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Engineering
Library



THEORY AND CALCULATION

OF

ALTERNATING-CURRENT PHENOMENA



McGraw-Hill BookCompany



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THEORY AND CALCULATION

OF

ALTERNATING CURRENT
PHENOMENA



BY

CHARLES PROTEUS STEINMETZ, A.M., PH.D.

\\



FIFTH EDITION
THOROUGHLY REVISED AND ENTIRELY RESET



McGRAW-HILL BOOK COMPANY, INC.

239 WEST 39TH STREET. NEW YORK



LONDON: HILL PUBLISHING CO., LTD.

6 & 8 BOUVERIE ST., E. C.

1916



-SP-

Engineering
Library



COPYRIGHT, 1916, BY THE
MCGRAW-HILL BOOK COMPANY, INC.

COPYRIGHT, 1908, BY THE
McGRAW PUBLISHING COMPANY

COPYRIGHT, 1900, BY
ELECTRICAL WORLD AND ENGINEER (INCORPORATED)



..,-, *,-,.'_ * > ,

?'U&;.. ;



THE . MAPLK. PRESS- YORK. PA



De&fcateD

TO THE
MEMORY OF MY FATHER

CARL HEINRICH STEINMETZ



338054



PREFACE TO FIFTH EDITION

When the first edition of " Alternating-Current Phenomena"
appeared nearly twenty years ago, it was a small volume of 424
pages. From this, it grew to a volume of 746 pages in the fourth
edition, which appeared eight years ago. Since that time, the
advance of electrical engineering has been more rapid than ever
before, and any attempt to treat adequately in one volume all
the new material developed in the last eight years, and the
material which during this time has become of such importance
as to require more extensive consideration, thus became out of the
question. It was found necessary, therefore, to divide the work
into three volumes. In the following, under the old title " Theory
and Calculation of Alternating-Current Phenomena," is included
only the discussion of the most common and general phenomena
and apparatus, old and new, revised and expanded so as to bring
it up to our present knowledge. All the material, some partly
old, but mostly new, which could not find place in the present
volume, will be presented in two supplementary volumes, under
the titles " Theory and Calculation of Electric Circuits," and
" Theory and Calculation of Electrical Apparatus."

In the study of electrical engineering theory, it is recommended
to read first Part I of " Theoretical Elements of Electrical Engi-
neering," and then the first three sections of "Alternating Cur-
rent Phenomena," but to parallel the reading with that of the
chapters of " Engineering Mathematics," which deal with the
mathematics involved. Then Sections IV to VII of "Alter-
nating-Current Phenomena" should be studied simultaneously
with the corresponding discussion of the apparatus in the Part
II of "Theoretical Elements." Following this should be taken
up the study of "Theory and Calculation of Electric Circuits/'
"Theory and Calculation of Electrical Apparatus," and the first
three sections of "Transient Phenomena," and, finally, the study
of "Electric Discharges, Waves and Impulses" and of the fourth
section of "Transient Phenomena."

In the present edition of "Alternating-Current Phenomena,"
the crank diagram of vector representation, and the symbolic
method based on it, which denotes the inductive reactance by

vii



viii PREFACE TO FIFTH EDITION

Z = r + jx } have been adopted in conformity with the decision
of the International Electrical Congress of Turin, but the time
diagram or polar coordinate system has been explained and dis-
cussed in Chapter VII, since the crank diagram is somewhat
inferior to the polar diagram, as it is limited to sine waves, and
the time diagram will thus remain in use when dealing with
general waves and their graphic reduction.

CHARLES P. STEINMETZ.

SCHENECTADY, N. Y.,

May, 1916.



PREFACE TO FIRST EDITION

THE following volume is intended as an exposition of the
methods which I have found useful in the theoretical investiga-
tion and calculation of the manifold phenomena taking place in
alternating-current circuits, and of their application to alternat-
ing-current apparatus.

While the book is not intended as first instruction for a begin-
ner, but presupposes some knowledge of electrical engineering, I
have endeavored to make it as elementary as possible, and have
therefore used only common algebra and trigonometry, practi-
cally excluding calculus, except in 144 to 151 and Appendix II;
and even 144 to 151 have been paralleled by the elementary
approximation of the same phenomenon in 140 to 143.

All the methods used in the book have been introduced and
explicitly discussed, with examples of their application, the first
part of the book being devoted to this. In the investigation 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 alternating-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 electrical engineering, read be-
fore the International Electrical Congress at Chicago, 1893. The
complex imaginary quantity is gradually introduced, with full
explanations, the algebraic operations with complex quantities
being discussed in Appendix I, so as not to require from the reader
any previous knowledge of the algebra of the complex imaginary
plane.

While those phenomena which are characteristic of polyphase
systems, as the resultant action of the phases, the effects of un-
balancing, the transformation of polyphase systems, etc., have
been discussed separately in the last chapters, many of the in-
vestigations in the previous parts of the book apply to poly-
phase systems as well as single-phase circuits, as the chapters on
induction motors, generators, synchronous motors, etc.

ix



x PREFACE TO FIRST EDITION

A part of the book is original investigation, either published
here for the first time, or collected from previous publications
and more fully explained. Other parts have been published be-
fore 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 expenditure of much
more time than is at my disposal, without offering sufficient com-
pensation; since I believe that the reader who wants informa-
tion on some phenomenon or apparatus is more interested in
the information than in knowinjg who first investigated the
phenomenon.

Special attention has been given to supply a complete and ex-
tensive index for easy reference, and to render the book as free
from errors as possible. Nevertheless, it probably 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 WOLCOTT, for the
interest they have taken in the book while in the course of pub-
lication, as well as for the valuable assistance given by them in
correcting and standardizing the notation to conform to the
international system, and numerous valuable suggestions regard-
ing desirable improvements.

Thanks are due to the publishers, who have spared no
effort or expense to make the book as creditable as possible
mechanically.

CHARLES PROTEUS STEINMETZ.

January, 1897.



CONTENTS

SECTION I

METHODS AND CONSTANTS
CHAPTER I. INTRODUCTION.

PAGE

1. Fundamental laws of continuous-current circuits. 1

2. Impedance, reactance, effective resistance. 2

3. Electromagnetism as source of reactance. 2

4. Capacity as source of reactance. 4

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

6. Fundamental wave and higher harmonics, alternating waves

with and without even harmonics. 5

7. Alternating waves as sine waves. 8

8. Experimental determination and calculation of reactances. 8

CHAPTER II. INSTANTANEOUS VALUES AND INTEGRAL VALUES.

9. Integral values of wave. 1 1

10. Ratio of mean to maximum to effective value of wave. 13

11. General alternating-current wave. 14

12. Measurement of values. 15

CHAPTER III. LAW OP ELECTROMAGNETIC INDUCTION.

13. Induced e.m.f. mean value. 16

14. Induced e.m.f. effective value. 17

15. Inductance and reactance. 17

CHAPTER IV. VECTOR REPRESENTATION.

16. Crank diagram of sine wave. 19

17. Representation of lag and lead. 20

18. Parallelogram of sine waves, Kirchhoff's laws, and energy

equation. 21

19. Non-inductive circuit fed over inductive line, example. 22

20. Counter e.m.f. and component of impressed e.m.f. 23

21. Example, continued. 24

22. Inductive circuit and circuit with leading current fed over in-

ductive line. Alternating-current generator. 25

23. Vector diagram of alternating-current transformer, example. 26

24. Continued. 28

CHAPTER V. SYMBOLIC METHOD.

25. Disadvantage of graphic method for numerical calculation. 30

26. Trigonometric calculation. 31 '

xi



xii CONTENTS

PAGE

27. Rectangular components of vectors. 31

28. Introduction of j as distinguishing index. 32

29. Rotation of vector by 180 and 90. j = \/~^~l. 32

30. Combination of sine waves in symbolic expression. 33

31. Resistance, reactance, impedance, in symbolic expression. 34

32. Capacity reactance in symbolic representation. 35

33. Kirchhoff's laws in symbolic representation. 36

34. Circuit supplied over inductive line, example. 37

35. Products and ratios of complex quantities. 37

CHAPTER VI. TOPOGRAPHIC METHOD.

36. Ambiguity of vectors. 39

37. Instance of a three-phase system. 39

38. Three-phase generator on balanced load. 41

39. Cable with distributed capacity and resistance. 42

40. Transmission line with inductance, capacity, resistance, and

leakage. 43

41. Line characteristic at 90 lag. 45

CHAPTER VII. POLAR COORDINATES AND POLAR DIAGRAM.

42. Polar Coordinates 46

43. Sine wave, vector representation or time diagram. 46

44. Parallelogram of sine waves, Kirchhoff's Laws and power

equation. 48

45. Comparison of time diagram and crank diagram. 49

46. Comparison of corresponding symbolic methods. 51

47. Disadvantage of crank diagram. General wave end its

equivalent sine wave in time diagram. 52



SECTION II
CIRCUITS
CHAPTER VIII. ADMITTANCE, CONDUCTANCE, SUSCEPTANCE.

48. Combination of resistances and conductances in series and

in parallel. 54

49. Combination of impedances. Admittance, conductance,

susceptance. 55

50. Relation between impedance, resistance, reactance, and ad-

mittance, conductance, susceptance. 56

51. Dependence of admittance, conductance, susceptance, upon

resistance and reactance. Combination of impedances and
admittances. 57

52. Measurements of admittance and impedance. 59



CONTENTS xiii

CHAPTER IX. CIRCUITS CONTAINING RESISTANCE, INDUCTANCE, AND
CAPACITY.

PAGE

53. Introduction. 60

54. Resistance in series with circuit. 60

55. Reactance in series with circuit. 63

56. Discussion of examples. 65

57. Reactance in series with circuit. 67

58. Impedance in series with circuit. 69

59. Continued. 70

60. Example. 70

61. Compensation for lagging currents by shunted condensance. 72

62. Complete balance by variation of shunted condensance. 73

63. Partial balance by constant shunted condensance. 75

64. Constant potential constant-current transformation. 76

CHAPTER X. RESISTANCE AND REACTANCE OF TRANSMISSION LINES.

65. Introduction. 78

66. Non-inductive receiver circuit supplied over inductive line. 79

67. Example. 81

68. Maximum power supplied over inductive line. 82

69. Dependence of output upon the susceptance of the receiver

circuit. 82

70. Dependence of output upon the conductance of the receiver

circuit. 84

71. Summary. 85

72. Example. 86

73. Condition of maximum efficiency. 88

74. Control of receiver voltage by shunted susceptance. 89

75. Compensation for line drop by shunted susceptance. 90

76. Maximum output and discussion. 91

77. Example. 92

78. Maximum rise of potential in receiver circuit. 94

79. Summary and examples. 96

CHAPTER XI. PHASE CONTROL.

80. Effect of the current phase in series reactance, on the voltage. 97

81. Production of reactive currents by variation of field of syn-

chronous machines. 98

82. Fundamental equations of phase control. 99

83. Phase control for unity power-factor supply. 100

84. Phase control for constant receiver-voltage. 102

85. Relations between supply voltage, no-load current, full-load

current and maximum output current. 104

86. Phase control by series field of converter. 105

87. Multiple-phase control for constant voltage. 107

88. Adjustment of converter field for phase control. 109



xiv CONTENTS

SECTION III
POWER AND EFFECTIVE CONSTANTS

CHAPTER XII. EFFECTIVE RESISTANCE AND REACTANCE.

PAGE

89. Effective resistance, reactance, conductance, and susceptance. Ill

90. Sources of energy losses in alternating-current circuits. 112

91. Magnetic hystersis. 113

92. Hysteretic cycles and corresponding current waves. 114

93. Wave-shape distortion not due to hysteresis. 117

94. Action of air-gap and of induced current on hysteretic

distortion. 119

95. Equivalent sine wave and wattless higher harmonics. 120

96. True and apparent magnetic characteristic. 121

97. Angle of hysteretic advance of phase. 122

98. Loss of energy by molecular magnetic friction. 123

99. Effective conductance, due to magnetic hysteresis. 126

100. Absolute admittance of iron-clad circuits and angle of hysteretic

advance. 129

101. Magnetic circuit containing air-gap. 131

102. Electric constants of circuit containing iron. 132

103. Conclusion. 133

104. Effective conductance of eddy currents. 135

CHAPTER XIII. FOUCAULT OR EDDY CURRENTS.

105. Advance angle of eddy currents. 136

106. Loss of power by eddy currents, and coefficient of eddy currents. 137

107. Laminated iron. 138

108. Iron wire. 140

109. Comparison of sheet iron and iron wire. 141

110. Demagnetizing or screening effect of eddy currents. 142

111. Continued. 143

112. Large eddy currents. 144

113. Eddy currents in conductor and unequal current distribution. 144

114. Continued. -145

115. Mutual inductance. 147

CHAPTER XIV. DIELECTRIC LOSSES.

116. Dielectric hysteresis. 150

117. Leakage. Dynamic current and displacement current.

Power factor. 151

118. Effect of frequency. Heterogeneous dielectric. 153

119. Power factor and stress in compound dielectric. 154

120. Variation of power factor with frequency. 156

121. Dielectric circuit and dynamic circuit. Admittance and im-

pedance. Admittivity. 158

122. Study of dielectric field. 160

123. Corona. Dielectric strength and gradient. 161



CONTENTS xv

PAGE

124. Corona and disruption. Numerical instance. 161

125. Energy distance, disruptive gradient and visual corona

gradient. 164

126. Law of corona on parallel conductors. 165

CHAPTER XV. DISTRIBUTED CAPACITY, INDUCTANCE, RESISTANCE,
AND LEAKAGE.

127. Energy components and wattless components. 168

128. Distributed capacity. 168

129. Magnitude of charging current of transmission lines. 170

130. Line capacity represented by one condenser shunted across

middle of line. 171

131. Distributed capacity, inductance, conductance and resistance. 172

132. Constants of transmission line. 174

133. Oscillating functions of distance. Approximate calculation. 175

134. Equations of transmission line. 177

CHAPTER XVI. POWER, AND DOUBLE-FREQUENCY QUANTITIES IN
GENERAL.

135. Double frequency of power. 179

136. Symbolic representation of power. 180

137. Extra-algebraic features thereof. 182

138. Polar coordinates. 183

139. Combination of powers. 184

140. Torque as double-frequency product. 185



SECTION IV
INDUCTION APPARATUS

CHAPTER XVII. THE ALTERNATING-CURRENT TRANSFORMER.

141. General. 187

142. Mutual inductance and self-inductance of transformer. 187

143. Magnetic circuit of transformer. 188

144. Continued. 189

145. Polar diagram of transformer. 190

146. Example. 192

147. Diagram for varying load. 196

148. Example. 197

149. Symbolic method, equations. 197

150. Continued. 199

151. Apparent impedance of transformer. Transformer equivalent

to divided circuit. 201

152. Continued. 202

153. Experimental determination of transformer constants. 205

154. Calculation of transformer constants. 207



xvi CONTENTS

CHAPTER XVIII. POLYPHASE INDUCTION MOTOR.

PAGE

155. Slip and secondary frequency. 208

156. Equations of induction motor. 209

157. Magnetic flux, admittance, and impedance. 210

158. E.m.f. 212

159. Graphic representation. 214

160. Continued. 215

161. Torque and power. 216

162. Power of induction motors. 217

163. Maximum torque. 219

164. Continued. 221

165. Maximum power. 222

166. Starting torque. 223

167. Equations of torque. 227

168. Synchronism. 229

169. Near synchronism. 229

170. Numerical example of induction motor. 230

171. Calculation of induction-motor curves. 232

172. Numerical example. 235

CHAPTER XIX. INDUCTION GENERATOR.

173. Induction generator. 237

174. Power-factor of induction generator. 237

175. Constant speed induction generator. 239

176. Induction generator and synchronous motor. 242

CHAPTER XX. SINGLE-PHASE INDUCTION MOTOR.

177. Single-phase induction motor. 245

178. Starting devices of single-phase motor. 246

179. Polyphase motor on single-phase circuit. 247

180. Condenser in tertiary circuit. 249

181. Speed curves with condenser. 250

182. Monocyclic starting device. 253

183. Resistance-reactance starter. 256

184. Discussion. 257

SECTION V

SYNCHRONOUS MACHINES
CHAPTER XXI. ALTERNATE-CURRENT GENERATOR.

185. Magnetic reaction of lag and lead. 259

186. Self-inductance and synchronous reactance. 261

187. Equations of alternator. 263

188. Numerical instance, field characteristic. 264

189. Dependence of terminal voltage on phase relation. 266

190. Constant potential regulation. 267

191. Constant current regulation, maximum output. 270



CONTENTS xvii
CHAPTER XXII. ARMATURE REACTIONS OF ALTERNATORS.

PAGE

192. Similarity and difference between armature reaction and self-

induction. 272

193. Graphic representation of armature reaction and self-induction. 273

194. Symbolic representation. 274

195. Discussion: synchronous reactance and norminal induced

e.m.f. 276

196. Variability, and quadrature components in space, of armature

reaction and self-induction. 278

197. Graphic representation of variable armature reaction and

self-induction. 279

198. Symbolic representation. 281

199. Continued. 284

200. Regulation curve of alternator. 287

201. Example. 288

202. Discussion. 290

CHAPTER XXIII. SYNCHRONIZING ALTERNATORS.

203. Introduction. 292

204. Rigid mechanical connection. 292

205. Uniformity of speed. 292

206. Synchronizing. 293

207. Running in synchronism. 293

208. Series operation of alternators. 294

209. Equations of synchronous-running alternators, synchronizing

power. 294

210. Special case of equal alternators at equal excitation. 297

211. Numerical example. 300

CHAPTER XXIV. SYNCHRONOUS MOTOR.

212. Graphic method. 301

213. Continued. 302

214. Example. 303

215. Constant impressed e.m.f. and constant current. 306

216. Constant impressed and counter e.m.f. 307

217. Constant impressed e.m.f. and maximum efficiency. 310

218. Constant impressed e.m.f. and constant output. 311

219. Analytical method. Fundamental equations and power

characteristic. 315

220. Maximum output. 318

221. No load. 319
. 222. Minimum current. 321

223. Maximum displacement of phase. 323

224. Constant counter e.m.f. 323

225. Numerical example. 324

226. Discussion of results. 326

227. Phase characteristics of synchronous motor. 328



xviii CONTENTS

PAGE

228. Example. 331

229. Load curves of synchronous motor. 334

230. Variable armature reaction and self-induction. 338

231. Synchronous condenser. 339

SECTION VI
GENERAL WAVES
CHAPTER XXV. DISTORTION OP WAVE-SHAPE, AND ITS CAUSES.

232. Equivalent sine wave. 341

233. Cause of distortion. 341

234. Lack of uniformity and pulsation of magnetic field. 342

235. Continued. 345

236. Pulsation of reactance. 348

237. Pulsation of reactance in reaction machine. 348

238. General discussion. 350

239. Pulsation of resistance, arc. 350

240. Example. 351

241. Distortion of wave-shape by arc. 353

242. Discussion. 353

243. Calculation of example. 354

244. Separation of overtones from distorted wave. 357

245. Resolution of exciting-current wave of transformer. 360

246. Distortion of e.m.f. wave with sine wave of current, in iron-clad

circuit. 361

247. Existence and absence of third harmonic in three-phase

system. 363

248. Suppression of third harmonics in transformers on three-phase

system. 364

249. Wave-shape distortion in Y-connected transformers. 385

250. Disappearance of distortion by delta connection, etc. 367

CHAPTER XXVI. EFFECTS OF HIGHER HARMONICS.

251. Distortion of wave-shape by triple and quintuple harmonics.

Some characteristic wave-shapes. 369

252. Effect of self-induction and capacity on higher harmonics. 372

253. Resonance due to higher harmonics in transmission lines. 373

254. Power of complex harmonic waves. 375

255. Three-phase generator. 375

256. Decrease of hysteresis by distortion of wave-shape. 377

257. Increase of hysteresis by distortion of wave-shape. 377

258. Eddy currents and effect of distorted waves on insulation. 377

CHAPTER XXVII. SYMBOLIC REPRESENTATION OF GENERAL ALTER-
NATING WAVE.

259. Symbolic representation. 379

260. Effective values. 381



CONTENTS xix

PAGE

261. Power, torque, etc. Circuit-factor. 381

262. Resistance, inductance, and capacity in series. 384

263. Apparent capacity of condenser. 386

264. Synchronous motor. 389

265. Induction motor. 392

SECTION VII
POLYPHASE SYSTEMS
CHAPTER XXVIII. GENERAL POLYPAASE SYSTEMS.

266. Definition of systems, symmetrical and unsymmetrical systems. 396

267. Flow of energy. Balanced and unbalanced systems. Inde-

pendent and interlinked systems. Star connection and ring
connection. 396

268. Classification of polyphase systems. 398

CHAPTER XXIX. SYMMETRICAL POLYPHASE SYSTEMS.

269. General equations of symmetrical systems. 399

270. Particular systems. 400

271. Resultant m.m.f. of symmetrical system. 401

272. Particular systems. 403

CHAPTER XXX. BALANCED AND UNBALANCED POLYPHASE SYSTEMS.

273. Flow of energy in single-phase system. 405

274. Flow of energy in polyphase systems, balance factor of system. 406

275. Balance factor. 406

276. Three-phase system, quarter-phase system. 407

277. Inverted three-phase system. 408

278. Diagrams of flow of energy. 408

279. Monocyclic and polycyclic systems. 409

280. Power characteristic of alternating-current system. 409

281. The same in rectangular coordinates. 409

282. Main power axes of alternating-current system. 414

CHAPTER XXXI. INTERLINKED POLYPHASE SYSTEMS.

283. Interlinked and independent systems. 415

284. Star connection and ring connection. Y-connection and delta

connection. 415

285. Continued. 417

286. Star potential and ring potential. Star current and ring

current. Y-potential and Y-current, delta potential and
delta current. 417

287. Equations of interlinked polyphase systems. 417

288. Continued. 419



xx CONTENTS

CHAPTER XXXII. TRANSFORMATION OF POLYPHASE SYSTEMS.

PAGE

289. Constancy of balance factor. 422

290. Equations of transformation of polyphase systems. 422

291. Three-phase quarter-phase transformation. 423

292. Some of the more common polyphase transformations. 425

293. Transformation with change of balance factor. 430

CHAPTER XXXIII. COPPER EFFICIENCY OF SYSTEMS.

294. General discussion. 431

295. Comparison on the basis of equality of minimum difference of

potential. 433

296. Comparison on the basis of equality of maximum difference of

potential between conductors. 437

297. Continued. 439

298. Comparison on the basis of equality of maximum difference of

potential between conductors and ground. 440

CHAPTER XXXIV. METERING OF POLYPHASE CIRCUIT.

299. General equations. 442

300. Continued. 443

301. Three-phase metering. 445

302. Discussion. 446



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