Theory and Calculation of Alternating Current Phenomena (1900) — part 1 of 19

Key topics: AC transformer equivalent circuit, impedance, admittance, complex quantities, resonance, distributed capacity/inductance, induction machines, synchronous machines, commutating machines, surges, harmonics, symbolic method of AC calculation.

Relevant to VIC research: transformer modeling, resonant circuits, voltage magnification, impedance matching, transient behavior in coupled inductors.

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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.