I aught er of William tuart Smith t US.Navy ENGINEERING LIBRARY ? 7 i "^T 1 THEORY AND CALCULATION ALTERNATING CURRENT PHENOMENA . BY CHARLES PROTEUS STEINMETZ WITH THE ASSISTANCE OF ERNST J. BERG NEW YORK THE W. J. JOHNSTON COMPANY 253 BROADWAY l8 97 COPYRIGHT, 1897, BY TUB W. J. JOHNSTON COMPANY. V-a ENGINEERING LIBRARY u TYPOGRAPHY BY C. J. PETERS & SON, BOSTON. DEDICATED TO THE MEMORY OF MY FATHER, CARL HEINRICH STEINMETZ. PREFACE. 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- VI 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 L, 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. Vll 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 rniake the book as creditable as pos- sible mechanically. CHARLES PROTEUS STEINMETZ. January, 1897. /CONTENTS. CHAP. I. Introduction. , , ,- j, 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 calculation. 24, p. 34. Trigonometric calculation. 25, p. 34. Rectangular components of vectors. 26, p. 36. Introduction of j as distinguishing index. 27, p. 36. Rotation of vector by 180 and 90. j = V^l. X 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. 48. Three-phase generator on unbalanced load. 37, p. 50. Quarter-phase system with common return. 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. 59, p. 86. Instance. CONTENTS. XI CHAP. IX. Resistance and Reactance of Transmission Lines Continued. 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 oi< output upon the conductance of the re- ceiver circuit. 63, p. 90. Summary. 64, p. 92. Instance. 65, p. 93. Condition of maximum efficiency. 66, 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. Maximum 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. Xll 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. 148. Energy components and wattless components. CHAP. XII. Distributed Capacity, Inductance, Resistance, and Leakage. 102, p. 150. Introduction. 103, p. 151. Magnitude of charging current of transmission lines. 104, p. 152. Line capacity represented by one condenser shunted 'across middle of line. 105, p. 153. Line capacity represented by three condensers. 106, p. 155. Complete investigation of distributed capacity, induc- tance, leakage, and resistance. 107, p. 157. Continued. 108, p. 158. Continued. 109, p. 158. Continued. 110, p. 159. Continued. 111, p. 161. Continued. 112, p. 161. Continued. 113, p. 162. Difference of phase at any point of line. 114, p. 163. Instance. 115, p. 165. Particular cases, open circuit at end of line, line grounded at end, infinitely long conductor, generator feeding into closed circuit. CHAP. XIII. The Alternating-Current Transformer. 116, p. 167. General. 117, p. 167. Mutual inductance and self-inductance of transformer. 118, p. 168. Magnetic circuit of transformer. 119, p. 169. Continued. 120, p. 170. Polar diagram of transformer. 121, p. 172. Instance. 122, p. 176. Diagram for varying load. 123, p. 177. Instance. 124, p. 178. Symbolic method, equations. 125, p. 180. Continued. 126, p. 182. Apparent impedance of transformer. Transformer equivalent to divided circuit. 127, p. 183. Continued. 128, p. 186. Transformer on non-inductive load. 129, p. 188. Constants of transformer on non-inductive load. 130, p. 191. Numerical instance. CONTENTS. xili CHAP. XIV. General Alternating-Current Transformer. 131, p. 194. Introduction. 132, p. 194. Magnetic cross-flux or self-induction of transformer. 133, p. 195. Mutual flux of transformer. 134, p. 195. Difference of frequency between primary and second- ary of general alternate*fcurrent transformer. 135, p. 195. Equations of general alternate-current transformer. 136, p. 201. Power, output and input, mechanical and electrical. 137, p. 202. Continued. 138, p. 203. Speed and output. 139, p. 205. Numerical instance. CHAP. XV. Induction Motor. 140, p. 207. Slip and secondary frequency. 141, p. 208. Equations of induction motor. 142, p. 209. Magnetic flux, admittance, and impedance. 143, p. 211. E.M.F. 144, p. 213. Graphic representation. 145, p. 214. Continued. 146, p. 216. Torque and power. 147, p. 218. Power of induction motors. 148, p. 219. Maximum torque. 149, p. 221. Continued. 150, p. 222. Maximum power. 151, p. 224. Starting torque. 152, p. 225.* Continued. 153, p. 227. Starting resistance. 154, p. 228. Synchronism. 155, p. 228. Near synchronism. 156, p. 229. Induction generator. 157, p. 229. Comparison of induction generator and synchronous generator. 158, p. 230. Numerical instance of induction motor. CHAP. XVI. Alternate-Current Generator. 159, p. 234. General. 1'60, p. 235. Magnetic reaction of lag and lead. 161, p. 237. Self-inductance of alternator. 162, p. 238. Synchronous reactance. 163, p. 238. Equations of alternator. 164, p. 239. Numerical instance, field characteristic. 165, p. 244. Dependence of terminal voltage on phase relation. 166, p. 244. Constant potential regulation. 167, p. 246. Constant current regulation, maximum output. CHAP. XVII. Synchronizing Alternators. 168, p. 248. Introduction. 169, p. 248. Rigid mechanical connection. XIV CONTENTS. CHAP. XVII. Synchronous Motor Continued. 170, p. 248. Uniformity of speed. 171, p. 249. 172, p. 250. Synchronizing. Running in synchronism. 173, p. 250. Series operation of alternators. 174, p. 251. Equations of synchronous running alternators, synchro- nizing power. 175, p. 254. Special case of equal alternators at equal excitation. 176, p. 257. Numerical instance. CHAP. XVIII. Synchronous Motor. 177, p. 258. Graphic method. 178, p. 179, p. 180, p. 181, p. 182, p. 183, p. 184, p. 260. Continued. 262. Instance. 263. -Constant impressed E.M.F. and constant current. 266. Constant impressed and counter E.M.F. 269. Constant impressed E.M.F. and maximum efficiency. 271. Constant impressed E.M.F. and constant output. 275. Analytical method. Fundamental equations and power characteristic. 185, p. 279. Maximum output. 186, p. 280. No load. 282. Minimum current. 284. Maximum displacement of phase. 286. Constant counter E.M.F. 286. Numerical instance. 288. Discussion of results. 187, 188, 189, 190, 191, p. CHAP. XIX. 192, p. 193, p. 194, p. 195, p. 196, p. 197, P : 198, p. 199, p. 200, p. 201, p. 202, p. 203, p. CHAP. XX. 204, p. 205, 206, 207, Commutator Motors. 291. Types of commutator motors. 291. Repulsion motor as induction motor. 293. Two types of repulsion mot9rs. 295. Definition of repulsion motor. 296. Equations of repulsion motor. 297. Continued. 298. Power of repulsion motor. Instance. 300. Series motor, shunt motor. 303. Equations of series motor. 304. Numerical instance. 305. Shunt motor. 307. Power factor of series motor. Reaction Machines. 308. General discussion. 309. Energy component of reactance. p. 309.* Hysteretic energy component of reactance, p. 310. Periodic variation of reactance. CONTENTS. XV CHAP. XX. Reaction Machines Continued. 208, p. 312. Distortion of wave-shape. 209, p. 314. Unsymmetrical distortion of wave-shape. 210, p. 315. Equations of reaction machines. 211, p. 317. Numerica> instance. CHAP. XXI. Distortion of Wave-Shape and its Causes. 212, p. 320. Equivalent sine wave. 213, p. 320. Cause of distortion. 214, p. 321. Lack of uniformity and pulsation of magnetic field. 215, p. 324. Continued. 216, p. 327. Pulsation of reactance. 217, p. 327. Pulsation of reactance in reaction machine. 218, p. 329. General discussion. 219, p. 329. Pulsation of resistance, arc. 220, p. 331. Instance. 221, p. 332. Distortion of wave-shape by arc. 222, p. 333. Discussion. CHAP. XXII. Effects of Higher Harmonics. 223, p. 334. Distortion of wave-shape by triple and quintuple har- monics. Some characteristic wave-shapes. 224, p. 337. Effect of self-induction and capacity on higher harmonics. 225, p. 338. Resonance due to higher harmonics in transmission lines. 226, p. 341. Power of complex harmonic waves. 227, p. 341. Three-phase generator. 228, p. 343. Decrease of hysteresis by distortion of wave-shape. 229, p. 343. Increase of hysteresis by distortion of wave-shape. 230, p. 344. Eddy currents. 231, p. 344. Effect of distorted waves on insulation. CHAP. XXIII. General Polyphase Systems. - 232, p. 346. Definition of systems, symmetrical and unsymnvetrical systems. 233, p. 346. Flow of power. Balanced and unbalanced systems. Independent and interlinked systems. Star connection and ring connection. 234, p. 348. Classification of polyphase systems. CHAP. XXIV. Symmetrical Polyphase Systems. 235, p. 350. General equations of symmetrical system. 236, p. 351. Particular syste'ms. 237, p. 352. Resultant M.M.F. of symmetrical system. 238, p. 355. Particular systems. CHAP. XXV. Balanced and Unbalanced Polyphase Systems. 239, p. 356. Flow of power in single-phase system. 240, p. 357. Flow of power in polyphase systems, balance factor of system. xvi CONTENTS. CHAP. XXV. Balanced and Unbalanced Polyphase Systems Con- tinued. 241, p. 358. Balance factor. 242, p. 358. Three-phase system, q-uarter-phase system. 243, p. 359. Inverted three-phase system. 244, p. 360. Diagrams of flow of power. 245, p. 363. Monocycjic and polycyclic systems. 246, p. 363. Power characteristic of alternating-current system. 247, p. 364. The same in rectangular coordinates. 248, p. 366. Main power axes of alternating-current system. CHAP. XXVI. Interlinked Polyphase Systems. 249, p. 368. Interlinked and independent systems. 250, p. 368. Star connection and ring connection. Y connection and delta connection. 251, p. 370. Continued. 252, p. 371. Star potential and ring potential. Star current and ring . I current. Y potential and Y current, delta potential and delta current. 253, p. 371. Equations of interlinked polyphase systems. 254, p. 373. Continued. CHAP. XXVII. Transformation of Polyphase Systems. 255, p. 376. Constancy of balance factor. 256, p. 376. Equations of transformation of polyphase systems. 257, p. 378. Three-phase, quarter-phase transformation. 258, p. 379. Transformation with change of balance factor. CHAP. XXVIII. Copper Efficiency of Systems. 259, p. 380. General discussion. 260, p. 381. Comparison on the basis of equality of minimum differ- ence of potential. 261, p. 386. Comparison on the basis of equality of maximum differ- ence of potential. 262, p. 388. Continued. CHAP. XXIX. Three-phase System. 263, p. 390. General equations. 264, p. 393. Special cases : balanced system, one branch loaded, two branches loaded. CHAP. XXX. Quarter-phase System. 265, p. 395. General equations. 266, p. 396. Special cases : balanced system, one branch loaded. APPENDIX I. Algebra of Complex Imaginary Quantities. 267, p. 401. Introduction. 268, p. 401. Numeration, addition, multiplication, involution. CONTENTS. xvii APPENDIX I. Algebra of Complex Imaginary Quantities Con- tinued. 269, p. 402. Subtraction, negative number. 270, p. 403. Division, fraction. 271, p. 403. Evolution and logarithmation. 272, p. 404. Imaginary uni<^ complex imaginary number. 273, p. 404. Review. 274, p. 405. Algebraic operations with complex quantities. 275, p. 406. Continued. 276, p. 407. Roots of the unit. 277, p. 407. Rotation. 278, p. 408. Complex imaginary plane. APPENDIX II. Oscillating Currents. 279, p. 409. Introduction. 280, p. 410. General equations. 281, p. 411. Polar coordinates. 282, p. 412. Loxodromic spiral. 283, p. 413. Impedance and admittance. 284, p. 414. Inductance. 285, p. 414. Capacity. 286, p. 415. Impedance. 287, p. 416. Admittance. 288, p. 417. Conductance and susceptance. 289, p. 418. Circuits of zero impedance. 290, p. 418. Continued. 291, p. 419. Origin of oscillating currents. 292, p. 420. Oscillating discharge. 293, p. 421. Oscillating discharge of condensers. 294, p. 422. Oscillating current transformer. 295, p. 424. Fundamental equations thereof. :.*'. ' Jfr 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 / r, where r, the resistance, is a constant of the circuit. 2.) Joule's law: P = i*r, where P is the rate at which energy is expended by the current, i, in the resistance, r. 3.) The power equation : P = ei, where P is the power expended in the circuit of E.M.F., e, and current, /". 4.) Kirchhoff's laws : a.) The sum of all the E.M.Fs. in a closed circuit = 0, if the E.M.F. consumed by the resistance, zr, 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 currents which rapidly and periodically change their 2 fc ' v ,Vt AL TJLRATA KING-CURRENT PHENOMENA. [ 2 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 j z, 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 t 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, *r, or z= v r' 2 -j- x*. 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 t 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 3] 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 ei^gy 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, / i y is denoted by Z, 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, Z, sometimes its ratio with the ohmic resistance, r, is used, and is called the Time- Constant of the circuit : 4 ALTERNATING-CURRENT PHENOMENA. [3 If a conductor surrounds with n turns a magnetic cir- cuit of reluctance, (R, the current, z, in the conductor repre- sents the M.M.F. of ni ampere-turns, and hence produces a magnetic flux of #z'/(R lines of magnetic force, sur- rounding each n turns of the conductor, and thereby giving 3> = ;/ 2 z/(R interlinkages between the magnetic and electric circuits. Hence the inductance is L = $/ i = n 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 i is the current, and L is the inductance of a circuit, the magnetic flux interlinked with a circuit of current, i, 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 TT / 2 -7- 1, the maximum rate of cutting is 2 TT TV, 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 / = effective value of alternating current, e = 2 TT NLi is the effective value of E.M.F. of self-inductance, and the ratio, e j i = 2 TT NL, is the magnetic reactance : x m = 2 TT NL. m Thus, if r = resistance, x m = reactance, z impedance, the E.M.F. consumed by resistance is : ^ = ir ; the E.M.F. consumed by reactance is : e 2 = ioc^ 4] 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, ^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 t the average rate of charge and discharge is 4 N, and 2 rr N the maximum rate of charge and discharge, sinusoidal waves sup- posed, hence, i = 2 ?r NCe the current passing into the con- denser, which is in quadrature to the E.M.F., and leading. It is then:- ^ = 7 = 2 the capacity reactance, or condensance. Polarization in electrolytic conductors acts to a certain extent like capacity. The capacity reactance is inversely proportional to the frequency, and represents the leading out-of-phase wave ; the magnetic reactance is directly proportional to the frequency, and represents the lagging out-of-phase wave. Hence both are of opposite sign with regard to each other, and the total reactance of the circuit is their difference, x = x m x c . 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 TING- CURRENT PHENOMENA. [5,6- A further discussion of these quantities will be found in the later chapters. 5. In Joule's law, P = fir, r is not the true ohmic resistance any more, but the "effective resistance;" that is, the ratio of the energy component of E.M.F. to the cur- rent. Since in alternating-current circuits, besides by the ohmic resistance of the conductor, energy is expended, partly outside, partly even inside, of the conductor, by magnetic hysteresis, mutual inductance, dielectric hystere- sis, etc., the effective resistance, r, is in general larger than the true resistance of the conductor, sometimes many times larger, as in transformers at open secondary circuit, and is not a constant of the circuit any more. It is more fully discussed in Chapter VII. In alternating-current circuits, the power equation con- tains a third term, which, in sine waves, is the cosine of the difference of phase between E.M.F. and current : P = ei cos <. Consequently, even if e and i are both large, P may be very small, if cos < is small, that is, 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 : - i = / sin (/ - A) = /sin 2 TT N (t - A) ; 6] INTRODUCTION. 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 pertods per second ; and ^ is the time, where the wave is y zero, o^the epocJi of the wave, generally called the phase* Obviously, "phase" or "epoch" attains a practical meaning only when several waves of different phases are considered, as "difference of phase." When dealing with one wave only, we may count the time from the moment f AN, \ '/ "~N \ / c \ / \ / \ \ 1 \ / 5 A V i N \ / \ / \ J \ v. 2 \ ^ Fig. 1. Sine Waue. where the wave is zero, or from the moment of its maxi- mum, and then represent it by : i = I 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 : / = A sin 2 TtN(t /i) + L sin irN(t / 2 ) 7 sin 6 * " 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 ALTERNATING-CURRENT PHENOMENA. [6 where 7j, 7 2 , 7 3 , . . . are the maximum values of the differ- ent components of the wave, t^ / 2 , / 3 . . . the times, where the respective components pass the zero value. The first term, 7 X sin 2 TT N (t tj, is called the fun- damental wave, or \.}\Q first harmonic; the further terms are called the higher harmonics, or "overtones," in analogy to the overtones of sound waves. 7 W sin 2 mr N (/ /) is the th harmonic. By resolving the sine functions of the time differences, / /j, t / 2 . . . , we reduce the general expression of the wave to the form : i = ^i sin 2 irNt -f ^ 2 sin 4 TT^V? + A z sin 6 TrTV 7 ? + . . . + l cos 2 -xNt + A cos 4 vNt + ^ 8 cos 6 irNt +. . . . /7g. 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 is expressed by : /=7 1 sin2 7 r^(/ A) + 7 3 sin TT N (t / 3 ) 5 sin 10 TT Nt + $ cos 10 irNt + or, i = A[ sin 2 TT Nt + A s sin 6 irNt + -f B cos 2 irNt -f B z cos 6 -xNt + 7] 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 thfe 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, and thereby causes, a general tendency towards simple sine shape, which has the effect, that, in general, the alternating currents in our light and power circuits are sufficiently near sine waves. to make the assumption of sine shape permissible. Hence, in the calculation of alternating-current phe- nomena, we can safely assume the alternating wave as a sine wave, without making any serious error ; and it will be sufficient to keep the distortion from sine shape in mind as a possible, though improbable, disturbing factor, which 10 ALTERNATING-CURRENT PHENOMENA. [7 generally, however, is in practice negligible perhaps with the only exception of low-resistance circuits containing large magnetic reactance, and large condensances in series with each other, so as to produce resonance effects of these higher harmonics. -8] INSTANTANEOUS AND INTEGRAL VALUES. 11 CH APTK R 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 vahte, which characterizes the wave as a whole. As such integral value, almost exclusively the effective i Fig. 4. Alternating Wave. vahte 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 cas"e, the wave is called an alternating wave, in the latter a pulsating ^vave. 12 ALTERNATING-CURRENT PHENOMENA. [S 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. Pulsating waves are produced only by commutating machines, and by unipolar machines (or by the superposi- tion of alternating waves upon continuous currents, etc.). All inductive apparatus without commutation give ex- clusively alternating waves, because, no matter what con- / s**~ X, \ / / x N / ' X \ / f / \ 7 / \ A VEF AQE VA LUE [ / \ 1 / I / \ s. / \ "v* +S / 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 9] 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 ?r / 2 from A to B, the sine varies from to OB = 1. Hence the average variation of the sine bears to that of the corresponding arc the ratio 1 -5- ir/2, or 2/7r -5- 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 -=- 2 / ^, and since the variations of a sine-function are sinusoidal also, we have, Mean value of sine wave maximum value = hi 7T = .63663. The quantities, "current," "E.M.F.," "magnetism," etc., are in reality mathematical fictions only, as the components 14 ALTERNATING-CURRENT PHENOMENA. [9 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 : The effective value of an alternating wave, or the value representing the same effect as the periodically varying wave, is the square root of the mean square. In a sine wave, its relation to the maximum value is found in the following way : Fig. 7. Let, in Fig. 7, AOB represent a quadrant of a circle with radius 1. Then, since the sines of any angle and its complemen- tary angle, 90 <, fulfill the condition, sin 2 < + sin 2 (90 <)=!, 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 : : 1O] INSTANTANEOUS AND INTEGRAL VALUES. 15 The effective value of a sine function bears to its maxi- mum value the ratio, A= -r 1 = -70711. Hence, we have for trfe sine curve the following rela- tions : MAX. EFF. ARITH. MEAN. Half Period. Whole Period. 1 1 V2 2 7T 1 .7071 .63663 1.4142 1 .90034 1.5708 1.1107 1 10. Coming now to the general alternating curve, i = ^ sin 27r Nt + A z sin 4rr Nt + A* sin 6ir Nt + . . . + .#! cos lieNt + BZ cos 47r7V? + ^ 3 cos GTT^W + . . . , we find, by squaring this expression and canceling all the products which give as mean square, the effective vahie, The mean value does not give a simple expression, and is of no general interest. 16 AL TERN A TING-CURRENT PHENOMENA. [11 CHAPTER III. LAW OF ELECTRO-MAGNETIC INDUCTION. 11. If an electric conductor moves relatively to a mag- netic field, an E.M.F. is induced in the conductor which is proportional to the intensity of the magnetic field, to the length of the conductor, and to the speed of its motion perpendicular to the magnetic field and the direction of the conductor ; or, in other words, proportional to the number of lines of magnetic force cut per second by the conductor. As a practical unit of E.M.F., the volt is defined as the E.M.F. induced in a conductor, which cuts 10 8 = 100,000,000 lines of magnetic force per second. If the conductor is closed upon itself, the induced E.M.F. produces a current. A closed conductor may be called a turn or a convolution. In such a turn, the number of lines of magnetic force cut per second is the increase or decrease of the number of lines inclosed by the turn, or ;/ times as large with ;/ 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, during each complete alternation or cycle, the total flux is cut four times, twice passing into, and twice out of, the turns. 12] LA W OF ELECTRO-MA GNE TIC 2ND UCTION. 1 7 Hence, if N = number of complete cycles per second, or the frequency of the relative alternation of flux <, the average E.M.F. induced In n turns is, ^avg. = ?iN 10 - volts. This is the fundamental equation of electrical engineer- ing, and applies to continuous-current, as well as to alter- nating-current, apparatus. 12. In continuous-current machines and in many alter- nators, the turns revolve through a constant magnetic field ; in other alternators and in induction motors, the mag- netic 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, = flux inclosed per turn, and N = frequency, the E.M.F. induced in the machine is E = 4;z3>^V10~ 8 volts, independent of the num- ber of poles, or 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 N the frequency, this formula gives, &VK = 4 n$> JV10 - 8 volts. Since the maximum E.M.F. is given by, ^max. = | avg. E, we have ^ma*. = 2 7T ff .AT 10 ~ 8 volts, which is the fundamental formula of alternating-current induction by sine waves. 18 AL TERN A TING-CURRENT PHENOMENA . [13 13. If, in a circuit of n turns, the magnetic flux, O, inclosed by the circuit is produced by the current flowing in the circuit, the ratio flux X number of turns X 10~ 8 current is called the inductance, L, of the circuit, in henrys. The product of the number of turns, n, into the maxi- mum flux, , produced by a current of /amperes effective,. or/V2 amperes maximum, is therefore n$> =Z/V2 10 8 ; and consequently the effective E.M.F. of self -inductance is: E = V27r/z4>^10- 8 = 2-,rNLI volts. The product, x = 2 WVZ, 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. 14] GRAPHIC REPRESENTA TION. 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. 1- are represented by the same 20 AL TERNA TING-CURRENT PHENOMENA. [15 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. 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. Fig. 10. 15. The sine wave, Fig. 1, is represented in polar coordinates by one. circle, as shown in Fig. 10. The diameter of the characteristic curve of the sine wave, /= OC, represents the intensity of the wave; and the am- plitude of the diameter, OC, Z w = AOC, is the phase of the 16] GRAPHIC REPRESENTATION. 21 wave, which, therefore, is represented analytically by the function : ,, i I cos ( (j) to), where = 2TI7 / T is the instantaneous value of the ampli- tude corresponding to the instantaneous value, i, of the wave. The instantaneous values are cut out on the movable ra- dius vector by its intersection with the characteristic circle. Thus, for instance, at the amplitude AOB l = ^ = 2 irt-^ / T (Fig. 10), the instantaneous value is OB '; at the amplitude AOB 2 = (/> 2 = 27T/ 2 / T t the instantaneous value is OB" , and negative, since in opposition to the vector OB 2 . 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. 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. 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. 16. To combine different sine waves, their graphical representations, or vectors, are combined by the parallelo- gram law. If, for instance, two sine waves, OB and OC (Fig. 11), are superposed, as, for instance, two E.M.Fs. acting in the same circuit, their resultant wave is represented by 22 AL TERN A TING-CURRENT PHENOMENA. [16 OD y the diagonal of a parallelogram with OB and OC as. sides. For at any time, /, represented by angle < = AOX, trie- 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 lengthy OC y 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 the 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 17] GRAPHIC REPRESENTA TION. 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 IE cos 17. Suppose, as an instance, that over a line having the resistance, r, and the reactance, x QirNL, 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, OL 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- s'ented by OE r in the diagram. The self-inductance of the line induces an E.M.F. which is proportional to the current / and reactance x, and lags a quarter of a period, or 90,, behind the current. To overcome this counter E.M.F, AL TERNA TING-CURRENT PHENOMENA. 18 of self-induction, an E.M.F. of the value Ix is required, in phase 90 ahead of the current, hence represented by vector OE X . Thus resistance consumes E.M.F. in phase, and reactance an E.M.F. 90 ahead of the current. The E.M.F. of the generator, E , has to give the three E.M.Fs., E, E r , and E x , hence it is determined as their resultant. Combining by the parallelogram law, OE r and OE X , give OE Z , the E.M.F. required to overcome the impedance of the line, and similarly OE Z and OE give OE , the E.M.F. required at the generator side of the line, to yield the E.M.F. E at the receiving end of the line. Algebraically, we get from Fig. 12 or, E = . _ In this instance we have considered the E.M.F. con- sumed by the resistance (in phase with the current) and the E.M.F. consumed by the reactance (90 ahead of the current) as parts, or components, of the impressed E.M.F., E y and have derived E by combining E r , E x , and E. Fig. 13. 18. We may, however, introduce the effect of the induc- tance directly as an E.M.F., E x , the counter E.M.F. of self-induction = Ig, and lagging 90 behind the current ; and the E.M.F. consumed by the resistance as a counter E.M.F., Ef = Ir, but in opposition to the current, as is done in Fig. 13 ; and combine the three E.M.Fs. E , EJ, E x , to form a resultant E.M.F., E, which is left at the end of the line. 18] GRAPHIC REPRESENTA TION, E and E x ' combine to form E z ', the counter E.M.F. of impedance ; and since Eg and E must combine to form E, E is found as the side of a parallelogram, OE EE^ t whose other side, OE Z , and" diagonal, OE, are given. Or we may say (Fig. 14, that to overcome the counter E.M.F. of impedance, OE Z ', of the line, the component, OE er of the impressed E.M.F., together with the other component OE, must give the impressed E.M.F., OE . As shown, we can represent the E.M.Fs. produced in a circuit in two ways either as counter E.M.Fs., which com- bine with the impressed E.M.F., or as parts, or components,. E.'r O Fig. 14. of the impressed E.M.F., in the latter case being of opposite phase. According to the nature of the problem, either the one or the other way may be preferable. As an example, the E.M.F. consumed by the resistance is Ir, and in phase with the current ; the counter E.M.F. of resistance is in opposition to. the current. The E.M.F. consumed by the reactance is Ix, and 90 ahead of the cur- rent, while the counter E.M.F. of reactance is 90 behind the current ; so that, if, in Fig. 15, OI, is the current, OE r = E.M.F. consumed by resistance, OE r ' = counter E.M.F. of resistance, OE X = E.M.F. consumed by inductance, OE X ' = counter E.M.F. of inductance, OE Z = E.M.F. consumed by impedance, OE Z ' = counter E.M.F. of impedance. ALTERNATING-CURRENT PHENOMENA. [19,2O 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. E'Z E'r O Ex |Er Ei 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 Irf E. This is different from, and less than, the E.M.F. of impedance E z = Iz -- Hence it is wrong to calculate the drop of potential in a circuit by multiplying the current by the impedance ; and the drop of potential in the line depends, with a given current fed over the line into a non-inductive circuit, not only upon the constants of the line, r and x, but also upon the E.M.F., Ej 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 w. The 20] GRAPHIC REPRESENTA TION, 27 E.M.F. consumed by the resistance, Ir, is in phase with the current, and represented by OE r \ the E.M.F. consumed by the reactance, f x , is 90 ahead of the current, and re- presented by OE X . Combining OE, OE ri and OE X , we get OE , the E.M.-F-. required at the generator end of the line. Comparing Fig. 16 with Fig. 13, we see that in the former OE is larger; or conversely, if E is the same, E will be less with an inductive load. In other words, the drop of potential in an inductive line is greater, if the receiving circuit is inductive, than if it is non-inductive. From Fig. 16, E = -V(J5 cos w + fr) 2 + (E sin & E r Fig. 16. If, however, the current in the receiving circuit is leading, as is the case when feeding condensers or syn- chronous motors whose counter E.M.F. is larger than the impressed E.M.F., then the E.M.F. will be represented, in Fig. 17, by a vector, OE, lagging behind the current, Of, by the angle of lead '; and in this case we get, by combining OE with OE r , in phase with the current, and ~OE X , 90 ahead of the current, the generator E.M.F., OE~ , which in this case is not only less than in Fig. 16 and in Fig. 13, but may be even less than E ; that is, the poten- tial rises in the line. In other words, in a circuit with leading current, the self-induction of the line raises the potential, so that the drop of potential is less than with 28 AL TERN A TING- CURRENT PHENOMENA. [21 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 same diagrams, Figs. 13 to 17, can be considered as polar diagrams of an alternating-current generator of- an E.M.F., E , a resistance E.M.F., E r = Ir, a reactance, E x Ix, 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 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,. 21 GRAPHIC REPRESENTA TION. 29 since the induced . E.M.F. lags 90 behind the inducing flux. Thus the secondary induced E.M.F., lt will be represented by a vector, Q lt in Fig. 18, at the phase 180. The secondary current, I v lags behind the E.M.F. E l by an angle G> lf 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 diagram by the vector OF lt of phase 180 -f 5. Fig. 18. Instead of the secondary current, f lt we plot, however, the secondary M.M.F., F l = n^ f lt where x is the number of secondary turns, and F 1 is given in ampere-turns. This makes us independent of the ratio of transformation. From the secondary induced E.M.F., E lt we get the flux, 3>, required to induce this E.M.F., from the equation I where EI = secondary induced E.M.F., in effective volts, JV frequency, in cycles per second. 7/x = number of secondary turns. < = maximum value of magnetic flux, in webers. The derivation of this equation has been given in a preceding chapter. This magnetic flux, <|>, is represented by a vector, O&, at 30 ALTERNATING-CURRENT PHENOMENA. [22 the phase 90, and to induce it an M.M.F., &, is required, 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 OM, by an angle a, the angle of hysteretic lead. (See Chapter on Hysteresis.) This M.M.F., &, is the resultant of the secondary M.M.F., & lt and the primary M.M.F., & \ or graphically, OF is the diagonal of a parallelogram with OF^ and OF as sides. OF l and OF being known, we find OF , the primary ampere- turns, n , and therefrom, the primary current, I = $ / n , which corresponds to the secondary current, I I . To overcome the resistance, r , of the primary coil, an E.M.F., E r = Ir , is required, in phase with the current, 7 , and represented by the vector OE r . To overcome the reactance, x = 2 TT n L , of the pri- mary coil, an E.M.F. E x = I x is required, 90 ahead of the current 7 , and represented by vector, OE X . The resultant magnetic flux, $, which in the secondary -coil induces the E.M.F., 1 , induces in the primary coil an E.M.F. proportional to 1 by the ratio of turns n j n^, and in phase with l , or, which is represented by the vector OE t ' . To overcome this counter E.M.F., E { ', a primary E.M.F., E lt is required, equal but opposite to t ' ' , and represented by the vector, OE V . The primary impressed E.M.F., E , must thus consist of the three components, OE^ OE r , and OE X , and is, there- fore, a resultant OE , while the difference of phase in the primary circuit is found to be ^f % = E OA. 22. Thus, in Figs. 18, 19, and 20, the diagram of an .alternating-current transformer is drawn for the same sec- 22] GRAPHIC REPRESENTATION. 31 ondary E.M.F., E lt and secondary current, I lt but with dif- ferent conditions of secondary displacement : In Fig. 18, the secondary current, I , lags 60 behind the sec- ondary E.M.F., jffj. . In Fig. 19, the secondary current, f l9 is in phase with the secondary E.M.F., JS l . In Fig. 20, the secondary current, 7i , leads by 60 the second- ary E.M.F., EI. Fig. 19. These diagrams show that lag in the secondary circuit in- creases and lead decreases, the primary current and primary E.M.F. required to produce in the secondary circuit the ,same E.M.F. and current ; or conversely, at a given primary Fig. 20. impressed E.M.F., E , the secondary E.M.F., E v 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 ALTERNATING-CURRENT PHENOMENA. [22 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. 23] SYMBOLIC METHOD. 33 CHAPTER V. SYMBOLIC METHOD. 23. The graphical method of representing alternating- , we can represent it by its rectangular coordinates,. a and b (Fig. 22), where a = /cos ui is the horizontal component, b = /sin co is the vertica^ 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 Fig. 23. law is reduced to the addition, or subtraction, of their rectangular components. That is, Sine waves are combined, or resolved, by adding, or subtracting, their rectangular components. For instance, if a and b are the rectangular components of a sine wave, /, and d and b' the components of another sine wave, /' (Fig. 23), their resultant sine wave, I , has the rectangular components a = (a + a'), and b = (b + b'). To get from the rectangular components, a and b, of a sine wave, its total intensity, i, and phase, to, we may com- bine a and b by the parallelogram, and derive, tan 36 ALTERNATING-CURRENT PHENOMENA. [26,27 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, 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, _ *=Va 2 + * 2 , and of phase, tan w = 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 -\- 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 algebraic expression, a + jb, of a sine wave by 1 means reversing the wave, or rotating it tJirough 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 28] SYMBOLIC METHOD. 3T component, b, and the vertical component, a, and is rep- resented algebraically by the expression, ja b. Multiplying, however, a rj- jb by j, we get : therefore, if we define the heretofore meaningless symbol, j, by the condition, / 2 = - 1, we have hence : Multiplying the algebraic expression, a -{-jb, of a sine wave by j means rotating the wave through 90, or one-quarter pe- riod ; that is, retarding the wave through one-quarter period, -6 Fig. 24. Similarly, Multiplying by j means advancing the wave through one-quarter period. since / 2 = 1, j = V^T ; that is, - j is the imaginary unit, and the sine wave is represented by a complex imaginary quantity, a -f 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 1 a + jb, 38 ALTERNATING-CURRENT PHENOMENA. [29 where a is the horizontal and b the vertical component of the wave ; the intensity is given by the phase by tan w = - , a and a = i cos o>, b = i sin to ; hence the wave a -\-jb can also be expressed by i (cos W +y sin w)> or, by substituting for cos w and sin their exponential expressions, we obtain ie^. 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, and combined give the sine wave f=(a + a')+j(b + b f ). It will thus be seen that the combination of sine waves is reduced to the elementary algebra of complex quantities. 29. If /= i -\-ji' 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 tr NL the reactance, the E.M.F. produced by the reactance, or the counter 29] SYMBOLIC METHOD. 39 E.M.F. of self-inductance, is the product of the current and reactance, and lags 90 behind the current; it is, therefore, represented by the expression j jxl =jit?ci xi r . 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 j x j~= jxi + xi'. Hence, the E.M.F. required to overcome the resistance, r, and the reactance, x, is (r-jx)I- 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, since j 2 = 1 E = (ri + xi') + j (ri r - xi) ; 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 ( e +X) ( r + /*) erJx _._ .Jr + ex T = or, if E = e -\-je' is the impressed E.M.F., and / = / -f- ji r the current flowing in the circuit, its impedance is z = _ = ' |* 4- /' 40 AL TERN A TING-CURRENT PHENOMENA. [ 30, 3 1 30. If C is the capacity of a condenser in series in a circuit of current / = i + /*', the E.M.F. impressed upon the terminals of the condenser is E = - - , 90 behind the current; and may be represented by 3- - , or jx^ /, 7T 2. V Cx where x^ = - - - is the capacity reactance or condensance j 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 irNL = magnetic reactance. If C = capacity, Xi = - = capacity reactance, or conden- 2i TT yvCx sance ; Z = r j (x Xi)j 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. 32] SYMBOLIC METHOD. 41 Since, if a complex quantity equals zero, the real part as- well as the imaginary part must be zero individually, if a +jb = 0, . a = 0, b = 0, and if both the E.M.Fs. amd currents are resolved, 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 towards 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 cannot be represented as a vector in the diagram. In what follows, complex quantities will always be de- noted by capitals, absolute quantities and real quantities by small letters. 32. Referring to the instance given in the fourth chapter, of a circuit supplied with an E.M.F., E, and a cur- rent, /, over an inductive line, we can now represent the impedance of the line by Z = r jx, where r = resistance, x = reactance of the line, and have thus as the E.M.F. at the beginning of the line, or at the generator, the expression E = E + ZL Assuming now again the current as the zero line, that is, / = 2, we have in general E = E + ir-jix-, hence, with non-inductive load, or E = e, EO = ( + t'r) fix, e = V( = values which easily permit calculation. e ix 33] TOPOGRAPHIC METHOD. 43 i - 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. Fig. 25. 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 OT (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 /j = i ji'. Or, if the difference of potential from terminal B to terminal A is denoted by the E = e + je' , the difference of potential from A to B is E l = e je'. 44 ALTERNATING-CURRENT PHENOMENA. [34 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, and consist- ing of three E.M.Fs., of equal intensity, differing in phase by one-third of a period. Let the E.M.Fs. in the direction Fig. 27. from the common connection O of the three branch circuits to the terminals A lt A 2 , A s , be represented by E ly E 2 , E B . Then the difference of potential from A 2 to A 1 is E% E ly since the two E.M.Fs., E and are connected in cir- cuit between the terminals A* and A*, in the direction,. 34] TOPOGRAPHIC METHOD. 45 A 1 O A 2 ; that is, the one, ^ 2 , in the direction OA Z , from the common connection to terminal, the other, E lt in the opposite direction, A^O, .from the terminal to common connection, and represented by E l . Conversely, the dif- ference of potential from A To A 2 is E^ E 2 . 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. E? 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 ALTERNATING-CURRENT PHENOMENA. [35 tial at one of the three-phase terminals, by point E^. The potentials at the two other terminals will then be given by the points E 2 and E z , which have the same distance from O as E lt and are equidistant from E 1 and from each other. Fig. 29. The difference of potential between any pair of termi- nals for instance E 1 and E^ is then the distance E^E^ , Z, according to the direction considered. or I. E 35. If, now, in Fig. 29, a current, I lt in phase with E.M.F., E lt passes through a circuit, the counter E.M.F. of resistance, r, is E r = fr, in opposition to /j or E 19 35] TOPOGRAPHIC METHOD. 47 .and hence represented in the diagram by point E r , and its combination with E^ by E. The counter E.M.F. of reactance, x, is E x = Ix, 90 behind the current f lt or E.M.F., E lf and therefore represented by point E x , and giving, by its combination with E^, the terminal potential of the generator E^, which, as seen, is less than the E.M.F., E lf If all the three branches are loaded equally by three currents flowing into a non-inductive circuit, and thus in Fig. 31. phase with the E.M.Fs. at the generator terminals (repre- sented in the diagram, Fig. 30, by the points E lt E z , E ZJ equidistant from each other, and equidistant from the zero point, O), the counter E.M.Fs. of resistance, Ir t are repre- sented by the distances EE 1 ', as E-^E^ etc., in phase with the currents, /; and the counter E.M.Fs. of reactance, f x , are represented by the distance, E 1 E in quadrature with the current, thereby giving, at the generator E.M.Fs., the points E{, E z , E. Thus, the triangle of generator E.M.Fs. EfE^Ej, pro- duces, with equal load on the three branches and non- 48 ALTERNATING-CURRENT PHENOMENA. [36 inductive circuit, the equilateral triangle, E 1 E^E Z9 of ter- minal potentials. If the load is inductive, and the currents, /, lag behind the terminal voltages, E, by, say, 40, we get the diagram shown in Fig. 31, which explains itself, and shows that the drop of potential in the generator is larger on an inductive load than on a non-inductive load. Conversely, if the currents lead the terminal E.M.Fs. by, say, 40, as shown in Fig. 32, the drop of potential in the generator is less, or a rise may even take place. 36. If, however, only one branch of the three-phase circuit is loaded, as, for instance, E^EJ, the E.M.F. pro- ducing the current (Fig. 33), is E^E^ ; and, if the current lags 20, it has the direction Of, where Of forms with E^EI the angle eo 20 ; that is, the current in E^ is Of lt and the current in E^ is 6>/ 2 , the return current of OI V Hence the potential at the first terminal is E^, as de- rived by combining with Ef the resistance E.M.F., E^E^ 9 in phase, and the reactance, E.M.F., EE 19 in quadrature, with the current ; and in the same way, the E.M.F. at the 36] TOPOGRAPHIC METHOD. 49 second terminal is -ZT 2 , derived by the combination of E< with E we get : hence, * 2 / = (^ + ^ 2 ) (g 2 + 2 ) = 1 ; 1 1 ) the absolute value of and, y yV 2 + b 2 ' i impedance ; ) the absolute value of. admittance. ALTERNATING-CURRENT PHENOMENA. [41 41. If, in a circuit, the reactance, x, is constant, and the resistance, r, is varied from r to r = oo , the susceptance, b, decreases from = 1 / .r at r = 0, to $ = at r = oo ; while the conductance, g = at r = 0, increases, reaches a maximum for r = x, where g = 1 / 2 r is equal to the susceptance, or g = b, and then decreases again, reaching = at r = oo . OHN 2.0 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1.0 .9 .8 .7 s ^ \ V \ RE/ crt NC CO NS1 AN! = .l OH MS / \ \ s \ \ / \ v \ / S \ \ / / \ 1 'X / > \ X ^ P \ \ ,# <<," \ ^ \/ / / r \ ^ >; ' XN X / \ / ' ^ \, X 7 / i t > ^\ "^x \ / s ^^ j < ^N "**** ^ ^ ^^^ r, 3 r ^ ^ \ -^, C^; ^ .4 o /? ^2 X I ^r ^ x ^ o i 3 ^ ^ .1 ( 8 ""*^^. ~ ~- RESISJAN DE: % O^MS 1 1 i - > .1 ~. 3 4 .5 .Q .7 .8 .9 1.0 1.1 1.2 1.3 1.1 1.5 1.0 1.7 1.8 Fig. 36. In Fig. 36, for constant reactance x = .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, 41] ADMITTANCE, CONDUCTANCE, SUSCEPTANCE. 57 Obviously, if the resistance, r, is constant, and the reac- tance, x, is varied, the values of conductance and susceptance are merely exchanged, the conductance decreasing steadily from g 1 / r to 0, and the susceptance passing from at x = to the maximum, b =4 I ^ r g =\ / ^x at x = r> and to b at x = oo . The resistance, r, and the reactance, x, vary as functions of the conductance, g, and the susceptance, b, varies, simi- larly to g and b, as functions of r and x. The sign in the complex expression of admittance is always opposite to that of impedance ;.it follows that if the current lags behind the E.M.F., the E.M.F. leads the cur- rent, and conversely. We can thus express Ohm's law in the two forms E = IZ, I = EY, and therefore The joint impedance of a number of series-connected im- pedances is equal to the sttm 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. [42,43 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, or, as expressed in their com- plex form, * E = ZI, or, / = YE, .and ^E = in a closed circuit, 27 = at a distributing point, where E y 7, Z, Y, are the expressions of E.M.F., current, Impedance, and admittance in complex quantities, these laws 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, 7, Z y 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 combinations will here be considered. 1.) Resistance in series with a circuit. 43. In a constant-potential system with impressed E.M.F., 43] RESISTANCE, INDUCTANCE, CAPACITY. 59 let the receiving circuit of impedance Z = r jxj z = -\/r' 2 -f- x' 2 > be connected in series with a resistance, r . The total impedance of tMe circuit is then Z + r = r + r jx; hence the current is Z + r r +r -jx (r + r )* -f x* ' .and the E.M.F. of the receiving circuit, becomes E = IZ = E ( r r + r -jx (r + r ) 2 + x* = EQ{# + rr jr x} t z* + 2 rr + r 2 or, in absolute values we have the following : Impressed E.M.F., current, V^T^y 2 !^ 2 V* 2 + 2 rr + r* ' E.M.F. at terminals of receiver circuit, ' o) ' V z -f- 4 rr c difference of phase in receiver circuit, tan c3 = - ; x difference of phase in supply circuit, tan w n = r *~ r x a.} If x is negligible with respect to r, as in a non-induc- tive receiving circuit, T E o 77 _. TT r and the current and E.M.F. at receiver terminals decrease .steadily with increasing r . 60 ALTERNATING-CURRENT PHENOMENA. [44 b.) If r is negligible compared with x, as in a wattless receiver circuit, T E or, for small values of r , that is, the current and E.M.F. at receiver terminals remain approximately constant for small values of r , and then de- crease with increasing rapidity. 44. In the general equations, x appears in the expres- sions for / and E only as .r 2 , 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, #, 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 are shown in dotted lines the current, /, and the E.M.F., E, at the receiver circuit, for E = const. = 100 volts, z = 1 ohm, and"- a.) r = .2 ohm (Curve I.) .) r = .8 ohm (Curve II.) with values of reactance, x= V^ 2 r 2 , for abscissae, from x = + 1.0 to x = + 1.0 ohm. As shown, / and E are smallest for x 0, r = 1.0, or for the non-inductive receiver circuit, and largest for x = zt 1.0, r = 0, or for the wattless circuit, in which latter a series resistance causes but a very small drop of potential. Hence the control of a circuit by series resistance de- pends upon the difference of phase in the circuit. For r = .8 and x = 0, x = + .8, and x = - .8, the polar diagrams are shown in Figs. 37, 38, 39. 45] RESISTANCE, INDUCTANCE, CAPACITY. 61 2.) Reactance in series with a circuit. 45. In a constant potential system of impressed E.M.F., let a reactance, x , be connected in series in a receiver cir- cuit of impedance 2 = Vr 2 + * 2 . = r x, [100 90 IMPE ^ IM DANC LIN PRESSED E.M.F. 3E OF RECEIVER B RESISTANCE C CONSTANT, CIRCUIT cor ONSTA.NT r r, EO " 4ST . IOO XNT, 2 8 z- 1.0 X VOLTS AND AMPER g S g 3 ^ ^ - ^ *B - ^i -. J - - _ ;= -= ^ It DUC TAf CE RE/ CT^ NCE CON DEN SAN CE- a:= OHI /IS o Fig. 37. +1. .0 .8 . 7 .6 .5 .4 .8 .2 .1 T-.l ' -.2 - -.3 - -.4 - -.5 - -.6 -.7 -.8 - -.9-1 Variation of Voltage at Constant Series Resistance with Phase Relation of Receiver Circuit. Then, the total impedance of the circuit is Z x = r x x. r Er Fig. 38. and the current is, f= Fig. 39. while the difference of potential at the receiver terminals is, 62 ALTERNATING-CURRENT PHENOMENA. [45 Or, in absolute quantities : Current, E.M.F. at receiver terminals, = J r ' + x ' 2 = f V r a + (x + x y v** 2 2 **. difference of phase in receiver circuit, tan = - ; x difference of phase in supply circuit, a.) If x is small compared with r, that is, if the receiver circuit is non-inductive, / and E change very little for small values of x ; but if x is large, that is, if the receiver circuit is of large reactance, / and E change much with a change of x . b.) If x is negative, that is, if the receiver circuit con- tains condensers, synchronous motors, or other apparatus which produce leading currents above a certain value of x the denominator in the expression of E, becomes < z, or E > E ; that is, the reactance, x , raises the potential. c.) E = E , or the insertion of a series inductance, x , does not affect the potential difference at the receiver ter- minals, if _ or, x = 2 x. That is, if the reactance which is connected in series in the circuit is of opposite sign, but twice as large as the reactance of the receiver circuit, the voltage is not affected, but E = E , /= E j z. If x > 2 x, it raises, if x < 2x, it lowers, the voltage. We see, then, that a reactance inserted in series in an alternating-current circuit will lower the voltage at the 45] 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^he 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 : = -*{ <-!(?)'+!(-)'- + ... (1 -\- x)~* expanded by the binomial theorem is 1 + nx + (-!) 1-2 is small compared with r : \Vi + * (1+*) Therefore, if g.-^g = !/*- *\r That is, the percentage drop of potential by the insertion of reactance in series in a non-inductive circuit is, for small E r , E ro 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. ALTERNA TING-CURRENT PHENOMENA. [46 46. As an instance, in Fig. 41 the changes of current, /, and of E.M.F. at receiver terminals, E, at constant im- pressed E.M.F., E , are shown for various conditions of a receiver circuit and amounts of reactance inserted in series. Fig. 41 gives for various values of reactance, x (if posi- tive, inductance if negative, condensance), the E.M.Fs., , at receiver terminals, for constant impressed E.M.F.,. VOLTS E OR AMPERES I 100 90 8. J70 |: > 30 20 10 Xo +3 IMPRESSED E.'M.F. CONSTANT, E -IMPEDANCE OF RECEIVER CIRC.UI II. r=.6 =+.8 III. r=.6 X = -.8 5=160 T CONS' a PAN >l r.z = 1. ni" 16 ^ / \ u o / * \ / \ 14 / \ / \ 13 / \ / \ 12 / \ / \H o/ / \ 7 / ^ ^ X ^ X / III / ^ / 8 \ S \ / / / y ? 7 > V^ x, / / \s / / / | > X X X ^ / . S i S 5 X ^ . ^ ^ *** 4 . . -- -" _-^- == - ^- ^~~ _3 2 ?_ 1 I c; HM S IN DUOTAr CE -RE AC, ANC E- -^c ONI )EN 3ANCE 2.8 2.6 2.4 2 2 2.0 1.8 1.6 1.4 1.2 1.0 .8 . Fig. 41. .4 -f .2 .2 .4 .6 .8 1.0 1.2-1. E Q = 100 volts, and the following conditions of receiver ; =1.0, r=1.0, *= (Curve I.) 2=1.0, r= .6,*= .8 (Curve II.) 0=1.0, r= .6, x= -.8 (Curve III.) As seen, curve / is symmetrical, and with increasing x the voltage E remains first almost constant, and then drops off with increasing rapidity. In the inductive circuit series inductance, or, in a con- denser circuit series condensance, causes the voltage to drop off very much faster than in a non-inductive circuit. 46] RESISTANCE, INDUCTANCE, CAPACITY. 65 Series inductance in a condenser circuit, and series con- densance in an inductive circuit, cause a rise of potential. This rise is a maximum for x = i .8, or, x x (the condition of resonance), and the E.M.F. reaches the value, E = 167 volts, or, E = E ^fr. This rise of potential by series reactance continues up to x = =t 1.6, or, X Q 2x, Fig. 42. where E = 100 volts again ; and for x > 1.6 the voltage drops again. At x = i .8, x = =p .8, the total impedance of the circuit is r j (x + x^ = r = .6, x + x = 0, and tan w = ; that is, the current and E.M.F. in the supply circuit are in phase with each other, or the circuit is in electrical resonance. E r Fig. 43. Since a synchronous motor in the condition of efficient working acts as a condensance, we get the remarkable result that, in synchronous motor circuits, choking coils, or reactive coils, can be used for raising the voltage. In Figs. 42 to 44, the polar diagrams are shown for the conditions E Q = 100, x = .6, x = (Fig/ 42) E = 85.7 x = + .4 (Fig. 43) E = 73.7 x == .4 (Fig. 44) E = 106.6 ALTERNA TING-CURRENT PHENOMENA. [47 47. In Fig. 45 the dependence of the potential, E, upon the difference of phase, 2 + x* we get the current, /= = * Z + Z (r + r.) -/( and the E.M.F. at receiver terminals, E = E - = E r .J* Or, in absolute quantities, the current is, /= E the E.M.F. at receiver terminals is, E = * - E z + r o y + ( X + ^ ) 2 V* 2 + * 2 + 2 (rr the difference of phase in receiver circuit is, tan to =^= - ; x .and the difference of phase in the supply circuit is, . x +x 49. In this case, the maximum drop of potential will not take place for either x = 0, as for resistance in series, or for r = 0, as for reactance in series, but at an intermediate point. The drop of voltage is a maximum ; that is, E is a minimum if the denominator of E is a maximum ; or, since z, z ot r , x are constant if rr + xx is a maximum, since x = V^ 2 r 2 if rr + x V^ 2 r^ is a maximum, 70 A L TERN A TING CURRENT-PHENOMENA. [ 49 a function, f = rr -f x V^r 2 r 2 is a maximum when its differential coefficient equals zero. For, plotting / as curve with r as abscissae, at the point where / is a maxi- mum or a minimum, this curve is for a short distance horizontal, hence "the tangens-function of its tangent equals zero. The tangens-function of the tangent of a curve, how- ever, is the ratio of the change of ordinates to the change: of abscissae, or is the differential coefficient of the func- tion represented by the curve. 150 110 / J30 / / 110 t / / S / 80 ZP 60 j ^--- ^-* ^ ^ i Zo = ^-"" 3^ - ' ^^^ J 60 ^ * -""^ ~ * - ~* . , - ^^ 30 20 H-, < i _ Z A -i.fc - 1 . 10 X- > 1. .9 .8 .7 .6 .5 .4 .3 .2 .1 - -.1 - -.2 - -.3 - -.4 - -. - -.6 -.7 -.8 T9-1. Fig. 48. Thus we have : f = rr -f- x -\/z 2 r 2 = maximum or minimum, if Differentiating, we get : 50] 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, x l 'r 0) of the series impedance. Fig. 49. I, l o Fig. 50. 50. As att example, Fig. 48 shows the E.M.F., E, at the receiver terminals, at a constant impressed E.M.F., E = 100, a constant impedance of the receiver circuit,. z = 1.0, and constant series impedances, Z = .3 -y.4 (Curve I.) Z = 1.2 yl.6 (Curve II.) as functions of the reactance, ;r, of the receiver circuit Fig. 57. Figs. 49 to 51 give the polar diagram for E = 100, x = .9, x = 0, ^r = - .9, and Z = .3 -7 .4. ALTERNATING-CURRENT PHENOMENA. 51 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. I , or approximately, / = . This potential control, however, causes the current taken from the mains to lag greatly behind the E.M.F., and thereby requires a much larger current than corresponds to the power consumed in the receiver circuit. Since a condenser draws from the mains a leading cur- rent, a condenser shunted across such a circuit with lagging current will compensate for the lag, the leading and the lagging current combining to form a resultant current more or less in phase with the E.M.F., and therefore propor- tional to the power expended. 52] 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,^, and the whole shunted by a condenser of condensance* c, entailing but a negligible loss. of energy. Then, if E = impressed E.M.F., - the current in receiver circuit is, the current in condenser circuit is, and the total current is r jx jc 2 or, in absolute terms, I = E \I ( ^ -} + ( ** 1 ' while the E.M.F. at receiver terminals is, E r ^ = 52. The main current, 7 , is in. phase with the impressed E.M.F., E , or the lagging current is completely balanced^ or supplied by, the condensance, if the imaginary term in the expression of I disappears ; that is, if r 2 + This gives, expanded : *o Hence the capacity required to compensate for the lagging current produced by the insertion of inductance in series to a non-inductive circuit depends upon the resis- tance and the inductance of the circuit. x being constant,. 74 ALTERNATING-CURRENT PHENOMENA. [52 with increasing resistance, r, the condensance has to be increased, or the capacity decreased, to keep the balance. ~2 _|_ T 2 Substituting c = - ^- , X o we get, as the equations of the inductive circuit balanced by condensance : r jx r 2 - -+- xj Vr 2 + x, j = _ jE x j = E x c I = r I = o r and for the power expended in the receiver circuit : 72 ^' _ 0, and the main current will become leading. We get in this case : The difference of phase in the main circuit is, tan a> rt = 76 AL TERN A TING-CURRENT PHENOMENA. [54 when r or at no load, ' and increases with increasing resistance, as the lead of the current. At the same time,,, the current in the receiver circuit, /, is approximately con- stant for small values of r, and then gradually decreases. / AMPERES 10 fl 8 7 6 6 4 3 2 ^l IMPRESSED E.M.F. CONSTANT, E =IOOO VOLTS. SERIES REACTANCE CONSTANT, X JOO OHMS. SHUNTED CONDENSANCE CONSTANT, C= IOO OH VARIABLE RESISTANCE. IN RECEIVER CIRCUIT- (.CURRENT IN RECEIVER CIRCUIT. II. CURRENT IN CONDENSER CIRCUIT. III. CURRENT IN MAIN CIRCUIT. IV.E.M.F. AT RECEIVER CIRCUIT. MS. VOL 6 II. 1 V f^~ - ^- -600 -500 ^ **"^' ^^**^ --, ~-^ IV^ / .. "*- ^ III, / -- *. -*-*. * "^ - . / / r-100 / RESISTANCE r OF RECEIVER CIRCUIT, OHMS. 7 1 1 1 1 10 20 30 40 50 60 .70 80 90 100 110 120 130 140 150 160 170 180 190 200 OH MS Fig. 54. In Fig. 54 are shown the values of /, /!,/<>, E, in Curves I., II., III., IV., similarly as in Fig. 50, for E = 1000 volts, c = X Q = 100 ohms, and r as abscissae. 5.) Constant Potential Constant Current Transformation. 54. In a constant potential circuit containing a large and constant reactance, x lt and a varying resistance, r, the current is approximately constant, and only gradually drops off with increasing resistance, r, that is, with increasing load, but the current lags greatly behind the E.M.F. This lagging current in the receiver circuit can be supplied by a shunted condensance. Leaving, however, the condensance constant, c = x , so as to balance the lagging current at no 54] RESISTANCE, INDUCTANCE, CAPACITY. 77 load, that is, at r = 0, it will overbalance with increasing load, that is, with increasing r, and thus the main current will become leading, while the receiver current decreases if the impressed E.M.F., E*, is kept constant. Hence, to keep the current in the receiver circuit entirely constant, the impressed E.M.F., E , has to be increased with increasing resistance, r; that is, with increasing lead of the main cur- rent. Since, as explained before, in a circuit with leading current, a series inductance raises the potential, to maintain the current in the receiver circuit constant under all loads, an inductance, x^ , inserted in the main circuit, as shown in the diagram, Fig. 55, can be used for raising the potential, E , with increasing load. Fig. 55. Let be the impressed E.M.F. of the generator, or of the mains, and let the condensance be x c = x ; then Current in receiver circuit, r x current in condenser circuit, Hence, the total current in main line is Er rjx 78 ALTERNATING-CURRENT PHENOMENA. [ 54 and the E.M.F. at receiver terminals, E = Ir = E ' r ; r /*<, E.M.F. at condenser terminals, ^o5 E.M.F. consumed in main line, hence, the E.M.F. at generator is E*=E + E'=E n )l AX and the E.M.F. at condenser terminals, current in receiver circuit, r jx r(x x z ) jx? This value of / contains the resistance, r, only as a fac- tor to the difference, x ;r 2 ; hence, if the reactance, ;r 2 , is chosen = x , r cancels altogether, and we find that if x^ = x , the current in the receiver circuit is constant, /=y- 2 , *0 and is independent of the resistance, r ; that is, of the load. Thus, by substituting, we have, ;r 2 = x , - Impressed E.M.F. at generator, 2 = ^2 + /V, ^2 = V^ 2 2 + * 2 ' 2 = constant ; current in receiver circuit, 7 =y^-, / = ^ = constant ; x x E.M.F. at receiver circuit, E =Sr=/^^-, E = E*L or proportional to load r; 55] RESISTANCE, INDUCTANCE, CAPACITY. 79 E.M.F. at condenser terminals, E = * ( x +J r ^ *0 . w 1+y rw.L \ -w current in condenser circuit, -l, hence > E z ; main current, E n r zr E r ( P ro P rtional to the load, = ~TT S = - , <. r , and in phase with (E.M.F, E,. The power of the receiver circuit is, IE the power of the main circuit, , hence the same. 55. This arrangement is entirely reversible ; that is, if EZ = constant, 7 = constant ; and if -4 = constant, E = constant. In the latter case we have, by expressing all the quanti- ties by f : - Current in main line, f Q = constant; E.M.F. at receiver circuit, = So*,, constant ; current in receiver circuit, , proportional to the load i; current in condenser circuit, 80 ALTERNATING-CURRENT PHENOMENA. [55 E.M.F. at condenser terminals, Impressed ElM.F. at generator terminals, x 2 1 E z = / , or proportional to the load - . From the above we have the following deduction : Connecting two reactances of equal value, x , in series to a non-inductive receiver circuit of variable resistance, r y and shunting across the circuit from midway between the inductances by a capacity of condensance, x c = x , trans- forms a constant potential main circuit into a constant cur- rent receiver circuit, and, inversely, transforms a constant current main circuit into a constant potential receiver cir- cuit. This combination of inductance and capacity acts as a transformer, and converts from constant potential to con- stant current and inversely, without introducing a displace- ment of phase between current and E.M.F. It is interesting to note here that a short circuit in the receiver circuit acts like a break in the supply circuit, and a break in the receiver circuit acts like a short circuit in the supply circuit. As an instance, in Fig. 56 are plotted the numerical values of a transformation from constant potential of 1,000 volts to constant current of 10 amperes. Since E z = 1,000, / = 10, we have : x = 100 ; hence the constants of the circuit are : E* = 1000 volts ; / = 10 amperes; E = 10 r, plotted as Curve I., with the resistances,/-, as abscissae; / / *- \ 2 E = 1000 V/l + ( -y- ), plotted as Curve II. ; V 100 I /! = 10 y/1 + ( Y, plotted as Curve III.- 7 = .1 r, plotted as Curve IV. 56] 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. CURRENT IN RECEIVER CIRCUIT CONSTANT, IMPRESSED E.M.F.CONSTANT, E 2 =IOOO VOL 2 REACTANCES OF X =IOO OHMS EACH. SH THE CONDENSANCE, Z C = IOO OHMS. VARIABLE RESISTANCE IN RECEIVER CIRCUI 1. E.M.F. AT RECEIVER CIRCUIT. 5 II, E.M-F. AT CONDENSER CIRCUIT. III. CURRENT IN CONDENSER CIRCUIT. IV. CURRENT IN MAIN LINE V. CURRENT IN MAIN LINE INCLUDING LC VI. EFFICIENCY OF TRANSFORMATION. 1 -=>IO AMPERES T.S, UNTED IN THE.IR IV r. 90 80 70 60 o IDC LE BY OLT 1400 x-** ^ 1300 5SSE S ^ ^-*- ' " / 12CO J ^^^ --^ w ^- > s ' 1100 _\\i ^ - *~*'^ 1000 900 i r=^ . - VI T: r^ - x^ ^L X* 800 ^ -^ "^ y '' ^ ^ 700 / Nl X'' ^ ^ 600/ f ^ ' \> X' 50/0 , ''' x^ x^ r .''' ^< ^ in M 3/00 x X ^ 30 20 10 200 ^-' '^X ^ 100 S ^ ^ =iES ST/S NCE r c IF R ECE IVE =? Cl RCL IT, OHf us ^X ^ Fig. 56. Constant-Potential Constant-Current Transformation. Let- r\ = 2 ohms = effective resistance of condensance ; r = 3 ohms = effective resistance of each of the inductances. We then have : Power consumed in condensance, I? r = 200 + .02 r 2 ; power consumed by first inductance, 7 2 r = 300 ; power consumed by second inductance, 7 2 r = .03 r 2 . Hence, the total loss of energy is 500 -f- -05 r 2 ; output of system, / 2 r = 100 r input, 500 + 100r + .05r 2 ; 100 r It follows that the main current, I , increases slightly by the amount necessary to supply the losses of energy in the apparatus. 82 ALTERNATING-CURRENT PHENOMENA. [56 This curve of current, 7 , 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. 57] RESISTANCE OF TRANSMISSION LINES. 83 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 +jb, of the receiver circuit. The conductance, g, 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. [58 receiver circuit can be assumed to consist of two branches, a conductance, g t the non-inductive part of the circuit, shunted by a susceptance, b, which can be varied without expenditure of energy. The two components of current can thus be considered separately, the energy component as determined by the load on the circuit, and the wattless component, which can be varied for the purpose of regu- lation. Obviously, in the same way, the E.M.F. at the receiver circuit may be considered as consisting of two components, the energy component, in phase with the current, and the wattless component, in quadrature with the current. This will correspond to the case of a reactance connected in series to the non-inductive part of the circuit. Since the effect of either resolution into components is the same so far as the line is concerned, we need not make any assump- tion as to whether the wattless part of the receiver circuit is in shunt, or in series, to the energy part. Let- Z = r jx = impedance of the line ; o- ~\/ }* ^ I -y 2 . "o "ft ~T~ "^o > y = g + jb = admittance of receiver circuit ; E = e -\- je ' = impressed E.M.F. at generator end of line ; -E = V^ 2 + e ' 2 ; E e -\-je' = E.M.F. at receiver end of line; E = Ve* + e' 2 ' f = i -f ji ' = current in the line ; I = V* 2 + i ' 2 - The simplest condition is that of a non-inductive receiver circuit, such as a lighting circuit. 1.) Non-inductive Receiver Circuit Stipplied over an Inductive Line. 58. In this case, the admittance of the receiver circuit is Y = g, since b = 0. $58] RESISTANCE OF TRANSMISSION LINES. 85 We have then : current, I = Eg', impressed E.M.F., .'= E + ZJ = E (1 + Z 9 g). Hence - E.M.F. at receiver circuit, E E = 1 + Z g current, f = * = E g Hence, in absolute values E.M.F. at receiver circuit, E = current, I = The ratio of E.M.Fs. at receiver circuit and at genera- tor, or supply circuit, is E 1 r'2 2 and the power delivered in the non-inductive receiver cir- cuit, or r? o _. output, P = I E = As a function of g, and with a given E , r , and x , this power is a maximum, if ^=0; dg- that is - 1 + g*r* + g*x* = ; hence conductance of receiver circuit for maximum output, = 1 = _ 1_ " Vr 2 + X* *o ' Resistance of receiver circuit, r m = = z ; 86 ALTERNATING-CURRENT PHENOMENA. [59 and, substituting this in P - Ma and ratio of E.M.F. at receiver and at generator end of line, = !L = 1 a ~ E~~ efficiency, That is, the output which can be transmitted over an inductive line of resistance, r , and reactance, x , that is, of impedance, z , into a non-inductive receiver circuit, is a maximum, if the resistance of the receiver circuit equals the impedance of the line, r = z , and is 77 2 P = ~2(r a + ,.) ' The output is transmitted at the efficiency of and with a ratio of E.M.Fs. of 1 59. We see from this, that the maximum output which can be delivered over an inductive line is less than the output delivered over a non-inductive line of the same resistance that is, which can be delivered by continuous currents with the same generator potential. In Fig. 57 are shown, for the constants E = 1000 volts, Z = 2.5 6/ ; that is, r = 2.5 ohms, x = 6 ohms, z = 6.5 ohms, with the current I n as abscissae, the values 60] 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 arili P, for a non-inductive line of resistance, r = 2.5 ohms, X Q 0, are shown in Curves IV.,, V., and VI. NON-INDUCTIVE RECEIV SUPPLIED OVER INDUCTIVE LI Z =2.5-6/ AND OVER NON-INDUCTIVE L^ r = 2.5 CURVE 1. E.M.F. AT RECEIVER CIRC || II.' OUTPUT IN " III. EFFICIENCY OF TRANSMISJ ' VU ER CIRCUIT ME OF IM.PEC E OF RESIS" UIT, INDUCTIV NON-INDUCTIV INDUCTIV NON-INDUCTIV ION, INDUCTIV NON-4WDyCTIV )AN FAN E L^ E ' E. ' E ( E ' E ' / :E CE E X u 100 90 80 70 60 50 UU K ^* ^ o / z Ul < w / / U- u_ -5- 1000 / .X'' ^ ^ <' UT 100^ -^ ^ ^_^ // / s , 903: 900 40 30 20 10 ^^ ^ , ~^~~^ \-\f \ 30* 800 // \, "^ l ^. -^ Jl/ \ 70tf TOO / > ^V . V fiOO ) / ^ \ \ 5(K --* 500 -~-^ / \ \ m 400 / \ \ 3W :?oo / \ \ 20^ :WO / \ \(H 100 / CUF RE MTI N L NE:| | \MP ERE 3 \ 10 20 30 40 60 60 70 80 90 100 110 120 130 140 150 160 170 180 ' 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, r in addition to the conductance, g, its admittance can be written thus: FZ ). Then- current, f = Impressed E.M.F., E = E + I Z = E (1 88 AL TERN A TING-CURRENT PHENOMENA. [61 Hence E.M.F. at receiver terminals, E = YZ (1 + r g + * b)- j ( Xo g _ r. J) ' current, 1 + YZ (1 + r.g + x.b)~ j ( Xo g -r b}' or, in absolute values E.M.F. at receiver circuit, current, ratio of E.M.Fs. at receiver circuit and at generator circuit, r og + ^ ^)* + (x 9 g - and the output in the receiver circuit is, 61. a.) Dependence of the output upon the susceptance of the receiver circuit. At a given conductance, g, of the receiver circuit, its output, P = E^a^g, is a maximum, if a 2 is a maximum ; that is, when /= ~ = (1 4- r og + * bY + (* g - r b? is a minimum. The condition necessary is or, expanding, Hence Susceptance of receiver circuit, b= - * -or ^ + b = 0, 62] RESISTANCE OF TRANSMISSION LINES. 89 s 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, ^ = ^t 1 . z (g + go} ' maximum output, P\ = ) *? (g + goY current, * b o y + (r b and, expanding, phase difference in receiver circuit, ~ b b tan o> = - = -- - ; ^ ^ phase difference in generator circuit, + ' 62. #.) Dependence of the output upon the conductance of the receiver circuit. At a given susceptance, b, of the receiver circuit, its output, P = E 2 a 2 g, is a maximum, if + r. g- + ^o^) 2 + (*.f - 90 ALTERNATING-CURRENT PHENOMENA. [63 that is, expanding, - (1 + r g + x l>Y + ( Xo g - r by - 2g(r + r *g + x *g) = 0; or, expanding, (b H- b Y = g* _ #; g = V^o 2 + (b + b o y. Substituting this value in the equation for a, page 88, we get - ratio of E.M.Fs., (b ^o V2 g (g + ^ ) V2 power, 2 2 { go 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 ^ = ^? 7=7o J hence: K= g + jb = g Jb ; x = x , r = r , ar = z , Z = r jx = r + jx ; substituting this value, we get ratio of E.M.Fs., a m = -^ = - ; 2^ 2r power, ^ = - ; "* r o 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- 63] RESISTANCE OF TRANSMISSION LINES. 91 dance, Z = r jx , into a non-inductive receiver circuit, is a maximum for the resistance, r = z , or conductance, g = y Q , of the receiver circuit, or 77 2 r> __ t ^o ~ 2 fro + *o) ' 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, g = Vfr a +(* + 4,) a , and is E*y* 2 (* + *,)' at the ratio of potentials, With a receiver circuit of constant conductance, g, the output, as a function of the reactance, b, is a maximum for the reactance, b b , and is p _ * at the ratio of potentials, y (g + go) The maximum output which can be delivered over an in- ductive line, as a function of the admittance or impedance of the receiver circuit, takes place when Z r -f jx , or Y= g jb \ that is, when the resistance or conductance of a 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 = 2 / 4 r , or independent of the reactances, but equal to the output of a continuous-current 92 AL TERN A TING-CURRENT PHENOMENA. [64 circuit of equal line resistance. The ratio of potentials is, in this case, a = z o j 2 r , while in a continuous-current circuit it is equal to ^. The efficiency is equal to 50 per cent. RATIO OF POTENTIAL nj ATlREqEIVJNG ANC .01 .02 .03 .Oi .05 .06 .07 .08 .09 .10 .11 .12 .13 .14 .15 J6 .17 Fig. 58. Variation of the Potential in Line at Different Loads. 64. As an instance, in Fig. 58 are shown, for the constants E = 1000 volts, and Z = 2.5 6/; that is, for r = 2.5 ohms, x = Gohms, z = 6.5 ohms, and with the variable conductances as abscissae, the values of the - output, in Curve I., Curve III., and Curve V. ; ratio of potentials, in Curve II., Curve IV., and Curve VI.;. Curves I. and II. refer to a non-inductive receiver circuit ; 65] RESISTANCE OF TRANSMISSION LINES. 93 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' ' ... . =.142; Curves VII. and VIII v refer to 4 non-inductive re- ceiver circuit of a 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 AND RATIO OF POTENTIAL d AT RECEIVING.AND SENDING END OF LINE OF IMPEDANCB 2 =-2.5-3l/ AT CONSTANT IMPRESSED E. M.F. E^lOOO #=.0592 I OUTPUT II RATIO OF POTENTIALS SUSCEPTANCE .OF RECEIVER CIRCUIT -3 -2 -1 +1 +2 +3 Fig. 59. Variation of Potential in Line at Various Loads. 3.) Maximum Efficiency. 65. The output, for a given conductance, g, of a receiver circuit, is a maximum if b = b . This, however, is gen- erally not the condition of maximum efficiency. 94 ALTERNATING-CURRENT PHENOMENA. [65 The loss of energy in the line is constant if the cur- rent is constant; the output of the generator for a given current and given generator E.M.F. is a maximum if the cur- rent is in phase with the E.M.F. at the generator terminals. Hence the condition of maximum output at given loss, or of maximum efficiency, is tan o> = 0. The current is I=E _ _ (1 + r og + x b) j(x g r b) ' multiplying numerator and denominator by (1 -f r g -\-x b) + j (x g r b~) y to eliminate the imaginary quantity from the denominator, we have /U(l + r g+*ob) -b(* g-r b)} +\ \J {b (1 + r g + x b) + g (x og - r b}} ) (1 + r g + x bY -f ( Xo g - r bY The current, 7 , is in phase with the E.M.F., E , if its quadrature component that is, the imaginary term dis- appears, or b (1 + r g+x b) +g(x g-r b) = 0. This, therefore, is the condition of maximum efficiency. Expanding, we have, Hence, the condition of maximum efficiency is, that the reactance of the receiver circuit shall be equal, but of oppo- site sign, to the reactance of the line. Substituting x = x > we have, ratio of KM.Fs., _ = E_ = _ 0_ _ = Vr 2 + * .01 .02 .03 .04 .05 .06 .07 .08 Fig. 60. Load Characteristic of Transmission Line. In Fig. 60 are shown, for the constants E = 1,000 volts, Z = 2.5 6y ; r = 2.5 ohms, x = 6 ohms, z 6.5 ohms, 96 ALTERNATING-CURRENT PHENOMENA. [66 and with the variable conductances, g t 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 4~ r o 4.) Control of Receiver Voltage by Shunted Susceptance. 66. By varying the susceptance of the receiver circuit,, the potential at the receiver terminals is varied greatly. Therefore, since the susceptance of the receiver circuit can be varied at will, it is possible, at a constant generator E.M.F., to adjust the receiver susceptance so as to keep the potential constant at the receiver end of the line, or to vary it in any desired manner, and independently of the generator potential, within certain limits. The ratio of E.M.Fs. is r g + x b? + (x og - r $ If at constant generator potential E , the receiver potential E shall be constant, a = constant ; hence, (1 + r g + * W + (x g ~ r by 2 = i ; or, expanding, b = - b which is the value of the susceptance, b, as a function of the receiver conductance, that is, of the load, which is required to yield constant potential, aE , at the receiver circuit. For increasing g, that is, for increasing load, a point is reached, where, in the expression 67, 68] RESISTANCE OF TRANSMISSION LINES. 97 the term under the .root becomes imaginary, and it thus becomes impossible to maintain a constant potential, aE . Therefore, the maximum output which can be transmitted at potential aE, is given by the expression a hence b = o , the susceptance of receiver circuit, and g = go + , the conductance of receiver circuit ; a ^- - g , the output 67. If a = 1, that is, if the voltage at the receiver cir- cuit equals the generator potential g = y -go', P=a*E*(y - go ). If a = 1 when g = 0, b = when g > 0, b < ; if a > 1 when g = 0, or g > 0, b < 0, that is, condensance ; if a < 1 when g = 0, b > 0, when g = - g + i/ ^ A - 2 , J = ; when g > - g + J(y\- b \ b < 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 I da = ; expanding : 2 a (&- - g \ - f!f' = ; \a ) a* hence, y = 2ag oj and * = -= l =^-' 98 AL TERN A TING-CURRENT PHENOMENA . [ 69 the maximum output is determined by & == >0 I == <0 j a and is, P = ^ . 4 /- From : a = -2- = -^_ , the line reactance, x , can be found, which delivers a maximum output into the receiver circuit at the ratio of potentials, a, and z = 2 r a, x o = r o V4 a 1 1 ; for a = 1, If, therefore, the line impedance equals 2 a times the line resistance, the maximum output, P = E 2 /4:r , is trans- mitted into the receiver circuit at the ratio of potentials, a. If z = 2 r , or JT O = r V3, the maximum output, P = E 2 /4r , can be supplied to the receiver circuit, without change of potential at the receiver terminals. Obviously, in an analogous manner, the law of variation of the susceptance of the receiver circuit can be found which is required to increase the receiver voltage proportionally to the load ; or, still more generally, to cause any desired variation of the potential at the receiver circuit indepen- dently of any variation of the generator potential, as, for in- stance, to keep the potential of a receiver circuit constant, even if the generator potential fluctuates widely. 69. In Figs. 61, 62, and 63, are shown, with the output, P = E* g a 2 , as abscissae, and a constant impressed E.M.F., E = 1,000 volts, and a constant line impedance, Z = 2.5 6/, or, r = 2.5 ohms, x = 6 ohms, z = 6.5 ohms, the following values : VOLTS 1000 1 RATIO OF RECEIVER VOLTAGE TO SENDER VOLTAGE: a =I.O J LINE IMPEDANCE: Z = 2.5. - CONSTANT GENERATOR POTENT I. ENERGY CURRENT If. REACTIVE CURRENT DI7 TOTAL CURRENT IV. CURRENT IN NON-INDUCTIVE RECEIVER CIRCUIT WITHOUT COMPENSATION "V. POTENTIAL " " ' r- " " ' OUTPUT IN RECEIVER CIRCUIT, K Fig, 61. Variation of Voltage Transmission Lines. RATIO OF RECEIVER VOLTA3E TO SENDER VOLTAGE: a =.7 _LINE IMPEDANCE: Z ^2.5. lit . ENERGY CURRENT CONSTANT GENERATOR POTENTIAL E =IOOO ^ REACTIVE CURRENT TOTAL CURRENT CURRENT IN NON-INDUCTIVE CIRCUIT WITHOUT COMPENSATIO 1 V. POTENTIAL " " " " " " VOLTS 1000 200 180 160 HO 120 100 80 60 10 300 *00 700 600 600 400 300 \ OUTPUT IN RECEIVER CIRCl IT, KILOWATTS 30 40 50 60 70 80 Fig. 62. Variation of Voltage Transmission Lines. 100 ALTERNATING-CURRENT PHENOMENA. [69 r i i i i i i i i i i i i i i i RATIO OF RECEIVER VOLTAGE'TO SENDER VOLTAQE: a =1 3 'LINE IMPEDANCE: z a =2.5. ej CONSTANT GENERATOR POTENTIAL E =IOOO III. TOTAL CURRENT " IV. CURRENT TN NON-INDVCTIVE RECEIVER CIRCUIT WITHOUT COMPEC V. POTENTIAL 10 20 30 40 60 60 70 80 90 Fig. 63. Variation of Voltage Transmission 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. 58) ; a= .7 (Fig. 59) ; a = 1.3 (Fig. 60). 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. 70] 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 g^rierator, it is of interest to determine what is the maximum value, of potential, E, that can be produced at the receiver circuit with a given generator potential, E . The condition is that a = maximum or = minimum; a' 2 that is, ^q/O - <*S substituting, i- = (1 + r and expanding, we get, (x og - a value which is impossible, since neither r nor g can be negative. The next possible value is g = 0, a wattless circuit. Substituting this value, we get, and by substituting, in b -f- b = ; that is, the sum of the susceptances = 0, or the condition of resonance is present. Substituting, we have Vr^eT r o b 102 ALTERNATING-CURRENT PHENOMENA. [71 The current in this case is, .or tiie ame 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. VOVT 2000 4900 1800 1700 1600 1500 1400 1300 1200 1100 1000 900 800 700 600 500 N s \ V ^ \ \ \ \ \ \ kN \ \ CONSTANT IMPRESSED E.M.F. E =IOOO " LINE IMPEDANCE Z =2.5- 6J \ MAXIMUM OUTPUT BY COMPENSATION II MAXIMUM EFFICIENCY BY COMPENSATION III NON-INDUCTIVE RECEIVER CIRCUIT IV NON-INDUCTIVE LINE AND NON-INDUCTIVE RECEIVER CIRCUIT \l ]j 1 1 a j -/L ^^ ^ ^-^ *^ m X ^ II- N^^ / M $ X \ n 1 / ^ \ I/ /( \ 400 ff/ \ j 800 '200 100 T) / ^ ^ ^i\ ^ ,< ' ^ s ^ ^ HCV) C ^ ^ OUT PUT K.W > 10 20 30 40 50 60 70 80 90 100 Fig. 64. Efficiency and Output of Transmission Line. 71. As summary to this chapter, in Fig. 64 are plotted, for a constant generator E.M.F., E = 1000 volts, and a line impedance, Z = 2.5 6/, or, r = 2.5 ohms, x = 6 ohms, z = 6.5 ohms ; and with the receiver output as 71] 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 rion-inductive receiver cir- cuit, (Curve III.) ; the condition b = 0, b = 0, or a non-inductive line and non- inductive receiver circuit, or a non-inductive receiver circuit and a non-inductive line. 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 or condensance, at will. 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. [72 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, z = r 2 + x\ 3.) By the ratio : r = Power consumed _ _ (E.M.F.) 2 m (Current) 2 Power consumed ' where, however, the "power" and the "E.M.F." do 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, (r _ Energy component of current Total E.M.F. is called the effective conductance of the circuit. 73] EFFECTIVE RESISTANCE AND REACTANCE. 105 In the same way, the value, _ Wattless component of E.M.F. Total current is the effective reactance, and * riit , _ Wattless component of current Total 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 ferric inductances, 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 ; If.) Dielectric hysteresis. 106 ALTERNATING-CURRENT PHENOMENA. [ 74 2.) Primary electric currents, as, a.) Leakage or escape of current through the in- sulation, 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 con- ducting materials ; c.) Secondary currents of mutual inductance in neighboring circuits. 4.) Induced electric charges, electrostatic influence. While all these losses can be included in the terms effective resistance, etc., only the magnetic hysteresis and the eddy currents in the iron will form the subject of what follows. 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 of force. 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 resistance is negligible, the counter E.M.F. equals the impressed E.M.F. ; hence, if the impressed E.M.F. is 75] EFFECTIVE RESISTANCE AND REACTANCE. 107 a sine wave, the counter E.M.F., and, therefore, the mag- netic flux which induces the counter E.M.F. must follow sine waves also. The alternating wave of current is not a .sine wave in this case, but i distorted by hysteresis. It is possible, however, to plot the cfurrent 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 y of the current, the maximum magnetic flux, <, is found by the formula : hence, - _ " V2 TT n N ' A maximum flux, $, and magnetic cross-section, S, give the maximum magnetic induction, (B = < / 5. If the magnetic induction varies periodically between + (B and (B, the M.M.F. varies between the correspond- ing values -f- if and if, 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., if, in ampere-turns per unit length of magnetic cir- cuit, the length, /, of the magnetic circuit, and the number of turns, n, of the electric circuit, are found the instantaneous values of current, /, corresponding to a M.M.F., if, that is, .as magnetic induction (B, and thus induced E.M.F. e, as : n 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. 75 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, rj = .0033,. and are given with ampere-turns per cm as abscissae, and kilo-lines of magnetic force as ordinates. Fig. 65. Hysteretic Cycle of Sheet Iron. In Figs. 66, 67, 68, and 69, the sine curve of magnetic induction as derived from the induced E.M.F. is plotted in dotted lines. 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. 66, 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. 75] 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 = = 2000 3-1.8 \ CB-6000 3 "2.9 X 7 Figs. 66 and 67. Distortion of Current Wave 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 ALTERNATING-CURRENT PHENOMENA. 75 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 & = 10,000 ; and at still /\\ NX f\ 16000 A 20 \ \ 13 \J \J Figs. 68 and 69. Distortion of Current Wave by //ysferes/s. 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 remains within the range of magnetic saturation, since in the proximity of satu- ration the current near the maximum point of magnetization has to rise abnormally to cause even a small increase of magnetization. 76,77] EFFECTIVE RESISTANCE AND REACTANCE. Ill The four curves, Figs. 66, 67, 68, and C9, are not drawn to the same scale. The maximum values of M.M.F., cor- responding to the maximimT values of magnetic induction, 10 - 8 , or * 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, F 1>6 l/"10 5 '8 7? 1.6 P = 77 ^M = 58 77 ^- , where ^ or, substituting 77 = .0033, we have A = 191.4 19 19 5 o ' /z or, substituting F= SL, where L = length of magnetic circuit, A = ryZlO 5 - 8 = 58 r, ZIP 3 = x Z and ^ = 58 9 ^ ] ' 6 Z 103 = 191 ' 4 - In Figs. 73, 74, and 75, is shown a curve of hysteretic loss, with the loss of power as abscissae, and in curve 73, with the E.M.F., E, as ordinates, for Z = 1, S = 1, N= 100, and n = 100 ; 118 AL TERNA TING-CURRENT PHENOMENA. [80 9000 / ' / / RELATI ON ET\ VEEN E AND P / 6000 F OR L-1 , s= -1, 4-1 00 71 = 10C / 7 / / / 6000 4000 LJ * / 2 i / Q- /< ^ / / / 1000 S P x /r ,X lx . ' ^ :.M F. Fig. 73. Hysteresis Loss as Function of E, M. F. 1.3 1.0 ETW ILA TION. EEN n A ND L T 1, S=1 N 100. E=100 = NUMBER OF 1 URNS 50 100 150 200 250 300 350 400 Fig. 74. Hysteresis Loss as Function of Number of Turns. 81] EFFECTIVE RESISTANCE AND REACTANCE, RELATION BETWEEN N AND P FOR S-l, L = l, U = IOO, E = IOO )0 200 300 75. Hysteresis Loss as Function of Cycles. in curve 74, with the number of turns as ordinates, for Z = 1, 5=1,^=100, and . = 100; in curve 75, with the frequency, JV t or the cross-section, S 9 as ordinates, for L = 1, n = 100, and E = 100. As shown, the hysteretic loss is proportional to the 1.6 th power of the E.M.F., inversely proportional to the 1.6 th power of the number of turns, and inversely proportional to the .6 th power of frequency, and of cross-section. 81. If g = effective conductance, the energy compo- nent of a current is / = Eg, and the energy consumed in a conductance, g t is P = IE = E^g. Since, however : Z7- 1.6 EM.6 7~ A ** 1_ A J-* Z7*2 ^ 6 = or AT" 6 ' iicivv^ - 58 77 ZIP 3 From this we have the following deduction : 120 ALTERNATING-CURRENT PHENOMENA. [81 The effective conductance due to magnetic hysteresis is proportional to the coefficient of hysteresis, ??, and to the length of the magnetic circuit, L, and inversely proportional to the Jj, th power of the E.M.F., to the .6 th power of the frequency, N, and of the cross-section of the magnetic circuit, S, and to the 1.6 th power of the number of turns, n. Hence, the effective hysteretic conductance increases with decreasing E.M.F., and decreases with increasing RELATION BETWEEN flfAND E FOR L = l,' N=|00. S\, n= 50 100 150 200 250 300 350 Fig. 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 = 1, E = 100, N= 100, 5 = 1, and n - 100, respectively, with the conductance, g, as ordinates, and with 81] EFFECTIVE RESISTANCE AND REACTANCE. 121 RELATION BETWEEN Q AND N FOR L-l. E = IOO. 8-1, n-\OO 50 100 150 200 250 300 350 .400 Fig. 77. Hysteresis Conductance as Function of Cycles. 1 j R .LA' ION BE' 'WE EN IAN Dfl FOF L= 1, E = 1( 30, N^ 100 , 8 = -1. \ \ 06 \ \ ^v V \ S. = NL ^ MB ^= :R o * FT ==^ JRN n 50 100 150 200 250 300 350 400 '. Fig. 78. Hysteresis Conductance as Function of Number of Turns. 122 ALTERNATING-CURRENT PHENOMENA. [82 E as abscissas 'in Curve 76. -A^as abscissae in Curve 77. n as abscissae 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 5 by 50 per cent causes a variation in g of 21 per cent. If (R = magnetic reluctance of a circuit, JF A = maximum M.M.F., / = 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, =-V2 we have ^2. (R hence, the absolute admittance of the circuit is (R1 8 E I-K^N N' 10 8 where a = , a constant. 2-n-n 2 Therefore, the absolute admittance, y, of a circuit of neg- ligible resistance is proportional to the magnetic rehictance, (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, varjes 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- 82] EFFECTIVE RESISTANCE AND REACTANCE. 123 former, (R is the reluctance of the iron circuit. Hence, if /A = permeability, since and <5^ = LF=--LW = M.M.F., 4t 7T 3> ,5" (B = ^ 5C = magnetic flux, and (R = ; 4: TT (JL S substituting this value in the equation of the admittance, (R IP 8 Z IP 9 z y = * s-Tr* we have zr^r-7 where * = M^ = ^^ Therefore, in art ironclad circuit, the absolute admittance, y, is inversely proportional to the frequency, N, to the perme- ability, /A, 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. A The conductance is g = - ; and the admittance, y = -? ; hence, the angle of hysteretic advance is Sin a = -- = ^ ; y zE* or, substituting for A and z (p. 117), N A rjZlO 5 ' 8 S Sin a a - '- - - , * 2- 8 7r 1 - 6 6 1 ' 6 /2 L6 Z10 9 ' or, substituting E = 2 we have sin a = 124 AL TERNA TING-CURRENT PHENOMENA. [ 83 which is independent of frequency, number of turns, and shape and size of the magnetic and electric circuit. Therefore, in an ironclad inductance, the angle of hysteretic advance, a, depends upon the magnetic constants, permeability and coefficient of hysteresis, and upon the maximum magnetic induction, but is entirely independent of the frequency, of the shape and other conditions of the magnetic and electric circtiit ; and, therefore, all ironclad magnetic circuits constructed of the same quality of iron and using the same magnetic density, give the same angle of hysteretic advance. The angle of hysteretic advance, a, in a closed circuit transformer, depends upon the quality of the iron, and upon the magnetic density only. The sine of the angle of hysteretic advance equals Jf, times the product of the permeability and coefficient of hysteresis, divided by the .Jj? h power of tJie magnetic density. 83. If the magnetic circuit is not entirely ironclad, and the magnetic structure contains air-gaps, the total re- luctance is the sum of the iron reluctance and of the air reluctance, or (R = (R . _|_ (R o . hence the admittance is Therefore, in a circuit containing iron, the admittance is the sum of the admittance due to the iron part of the circuit, y { = a I N(S(. iy and of the admittance due to the air part of tJie circuit, y a = a I N(& a , 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 sin a = 84] EFFECTIVE RESISTANCE AND REACTANCE. 125 and a maximum, gjy iy for the ironclad circuit, but decreases with increasing width of the air-gap. The introduction of the air-gap of reluctance, (R a , decreases sin a in the ratio, In the range of practical application, from (B = 2,000 to (B = 12,000, the permeability of iron varies between 900 and 2,000 approximately, while sin a in an ironclad circuit varies in this range from .51 to .69. In air, /* = 1. If, consequently, one per* cent of the length of the iron is replaced by an air-gap, the total reluctance only varies in the proportion of 1^ to l^o* r 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 y. Therefore, in an electric circuit containing iron, but forming an open magnetic circuit whose air-gap is not less than T ^o 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, t G\.a N b - , or : x = . N (Ra The angle of hysteretic advance is small, below 4, and the hysteretic conductance is, A g ~ E A N A ' 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. [84 then from the equation, E= V27ryV10- 8 , where, JV = frequency, n = number of turns, we get the magnetism, <, and by means of the magnetic cross section, S, the maximum magnetic induction : (B = < / .S. From (&, we get, by means of the magnetic characteristic of the iron, the M.M.F., = SF ampere-turns per cm length, where SF^ae, 4 IT if OC = M.M.F. in C.G.S. units. Hence, if Li length of iron circuit, 9^ = Z, $ = ampere-turns re- quired in the iron ; if L a = length of air circuit, SF a = - - = ampere-turns re- , . ^ 4 TT quired m the air ; hence, &= & t -\- $ a = total ampere-turns, maximum value, and $ I V2 = effective value. The exciting current is and the absolute admittance, y = VPTT 2 = L . L If SF t - is not negligible as compared with CF a , this admit- tance, y, 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 = W ' j E?, and vari- able with the E.M.F., E. 85] EFFECTIVE RESISTANCE AND REACTANCE. 127 The angle of hysteretic advance is, sin a = - ; , y _____ the susceptance, b = Vy 2 ' g* ; the effective resistance, . and the reactance, , x = y 2 85. 'As conclusions, we derive from this chapter the following : 1.) Iii 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. 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, /*, the coefficient of hys- teresis, -r), and the maximum magnetic induction, as shown in the equation, 4 sin a = f L . (B- 4 6.) The effect of hysteresis can be represented by an admittance, Y = g +j b, or an impedance, Z = r jx. 7.) The hysteretic admittance, or impedance, varies with the magnetic induction; that is, with the E.M.F., etc. 128 ALTERNATING-CURRENT PHENOMENA. [85 8.) The hysteretic conductance, -, is proportional to the coefficient of hysteresis, 17, and to the length of the magnetic circuit, L, inversely proportional to the .4 th power of the E.M.F., E, to the .6 th power of frequency, N, and of the cross-section of the magnetic circuit, S, and to the 1.6 th power of the number of turns of the electric circuit, n, as expressed in the equation, 58 r, L 10 8 g = A JV- G S- 6 n 1 - 6 ' 9.) The absolute value of hysteretic admittance, is proportional to the magnetic reluctance : (R = _(, io' 10.) In an ironclad circuit, the absolute value of admit- tance is proportional to the length of the magnetic circuit, and inversely proportional to cross-section, S, frequency, N, permeability, /x, and square of 'the number of turns, n, or 127 L 10 6 11.) In an open magnetic circuit, the conductance, g, is the same as in a closed magnetic circuit of the same iron part. 12.) In an open magnetic circuit, the admittance, y, is practically constant, if the length of the air-gap is at least T 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 = 8df=2 7 r*lV 2 W y xVx, in C.G.S. units or ergs per second, and, consequently, the total power consumed in one cm 2 of the sheet of thickness, d, is , d , d / T /* 3F d J d V ~ 2 7T 2 J?V 2 / 2 10 - 8 ampere-turns per cm. For example, if / = .1 cm, N= 100, ffi = 5,000, then f= 1,338 ampere-turns per cm; that is, half as much as in a lamina of the thickness /. 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 easily be avoided. 95] 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, inputting 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 in Maxwell. 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 stranded conductor, or several conductors in parallel. 140 ALTERNATING-CURRENT PHENOMENA. [96 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, 7 * where / = total current in conductor. Fig. 82. The magnetic reluctance of a tubular zone of unit length and thickness dx, of radius x, is dx 96] FOUCAULT OR EDDY CURRENTS. 141 The current inclosed by this zone is I x = ix z ir, and there- fore, the M.M.F. acting upon this zone is Cc _ 4:7T , 4 7T 2 I ' X 2 *~"io* : ~ir~' and the magnetic flux in this^zone is & 2 + p 2 = .95 -*- 1, we have, k = .51 X 10 ~ 6 ; 142 ALTERNATING-CURRENT PHENOMENA. [97 hence .51 X 10 ~ 6 = V2 ^ NX* 10 ~ 9 or NR* = 36.6 ; hence, when 7V= 125 100 60 33.3 = .541 .605 .781 1.05 cm. D = 2 = 1.08 1.21 1.56 2.1 cm. Hence, even at a frequency of 125 cycles, the effect of unequal current distribution is still negligible at one cm diameter of the conductor. Conductors of this size are, however, excluded from use at this frequency by the exter- nal self-induction, which is several times larger than the resistance. We thus see that unequal current distribution is usually negligible in practice. 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- 97] FOUCAULT OR EDDY CURRENTS. 143 ponent, representing the loss of power, and a wattless component, representing the decrease of self-inductance. Let- j x = 2 TT N L = reactance 'of main circuit; that is, L = HI total number of interiinkages with the main conductor, of the lines of magnetic force produced by unit current in that conductor ; x^ = 27rA r Zj = reactance of secondary circuit ; that is, L l = total number of interlinkages with the secondary conductor, of the lines of magnetic force produced by unit current in that conductor; x m = 2 TT N L m mutual inductance of circuits ; that is, L m = total number of interlinkages with the secondary conductor, of the lines of magnetic force produced by unit current in the main conductor, or total number of inter- linkages with the main conductor of the lines of magnetic force produced by unit current in the secondary conductor. Obviously : x * As coefficient of self-inductance L, L', the total flux surrounding the conductor is here meant. Quite frequently in the discussion of inductive apparatus, especially of transformers, that part of the magnetic flux is denoted 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, Z, is in this case equal to the sum of the self-inductance, L ', ;and the mutual inductance, L m . The object of this distinction is to separate the wattless part, L', of the total self-inductance, Z, from that part, L m , which represents the transfer of E.M.F. into the secondary circuit, since the action of these two components is essentially different. Thus, in alternating-current transformers it is customary and will be done later in this book to denote as the self-inductance, Z, of each circuit ^nly that part of the magnetic flux produced by the circuit which passes between both circuits, and thus acts in " choking " only, but not in transform- ing; while the flux surrounding both circuits is called mutual inductance, or useful magnetic flux. With this denotation, in transformers the mutual inductance, L m , is usu- .ally very much greater than the self-inductances, //, and Z/, while, if the self-inductances, L and Zj , represent the total flux, their product is larger than the square of the mutual inductance, L m ; or 144 ALTERNATING-CURRENT PHENOMENA. [ 9S Let r x = resistance of secondary circuit. Then the im- pedance of secondary circuit is z l = x ; E.M.F. induced in the secondary circuit, EI = /#,/, where / = primary current. Hence, the secondary current is and the E.M.F. induced in the primary circuit by the secon dary current, 7 X is or, expanded, Hence, = effective conductance of mutual inductance ; . r? + b = ~ Xm * l = effective susceptance of mutual inductance. r^ + x^ The susceptance of mutual inductance is negative, or of opposite sign from the susceptance of self-inductance. Or, Mutual inductance consumes energy and decreases tJie 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, called 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 dielectric hys- teresis. $99] FOUCAULT OR EDDY CURRENTS. 145 While the laws of the loss of energy by magnetic hys- teresis are fairly well understood, and the magnitude of the effect known, the phenomenon of dielectric hysteresis is .still almost entirely unknown as concerns its laws and the magnitude of the effect. It is quite probable that the loss of power in the dielec- tric in an alternating electrostatic field consists of two dis- tinctly different components, of which the one is directly proportional to the frequency, analogous to magnetic hysteresis, and thus a constant loss of energy per cycle, independent of the frequency ; while the other component is proportional to the square of the frequency, analogous to the loss of power by eddy currents in the iron, and thus a loss of energy per cycle proportional to the frequency. The existence of a loss of power in the dielectric, pro- portional to the square of the frequency, I observed some time ago in paraffined paper in a high electrostatic field and at high frequency, by the electro-dynamometer method, .and other observers under similar conditions have found the same result. Arno of Turin found at low frequencies and low field strength in a larger number of dielectrics, a loss of energy per cycle independent of the frequency, but proportional to the 1.6 th power of the field strength, that is, following the same law as the magnetic hysteresis, 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-& = 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 ALTERNATING-CURRENT PHENOMENA. [99 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 = ', 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 admit- tance of a condenser, = g c, where g = hysteretic conductance, // = hysteretic susceptance ; and the dielectric hysteretic impedance of a condenser, Z = r jb' = r+jx c , where : r = hysteretic resistance, oc c = hysteretic condensance ; and the angle of dielectric hysteretic lag, tan a = f = - , ' oc are constants of the circuit, independent of E.M.F. and fre- quency. The E.M.F. is obviously inversely proportional to the frequency. The true static dielectric hysteresis, observed by Arno as proportional to the 1.6 th power of the density, will enter the admittance and the impedance as a term variable and dependent upon E.M.F. and frequency, in the same manner as discussed in the chapter on magnetic hysteresis. To the magnetic hysteresis corresponds, in the electro- static field, the static component of dielectric hysteresis,, following, probably, the same law of 1.6 th power. To the eddy currents in the iron corresponds, in the electrostatic field, the viscous component of dielectric hys- teresis, following the square law. 1OO] FOUCAULT OR EDDY CURRENTS. 14T To the phenomenon of mutual inductance corresponds, in the electrostatic field, the electrostatic induction, or in- fluence. t 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 susceptance, or condensance. Thus, the electrostatic influence introduces an effective conductance, g, and an effective susceptance, b, of oppo- site 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 these high densities. 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, [ 101 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 synchronous 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. ~1O1] 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^; 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. [102 CHAPTER XII. DISTRIBUTED CAPACITY, INDUCTANCE, RESISTANCE, AND LEAKAGE. 102. 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. f.mimillllJlllllllliJi ITTTTTTTmTTTTTTTrTTTTTi 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, especially concentric cables, and to a certain degree in the high-potential coils of alternating- current transformers. 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- 1O3] DISTRIBUTED CAPACITY. 151 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 condenser of \ the line capacity. This approximation, based on Simpson's rule, assumes the variation of the elec- tric 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 approximation is not permissible, but each line element has to be considered as an infinitely small condenser, and the differential equations based thereon integrated. 103. 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 l.llXlO- 6 e/ . , , c = microfarads, 4 log nat where K = dielectric constant of the surrounding medium = 1 in air ;; / = length of conductor = 5 X 10 6 cm. ; d = distance of conductors from each other = 50 cm. ; B = diameter of conductor = 1 cm. Since C = .3 microfarads, the capacity reactance is 10 6 152 ALTERNATING-CURRENT PHENOMENA. [ 1O4 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, g i = = 2.25 amperes. x 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 7 = - = 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, however, the length of transmission is 150 km and the voltage 30,000, capacity reactance at 60 cycles, x = 2,970 ohms ; charging current, i = 10.1 amperes ; line resistance, r = 66 ohms ; main current at 10 p>er cent loss, / = 45.5 amperes. The condenser current is thus about 22 per cent, of the main current. 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. 104. A.) Line capacity represented by one condenser .shunted across middle of line. Let Y = g -\- jb = admittance of receiving circuit; z = r j x = impedance of line ; b c = condenser susceptance of line. 1O5] DISTRIBUTED CAPACITY, 15& Denoting, in Fig. 84, the E.M.F., viz., current in receiving circuit by E, 7, the E.M.F. at middle of line by E', the E.M.F., viz., current at generator by E ,J ; if Fig. 84. Capacity Shunted across Middle of Line. We have, f = I-jb c E c ' \l-\ ( r ->^) \ 2 jb e (r-jx) ~~ I or, expanding, / =^{[^+^(r* = 1 + (r -/*) (*+>*) -(r- jx) = ^ 1 1 + (r -jx) g + jt -i-l^ (r -j x 105. j5.) Z/^ 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 E, 7, the E.M.F. at middle of line by E r , 154 ALTERNATING-CURRENT PHENOMENA, [ 1O5 the current on receiving side of line by 7', the current on generator side of line by /", the E.M.F., viz., current at generator by , S , I Fig. 85. Distributed Capacity. otherwise retaining the same denotations as in A.), We have, r = i-* -jx) ff + jb -- (r _ 7. - As will be seen, the first terms in the expression of E and of I are the same in A.) and in B.). 1O6] DISTRIBUTED CAPACITY. 155 106. 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 b^ 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, 156 ALTERNATING-CURRENT PHENOMENA. [106 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 by no means 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 far greater amounts of energy than the resistance does. The dielectric hysteresis 1O7] DISTRIBUTED CAPACITY. 157 appears in the circuit .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. 107. 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 s elf -induct am e ; 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 pJiase 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, beingx wattless, and due to : Capacity and Electrostatic influence. Hence we get four constants : Effective resistance, r, Effective reactance, x, Effective conductance, g, Effective susceptance, b = b r> 158 ALTERNATING-CURRENT PHENOMENA. [ 1O8, 1O9 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. 108. 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, ^i = r^ jx\ the impedance of receiver circuit, or admittance, ^1=^1 +>i, and E.M.F., E , at generator terminals are given. Current and E.M.F. at any point of circuit to be determined, etc. 109. Counting now the distance, x, from a point, 0, of the line which has the E.M.F., fit 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, and substituting (3) in (1) and (2), we get : (4) (5) the differential equations of E and I. 110. 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 = # e rx , where e is the basis of natural logarithms ; for, differen- tiating this, we get, 160 A L TERN A TING-CURRENT PHENOMENA . [ 1 1 hence, v* = (g - j b c ) (r - J x) ; or, v = i V (g - jb c ) (r - j x) ; hence, the general integral is : -++ J.--T (8) where a and b are the two constants of integration ; sub- stituting * = a-y/3 (9) into (7), we have, or, c?-p=gr-xb e ; 2aft=gx + b c r-, therefore, (10) = Vl/2 / - ====== - f v 11 / = Vl/2 substituting (9) into (8) : w = af.(-~J^* -f ^ +y sin/3x) ; w = (^e ax + ^~ ax ) cos/?x > (rtc ax ^c- ax ) sin /?x (12) which is the general solution of differential equations (4) and (5) Differentiating (8) gives : hence, substituting, (9) : = (a //?) {(#c ax e~ ax ) (cos /?x / Substituting now / for w, and substituting (13) in (1), and writing, 111, 112] DISTRIBUTED CAPACITY we get : 161 (l- * {(At"+l > * *~/0 sin /?x}; 1 sin R*\- (14) where ^4 and B are the constants of integration. These are the general integral equations of the problem. 111. If 7 t = /! -f y // is the current . = by substituting (15) in (14), we get : 2A= {(a ,\ + ft //) + (^ + b c ^' H- / {( ''/ - ^ *i) + (^ V - 2 ^ = {(a 4 + /8 //) (16) a and ft being determined by equations (11). 112. If Z = J? j X is the impedance of the receiver circuit, E e -f j ' ej is the E.M.F. at dynamo terminals (17), and / = length of line, we get at x = 0, K _ hence or At x = /, A- B - ^T g - Jb c E A B a j $ A + g-jb c ' E.= 1 g {(At* 1 - &t-**ivQ*ptJ(A** l +-JB'j sin/3/}. (18) (19) 162 ALTERNATING-CURRENT PHENOMENA. [113 Equations (18) and (19) determine the constants^ and B, which, substituted .in (14), give the final integral equations. The length, X = %-*/ ft is a complete wave length (20), which means, that in the distance 27T//3 the phases of cur- rent 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. 113. The difference of phase, , between current, /, and E.M.F., E, at any point, x, of the line, is determined by the equation, D (cos o> -f- / sin u>) = , where D is a constant. Hence, varies from point to point, oscillating around a medium position, w x , which it approaches at infinity. This difference of phase, ><*, towards which current and E.M.F. tend at infinity, is determined by the expression, D (cos w* + /sin *) = / // or, substituting for E and /their values, and since e~ a * = 0, and A c ax (cos ft x / sin ft x), a cancels, and >(cosZ>* + / sin woe) = a ~ //? g - j c = (ag + fib e ) -/(a b e - ftg) . hence, tanS, = - af ,*\ (21) ^H- P&c This angle, o>x, = ; that is, current and E.M.F. come more and more in phase with each other, when ^c ~ $g= 0; that is, a -r- ft = g -^ b c , or, 114] DISTRIBUTED CAPACITY. 163 substituting (10), gives, hence, expanding, r -4- x g -f- b c ; (22) * that is, the ratio of resistance to inductance equals the ratio of leakage to capacity. This angle, woe, = 45 ; that is, current and E.M.F. differ by Jth period, if a b c + $g = <*. + ft b c ; or, a = ^ + g . (3 b c +g' which gives : rg+ xb c = 0. (23) That is, two of the four line constants must be zero ; either g and x, or g and b c . 114. 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- ence of phase between both as function of the distance from receiver circuit ; under the conditions, E.M.F. at receiving end, 10,000 volts ; hence, E^ = e l = 10,000; current at receiving end, 65 amperes, with an energy co- efficient of .385, that is, 1= h+jij = 25 + 60 y; line constants per unit length, g = 2 x 10- 5 , b c = 20 x 10- 5 ; length of line corresponding to x = L = - = 221.5 = one complete period of the wave * [ of propagation. A = 1.012 1.206 y, B = .812 + .794 / 164 AL TERNA TING-CURRENT PHENOMENA. [115 These values, substituted, give, 7 = { x (47.3 cos /3x -f 27.4 sin fix) e- ax (22.3 cos fix + 32.6 sin fix)} + j {e ax (27.4 cos fix 47.3 sin /?x) -f e- ax (32.6 cos /3x 22.3 sin /3x)}; E = {c ax (6450 cos /3x + 4410 sin /?x) + c- a) (3530 cos /3x 4410 sin /3x)} _j_y| c ax (4410 cos fix 6450 sin J3x) c - a; (4410 cos fix + 3530 sin 0x)}; tan oi, = = - .073, = - 4.2. ^ + 30 * E s N e C f \ \ 1 OLT? 0,000 *20 i \ ..J. + 10' / i ^ ' "\ L (i \h /* \ X 'e ^*"" ^~- 3 7000 10 \ s -.- / 2,000 -20 i \ /' / _! 23,000 30 / ^** ' ^ ^ " 40 ^ i f M W 260 2 f.'ooo 4000 2.000 1 -*? / 240 3 I ^ /' *?0 2 / / 200 20,000 x *>. / 9 * 190 18.000 / *^ ^^ _^ . / 80 l.,000 / 1 s ,,0 4 000 / 1 ~ V / 7 120 ,.000 / \ ^ / too 0,000 X y x ^ ^ r = l s =4 80 8.000 / [= 5 o.oo It 60 6,000 \ / t! .000 40 4,000 \ 20 2,000 o ^ i 3L T 5L 4~ 3L T Fig. 86. -115] DISTRIBUTED CAPACITY. 165 115. The following are some particular cases : A.) Open circuit at end of lines : x = :" /! = 0. hence, E= x _ -ax) cos x _y *r-*fft '.) Line grounded at end: x = : E, = 0. 'X _ -x) cos x _y( ax + -ax) s ' m ft x IX + e~ ax ) cos/3x y(c ax c- ax )sin/?x}. C.) Infinitely long conductor : Replacing x by x, that is, counting the distance posi- tive in the direction of decreasing energy, we have, x = oo : 1= 0, E = 0; hence B = 0, and E = 1_ A _ ax (CQS ^ x gin .r y^ / = i ^e- ax (cos/3x +y s in/3x), involving decay of the electric wave. The total impedance of the infinitely long conductor is 8- __ (a -j 166 ALTERNATING-CURRENT PHENOMENA. [115 The infinitely long conductor acts like an impedance that is, like a resistance J\. = , combined with a reactance g* + V 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 oi = l, or, o> = 45 ; that is, current and E.M.F. differ. by Jth period. D.) Generator feeding into closed circuit : Let x = be the center of cable ; then, E^ = E_* ; hence : E = at x = ; /x =/-*; which equations are the same as in B y where the line is grounded at x = 0. 116,117] ALTERNATING-CURRENT TRANSFORMER. 16T CHAPTER XIII. THE ALTERNATING-CURRENT TRANSFORMER. 116. 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. 117. 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 168 ALTERNATING-CURRENT PHENOMENA. [118 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 ; and, therefore, in the constant potential transformer the primary and sec- ondary 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. 118. 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 = 27r7Y^10- 8 volts, = 4.447W/$10- 8 volts; hence, if the E.M.F., frequency, and number of turns are determined, the maximum magnetic flux is To produce the magnetism, $, of the transformer, a M.M.F. of <5 ampere-turns is required, which is determined by the shape and the magnetic characteristic of the iron, in the manner discussed in Chapter X. 119] ALTERNATING-CURRENT TRANSFORMER. 169 For instance, in the closed magnetic circuit transformer, the maximum magnetic induction is 1 "4 ' ? j, where 5 = the cross-section o'F magnetic circuit. 119. To induce a magnetic density, , by an angle, a, the angle of hysteretic advance of phase, and consists of two components, the hysteretic energy current, in quadrature with the magnetic flux, and therefore in phase with the induced E.M.F. = I 00 sin a; and the magnetizing current, in phase with the magnetic flux, and therefore in quadrature with the induced E.M.F., and so wattless, = I 00 cos a. The exciting current, 7 00 , is determined from the shape and magnetic characteristic of the iron, and number of turns ; the hysteretic energy current is T . Power consumed in the iron J. nn sin a = . Induced E.M.F. 120. Graphically, the polar diagram of M.M.Fs. ot a transformer is constructed thus : Let, in Fig. 87, O<& = the magnetic flux in intensity and phase (for convenience, as intensities, the effective values are used throughout), assuming its phase as the vertical ; 12O] ALTERNATING-CURRENT TRANSFORMER. 171 that is, counting the time from the moment where the rising magnetism passes its zero value. Then the resultant M.M.F. is represented by the vector O$, leading O<$> by the anglfe $O3> = 03. The induced E.M..Fs. have^the phase 180, that is, are plotted towards the left, and represented by the vectors and If, now, wj' = angle of lag in the secondary circuit, due to the total (internal and external) secondary reactance, the secondary current 7 t , and hence the secondary M.M.F., 9 r 1 = n^ / x , will lag behind E by an angle ft', and have the phase, 180 -f /?', represented by the vector O^ l . Con- structing a parallelogram of M.M.Fs., with O$ as a diag- onal and O&i as one side, the other side or O$ is the primary M.M.F., in intensity and phase, and hence, dividing by the number of primary turns, n , the primary current is Io = $ol n o To complete the diagram of E.M.Fs. , we have now, In the primary circuit : E.M.F. consumed by resistance is I r ot in phase with I , and represented by the vector OE or ; E.M.F. consumed by reactance is I x , 90 ahead of I , and represented by the vector OE x ; E.M.F. consumed by induced E.M.F. is , equal and oppo- site thereto, and represented by the vector OE f Hence, the total primary impressed E.M.F. by combina- tion of OE or , OE OK , and OEJ 1 by means of the parallelo- gram of E.M.Fs. is, = OE , and the difference of phase between the primary impressed E.M.F. and the primary current is ft = E O$ . " In the secondary circuit : Counter E.M.F. of resistance is /i^ in opposition \vith/j, and represented by the vector OE r k 172 ALTERNATING-CURRENT PHENOMENA. 121 Counter E.M.F. of reactance is /^i, 90 behind f lt and represented by the vector OE\ K ' ; Induced E.M.Fs., E{ represented by the vector OE{. Hence, the secondary terminal voltage, by combination of OE^ y OE^ and OE by means of the parallelogram of E.M.Fs. is ! = OE ly 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." Fig. 88. Transformer Diagram with 80 Lag in Secondary Circuit. 121. In the construction of the transformer diagram, it. is usually preferable not to plot the secondary quantities, current and E.M.F., direct, but to reduce them to corre- spondence with the primary circuit by multiplying by the ratio of turns, a = n / n v for the reason that frequently primary and secondary E.M.Fs., etc., are of such different 121] ALTERNATING-CURRENT TRANSFORMER. 173 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.lVl.Fs. coincide, or Fig. 89. Transformer Diagram with 50" Lag in Secondary Circuit. Figs. 88 to 94 give the polar diagram of a transformer having the constants r = .2 ohms, b = .0173 mhos, x = .33 ohms, Ejf = 100 volts, r = .00167 ohms, /i = 60 amperes, x 1 = .0025 ohms, a = 10. g = .0100 mhos, for the conditions of secondary circuit, S = 80 lag in Fig. 88. 50 lag " 89. 20 lag 90. O, -or in phase, " 91. ft' = 20 P lead in Fig. 92, 50 lead " 93. 80 lead " 94. As shown with a change of /?/, E ,g l ,g , etc., change in intensity and direction. The locus described by them are circles, and are shown in Fig. 95, with the point corre- sponding to non-inductive load marked. The part of the locus corresponding to a lagging secondary current is 174 ALTERNATING-CURRENT PHENOMENA. [121 Fig. 90. Transformer Diagram with 20 Lag in Secondary Circuit. Eo Fig. 91. Transformer Diagram with Secondary Current in Phase with E.M.F. Fig. 92. Transformer Diagram with 20 Lead in Secondary Current. 121] ALTERNATING-CURRENT TRANSFORMER. 175 E, E,' Fig. 93. Transformer Diagram with 50 Lead in Secondary Circuit Fig. 94. Transformer Diagram with 80 Lead in Secondary Circuit. Fig. 95. 176 ALTERNATING-CURRENT PHENOMENA. [ 122 shown in thick full lines, and the part corresponding to leading current in thin full lines. 122. This diagram represents the condition of con- stant secondary induced E.M.F., E^, that is, corresponding to a constant maximum magnetic flux. By changing all the quantities proportionally from the diagram of Fig. 95, the diagram for the constant primary impressed E.M.F. (Fig. 96), and for constant secondary ter- minal voltage (Fig. 97), are derived. In these cases, the locus gives curves of higher order. Fig. 96. Fig. 98 gives the locus of the various quantities when the load is changed from fulLload, 7 X = 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, E 19 and thus of E , etc., are straight lines; and in b.) and c.}, parts of one and the same circle a.) is shown 123] ALTERNATING-CURRENT TRANSFORMER. 177 in full lines, b.) in heavy full lines, and c.) in light fiill 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. 97. 123. It must be understood, however, that for the pur- pose of making the diagrams plainer, by bringing the dif- ferent values to somewhat nearer the same magnitude, the constants chosen for these diagrams represent, not the mag- nitudes found in actual transformers, but refer to greatly exaggerated internal losses. In practice, about the following magnitudes would be found : r = .01 x = .033 ohms; ohms ; = .00008 ohms ; Xi = .00025 ohms ; g = .001 ohms ; b = .00173 ohms ; that is, about one-tenth as large as assumed. Thus the E-. , etc., under the different changes of the values of E , conditions will be very much smaller. 178 ALTERNATING-CURRENT PHENOMENA. [124 Symbolic Method. 124. In symbolic representation by complex quantities the. transformer problem appears as follows : The exciting current, 7 00 , of the transformer depends upon the primary E.M.F., which dependance can be rep- resented by an admittance, the " primary admittance," Y =. g -\-jb oJ of the transformer. Fig. 98. The resistance and reactance of the primary and the secondary circuit are represented in the impedance by Z = r jx , and Z l = r j x^. Within the limited range of variation of the magnetic density in a constant potential transformer, admittance and impedance can usually, and with sufficient exactness, be considered as constant. Let n = number of primary turns in series ; i = number of secondary turns in series; a = = ratio of turns ; Y = go jb = primary admittance Exciting current Primary counter E.M.F. ' 124] ALTERNATING-CURRENT TRANSFORMER. Z = r j X Q = primary impedance E.M.F. consumed in primary coil by resistance and reactance. Primary current Z 1 = ri j x = secondary impedance E.M.F. consumed in sedbndary coil by resistance and reactance . Secondary current where the reactances, x and x^ , refer to the true self -induc- tance only, or to the cross-flux passing between primary and secondary coils ; that is, interlinked with one coil only. Let also Y g^jb total admittance of secondary circuit, including the internal impedance ; E = primary impressed E.M.F. ; E ' = E.M.F. consumed by primary counter E.M.F. ; EI. = secondary terminal voltage; EI = secondary induced E.M.F. ; I = primary current, total ; I 00 = primary exciting current ; ^ secondary current. Since the primary counter E.M.F., EJ, and the second- ary induced E.M.F., E^, are proportional by the ratio of turns, a, .' = - *{. (1) The secondary current is : 7, = KE/, (2) consisting of an energy component, gE^, and a reactive component, g E-[. To this secondary current corresponds the component of primary current, / f The primary exciting current is f 00 =y Ej. (4) Hence, the total primary current is : 180 ALTERNATING-CURRENT PHENOMENA. [ 125 or, The E.M.F. consumed in the secondary coil by the internal impedance is Z^I^. The E.M.F. induced in the secondary coil by the mag- netic flux is E-[. Therefore, the secondary terminal voltage is or, substituting (2), we have The E.M.F. consumed in the primary coil by the inter- nal impedance is Z 1 . The E.M.F. consumed in the primary coil by the counter E.M.F. is El. Therefore, the primary impressed E.M.F. is or, substituting (6), "* Z Y * (8) + Z.y. + ^? 125. We thus have, primary E.M.F., E = - aE{ j 1 + Z Y + ^1 j , (8) secondary E.M.F., E l = E{ { 1 - Z, Y}, (7) primary current, f = - ^ { Y -f a 2 Y } , (6) a secondary current, 7 T = YE{, (2) as functions of the secondary induced E.M.F., E^ as pa- rameter. 125] ALTERNATING-CURRENT TRANSFORMER. 181 From the above we derive Ratio of transformation of E.M.Fs. : (9) Ratio of transformations of currents : From this we get, at constant primary impressed E.M.F., E = constant ; secondary induced E.M.F., a a- E.M.F. induced per turn, a secondary terminal voltage, primary current, E secondary current, 77 \r T -fiin -I At constant secondary terminal voltage, EI = const. ; 182 ALTERNATING-CURRENT PHENOMENA. [126 secondary induced E.M.F., E.M.F. induced per turn, * l-Z,y' primary impressed E.M.F., E = aE^ 1 - Z^ primary current, E l ;1 secondary current, (12) 126. Some interesting conclusions can be drawn from these equations. The apparent impedance of the total transformer is J (13) Substituting now, = V, the total secondary admit- tance, reduced to the primary circuit by the ratio of turns, ' Y -\- Y 1 is the total admittance of a divided circuit with the exciting current, of admittance Y , and the secondary 127] ALTERNATING-CURRENT TRANSFORMER. 183 current, of admittance Y' (reduced to primary), as branches. Thus : is the impedance of this divided circuit, and Z = Z ' + Z . That, is : . (17) The alternate-current transformer, of primary admittance Y , total secondary admittance V, and primary impedance Z , is equivalent to,. and can be replaced by, a divided circuit with the branches of admittance V , the exciting current, and admittance Y' = Y/a 2 , the secondary current, fed over mains of the impedance Z , tJie internal primary impedance. This is shown diagrammatically in Fig. 99. Generator Transformer I E. Receiving: Circuit I Fig. 99. 127. Separating now the internal secondary impedance from the external secondary impedance, or the impedance of the consumer circuit, it is 1 where Z = external secondary impedance, (18) (19) 184 ALTERNATING-CURRENT PHENOMENA. [127 Reduced to primary circuit, it is That is : Y = Z/ + Z'. (20) An alternate-current transformer, of primary admittance Y , primary impedance Z , secondary impedance Z^, and ratio of turns a, can, when the secondary circtiit is closed by an impedance Z (the impedance of the receiver circuit], be replaced, and is equivalent to a circuit of impedance Z ' = T, fed over mains of the impedance Z -\- Z^, where Z^ = , shunted by a circuit of admittance Y , which latter circuit branches off at the points a b, between the impe- dances Z n and Zl . Generator Transformer Receiving Circuit 7. Zo cJ Zfa 2 z, Q) yJ fz-Vz \ 1 * b Fig. 100. This is represented diagrammatically in Fig. 100. 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 b, in diagram Fig. 100, as shown in diagram Fig. 101. It therefore follows : An alternate-ctirrent transformer, of x secondary coils, of the internal impedances Z^, Z^ 1 , . . . Z-f, closed by external secondary circuits of the impedances Z 1 , Z 11 , . . . Z x , is equiv- alent to a divided circuit of x -f- 1 branches, one branch of 127] AL TERN A TING-CURRENT TRANSFORMER. 185 Generator Transformer Receiving Circuits Z 1 Fig. 101. admittance Y , the exciting current, the other branches of the impedances Z/ + Z 1 , Z^ n + Z n , . . . Zf + Z* the latter impedances being reduced to the primary circuit by tJie ratio of tiirns, and the whole divided circuit being fed by the primary impressed E.M.F. E , over mains of the impedance z. Consequently, transformation of a circuit merely changes all the quantities proportionally, introduces in the mains the impedance Z -f Z^, and a branch circuit between Z and Z, of admittance Y . Thus, double transformation will be represented by dia- gram, Fig. 102. Transformer Transformer Receiving Circuits z UJ) Yoc Fig. 102. 183 ALTERNATING-CURRENT PHENOMENA. [128 With this the discussion, of the alternate-current trans- former ends, by becoming identical with that of a divided circuit containing resistances and reactances. Such circuits have explicitly been discussed in Chapter VIII., and the results derived there are now directly appli- cable to the transformer, giving the variation and the con- trol of secondary terminal voltage, resonance phenomena, etc. Thus, for instance, if Z^ = Z , and the transformer con- tains a secondary coil, constantly closed by a condenser reactance of; such size that this auxiliary circuit, together with the exciting circuit, gives the reactance x , with a non-inductive secondary circuit Z l = 1\, we get the condi- tion of transformation from constant primary potential to constant secondary current, and inversely, as previously discussed. Non-inductive Secondary Circuit. 128. In a non-inductive secondary circuit, the external secondary impedance is, Z = Ri, or, reduced to primary circuit, Z - ^ - p ^~^~ Assuming the secondary impedance, reduced to primary circuit, as equal to the primary impedance, J XQ itis ' Substituting these values in Equations (9), (10), and (13), we have Ratio of E.M.Fs. : r jx 128] ALTERNATING-CURRENT TRANSFORMER. 187 . K -f- r j x , r jx f r jx r or, expanding, and neglecting terms of higher than third order, R or, expanded, Neglecting terms of tertiary order also, = ^ | 1 + 2 i ( Ratio of currents : or, expanded, Neglecting terms of tertiary order also, Total apparent primary admittance : 1 + r - jX \ + (r -jx } ( go r Jx {R + (r -Jx ) + R (r -jx^ (g +/^)} {1- (g (r - jx ) - ^ (g +jb )-1R ( r 188 ALTERNATING-CURRENT PHENOMENA. [129 or, Neglecting terms of tertiary order also : < = R 1 + 2 /o _ R (go + , Angle of lag in primary circuit : tan o> = J ? , hence, 2 r b - tan = -- 1 + ~ - - & ~ 2 r ogo -2x b + &g* + JR* Neglecting terms of tertiary order also : -24. jf^j tan B h Xg. - ^, **.-* a Substituting these values we get, as the equations of the transformer on non-inductive load, Ratio of E.M.Fs. : \ 1 + rf(p -/q) ( or, eliminating imaginary quantities, Ratio of currents : LL= 1 fi ... ( h +/g) , (P -y /i ( X = 120; P, =-02- Substituting these values, gives : 100 h .= .02 ; g = .04. V(1.0014 + .02 d^f + (.0002 + .06 d)* 1.0014 + --. 0002; = . 1 ^ .06 ^ + --. tan Co,, = Fig. 103. Load Diagram of Transformer. 192 ALTERNATING-CURRENT PHENOMENA. [13(> In diagram Fig. 103 are shown, for the values from d = to d= 1.5, with the secondary current c 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 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 n Q = number of primary turns in series per circuit ; MI = number of secondary turns in series per circuit ; a = = ratio of turns ; i Y Q = g Q -\-jb Q primary admittance per circuit ; where g Q = effective conductance ; = susceptance ; ZQ ' = r o J x o = internal primary impedance per circuit, where r = effective resistance of primary circuit ; x = reactance of primary circuit ; Z n = /i j'xi = internal secondary impedance per circuit at standstill, or for s = 1, where ri = effective resistance of secondary coil ; Xi = reactance of secondary coil at standstill, or full fre- quency, s = 1. Since the reactance is proportional to the frequency, at the slip s, or the secondary frequency s N, the secondary impedance is ; Zi = r \ J sx \- Let the secondary circuit be closed by an external re- sistance r, and an external reactance, and denote the latter $135] ALTERNATING-CURRENT TRANSFORMER. 197 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 ^ EQ primary impressed E.M.F. per circuit, EQ = E.M.F. consumed by primary counter E.M.F., E^ = secondary terminal E.M.F., E{ = secondary induced E.M.F., e = E.M.F. induced per turn by the mutual magnetic flux, at full frequency W, I Q = primary current, f QO = primary exciting current, /! = secondary current. It is then : Secondary induced E.M.F. EI = sn^e. Total secondary impedance Z L + Z = (r, + r) -js (*, + x) ; hence, secondary current _ EI _ kn^e ~ Z x + Z ~ (r, + r) -js ( Xl + x) ' Secondary terminal voltage i -, fi /o^i I sn^e(r / 's x) sn^e \ 1 ^ ^ J- *- (r, + r) -js (x, + *>) ) (rj + r) -js ( Xl + x) * 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" -f x'" ', where x' is that part of the reactance which is proportional to the frequency, x 1 ' that part of the reac- tance independent of the frequency, and x'" that part of the reactance which is inversely proportional to the frequency ; and have thus, at slip s, or frequency sN t the external secondary reactance sx' + x" + . 198 ALTERNATING-CURRENT PHENOMENA. [135 E.M.F. consumed by primary counter E.M.F. EQ = * J hence, primary exciting current : 7 00 = jE 'Y = n Q e(g Q + /<>) Component of primary current corresponding to second- ary current I I : tf = E, I | = . 7 cos (^, 7) = ac + y8^. 202 AL TERN A TING-CURRENT PHENOMENA. [. 1 37 Making use of this, and denoting, *t* gives : Secondary output of the transformer Internal loss in secondary circuit, ** Total secondary power, Internal loss in primary circuit, \ . Total electrical output, plus loss, = P l + tf + PJ = ^Y (r + 2 ri ) = f w (r + 2 ^). V V 7 . Total electrical input of primary, YV + >i + ^0 = / (r + n + J^). Hence, mechanical output of transformer, P= P - P l = w (1 - s) (r+rj. Ratio, mechanical output P _ 1 S _ speed total secondary power p ,\ v I ~ ~ ~ slip *. _ . - 137. 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 _ snfe^r+^ + sr^ 138] AL TERN A TING-CURRENT TRA NSFORMER. 203 Mechanical output, p = s (* *] Electrical output of secondary V o O O s 2 n? e 2 - r p *i - Losses in transformer, 2s a n l *e 2 .r l i _i pi pi _ t -h /i = ^ = Of these quantities, P 1 and P 1 are always positive ; P Q and .P can be positive or negative, according to the value of s. Thus the apparatus can either produce mechanical power, acting as a motor, or consume mechanical power; and it can either consume electrical power or produce electrical power, as a generator. 138. At j = 0, synchronism, P Q = 0, P = 0, P l = 0. At < s < 1, between synchronism and standstill. P l , P and P are positive ; that is, the apparatus con- sumes electrical power P Q in the primary, and produces mechanical power P and electrical power P l + P^ in the secondary, which is partly, P^ , consumed by the internal secondary resistance, partly, P 1 , available at the secondary terminals. In this case it is : PI + PS _ s ~P~ = l-s' that is, of the electrical power consumed in the primary circuit, P Q , a part P Q l is consumed by the internal pri- mary resistance, the remainder transmitted to the secon- dary, and divides between electrical power, P 1 -f P^ t and mechanical power, P, in the proportion of the slip, or -drop below synchronism, s, to the speed : 1 s. 204 AL TERN A TING-CURRENT PHENOMENA , [138 In this range, the apparatus is a motor. At s > 1 ; or, backwards driving, P < 0, or negative ; that is, the apparatus requires mechanical power for driving. It is then : P Q - P l - J\ l < P 1 ; 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 P < 0, beyond synchronism, P < ; that is, the apparatus has to be driven by mechanical power. P Q < ; that is, the primary circuit produces electrical power from the mechanical input. At r'+ri+ Jfi = 0, or, s < - ^t^ 1 ; r the electrical power produced in the primary becomes less than required to cover the losses of power, and P Q becomes positive again. We have thus : consumes mechanical and primary electric power ; produces secondary electric power. _ L+l! < , < o r consumes mechanical, and produces electrical power in primary and in secondary circuit. 139] ALTERNATING-CURRENT TRANSFORMER. 205 consumes primary electric power, and produces mechanical and secondary electrical power. Vf< consumes mechanical^ and primary electrical power ; pro- duces secondary electrical power. GENERAL ALTERNATE CURRENT TRANSFORMER ^> ^ ^^ i 5g ^> v . * x 1VVTP UT(J 2 ^ i J 4 L lO<-r ^ ^ r g iiio ^~, ii L^o, *^ -il^ UJ . . ^ X^| ^ , ^^ v ^ , ^ 2 / ^ f NN ^S A Li i^- ^ J / / \ W = total maximum flux of the magnetic field per motor pole. It is then E = V27r;*7v~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 <2> (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., E= ~e-, where e = V2 irnN$ 10~ 8 may be considered as the " Active E.M.F. of the motor." Since the secondary frequency is s N, the secondary induced E.M.F. (reduced to primary system) is E l = se. 210 AL TERN A TING-CURRENT PHENOMENA. [ 142 Let I = exciting current, or current passing through the motor, per primary circuit, when doing no work (at synchronism), and primary admittance per circuit = . It is thus ge = magnetic energy current, ge* = loss of power by hysteresis (and eddy currents) per primary coil. Hence pge i = total loss of energy by hysteresis and eddys, as calculated according to Chapter X. be = magnetizing current, and nb e = effective M.M.F. per primary circuit ; hence , P. nbe = total effective M.M.F. ; and - nbe = total maximum M.M.F., as resultant of the M.M.Fs. of v^ the /-phases, combined by the parallelogram of M.M.Fs.* If (31 = reluctance of magnetic circuit per pole, as dis- cussed in Chapter X., it is V2 Thus, from the hysteretic loss, and the reluctance, the constants, g and ,'and thus the admittance, Y. are derived. Let r = resistance per primary circuit ; z = reactance per primary circuit ; thus, Z r jx = impedance per primary circuit ; * Complete discussion hereof, see Chapter XXIII. 143] INDUCTION MOTOR. 211 /'! = resistance per secondary circuit reduced to primary sys- tem ; x l = reactance per secondary circuit reduced to primary system, at full frequency, N; hence, - j^ sxi = reactance per secondary circuit at slip s\ and Z l = ri jsx l = secondary internal impedance. 143. It is now, Primary induced E.M.F., E = -e. Secondary induced E.M.F., EI = se. Hence, Secondary current, ^i n jsxi Component of primary current, corresponding thereto, // = _ /. = ". ; Primary exciting current, 7 =eY= hence, Total primary current, E.M.F. consumed by primary impedance, ,= ZI e( r -joe) \ S- + (g+jb) \ ; \r^-jsx \ 212 ALTERNATING-CURRENT PHENOMENA. [143 E.M.F. required to overcome the primary induced E.M.F., - = e; hence, Primary terminal voltage, We get thus, in an induction motor, at slip s and active E.M.F. e, it is Primary terminal voltage, . = e \l + s (r -/*> + (r-j*) r \ ~ J SX \ Primary current, or, in complex expression, Primary terminal voltage, Primary current, To eliminate e, we divide, and get, Primary current, at slip j, and impressed E.M.F., E : T = s + Z 1 Y E . ZT^ + SZ + ZZ^Y ' or, jx) + (r- jx) fa - jsx^ (g + jb) Neglecting, in the denominator, the small quantity ^Y, it is 144] INDUCTION MOTOR, 213 l + sZ (rjx) r> or, expanded, ) + r x V + jr t (r^- xb) + ^ 2 ^ rbj] + /== Hence, displacement of phase between current and E.M.F., tanoi = (rr + JV) Neglecting the exciting current, I ot altogether, that is, setting Y = 0, it is, I=sE ( r i + J 'o tan s (x + sr 144. In graphic representation, the induction motor diagram appears as follows : \ 214 ALTERNATING-CURRENT PHENOMENA. Denoting the magnetism by the vertical vector OQ in Fig. 105, 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 O l at the amplitude of 180. Dividing OE^ by a in the proportion of r^-^sx^, and connecting a with the middle b of the upper arc of the circle OE^ , this line inter- sects the lower arc of the circle at the point 1^ r^ . Thus, Ol^r^ is the E.M.F. consumed by the secondary resistance, and OI^x^ equal and parallel to E^I^r^ is the E.M.F. con- sumed by the secondary reactance. The angle, E l OI^ i\ = Wj is the angle of secondary lag. The secondary M.M.F. OG l is in the direction of the vector Ol^r^. Completing the parallelogram of M.M.Fs. with OF as diagonal and OG 1 as one side, gives the primary M.M.F. OG as other side. The primary current and the E.M.F. consumed by the primary resistance, represented by Olr, is in line with OG, the E.M.F. consumed by the pri- mary reactance 90 ahead of OG, and represented by OIx, and their resultant OIz is the E.M.F. consumed by the primary impedance. The E.M.F. induced in the primary circuit is OE ly and the E.M.F. required to overcome this counter E.M.F. is OE equal and opposite to OE 1 . Com- bining OE with OIz gives the primary terminal voltage represented by vector OE , and the angle of primary lag, 145. 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 : It is l = s l f O A -r- /! r = E -r- /! s x 145] INDUCTION MOTOR. 215 constant. X 1 That is, /]_ r lies on a half-circle with 1 E^ as diameter. Fig. 106. That means G l lies on a half-circle g^ in Fig. 106 with OA as diameter. In consequence hereof, G lies on half- circle g with FB equal and parallel to OA as diameter. Thus Ir lies on a half-circle with Z>//" as diameter, which circle is perspective to the circle FB, and Ix lies on a half- circle with IK as diameter, and Is on a half-circle with LN as diameter, which circle is derived by the combination of the circles Ir and Ix. 216 AL TERNA TING-CURRENT PHENOMENA. [ 146 The primary terminal voltage E lies thus on a half- circle eo equal to the half-circle 1 'z, 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 voltage E is kept constant, the circle e o of the primary impressed voltage changes to an arc with O as center, and all the cor- responding 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 conveniently are constructed point for point by reduction from the circle of the locus in Fig. 106. Torque and Power. 146. The torque developed per pole by an eiectnc motor equals the product of effective magnetism, j , the space displacement between armature current and field magnetism is f, <) = 90+ w, hence, sin ( / : ) = cos oij It is, however, cos ! = r * , thus, 9 = substituting these values in the equation of the torque, it is dp s ^ e* 10 8 or, in practical (C.G.S.) units, _ dp s r The Torque of the Induction Motor. At the slip s, the frequency N, and the number of poles d, the linear speed at unit radius is hence the output of the motor, P= TS or, substituted, 218 A L TERN A TING-CURRENT PHENOMENA. [ 147 The Power of the Induction Motor. 147. We can arrive at the same results in a different way : By the counter E.M.F. e of the primary circuit with current / = 7 + 7 X the power is consumed, el = eI -+- el. The power eI is that consumed by the primary hysteresis and eddys. The power e 7 X 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 ='/i Of this power a part, -fi^/j, is consumed in the secondary circuit by resistance. The remainder, P' = I l (e- E& 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 (included mechanical and secondary magnetic friction). Thus the mechanical output per motor circuit is P' = /!( 1, that is, the maximum torque falls below standstill, and the torque constantly increases from synchronism down to standstill. It is evident that the position of the maximum torque point, s t) can be varied by varying the resistance of the secondary circuit, or the motor armature. Since the slip at the maximum torque point, s t , is directly proportional to the armature resistance, r\ , it follows that very constant speed and high efficiency will bring the maximum torque point near synchronism, and give small starting torque, while good starting torque means a maximum torque point at low speed ; that is, a motor with poor speed regulation and low efficiency. Thus, to combine high efficiency and close speed regu- lation with large starting torque, the armature resistance has 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. 149. If s t = I, it is r = r 2 + (x l + x)*. 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. 222 ALTERNATING-CURRENT PHENOMENA. [15O If, s t > 1, it is T-i > Vr 2 + ( Xl + *) 2 . In this case, the maximum torque point is reached only by driving the motor backwards, as countertorque. r As seen above, the maximum torque, r t , is entirely inde- pendent of the armature resistance, and the same 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, analogous as resistance in the armature circuit of a continu- ous-current shunt motor. Further discussion on the effect of armature resistance is found under "Starting Torque." Maximum Power. 150. The power of an induction motor is a maximum for that slip, s , where , _ = 0; ds or, since 7>= j(s rf/| expanded, this gives J * ~ n + Vfa + r) 2 + (^ + ^] 15O] INDUCTION MOTOR. 223 substituted in P, we get the maximum power, PE? This result has a simple physical meaning : (r -+- r) = R is the total resistance of the motor, primary plus secondary (the latter reduced to the primary). (x^ -f x) is the total reactance, and thus V(r L + rf + (x l + ^r) 2 = Z is the total impedance of the motor. Hence it 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 tan be transmitted over an inductive line into a non-inductive receiver circuit. The torque corresponding to the maximum output P p is, This is not the maximum torque, but the maximum torque, r t , takes place at a lower speed, that is, greater slip, that is, s t > 224 AL TERN A TING-CURRENT PHENOMENA. [ 151 It is obvious from these equations, that, to reach as large an output as possible, R and Z should be as small as possi- ble ; that is, the resistances ^ + r, and the impedances, Z, and thus the reactances, x -f x, should be small. Since r\ -+- r is usually small compared with x l + x, it follows, that the problem of induction motor design consists in con- structing the motor so as to give the minimum possible reactances, x l -f- x. Starting Torque. 151. In the moment of starting an induction motor, * is s = 1 ; hence, starting current : (ri -y*i) + (r -joe} + ( ri -/*i) (r or, expanded, with the neglection of the last term in the denominator, as insignificant : r and, displacement of phase, or angle of lag, tan<3 = (-^i + [r x + r] + ^ [^ + ^]) + (r*! - flFf,) Neglecting the exciting current, g = = b, these equa- tions assume the form : r_ or, eliminating imaginary quantities, and 152] INDUCTION MOTOR. 225 That means, that in starting the induction motor without additional resistance in the armature circuit, in which case ^ -f- x is large compared with r v -f r, and the total impe- dance, Z, small, the motor takes excessive and greatly lagging curients. The starting torque is = dp * r^ ~ ItrN Z 2 ' That is, the starting torque is proportional to the armature resistance, and inverse 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. 152. Since, necessarily, and since the starting current is, approximately, r _o ~~ z' it is, 4 IT N oo - 47T 226 ALTERNATING-CURRENT PHENOMENA. [ 152 Would be the theoretical torque developed at 100 per cent .efficiency, and power factor by E.M.F., E , and current, /> at synchronous speed. It is thus, TO < T OO , and the ratio between the starting torque, T O , and the theo- retical maximum torque, T OO , gives a means to judge the perfection of a motor regarding its starting torque. This ratio, T O j r oo , reaches .8 to .9 in the best motors. Substituting I=E/Z in the equation of starting torque, it assumes the form, . dp T - Since 4 TT N I d = 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 77 = ':','. dp Ef r, or,smce r a = ^-^ (r+r y +(Xl+x y dr expanded, this gives, the same value as derived in the paragraph on " maximum torque." Thus, adding to the internal armature resistance i\ in starting the additional resistance, makes the motor start with maximum torque, while with in- creasing speed the torque constantly decreases, and reaches 153] INDUCTION MOTOR. 227 zero at synchronism. Under these conditions, the induc- tion motor behaves similar 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, wmle with the series motor the speed indefinitely increases with decreasing load. 153. The additional armature resistance, /Y', required to give a certain starting torque, is found from the equation of starting torque : Denoting the internal armature resistance by r, total armature resistance is r^ = r-l + 1\' , and thus, _ d p E* r{ -\- r" _ ~~~ hence, This gives two values, one above, the other below, the maximum torque point. 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. 107, giving the desired torque T . The smaller value of r will give fairly good speed reg- ulation, and thus in small motors, where the comparatively large starting current is no objection, the permanent arma- ture resistance may be chosen to represent this value. The larger value of r" allows to start with minimum current, but requires cutting out of the resistance after the start, to secure speed regulation and efficiency. 228 ALTERNATING-CURRENT PHENOMENA. [154,155 Synchronism. 154. At synchronism, s = 0, it is : f. = or, rationalized : 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. 155. 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. It is, current : r *+r*(g+jl>) F . 1 = - JL , r\ or, eliminating imaginary quantities : j- i i S r\ angle of lag : x l + -}- rfg s + or, inversely : s - ~ 156,157] INDUCTION MOTOR. 229 that is : Near synchronism, 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. Induction Generator. 156. 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 s 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 r become negative, and the machine becomes an electric generator, converting me- chanical into electric energy. Substituting in this case : s l = s, where k is the acceleration, or the slip of the machine above synchronism, we derive the equations of the induction generator, which are the same as those of the induction motor, except that the sign before the " slip " is reversed. 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. 157. The induction generator differs from the standard alternating-current generator essentially, in so far as it has no definite frequency of its own, but can operate at any frequency above that corresponding to its speed. But it can generate electric energy only when in circuit with an alternating-current apparatus of definite frequency, as an 230 AL TERN A TING-CURRENT PHENOMENA. [158 alternator or synchronous motor. That is, the induction, generator requires- a "frequency setter" for its operation. When operating in parallel with standard alternators, the phase relation of the current issuing from the induction generator mainly depends besides upon the slip upon the self-induction of the induction generator, and can be varied thereby. v " Hence the induction generator can be used to control the phase relation in an alternating-current circuit. When connected in series in a circuit, the E.M.F. of the induction generator is approximately proportional to the current. Thus it can be used as booster, to add voltage to a line in proportion to the current passing therein. Example. 158. As an instance are shown, in Fig. 107, charac- teristic curves of a 20 horse-power three-phase induction Amperes 140 160 ISO 200 220 240 260 280 300 320 310 Fig. 107. Speed Characteristics of Induction Motor. 158] INDUCTION MOTOR. 231 20 H.P Thr ee-p ia ^e ffidu :tion Mot or. 110 Volt ,.8 00 ^evo utioi ( is. e 5OC ; /cles Cur Y rent - .H Diag'ram -.4.il 33 L z, -.oi -JDJ; J-.C * 9j (fifij Hn serp^ pwer F 30 X x- \ i 28 / .' ^" 'orqi - le -.^ \ .c 2 26 / / \ s \ CO q- O 2_L_ t 22 [ / / A C O o / / \ S Q_ S 20 2 / / \\ "g 100 3 ^ -- 1 =2 '- . - . -, De ec/ \\ 11 be 90 80 J / / ^ > **^ t -- A 1 " ^. \ y 2 12" 7 / 5*5* "-V. s \\ 70 / V x s * v \\ 60 1 i_ 1, 7 x x. >^ \\\ 50 H B I/ \40 6 i / | 3 4 I 1 2 2 U 10 1 50 K 150 Amperes 1 2C 250 3( lasso Fig. 108. Current Characteristics of Induction Motor. motor, of 900 revolutions synchronous speed, 8 poles, fre- quency 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 : n() - = 60.0 : or E = 63.5. The constants of the motor are : Primary admittance, Y = .1 -|" .4 /. Primary impedance, Z = .03 .09 /. Secondary impedance, Z^ = .02 .085 /. 232 AL TERN A TING-CURRENT PHENOMENA. [ 158 In Fig. 107 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 : r^ .02 : short circuit of armature, full speed. r^ = .045 : .025 ohms additional resistance. r = .16 : .16 ohms additional resistance, maximum starting torque. TI = .75 : .73 ohms additional resistance, same starting torque as = .045. Fig. 109. Speed Characteristics of Induction Motor. 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. On Fig. 108 is shown, with the current input per line as abscissae, the torque in kilogrammetres and the output 158] .INDUCTION MOTOR. 23S in kilowatts as ordinates in drawn lines, and the speed and the magnetism, in per cent of their synchronous values, as ordinates in dotted lines, for the armature resistance r = .02, or short circuit. * .<| In Fig. 109 is shown, wifcii 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 resis- tance ;-! = .045, for the whole range of speed from 120 per 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. 234 ALTERNATING-CURRENT PHENOMENA. [159 CHAPTER XVI. ALTERNATING-CURRENT GENERATOR. 159. 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 mas:- o 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. 110, 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, 160] ALTERNATING-CURRENT GENERATOR. 235 during the synchronous rotation of the armature, a pulsa- tion of the magnetic flux passing through it is produced. This pulsation of the magnetic flux induces E.M.F. in the field spools, and thereby makes the field current pulsating also. Thus, we have, in this ga.se, even on open circuit, no Fig. 770. 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. 160. The relative position of the armature M.M.F. with respect 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. 110, 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 236 A L TERN A TING-CURRENT PHENOMENA . [ 160 density at the trailing-pole corner. Since the internal self- inductance of the alternator alone 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. 111. Since the armature current flows Fig. 777. 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. 112. preceding-field pole, as shown in diagram Fig. 112, 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. 161] ALTERNATING-CURRENT GENERATOR. 237 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 direction of rotation is opposite ; and thus a lagging current tends to magnetize, a leading current to demagnetize, the field. 161. 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. This means that, if the armature current lags, the E.M.F. of self-inductance will be more than 90 behind the induced E.M.F., and therefore in partial opposition, and will 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- 238 ALTERNATING-CURRENT PHENOMENA. [162,163 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 synchronous 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 magnetizing action of the armature current, and the electric reaction due to the self-induction of the armature current, it is in general sufficiently near for prac- tical purposes, and well suited to explain the phenomena taking place in the alternator under the various conditions of load. 162. This synchronous reactance, x, is frequently not constant, but is pulsating, owing to the synchronously vary- ing reluctance of the armature magnetic circuit, and tke 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. 163. Let E = induced E.M.F. of the alternator, or the E.M.F. induced in the armature coils by their rotation through the constant magnetic field produced by the cur- rent in the field spools, or the open circuit voltage of the alternator. 164] ALTERNATING-CURRENT GENERATOR. 239 Then E = V2 TT .V J/ 10 ~ 8 ; where n = total number of turn's in series on the armature, N = frequency, ^ M = total magnetic flux per field pole. Let oc = synchronous reactance, r = internal resistance of alternatof ; then Z = r j x = internal impedance. If the circuit of the alternator is closed by the external impedance, and current or, E and, terminal voltage, E = IZ = E - IZ E (r jx) (r + r) j (x + x) ' or, H- (x 1 i/1 + 2 r r 2 + ^Lf 4- 5J| or, expanded in a series, ~ (, r r+ * .y 2 (r r + x x) (r x + x r) ^ \ j J r 2 i ^2 + 7 2 (r 2 + jc 2 ) " j * As shown, the terminal voltage varies with the condi- tions of the external circuit. 164. As an instance, in Figs. 113-118, at constant induced, the E.M.F., E = 2500 ; 240 AL TERN A TING-CURRENT PHENOMENA. [ 164 and the values of the internal impedance, Z = r -jx = 1 - lOy. With the current / as abscissae, the terminal voltages E as ordinates in drawn line, and the kilowatts output, = 7 2 r, in dotted lines, the kilovolt-amperes output, = IE, in dash- ^ ^ , 1* S \ \ -^ ^^.^^ j ' \ \ N. ^^""l *^ ^ / \ 1 \ / \ N \ ^ \ \ \ / S 9 \ \ \ 4 \ t \ \ 45* Jj \ I > II II O-O '/ \ o o X X V \ \ \ / F ELD CHA 1AC1 ERIS TIC \ \ I 1 l 250( R = 3 f z = E. x MOj, = J \\ / \ j Arr P 3. \ .40 60 80 100 120 140 160 180 200 20 240 2 Fig. 113. Field Characteristic of Alternator on Non-inductive Load. dotted lines, we have, for the following conditions of external circuit : In Fig. 113, non-inductive external circuit, x 0. In Fig. 114, inductive external circuit, of the condition, r/x = -|- .75, with a power factor, .6. In Fig. 115, inductive external circuit, of the condition, r= 0,, with a power factor, 0. 164] ALTERNATING-CURRENT GENERATOR. 241 In Fig. 116, external circuit with leading current, of the condi- tion, r / x = .75, with a power factor, .6. In Fig. 117, external circuit with leading current, of the condi- tion, r = 0, with a power factor, 0. 20 21 22 20 18 16 M 12 10 8 6 4 J 1 \ | FIELD CHARA E72500, Z^MOj. i CTERISTIC \ \ 7 \ N N ^ - ^ A \ X \ ~o / ^ ^ \ 08 y X \ " / 9 /' N \ /. > \> s \ 1 X "\\ >, \ u / \ \ 1 \ [4 20 40 60 80 100 120 UO 160 180 200 220 240 260 Amps. Fig. 114. 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 ellipsis. With reactive load the curves are more nearly straight lines. The voltage drops rapidly on inductive, rises on capacity load. The output increases from zero at open circuit to a max- imum, and then decreases again to zero at short circuit. 242 ALTERNATING-CURRENT PHENOMENA. 164 1 \ FIELD C 0=2500, JHARACTE Zo-1-^j, r^ :RISTIC o, 90 Lag \ \ \ \ \ \ \ \ "o uj \ <^ X S| s. 11 II O 00 - x ^ \ v^ \ V o o ^ N f .. \ j / \ \ \ / \ \ \ y \ \ \ \ , \ \ Fig. 115. Field Characteristic of Alternator, on Wattless Inductive Load. Volts 1200 400 3000 E =2500 FIELD CHARACTERIST Ampe = -.75 OP 60^ P. F 200 210 >*' 280 >/> >*' 2 \i Fig. 116. Field Characteristic of Alternator, at 60% Power-factor on Condenser Load. 164] AL TERN A TING-CURRENT GENERA TOR. 243 'i FIE LD C PTARACTE i RIST C I i i r E^2'pOO,|Zo=1-iOj, = o, 90 Lagging Curre it 1 i i I*R = / . V / / / / i / ' Itj i / "I / / ^/ / / 7 i 1 / \/ s ***i O/ '// T >i J. r K / / / / /^ / / / ''/. / / ^ '/ ^'" 4 '/ x 10 D=Ar nps. 2 ' Fig. 117.. 6 8 10 12 II 10 IS 20 22 24 26 F/'e/rf Characteristic of Alternator, on Wattless Condenser Load. 244 AL TERN A TING-CURRENT PHENOMENA. [ 1 65 4oO 400 350 300 100 50 50 7 100 100 50 100 loO 200 2500, Z=1 \ 300 350 100 150500 \\ Fig. 118. Field Characteristic of Alternator. 165. The dependence of the terminal voltage, E, upon the phase relation of the external circuit is shown in Fig. 119, which gives, at impressed E.M.F., E = 2,500 volts, for the currents, /= 50, 100, 150, 200, 250 amperes, the terminal voltages, E, as ordinates, with the inductance factor of the external circuit, s as abscissae. V; 166. If the internal impedance is negligible- compared with the external impedance, then, approximately, E + 166] AL TERN A TING-CURRENT GENERA TOR. 245 1 .9 .7 .6 .5 .4 .3 .2 .1 -.1 -.2 -.3 -.4 -.5 %6 -.7 -.8 Fig. 119. Regulation of Alternator on Various Loads. -.9 -i that is, an alternator with small internal resistance and syn- chronous reactance tends to regulate for constant terminal voltage. Every alternator does this near open circuit, especially on non-inductive load. Even if the synchronous reactance, x , is not quite neg- ligible, this regulation takes place, to a certain extent, on non-inductive circuit, since for x = 0, E = 2 -4- - r r* and thus the expression of the terminal voltage, E, contains the synchronous reactance, x , only as a term of second order in the denominator. On inductive circuit, however, x appears in the denom- inator as a term of first order, and therefore constant poten- tial regulation does not take place as well. 246 ALTERNATING-CURRENT PHENOMENA, [ 167 With a non-inductive external circuit, if the synchronous reactance, ;r , of the alternator is yery large compared with the external resistance, r, T E. 1_ current * , Xi approximately, or constant ; or, if the external circuit con tains the reactance, x, approximately, or constant. The terminal voltage of a non-inductive circuit is XQ approximately, or proportional to the external resistance. If an inductive circuit, E = yV 2 -j- x* > dC ~~j~~ OC approximately, or proportional to the external impedance. 167. 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. The modern alternators are generally more or less ma- 167] ALTERNATING-CURRENT GEA?ERATOR. 247 chines of the first class ; the old alternators, as built by Jablockkoff, Gramme, etc., were machines of the second class, used for arc lighting, where constant-current regula- tion is an advantage. Obviously, large external reactances cause the same reg- ulation for constant current independently of the resistance, r, as a large internal reactance, ;r . On non-inductive circuit, if /"O _ the output is P = IE = ^- 2 (T + r o) + X Q ^ D 2 *- 2 _|_ *- 2 77 : = {(r ' "" ' ~* ^ 2; hence, if XQ = or r - then That is, the power is a maximum, and + r and y V2 s {% + /' | Therefore, with an external resistance equal to the inter- nal impedance, or, r = Z Q = Vr 2 -f :r 2 , the output of an alternator is a maximum, and near this point it regulates for constant output ; that is, an increase of current causes a proportional decrease of terminal voltage, and inversely. The field characteristic of the alternator shows this effect plainly. 248 ALTERNATING-CURRENT PHENOMENA. [168-170 CHAPTER XVII. SYNCHRONIZING ALTERNATORS. 168. 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. 169. 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 alter'nators. 170. The second important condition of parallel opera- tion is uniformity of speed ; that is, constancy of frequency. 171] SYNCHRONIZING ALTERNATORS. 24 & 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 willfctend 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: This difficulty does not exist with turbine or water-wheel driving. 171. In synchronizing alternators, we have to distin- guish the phenomenon taking place when throwing the machines in parallel or out of parallel, and the phenome- non 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 adjustment before synchronizing, sufficiently limited not to endanger the machines mechanically ; since the cross- currents, and thus the interchange of power, are limited by self-induction and armature reaction. 250 AL TERNA TING-CURRENT PHENOMENA. [ 172 , 1^ 3 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. 172. 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. 173. .Series operation of alternators is possible only by rigid mechanical connection, or by some means whereby the machines, with regard to their synchronizing power, act essentially in parallel ; as, for instance, by the arrange- ment shown in Fig. 120, where the two alternators, A lf A 2 , are connected in series, but interlinked by the two coils of a large transformer, T, of which the one is connected across the terminals of one alternator, and the other across the terminals of the other alternator in such a way that, when operating in series, the coils of the transformer will be without current. In this case, by interchange of power 174] SYNCHRONIZING ALTERNATORS. 251 through the transformers, the series connection will be maintained stable. SULQJUUUL) r Fig. 120. 174. In two parallel operating alternators, as shown in Fig. 121, let the voltage at the common bus bars be assumed Fig. 121. as zero line, or real axis of coordinates of the complex method ; and let 252 ALTERNATING-CURRENT PHENOMENA. [ 174r 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 r jx Let E l = e / 2 ) = induced E.M.F. of sec- ond machine ; /! = 4 +/// = current of first machine ; / 2 = / 2 -{-yV/ = current of second machine ; Z x = r t yX = internal impedance, and F x =^ t -\- jb^ = inter- nal admittance, of first machine ; ' Z 2 = r% jx =F= internal impedance, and Y z = g z -\- jb^ = inter- nal admittance, of second machine. Then, = ^ + 7^!, or e 1 jei = (e -} This gives the equations or eight equations with nine variables: e lt e^, e 2 , e 2 ', i l , 174] SYNCHRONIZING ALTERNATORS. 253 Combining these equations by twos, substituted in . , . - *1 -f* 2 =* = c(gi + g* + g) ; and analogously, ^1 ^iVl + ^2^2 ^2^2 = * (^1 + ^2 + ^) 5 dividing, substituting g = v cos a 2 ^ = v sin a */ = tfj sin wj ^ 2 ' = a 2 sin w 2 gives ^ + 1 + ^2 __ a\ v\ cos (^ (uQ + ^2 ^2 cos (a 2 ^ 2 ) /^ + b^ -\- b-L a v^ sin (a A wj) + ^ 2 ^ 2 sin (a 2 a> 2 ) as the equation between the phase displacement angles and o> 2 in parallel operation. The power supplied to the external circuit is of which that supplied by the first machine is, P\ = i ; by the second machine, / a == a . The total electrical work done by both machines is, P=P l + P,, of which that done by the first machine is, PI = *i h e{ i{ ; by the second machine, P. 2 = g z 4 _ e 2 ' // - 254 ALTERNATING-CURRENT PHENOMENA. [175 The difference of output of the two machines is, A p =/i / 2 = e (h 4) 5 denoting oij -f- to 2 a>! o> 2 cj ~^r ~^r 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. 175. SPECIAL CASE. Two equal alternators of equal excitation. a l = a% = a z = Z 2 , we have (cos w : + cos oi 2 ) = e (2 + r 0< - + X Q b) a (sin A! + sin o> 2 ) = expanding and substituting oil -f- o> 2 175] SYNCHRONIZING ALTERNATORS. 255 or , a cos e cos 8 = e I 1 -f- ^ ^* ^ a sin e cos 8 = e ; 2 hence tan c = -> y o^'~*o = constant. That is >! + w a = constant; and cos 8 - -1 + ^ + a cos 8 or, ^ = v/(i + ^^) 2 +(^v^) at no-phase displacement between the alternators, or, we have e = | From the eight initial equations we get, by combina- subtracted and expanded or, since e l 1 M2 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 h 5 the value of interchange of power between the alternators, A p =A-A; the value of synchronizing power, ^ , in dash-dot line, Curve V. 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. 258 ALTERNATING-CURRENT PHENOMENA. [177 CHAPTER XVIiL SYNCHRONOUS MOTOR. 177. 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 lt be connected as synchronous motor with a supply circuit of E.M.F., E Q , by a circuit of the impedance Z. If E Q is the E.M.F. impressed upon the motor termi- nals, Z is the impedance of the motor of induced E.M.F., E l . If E Q is the E.M.F. at the generator terminals, Z is the impedance of motor and line, including transformers and other intermediate apparatus. If E Q is the induced E.M.F. of the generator, Z is the sum of the impedances of motor, line, and generator, and thus we have the prob- lem, generator of induced E.M.F. E Q , and motor of induced E.M.F. E l ; or, more general, two alternators of induced E.M.Fs., E Q , E lt connected together into a circuit of total impedance, Z. Since in this case several E.M.Fs. are acting in circuit 177] SYNCHRONOUS MOTOR. 259 with the same current, it is convenient to use the current, /, as zero line OI of the polar diagram. If / = i = current, and Z = impedance, r = effective resistance, x = effective reaqtance, and z = Vr 2 + x* = absolute value of impedance, then the E.M.F. consumed by the resistance is E l = ri, and in phase with the cur- rent, hence represented by vector OE l ; and the E.M.F. consumed by the reactance is 2 = xi, and 90 ahead of the current, hence the E.M.F. consumed by the impedance is E = ^(E v f + (j 2 , or = i Vr 2 -f- x^ = iz, and ahead of the current by the angle 8, where tan 5 = x / r. We have now acting in circuit the E.M.Fs., E, E l , E Q ' t or E l and E are components of E Q ; that is, E Q is the diagonal of a parallelogram, with E l and E as sides. Since the E.M.Fs. E lt E 2 , E, are represented in the diagram, Fig. 122, by the vectors OE lt OE 2 , OE, to get the parallelogram of E Q , E l , E, we draw arc of circle around with E Qt and around E with E^. Their point of intersection gives the impressed E.M.F., OE Q = E Q , and completing the parallelogram OE, E Q , E lt we get OE l = E lt the induced E.M.F. of the motor. IOE Q is the difference of phase between current and im- pressed E.M.F., or induced E.M.F. of the generator. IOE^ 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 OL Hence, dropping perpendiculars, E Q E^ and E 1 E 1 \ from Q and E l upon Of, it is P Q = i X OE^ = power supplied by induced E.M.F. of gen- erator. PI = i X Oi l = electric power transformed in mechanical power by the motor. P = i x OE l = power consumed in the circuit by effective resistance. 260 ALTERNATING-CURRENT PHENOMENA. [178 Obviously P = P l + P. Since the circles drawn with E Q and E 1 around O and E respectively intersect twice, two diagrams exist. In gen- eral, in one of these diagrams shown in Fig. 122 in drawn Fig. 122. lines, current and E.M.F. are in the same direction, repre- senting mechanical work done by the machine as motor. In the other, shown in dotted lines, current and E.M.F. are in opposite direction, representing mechanical work con- sumed by the machine as generator. Under certain conditions, however, E Q is in the same, E in opposite direction, with the current ; that is, both ma- chines are generators. 178. It is seen that in these diagrams the E.M.Fs. are considered from the point of view of the motor ; that is, 178] SYNCHRONOUS MOTOR. 261 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. 123 to 125, con- sists of three components ; the E.M.F. OE% E z , consumed Fig. 123. by the impedance of the motor, the E.M.F. E^ E% = E z consumed by the impedance of the line, and the E.M.F. E% E = E consumed by the impedance of the generator. Hence, dividing the opposite side of the parallelogram 1 E Qy in the same way, we have : OE l = E l = induced E.M.F. of the motor, OE Z = E^ = E.M.F. at motor terminals or at end of line, OE% = _E 3 = E.M.F. at generator terminals, or at beginning of line. OE Q = E^ = induced E.M.F. of generator. 262 ALTERNA TING-CURRENT PHENOMENA. 179 The phase relation of the current with the E.M.Fs. E I} ) depends upon the current strength and the E.M.Fs. E 1 and E Q . 179. Figs. 123 to 125 show several such diagrams for different values of E l9 but the same value of / and E Q . The motor diagram being given in drawn line, the genera- tor diagram in dotted line. Fig. 124. As seen, for small values of E l the potential drops in the alternator and in the line. For the value of E 1 = E Q the potential rises in the generator, drops in the line, and rises again in the motor. For larger values of E lt the potential rises in the alternator as well as in the line, so that the highest potential is the induced E.M.F. of the motor, the lowest potential the induced E.M.F. of the gen- erator. ISO] SYNCHRONOUS MOTOR. It is of interest now to investigate how the values of these quantities change with a change of the constants. Fig. 125. 180. A. Constant impressed E.M.F. E Q , constant current strength I = z, variable motor excitation E. (Fig. 126.) If the current is constant, = z; OE, the E.M.F. con- sumed by the .impedance, and therefore point E, are con- stant. Since the intensity, but not the phase of E Q is constant, E Q lies on a circle E Q . In the first case, E l = Jj^ (Fig. 127), we see that at Fig. 128. very small curren., that is very small OE, the current / leads the impressed E.M.F. E Q by an angle E OC = o . This lead decreases with increasing current, becomes zero, and afterwards for larger current, the current lags. Taking now any pair of corresponding points E, E Q , and producing until it intersects e if in E { we have ^^ E i OE = 90, r/f , thus: OE 1 = EE Q =OE Q = . L ; that is, EE i 268 ALTERNA TING-CURRENT PHENOMENA. 181 That means the characteristic curve ^ is the enve- lope of lines EE i , of constant lengths 2 E^ , sliding between the legs of the right angle E { OE\ hence, it is the sextic hypocyloid osculating circle e Q , which has the general equa- tion, with e, e { as axes of coordinates : V> 2 = V 4 Q 2 In the next case, E^ < E^ (Fig. 128) we see first, that the current can never become zero like in the first case,. Fig. 129. E l = E Q , but has a minimum value corresponding to the E E minimum value of OE^ : //= - , and a maximum value : // = . Furthermore, the current can never Jo lead the impressed E.M.F. E^ 9 but always lags. The mini- 182] SYNCHRONOUS MOTOR. 269 mum lag is at the point H. The locus e v as envelope of the lines EE, is a finite sextic curve, shown in Fig. 128. In the case E l > E Q (Fig. 129) the current cannot equal zero either, but begins at a? finite value C^, corresponding E E to the minimum value of OE^ : // = . At this value however, the alternator E l is still generator and changes to a motor, its power passing through zero, at the point corresponding to the vertical tangent, onto e v 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 : 1st. 7=0, 7=^. z *>H / -^o E v T _ EQ + E\ -U. 2 - 2 - OJ r J -^0 r -L^Q "f f*I OU. Y 5 Y - , Z Z Since the current passing over the line at E l = O, that is, when the motor stands still, is 7 = E^j z t 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. Hence in such a case, if the synchronous motor drops out of step, the current passing over the line goes down to one-half or less ; or, in other words, in such a system the motor, under certain conditions of running, is more liable to burn up than when getting out of step. 182. C. EQ = constant, E v varied so that the efficiency is a maximum for all currents. (Fig. 130.) Since we have seen that the output at a given current strength, that is, a given loss, is a maximum, and therefore 270 AL TERN A TING-CURRENT PHENOMENA. 182 the efficiency a maximum, when the current is in phase with the induced E.M.F. E Q of the generator, we have as the locus of E the point E (Fig. 130), and when E with increasing current varies on *?, E l must vary on the straight line e l parallel to e. Hence, at no-load or zero current, E l = E Q , decreases with increasing load, reaches a minimum at OEf perpen- dicular to e lt and then increases again, reaches once more Fig. 130. E l = E Q at Ef, and then increases beyond E Q . The cur- rent is always ahead of the induced E.M.F. E l of the motor, and by its lead compensates for the self-induction of the system, making the total circuit non-inductive. The power is a maximum at Ef, where OEf = EfE Q = 1/2 x 02f , and is then = / x EO/%. Hence, since OE* = Ir = E Q /2, 1= E Q /2randP = E 2 /4:r, 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. 183] SYNCHRONOUS MOTOR. In this case, 271 In general, it is, taken fgpm 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. EI here acting in the same way as the condenser capacity in Chapter IX. Fig. 131. 183. D. E Q = constant; P = constant. If the power of a synchronous motor remains constant, we have (Fig. 131) / x OEJ- = constant, or, since OE 1 = 272 A L TERN A TING-CURRENT PHENOMENA . 183 Ir, I = OE l /r, and: OE 1 x OE? = OE 1 x E l E Q l = constant. Hence we get the diagram for any value of the current /, at constant power P^ , by making OE 1 = Ir, E 1 EJ* = P 1 / C erecting in E Q l a perpendicular, which gives two points of, intersection with circle e Qt E Q , one leading, the other lagging. Hence, at a given impressed E.M.F. E Q , the same power P l E, | 1250 7 1100/1580 3116.7 1480 32 1050/1840 2/25 2120 2170 37.5 40 45.5 Ef 1000 P=IOOO 46 < E^2200 2/25 22 3/16.7 Fig. 132. can be transmitted by the same current I with two different induced E.M.Fs. E } of the motor; one, = EE Q small, corresponding to a lagging current ; and the other, OE l = EE Q large, corresponding to a leading current. The former is shown in dotted lines, the latter in drawn lines, in the diagram, Fig. 131. Hence a synchronous motor can work with a given out- put, at the same current with two different counter E.M.Fs. 183] SYNCHRONOUS MOTOR. 273 19 and at the same counter E.M.F. E lt at two different currents /. In one of the cases the current is leading, in the other lagging. ^, In Figs. 132 to 135 are shown diagrams, giving the points E = impressed E.M.F., assumed as constant = 1000 volts, E E.M.F. consumed by impedance, E' = E.M.F. consumed by resistance. 1450 17.3 1170/1910 10/30 1040/1930 8/37.5 E=1000 P=6000 340 < E,<1920 7< I < 43 Fig, 133. of the motor, lf is O l} equal and shown in the diagrams, to avoid The counter E.M.F. parallel EE Q , but not complication. The four diagrams correspond to the values of power, or motor output, P = 1,000, 6,000, 9,000, P = 1.000 46 < E < 2,200, P = 6.000 340 < E l < 1,920, P = 9,000 540 ) = - . ... siri (t e) Since the three E.M.Fs. acting in the closed circuit : e Q = E.M.F. of generator, e l = C.E.M.F. of synchronous motor, e = zi= E.M.F. consumed by impedance, form a triangle, that is, e 1 and e are components of three variables are left, ^j, /, /, of which two are independent. Hence, at given e^ and z, 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 /, according to the excita- tion ; and with the same load, the current i can be varied in a wide range, by varying the field excitation e l . The meaning of equation (7) is made more perspicuous 184] SYNCHRONOUS MOTOR. 277 by some transformations, which separate e l and z, as .func- tion of/ and of an angular parameter <. Substituting in (7) the new coordinates : (8) we get substituting again, F/g. 73(5. . 737. we get (9) - V2 - 2r = 2 e a - a V2 - e b = V(l - e 2 ) (2 a 2 - 2 j3 2 - P), (11) and, squared, 2 2 substituting ca (a - e /;) V2 _ /? Vl - e 2 = a/, gives, after some transposition, (13) (14) 278 ALTERNATING-CURRENT PHENOMENA. [184 hence, if r - R = /(I -<*)(<*- 2*6) a . (15) V' 2c 2 the equation of a circle with radius R. Substituting now backwards, we get, with some trans- positions : {r* (e* + * 2 / 2 ) - z* (e* - 2 r/)} 2 + {r x (e* - * 2 / 2 )} 2 = ^V(^ 2 -4r/) (17) the Fundamental Eqtiation of the SyncJironous Motor in a modified form. The separation of e l and i can be effected by the intro- duction of a parameter by the equations: r x (e z 2 i 2 ) = x z e^ V^ 2 krp sin 9 These equations (18), transposed, give ( -cos4> + sinc^ W ^ r P ( / The parameter 9 has no direct physical meaning, appar- ently. These equations (19) and (20), by giving the values of ^ and i as functions of / and the parameter 9 enable us to construct the Power Characteristics of the Synchronous Motor, as the curves relating ^ and z, for a given power /> by attributing to 9 all different values. 185] SYNCHRONOUS MOTOR. 279 Since the variables.?' and iv in the equation of the circle (16) are quadratic functions of e 1 and i, the Power Charac- teristics of tJie SyncJironous 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. 185. A. Maximum Output. Since the expression of ^ and i [equations (19) and (20)] contain the square root, V^ 2 4r/, it is obvious that the maximum value of / corresponds to the 9 moment where this square root disappears by passing from real to- imaginary ; that is, e * _ 4 r p = 0, This is the same value which represents the maximum power transmissible by E.M.F., e Q , over a non-inductive line of resistance, r\ or, more generally, the maximum power which can be transmitted over a line of impedance, z = -Vr 2 + x 2 , into any circuit, shunted by a condenser of suitable capacity.. Substituting (21) in (19) and (20), we get, (22) and the displacement of phase in the synchronous motor. / -\ p r cos (e 1 , / ) = ^_ = - ; te\ z hence, / -\ X /OQY tan (*!, i) , (Z6), 280 ALTERNATIA T G-CURRENT PHENOMENA. [186 that is, the angle of internal displacement in the synchron- ous motor is equal, but opposite to, the angle of displace- ment of line impedance, fa, = - (', 0, = - (*, r\ (24) and consequently, (25) that is, the current, i t is in phase with the impressed E.M.F., * . If 5 < 2 r, ^i < ^ ; that is, motor E.M.F. < generator E.M.F. If z = 2 r, e l = e ; that is, motor E.M.F. = generator E.M.F. If z > 2 r, e l > generator E.M.F. In either case, the current in the synchronous motor is leading. 186. B. Running Light, p = 0. When running light, or for p = 0, we get, by substitut- ing in (19) and (20), + - 2 ( z (26) Obviously this condition can never be fulfilled absolutely, since/ must at least equal the power consumed by friction, etc. ; and thus the true no-load curve merely approaches the curve/ = 0, being, however, rounded off, where curve (26) gives sharp corners. Substituting / = into equation (7) gives, after squar- ing and transposing, f * + e( ? + W 4 - 2 eftf - 2 z*i 2 e, = ; by the condition, */^i d f I dl f\ i rv "7^ = ~~ ~77TT = as z l xe ^ = ' dt df I dei hence, i = e, ^i = =F^o- (31) rsr r The maximum value of current, i, is given by equation (28) by at A - = 0, as ( 32 > If, as abscissae, e lf 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 curves of synchronous motors running light, published by Mordey and others, the two sides of the V-shaped 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. 138, 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 1 , whose areas contain the quartic curves of generator, and the interior space C and exterior space D, whose points do not represent any actual condition of the alternator circuit, but make e lf i imaginary. A and A' and the same B and !>' , are identical condi- tions of the alternator circuit, differing merely by a simul- 282 ALTERNA TING-CURRENT PHENOMENA. 187 200- 160- \ r.000 4000 3000X^2000 loop Volt3 1000 2000\/ 3000 4000 5000 \ /A' \ B' F/g. 738. 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. 187. C. Minimum Current at Given Power. The condition of minimum current, i, at given power, /, is determined by the absence of a phase displacement at the impressed E.M.F. e Q , -: 187] SYNCHRONOUS MOTOR. 283 This gives from diagram Fig. 132, ^ = ^ + /V>_2*> ^ (33) > ' z or, transposed, e, *= V(*o-'>) a + ' a * 2 . (34) This quadratic curve passes through the point of zero current and zero power, / = 0, e\ = e*, through the point of maximum power (22), /_ ^ _e*z 1 7T~ > *i 77~ > 2 r 2 r and through the point of maximum current and zero power, ,, _ e Q X /QPIN ) 2 z' 2 .r 2 , which represents the output transmitted over an inductive line of impedance, z = vV' 2 -f jr 2 into a non-inductive circuit. Equation (34) is identical with the equation giving the maximum voltage, ^1 , at current, z, which can be produced by shunting the receiving circuit with a condenser; that is, the condition of " complete resonance " of the line, z VV 2 + jr 2 , with current, /. Hence, referring to equation (35), e^ = ^ O or < * . 190. F. Numerical Instance. Figs. 138 and 139 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. 19O] SYNCHRONOUS MOTOR. 287 In this case we have, 195. 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. 144. space by 45 in a bipolar motor from the direction of the magnetic flux, as shown diagrammatically in Fig. 144. 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- 296 ALTERNATING-CURRENT PHENOMENA. [196 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. 196. Let 4> = maximum magnetic flux per field pole ; e = effective E.M.F. induced thereby in the field turns; thus : e n = number of turns, JV= frequency. thus: -^-xnN The instantaneous value of magnetism is sin /? ; and the flux interlinked with the armature circuit ! = 3> sin (3, sin A ; when A is the angle between the plane of the armature coil and the direction of the magnetic flux. The E.M.F. induced in the armature circuit, of n turns, as reduced to primary circuit, is thus : = n 3> ) sin A cos (3 i + sin /? cos A ( ( dt dt ) 10 -8 If N = frequency in cycles per second, N^ speed in cycles per second (equal revolutions per second times num- ber of pairs of poles), it is : Illf \ f ->" .-U .. .": .. i ; 197] COMMUTATOR MOTORS. 297 and since X = 45, or sin X = cos X = 1/V2, it is, sub- stituted : or, since e l = e { cos (3 -{- k sin /? } ; where ^ = A = rat io s P eed ; N frequency or the effective value of secondary induced E.M.F., 197. Introducing now complex quantities, and counting the time from the zero value of rising magnetism, the mag- netism is represented by /*; the primary induced E.M.F., E =-e; the secondary induced E.M.F. : V2 hence, if Z x = /i j'xi = secondary impedance reduced to primary circuit, Z = r j x = primary impedance, Y = g + j b = primary admittance, it is, secondary current, ^i V2 n- primary exciting current, 298 AL TERN A TING-CURRENT PHENOMENA. [ 198 hence, total primary current, or (* V2 > Primary impressed E.M.F., or Neglecting in Q the last term, as of higher order, ( V2 ^i-7^i> or, eliminating imaginary quantities, V2 198. The power consumed by the primary counter E.M.F., e, that is, transferred into the secondary circuit, is 1 + j V2 n y^i or, eliminating the imaginary quantities, J V2 The power consumed by the secondary resistance is 2 r? + oc* ' Hence, the difference, or the mechanical power at the motor shaft 2 (r? -f X f) 198] COMMUTATOR MOTORS. 299 and, substituting for e, p = {r, (V2 - 1) -f k Xl V2 - (r + r : V2 (x V2 - If r and r x are "small compared with x and ^ , this is approximately, Thus the power is a maximum for dP jdk = 0, that is, _* + *,V2 1000 ^, _. 900 / ^^ """""""'"'' 300 ^ / 700 I / / R IPL LS ON M 3TC )R COO 500 o en / r e = OC nJ / / r = .1 r,= 05 400 ^ / X s 2. x i = 300 / P = 1UU (1 7 E? 02 ^ K) 1.4)1 K + (3.14 .0 -1 'K^ 200 / ieo/ K- Spe -"rp ed 3UfiJ / .2 .4 .G .8 1. 1. 2 !. 1 1. 6 !. 8 2. r. 745. Repulsion Motor. As an instance is shown, in Fig. 145, the power output as ordinates, with the speed k = A\ / N as abscissae, of a repulsion motor of the constants, e n = 100. r =- .1 r, = .05 x == 2.0 Xl = 1.0 giving the power, == 10,000 {.02 + 1.411 - .05 I 2 } (.171 + 2) 2 + (3.14- ,\Kf 300 AL TERN A TING-CURRENT PHENOMENA. [199 SERIES MOTOR. SHUNT MOTOR. 199. 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. 146. Series Motor. uct, the torque, thus maintained the same sign. There- fore such a motor, when supplied by an alternating current, will operate also, provided that the reversals in field and in armature take place simultaneously. In the series motor this is necessarily the case, the same current passing through field and through armature. With an alternating current in the field, obviously the 199] COMMUTATOR MOTORS. 301 magnetic circuit has to be laminated to exclude eddy cur- rents. Let, in a series motor, Fig. 146, i 3> = effective magnetism per ole, n = number of field turns per pole in series, n = number of armature turns in series between brushes, / = number of poles, (R = magnetic reluctance of field circuit, (Rj = magnetic reluctance of armature circuit, <&! = effective magnetic flux produced by armature current (cross magnetization), r resistance of field (effective resistance, including hys- teresis), t\ = resistance of armature (effective resistance, including hys- teresis), N = frequency of alternations, JV-L = 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). = 4;z 1 7Vi10- 8 E.M.F. of self-induction of field, E r = 2 E.M.F. of self-induction of armature, E{ = 27r/; 1 ^ 1 10- 8 , E.M.F. consumed by resistance, E r = (r + n) I, where / = current passing through motor, in amperes effective. Further, it is : Field magnetism : 7 10 8 302 ALTERNATING-CURRENT PHENOMENA. [199 Armature magnetism : Substituting these values, ,-, 4 n Thus the impressed E.M.F., rf + (E r or, since *2 = reactance of field ; = 2 TT N = reactance of armature ; and 20O] COMMUTATOR MOTORS. 303 200. The power output at armature shaft is, P= El knn^N, F2 ^^ 5 , 2 n, pn N - ^ ^ + ^+ TT pn N The displacement of phase between current and E.M.F. is 2 i_ N^ x r te pn N * Neglecting, as approximation, the resistances r + r lt it is, tan o> = 2 _ t 2 i. ^ , TT pn N 2 n TT pn 304 ALTERNATING-CURRENT PHENOMENA. [201 hence a maximum for, 2 n^ N = \_ ic pn N ~'' 2 pn N or, 2 n ' ~TT P^l substituting this in tan , it is : tan o> = 1, or, w = 45. 201. Instance of such an alternating-current motor, 9 = 100 A T = 60 / = 2. r - .03 T! = .12 x = .9 jq = .5 = 10 % = 48 Special provisions were made to keep the armature re- actance a minimum, and overcome the distortion of the field by the armature M.M.F., by means of a coil closely surrounding the armature and excited by a current of equal phase but opposite direction with the armature current (Eickemeyer). Thereby it was possible to operate a two- circuit, 96-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, 7 = 100 ~ V(.023 NI + .15) 2 + 1.96 P= 2307V1 (.023 JVi -f .15) 2 + 1.96 1.4 .023 TV, -f .15 tan " = .023^ + .15' 0r ' C S "- 202] COMMUTATOR MOTORS. 305 In Fig. 147 are given, with the speed TVj as abscissae, the values of current /, power P, and power factor cos w of this motor. Amp. Watts SER Cos 03 = 30 40 50 60 |70 ES 60 MOTOR 00 A/C023 A/IP23 A/CQ23 '3D 15)' 1.96 1.9 15) 2 +1.9 Fig. 147. Series Motor. 202. 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 counter E.M.F. of the armature should be in phase with the impressed E.M.F., and thereby the armature current lag less, to represent power. Since how- ever, the field current, and thus the field magnetism, lag nearly 90, the induced E.M.F. of the armature will lag nearly 90, and thus not represent power. 306 ALTERNATING-CURRENT PHENOMENA. [202 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. 146, 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 armature. To overcome this difficulty various arrangements /have been proposed, but have not found an application. 203] COMMUTATOR MOTORS. 307 203. 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 s.een from Fig. 147, 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 TT N / 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. 308 ALTERNATING-CURRENT PHENOMENA. [204 CHAPTER XX. REACTION MACHINES. 204. 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 l reduced to zero, and sometimes even if the field circuit is reversed and the counter E.M.F. E l made negative. Inversely, under certain conditions of load, the current and the E.M.F. of a generator do not disappear if the gene- rator field is broken, or even reversed to a small negative value, in which latter case the current flows against the E.M.F. E Q of the generator. , Furthermore, a shuttle armature without any winding will in an alternating magnetic field revolve when once brought up to synchronism, and do considerable work as a motor. These phenomena are not due to remanant magnetism nor to the magnetizing effect of Foucault currents, because 205, 206] REACTION MACHINES. 309 they exist also in machines with laminated fields, and exist if the alternator is brought up to synchronism by external means and the remanant magnetism of the field poles de- stroyed beforehand by application of an alternating current. 205. 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. 206. A case of this nature has been discussed already in the chapter on Hysteresis, from a different point of view. 310 ALTERNATING-CURRENT PHENOMENA. [ 2O7 There the effect of magnetic hysteresis was found to distort the current wave in such a way that the equivalent sine wave, that is, the sine wave of equal effective strength and equal power with the distorted wave, is in advance of the wave of magnetism by what is called the angle of hysteretic advance of phase a. Since the E.M.F. induced by the magnetism, or counter E.M.F. of self-induction, lags 90 behind the magnetism, it lags 90 + a behind the current ; that is, the self-induction in a circuit containing iron is not in quadrature with the current and thereby wattless, but lags more than 90 and thereby consumes power, so that the reactance has to be represented by X = h 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. 207. 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. 148 and 149. 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 M, 90 ahead of E. The reactance is represented by the sine wave x, varying with tfie double frequency of E, and shown in Fig. 148 to reach the maximum value during the rise of magnetism, in Fig. 149 during the decrease of magnetism. The current / re- quired to produce the magnetism < is found from <$ and x in combination with the cycle of molecular magnetic friction of the material, and the power P is the product IE As seen in Fig. 148, the positive part of P is larger than the 2O7] REACTION MACHINES. 311 , V V /\ -w \ 74S. Variable Reactance, Reaction Machine. \ Fig. 149. Variable Reactance, Reaction Machine. 312 AL TERN A TING-CURRENT PHENOMENA. [ 208 negative part ; that is, the machine produces electrical energy as generator. In Fig. 149 the negative part of P is larger than the positive ; that is, the machine consumes electrical energy and produces mechanical energy as synchronous motor. In Figs. 150 and 151 are given the two hysteretic cycles or looped curves 3>, 7 under the two conditions. They show that, due to the variation of reactance x t in the first case the hysteretic cycle has been overturned so as to represent not consumption, but production of electrical Fig. 150. 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. Hence, such a synchronous motor can be called " hyste- resis motor," since the mechanical work is done by an ex- tension of the loop of hysteresis. 208. 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- 2O8] REACTION MACHINES. 313 mum, will take place at the moment where the armature 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-fF., 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 Z 7 Fig. 151. Hysteretic Loop of Reaction Machine. armature current which magnetism induces in the arma- ture conductor the E.M.F. of self-induction is propor- tional to the instantaneous value of the current, divided by the instantaneous value of the reluctance. Since the extreme values of the reluctance coincide with the sym- metrical positions of the armature with regard to the field poles, that is, with zero and maximum value of the in- duced E.M.F., E Q , of the machine, it follows that, if the current is in phase or in quadrature with the E.M.F. E Q) the reluctance wave is symmetrical to the current wave, and the wave of magnetism therefore symmetrical to the 314 ALTERNATING-CURRENT PHENOMENA. [209 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. 209. 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 current 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. 210] RE A C TION MA CHINES. 315 210. 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 s&e 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 -f- b cos 2 <. Let the inductance, or the coefficient of self-induction, be represented by L = I + $ cos 2 < = /(I _j_ y cos 2 <) where y = amplitude of variation of inductance. Let cos (3 + fl + |\ sin u> sin ft and the effective value of E.M.F., = 2 TT AT//4/1 -f ^ y cos 2 w. Hence, the apparent power, or the voltamperes y = IE = 2 E 2-rrNl /I .I? V" ~T~ The instantaneous value of power is co sn in/? 1 and, expanded -I - sin 2 co cos 2 /? + sin 2 ft I cos 2 co Integrated, the effective value of power is P= 7r 211] RE A C TION MA CHINES. 317 hence, negative ; that is, the machine consumes electrical, and produces mechanical, power, as synchronous motor, if o> > ; that is, with lagging current. Positive ; that is, the machine produces electrical, and consumes mechanical, powe# as generator, if o> < ; that is, with leading current. The power factor is P y sin 2 w f P V+J- hence, a maximum, if, y cos 2 oi ; cos 2w = -__ 8 + y 2 . or, expanded, COS 2 *= - -r 7 zsr 7 y 4 4 The power, P, is a maximum at given current, /, If sin 2 o> = 1 ; that is, = 45 at given E.M.F., E, the power is P= - f 1 -f ^- y COS 2 w 4 V hence, a maximum at or, expanded, 211. We have thus, at impressed E.M.F., E, and negli- gible resistance, if we denote the mean value of reactance, x= 2TrNl. Current ,., 7= _ ^ _ y COS 2 o>. 318 AL TERN A TING-CURRENT PHENOMENA. [211 Voltamperes, Power, ' 2 y sin 2 cu 2^1+f-yc Power factor, / T- T\ y sin 2 /= cos G,/) = 7 =^= Maximum power at cos 2 w = ...V' Maximum power factor at cos o> > : synchronous motor, with lagging current, w < : generator, with leading current. As an instance is shown in Fig. 152, with angle as abscissae, the values of current, power, and power factor, for the constants, E = 110 x = 3 y =.8 hence, - 41 p /= cos (,!) = Vl.45 - cos 2 2017 sin 2o> 1.45 cos 2 cos /? { 1 + e cos [2 (3 o>]} where, = average magnetic flux, e = amplitude of pulsation, and co = phase of pulsation. In a machine with y slots per pole, the instantaneous flux interlinked with the armature conductors will be : + ecos[2 r /?- o>]}, if the assumption is made that the pulsation of the magnetic flux follows a simple sine law, as can be done approximately. In general the instantaneous magnetic flux interlinked with the armature conductors will be : = 3> cos /3 {1 + e l cos (2 /? wj) + 2 cos (4 (3 o> 2 ) + . . . }, where the terms e v is predominating if y = number of armature slots per pole. This general equation includes also the effect of lack of uniformity of the magnetic flux. 214] DISTORTION OF WAVE-SHAPE. 323 140 N6 Lo ad ^* 'x, 130 x, ^14 i.5 *= 2.6 // S N 120 ^ r\ 1 110 ^ i V 100 / \ \ 00 i \ SO i \ 70 '/ \ GO \ .50 | \ 10 /I \ 30 // \ 20 I V 10 l! \ 1-10 /-- -\ ^=- 20 M 10 V) 5" GO 70 - M _ 90 .! 100 -=-^; 110 5 ^ 140 ^^ 150 ^^ 1GO -^ 170 ISO Fig. 153. No-load Waue of E.M.F. of Multitooth Three-phaser. 130 w th L oad 120 *t -12 7.0 J% 5 3,2 ^ ' :--- >x 110 \ 100 / \ 00 , / \x 80 / 1 70 / t 60 / \ 50 / V 40 f , f \ \ 30 I \ 20 1 \ 10 1 5 // ^ ~^. /-" >v 10 10 20 30 10 50 CO 70 80 W 100 110 120 -_ 130 ^- 140 150 1GO 170 ISO Fig. 154. Full-Load Waue of E.M.F. of Multitooth Three-phaser. 324 AL TERN A TING-CURRENT PHENOMENA. [215 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 : < = 3> COS ft {1 + e COS [2 yft o>]}. Hence the E.M.F. induced thereby : e = n^- dt = - V2irJVfc { COS /? (1 + e COS [2y -])}. ff-ft And, expanded : e = V2 TT Nn* {sin /? + e 2y ~ 1 sin [ (2 y - 1) - a] 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 y+1). 215. If y= 1 it is: e = V2 TT Nn 3> {sin ft + 1 sin (ft - fi) + |f sin (3 /? - a) } ; ^ 2 that is : In a unitooth single-phaser a pronounced triple harmonic may be expected, but no pronounced higher harmonics. Fig. 155 shows the wave of E.M.F. of the main coil of a standard monocyclic alternator at no load, represented by : e = E {sin ft .242 sin ( 3 ft 6.3) .046 sin (5ft 2.6) + .068 sin (7 ft - 3.3) - .027 sin' (9 ft - 10.0) - .018 sin (11 ft - 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 <$> j sin ft + sin (3 ft - A) + ^ sin (5 - fi) 215] DISTORTION OF WAVE-SHAPE. 325 the no-load wave of a unitooth quarter-phase machine, hav- ing pronounced triple and quintuple harmonics. If y = 3, it is : ^ sin (7 - o> 2 That is : In a unitooth three-phaser, a pronounced quin- tuple and septuple harmonic may be expected, but no pro- nounced triple harmonic. 130 110 100 Sine c =/ taUsu cMac f A ternato at ro I -.027 sin(9 )-.018 n sin (11 ^-T A.M.-12-150-eOO ad 2n S Co V ~ 'n 3L2 -3,86 4^! .72 = -1,4 . 755. No-load Wave of E.M.F. of Unitooth Monocyclic Alternator. Fig. 156 shows the wave of E.M.F. of a standard unitooth three-phaser at no load, represented by : e = E {sin - .12 sin (30- 2.3) - .23 sin (5 /3 - 1.5) + .134 sin (7 (3 - 6.2) - .002 sin (90 + 27.7) - .046 sin (11 - 5.5) + .031 sin (13 - 61.5)}. Thus giving a pronounced quintuple and septuple and a lesser triple harmonic, probably due to the deviation of the field from uniformity, and deviation of the pulsation of reluctance from sine shape. 326 AL TERN A TING-CURRENT PHENOMENA. [215 110 130 ., 1-20 ne cor -po ien S / ^ f ^ Co . c| mp 3 ne ts 110 of v avi / 1 A / Jfw we 100 f ] \ ^ / y*- 0,5 -10, | 22,8 2,4 I \ ) -. 05J5 -2,95 1 \ y^ 7,87 ^ 60 I, .5,S 5 jy 1 > 50 9 \ y,/ : 40 A nal) SIS ( f A ter lato S V a vc S \ so ^ u T f (/>) "ZL" (xi n i ^-t- yi c OS i <7>) i=2 T-1] \ 20 / U TltO oth Thr ep last Ma chi le /! T. 12- 15C -60 ) \ 10 / Y.E.M F. at r o lo ad \ | *i** C, *^- 32; -10 10 * > *-^ 20 ^*s* N 30 40 50 C 70 E 00 1 00 110 20 130 140 150 1GO 170 180 Fig. 156. 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 : GO T the instantaneous magnetic flux is : hence, the E.M.F. e= o. ) e y cos ( 2 7^ S Y) C ' 1 oo r- os(^-^i)+SI| ^ cos((2 7 + 00 2y + l 2 sin ((2 y + 1) - Y ) + v+ i sin ((2 216,217] DISTORTION- OF WAVE-SHAPE. 327 Pulsation of Reactance. 216. The main causes f 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 of the permea- bility 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. 217. 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, i = /cos 3 328 A L TERN A TING-CURRENT PHENOMENA . [217 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 (2 -)}; or, more general, X = x \ 1 + 57; s cos (2 y ft - 3> y ) } ; 1 the wave of magnetism is :OS /3 - < COS , ^~1 QO ft - "v) + ^^ cos ((2 y + 1) ft - "y + 1)1 1 ; hence the wave of induced E.M.F. if ft in ((2 y + 1) - fi v + 1)] sn that is, the pulsation of reactance of frequency, 2y, intro- duces two higher harmonics of the order (2 y 1), and If X=x{\. + e cos (2/3 o>)}, icos (/3 - A) + icos 2 2 ^ = ^ | sin ^ + I sin (/3 - S>) + ^ sin (3 (3 - co) | ^ Since the pulsation of reactance due to magnetic satu ration and hysteresis is essentially of the frequency, 218,219] DISTORTION OF WAVE-SHAPE. 329 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 z, and thus the power consumed or produced in the electric circuit, depend upon the angle, o>, as discussed before. 218. 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. 219. 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 ^ ^ ^ current for small currents, low for large currents. Thus in an alternating arc the apparent resistance will vary during 330 AL TERNA TING-CURRENT PHENOMENA. [219 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 / = V2 sin ft and the apparent resistance of the arc, in first approxima- tion, by X = r (1 -f e cos 2 <) ; thus the potential difference at the arc is _ ec os2 <) = rl V2 1 - i sin $ -f 1 sin 3 4 I. Hence the -effective value of potential difference, and the apparent resistance of the arc, E ro = _ = The instantaneous power consumed in the arc is, / = ie = 2 r/ 2 j ^1 - -\ sin 2 <#> + | sin <#> sin 3 Hence the effective power, 22O] DISTORTION OF WAVE-SHAPE. 331 The apparent power, or volt amperes consumed by the arc, is, thus the power factor of tke arc, 71 that is, less than unity. 220. 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 co =f of an angle, to, whose sign is indefinite. As an instance are shown, in Fig. 157 for the constants, /= 12 r= 3 e = .9 the resistance, a = 3 {1 + .9 cos 2 /?) ; the current, / = 17 sin (3 the potential difference, e = 28 (sin j3 + .82 sin 3 (3). In this case the effective E.M.F. is =25.5; 332 AL TERN A TING-CURRENT PHENOMENA. [221 the apparent resistance, the power, the apparent power, the power factor, r = 2.13; P = 244 ; El =307; / = .796. r\ ARIABLE = 28( RESISTANCE 9 cis 2 fi) in/5+. 82 s n 3 K \ \ \ F/g. 757. 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. 221. In reality the distortion is of more complex nature ; since the pulsation of resistance in the arc does- not follow 222] DISTORTION OF WAVE-SHAPE. 333 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. 158.* ONE PAIR CARBONS EGULATED BY HANDm At Ci dynamo e, m. f,|| \ II. lf " " current, II, ' watts, 758. Electric Arc, 222. 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. 334 AL TERNA TING-CURRENT PHENOMENA. 223 CHAPTER XXII. EFFECTS OF HIGHER HARMONICS. 223. To elucidate the variation in the shape of alternat- ing waves caused by various harmonics, in Figs. 159 and Distortion of Wave Shape by Triple Harmonic Sin.tf-3 sin.(3/?-Gfl) Fig. 159. Effect of Triple Harmonic. 160 are shown the wave-forms produced by the superposi- tion of the triple and the quintuple harmonic upon the fundamental sine wave. $223] EFFECTS OF HIGHER HARMONICS. 335 In Fig. 159 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 #ie equations : sin ft sin p .3 sin 3 p sin p .3 sin (3 p 45) sin P - .3 sin (3 ft - 90) sin p - .3 sin (3 p - 135) sin p .3 sin (3 p 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. 160 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 p sin p - .2 sin 5 p sin P - .2 sin (5 - 45) sin P .2 sin (5 p 90) sin p - .2 sin (50- 135) sin p - .2 sin (5 p 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 336 AL TERN A TING- CURRENT PHENOMENA. [223 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 Shape by Quintuple Harmonic Sin./? -.2 sin. (5/2-55; > J \J Fig. 160. Effect of Quintuple Harmonic. Sharp peak coincides with flat zero in the triple, with sharp zero in the quintuple harmonic. Thus in general, from simple inspection of the wave shape, the existence of these first harmonics can be dis- covered. Some characteristic shapes of curves are shown in Fig. 161: 224] EFFECTS OF HIGHER HARMONICS. 337 Fig. 161. Some Characteristic Wave Shapes. Flat top with flat zero : sin (3 .15 sin 3 ft .10 sin 5 p. Flat top with sharp zero : sin ft - .225 sin (3ft 180) - .05 sin (5 ft - 180). Double peak, with sharp zero : sin ft - .15 sin (3 ft 180) .10 sin 5 ft. Sharp peak with sharp zero : sin ft .15 sin 3 ft .10 sin (5 ft 180). 224. 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 338 A L TERN A TING-CURRENT PHENOMENA . [225 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. 225. In long-distance transmission over lines of notice- able inductance and capacity, rise of voltage due to reso- nance may under circumstances be expected with higher harmonics, as waves of higher frequency, while the funda- mental 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 : The resistance drop in the transformers at full load = 1 per cent. The inductance drop in the transformers at full load = 5 per cent with the fundamental wave. The resistance drop in the line at full load 10 per cent. 225] EFFECTS OF HIGHER HARMONICS. 339 The inductance drop in the line at full load = 20 per cent with the fundamental wave. The capacity or charging current of the line =20 per cent 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 E, or, r = .06 . The reactance E.M.F. between generator terminals and condenser is, for the fundamental frequency, Ix = .15 E, or, x = .15 , thus the reactance corresponding to the frequency (2>& 1) N of the higher harmonic is : x (2k- 1) = .15 (2k - 1) . The capacity current at fundamental frequency is : / = .2 I, hence, at the frequency : (2 k 1) N\ E ' if: e' = E.M.F. of the (2 k l) th harmonic at the condenser, e = E.M.F. of the (2k l) th harmonic at the generator terminals. The E.M.F. at the condenser is : + /.*(2-l) 2 ; 340 ALTERNATING-CURRENT PHENOMENA. [225 hence, substituted : VI - .059856 (2 k - I) 2 + .0009 (2 k - I) 4 the rise of voltage by inductance and capacity. Substituting : = 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 : da . 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 : 342 A L TERNA TING-CURRENT PHENOMENA. [227 while the same E.M.F. was found by direct calculation from number of turns, magnetic flux, and frequency to be equal to 2 e ; that is the two values found for the same E.M.F. have the proportion V3 : 2 = 1 : 1.154. Fig. 162. 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 : e = Hence it follows that, while the E.M.F. between two col- lector rings in the machine shown diagrammatically in Fig. 162 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. 228,229] EFFECTS OF HIGHER HARMONICS. 343 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 producecP with the same number of arma- ture turns if the magnetic disposition were such as to pro- duce a sine wave. 228. Inversely, if such 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. 229. 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 344 ALTERNATING-CURRENT PHENOMENA. [ 23O, 231 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 belongs to the former class ; the waves derived from continuous-current machines, tapped at two equi-distant points of the armature, in gen- eral to the latter class. 230. 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. 231. 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 the effective value. Since disruptive phenomena do not 231] EFFECTS OF HIGHER HARMONICS. 345 always take place immediately after application of the potential, but the time element plays an important part, it is possible that insulation-strain and striking-distance is, in a certain range, dependent upon the effective potential, and thus independent of th& wave-shape. 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. 346 ALTERNATING-CURRENT PHENOMENA. [232,233 CHAPTER XXIII. GENERAL POLYPHASE SYSTEMS. 232. 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. 233. The flow of power in a single-phase system is pulsating ; that is, the watt curve of the circuit is a sine 233] GENERAL POLYPHASE SYSTEMS. 347 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^ystem, 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. 163. Fig. 164. 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. 163 and 164, is an interlinked system. 348 ALTERNATING-CURRENT PHENOMENA. [234 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. 165, ~ J'E Fig. 165. 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. 166, as star-connected four-phase system. E -HE Fig. 166. 234. 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- 234] GENERAL POLYPHASE SYSTEMS. 349 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. 350 ALTERNATING-CURRENT PHENOMENA. [235 CHAPTER XXIV. SYMMETRICAL POLYPHASE SYSTEMS. 235. If all the E.M.Fs. of a polyphase system are equal in intensity, and differ from each other by the same angle of difference of phase, the system is called a symmetrical polyphase system. Hence, a symmetrical %-phase system is a system of n E.M.Fs. of equal intensity, differing from each other in phase by I./ ;/ of a period : e 1 = E sin e 2 = E sin ( (3 n 2 (n - 1) TT e n = E sin ( (3 n The next E.M.F. is again : *! = ^ sin 08 - 2 TT) = ^ sin ft. In the polar diagram the n E.M.Fs. of the symmetrical 7Z-phase system are represented by n equal vectors, follow- ing each other under equal angles. Since in symbolic writing, rotation by 1 / n of a period, or angle 2-n-/ n, is represented by multiplication with : n n the KM.Fs. of the symmetrical polyphase system are: 236] SYMMETRICAL POLYPHASE SYSTEMS. 351 20 \ 7T , . . Zi 7T \ 7-. , I cos . 4" / sin - \ = j& e ; n n n / 2 In 1) 7T . . . 2 ( 1) 7T\ rr n _l I cos ^ - 1 -- \-j sin ^ - 1 ] =E e n \ The next E.M.F. is again : E ( cos 2 TT +/ sin 2 TT) = E t n = E. Hence, it is 2,T - . 27T n /T = cos + J sm BBS vl. Or in other words : In a symmetrical ^-phase system any E.M.F. of the system is expressed by : where : e = 236. Substituting now for n different values, we get the different symmetrical polyphase systems, represented by /'*, 1 n/q- 2 7T . . 2 7T where, e = vl = cos -- \-j sm - . n n 1.) = 1 c = 1 ^ = E, the ordinary single-phase system. 2.) = 2 e = - 1 *E=a.nd - E. Since ^ is the return of E, n = 2 gives again the single-phase system. ON o 2 7T ..27T 3.) * = 3 c = cos - - + j sm - = o o 352 ALTERNATING-CURRENT PHENOMENA. [237 The three E.M.Fs. of the three-phase system are : Consequently the three-phase system is the lowest sym metrical polyphase system. 4.) // = 4, e = cos^- + /sin^- =j, * 2 = 1, C 3 =-/. The four E.M.Fs. of the four-phase system are : J The two rectangular components of this M.M.F. are ZiTTt i' =//cos n and /*" = // sin n Hence the M.M.F. of this coil can be expressed by the symbolic formula : fi = ;//V2 sin ft - cos + /smL . V n 1 \ n n J Thus the total or resultant M.M.F. of the n coils dis- placed under the n equal angles is : 1 or, expanded : jin 5r cos 2 - -+/ sin --cos 1~ \ /z 11 n n x ."" / 2 77- 2 z/ TT * . ... cos p ?j sin cos + i sm 4 Vv ; It is, however : f- j sin cos = i/l-[- cos - + / sin 354 ALTERNATING-CURRENT PHENOMENA. [237 ". 2iri 2?r/ . . . o27rz // H 4?r/ sm - cos -- hysm 2 - = [ 1 cos - - / sin n and, since: n n n 2 ;/ - 2 < = o, 1 1 as the sum of all the roots of Vl, it is, /= ; * ;/ ( V2 (sin /? + j cos )8). or, ,. n n r I , , n . . n . f= - (sm ft + j cos (3) v 2 = ^(sin/?+/cos/3); V2 the symbolic expression of the M.M.F. produced by the n circuits of the symmetrical /z-phase system, when exciting n equal magnetizing coils displaced in space under equal angles. The absolute value of this M.M.F. is : V2 ;v5 2 Hence constant and equal /V2 times the effect've M.M.F. of each coil or n/2 times the maximum M.M.F. of each coil. The phase of the resultant M.M.F. at the time repre- sented by the angle (3 is : tan o> cot ft ; That is, the M.M.F. produced by a symmetrical //-phase system revolves with constant intensity : ^, n^ ^ and constant speed, in synchronism with the frequency of the system ; and, if the reluctance of the magnetic circuit 238] SYMMETRICAL POLYPHASE SYSTEMS. 355 is constant, the magnetism revolves with constant intensity and constant speed also, at the point acted upon symmetri- cally by the n M.M.Fs. of the //-phase system. This is a characteristic'" feature of the symmetrical poly- phase system. jb 238. In the three-phase system, n = 3, F 1.5 ^ max where $ max is the maximum M.M.F. of each of the magne- tizing coils. In a symmetrical quarter-phase system, n = 4, F = 2 &max> where ^ max 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.F. 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, of an even number of phases, consisting of one-half of a symmetrical system, are called hemisym- metrical systems. 356 ALTERNATING-CURRENT PHENOMENA. [239 CHAPTER XXV. BALANCED AND UNBALANCED POLYPHASE SYSTEMS. 239. If an alternating E.M.F. : e = E V2 sin ft produces a current : i = 7 V2 sin OB- w), where o> is the angle of lag, the power is : / = ei = 2 Ssm ft sin (0 G) = S(cos A sin (2 /? w)), and the average value of power : P = JSfCQS w. Substituting this, the instantaneous value of power is found as : p = p(\. sin (2 ff -a)' I COS w 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 : p = et. If the angle of lag w = it is : / = P (I - sin 2 ft) hence the flow of power varies between zero and 2 P, where P is the average flow of energy or the effective power of the circuit. 24O] BALANCED POLYPHASE SYSTEMS. 357 If the current lags or leads the E.M.F. by angle the power varies between -- LA- and COSw'y COSo> 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 s> = 90, it is : / = El cos 2/3; that is, the effective power :/> = (), 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. 240. If in a polyphase system e ii e 2> 3? - - - = instantaneous values of E.M.F. ; *\> *2j *sj = instantaneous values of current pro- duced thereby ; the total flow of power in the system is : The average flow of power is : P = Eif-i cos ! -f 2 f 2 cos w 2 -f . . . . The polyphase system is called a balanced system, if the flow of energy : p = e^ + * 2 /2 + * 8 /8 -f . . . . 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. 358 AL TERN A TING-CURRENT PHENOMENA. [ 241 , 242 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. 241. 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. 242. 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 6l = E V2 sin ft and t\ = 7 V2 sin (ft - 240)) = 3 JZfcos ui = P, or constant. Hence the symmetrical three-phase system is a balanced system. 2.) Quarter-phase system : Let fl = E V2 sin ft t\ = 7 V2 sin (ft - A) ; e 2 = E V2 cosft 4 = 7 V2 cos (ft-&)> 243] BALANCED POLYPHASE SYSTEMS. 359 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 Ef(sm ft sin (ft w) + cos ft cos (ft o>)) = 2 EScos 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. ; /", = I V2 sin f ft w - - ) = current the instantaneous flow of power is : 1 2 El yTsin ( ft - *L] sm(ft-Z- 27r * 1 or / = n E I = P t or constant. 243. 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- 360 AL TERNA TING-CURRENT PHENOMENA. 244 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 320, 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. 244. In Figs. 167 to 174 are shown the E.M.Fs. as and currents as i in drawn lines, and the power as / in dotted lines, for : Fig. 167. Single-phase System on Non-inductiue Load. 244] BALANCED POLYPHASE SYSTEMS. 361 Fig. 168. Single-phase System on Inductive Load of 60 Lag. "7"^ >-*=- >-, />V'A / A \A Fig. 173. Inverted Three-phase System on Non-inductive Load. 245,246] BALANCED POLYPHASE SYSTEMS. 363 Fig. 174. Inverted Three-phase System on Inductive Load of 60 Lag. 245. 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. 246. 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 ,/J 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, 364 ALTERNATING-CURRENT PHENOMENA. [247 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 : sin (2 ff - & COS 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 : 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. 247. To derive the equation in rectangular coordinates we introduce a substitution which revolves the system of coordinates by an angle w/2, so as to make the symmetry axes of the power characteristic of the coordinate axes. / = V^T7 a , tan(0-AW. x hence, sin (2/3 - *) = 2 sin (ft - f^cos |> - 1 = \ / L ^ J substituted, or, expanded : the sextic equation of the power characteristic. Introducing : a = (1 + c) P = maximum value of power, b = (1 e) P = minimum value of power; 247] BALANCED POLYPHASE SYSTEMS. 365 it is P = ^-t > I A ' d ~T~ ' J&* hence, substituted, and expanded : <* + /) _ i { a ( the equation of the power characteristic, with the main power axes a and b, and the balance factor : b / a. It is thus : Single-phase non-inductive circuit : p = P (1 -j- sin 2 <), b = 0, a = 2P (*"+/) - ^ 2 + (* + j) 4 = 0, /* = 0. . Single-phase circuit, 60 lag : p = P (1 + 2 sin 2 <), ^ = - /> tf = + 3 /> Single-phase circuit, 90 lag \ p = E I sin 2 <, b = E I r a = + El Three-phase non-inductive circuit : / = P, b = 1, a = 1 * 2 +/ ^ 2 = : circle, b / a = + 1. Three-phase circuit, 60 lag : p = P, b = 1, # = 1 Quarter-phase non-inductive circuit :p = P, b = 1, # = **-+y - ^ 2 = : circle, bja = + 1. Quarter-phase circuit, 60 lag :/=/>, ^ = 1, # = 1 ^ 2 +/ P* = : circle. /a = + 1. 366 ALTERNATING-CURRENT PHENOMENA. [248 Inverted three-phase non-inductive circuit : Inverted three-phase circuit 60 lag :/ = P (\. -\- sin 2 <), = 0, a = 2P a and ^ are called the main power axes of the alternating- current system, and the ratio b j a is the balance factor of the system. Figs. 175 and 176. 248. 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- 248] BALANCED POLYPHASE SYSTEM. 867 spending to the Figs. 167, 168, 173, and 174 are shown in Figs. 175, 176, 177, and 178. Figs. 177 and 178. The balanced quarter-phase and three-phase systems give as polar characteristics concentric circles. 368 ALTERNATING-CURRENT PHENOMENA. [249,250 CHAPTER XXVI. POLYPHASE SYSTEMS. 249. 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. 250. In an interlinked polyphase system two ways exist of connecting apparatus into the system. 250] INTERLINKED POLYPHASE SYSTEMS. 369 1st. The star connection, represented diagram mat ically in Fig. 179. In this connection the n circuits excited by currents differ from each other by 1 / n of a period, and are connected with their one* end together into a neutral point or common connection, wftich 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. 163. 2d. The ring connection, represented diagrammatically in Fig. 180, 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. 164. In consequence hereof we distinguish between star- connected and ring-connected generators, motors, etc., or 370 ALTERNATING-CURRENT PHENOMENA. [251 ... 6 Fig. 180. in three-phase systems Y- connected and delta-connected apparatus. 251. 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/;z 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, Jhat is, between two lines. In general some of the apparatus will be arranged in star connection, some in ring connection, as the occasion may require. 252,253] INTERLINKED POLYPHASE SYSTEMS. 371 252. 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 poteijial, 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. The total power of the polyphase system is equal to the sum of all the star or Y powers, or to the sum of all the ring or delta powers. 253. 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 : e* E, at the z th terminal, where : * = 0, 1, 2. . . . n- 1, e = cos +/sin = VI 372 ALTERNATING-CURRENT PHENOMENA. [253 Hence the E.M.F. in the circuit from the z th to the th terminal is : E ki = JE - ?E = (e* - e) E. The E.M.F. between adjacent terminals i and i + 1 is : ( e ' + i _ t) E = e (e - 1) . In a generator with ring-connected circuits, the E.M.F. per circuit : is the ring E.M.F., and takes the place of while the E.M.F. between terminal and neutral point, or the star E.M.F., is : iff* Hence in a star-connected generator with the E.M.F. E per circuit, it is : Star E.M.F., j E. Ring E.M.F., e* (e - 1) E. E.M.F. between terminal i 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. ,lc _ fi E.M.F. between terminals i and /, - 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 : e* E are the E.M.Fs. between the n terminals and the neutral point, or star E.M.Fs., - 254] INTERLINKED POLYPHASE SYSTEMS. 373 /,- = 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 : JE - Z^. Difference of potential between terminals k and i : ( *-c*)- (Z k l k -Z,/,), where 7 2 - is the star current of the system, Z { the star im- pedance. The ring potential at the end of the line between ter- minals i and k is E ikt and it is : E ik = -E ki . If now I ik denotes the current passing from terminal i to terminal k, and Z ik impedance of the circuit between ter- minal i and terminal k, where : fit = ~ /, Z ik = Z ki , it is E ik = Z ik I ik . If I io denotes the current passing from terminal i to a ground or neutral point, and Z io is the impedance of this circuit between terminal i and neutral point, it is : 254. We have thus, by Ohm's law and KirchhorFs 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. I { = the current issuing from the terminal i of the gen- erator, or the star current. Z { = the impedance of the line connected to a terminal i of the generator, including generator impedance. E i = the E.M.F. at the end of line connected to a ter- minal i of the generator. 374 ALTERNATING-CURRENT PHENOMENA. [254 E ik = the difference of potential between the ends of the lines i and k. I ik = the current passing from line i to line k. Z ik = the impedance of the circuit between lines i and k. Iio> two - - - = the current passing from line's" to neu- tral points 0, 00, . . ., . Z io , Z ioo . . . . = the impedance of the circuits between line i and neutral points 0, 00, .... It is then : 1.) E ilc = MJ fit = fid, Z ilc = ZM, I io = / ot> Z io = Z oi , etc. 2.) Ei = SE-Z i f i . 3.) Ei = Z io l io = Z ioo l ioo = . . . . 4.) E ik = E k - E i = (e* - e<) E - (Z k l k - Z/). 5-) -Eik.~ 6.) I, = 7.) If the neutral point of the generator does not exist, as in ring connection, or is insulated from the other neutral points : = ; =0, etc, 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 : 254] INTERLINKED POLYPHASE SYSTEMS. 375. 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 :* Eli cos fa at the generator 1 n n = y* y* E ik I ik cos ijk in the receiving circuits. i 376 ALTERNATING-CURRENT PHENOMENA. [255,256 CHAPTER XXVII. TRANSFORMATION OF POLYPHASE SYSTEMS. 255. 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. 256. Let E lt E z , E z . . . . be the E.M.Fs. of the primary system which shall be transformed into EI, EI, ^ .... the E.M.Fs. of the secondary system. Choosing two magnetic fluxes, and ^, of different 256] TRANSFORMATION OF POLYPHASE SYSTEMS. 377 phases, as magnetic circuits of the two transformers, which induce the E.M.Fs., e and F, per turn, by the law of paral- lelogram the E.M.Fs., E l , E 2 , .... can be dissolved into two components, E l and E^ , E 2 and E 2 , . . . . of the phases, e and J. Then, - 1, 2 , . . . . are the counter E.M.Fs. which have to be induced in the primary polyphase circuits of the first transformer ; hence EI I e, E% 1 ~e . . . . are the numbers of turns of the primary coils of the first transformer. Analogously EI IT E z /J . . . . are the number of turns of the primary coils in the second transformer. In the same manner as the E.M.Fs. of the primary system have been resolved into components in phase with e and e, the E.M.Fs. of the secondary system, E^ 9 E z \ .... are produced from components, E^ and E^, E^- and E^ 1 , .... in phase with e and 7, and give as numbers of second- ary turns, EI / cos ft and cos (ft 90). Then the E.M.Fs. induced per turn in the transformers are sin ft and e sin (/3 - 90) ; hence, in the primary circuit the first phase, E sin ft will give, in the first transformer, Ej e primary turns; in the second transformer, primary turns. The second phase, E sin (ft 120), will give, in the first transformer, E / 2 e primary turns; in the second transformer, 2- - primary turns. The third phase, E sin (ft 240), will give, in the first transformer, E J % e primary turns; in the second trans- - E xV3 former, primary turns. 2 e In the secondary circuit the first phase E' sin ft will give in the first transformer: E' I e secondary turns; in the second transformer : secondary turns. The second phase : E' sin (ft 90) will give in the first transformer : secondary turns ; in the second transformer, E' I e secondary turns. Or, if : E = 5,000 E' = 100, e = 10. PRIMARY. SECONDARY. 1st. 2d. 3d. 1st. 2d. Phase. first transformer + 500 - 250 - 250 10 second transformer + 433 433 10 turns. 258] TRANSFORMATION OF POLYPHASE SYSTEMS. 379 That means : Any balanced polyphase system can be transformed by two transformers only, without, storage of energy, into any other balanced polyphase system. { 258. Transformation with a change of the balance factor of the system is possible only by means of apparatus 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 the secondary. The most efficient storing device of electric energy is mechanical momentum in revolving machinery. It has, however, the disadvantage of requiring attendance. 380 ALTERNATING-CURRENT PHENOMENA. [259 CHAPTER XXVIII. COPPER EFFICIENCY OF SYSTEMS. 259. 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. Comparing now the different systems, as two-wire single-phase, single-phase three-wire, three-phase and quar- ter-phase, as basis of comparison equality of the potential is used. Some systems, however, as for instance, the Edison- three-wire system, or the inverted three-phase system, have- 26OJ COPPER EFFICIENCY OF SYSTEMS. 381 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 ba^sis of tJie 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 wires or high differences of potential 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. 260. 1st. Comparison on the basis of equality of the minimum difference of potential, in low potential lighting circuits : 382 ALTERNATING-CURRENT PHENOMENA. [ 26O 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' 2 r. Using, however, a three-wire system, the potential be- tween outside wires and neutral being given = e, the potential between the outside wires is = 2 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, f of the copper of the single-phase system is needed ; if the neutral wire is ^ the cross-section of each of the outside wires, ^\ of the copper is needed. Obviously, a single- phase five-wire system will be a system of distribution at the potential 4 e, and therefore require only T V of the copper of the single-phase system in the outside wires ; and if each of the three neutral wires is of ^ the cross-section of the outside wires, / T == 10.93 per cent of the copper of a single- phase system. 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 = ^ / V3 ; and since three branches are used, the total power is 3 e ^ / V3 = e i^ VS. Hence if the same power has to be transmitted by the three-phase system as with the single-phase system, the three-phase line current must be ^ = i j VS ; hence if r^ = resistance of each of the three wires, the loss per wire is if r^ = z'Vj/3, and the total loss is i 2 r\ , while in the single-phase system it is 2 i\. Hence, to get the same loss, it must be : r^ = r t that is, each of the three three-phase lines has twice the resis- tance that is, half the copper 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. 26O] COPPER EFFICIENCY OF SYSTEMS. 383 Introducing, however, a fourth 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 V5, since the Y potential = e, and the potential of the system is raised thereby from e to e V3 ; that is, only \ 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 ^ 7 = 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 T 5 g = 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 / 2 , the current in the central wire is 2 2 V2 ; and if the same current density is chosen for all 384 ALTERNATING-CURRENT PHENOMENA. [ 26Q three wires, as the condition of maximum efficiency, and the resistance of each outside wire denoted by r 2 , the re- sistance of the central wire = r z /V2, anc * the loss of power per outside wire is if r 2 , in the central wire 2 z' 2 2 r 2 / V2 = ifr^ V2; hence the total loss of power is 2 z' 2 2 r 2 + z' 2 2 ; 2 V2 = z' 2 2 r 2 (2 -f V2). The power transmitted per branch is z' 2 e, hence the total power 2 z' 2 e. To transmit the same power as by a single-phase system of power, ei y it must be z 2 =7/2; hence the loss, * V 2< 2 + ^) . Since this loss shall be the same as the loss 2z' 2 r in the single- phase system, it must be 2 r = , or r~ = fcv r 2 + V2 9 _L A/^ Therefore each of the outside wires must be - - times 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 : >) | 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 t s in the three lines of equal resistance r 3 , gives the out- put 2 or, I, + 7 2 ' - /,' = O (2) These three equations (2) added, give (1) as dependent equation. At the ends of the lines 1, 2, 3, it is : V = ^ 2 _Z 3 / 3 + Z 1 / (3) ^ / = ^ 3 _ Zi/i + Z2/2 J the differences of potential, and ^ (4) the currents in the receiver circuits. These nine equations (2), (3), (4), determine the nine quantities : 7 15 7 2 , 7 3 , 7/, 7 2 ', 7 3 ', "/, ^ 2 ', ^ 3 '. Equations (4) substituted in (2) give : (5) These equations (5) substituted in (3), and transposed, give, since E l = e E 2 = E as E.M.Fs. at the generator terminals. e E - E^ (1 + FiZg + r^) + ^/ F 2 Z 8 + ^ 1", Z 2 = 0] #E - E, f (1 + F 2 Z 8 + r.ZO + ^/ FaZ! + ^/ Fj Z 3 = I (6) - ^/ (i + Y S Z, + r 3 z 2 ) + ^/ F^ + ^/ F 2 z x = o I 392 AL TERNA TING-CURRENT PHENOMENA. [263 as three linear equations with the three quantities \', F ' F ' J-^n ) *-'& Substituting the abbreviations : - (1 + FjZ, + PiZ 8 ), F 2 Z 3 , P1Z 8 , -(1 + FaZa + PiZ,), ^, KsZ!, - (1 + F 3 Z! + Pg FZ ., F 2 Z 1? - -(1 + F 3 Z 1 + F 3 Z 2 ) , Z 2 -f- FiZg), F 2 Z 3? e = it is: (8) D // = -/ Y. A hence, +/ 2 -f/a = =0 (10) (11) 264] THREE-PHASE SYSTEM. 264. SPECIAL CASES. A. Balanced System Substituting this in (6), and transposing : *E 3FZ 3FZ *'=* 1 + 3KZ EY 1 + 3FZ f 3FZ EY f 3 FZJ 1 + 3FZ _ (c-l)^F 1 4-3FZ 1+3FZ (12) The equations of the symmetrical balanced three-phase system. B. One circuit loaded, two unloaded: Substituted in equations (6) : e E - E{ + E{ FZ = unloaded branches. E - E z r (l + 2 FZ) = 0, loaded branch. hence : 2FZ 1 + 2FZ 1 + 2 unloaded ; loaded ; all three E.M.FS. (13) unequal. 394 AL TERN A TING-CURRENT PHENOMENA . [264 * Si 4 = /2 ';; ] 1 + 2FZ J EY \ 1 + 2 YZ EY = 1 + 2FZ (13) (13) C. Two circuits loaded, one unloaded, Y 1 = Y 2 = Y, Y 3 = 0, Z, = Z 2 = Z 3 = Z. Substituting this in equations (6), it is : e^ - E{ (1 + 2 YZ) + EJYZ = 01 . *E - El (1 + 2 YZ) + E{ YZ = o} loaded branches ' E - E{ + (E{ + ^/) FZ = unloaded branch. or, since : E - E - E' = thus: 1 +YZ 4 FZ + 3 F 2 Z 2 loaded branches. unloaded branch. - (W) As seen, with unsymmetrical distribution of load, all three branches become more or less unequal. 265] QUARTER-PHASE SYSTEM. 395 CHAPTER XXX. QTJAKTER-PHASE SYSTEM. 265. In a three-wire quarter-phase system, or quarter- phase system with common return wire of both phases, let the two outside terminals and 'wires be denoted by 1 and 2, the middle wire or common return by 0. It is then : EI = E = E.M.F. between and 1 in the generator. 2 = j E = E.M.F. between and 2 in the generator. Let: 7j and 7 2 currents in 1 and in 2, f = current in 0, Z l and Z 2 = impedances of lines 1 and 2, Z = impedance of line 0. Y l and Y 2 = admittances of circuits to 1, and to 2, // and 7 2 '= currents in circuits to 1, and to 2, E^ndE^ = potential differences at circuit to 1, and Oto2. it is then, 7 : + 7 a + 7 = ) ~, or, 7 = - (7, + 7 2 ) ) that is, f is common return of I and 7 2 . Further, let : *' =JE - 7 2 Z + 7 Z =JE - 7 2 (Z 2 + Z ) - and (3) 396 ALTERNATING-CURRENT PHENOMENA. [266 Substituting (3) in (2) ; and expanding : ' = _ 1 + F 2 Z 2 + F 2 Z (1 -y) F 2 z + F 2 z 2 ) - K.F^ (1 + F^ + PiZ,) (1 + F 2 Z + F 2 Z 2 ) - F! F 2 Z 2 Hence, the two E.M.Fs. at the end of the line are un- equal in magnitude, and not in quadrature any more. 266. SPECIAL CASES : A. Balanced System. Z ] = Z 2 = Z; Z = Z/V2; Y, = F 2 = F Substituting these values in (4), gives : = E 1 + V2 (1 + V2) YZ + (1 + V2) F 2 Z 2 1 + (1.707 - .707/) FZ 1 + 3.414 FZ + 2.414 F 2 Z 2 (5), 1 + V2 (1 + V2) FZ + (1 + V2) F 2 Z : 1 + (1.7Q7 + .7Q7y) Y Z 1 + 3.414 FZ + 2.414 F 2 Z 2 Hence, the balanced quarter-phase system with common return is unbalanced with regard to voltage and phase rela- tion, or in other words, even if in a quarter-phase system with common return both branches or phases are loaded equally, with a load of the same- phase displacement, nevertheless the system becomes unbalanced, and the two E.M.Fs. at the end of the line are neither equal in magnitude, nor in quadrature with each other. 266] QUARTER-PHASE SYSTEM. B. One branch loaded, one unloaded. 397 Y, = j), F 2 = F F x = F, F 2 = 0. Substituting these values in (4), gives : 1 + FZ 1 + V ^-y .1 = v ~ """ ~~ ^^~ *) V2 FZ V2 V2 1 - f~JJB = E\ 2.414 + 1^1 FZ 1 1-Fz =y^ 1 + V2 V2 1 1.707 FZ ?/ = ^ 1 + FZ 1 + V2 /=y^ 1 + 1.707 FZ + FZ i + V2+y V2 1 + FZ M^V2 V2 FZ 1 + = ^^ 2.414 -ft FZ (6) 398 ALTERNATING-CURRENT PHENOMENA. [266 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. 267. 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. 268. 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, 401 402 APPENDIX I. [269 that is, the terms of addition, or addenda, are interchange- able. Multiple addition of the same terms : a -f a + a + + a = c b equal numbers introduces the operation of multiplication : a X b = e. 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 : a X a X a X . . . X a = c b equal numbers introduces the operation of involution : a b = c Since a b is not equal to b a , the terms of involution are not interchangeable. 269. The reverse operation of addition introduces the operation of subtraction : If a + b = c, it is c b = a. This operation cannot be carried out in the system of absolute numbers, if : 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: - a = (- 1) X a, where ( 1) is the negative unit. Thereby the system of numbers is subdivided in the 270,271] COMPLEX IMAGIA T ARY QUANTITIES. 403 positive and negative numbers, and the operation of sub- traction possible for all values of subtrahend and minuend. or (- 1,) -X (- 1) = 1 ; that is," the negative unit TS defined by : 270. 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 the fraction : where b is the integer fraction, and is denned by : 271. The reverse operation of involution introduces two new operations, since in the involution : #*=c, the quantities a and b are not reversible. Thus ^ = a, the evolution. log a c = b, the logarithmation. The operation of evolution of terms c, which are not complete powers, makes a further expansion of the system 404 APPENDIX I. [272, 273 of numbers necessary, by the introduction of the irrational number (endless decimal fraction), as for instance : V2 = 1.414213. 272. The operation of evolution of negative quantities c with even exponents b, as for instance makes a further expansion of the system of numbers neces- sary, by the introduction of the imaginary unit. Thus where : V 1 is denoted by /. Thus, the imaginary unity is defined by : y 2 = - 1. By addition and subtraction of real and imaginary units, compound numbers are derived of the form : which are denoted as complex imaginary numbers. No further system of numbers is introduced by the operation of evolution. The operation of logarithmation introduces the irrational and imaginary and complex imaginary numbers also, but no further system of numbers. 273. 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, 274] COMPLEX IMAGINARY QUANTITIES. 405 Real and imaginary numbers and complex imaginary numbers. Therewith closes the Afield 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, arid 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. 274. Definition of imaginary unit: f 2 . = 1. Complex imaginary number: A = a + jb. Substituting : a = r cos ft b = r sin ft, it is A = r (cos ft -\- j sin ft), where r V# 2 -f b\ a r = vector, ft = amplitude of complex imaginary number A. Substituting : cos ft - 27 406 APPENDIX L [ 275 it is A = I \\n <*>_ -I where e = lim 1 + -} = ^ - - w= oo \ r nl o 1 X 2 X3 X . . . Xk is the basis of the natural logarithms. Conjugate numbers : a -f- j b = r (cos (3 -\- j sin /?) = reJ& and a -jb = r(cos [- /?] +7 sin [- ]) = r*-./* it is ( +jb) (a -jb) = a* + l>* = r 2 . Associate numbers : a+jb = r(cos/8+/sin)8) = ?V and * +/ = r (cos [| - )8] +/sin g -/?]) = r tf y it is (a + / b) (b +ja)= j (a 2 + 2 ) = y r a . If it is If it is a 0, 6 = 0. 275. Addition and Subtraction : Multiplication : or r (cos ft + j sin ft) X r' (cos /?' -f- j sin /3') = rr r (cos or r^ X r e = r Division : Expansion of complex imaginary fraction, for rationaliza- tion of denominator or numerator, by multiplication with the conjugate quantity : 276,27-7] COMPLEX IMAGINARY QUANTITIES. 40T a+jb (*+jfy( r , r'(cos(3 +j smfi') r' = r_ . ( ^_^ r ' ' r' involution : (a +jb)* = {r (cos ($ +J sin n n ) = evolution: IJa +jb = ^r (cos p+j sin p) = ^re^ = 3/r{ cos^ + /sin^ ) = tyreJ* \ 11 nj 276. Roots of the Unit : =l -1 U1 lj w) represent the oscillating wave. We have then dE tan 3 = sin (< o>) a cos ( w cos (< o>) -{tan (<_&) + }; 283] OSCILLATING CURREiVTS. 413 that is, while the slope .of the sine -wave, s = e cos (< ), is represented by tan y = tan (< w), the slope of the exponential spiral y = ^e~ a< Ms tan a = a = constant. That of the oscillating wave E = ec.~ a *> cos (< ) is tan j3 = {tan (< u>) -f 0}. 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 E = ? - C os (< to) = / (cos to +/ sin to) dec a = (h + / 4) dec a, where /\ = z cos to, / 2 = / sin to, # = tan a. We have then, The electromotive force consumed by the resistance r of the circuit J? r=r /deca. The electromotive force' consumed by the inductance L of the circuit, dt d$ d<$> Hence E x = xif.~ a ^ {sin (< to) -+- a cos ( o>)} . . . r / a (^ *^v I- C /. ^ \ = -- - sm ( to -f- a). COS a Thus, in symbolic expression, E*=* -- ^{ sin (w a) +y cos (w a)} dec a COS a = ^ / (a -\- j) (cos w -f j sin to) dec a ; that is, E^xI(a-\-j) dec a . Hence the apparent reactance of the oscillating current circuit is, in symbolic expression, X = x (a +/) dec a. Hence it contains an energy component ax, and the impedance is Z = (r X) dec a = {r x (a +_/)} dec a = (r a x jx) dec a. Capacity. 285. Let r = resistance, C= capacity, and x c = I/ 2 TT C = capacity reactance. In a circuit excited by the oscillating 286] OSCILLATING CURRENTS. 415 current /, the electromotive force consumed by the capacity C is or, by substitution, E x = x \ if.~ a ^ cos (< - = - z e~ a * {sin (< w) a cos (< >' !+<< -v * & Q> sin (< - - a) ; (1 + # 2 ) cos a hence, in symbolic expression, ' - 2 ( a 4~ /) ( cos & +/ sin ^ ( a -\- f) dec a. rf^ ( ~ a+J ' 286. We have then : In an oscillating current circuit of resistance r, induc- tive reactance x, and capacity reactance x c , with an expo- nential decrement a, the apparent impedance, in symbolic expression, is : = j r - a ( x + -^ ) -j(x - -^) } dec a, C V 1 + a- 1 V 1 + a- 1 ) 416 APPENDIX II. [287 and, absolute, Admittance. 287. Let /== / _ a cQs ^ _ ^ Then from the preceding discussion, the electromotive force consumed by resistance r, inductive reactance .r, and capa- city reactance x c , is E = ie-*4> | cos(<- - * *c sin < where tan 8 a r ax - - x . a x. substituting w + 8 for w, and ^ = iz a we- have = ee -"<> cos <> o>, = ~ a ^ COS (<^> W 3) ^ . cos 8 / , N . sin a } cos ( <^> o>) H hence in complex quantities, = e (cos o> -J- / sin 288] OSCILLA TING CURRENTS. 417 or, substituting, a' r ax -- x X 1 + a< 1-M 288. Thus in complex quantities, for oscillating cur- rents, we have : conductance, a r ax ' susceptance, admittance, in absolute values, ~ ~ in symbolic expression, 1 + a 2 1 + 1 + a * \ 1 + a Since the impedance is we have 418 APPENDIX II. [ 289, 29O that is, the same relations as in the complex quantities in alternating-current circuits, except that in the present case all the constants r af x a , s at g, 2, y, depend upon the dec- rement a. Circuits of Zero Impedance. 289. In an oscillating-current circuit of decrement a, of resistance r, inductive reactance x, and capacity reactance x c , the impedance was represented in symbolic expression by Z = r. -/* = (r -a X - ^ or numerically by = r - ax - -^ '+ ( X - Thus the inductive reactance x, as well as the capacity reactance x c , do not represent wattless electromotive forces as in an alternating-current circuit, but introduce energy components of negative sign a x 11 * - * 4- a 2 that means, 11 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." 290. 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 : = Q In this case we have 291] OSCILLATING CURRENTS. 419 substituting in this equation x = l-nNL x c = 3 : 2 and expanding, we have* C 2aZ.' That is, " If in an oscillating-current circuit, the decrement 1 and the frequency N = rj^traL, 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." 291. 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 a - 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 420 APPENDIX II. [292 those flowing in a circuit of zero impedance, representing oscillating discharges of the circuit. Lightning strokes usually belong to this class. Oscillating Discharges. 292. The condition of an oscillating discharge is Z = 0, that is, If r = 0, that is, in a circuit without resistance, we have <2 = 0, N = 1 / 2 TT VZ C ; that is, the currents are alter- nating with no decrement, and the frequency is that of resonance. If 4 H r* C - 1 < 0, that is, r > 2 VZyT, 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 = 0. At this moment the 1 current is zero ; that is, Z=// 2 , t\ = 0. Since E xc = xl( a -f /) dec a e at = 0, e we have x 4 Vl + a 2 = e or 4 = - x Vl + a Substituting this, we have, /> I = j dec a, E r -=je x Vl + a? E x = =-(1 ja) dec a, E xc = Vl + a* Vl + a the equations of the oscillating discharge of a condenser of initial voltage e. Since x = 2 TT N Z, a = 2aL we have * - JL - r -i / 4Z _ 1 2 * " 2 V r 2 C hence, by substitution, 1C 1C 1 =J ' e \'J / deca ' E r=Je r \J-j- dec a, E***^ 422 APPENDIX II. [294 a = , = > r v /_^_i v/ V^C the final equations of the oscillating discharge, in symbolic expression. - Oscillating Current Transformer. 294. 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,. r l = r -\- 7-j", where 1\ = external, 1\' internal, resistance, The total inductance L l = L^ -f L^', where Z/ = external,. LI' internal, inductance ; total capacity, C v Then the total admittance of the secondary circuit is >.) dec a = where x 1 = 2?rA r Z 1 = inductive reactance: x cl = 1/2 F dec a. E ' E' = - dec a = induced primary electromotive force. / Hence the total primary electromotive force is =(' + IZ) dec a = ^L {1 + Z F +/ 2 Z KJ dec a. * P In an oscillating discharge the total primary electro- motive force E = ; that is, or, the substitution r a x -- X(.\ / x o. 424 APPENDIX II. [295 Substituting in this equation, x=%irN ' C, ;r c = l/2ir JVC, 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. 295. If the exciting current of the transformer is neg- ligible, that is, if Y Q = 0, the equation becomes essentially simplified, a \ -I *c \ a x -- x c \ / \x -- * !+*>') '\ 1+^_Q. - * - ITP*" ) ~ j (*> - r??) that is, or, combined, (r l -2ax 1 )+^(r-2ax) = 0, = 2* Substituting for x l , x, x c l , x c , we have a = c , 1 2 (A irjffQ (r 1 +/V) 2 (C 1 +/ 2 C) ;} dec a, _-, P I =pES Y l dec a, 7i = ^/ Fj dec a, the equations of the oscillating-current transformer, with [ as parameter. INDEX. PAGE . 130 . 123 52, 53 . 122 . 57 Advance Angle, of eddy currents hysteretic, of ironclad circuit Admittance, Chap. vn. . . . absolute, of ironclad circuit . combination in parallel . . combination in series 57 primary, of transformer .... 178 Air-gap in magnetic circuit .... 124 in magnetic circuit, in its action on the hysteretic distortion . . Ill Algebra of complex imaginary quan- tities, Appendix 1 401 Alternating wave . ' 11 Alternators, synchronizing, Chap. xvii 248 Ambiguity of vectors . . . . . .43 Amplitude or maximum value of wave, 7 Analytic method of treating synchro- nous motor 275 Apparent impedance of transformer . 182 Approximations of line capacity, 152, 153 Arc, power factor of 331 pulsating, resistance of .... 329 Armature resistance of induction mo- tor, effect on starting torque . 226 Asynchronous motor, see induction motor. Average value of wave 11 Balanced polyphase system, Chap. xxv. . . . 356, 357, 347 quarter-phase system, equations . 396 three-phase system, equations . . 393 Balance factor of polyphase system . 357 unchanged by transformation of polyphase system 376 Balance of polyphase system, in topo- graphic representation ... 46 Capacity 5 effect on higher harmonics . . .337 PAGE Capacity load, effect on alternator field characteristic 239 of line, approximation . . 152, 153- Characteristic, field, of alternator . . 240 magnetic, true and apparent . . 11& wave-shapes 336 Charging current of transmission lines, magnitude 151 Coefficient, of eddy currents . . 133, 131 of hysteresis 116 Combination of admittances in par- allel 57 of conductances 52 of impedances in series .... 53 of resistances 52" of sine waves, by parallelogram or polygon on vectors .... 21, 22 of sine waves in symbolic expres- sion 38 Commutator motors, Chap. xix. . . 291 Commutator, sparking at, in series and shunt motor 30& Comparison of systems regarding cop- per efficiency, Chap, xxvni . 380 Compensation of line drop by shunted susceptance 97 for lagging currents by shunted condensance 72 Complete resonance in transmission lines 340 Complex harmonic wave 9 Complex imaginary quantities, Ap- pendix 1 401 Condensance or capacity reactance . 5 reactance for constant potential, constant current transforma- tion 76- shunted, compensation for lagging current 72 shunted, control of receiver voltage in transmission lines .... 96 425 426 INDEX. Condensance, symbolic or complex imaginary representation . . 40 Condenser current of transmission lines, magnitude 151 Conductance, Chap, vn 52, 53 combination of- . . 52 effective 104 of magnetic hysteresis . . . 119 of mutual inductance .... 142 equivalent, of eddy currents . . . 130 of receiver circuit supplied over in- ductive line 89 Conductor, eddy currents in .... 138 Constant current alternator .... 246 Constant current, constant potential transformation 76 Constant potential regulation, of al- ternator 245 Constants, electric, of magnetic circuit containing iron 125 of alternating-current transformer, 190 Control of receiver potential in trans- mission lines by shunted sus- ceptance 97 Converter, see transformer. Converter motor, see induction motor. Coordinates, rectangular and polar . 19 Copper efficiency of polyphase sys- tems, Chap, xxvin 380 Counter E.M.F. and component of impressed E.M.F 24 Creeping, magnetic 114 Cross currents, between synchronizing alternators 249 Cross flux, magnetic, of transformer, 167, 194 Current, minimum, at given power, in synchronous motor 282 Cycles, magnetic, of hysteresis . . . 107 Delta connection of three-phase sys- tem 369 current of three-phase system . . 371 potential of three-phase system . 371 Demagnetizing effect of eddy currents, 136 Dielectric hysteresis 144, 105 Dielectric hysteretic admittance, im- pedance, etc 145 Dielectric hysteretic lag angle . . . 145 Displacement of phase, maximum, in synchronous motor . . . . 284 Disruptive phenomena 147 Distortion of wave-shape and eddy currents, Chap, xxi 320 Distortion of wave-shape by arc . . 329 causing decrease of hysteresis . . 343 causing increase of hysteresis . . 343 in reaction machines 312 of current wave by hysteresis . . 107 by quintuple harmonic . . . 335 by triple harmonic 335 Distributed capacity, approximation, 152, 153 complete investigation .... 155 instance and particular cases . . 162 Divided circuit, equivalent to alternat- ing-current transformer . . . 183 Double transformation 186 Eddy currents, Chap. xi. . . 129, 106 not affected by wave-shape . . . 344 Effect, see power, and output. Effective resistance and reactance, Chap, x 104 resistance, reactance, conductance, susceptance 104 resistance 2, 6 value of wave 11, 14 Efficiency of constant potential, con- stant current transformation by resistance reactance .... 81 Electro-magnetic induction, law of, Chap, in 16 Electro-magnetism, as source of re- actance Electrostatic hysteresis, see dielectric hysteresis. induction, or influence 147 Energy components of current and of E.M.F 148 component of reactance .... 309 equation of alternating currents . 23 Equations, of alternator 238 of general alternating-current trans- former . 195 Equivalent sine wave . . 320,111,115 Exciting current of transformer . . 169 Field characteristic of alternator . 240 Flow of power in single-phase system, 356 in polyphase system - . . 357, 360 Flux of alternating-current generator, 234 INDEX. 427 Flux, pulsation of, in alternating-cur- rent generator 234 Foucault currents, Chap, xi . . 129, 106 Fourier series, univalent function f>f time 7 Four-phase, see quarter-phase. Four-wire quarter-phase system, see independent quarter-phase sys- tem. Frequency of wave 7 Frequency setter, with induction gen- erator 230 Friction, molecular 105 Fundamental wave, or first harmonic, 8 General alternating-current trans- former. Chap. xiv. . . 193, 195 Generator action of general alternat- ing-current transformer . . . 203 Generator, alternating-current, Chap, xvi 234 induction 229 on unbalanced load, in topographic representation 48 polar diagram on non-inductive, inductive, and capacity circuit, 28 quarter-phase, in topographic rep- resentation 49 three-phase, on balanced load, in topographic representation . 46 Graphic method, disadvantage for nu- merical calculation 33 of induction motor 213 of synchronous motor 258 Graphic representation, Chap. iv. . . 19 Harmonics, even 8 first 8 higher, effect of, Chap. xxn. . . 334 causing resonance in transmis- sion lines 338 effect of self-induction and ca- pacity 338 of distorted wave Ill or overtones 8 Hedgehog transformer 169 Huntrng of parallel running alterna- tors 249 Hysteresis, coefficient of 116 decrease due to distorted wave . . 343 dielectric 105, 144 PACK Hysteresis, effect of, on electric cir- cuit, conclusions 127 increase due to distorted wave . . 343 magnetic .' . 105, 106 motor 312 Hysteretic advance of phase .... 115 energy current . , 115 energy current of transformer . . 170 loss, curve of 114 Imaginary quantities, complex, Ap- pendix 1 401 Impedance 2, 5 apparent, of transformer .... 182 combination in parallel .... 57 combination in series 53 in series to circuit 68 in symbolic or complex imaginary representation 39 internal, of transformer .... 178 Increase of output in transmission lines, by shunted susceptance . 100 Independent polyphase system . . . 347 quarter-phase system 398 Induced E.M.F., derivation of formula, 16 Inductance, introduction and deriva- tion of formula 18 mutual and self 142 Induction, electrostatic, or influence . 146 generator 229 motor as special case of general al- ternating-current transformer, 200,207 motor, Chap xv 207 mojtor, graphic method .... 213 motor, numerical instance . . . 230 motor, symbolic method .... 208 Inductive line, maximum power sup- plied over 87 feeding non-inductive, inductive and capacity circuit, graphic, 23, 26, 27 maximum efficiency of transmitted power 93 supplying non-inductive receiver circuit 84 symbolic method 40 Inductive load, effect on alternator field characteristic 239 Influence or electrostatic induction . 146 Instantaneous values of wave, Chap. II. 11 428 INDEX. Insulation strain in high potential line, due to distorted wave-shape . 344 Integral values of wave, Chap. n. . . 11 Internal impedance of transformer . 178 Intensity of wave 20 Interlinked polyphase systems, Chap, xxvi 368, 347 quarter-phase system 395 Inverted three-phase system, as unbal- anced system 359 flow of power 363 Ironclad inductance 124 magnetic circuit 106 Iron wire 133, 135 J, introduction as a symbol .... 36 definition as \/ 1 37 Joule's Law of alternating circuits . 6 of continuous currents .... 1 Kirchhoff' s Laws of alternating cir- cuits 6, 22 in symbolic or complex imaginary representation 40 of continuous current circuits . . 1 Lag angle of dielectric hysteresis . . 145 demagnetizing alternator and mag- netizing synchronous motor . 235 Laminated iron ... . . . 131,135 Lead, magnetizing alternator and de- magnetizing synchronous motor 235 Leakage 106 current of transformer .... 169 Light running of synchronous motor, 280 Line capacity, approximation . 152, 153 complete investigation .... 155 Line, inductive, feeding non-inductive, inductive and capacity circuit, graphic 23, 26, 27 load characteristic of ..... 95 symbolic method 41 Load characteristic of transmission line *..... 95 Loop, hysteretic 107 Magnetic hysteresis . . . 105, 106 Magnetizing current 115 of transformer 170 Magneto-motive force of alternating- current generator 234 resultant of symmetrical polyphase system 352 Maximum output, or power, of alter- nator 247 of induction motor 222 of synchronous motor 279 supplied over inductive line . . 87 Maximum rise of potential in receiver circuit fed by inductive line . 101 torque of induction motor . . . 219 value of wave 11 Mean value of wave 12 Mechanical power of induction motor, 216, 218 of general alternating-current trans- former . * 201, 202* Minimum current at given power in synchronous motor .... 282 Molecular friction 105 Monocyclic systems 363 power characteristic 363 Monophase, see single-phase. Motor action of general alternating- current transformer .... 203 commutator, Chap, xix 291 repulsion 291 series 300 shunt 305 synchronous, Chap. xvm. . . . 258 Mutual flux of transformer .... 195 inductance 142" induction of transformers . . . 167 Non-inductive load, effect on alterna- tor field characteristic . . .239 receiver circuit supplied over in- ductive line 84 Non-synchronous Motor, see induc- tion motor. Ohm's Law of alternating-current circuit 2, 5 of continuous currents .... 1 Oscillating currents and E.M.Fs., Appendix II 409 Output of general alternating-current transformer 201, 202: increase of, in transmission lines by shunted susceptance . . . 100' maximum of alternator . . . . 247 maximum of non-inductive receiver circuit supplied over inductive line 85 maximum of synchronous motor . 279- INDEX. 429 Output of receiver circuit supplied over inductive line .... 87 Overtones or higher harmonics ... 8 see power , 21 Parallelogram of vectors waves Parallel operation of alternators, Chap, xvn ........ 248 Period of a wave ... t ... 7 Phase of a wave ...... 7, 20 Phase displacement, maximum, in synchronous motor ..... 284 Phase relation of current in alternator, reacting on field ..... 235 Polar coordinates ....... 19 Polar diagram of transformer . . . 170 Polarization, equivalent to capacity, 5, 147 Polycyclic systems . . .. . . : . . 363 power characteristic ..... 363 Polygon of sine waves ..... 22 Polyphase systems, general, Chap, xxiii ....... 346 interlinked, Chap. xxvi. . . . 368 power characteristic ..... 363 symmetrical, Chap. xxiv. . . . 350 balance factor of ..... 357 transformation of, Chap. xxvu. . 376 Power axes of alternating-current sys- tem ....... . . 366 characteristics of alternating-cur- rent systems .... 363, 364 characteristics of synchronous mo- tor .......... 278 equation of alternating circuits . 6 equation of continuous currents . 1 factor of arc ........ 331 factor of series motor .... 307 flow of, in single-phase system . . 356 in polyphase system . . 357, 360 of complex harmonic wave . . . 341 of general alternating-current trans- former ...... 201, 202 of induction motor . . . 217, 219 maximum, of alternator .... 247 maximum, of induction motor . . 222 of repulsion motor ..... 298 of synchronous motor .... 303 see output. Primary admittance of transformer . 178 Pulsating wave ........ 11 Pulsation of flux of alternate-current generator 234 Pulsation of magnetic flux distorting the wave-shape 324 of reactance distorting the wave- shape 328 of reluctance distorting the wave- shape 327 of resistance distorting the wave- shape 329 Quarter-phase system, Chap. xxx. 395, 348, 352 as balanced system 358 copper efficiency 380 equations 395 flow of power . \ . . ^ . . 361 resultant M.M.F. of .,'... 355 tcansformation to three-phase . . 378 with common return, unbalancing, 50 Quintuple harmonic, distortion of wave-shape by 335 Reactance, capacity, or condensance, 5 and resistance of transmission lines, Chap. ix. ... .;. ... . . . 83 as component of impedance . . 2 capacity in symbolic representation 40 condensance for constant potential, constant current transformation, 76 effective, Chap. x. . .'. . . .104 in series to circuit . . . ". . .61 in symbolic representation ... 38 introduction and derivation of for- mula 17 magnetic 4 synchronous, of alternator . . . 238 synchronous, of synchronous mo- tor 290 variable in reaction machines, 309, 310 Reaction of lead or lag in alternator and synchronous motor . . . 236 machines, Chap, xx 308 machines, distortion of wave-shape, 312 machines, equations 315 Reactive component of pulsating re- sistance 329 Rectangular coordinates 19 of vectors in graphic representation by polar diagram 35 Reduction of transformer to divided circuit . . 182 430 INDEX. Regulation of alternator for constant potential 245 Reluctance of magnetic circuit of alter- nator 234 variable in reaction machines, 309, 310 Repulsion motor 291 equations 296 power of 298 Resistance, combination of .... 52 and reactance of transmission lines, Chap, ix 83 as component of impedance ... 2 effective, Chap, x 104, 2, 6 in series to circuit 58 in symbolic representation ... 38 of induction motor armature, af- fecting starting torque . . . 226 Resolving sine waves in symbolic ex- pression 38 Resonance 65 by higher harmonics 115 by higher harmonics in transmis- sion lines 338 complete, in transmission lines . 340 in transmission lines 101 Resultant magnetic flux of alternator, 234 Ring connection of interlinked poly- phase system 368 of polyphase system 347 Ring current of interlinked polyphase system 371 Ring potential of interlinked poly- phase system 371 Rise of potential, maximum, in re- ceiver circuit fed by inductive line 101 Rotation of vectors 36 Ruhmkorff coil, wave-shape .... 9 Screening effect of eddy currents in iron 129 of eddy current 136 of electric conductor 138 Secondary load on its action on hys- teretic distortion Ill Self-inductance 3 and mutual inductance .... 142 of alternator 237 Self-induction, coefficient of, or self- inductance 3 effect of higher harmonics . . . 337 PACK Self-induction, E.M.F. of 3. of transformer 167, 194 Series motor 300 equations of 301 phase displacement 303 power ' 303 Series operation of alternators . . . 250 Sheet iron 131, 135 Shunted condensance, compensation for lagging current 72 Shunt motor 300, 305 Sine wave, equivalent .... Ill, 115 as simplest wave 6 in polar coordinates 20 Single-phase system 351 copper efficiency 380 flow of power 356, 360 power characteristic 363 Slip of general alternating-current transformer 195 in induction motor 207 Sparking in series and shunt motors . 306 Star connection of interlinked poly- phase system 368 of polyphase system 347 Star current of interlinked polyphase system 371 Star potential of interlinked poly- phase system 371 Starting resistance of induction motor, 227 torque of induction motor . . . 224 Susceptance, Chap, vn 52, 53 effective 104 effective of mutual inductance . . 143 of receiver circuit supplied over inductive line, as varying out- put 88 shunted, controlling receiver volt- age in transmission line ... 96 Symbolic method, Chap, v 33 applied to transformer .... 178 representation 208 Symmetrical polyphase system, Chap, xxiv 350, 346 Synchronizing alternators, Chap, xvn 248 as condensance or inductance . . 147 Synchronizing power of alternators, 250, 254, 256 Synchronism of induction motor . . 228 running near, of induction motor, 228 INDEX. 431 Synchronous motor, Chap. xvm. . . 258 analytic method 275 discussion of results 288 fundamental equations . . . *. 276 graphic method JB8 maximum output . . * . . . . 1:79 power characteristic 278 Synchronous reactance of alternator . 238 of synchronous motor .... 290 Three-phase generator on balanced load .- . . . 46 on unbalanced load 48 Three-phase system, Chap, xxix., 390, 348, 351 as balanced system 358 copper efficiency 380 flow of power . . . . . . .362 inverted, as unbalanced system . 359 flow of power 352 resultant M.M.F 355 topographic representation ... 44 transformation to quarter-phase . 378 Three - phase system, unbalanced, 'equations 390 Three-wire system, copper efficiency . 381 Time-constant 3 Torque of induction motor .... 216 maximum, of induction motor . . 219 starting, of induction motor . . 224 Transformation, constant potential, constant current 76 double. ... - N v - 186 from three-phase to quarter-phase and inversely 378 of polyphase systems, Chap, xxvu., 376 Transformer, Chap, xm 167 constants on non-inductive load . 190 equations 195 equivalent to divided circuit, 183, 184 general alternating-current, Chap, xiv 193 magnetic cross-flux 194 mo'tor, see induction motor. numerical instance 191 phase difference in secondary and in primary 31 polar diagram .... 170, 28, 31 stationary, as special case of gen- eral alternating-current trans- former . 199 PAGB Transformer, symbolic method . . 178 Transmission line, load characteristic, 95 resistance and reactance, Chap, ix., 83 resonance by higher harmonics . 338 Trigonometric calculation and graphic representation 34 Triple harmonic, distortion of wave- shape 334 Two-phase, see quarter-phase. Unbalanced polyphase system, Chap, xxv 356, 347 quarter-phase system, equations . 396 three-phase system, equations, 390, 393 Unbalancing of quarter-phase system with common return . . . . 5"0 of three-phase generator or system on unequal load 48 Unequal current distribution in con- ductor 139 Uniformity of speed as condition of parallel operation of alternators, 248 Univalent function of time and Fou- rier series 7 Unsymmetrical polyphase system. .346 Variable reactance and reluctance in alternators 309, 310 Variation of receiver potential in transmission lines by shunted susceptance 96 Vectors 19 ambiguity of 43 Wattless component of current and of E.M.F 148 Wave-shape, characteristic .... 336 distortion and its caus.es, Chap, xxi 320 distortion by arc 329 distortion by quintuple harmonics, 335 distortion by triple harmonics . . 335 distortion causing decrease of hys- teresis 343 distortion causing increase of hys- teresis ........ 343 distortion in reaction machines . 312 Wire, iron 133, 135 Y connection of three-phase system . 369 Y current of three-phase system . . 371 Y potential of three-phase system . . 371 THIRD EDITION. . GREA TL Y ENLARGED. A DICTIONARY OF Electrical Words, Terms, and Phrases. BY EDWIN J. HOUSTON, PH.D. (Princeton}, AUTHOR OF Advanced Primers of Electricity / Electricity One Hundred Years Ago a/id To-day, etc., etc., etc. Cloth. 667 large octavo pages, 582 Illustrations. Price, $5.00. Some idea of the scope of this important work and of the immense amount of labor involved in it, may be formed when it is stated that it contains defini- tions of about 6000 distinct words, terms, or phrases. The dictionary is not a mere word-book ; the words, terms, and phrases are invariably followed by a short, concise definition, giving the sense in which they are correctly employed, and a general statement of the principles of electrical science on which the defi- nition is founded. Each of the great classes or divisions of electrical investiga- tion or utilization comes under careful and exhaustive treatment ; and while close attention is given to the more settled and hackneyed phraseology of the older branches of work, the newer words and the novel departments they belong to are not less thoroughly handled. Every source of information has been re- ferred to, and while libraries have been ransacked, the note-book of the labora- tory and the catalogue of the wareroom have not been forgotten or neglected. So far has the work been carried in respect to the policy of inclusion that the book has been brought down to date by means of an appendix, in which are placed the very newest words, as well as many whose rareness of use had con- signed them to obscurity and oblivion. As one feature, an elaborate system of cross-references has been adopted, so that it is as easy to find the definitions as the words, and atiases are readily detected and traced. The typography is ex- cellent, being large and bold, and so arranged that each word catches the ej-e at a glance by standing out in sharp relief from the page. Copies of this or any other electrical book published will be sent by mail, POST- AGE PREPAID, to any address in the world, on receipt of price. The W. J. Johnston Company, Publishers, 253 BROADWAY, NEW YORK. Elementary Electro=Technical Series. BY EDWIN J. HOUSTON, PH.D., AND A. E. KENNELLY, Sc.D t Alternating Electric Currents. Electric Incandescent Lighting^ Electric Heating. Electric Motor. Electromagnetism. Electric Street Railways. Electricity in Electro=Therapeutics. Electric Telephony. Electric Arc Lighting. Electric Telegraphy. Cloth. Price per volume, $1.00. The publication of this series of elementary electro-technical treatises on applied electricity has been undertaken to meet a demand which is believed to exist on the part of the public and others for reliable informa- tion regarding such matters in electricity as cannot be readily understood by those not specially trained in electro-technics. The general public, students of elementary electricity and the many interested in the subject from a financial or other indirect connection, as well as electricians desiring information in other branches than their own, will find in these works precise and authoritative statements concerning the several branches of applied electrical science of which the separate volumes treat. The repu- tation of the authors and their recognized abilities as writers, are a sufficient guarantee for the accuracy and reliability of the statements con- tained. The entire issue, though published in a series of ten volumes, is nevertheless so prepared that each book is complete in itself and can be understood independently of the others. The volumes are profusely illus- trated, printed on a superior quality of paper, and handsomely bound in covers of a special design. Copies of this or any other electrical book published will be sent by mail, POSTAGE PREPAID, to any address in the world on receipt of price. The W. J. Johnston Company, Publishers, 253 BROADWAY, NEW YORK. Electrical Power Transmission. -By LofolS BELL, Pn.D. Uniform in size with " The Electric Hailway in Theory and Practice. " Price, $2.5O. The plan of the work is similar to that of " The Electric Railway in Theory and Practice," the treatment of the subject being non-mathematical and not involving on the part of the reader a knowledge of the purely scientific theories relating to electrical currents. The book is essentially practical in its character, and while primarily an engineering treatise, is also intended for the information of those interested in electrical trans- mission of power, financially or in a general way. The author has a practical acquaintance with the problems met with in the electrical trans- mission of energy from his connection with many of the most important installations yet made in America, and in these pages the subject is devel- oped for the first time with respect to its practical aspects as met with in actual work. The first two chapters review the fundamental principles relating to the generation and distribution of electrical energy, and in the three succeeding ones their methods of application with both continuous and alternating currents are described. The sixth chapter gives a general discussion of the methods of transformation, the various considerations applying to converters and rotary transformers being developed and these apparatus described. In the chapter on prime movers various forms of water-wheels, gas and steam engines are discussed with respect to their applicability to the purpose in view, and in the chapter on hydraulic development the limitations that decide the commercial availability 1 of water power for electrical transmission of power are pointed out in de- tail. The five succeeding chapters deal with practical design and with construction work the power-house, line, and centres of distribution being taken up in turn. The chapter on the latter subject will be found of par- ticular value, as it treats for the first time in a thorough and practical manner one of the most difficult points in electrical transmission. The chapter on commercial data contains the first information given as to costs, and will, therefore, be much appreciated by engineers and others in decid- ing as to the commercial practicability of proposed transmission projects. This is the first work covering the entire ground of the electrical trans- mission of power that has been written by an engineer of wide practical experience in all of the details included in the subject, and thus forms a valuable and much-needed addition to electrical engineering literature. Copies of this or any other electrical book published will be sent by mail, POSTAGE PREPAID, to any address in the world, on receipt of price. The W. J. Johnston Company, Publishers, 253 BROADWAY, NEW YORK. THIRD EDITION. EXTENSIVELY REVISED AND ENLARGED. THE ELECTRIC RAILWAY IN THEORY AND PRACTICE. BY O. T. CROSBY AND DR. LOUIS BELL. Large Octavo. Profusely Illustrated. Price, $2.5O. Few technical books have met with so wide an appreciation as "The Elec- tric Railway," which has had the distinction of being accepted throughout the world as the standard authority on the subject of which it treats. The advances in electric traction made since the second edition of the work have been so notable, that the authors, in undertaking the preparation of a new edition, found it necessary to practically rewrite the book, so that the present edition conforms to the very latest knowledge on the subject, both in the domain of theory and of practice. The original purpose Of the book has', however, been strictly adhered to namely, to place before those interested in electric rail- ways, whether in a technical, financial or general way, the principles upon which electric traction is founded and the standard methods employed in their appli- cation. In view of the probable application in the near future of altejnating currents to electric traction, the present edition includes their consideration in this relation, thus largely extending the value of the treatise. The recent developments in electric locomotives and high-speed electric traction, and the application of electricity to elevated railways and to passenger traffic on steam roads, are in this work considered for the first time connectedly with reference to their engineering and commercial aspects. In the first section of the book are developed the fundamental principles of electricity upon which the apparatus and methods of operation are based. The following section is devoted to prime movers, steam, hydraulic and gas the modern gas engine here receiving the full treatment which its growing importance calls for. The remainder of the work is devoted to the engineering, practical and commercial features of electric traction, all of the factors that enter being considered from the standpoint of the best practice, and the more important ones elaborated in detail. The plan of the book, in fact, includes the consideration of everything relating to the electrical and mechanical principles and details which enter into electric railway design, construction and operation, the whole being treated from the engineering and commercial standpoint, and without the use of mathematics or resort to purely scientific theory. Copies of this or any other electrical book published will be sent by mail, POSTAGE PREPAID, to any address in the world on receipt of price. The W. J. Johnston Company, Publishers, 253 BROADWAY, NEW YORK. PRACTICAL CALCULATION * OF Dynamo=Electric Machines. A MANUAL FOR ELECTRICAL AND MECHANICAL EN- GINEERS, AND A TEXT-BOOK FOR STUDENTS OF ELECTRO-TECHNICS. BY ALFRED E. WIENER, E.E., M.E., Member of the American Institute of Electrical Engineers. Cloth, Illustrated. Price, $2.50. Based upon the practical data and tests of nearly two hundred of the best modern dynamos, including the machines used at the recent World's Fair and those in the largest and most modern central-stations, a complete and entirely practical method of dynamo-calculations is developed. Differ- ing from the usual text-book methods, in which the application of the vari- ous formulas requires more or less experience in dynamo-design, the present treatise gives such practical information in the form of original tables and formulas derived from the result of practical machines of American as well as European make, comprising all the usual types of field-magnets and armatures, and ranging through all commercial sizes. The book contains nearly a hundred of such tables, giving the values of the various constants, etc., which enter into the formulas of dynamo-design, and for all capacities, from one-tenth to 2000 kilowatts, for high and slow speed, for bipolar and multipolar fields, and for smooth and toothed drum and ring armatures. Although intended as a text-book for students and a manual for practical dynamo-designers, any one possessing but a fundamental knowledge of algebra will be able to apply the information contained in the book to the calculation and design of any kind of a continuous-current dynamo. Copies of this or any other electrical book published will be sent by mail, POSTAGE PREPAID, to any address in the world on receipt of the price. The W. J. Johnston Company, Publishers, 253 BROADWAY. NEW YORK. THIRD ED'ITION. ALTERNATING CURRENTS. An Analytical and Graphical Treatment for Students and Engineers. BY FREDERICK BEDELL, PH.D., AND A. C. CREHORE, PH.D. Cloth. 325 pages, 112 Illustrations. Price, $2.5O. The present work is the first book that treats the subject of alternating cur- rents in a connected, logical, and complete manner. The principles are gradu- ally and logically developed from the elementary experiments upon which they are based, and in a manner so clear and simple as to make the book easily read by any one having even a limited knowledge of the mathematics involved. By this method the student becomes familiar with every step of the process of development, and the mysteries usually associated with the theory of alternat- ing currents are found to be rather the result of unsatistactory treatment than due to any inherent difficulty. The first fourteen chapters contain the analytical development, commencing with circuits containing resistance and self-induc- tion only, resistance and capacity only, and proceeding to more complex cir- cuits containing resistance, self-induction, and capacity, and resistance and dis- tributed capacity. A feature is the numerical calculations given as illustrations. The remaining chapters are devoted to the graphical consideration of the same subjects, enabling a reader with little mathematical knowledge to follow the* authors, and with extensions to cases that are better treated by the graphical than by the analytical method. CONTENTS. Part I. Analytical Treatment. Chapter I. Introductory to Circuits Containing Re- sistance and Self-induction. Chapter II. On Harmonic Functions. Chap* ^r III. Circuits Containing Resistance and Self-induction. Chapter IV. Introductory to Circuits Containing Resistance and Capacity. Chapter V. Circuits Containing Resistance and Capacity. Chap- ters VI, VII, VIII, IX, X, XI. Circuits Containing Resistance, Self-induction, and Capacity. Chapters XII, XIII. Circuits Containing Distributed Capacity and Self-induction. Part II. Graphical Treatment.-Chapters XIV, XV, XVI, XVII. Circuits Containing Resistance and Self-induction. Chapters XVIII, XIX. Circuits Containing Resistance and Capacity. Chapter XX. Circuits Containing Resistance, Self-induction, and Capacity. Appendix A. Relation between Practical and C. G. S. Limits. Appendix B. Some Me- chanical and Electrical Analogies. Appendix C. System of Notation Adopted. Copies of this or any other electrical book published will be sent by mail, POSTAGE PREPAID, to any address in the world, on receipt of price. The W. J. Johnston Company, Publishers, 253 BROADWAY, NEW YORK, Electrodynamic Machinery, BY EDWIN J. HOUSTON, PH.D., AND A. E. KENNELLY, Sc.D. Cloth. 322 Pages. 232 Illustrations. Price, $2.5O. The advent of the commercially perfected dynamo has marked such an era in electric progress, that the desire naturally exists on the part of students generally, and of electrical engineers in particular, to grasp thoroughly the principles underlying its constitution and operation. While excellent treatises on dynamo-electric machinery are in existence, yet a want has been experienced of a book which shall approach the sub- ject, not from the mathematical standpoint, but from an engineering stand- point, and especially from such an engineering standpoint as would arise in actual working with the apparatus. A book of this latter type will be found in the present work, which is the outcome of a series of articles on the same subject by these authors appearing in the Electrical World during 1894-1895. Elettrodynamic Machinery, as it is described in this volume, is limited to the consideration of continuous-current apparatus. The authors have endeavored to explain thoroughly the principles involved by the use of as simple mathematics as the case will permit. In order to fix these principles in the mind of the student, and to test his understanding of the same, they have been accompanied by numerous examples taken from practical work, so that a student who will intelligently follow the treat- ment found in these pages will obtain a grasp of the subject that it would be very difficult for him to obtain otherwise without personal experience. For this reason it is believed that the book will serve as a text-book on continuous-current dynamo-electric machinery for electric-engineering students of all grades. Copies of this or any other electrical book published will be sent by mail, POST \v PREPAID, to any address in the world, on receipt* oj price. The W. J. Johnston Company, Publishers, 253 BROADWAY, NEW YORK. ALTERNATING CURRENTS OF ELECTRICITY: Their Generation, Measurement, Distribution, and Application. BY GISBERT KAPP, M.I.C.E., M.I.E.E. {Authorized American Edition?) With an Introduction by William Stanley, Jr. Cloth. 164 pages, 37 Illustrations, and 2 plates. $1.00. The rapid development of alternating currents and the great part they are destined to play in the transmission of power have caused an increased interest in the subject, but unfortunately it has heretofore been presented in such a manner as to be beyond the reach of readers without a mathematical education. In the present work, the principles are developed in a simple manner that can be followed by any reader, and the various applications are sketched in a broad and instructive way that renders their understanding an easy task. The few mathematical formulas in the book are confined to appendices. Several chap- ters treat of various forms of alternating motors, especial attention being paid to the explanation and discussion of multiphase motors. This difficult subject is treated so lucidly that the reader is enabled to form as clear an idea of these new forms of motors as of the simpler continuous current machines. The treatment throughout is thoroughly practical, and the data and discussion on the design and construction of apparatus are invaluable to the electrician and designer. To the student and the general public this work will be a particular boon, bringing within their grasp a subject of the greatest importance and interest. CONTENTS. Introduction by William Stanley, Jr. Chap. I. Introductory. Chap. II. Measurement of Pressure, Current, and Power. Chap. III. Conditions of Maximum Power. Chap. IV. Alter, nating-current Machines. Chap. V. Mechanical Construction of Alternators. Chap. VI. Description of Some Alternators. Chap. VII Transformers. Chap. VIII. Central Stations and Distribution of Power. Chap. IX. Examples of. Central Stations. Chap. X. Parallel Coupling of Alternators. Chap. XI. Alternating-current Motors. Chap. XII. Self-starting Motors. Chap. XIII. Multiphase Currents. Copies of this or any other electrical book published will be sent by mail, POSTAGE PREPAID, to any address in the world, on receipt of price. The W. J. Johnston Company, Publishers, 253 BROADWAY, NEW YORK. EXPERIMENTS WITH Alternate Currents Of High Potential and High Frequency. BY NIKOLA TESLA. Cloth. 156 pages, with Portrait and 35 Illustrations* Price, $1.OO. Since the discovery of the telephone few researches in electricity have created as widespread an interest as those of Nikola Tesla into alternate currents of high potential and high frequency. Mr. Tesla was accorded the unusual honor of an invitation to repeat his experiments before distinguished scientific bodies of London and Paris, and the lecture delivered before the Institution of Elec- trical Engineers, London, is here presented in book-form. The field opened up, to which this book acts as a guide, is one in which the future may develop results of the most remarkable character, and perhaps lead to an entire revision cf our present scientific conceptions, with correspondingly broad practical re- sults. The currents of enormously high frequency and voltage generated by Mr. Tesla developed properties previously entirely unsuspected, and which pro- duced phenomena of startling character. No injurious effects were experi- enced when the human body was subjected to the highest voltages generated. Lamps with only one conductor v/ere rendered incandescent, and others with no connection whatever to the conducting circuit glowed when merely brought into proximity to the same. The book in which Mr. Tesla describes his mar- vellous experiments is one that every one who takes an interest in electricity or its future should read. The subject is popularly treated, and as the author is the master of a simple and agreeable style the book is fascinating reading. A portrait and biographical sketch of the author is included. Copies of this or any other electrical booK published will be sent by mail, POSTAGE PREPAID, to any address in the world, on receipt of price. The W. J. Johnston Company, Publishers, 253 BROADWAY, NEW YORK. PUBLICATIONS OF THEW. J.JOHNSTON COHPANY. The Electrical World. An Illustrated Weekly Review of Current Progress in Electricity and its Practical Applications. Annual subscription $3.00 General Index to The Electrical World, 1883-1896 (In Press) 8.0O Atkinson, Philip, Ph.D. The Elements of Static Electricity 1.5O Bedell, Frederick, Ph.D., and Crehore, Albert C., Ph.D. Alternating Currents. 2.5O Bell, Dr. Louis. Electrical Transmission of Power 2.5O Cox, Frank P., B.S. Continuous- Current Dynamos and Motors 2.OO Crosby, O. T., and Bell, Dr. Louis. The Electric Railway in Theory and Practice. 2.5O Davis, Charles M. Standard Tables for Electric Wiremen l.OO Foster, H. A. Central-Station Bookkeeping 2.5O Gerard's Electricity. Translated under the direction of Dr. Louis Duncan. With chapters by Messrs. Duncan, Steinmetz, Kennelly and Hutchinson 2.5O tiering, Carl. Universal Wiring Computer 1 .OO " " Recent Progress in Electric Railways l.OO " " Electricity at the Paris Exposition of 1889 2 OO " Tables of Equivalents of Units of Measurement >O Hopkinson, John, F.R.S. Original Papers on Dynamo Machinery and Allied Subjects 1.00 Houston, Edwin J., Ph.D. Electrical Measurements ,. 1 .OO " " " Electricity and Magnetism l.OO " " " Electricity 100 Years Ago and To-Day l.OO " " " Electrical Transmission of Intelligence l.OO " " " Dictionary of Electrical Words, Terms and Phrases. 5.0 O Houston, Edwin J., and Kennelly, A. E. The Electric Motor l.OO " " " " Electric Heating l.OO " " " " * Magnetism l.OO " " " " The Electric Telephone l.OO ' " " * " Electric Telegraphy 1.00 " * " Electric Arc Lighting l.OO 44 " M Electric Street Railways l.OO Houston, Edwin J., and Kennelly, A. E. Electrodynamic Machinery *2.5O * ** * u * " Electric Incandescent Lighting l.OO " "^ " -< " Alternating Electrical Currents l.OO " " " * Electricity in Electro-Therapeutics.. l.OO Johnston's Electrical and Street Railway Directory, Published annually.... 5.OO Kapp, Gisbert. Alternating Currents of Electricity 1 .OO Lightning Flashes and Electric Dashes ^Humorous Sketches, Illustrated) 1.5O Lock wood. T. D. Practical Information for Telephonists l.OO flartin, T. C., and Wetzler, Joseph. The Electric Motor and its Applications.. 3.OO Haver, William, Jr., and Davis, Minor. The Quadruplex 1.5O rierrill, E. A. Electric Lighting Specifications 1 .50 " Reference Book of Tables and Formulae for Electric Railway Engineers. 1 .OO Parham and Shedd. Shop and Road Testing of Dynamos and Motors. (InPress). 2.OO Parkhui st , Lieut. C. D. Dynamo and Motor Building for Amateurs 1 .OO Perry , Nelson W. Electric Railway Motors 1 .OO Proceedings of the National Conference of Electricians 75 Reid, James D. The Telegraph in America 7.OO Steinmetz, C. P. Theory and Calculation of Alternating-Current Phenomena. 2.5O Tesla, Nikola. Experiments with Alternating Currents of High Potential and High Frequency 1 .OO Thompson, Prof. Silvanus P. Lectures on the Electromagnet l.OO Wiener, Alfred E. Practical Calculation of Dynamo-Electric Machines 2. SO Wheeler, Schuyler S. Chart of Wire Gauges l.OO All the books here mentioned are bound in cloth, and with hardly an exception are handsomely and copiously illustrated. Copies of these or of any other electrical works published will be mailed to any address in the world, POSTAGE PREPAID, on receipt of price: The W. J. Johnston Company, Publishers, 253 BROADWAY, NEW YORK. THE PIONEER ELECTRICAL JOURNAL OF AMERICA. READ WHEREVER THE ENGLISH LANGUAGE IS SPOKEN. THE ELECTRICAL WORLD is the largest, most handsomely illustrated, and most widely circulated electrical journal in the world. It should be read not only by every ambitious electrician anxious to rise in his profession, but by every intelligent American. It is noted for its ability, enterprise, independence and honesty. For thoroughness, candor and progressive spirit it stands in the foremost rank of special journalism. Always abreast of the times, its treatment of everything relating to the prac- tical and scientific development of electrical knowledge is comprehensive and authoritative. Among its many features is a weekly Digest of Current Technical Electrical Literature^ which gives a complete re'sume' of current original contri- butions to electrical literature appearing in other journals the world over. a Year. May be ordered of any Newsdealer at 10 cents a week. Cloth Binders for THE ELECTRICAL WORLD postpaid, $1.00. The W. J. Johnston Company, Publishers, 253 BROADWAY, NEW YORK. UNIVERSITY OF CALIFORNIA LIBRARY This book is DUE on the last date stamped below. Fine schedule: 25 cents on first day overdue 50 cents on fourth day overdue One dollar on seventh day overdue. GINEERING LI MOV i 5 1949 19 4/953 LD 21-100m-12,'46(A2012sl6)4120 RARY VC 32599 866778 <5 C, UNIVERSITY OF CALIFORNIA UBRARY