
where,
d -— re x d -re -x
ac
or absolute,
e0 = e vX2 + 42
. = e° -
V^2 + 4«
substituting this value of e in the preceding equations,
gives, as function of the primary impressed E.M.F., e0:
secondary current,
7 = > absolu 7 = vi
V4» + 42 v ^i2 +
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 +
Lr°: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
°.RcV5
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
0
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 el, 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 /0 equal circuits,
displaced angulary in space by 1 //0 of a period, that is,
1 //„ of the width of two poles, and excited by /»0 E.M.Fs.
displaced in phase by 1 //0 of a period ; that is, in other
words, let the field circuits consist of a symmetrical /0-phase
system. Analogously, let the armature or secondary circuits
consist of a symmetrical /rphase system.
Let
n0 = 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, El = aE{
= secondary E.M.F. per circuit reduced to primary system;
if // = secondary current per circuit, fl= —
= secondary current per circuit reduced to primary system ;
if r^ = secondary resistance per circuit, rt = a2 r{
= secondary resistance per circuit reduced to primary system ;
if x± = secondary reactance per circuit, xt = a2 x\
= secondary reactance per circuit reduced to primary system ;
if £/ = secondary impedance per circuit, z1 = azz\
= 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 El = — se.
Let
I0 = 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
n0be = effective M.M.F. per primary circuit;
hence ^n0be = total effective M.M.F. ;
z
and
l^-n^be = total maximum M.M.F., as resultant of the M.M.Fs.
of the /0-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 rQ = resistance per primary circuit ;
XQ = reactance per primary circuit ;
thus,
•^o = ro — j XQ = impedance per primary circuit;
rv = resistance per secondary circuit reduced to pri-
mary system ;
xv = reactance per secondary circuit reduced to primary
system, at full frequency, .A7";
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,
/0 =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 + Ez
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., £0;
f=^—
or,
/= _ j + (>i-yji _ E
" (
Neglecting, in the denominator, the small quantity
F, it is
Z, F
0 + r\
or, expanded,
[(j^ + A'0) + r^ -f s^ (rog -
+/ [J3 (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= sEn^-
„ S
tan <D0 =
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
~OEl at the amplitude of 180°. Dividing ~OEl by a in the
proportion of rt -*- sxv and connecting a with the middle b
of the upper arc of the circle OEV this line intersects the
lower arc of the circle at the point 7X rr 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. OGl is in the direction of the
vector OIfv Completing the parallelogram of M.M.Fs.
with OF as diagonal and OGl 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
OIry is in line with OG, the E.M.F. consumed by the pri-
mary reactance 90° ahead of OG, and represented by OIxv
and their resultant Ofz0 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 OE1. Com-
bining OE with OIzQ gives the primary terminal voltage
represented by vector OEy 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- 7X rx = Ev -*• /! J.*!, then
/ir, x .£", /ir, x sE r, _,
= - ^ = constant.
That is, /^ lies on a half-circle with OA = — E' as
diameter.
That means Gl lies on a half-circle ^ in Fig. 115 with
OC as diameter. In consequence hereof, G0 lies on half-
circle^ with FB equal and parallel to OCas diameter.
Thus Ir0 lies on a half -circle with DH as diameter, which
circle is perspective to the circle FB, and Ix0 lies on a half-
circle with IK as diameter, and IzQ on a half-circle with LN
as diameter, which circle is derived by the combination of
the circles Ir0 and Ixv
The primary terminal voltage EQ lies thus on a half-
circle e0 equal to the half-circle Iz9 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 EQ is kept constant, the circle e0 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, 7t = current, per circuit, with
/rarmature 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., Ev 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, Iv lags behind its E.M.F.,
Ev by angle, <av 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 / ' = f0 + 7X the power is consumed, e I = e I0 + e 7r
The power e I0 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, £1fl, is consumed in the secondary
circuit by resistance. The remainder,
P' = fl(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
£0 = 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,
r2
"7
or, st =
Substituting this in the equation of torque, we get the
value of maximum torque,
That is, independent of the secondary resistance, rr
The power corresponding hereto is, by substitution of st
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 st,
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 st, 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, st, is between
synchronism and standstill, rather nearer to synchronism.
Only in motors of very large armature resistance, that is
low efficiency, st > 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, st can be varied by varying the resistance of the
secondary circuit, or the motor armature. Since the slip
of the maximum torque point, st, is directly proportional to
the armature resistance, rlf 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 ^ = Vr02 + (xl + *0)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, st>l,
then ^ > Vr02 + (^ + *0)2.
In this case, the maximum torque point is reached only
by driving the motor backwards, as countertorque.
As seen above, the maximum torque Tt, 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, sv, where
INDUCTION MOTOR. 253
expanded, this gives
sn — -
substituted in P, we get the maximum power,
2 {('i + ''o) + (^ + r0)2 + (^i + *o)2}
This result has a simple physical meaning : (i\ + r0) = 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 Vrx + r0)2 + (x^ + x0}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 Pp is,.
254 ALTERNATING-CURRENT PHENOMENA.
This is not the maximum torque ; but the maximum
torque, Tt, takes place at a lower speed, that is, greater slip,
• since,
-that is, st > sp.
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 ^ + r0, and the impedances, z,
and thus the reactances, x± + x0, should be small. Since
r± + r0 is usually small compared with x^ -f- x0 it follows, that
the problem of induction motor design consists in con-
structing the motor so as to give the minimum possible
reactances, x^ + x0.
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 _io11 010,io1 .
- 8
and, displacement of phase, or angle of lag,
fi + r0] + *! [Jfx 4- Jf0]) - jf (r0 ^ - *0 rt)
„ _
1 W°
r0)
INDUCTION MOTOR. 255
Neglecting the exciting current, g = 0 = b, these equa-
tions assume the form,
or, eliminating imaginary quantities,
and tan w0 =
+ 'o
That means, that in starting the induction motor without
additional resistance in the armature circuit, — in which case
^ + x0 is large compared with t\ •+• r0, and the total impe-
dance, z, small, — the motor takes excessive and greatly
lagging currents.
The starting torque is
T0=
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, Ta <
would be the theoretical torque developed at 100 per cent
efficiency and power factor, by E.M.F., E0, and current, /,
at synchronous speed.
Thus, T0<T00,
and the ratio between the starting torque T0, and the theo-
retical maximum torque, T^, gives a means to judge the
perfection of a motor regarding its starting torque.
This ratio, T0 / Tw, exceeds .9 in the best motors.
Substituting 7 = E0 / 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^ + r0)2 + (Xl + *0)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 T0.
The smaller value of r1" 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 w0
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 EQ = 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, Zx = .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.
rt = .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 rt == .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
z£0=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,
Y0 = g0 +jbw the primary exciting admittance,
Z0 = r0 —jx0, 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. cot 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,
/0 = /! + /oo = * (^i + A)
where,
The E.M.F. consumed by the primary impedance is,
^ = /oZ0 = * (r0 ->0) (^
the primary counter E.M.F. is e, thus the primary impressed
E.M.F.,
£,
where,
c\ —
or, absolute,
^0 =
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* + r22) W
where N= frequency.
The power output is torque times speed, thus :
The power input is,
^•l2 +
The voltampere input,
o2 ( Vi + V,) /o2 ( Vi - V8)
hence,
efficiency,
J\ _ a, (I - s)
J? Vi + V2
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
~Q0 ~ 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,
K0 = .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, slt is the same as that of the induction motor
except that sl 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,
Y0 = gQ + j\ — primary exciting admittance,
Z0 = r0 — jxQ = 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\ — Jsx\
270 ALTERNATING-CURRENT PHENOMENA.
where,
primary exciting current,
In = EY0 = e
thus, total primary current,
/0 = /i + foo
where,
^1 = <*\ + £b
primary impedance voltage,
& = S0(r0-
primary induced E.M.F.,
thus, primary terminal voltage,
£0 = e(l-s) -S0(r0-j[l- s] x0) = e
where,
fi = ! - s ~ rA - (1 - s
hence, absolute,
e0 = e V^
and,
Thus,
Secondary current,
T eO (ai
Primary current,
j _ eo (A + A)
Primary terminal voltage,
j-. ^0 \^"l
£« = —T-,
INDUCTION MOTOR.
Torque and mechanical power input,
T— P —\f nl — 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 0 = 110, and the constants,
K0 = .01 + .!/
171. As instance may be considered a power trans-
mission from an induction generator of constants Y0, Z0,
Zj, over a line of impedance Z = r —jx, into a synchron-
ous motor of synchronous impedance Zz = rz — jxz, operat-
ing at constant field excitation.
Let, e0 = 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,
E0 = e
thus, terminal voltage at synchronous motor terminals,
where,
4 = fi ~ rA ~ C1 - J) *A 4 =
Counter E.M.F. of synchronous motor,
E2
'
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, _ e2 (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.
eQ = 125 volts (at full frequency) and synchronous impe-
dance Z2 = .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.
Z0 = r0 — jxQ = primary self-inductive impedance.
Z^ = i\ —jxv = 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)r1
i ~ r*+ (2J-1)2^12 z ~ r*+ (2s-
primary exciting current,
4 = * (g +JI>}
thus, total primary current,
72 = 7, + 70 = e (
where,
primary induced E.M.F.,
se
primary impedance voltage,
ft (ro — >^o)
thus, primary impressed E.M.F.,
£3 = se + 72 (r0 -jsx0) = 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.,
£0 = E, + S(r0 ->0
where,
^i =/i + ^o5i + *b£a
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 = 0 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,
T2 = [,/J = ^
hence, its power output,
/»,= (!- s) r2 = (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,
P1 =
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 :
Z0 = 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 0
' SECO
NO MC
TOR.
|
H
8000
6000
-
4000
\
2000
1
—
—
—
""-s.
\
I
0
M
\\
\
-2000
\\
X
^
-4000
£
/
f
-60C(
./
-8000
1
0
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>0 at synchronism, and falls off with decreasing speed to
zero at standstill, if no starting device is used or to 4^ = /<£0
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] Fo1
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 Z1 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 Zl / 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 ;
TQ = /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 YQ = g0 -\-jb0 = primary exciting admittance of the
motor per delta circuit.
Z0 = r0 — jxQ = primary self-inductive impedance per
delta circuit.
Z^ = i\ —jx^ = secondary self-inductive impedance per
delta circuit reduced to primary.
Let
Ys = gs — jb9 = 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
Fo1 = 1.5 Y0 = 1.5 (£•„ +/£0) = primary exciting admit-
tance,
Y? = 1.5 Y0 [1 - (1 - 0 s]
= 1.5 (g0 +/£<)) [1 — (1 — 0 J] = secondary exciting
admittance at slip s.
Z0l = ?^° = 2fo~^*o) = primary self-inductive impe-
o o
dance.
Zxi = £L±^ Zi = ^L + ^ (ri -jsxj = secondary self-
o o
inductive impedance.
Z,1 = ^ = 2 (r° ~ ***> = tertiary self-inductive impe-
o o
dance of motor.
Thus,
Y4 = -^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] Y4
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 = 71 + /o1
= /, + /, +
= ' (*i + A)
primary impressed E.M.F.
thus, main counter E.M.F.
or,
and, absolute
V^2 + c*
hence, primary current
T_slW + %
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^= r1 [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 <?0 = single-phase impressed E.M.F.,
Y — total stationary admittance of motor per delta cir-
cuit,
Ez = 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 + Y8, and Y,
are connected in series, and have the respective E.M.Fs. E^
and e0 - Ey It is thus,
Y+ Ys+ Y=e0-£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 Aa
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 havet 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 E0 = 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 E0
where
n = total number of turns in series on the armature,
JV = frequency,
M = total magnetic flux per field pole.
Let x0 = synchronous reactance,
r0 = internal resistance of alternator ;
then Z0 — r0 — j x0 = internal impedance.
If the circuit of the alternator is closed by the external
impedance,
Z = r-jx,
the current is
E0 E0
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
E0=
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
0 210 2
0
Fig. 129. Field Characteristic of Alternator on Non-inductive Load.
' +
and the values of the internal impedance,
z0 = r0 -jXo = 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
0 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
:0=25OO, Z?1-10j, r =
RISTIC
o, 90° Lag
\
\
1 R =
0.
\
\
\
\
\
\
k
o »
>C"
-X
A
/
S
\%<
\
o 2"
X X
t
s
%
\
/
/
\
\
\
/
\
\
>
/
\
\
\
/
\
0
/
s,
\
Fig. 131. Field Characteristic of Alternator, on Wattless Inductive Load.
5
I
li'.'U
1000
HM
^
^
Ns
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^
**•!••
0
•
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
E0-2500, Zo-1-IOj,
= o. 90°Leading 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.,
E0 = 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 0 -.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, x0 , 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, x0, only as a term of second
order in the denominator.
On inductive circuit, however, x0 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, XQ, 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, .r0.
On non-inductive circuit, if
theoutputis
hence, if
or
then
dr
That is, the power is a maximum, and
£
and
7 =
V2 So {so + r0)
Therefore, with an external resistance equal to the inter-
nal impedance, or, r — ^0 = VV02 + 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 reaction1.
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, Al} A2,
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 — _/>•/ = a2 (cos w2 — j sin w2) = induced E.M.F. of sec-
ond machine ;
/! = /! -f-//i' = current of first machine ;
/2 = /2 -j-yY2' = current of second machine ;
Z^ = T! — jxi = internal impedance, and Yv = gi -\- jbl = inter-
nal admittance, of first machine ;
Z2 = r2 — jxz = internal impedance, and K2 =gz ~\~ jb<i = inter-
nal admittance, of second machine.
Then,
i^! , or ^ —je^= (e
2Z2, or <?2 —jej= (e
72 , or
This gives the equations —
4* + *"-**;
or eight equations with nine variables: ^, ^', ^2, ^/, /lf
316 ALTERNATING-CURRENT PHENOMENA.
Combining these equations by twos,
elrl -f eSxj. = er^ + t\2l2-
e*r9 + ^/^2 = e
substituted in
'i + H =
we have
and analogously,
'1^1 — ^iVi + 'a *a — <?aVa = ' (^ + ^2 +
dividing,
b + ^i + ^2 ^i ^;i + <?a ^ — ^iVi — ^a' ^2 '
substituting
g = V COS a Cl = tfj COS Wj ^2 = ^2 COS d)2
^ = z/ sin a ^/ = ^ sin oJj ^2' = a2 sin <o2
gives
a\ v\ cos (en — aQ + a2z>2 cos (a2 — a2)
tfj z/! sin (ai — w^) -\- a^Vs sin (a2 — a>2)
as the equation between the phase displacement angles
and oi2 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 = Pl + 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 x0 — t r0
e2' = /2 .r0 — // r0 .
*g=i\ +'a
eb = i{ + /a'
4* + 4" -</ + *"-<
Combining these equations by twos,
substituting el = a cos o^
e{ = a sin o^
^2 = a cos 0)2
e2f = a sin o)2,
we have a (cos wx + cos wa) = * (2 + r0^ +
a (sin Si + sin w2) = e (x^g — r0 li)
expanding and substituting —
8 =
318 AL TERN A TING-CURRENT PHENOMENA.
a cos e cos 8 = e ( 1 +
rQg -\-Xzb
a sin e cos 8 = ^^ ^
hence
That is
-and cos 8 = -
tan e = — ^ ^ — = constant.
+ A2 = constant;
-M1*5
z±aiy , /^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-
(''o2
subtracted and expanded —
.or, since
<?! — <?2 = ^ (cos wj — cos G2) = — 2 tf2 sin c sin 8
^/ — <?/ = a (sin wx — sin w2) = 2 a cos e sin 8 ;
we have
2 a s*n 8 - r0sin c}
— 2 ay0 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 £ - r0 sin c};
,
H
XQ£ — r*b
2
i 'of, T •*<) " \ i
2 J + V 2
we have,
2a2 sin 8 cos 8 j *0( 1 + r°* + x°*\ — r0
expanding,
A/ = nr
Hence, the transfer of power between the alternators,
A pt is a maximum, if 8 = 45° ; or Wj — w2 = 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 = 0 ; 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., EQ,
by a circuit of the impedance Z.
If E0 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 E0 is the E.M.F. at the generator terminals, Z is
the impedance of motor and line, including transformers
and other intermediate apparatus. If EQ 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. EQ, and motor of induced'
E.M.F. El; or, more general, two alternators of induced
E.M.Fs., E0, Elf 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 = Vr2 -f x2 =
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 E2 = xi, and 90° ahead of
the current, hence the E.M.F. consumed by the impedance
is E = V(£,,)2 + (E2f, or = i Vr2 + 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, Elf EQ;
or El and E are components of EQ ; that is, EQ is the
diagonal of a parallelogram, with El and E as sides.
Since the E.M.Fs. Elf Ez, E, are represented in the
diagram, Fig. 138, by the vectors OE~lf OE2, OE, to get
the parallelogram of £Q, Elt E, we draw arcs of circles
around 0 with EQ , and around E with El . Their point of
intersection gives the impressed E.M.F., OEQ = EQ, and
completing the parallelogram OE EQ E± we get, OE± = E± ,
the induced E.M.F. of the motor.
IOE0 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
EQ and E! upon OI, it is —
P0 = 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 OEl = power consumed in the circuit by effective
resistance.
SYNCHRONOUS MOTOR.
323
Since the circles drawn with EQ 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£ — Ez, 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 E1E(),
in the same way, we have : OEl = E1 = induced E.M.F. of
the motor, OEZ = 2?a = E.M.F. at motor terminals or at
end of line, OE3 = E3 = E.M.F. at generator terminals,
or at beginning of line. OEQ = EQ = 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. El
and
200. Figs. 139 to 141 show several such diagrams for
different values of Elf but the same value of / and EQ.
The motor diagram being given in drawn line, the genera-
tor diagram in dotted line.
Fig. 140.
As seen, for small values of E1 the potential drops in
the alternator and in the line. For the value of E1 = E0
the potential rises in the generator, drops in the line, and
rises again in the motor. For larger values of Ely 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. Ev, constant current
strength I = i, variable motor excitation Ev (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 EQ is
constant, EQ lies on a circle eQ with EQ as radius. From
the parallelogram, OE EQ El follows, since E1 EQ parallel
and = OE, that El lies on a circle el congruent to the circle
eQ, but with Ei} the image of E, as center : OEi = OE.
We can construct now the variation of the diagram with
the variation of El ; in the parallelogram OE EQ E1 , O and
E are fixed, and E0 and El move on the circles <?0 el so that
EQ E^ is parallel to OE.
SYNCHRONOUS MOTOR.
327
The smallest value of El consistent with current strength
/ is Olj = E^, 01 = EQ. In this case the power of the
motor is Olj1 x /, hence already considerable. Increasing
El 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 515 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 61} 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 1l, 7, the maximum value
of Elt consistent with the current /, has been reached, and
passing still further the E.M.F. El decreases again, while
the power still increases up to the maximum at Slt 8, and
then decreases again, but still El remaining generator, EQ
motor, until at 11^ 11, the power of EQ becomes zero; that
is, EQ changes again to a generator, and both machines are
generators, up to 12lf 12, where the power of El is zero, El
changes from generator to motor, and we come again to
the motor side of the diagram, and while El still decreases,
the power of the motor increases until lu 1, is reached.
Hence, there are two regions, for very large El from
5 to 6, and for very small El from 11 to 12, where both
machines are generators ; otherwise the one is generator,
the other motor.
For small values of El the current is lagging, begins,
however, at 2 to lead the induced E.M.F. of the motor Elf
at 3 the induced E.M.F. of the generator E0.
It is of interest to note that at the smallest possible
value of EI} 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. EQ and El constant, I variable.
Obviously EQ lies again on the circle eQ with EQ as radius
and O as center.
Fig. 143.
E lies on a straight line e, passing throtigh the origin;
Since in the parallelogram OE E0 Ev EEQ = E^ we
derive EQ by laying a line EEQ = E± from any point E
in the circle eQ, and complete the parallelogram.
All these lines EEQ envelop a certain curve elt 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, El = EQ ; 2d, El < EQ ; 3d, £1>£Q.
In the first case, El = EQ (Fig. 127), we see that at
Fig. 144.
very small current, that is very small OE, the current /
leads the impressed E.M.F. EQ by an angle EQOf = WQ.
This lead decreases with increasing current, becomes zero,
and afterwards for larger current, the current lags. Taking
now any pair of corresponding points E, EQ, and producing
EEQ until it intersects eit in Eif we have ^^ Ei OE — 90°,
El = EQ , thus : OE1 = EEQ=OEQ = EQEt ; that is, EE{ =
SYNCHRONOUS MOTOR.
331
2EQ. That means the characteristic curve el is the enve-
lope of lines EEiy of constant lengths 2EQ, sliding between
the legs of the right angle Et OE; hence, it is the sextic
hypocyloid osculating circle <?0, which has the general equa-
tion, with e, ei as axes of coordinates :
In the next case, E1 < EQ (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 OEl : 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 ev as envelope of the
lines EEty is a finite sextic curve, shown in Fig. 144.
If El < EQ , at small EQ — El , H can be above the zero
line, and a range of leading current exist between two ranges
of lagging current.
In the case E1 > EQ (Fig. 145) the current cannot equal
zero either, but begins at a finite value C±, corresponding
to the minimum value of OEQ : // = * — -. At this
value however, the alternator E1 is still generator and
changes to a motor, its power passing through zero, at the
point corresponding to the vertical tangent, onto elf 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 El = O, that
is, when the motor stands still, is 70 = EQj 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, El 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. EQ of the generator, we have as
the locus of EQ the point EQ (Fig. 146), and when E with
increasing current varies on <?, E± must vary on the straight
line ev parallel to c.
Hence, at no-load or zero current, El = E0, decreases
with increasing load, reaches a minimum at OE^ perpen-
dicular to clt and then increases again, reaches once more
Fig. 146.
El = EQ at E?, and then increases beyond E0. The cur-
rent is always ahead of the induced E.M.F. El 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 = EfEQ =
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.
El here acting in the same way as the condenser capacity
in Chapter IX.
204.
Fig. 147.
D. En = constant ; P = constant.
If the power of a synchronous motor remains constant,
we have (Fig. 147) / x OE^ = constant, or, since OE1 —
SYNCHRONOUS MOTOR.
335
Ir, I = OE1/ r, and: OE1 x OE? = O£l X E1EJ =
constant.
Hence we get the diagram for any value of the current
/, at constant power Plt by making OE1 = I r, E1E01 = Pl j I
erecting in EQl a perpendicular, which gives two points of
intersection with circle eQ, EQ, one leading, the other lagging.
Hence, at a given impressed E.M.F. EQ, 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, OEl = EEQ small,
corresponding to a lagging current ; and the other, OEl =
EEQ large, corresponding to a leading current. The former
is shown in dotted lines, the latter in drawn lines, in the
diagram, Fig. 147.
Hence a synchronous motor can work with a given out-
put, at the same current with two different counter E.M.Fs.
336
ALTERNATING-CURRENT PHENOMENA.
E1. In one of the cases the current is leading, in the
Dther lagging.
In Figs. 148 to 151 are shown diagrams, giving the points
E0 = 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, Elt is OElt equal and
shown in the diagrams, to avoid
The counter E.M.F.
parallel EEQ, 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 < El < 2,200,
P = 6,000 340 < £, < 1,920, 7 < I < 43 Fig. 133.
P = 9,000 540 < El < 1,750, 11.8 < / < 38.2 Fig. 134.
P = 12,000 920 < El < 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. Ev and
of current /, becomes narrower with increasing output.
338 ALTERNATING-CURRENT PHENOMENA.
In the diagrams, different points of EQ are marked with
1, 2, 3 . . . , when corresponding to leading current, with
21, 31, . . . , when corresponding to lagging current.
The values of counter E.M.F. Ev 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. El 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 = Vr2 -j- x2 = impedance of the circuit of (equivalent)
resistance r and (equivalent) reactance x = 2 TT NL, containing
the impressed E.M.F. e0* and the counter E.M.F. et 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. e1; hence —
p = *>! cos ft,^), (1)
thus, —
* If f0 = 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 t0= 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 :
e0 = 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 ^0, it is
(Fig. 152) :
e1 „ 2 eZ .1 „?. ^2 ,'2
hence, cos (*,.#) = •*- — — = -0 — - — . (5)
2 e^e '2,zie^
since, however, by diagram :
cos (el , 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., <?0 ; 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 = Vr2 + x*, three variables are left,
ev 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 e1.
The meaning of equation (7) is made more perspicuous
340 ALTERNATING-CURRENT PHENOMENA.
by some transformations, which separate ev 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 — e2
jr. 753.
we jret
a — a V2 — e b = V(l — e2) (2 a2 — 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*i2) - z* (Vo2 - 2 r/)}2 + {r x (e? - z*i2)}2 =
*2.sV(^02-4r/) (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 :
r3- (e? — z2 /2) - z2 (ef — 2rp)=xze() V<r0a — ±rp cos <£
rx (e? - z2/2) =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
el and i as functions of / and the parameter <£ enable us
to construct the Power Characteristics of the Synchronous
Motor, as the curves relating ev 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 e1 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 el and i [equations (19) and
(20)] contain the square root, W02 — 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 rp = 0,
°r>
/ = £.. (21)
This is the same value which represents the maximum
power transmissible by E.M.F., eQ, 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 = - (', 0,
= ~ <X '), (24)
and consequently,
(.-0,0=0; (25)
that is, the current, z, is in phase with the impressed
E.M.F., *0.
If 2 < 2 r, el < <?0; that is, motor E.M.F. < generator E.M.F.
If z = 2 r, el = e0 ; that is, motor E.M.F. = generator E.M.F.
If z > 2 r, <?! > r0; 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 / = 0 into equation (7) gives, after squar-
ing and transposing,
e* + e<* 4- 3*,-« - 2 ^V - 2 22rV + 2 ra*'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., eit is ^ = 0 at / = ^2. (29)
The minimum value of current, z, is / = 0 at et = e0 . (30)
The maximum value of E.M.F., elt is given by Equation (28)',
/= e* + 22z2 -e<?±2 xiel = 0 ;
by the condition,
hence,
The maximum value of current, z, is given by equation
(28) by
— = 0, as
del
(32)
If, as abscissas, elt 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 el , 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. eQ,
346 AL TERNA TING-CURRENT PHENOMENA.
This gives from diagram Fig. 153,
e1* = e(? + i*z*-2ie0r, (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,
enx
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. el} at which coincidence
SYNCHRONOUS MOTOR. 347
of phase (eQ , -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 (*0, *) ; (38)
hence,
cosfo, 0 = /+/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,
<?! = \/(^0 — 2 r)2 — z'2jr2, which represents the output transmitted over an
inductive line of impedance, z = vV2 + jr2 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), el = t0 ~ 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,
<>02 - e*y + **z2 (s2 + 8 r2) + 2 j*e* (5 r2 - 22) - 2 / V
(32 + 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., el = 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 eQ , 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 eQ < <?! , 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
> <?0 or < *0.
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,
<?0 = 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,
25002 - e* - 500 i* - 20 / = 40 V*V -/2 (7)
{^2 + 500 ?2 - 31.25 X 106 + 100 /}2 + (2 ^2 - 1000 /2}2 =
7.8125 x 1015 - 5 + 109/. (17)
el = 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 104 =p 40 /^ = 0
^ = 20 / ± V6.25 X 104 — 100 i*
At the" minimum value of C.E.M.F. e1 = 0 is / = 112 (29)
At the minimum value of current, / = 0 is el = 2500 (30)
At the maximum value of C.E.M.F. ev = 5590 is / = 223.5 (31)
At the maximum value of current i — 250 is el = 5000 (32)
Curve of zero displacement of phase,
€l = 10 V(250 - O2 + 4 *a (34)
= 10 V6.25 x 104 — 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 106-^2)2 + .65 X 106 /« - 1010/2 = 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. el.
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 el can be varied unrestrictedly,
while in reality the possible increase of el is limited by
magnetic saturation. Thus in Fig. 155, at an impressed
E.M.F., eQ = 2,500 volts, el 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 elt 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 elf 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., elt 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.
<?108
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, gl = — 2-TrnJV® {sin X cos /? + k cos X sin B\ 10~8,
or, since $ = — — — — ,
et = 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
Zl = r1—jx1= 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, 7X = — L = - e - _ - ,
primary exciting current, I0 = eY= e (g +jb},
hence, total primary current,
Primary impressed E.M.F., E0= — E + IZ\
= e 1 + (sinX
Neglecting in E0 the last term, as of higher order,
£0 = 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 _ **ri (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 + (xl + x sin \ — kr cos X)2 '
and the torque in synchronous watts,
P <?02 cos X (x1 sin X + r^k cos X)
~~ /£ ~~ (/i + ?" sin A + £# cos X)2 + (xt + x sin X — kr cos X)2
or T= V27r^lO-8 [/!<!> sin X 7X cos A]' = [^/! cos X}>
_ ^ cos X (xl sin X + r^k cos X)
r2 + x2
The stationary torque is, k = 0,
_ ifo2^ sin X cos X
0 = (rx + r sin X)2 + (^ + * sin X)2 '
and neglecting the primary impedance, r = 0 = x,
_ e^x^ sin X cos X _ (fo2^ sin2 X
which is a maximum at X = 45°.
At speed k, neglecting r = 0 = x,
<?02 cos X (X sin X + r^k cos X)
— r2 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,
(rx 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 = 0 we have,
T--
fa + kxf + (*t - krf
that is, T = 0 at k = 0, or, the motor is not self-starting,
when X = 0.
P =
dP
which is a maximum at constant X = 0 for, -— = 0, which
dk
gives,
rx-, — xr-.
MOO
--
'..i i'j
S
•^
"~
m
-t^>
,
/
no
1
/
/
R
:PL
LS
ON
M(
5TC
3R
;••')
m
0
/
V
OC
rt
/
/
r=
.!
r, '
05
joa
>
/
X
2.
x.
1.
M
/
p-
DO
1.17
0 1
j^ ^
<)
g
k
14
— i,
-,]
I
21 K I
/
UW
K1
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° e0 = 100.
r= .1 r1= .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,
^/ = 27r«1^V<I>110-8,
E.M.F. consumed by resistance,
Er = (r + *i) I,
where
/ = current passing through motor, in amperes effective.
Further, it is :
Field magnetism : $ = n 7108 / (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/108
1 = "V";
Substituting these values,
(R
ptfNI
E' =
(R
E1 = ^^niNI .
Er = (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 + rlf it
1 + |!
lan W = ? «j ^
7T /« 7V
^n2
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 AT=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 Nv 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
0
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. EQ of the generator.
Furthermore, a shuttle armature without any winding
will in an alternating magnetic field revolve when once
brought up to synchronism, and do considerable work as
a motor.
These phenomena are not due to remanent magnetism
nor to the magnetizing effect of Foucault currents, because
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., EQ, of the machine, — it follows that, if the
current is in phase or in quadrature with the E.M.F. EQ,
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 ZJ \ 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 —
+ -J2 — y COS 2 u>
The instantaneous value of power is
2sin(/? — c(,)f/l — |\ cos w cos y3 +
sin eo sin /3 [. .
7
and, expanded —
sin 2 eo cos2 /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> > 0 ; that is, with lagging current; positive, that is, the
machine produces electrical, and consumes mechanical,
power, as generator, if to > 0 ; 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,
^g2 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 > 0 : synchronous motor, with lagging current,
w < 0 : 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> = A0 + A, cos /8 + A* cos 2(3 + Az cos 3/8 + . . .
+ ^ sin £ + -#2 sin 2 0 + .#, 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 A0, Az, At . . . By 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 = $0 - 3>180
= 2 \Ai cos /3 + Aa cos 3 (3 + A6 cos 5 0 + . . .
+ BI sin j3 + Bz sin 3 ft + jB6 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 0 {1 + 6! cos (2 0 - SO + e, cos (4 £ - oV,) + . . . f ,
where the term ey 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
//
\\
0
•'/
/-
"--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) 0 - «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 + ^"cYcos(2yj8-aY),
1
the instantaneous magnetic flux is :
00
= $ cos 13
ey 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 jg 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,
r0 = 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 0 - .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 0 — .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 — iar2 + 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 = a1 +y<zu = a (cos a -\-j sin d)
of the alternating wave,
A — a0 cos (<£ — a)
applies to the sine wave only.
The general alternating wave, however, contains an in-
finite series of terms, of odd frequencies,
A = Al cos (<£ — #1) 4- Az 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* +jn */)
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
xm is that part of the reactance which is proportional to
the frequency (inductance, etc.).
x0 is that part of the reactance which is independent of
the frequency (mutual induction, synchronous motion, etc.).
xc 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 xm
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 (<£ — «,) + As cos (3 <£ — a8) +^5 cos (5 <f> — #6) +. .
is the square root of the sum of mean squares of individual
harmonics,
A= V i { A? + A82 + 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 :
= Pl +JPJ
1
where,
41-4 ALTERNATING-CURRENT PHENOMENA.
The jn 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~17
where,
consists of a series of inductance factors qn of the individual
harmonics.
As a rule, if <f = 2^-1 ^n2,
for the general alternating wave, that is q differs from
fo=vr^72
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 jn qn, 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 • ( *CN
Z = *•—./„ \nxm --
that is, containing resistance r, inductive reactance xm and
capacity reactance xc 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 w5 = - 1.33
It is thus in symbolic expression,
Zj = 10 + 80/; *! = 80.6
Z3 = 10 zz = 10
ZB = 10 - 32/; 25 = 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 10-32y5
= (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, ll = 11.15
" " " " triple " I3 = 40
« « « quintuple " I5 = 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/; - 850y5
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 = I2 r, as obvious.
The circuit factor is,
• Q El
= .418 + .239 j\ - .0203/5
or, absolute,
u = V.4182+ .2392 + .02032
= .482
The power factor is,
p = .418
418 ALTERNATING-CURRENT PHENOMENA.
The inductance factor of the first harmonic is : ql = .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 C0 = 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, - .085 + .067)
The synchronous impedance of the alternator is,
ZQ = r0 —jnnx0 = .3 — 5 njn
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,
106
or, in symbolic expression,
Let
Z^ =.r — jn 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) -j3 (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 js
(.3 + r) -A (x - 132) (.3 + r) - jz (3 * - 29)
__ 2.iiy5 1.13;; -i
(.3 + r) -j6 (5x- 1.4) ^ (.3 + r) -/7 (7 x + 16.1) J
thus the apparent capacity reactance of the condenser is,
and the apparent capacity,
106
^.) ^r = 0 : 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 = 0 : 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 0 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 TV 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 C0, 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 [lt + .12, - .23B - .13,]
and of synchronous impedance,
Z0 = .3-5*/;
feeds over a line of impedance,
C4PJ
CITV
Co =
= 20
mf i
CM
CL'IT
OF
r,E\
HAT
R
1
8
= EI O-J--I.L— .ya-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
/\
0
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
0
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,
Z2 = .3 - C «/;
The total impedance of the system is then,
Z = ZQ + Zl + Z2
= 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 - 45y8
460 260
~~ 2.6 - 75 j\ 2.6 - 105 jj
= «
where,
aj1 = 22.5 - 25.2 cos co + 146 sin a>
ag1 = .306 - .69 cos 3 to + 11.9 sin 3
a,1 = - .213
«7i = - .061
V1 = 130 - 146 cos w - 25.2 sin a>
^8« = 5.3 - 11.9 cos 3 o> - .69 sin 3 o>
a* = - 6.12
a7u = - 2.48
or, absolute,
1st harmonic,
3d harmonic,
5th harmonic,
a6 = 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 asn sin 3o>)
= /V + /'s1
^ = 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 + as11 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
/ 70, 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 I0, Curve IV.,
the difference between the total current and the current of
fundamental frequency, / — alt in percentage of the total
current /, and V the power of the third harmonic, Pj, in
percentage of the total power P1.
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 ZQ = Zl = .6 — 2.4/
and a sine wave of E.M.F. e0 = 110 volts is impressed upon
the motor.
The power output P, current input 7S, 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 Is = 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 = ls - 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, (lx + .12. - .235 - .134,)
to give the same output, the fundamental wave must be the
same : e0 = 110 volts, when assuming the higher harmonics
in the motor as wattless, that is
£0 = 110, + 13.2, - 25.3B - 14.7,
= *o + £<?
where £0l = 13.2, - 25.3B - 14.7T
= component of impressed E.M.F. of higher frequency-
REPRESENTATION^ Of ALTERNATING WAVES. 427
The effective value is :
EQ = 114.5 volts.
The condenser admittance for the general alternating
wave is
Yc= -.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
Z1 = - njn (XQ + XJ
= - 4.8 njn
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, Yc
= - 4.28/i - 1.54/3 + 4.93/5 + 4.02/7
High frequency component of motor impedance current,
|£ = .92/3 - 1.06y5 - .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,
/o1 = |f + & y1
= .99y3 - 1.14,; - .47/7
and total current,
without condenser,
4 = 4 + 41
= Is + .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
0
.01
.02
.035
.05
.07
.10
.13
.15
P
0
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 ;
e2=£sm((3-^L\',
en = 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, e3 = - /.
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£+ysin«2=£= ^Yl - cos i^'-ysin4^'
« » • « 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= — •
V25
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 <5max
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 £ = 0 it is :
p = P (1 — cos 2 0) ;
hence the flow of power varies between zero and 2 Pt 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 ez> *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 = glt\ -f <?2/2 -j- e,j, -f . . . .
The average flow of power is :
P = £i /i cos £>! -(- E<i /2 cos w2 -f- . . . .
The polyphase system is called a balanced system, if the
flow of energy :
/ = e\i\ + <V2 + 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) ;
ez = E V2 sin (ft - 120), /2 = / V2 sin (0 - « - 120) ;
ez = 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 = T5, 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) ;
e2 = 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 = T7, 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>0) = 2 sin ^ - ^ ) cos (/? - ^ j =
substituted,
^M' + ^j.
or, expanded :
— P2 (x2 + /* + 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> = 0
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* — P2 = 0: circle. & / a = + 1.
Three-phase circuit, 60° lag : / = P, 6 = 1, a = 1
a? +/- 7>a = 0 : 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 _|_ yy = 0< fila = Qf
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 0° 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 0° 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 zth to the £*
terminal is :
Eki = ** 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 :
cl 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, Zt the star im-
pedance.
The ring potential at the end of the line between ter-
minals i and k is Eik, and it is :
Eile = — Eti.
If now Iik denotes the current passing from terminal i to
terminal k, and Zik impedance of the circuit between ter-
minal i and terminal k, where :
fit = ~ /*,,
Zt* = Zti,
it is Eik = ZitIik.
If Iio denotes the current passing from terminal i to a
ground or neutral point, and Zio is the impedance of this
circuit between terminal i and neutral point, it is :
Eio = €*£- ZiSi = Ziolio.
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.
Zt = the impedance of the line connected to a terminal
i of the generator, including generator impedance.
EL = the E.M.F. at the end of line connected to a ter-
minal i of the generator.
458 ALTERNATING-CURRENT PHENOMENA.
Eik = the difference of potential between the ends of
the lines i and k.
Iik = the current passing from line i to line k.
Zik = the impedance of the circuit between lines i and k.
Iio, Iioo . . . . = the current passing from line i to neu-
tral points 0, 00, ....
Zio, Zioo . . . . = the impedance of the circuits between
line i and neutral points 0, 00, ....
It is then :
Zio = Zoi, etc.
2.) Et =JE-ZiIi.
3.) Ei = Zi0fi0 = Zioofj00 = . . . .
4.) Eik = Et'- E{ = (t* - e') E - (Zklk - ZJ^).
5.) Eik = ZikIik.
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 e1' E Ii cos $i at the generator
1
•f = "^i ^* Eik Iik 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 Elt E2, Ez . . . . 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., Elf E^, . . . . can be dissolved into
two components, El and Elt E^ and Ez, .... 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;.
Ev E2, .... 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, —
£il / J, £2l /?»•••• 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, 0 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 : 0 secondary turns.
The second phase : E' sin (ft — 90) will give in the first
transformer : 0 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
0
- 250 - 250
4- 433 - 433
10
0
0
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 TV °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? rt = iz rt /.3, and the total loss is z2 1\, while in
the single-phase system it is 2 t*r. Hence, to get the same
loss, it must be : rv = 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 rz, the re-
sistance of the central wire = r2/V2, and the loss of power
per outside wire is z'22 r2 , in the central wire 2 z'22 r2 / V2
= z'22 r2 V2 ; hence the total loss of power is 2 z'22 r2 + z'22 r2
V2 = z'22 r2 (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 z2 = z'/2; hence the loss, *2;a(2 + ^ . Since this
loss shall be the same as the loss 2z'2r in the single-
phase system, it must be 2 r = - — — r2 , or r2 = ~ . .
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 r3, 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'32 r3 = | z2 rs.
Hence, to give the same loss 2 z'2 r as the single-phase sys-
tem, it must be rs = 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 Z2, Zs = impedances of the lines issuing from genera-
tor terminals 1, 2, 3,
and Yl} Y2, Ys = 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 //, 72', 7/ = currents flowing through the admittances Y1,
F2, F3, 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 + ztIt) •
the differences of potential, and
ti
(4)
the currents in the receiver circuits.
These nine equations (2), (3), (4), determine the nine
quantities : flt 72, /3, //, 7a', 73', ^', Ti^ £&•
Equations (4) substituted in (2) give :
(5)
These equations (5) substituted in (3), and transposed,
give,
since £l = c E
Ez = £ 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_V7N>/
^zt y 2-^D — V*1 ~r -^s^i T *»^V /
A
c, F2Z3, F3Z2
a, - (1 + ^^3 +
, Y,Zlt -(1 + F3Z1+F3Z2)
- (1 + Y,Z2 + FiZ,), c, F3Z2
F.Z3, c2, YtZ,
Y.Z,, 1, - (1 + F3ZX + F3Z2)
(i + ^iz. + yiz,), F2z3, £
A = / FIZS, - (i
FaZ2, F2ZX,
it is:
D
72 = i
__ F2Z>2-
hence,
(8)
(9)
(10)
(11)
THREE-PHASE SYSTEM.
293. SPECIAL CASES.
A. Balanced System
Y, = F2 = F8 = F
Z, = Z2 = Z3 = 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! = F2 = 0, F8 = F
Zj = Z2 = Z3 = Z.
Substituted in equations (6) :
= ( unloaded branches.
E — E3'(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! = F2 = F, F8 = 0,
Zt = Z2 = Z3 = Z.
Substituting this in equations (6), it is :
e E — E{ (1 + 2 FZ) + .£/ FZ = 0)
£E — El (1 + 2 FZ) + E{ FZ = 0 J
E — £s' + (,£,' + ^2') FZ = 0 unloaded branch,
or, since :
E — Ez''— EZ'Y2 :'= 0,
E1 = ?
\ + FZ
thus:
1 + 4 FZ + 3 F2Z2
1 + 4 FZ + 3 F2Z2
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 0 and 1 in the generator.
Ez=jE = E.M.F. between 0 and 2 in the generator.
Let:
./i and 72 = currents in 1 and in 2,
70 = current in 0,
Z-L and Zz = impedances of lines 1 and 2,
Z0 = impedance of line 0.
Yl and Y2 = admittances of circuits 0 to 1, and 0 to 2,
// and //= currents in circuits 0 to 1, and 0 to 2,
Eia.-ndE2'= potential differences at circuit 0 to 1, and
0 to 2.
it is then, 7, -f 78 + 70 = 0 ) «v
or, I0 =-(/; + 72) j
that is, 70 is common return of 7: and 72.
Further, we have,
El =JE - 72 Z0 + 0Z0 =jE - 72 (Z2 + Z0) - A
and
A = K, E{
(3)
484 AL TERNA TING-CURRENT PHENOMENA.
Substituting (3) in (2) ; and expanding :
*•/ - *• _ l + F2Z2 + F2Z0(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.
Z0 = Z / V2 ;
F, = F2 = F
Substituting these values in (4), gives :
i + 1 + V2-yrz
' 1 + V2 (1 + V2) FZ + (1 + V2) F2Z;
_ E 1 + (1.707 - .707/) FZ
• 1 + 3.414 FZ + 2.414 F2Z2
(5)
V2
~J • 1 + V2 (1 + V2) FZ + (1 + V2) F2Z2
_ . ^ 1 + (1.707 + .707.;) FZ
' 1 + 3.414 FZ + 2.414 F2Z2
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 ab 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:
f2 = - 1.
Complex imaginary number:
Substituting :
a = r cos (3
b = r sin (3,
it is A = r (cos /3 -f / sin /?),
where r = a2 -- \
a
r = vector,
/3 = amplitude of complex imaginary number A.
Substituting :
eJft 4- c-JP
H
cos
sin/? =
494 APPENDIX I.
it is A = reJP,
where c = lim (l + -}"= yJT _ 1
„=» V n) o~lx2X3x
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 = af
If
a +J/= 0 ;
it is
a = 0,
304. Addition and Subtraction :
Multiplication :
(a +jb) (a' +jb') = (aa1 - 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'^'07 = 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'} = (aar+ 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 Br 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
j»W8Q
\
/
\
.
___
^.
•^-i
\
/
_^
—
->T=-
Vy
/.\
-'
-~
t
0
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 Af~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 = etra* 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'0* 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 -\-jez .
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,
Ef**L—~*iNI&t = *—.
dt d<$> d<$>
Hence Ex = — 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, Ex = — 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 xc = 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,
Ex = x I * e~a* cos (<£
{sin (<£ — w) — a COS (<£ — oi
2
(1 + 02) 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 xc , with an expo-
nential decrement a, the apparent impedance, in symbolic
expression, is :
*'
1 +a2/ V 1 +**
= ra — jxa;
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 xc , is
cos $ — r — ax — a*e — sin (<£ —
= iza(.~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 + a2/ \ 1 + a2 '
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 ra , xa , za , 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 xc,
the impedance was represented in symbolic expression by
-jxa =
! + «»
or numerically by
Thus the inductive reactance x, as well as the capacity
reactance xc, do not represent wattless electromotive forces
as in an alternating-current circuit, but introduce energy
components of negative sign
a
— ax — - - x :
1 + a2
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
0 ; x -- ^— = 0,
- — c
1 + a2 1 + fla
OSCILLATING CURRENTS. 507
substituting in this equation
x = 2 TT NL • xc =
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 r2 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 = 0 ; 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 = 0 dec a.
Since
/ = (/V+y/a) dec a, Er = Ir dec a,
Ex = -x I (a +/) dec a, Exc= _^L_/(- a +/) dec a,
we have r-aX--^—Xc = ^
I + a?
hence, by substitution,
Exc= 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 — 0 or <£ = 0.
A.t this moment the current is zero ; that is,
7=//2, /1==0.
Since Exe= •*/(— a +/) dec a = e at <f> = 0,
we have x /2 Vl + a2 = e or /2 = = .
x V 1 + a2
Substituting this, we have,
I —j — e dec a, Er =je r dec a,
x Vl + a2 x Vl + az
Ex = e (1 -ja) dec a, ^c= e (1 +/ «) dec a,
Vl + «8 Vl + a2
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 Ll = Z/ -f /,/', where Z/ = external,
Zj" = internal, inductance ; total capacity, Cv Then the
total admittance of the secondary circuit is
) dec a =
where xl= 2irJVLl= inductive reactance: xcl = \l1-jrNC ' =
capacity reactance. Let rQ = effecive hysteretic resistance,
Z = inductance ; hence, x^ = Z-n-N LQ = 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,
xc = 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 Iv = E^ Yl
dec a = secondary current.
Hence, // = / 7X dec a = pEJ Yl dec a = primary load
current, or component of primary current corresponding to
secondary current. Also, 70 = - 2j/ F0 dec a = primary
/ '
exciting current ; hence, the total primary current is
/= // + 70 = £-'{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 F0 +/2Z Y,} dec a.
P
In an oscillating discharge the total primary electro-
motive force E = 0 ; that is,
or, the substitution
a
1 +
(r0 - ax0) -.
. 0.
512 APPENDIX II.
Substituting in this equation, ^r=2 it N C, xc = ~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 YQ = 0, the equation becomes essentially
simplified, —
I a \ . I x \
(r — a x xc 1 — j ( x — I
1+/2v 1 + *8 i v Ljt^l=0;
that is,
or, combined, —
(r, -2aXl) +/2 (r-2 ax) = 0,
Substituting for xlt x, xel, xei we have
+/aZ)
i+/V / 4(A+/
+/2Z) 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. 498
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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