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UNIVERSITY LIBRARIES

THEORY AND CALCULATIONS
OF
ELECTRICAL APPARATUS

THEORY AND CALCULATIONS
OF
ELECTRICAL APPARATUS
BY
CHARLES PROTEUS STEINMETZ, A. M., PH. D,
KDITION SIXTH IMPUEHSION
McGRAW-HILL BOOK COMPANY, INC. NEW YORK; 370 SEVENTH AVENUE
LONDON: 6 & 8 BOUVEEIE ST., E. C. 4 1917

COPYRIGHT, 1917, BY THE MCGRAW-HILL BOOK COMPANY, INC. PBINTB0 IN THE UNITED HTATEB OF AMBHICA
MAPLE PRESS - YORK

PREFACE

In

the

twenty

years

since

the

first

edition

of

"
Theory

and

Cal-

culation of Alternating Current Phenomena" appeared, elec-
trical engineering has risen from a small beginning to the world's

greatest industry; electricity has found its field, as the means of

universal energy transmission, distribution and supply, and our

knowledge of electrophysics and electrical engineering has in-

creased many fold, so that subjects, which twenty years ago could be dismissed with a few pages discussion, now have expanded

and require tin extensive knowledge by every electrical engineer. In the following volume I have discussed the most important

characteristics of the numerous electrical apparatus, which have

been devised and have found their place in the theory of electrical

engineering. While many of them have not yet reached any

industrial importance, experience has shown, that not infre-

quently apparatus, which had been known for many years but

had

not

found

any

extensive

4 ,

practical

use,

become,

with

changes

of industrial conditions, highly important. It is therefore

necessary for the electrical engineer to be familiar, in a general

way, with the characteristics of the less frequently used types

of apparatus.

In some; respects, the following work, and its companion vol-

ume, "Theory and Calculation of Electric Circuits," may be

considered as continuations, or rather as parts of "Theory and

Calculation of Alternating Current Phenomena." With the 4th

edition, which appeared nine years ago, "Alternating Current

Phenomena" had reached about the largest practical bulk, and

who n rewriting it recently for the /)th edition, it became necessary

to subdivide it into three volumes, to include at least the most

necessary structural elements of our knowledge of electrical
engineering. The subject matter thus has been distributed into

three volumes: "Alternating Current Phenomena," "Electric

Circuits," and "Electrical Apparatus," CHARLES PROTEUS STEINMETZ,

CAMP MOHAWK, VIBLK'B CKKKK,
July, 1017.

CONTENTS

PREFACE

PAOE

CHAPTER T. SPEED CONTROL OP INDUCTION MOTORS.

/. Starting and Acceleration

1. The problems of high torque over wide range of speed, and of

....................... constant speed over wide range of load Starting by armature

rheostat *

1

2. A, Temperature starting device Temperature rise increasing

....................... secondary resistance with increase of current Calculation of

motor

2

3. Calculation of numerical instance Its discussion -Estimation

.............. of required temperature rise

4

4. B. Hysteresis starting device Admittance of a closed mag-

netic circuit \vith negligible eddy current loss Total secondary

..... impedance of motor with hysteresis starting device

5

5. Calculation of numerical instance Discussion- Similarity of

torque curve with that of temperature starting device Close

speed regulation Disadvantage of impairment of power factor

........... and apparent efficiency, due to introduction of reactance Re-

quired Increase of magnetic density

6

6. (L Eddy current starting device- Admittance of magnetic cir-

cuit with high eddy current losses and negligible hysteresis

............... Total secondary impedance of motor with eddy current starting

deviceNumerical instance

8

7. Double maximum of torque curve Close speed regulation-

High torque efficiency -Poor power factor, requiring increase

............ of magnetic density to get output Relation to double squirrel

cage motor and deep bar motor

10

//. Constant Speed Operation

H. Speed control by armature resistance Disadvantage of in-

......... eoiwUncy of speed with load Use of condenser in armature or

secondary- -Use of pyro-eleetric resistance

12

9, Speed control by variation of the effective frequency: con-

................ catenationBy changing the number of poles: rnultispeed

motors ........

13

10, A. Pyro-electric speed control Characteristic of pyro-

.............. olectric conductor Close speed regulation of motor Limita-

tion of pyro-eloctrio conductors

14

11, B. Condenser speed control Effect of condenser in secondary,

viii

CONTENTS

PAGE

giving high current and torque at resonarxce speed Calcula-

tion of motor .

16

12. Equations of motor Equation of torque Speed range of

maximum torque .

17

13. Numerical instance Voltampere capacity of required con-

denser

18

14. C. Multispeed motors Fractional pitch winding, and switch-

ing of six groups of coils in each phase, at a change of the num-

ber of poles .

.

.

20

15. Discussion of the change of motor constants due to a change of

the number of poles, with series connection of all primary turns

Magnetic density and inferior performance curves at lower

speeds

.

21

16. Change of constants for approximately constant maximum

torque at all speeds Magnetic density and change of coil

connection

22

17. Instance of 4 6 -=- -f- 8 pole motor Numerical calculation and

discussion

23

CHAPTER II. MULTIPLE SQUIRREL CAGE INDUCTION MOTOR.

18. Superposition of torque curves of high resistance low reactance,

and low resistance high reactance squirrel cage to a torque

curve with two maxima, at high and at low speed

27

19. Theory of multiple squirrel cage based on the use of the true

induced voltage, corresponding to the resultant flux which

passes beyond the squirrel cage Double squirrel cage induc-

tion motor

28

20. Relations of voltages and currents in the double squirrel cage

induction motor

29

21. Equations, and method of calculation

30

22. Continued: torque and power equation

31

23. Calculation of numerical instance of double squirrel cage

motor, speed and load curves Triple squirrel cage induction

motor

32

24. Equation between the voltages and currents in the triple

squirrel cage induction motor

34

.... 25. Calculation of voltages and currents

. ... 35

26. Equation of torque and power of the three squirrel cages, and

their resultant

37

27. Calculation of numerical instance of triple squirrel cage induc-

tion motor Speed and load curves

37

CHAPTER III. CONCATENATION.

Cascade or Tandem Control of Induction Motors

28. Synchronizing of concatenated couple at half synchronism The two speeds of a couple of equal motors and the three

CONTENTS

ix

PAGE

speeds of a couple of unequal motors Internally concatenated

motor .

... 40

29. Generator equation of concatenated couple above half syn-

chronism Second range of motor torque near full synchron-

ism Generator equation above full synchronism Ineffi-

ciency of second motor speed range Its suppression by

resistance in the secondary of the second motor

41

30. General equation and calculation of speed and slip of con-

catenated couple

42

31. Calculation of numerical instances

44

32. Calculation of general concatenated couple

45

33. Continued

46

34. Calculation of torque and power of the two motors, and of the

couple

47

35. Numerical instance

48

36. Internally concatenated motor Continuation of windings into
one stator and one rotor winding Fractional pitch No inter-

ference of magnetic flux required Limitation of available

speed Hunt motor

49

37. Effect of continuation of two or more motors on the character-

istic constant and the performance of the motor.'

50

CHAPTER IV. INDUCTION MOTOR WITH SECONDARY EXCITATION.

38. Large exciting current and low power factor of low speed induction motors and motors of high overload capacity

Instance

52

39. Induction machine corresponding to synchronous machine ex-

cited by armature reaction, induction machine secondary corresponding to synchronous machine field Methods of secondary

excitation : direct current, commutator, synchronous machine,

commutating machine, condenser

53

40. Discussion of the effect of the various methods of secondary

excitation on the speed characteristic of the induction motor . 55

Induction Motor Converted to Synchronous

41. Conversion of induction to synchronous motor Relation of

exciting admittance and self-inductive impedance as induction motor, to synchronous impedance and coreloss as synchronous

motor Danielson motor

-57

42. Fundamental equation of synchronous motor Condition of

unity power factor Condition of constant field excitation . . 60

.... 43. Equations of power input and output, and efficiency

61

44. Numerical instance of standard induction motor converted to

synchronous Load curves at unity power factor excitation and

at constant excitation

62

45. Numerical instance of low speed high excitation induction

motor converted to synchronous motor Load curves at unity

CONTENTS

PAGE

power factor and at constant field excitation Comparison

with induction motor

67

46. Comparison of induction motor and synchronous motor regard-

Ing armature reaction and synchronous impedance Poor induction motor makes good, and good induction motor makes

poor synchronous motor

69

Induction Motor Concatenated with Synchronous

47. Synchronous characteristic and synchronizing speed of concatenated couple Division of load between machines The

synchronous machine as small exciter .

.

71

48. Equation of concatenated couple of synchronous and induction

motor Reduction to standard synchronous motor equation . 72

49. Equation of power output and input of concatenated couple . 74

50. Calculation of numerical instance of 56 polar high excitation

induction motor concatenated to 4 polar synchronous . .

75

51. Discussion. High power factor at all loads, at constant

synchronous motor excitation

76

Induction Motor Concatenated with Commutating Machine

52. Concatenated couple with commutating machine asynchronous

Series and shunt excitation Phase relation adjustable

Speed control and power factor control Two independent

variables with concatenated commutating machine, against one

with synchronous machine Therefore greater variety of speed

and load curves

78

53. Representation of the commutating machine by an effective

impedance, in which both components may be positive or
negative, depending on position of commutator brushes ... 80

54. Calculation of numerical instance, with commutating machine

series excited for reactive anti-inductive voltage Load curves

and their discussion

82

Induction Motor with Condenser in Secondary Circuit

55.

Shunted

r
capacit3

neutralizing lagging current

of

induction

motor Numerical instance Effect of wave shape distortion

Condenser in tertiary circuit of single-phase induction motor

Condensers in secondary circuit Large amount of capacity

required by low frequency

84

56. Numerical instance of low speed high excitation induction

motor with capacity in secondary Discussion of load curves

and of speed

86

57. Comparison of different methods of secondary excitation, by

power factor curves: low at all loads; high at all loads, low at

light, high at heavy loads By speed: synchronous or constant

speed motors and asynchronous motors in which the speed

decreases with increasing load , , ,

88

CONTENTS

xi

Induction Motor with Commutator

PAGE

58. Wave shape of commutated full frequency current in induction

motor secondary Its low frequency component Full fre-

quency reactance for rotor winding The two independent

variables: voltage and phase Speed control and power factor

correction, depending on brush position

89

59. Squirrel cage winding combined with commutated winding

Heyland motor Available only for power factor control Its

limitation

91

CHAPTER V. SINGLE-PHASE INDUCTION MOTOK.

60. Quadrature magnetic flux of single-phase induction motor pro-

duced by armature currents The torque produced by it

The exciting ampere-turns and their change between synchron-

ism and standstill

93

61. Relations between constants per circuit, and constants of the

total polyphase motor Relation thereto of the constants of

the motor on single-phase supply Derivation of the single-

phase motor constants from those of the motor as three-phase or

quarter-phase motor

94

62. Calculation of performance curves of single-phase induction

motor Torque and power

96

63. The different methods of starting single-phase induction motors

Phase splitting devices; inductive devices; monocyclic de-

vices; phase converter

96

64. Equations of the starting torque, starting torque ratio, volt-

ampere ratio and apparent starting torque efficiency of the

single-phase induction motor starting device

98

65. The constants of the single-phase induction motor with starting

device

100

66. The effective starting impedance of the single-phase induction

motor Its approximation Numerical instance

101

67. Phase splitting devices Series impedances with parallel con-

nections of the two circuits of a quarter-phase motor Equa-
'

tions

103

68. Numerical instance of resistance in one motor circuit, with

motor of high and of low resistance armature

104

69. Capacity and inductance as starting device Calculation of

values to give true quarter-phase relation

106

70. Numerical instance, applied to motor of low, and of high arma-

ture resistance

108

71. Series connection of motor circuits with shunted impedance

Equations, calculations of conditions of maximum torque

ratio Numerical instance

109

72. Inductive devices External inductive devices Internal in-

ductive devices

Ill

73. Shading coil Calculations of voltage ratio and phase angle . 112

xil

CONTENTS

PAGE

74. Calculations of voltages, torque, torque ratio and efficiency . . 114 75. Numerical instance of shading coil of low, medium and high

resistances, with motors of low, medium and high armature

resistance

116

76. Monocyclic starting device Applied to three-phase motor

Equations of voltages, currents, torque, and torque efficiency . 117

77. Instance of resistance inductance starting device, of condenser

motor, and of production of balanced three-phase triangle by

capacity and inductance

120

78. Numerical instance of motor with low resistance, and with

high resistance armature Discussion of acceleration . .

.121

CHAPTER VI. INDUCTION MOTOR REGULATION AND STABILITY.

1. Voltage Regulation and Output

79. Effect of the voltage drop in the line and transformer im-

pedance on the motor Calculation of motor curves as affected

by line impedance, at low, medium and high line impedance . 123

80. Load curves and speed curves Decrease of maximum torque

and of power factor by line impedance Increase of exciting

current and decrease of starting torque Increase of resistance

required for maximum starting torque

126

2. Frequency Pulsation

81. Effect of frequency pulsation Slight decrease of maximum

torque Great increase of current at light load

131

3. Load and Stability

82. The two motor speed at constant torque load One unstable

and one stable point Instability of motor, on constant torque

load, below maximum torque point

132

83. Stability at all speeds, at load requiring torque proportional to

square of speed: ship propellor, centrifugal pump Three speeds at load requiring torque proportional to speed Two

stable and one unstable speed The two stable and one un-

stable branch of the speed curve on torque proportional to

speed

... 134

84. Motor stability function of the character of the load General

conditions of stability and instability Single-phase motor . , 136

4. Generator Regulation and Stability

85. Effect of the speed of generator regulation on maximum output

of induction motor, at constant voltage Stability coefficient

of motor Instance

137

CONTENTS

xiii

PAGE

86. Relation of motor torque curve to voltage regulation of system

Regulation coefficient of system Stability coefficient of

system

138

87. Effect of momentum on the stability of the motor Regulation

of overload capacity Gradual approach to instability

.

.

141

CHAPTER VII. HIGHER HARMONICS IN INDUCTION MOTORS.

88. Component torque curves due to the higher harmonics of the

impressed voltage wave, in a quarter-phase induction motor;

their synchronous speed and their direction, and the resultant

torque curve .

.

.

144

89. The component torque curves due to the higher harmonics of

the impressed voltage wave, in a three-phase induction motor

True three-phase and six-phase winding The single-phase

torque curve of the third harmonic

147

90. Component torque curves of normal frequency, but higher number of poles, due to the harmonics of the space distribu-

tion of the winding in the air-gap of a quarter-phase motor

Their direction and synchronous speeds

150

91. The same in a three-phase motor Discussion of the torque

components due to the time harmonics of higher frequency

and normal number of poles, and the space harmonics of normal

frequency and higher number of poles

154

92. Calculation of the coefficients of the trigonometric series repre-

senting the space distribution of quarter-phase, six-phase and

three-phase, full pitch and fractional pitch windings

155

M 93. Calculation of numerical values for 0, J, MJ

pitch defi-

ciency, up to the 21st harmonic

157

CHAPTER VII. SYNCHRONIZING INDUCTION MOTORS.

94. Synchronizing induction motors when using common secondary

resistance

159

95. Equation of motor torque, total torque and synchronizing
torque of two induction motors with common secondary rheo-

stat

160

96. Discussion of equations Stable and unstable position Maxi-

mum synchronizing power at 45 phase angle Numerical

instance

163

CHAPTER IX. SYNCHRONOUS INDUCTION MOTOR.

97. Tendency to drop into synchronism, of single circuit induction

motor secondary Motor or generator action at synchronism

Motor acting as periodically varying reactance, that is, as

reaction machine Low power factor Pulsating torque below

synchronism, due to induction motor and reaction machine

torque superposition

166

xlv

CONTENTS

CHAPTEK X. HYSTERESIS MOTOR.

PAGE

98. Rotation of iron disc in rotating magnetic field Equations

Motor below, generator above synchronism

168

99. Derivation of equations from hysteresis law Hysteresis torque

of standard induction motor, and relation to size

169

100. General discussion of hysteresis motor Hysteresis loop

collapsing or expanding

170

CHAPTER XI. ROTARY TERMINAL SINGLE-PHASE INDUCTION MOTORS.

101. Performance and method of operation of rotary terminal

single-phase induction Motor Relation of motor speed to

brush speed and slip corresponding to the load

172

102. Application of the principle to a self-starting single-phase power

motor with high starting and accelerating torque, by auxiliary

motor carrying brushes

. 173

CHAPTER XII. FREQUENCY CONVERTER OR GENERAL ALTERNATING CURRENT TRANSFORMER.

103. The principle of the frequency converter or general alternating

current transformer Induction motor and transformer special

cases Simultaneous transformation between primary elec-

trical and secondary electrical power, and between electrical

and mechanical power Transformation of voltage and of fre-

quency The air-gap and its effect

176

104. Relation of e.m.f., frequency, number of turns and exciting

current

177

105. Derivation of the general alternating current transformer

Transformer equations and induction motor equations, special

cases thereof

178

106. Equation of power of general alternating current transformer . 182

107. Discussion: between synchronism and standstill Backward

driving Beyond synchronism Relation between primary

electrical, secondary electrical and mechanical power . . . .184

108. Calculation of numerical instance

185

109. The characteristic curves: regulation curve, compounding

curve Connection of frequency converter with synchronous

machine, and compensation for lagging current Derivation of

equation and numerical instance

186

110. Over-synchronous operation Two applications, as double

synchronous generator, and as induction generator with low

frequency exciter

190

111. Use as frequency converter Use of synchronous machine or

induction machine as second machine Slip of frequency

Advantage of frequency converter over motor generator . . . 191

112. Use of frequency converter Motor converter, its advantages

and disadvantages Concatenation for multispeed operation . 192

CONTENTS

xv

CHAPTER XIII. SYNCHRONOUS INDUCTION GENERATOR.

PAGE 113. Induction machine as asynchronous motor and asynchronous

generator

194

114. Excitation of induction machine by constant low frequency

voltage in secondary Operation below synchronism, and

above synchronism

195

115. Frequency and power relation Frequency converter and syn-

chronous induction generator

196

1 16. Generation of two different frequencies, by stator and by rotor . 198

117. Power relation of the two frequencies Equality of stator and

rotor frequency: double synchronous generator Low rotor

frequency: induction generator with low frequency exciter,

Stanley induction generator

198

118. Connection of rotor to stator by commutator Relation of fre-

quencies and powers to ratio of number of turns of stator and

rotor

199

119. Double synchronous alternator General equation Its arma-

ture reaction

201

120. Synchronous induction generator with low frequency excita-

tion (a) Stator and rotor fields revolving in opposite direc-

tion (&) In the same direction Equations

203

121. Calculation of instance, and regulation of synchronous induc-

tion generator with oppositely revolving fields

204

122. Synchronous induction generator with stator and rotor fields

revolving in the same direction Automatic compounding and

over-compounding, on non-inductive load Effect of inductive

load

205

123. Equations of synchronous induction generator with fields re-

volving in the same direction

207

124. Calculation of numerical instance

209 .

.

.

CHAPTER XIV. PHASE CONVERSION AND SINGLE-PHASE GENERATION.

125. Conversion between single-phase and polyphase requires energy

atorage Capacity, inductance and momentum for energy

storage Their size and cost per Kva

212

126. Industrial importance of phase conversion from single-phase to polyphase, and from balanced polyphase to single-phase . . . 213
127. Monocyclic devices Definition of monocyclic as a system of

polyphase voltages with essentially single-phase flow of energy

Relativity of the term The monocyclic triangle for single-

phase motor starting

214

128. General equations of the monocyclic square

216

129. Resistance inductance monocyclic square Numerical in-

stance on inductive and on non-inductive load Discussion . 218

130. Induction phase converter Reduction of the device to the

simplified diagram of a double transformation

220

131. General equation of the induction phase converter

222

xvi

CONTENTS

PAGE 132. Numerical instance Inductive load Discussion and com-

parisons with monocyclic square

223

133. Series connection of induction phase converter in single-phase

.... induction motor railway Discussion of its regulation

226

134. Synchronous phase converter and single-phase generation

Control of the unbalancing of voltage due to single-phase load,

by stationary induction phase balancing with reverse rotation

of its polyphase system Synchronous phase balancer. . . . 227

135. Limitation of single-phase generator by heating of armature coils By double frequency pulsation of armature reaction Use of squirrel cage winding in field Its size Its effect on the

momentary short circuit current

229

136. Limitation of the phase converter in distributing single-phase load into a balanced polyphase system Solution of the

problem by the addition of a synchronous phase balancer to the

synchronous phase converter Its construction

230

137. The various methods of taking care of large single-phase loads

Comparison of single-phase generator with polyphase generator

and phase converter Apparatus economy

232

CHAPTER XV. SYNCHRONOUS RECTIFIERS.

138. Rectifiers for battery charging For arc lighting The arc ma-

chine as rectifier Rectifiers for compounding alternators

For starting synchronous motors Rectifying commutator

Differential current and sparking on inductive load Re-

sistance bipass Application to alternator and synchronous

motor

234

139. Open circuit and short circuit rectification Sparking with

open circuit rectification on inductive load, and shift of

brushes

237

140. Short circuit rectification on non-inductive and on inductive

load, and shift of brushes Rising differential current and flash-

Ing around the commutatorStability limit of brush position,
between sparking and flashing Commutating e.m.f . resulting from unsymmetrical short circuit voltage at brush shift

Sparkless rectification .

239

141. Short circuit commutation in high inductance, open circuit commutation in low inductance circuits Use of double brush

to vary short circuit Effect of loadThomson Houston arc

machine Brush arc machine Storage battery charging . , 243 142. Reversing and contact making rectifier Half wave rectifier

and its disadvantage by unidirectional magnetization of transformer The two connections full wave contact making recti-

fiers Discussion of the two types of full wave rectifiers The mercury arc rectifier

245

143. Rectifier with intermediary segments Polyphase rectifica-
tionStar connected, ring connected and independent phase

CONTENTS

xvii

PAGE

Y rectifiers

connected three-phase rectifier Delta connected

three-phase rectifier Star connected quarter-phase rectifier
Quarter-phase rectifier with independent phases Ring con-
nected quarter-phase rectifier Wave shapes and their discus-

sion Six-phase rectifier

250

144. Ring connection or independent phases preferable with a large number of phases Thomson Houston arc machine as con-

stant current alternator with three-phase star connected rectifier Brush arc machine as constant current alternator with

quarter-phase rectifiers in series connection

254

145. Counter e.m.f. shunt at gaps of polyphase ring connected

rectifier Derivation of counter e.m f . from synchronous mo-

tor Leblanc's Panchahuteur Increase of rectifier output with

increasing number of phases

. . 255

146. Discussion: stationary rectifying commutator with revolving

brushes Permutator Rectifier with revolving transformer

Use of synchronous motor for phase splitting in feeding

rectifying commutator: synchronous converter Conclusion . 257

CHAPTER XVI. REACTION MACHINES.

147. Synchronous machines operating without field excitation . . 260 148. Operation of synchronous motor without field excitation de-
pending on phase angle between resultant rn.m.f. and magnetic

flux, caused by polar field structure Energy component of

reactance . .

.

.

... .

261

149. Magnetic hysteresis as instance giving energy component of

reactance, as effective hysteretie resistance . .

.

262

150. Make and break of magnetic circuit Types of reaction

machines Synchronous induction motor Reaction machine

as converter from d.-c. to a.-c

263

151. Wave shape distortion in reaction machine, due to variable

reactance, and corresponding hysteresis cycles

264

152. Condition of generator and of motor action of the reactance

.... machine, as function of the current phase

... 267

153. Calculation of reaction machine equation Power factor and

maximum power

268

154. Current, power and power factor

Numerical instance

271 .

,

.

155. Discussion Structural similarity with inductor machine . 272

CHAPTER XVII. INDUCTOR MACHINES.

156. Description of inductor machine type Induction by pulsating

unidirectional magnetic flux

274

157. Advantages and disadvantages of inductor type, with regards

to field and to armature

275

158. The magnetic circuit of the inductor machine, calculation of

magnetic flux and hysteresis loss

276

xviii

CONTENTS

PAGE

159. The Stanley type of inductor alternator The Alexanderson

high frequency inductor alternator for frequencies of 100,000

cycles and over

. 279

160. The Eickemeyer type of inductor machine with bipolar field

The converter from direct current to high frequency alternating

current of the inductor type .

280

161. Alternating current excitation of inductor machine, and high

frequency generation of pulsating amplitude. Its use as

amplifier Amplification of telephone currents by high fre-

quency inductor in radio communication

281

162. Polyphase excitation of inductor, and the induction motor

inductor frequency converter

282

163. Inductor machine with reversing flux, and magneto communi-

cation Transformer potential regulator with magnetic com-

mutation

284

164. The interlocking pole type of field design in alternators and

commutating machines

286

165. Relation of inductor machine to reaction machine Half syn-

chronous operation of standard synchronous machine as

inductor machine

287

CHAPTER XVIII. SURGING OF SYNCHRONOUS MOTORS.

166. Oscillatory adjustment of synchronous motor to changed condition of load Decrement of oscillation Cumulative oscil-

lation by negative decrement

288

167. Calculation of equation of electromechanical resonance . . . 289

168. Special cases and example

... 292

169. Anti-surging devices and pulsation of power

293

170. Cumulative surging Due to the lag of some effect behind its

cause

Involving a frequency transformation of power

296 .

.

.

CHAPTER XIX. ALTERNATING CURRENT MOTOES IN GENERAL.

171. Types of alternating-current motors

300

172. Equations of coil revolving in an alternating field

302

173. General equations of alteraating-curreat motor

304

174. Polyphase induction motor, equations

307

175. Polyphase induction motor, slip, power, torque

310

176. Polyphase induction motor, characteristic constants . . . .312

177. Polyphase induction motor, example

313

178. Singlephase induction motor, equations

314

179. Singlephase induction motor, continued

316

180. Singlephase induction motor, example

318

181. Polyphase shunt motor, general

319

182. Polyphase shunt motor, equations

320

183. Polyphase shunt motor, adjustable speed motor

321

184. Polyphase shunt motor, synchronous speed motor

323

185. Polyphase shunt motor, phase control by it

324

186. Polyphase shunt motor, short-circuit current under brushes . 327

187. Polyphase series motor, equations

327

188. Polyphase series motor, example

330

CONTENTS

xix

CHAPTER XX. SINGLE-PHASE COMMUTATOR MOTORS.

PAGE

189. General: proportioning of parts of a.-c. commutator motor

different from d.-c

331

190. Power factor: low field flux and high armature reaction re-

quired Compensating winding necessary to reduce armature

self-induction

332

191. The three circuits of the single-phase commutator motor Compensation and over-compensation Inductive compen-

sation Possible power factors

336

192. Field winding and compensating winding: massed field winding and distributed compensating winding Under-com-

pensation at brushes, due to incomplete distribution of com-

pensating winding

338

193. Fractional pitch armature winding to secure complete local compensation Thomson's repulsion motor Eickemeyer in-

ductively compensated series motor

339

194. Types of varying speed single-phase commutator motors: con-

ductive and inductive compensation; primary and secondary

excitation; series and repulsion motors Winter Eichberg Latour motor Motor control by voltage variation and by

change of type

341

195. The quadrature magnetic flux and its values and phases in the

different motor types

.... 345

196. Commutation: e.m.f. of rotation and e.m.f. of alternation

Polyphase system of voltages Effect of speed

347

197. Commutation determined by value and phase of short circuit

current High brush contact resistance and narrow brushes . 349

198. Commutator leads Advantages and disadvantages of resist-

ance leads in running and in starting

351

199. Counter e.m.f. in commutated coil: partial, but not com-

plete neutralization possible

354

200. Commutating field Its required intensity and phase rela-

tions: quadrature field

356

201. Local commutating pole Neutralizing component and revers-

ing component of commutating field Discussion of motor

types regarding commutation

358

202. Motor characteristics: calculation of motor Equation of cur-

rent, torque, power

361

203. Speed curves and current curves of motor Numerical instance

Hysteresis loss increases, short circuit current decreases

power factor

364

204. Increase of power factor by lagging field magnetism, by

resistance shunt across field

366

205. Compensation for phase displacement and control of power

factor by alternating current commutator motor with lagging

field flux, as effective capacity Its use in induction motors and

other apparatus

370

206. Efficiency and losses: the two kinds of core loss

370

xx

CONTENTS

PAGE

207. Discussion of motor types: compensated series motors: conductive and inductive compensation Their relative advan-

tages and disadvantages

371

208. Repulsion motors: lagging quadrature flux Not adapted to

speeds much above synchronism Combination type: series

repulsion motor

373

209. Constructive differences Possibility of changing from type to

type, with change of speed or load

375

210. Other commutator motors: shunt motor Adjustable speed

polyphase induction motor Power factor compensation:

Heyland motor Winter-Eichberg motor

377

211. Most general form of single-phase commutator motor, with two

stator and two rotor circuits and two brush short circuits . . 381

212. General equation of motor

382

213. Their application to the different types of single-phase motor

with series characteristic

383

214. Repulsion motor: Equations

385

215. Continued

388

216. Discussion of commutation current and commutation factor . 391

217. Repulsion motor and repulsion generator

394

218. Numerical instance

395

219. Series repulsion motor: equations 220. Continued

397 398

221. Study of commutation Short circuit current underbrushes . 403

222. Commutation current

404

223. Effect of voltage ratio and phase, on commutation

406

224. Condition of vanishing commutation current

408

225. Numerical example

411

226. Comparison of repulsion motor and various series repulsion

motor

414

227. Further example Commutation factors

415

228. Over-compensation Equations

4 IS

229. Limitation of preceding discussion Effect and importance of

transient in short circuit current

419

CHAPTER XXI. REGULATING POLE CONVERTER.

230. Change of converter ratio by changing position angle between

brushes and magnetic flux, and by change of wave shape . . 422

A. Variable ratio by change of position angle between com-

mutator brushes and resultant magnetic flux

422

231. Decrease of a.-c. voltage by shifting the brushes By shifting

the magnetic flux Electrical shifting of the magnetic flux by varying the excitation of the several sections of the field pole . 422 232. Armature reaction and commutation Calculation of the re-

sultant armature reaction of the converter with shifted mag-

netic flux

426

233. The two directions of shift flux, the one spoiling, the other

CONTENTS

xxi

PAGE

improving commutation Demagnetizing armature reaction

and need of compounding by series field

429

Y B. Variable ratio by change of the wave shape of the voltage 429

234. Increase and decrease of d.-c. voltage by increase or decrease

of maximum a.-e. voltage by higher harmonic Illustration

by third and fifth harmonic

430

235. Use of the third harmonic in the three-phase system Transformer connection required to limit it to the local converter

circuit Calculation of converter wave as function of the pole

arc

432

236. Calculation of converter wave resulting from reversal of

middle of pole arc

.

.

. 435

237. Discussion

436

238. Armature reaction and commutation Proportionality of

resultant armature reaction to deviation of voltage ratio from

normal

437

239. Commutating flux of armature reaction of high a.-c. voltage Combination of both converter types, the wave shape distor-

tion for raising, the flux shift for lowering the a.-c. voltage Use of two pole section, the main pole and the regulating pole . 437 240. Heating and rating Relation of currents and voltages in

standard converter

439

241. Calculation of the voltages and currents in the regulating pole

converter

440

242. Calculating of differential current, and of relative heating of

armature coil

442

243. Average armature heating of n phase converter

444

244. Armature heating and rating of three-phase and of six-phase

regulating pole converter

445

245. Calculation of phase angle giving minimum heating or maxi-

mum rating

446

246. Discussion of conditions giving minimum heating Design

Numerical instance

448

CHAPTER XXII. UNIPOLAR MACHINES.

Homopolar Machines Acyclic Machines

247. Principle of unipolar, homopolar or acyclic machine The

problem of high speed current collection Fallacy of unipolar

induction in stationary conductor Immaterial whether mag-

net standstill or revolves The conception of lines of magnetic

force

450

248. Impossibility of the coil wound unipolar machine All electro-

magnetic induction in turn must be alternating Illustration

of unipolar induction by motion on circular track

452

249. Discussion of unipolar machine design Drum type and disc

type Auxiliary air-gap Double structure Series connection of conductors with separate pairs of collector rings . . . 454

xxii

CONTENTS

PAGE

250. Unipolar machine adapted for low voltage, or for large size high

speed machines Theoretical absence of core loss Possibility

of large core loss by eddies, in core and in collector rings, by

pulsating armature reaction

456

251. Circular magnetization produced by armature reaction

Liability to magnetic saturation and poor voltage regulation

Compensating winding Most serious problem the high speed

collector rings

457

252. Description of unipolar motor meter

.

.

,

. 458

CHAPTER XXIII. REVIEW.

253. Alphabetical list of machines: name, definition, principal

characteristics, advantages and disadvantages . .

... 459

CHAPTER XXIV. CONCLUSION.

254. Little used and unused types of apparatus Their knowledge important due to the possibility of becoming of great industrial importance Illustration by commutating pole machine . . 472
255. Change of industrial condition may make new machine types

important Example of induction generator for collecting

numerous small water powers

473

256. Relative importance of standard types and of special types of

machines

474

257. Classification of machine types into induction, synchronous,

commutating and unipolar machines Machine belonging to

two and even three types

474

INDEX

477

THEORY AND CALCULATION OF
ELECTRICAL APPARATUS
CHAPTER I
SPEED CONTROL OF INDUCTION MOTORS
I. STARTING AND ACCELERATION
1. Speed control of induction motors deals with two problems: to produce a high torque over a wide range of speed down to standstill, for starting and acceleration; and to produce an approximately constant speed for a wide range of load, for
constant-speed operation. In its characteristics, the induction motor is a shunt motor,
that is, it runs at approximately constant speed for all loads, and this speed is synchronism at no-load. At speeds below full speed, and at standstill, the torque of the motor is low and the current high, that is, the starting-torque efficiency and especially the apparent starting-torque efficiency are low.
Where starting with considerable load, and without excessive current, is necessary, the induction motor thus requires the use
of a resistance in the armature or secondary, just as the directcurrent shunt motor, and this resistance must be a rheostat,
that is, variable, so as to have maximum resistance in starting,
and gradually, or at least in a number of successive steps, cut
out the resistance during acceleration.
This, however, requires a wound secondary, and the squirrelcage type of rotor, which is the simplest, most reliable and therefore most generally used, is not adapted for the use of a starting rheostat. With the squirrel-cage type of induction motor, starting thus is usually done and always with large motors by lowering the impressed voltage by autotransformer, often in a number of successive steps. This reduces the starting
current, but correspondingly reduces the starting torque, as it does not change the apparent starting-torque efficiency.
The higher the rotor resistance, the greater is the starting torque, and the less, therefore, the starting current required for
1

2

ELECTRICAL APPARATUS

a given torque when starting by autotransformer. However, high rotor resistance means lower efficiency and poorer speed
regulation, and this limits the economically permissible resistance

in the rotor or secondary.
Discussion of the starting of the induction motor by arma-

ture rheostat, and of the various speed-torque curves produced

by various values of starting resistance in the induction-motor

secondary,

are

given

in

"
Theory

and

Calculation

of

Alternating-

current Phenomena" and in "Theoretical Elements of Electrical

7
Engineering.'
As seen, in the induction motor, the (effective) secondary resistance should be as low as possible at full speed, but should
be high at standstill very high compared to the full-speed value and gradually decrease during acceleration, to maintain constant high torque from standstill to speed. To avoid the inconvenience and complication of operating a starting rheostat,
various devices have been proposed and to some extent used, to
produce a resistance, which automatically increases with increasing slip, and thus is low at full speed, and higher at standstill.

A. Temperature Starting Device
A 2. resistance material of high positive temperature coeffi-
cient of resistance, such as iron and other pure metals, operated at high temperature, gives this effect to a considerable extent: with increasing slip, that is, decreasing speed of the motor, the secondary current increases. If the dimensions of the secondary resistance are chosen so that it rises considerably in temperature, by the increase of secondary current, the temperature and therewith the resistance increases.
Approximately, the temperature rise, and thus the resistance
rise of the secondary resistance, may be considered as propor-
tional to the square of the secondary-current, ii, that is, repre-

sented by:

= + r

r (1

2
aii ).

(1)

As illustration, consider a typical induction motor, of the

constants :

e = 110;

7

- - = g jb = 0.01

0.1 j;

+ ZQ - rQ +jxQ = 0.1

0.3 j;

+ + Zi = ri jxi = 0.1

0.3 j;

A the speed-torque curve of this motor is shown as in Fig. 1,

SPEED CONTROL
Suppose now a resistance, r, is inserted in series into the secondary circuit, which when cold that is, at light-load equals
the internal secondary resistance:

but increases so as to double with 100 amp, passing through it. This resistance can then be represented by:

+ r = r (1

if 10-4)

+ = 0.1 (1

if

10~

4 ),

FIG. 1. High-starting and acceleration torque of induction motor by posi-
tive temperature coefficient of secondary resistance.

and the total secondary resistance of the motor then is:

n + + =
r'i

TQ (1 a if)

(2)

+ = 0.2 (1

0.5

if

10-

4 ).

To calculate the motor characteristics for this varying resist-

ance, r'i, we use the feature, that a change of the secondary resistance of the induction motor changes the slip, s, in proportion

to the change of resistance, but leaves the torque, current, powerfactor, torque efficiency, etc., unchanged, as shown on page
We 322 of " Theoretical Elements of Electrical Engineering."

thus

calculate

the

motor

for

constant secondary resistance,

r 3,

but otherwise the same constants, in the manner discussed on

page

318

of

" Theoretical

Elements

of

Electrical

7
Engineering/

4

ELECTRICAL APPARATUS

A This gives curve of Fig. 1. At any value of torque, T, corre-
sponding to slip, s, the secondary current is:

+ ii - e \/&i2

2
&2 >

herefrom follows by (2) the value of r'i, and from this the new

value of slip :

+ = + s'

s

r'i

ri.

(3)

The torque, T, then is plotted against the value of slip, s', and

B B gives curve

of Fig. 1. As seen,

gives practically constant

torque over the entire range from near full speed, to standstill.

B Curve has twice the slip at load, as A, as its resistance has

been doubled.

3. Assuming, now, that the internal resistance, r x, were made

as low as
resistance

n possible,

= 0.05,

of high temperature

and the rest
coefficient: r

added as external
= 0.05, giving the

total resistance :

(4)

This gives the same resistance as curve A: r\ = 0.1, at light-
load, where ii is small and the external part of the resistance cold. But with increasing load the resistance, r'i, increases, and the
motor gives the curve shown as C in Fig. 1. As seen, curve C is the same near synchronism as A, but in
starting gives twice as much torque as A, due to the increased
resistance.
C and A thus are directly comparable: both have the same
constants and same speed regulation and other performance at
speed, but C gives much higher torque at standstill and during
acceleration.
For comparison, curve A! has been plotted with constant resistance r\ 0.2, so as to compare with B.
Instead of inserting an external resistance, it would be pref-
erable to use the internal resistance of the squirrel cage, to in-
crease in value by temperature rise, and thereby improve the

starting torque.

Considering in this respect the motor shown as curve C, At

= standstill, it is: ii

153; thus r\ = 0.217; while cold, the re-

= sistance is: r'i 0.1. This represents a resistance rise of 117

per cent. At a temperature coefficient of the resistance of 0.35,
this represents a maximum temperature rise of 335C. As seen,

SPEED CONTROL

5

by going to temperature of about 350C. in the rotor conductors which naturally would require fireproof construction it be-
A A comes possible to convert curve into C, or f into B, in Fig. 1.
Probably, the high temperature would be permissible only in the end connections, or the squirrel-cage end ring, but then, iron could be used as resistance material, which has a materially higher temperature coefficient, and the required temperature rise thus would probably be no higher.

B. Hysteresis Starting Device

4. Instead of increasing the secondary resistance with increasing slip, to get high torque at low speeds, the same result can be
produced by the use of an effective resistance, such as the effective or equivalent resistance of hysteresis, or of eddy currents.
As the frequency of the secondary current varies, a magnetic circuit energized by the secondary current operates at the varying frequency of the slip, s.
At a given current, ii, the voltage required to send the current through the magnetic circuit is proportional to the frequency, that is, to s. Hence, the susceptance is inverse proportional
to s:

y-J-

(q

The angle of hysteretic advance of phase, a, and the power-

factor, in a closed magnetic circuit, are independent of the

frequency, and vary relatively little with the magnetic density

and

thus

the

current,

over

a

wide

1
range,

thus

may

approxi-

mately be assumed as constant. That is, the hysteretic con-

ductance is proportional to the susceptance :

g' = V tan a.

(6)

Thus, the exciting admittance, of a closed magnetic circuit of negligible resistance and negligible eddy-current losses, at the
frequency of slip, s, is given by:

Y'

=

r
g

-1 jb

=

V (tan

a

j)

a

.6

-J-,--

6 ;

(, tan-j).x

frt\
(7)

l " Theory and Calculation of Alternating-current Phenomena," Chapter XII.

6

ELECTRICAL APPARATUS

Assuming tan a = 0.6, which is a fair value for a closed mag-
netic circuit of high hysteresis loss, it is :

r =~

-

(0.6 J),

the exciting admittance at slip, s.
Assume then, that such an admittance, F', is connected in series
into the secondary circuit of the induction motor, for the purpose of using the effective resistance of hysteresis, which increases with the frequency, to control the motor torque curve.
The total secondary impedance then is :

r7f Z/ 1

. *7 JL. l\ -f-yf

i+S

where : Y = g jb is the admittance of the magnetic circuit at

full frequency, and

V0 + y =

2

b2.

5. For illustration, assume that in the induction motor of the

constants :

= e Q

100;

- y = o.02 0.2 j;

+ Z = 0.05 0.15 j;

+ Z l = 0.05

0.15 j;

a closed magnetic circuit is connected into the secondary, of full

frequency admittance, and assume:

Y = g-jb;
g = 0.666 = 4;

thus, by (8) :

+ + Z\ = (0.05

0.11 a)

0.335 js.

(9)

The characteristic curves of this induction motor with hysteresis

starting device can now be calculated in the usual manner, dif-

fering from the standard motor only in that Z\ is not constant,

m and

the

proper

value

of

r i;

a? 3

and

has to be used for every

slip, s.
Fig. 2 gives the speed-torque curve, and Fig. 3 the load curves of this motor.

SPEED CONTROL

For

comparison

is

shown,

as

Tf ,

in

dotted

lines,

the

torque

curve of the motor of constant secondary resistance, and of the

constants:

- 7o = 0.01

0.1 j;

Z = + Q

0.01

0.3 j',

+ Zi = 0.1

0.3 j;

As seen, the hysteresis starting device gives higher torque at standstill and low speeds, with less slip at full speed, thus a
materially superior torque curve.

INDUCTION MOTOR Y =.02-.2j; Z =.05-K15;? ; 6 =100
+ Z 1 = (.05 .11s)-K335;7*s
SPEED CONTROL BY HYSTERESIS SPEED CURVES

FIG. 2. Speed curves of induction motor with hysteresis starting device.

p represents the power-factor, T? the efficiency, 7 the apparent efficiency, 77' the torque efficiency and 7' the apparent torqiie

efficiency.

However, T corresponds to a motor of twice the admittance

and

half

the

impedance

of

Te .

That is, to get approximately

the same output, with the hysteresis device inserted, as without

it, requires a rewinding of the motor for higher magnetic density,
the same as would be produced in Tf by increasing the voltage

-\/2 times.

It is interesting to note in comparing Fig. 2 with Fig. 1, that
the change in the torque curve at low and medium speed, pro-

duced by the hysteresis starting device, is very similar to that

produced by temperature rise of the secondary resistance; at

8

ELECTRICAL APPARATUS

speed, however, the hysteresis device reduces the slip, while the

temperature device leaves it unchanged.
The foremost disadvantage of the use of the hysteresis device

is the impairment of the power-factor, as seen in Fig. 3 as p.
The introduction of the effective resistance representing the

hysteresis of necessity introduces a reactance, which is higher

than

the

resistance,

and

thereby

impairs

the

motor

characteristics. "

Comparing Fig. 3 with Fig. 176, page 319 of Theoretical

FIG. 3. Load curves of induction motor with hysteresis starting device.

Elements

of

Electrical

7
Engineering/

which

gives

the

load

curves

of Tf of Fig. 2, it is seen that the hysteresis starting device reduced

the maximum power-factor, p, from 91 per cent, to 84 per cent.,

and the apparent efficiency, 7, correspondingly.

This seriously limits the usefulness of the device.

C. Eddy-current Starting Device

6. Assuming that, instead of using a well-laminated magnetic circuit, and utilizing hysteresis to give the increase of effective resistance with increasing slip, we use a magnetic circuit having very high eddy-current losses: very thick laminations or solid iron, or we directly provide a closed high-resistance secondary winding around the magnetic circuit, which is inserted into the induction-motor secondary for increasing the starting torque.

SPEED CONTROL

9

The susceptance of the magnetic circuit obviously follows the same law as when there are no eddy currents. That is:

At a given current, ii, energizing the magnetic circuit, the induced voltage, and thus also the voltage producing the eddy currents, is proportional to the frequency. The currents are proportional to the voltage, and the eddy-current losses, therefore, are proportional to the square of the voltage. The eddy-
current conductance, g, thus is independent of the frequency.
The admittance of a magnetic circuit consuming energy by eddy currents (and other secondary currents in permanent closed
circuits), of negligible hysteresis loss, thus is represented, as
function of the slip, by the expression:

Y' = g-j~ -

(11)

s

Connecting such an admittance in series to the inductionmotor secondary, gives the total secondary impedance:

+ Z = ',

Z-,

+ + = Ax

+ 2-j3\ j /*!

*
,

\'

(12)

Assuming :

= g

&.

(13)

That is, 45 phase angle of the exciting circuit of the magnetic circuit at full frequency which corresponds to complete screening of the center of the magnet core we get:

Fig. 4 shows the speed curves, and Fig. 5 the load curves, calculated in the standard manner, of a motor with eddy-current

starting device in the secondary, of the constants:

6 = 100;
- F = o.03 0.3 j;

+ Z = 0.033

0.1 jf;

+ Zi = 0.033 0.1 j;

6-3;

10 thus:

ELECTRICAL APPARATUS

7. As seen, the torque curve has a very curious shape: a
maximum at 7 per cent, slip, and a second higher maximum at
standstill.
The torque efficiency is very high at all speeds, and practically constant at 82 per cent, from standstill to fairly close of full speed, when it increases.

INDUCTION MOTOR

.Sj; 2 ~ -033+ 1j :

100

SPEED CONTROL BY EDDIES

.o

SPEED CURVES

FIG. 4. Speed curves of induction motor with eddy-current starting device.
But the power-factor is very poor, reaching a maximum of
78 per cent, only, and to get the output from the motor, required rewinding it to give the equivalent of a \/3 times as high voltage.
For comparison, in dotted lines as T1 is shown the torque curves of the standard motor, of same maximum torque. As seen, in
the motor with eddy-current starting device, the slip at load is very small, that is, the speed regulation very good. Aside from the poor power-factor, the motor constants would be very
satisfactory.
The low power-factor seriously limits the usefulness of the
device.
By differently proportioning the eddy-current device to the
secondary circuit, obviously the torque curve can be modified

SPEED CONTROL

11

and the starting torque reduced, the depression in the torque curve between full-speed torque and starting torque eliminated,
etc.
Instead of using an external magnetic circuit, the magnetic
circuit of the rotor or induction-motor secondary may be used,
and in this case, instead of relying on eddy currents, a definite secondary circuit could be utilized, in the form of a second squirrel cage embedded deeply in the rotor iron, that is, a double squirrel-cage motor.

FIG. 5. Load curves of induction motor with eddy-current starting device.
In the discussion of the multiple squirrel-cage induction motor,
Chapter II, we shall see speed-torque curves of the character as shown in Fig. 4. By the use of the rotor iron as magnetic circuit, the impairment of the power-factor is somewhat reduced, so that the multiple squirrel-cage motor becomes industrially
important.
A further way of utilizing eddy currents for increasing the
effective resistance at low speeds, is by the use of deep rotor
bars. By building the rotor with narrow and deep slots filled
with solid deep bars, eddy currents in these bars occur at higher frequencies, or unequal current distribution. That is, the current flows practically all through the top of the bars at the high

12

ELECTRICAL APPARATUS

frequency of low motor speeds, thus meeting with, a high resistance. With increasing motor speed and thus decreasing secondary frequency, the current penetrates deeper into the bar, until at full speed it passes practically uniformly throughout the entire bar, in a circuit of low resistance but somewhat
increased reactance.
The deep-bar construction, the eddy-current starting device and the double squirrel-cage construction thus are very similar in the motor-performance curves, and the double squirrel cage, which usually is the most economical arrangement, thus will be discussed more fully in Chapter II.
II. CONSTANT-SPEED OPERATION
8. The standard induction motor is essentially a constant-speed
motor, that is, its speed is practically constant for all loads, decreasing slightly with increasing load, from synchronism at no-load. It thus has the same speed characteristics as the directcurrent shunt motor, and in principle is a shunt motor.
In the direct-current shunt motor, the speed may be changed
by: resistance in the armature, resistance in the field, change of the voltage supply to the armature by a multivolt supply circuit,
as a three-wire system, etc.
In the induction motor, the speed can be reduced by inserting resistance into the armature or secondary, just as in the directcurrent shunt motor, and involving the same disadvantages: the reduction of speed by armature resistance takes place at a sacrifice of efficiency, and at the lower speed produced by armature resistance, the power input is the same as it would be with the same motor torque at full speed, while the power output is reduced by the reduced speed. That is, speed reduction by armature resistance lowers the efficiency in proportion to the
lowering of speed. The foremost disadvantage of speed control by armature resistance is, however, that the motor ceases to be a constant-speed motor, and the speed varies with the load: with a given value of armature resistance, if the load and with it
the armature current drops to one-half, the speed reduction of
the motor, from full speed, also decreases to one-half, that is, the motor speeds up, and if the load comes off, the motor runs up to practically full speed. Inversely, if the load increases, the speed slows down proportional to the load.
With considerable resistance in the armature, the induction

SPEED CONTROL

13

motor thus has rather series characteristic than shunt character-
istic, except that its speed is limited by synchronism.
Series resistance in the armature thus is not suitable to produce steady running at low speeds.
To a considerable extent, this disadvantage of inconstancy of
speed can be overcome:
(a) By the use of capacity or effective capacity in the motor
secondary, which contracts the range of torque into that of approximate resonance of the capacity with the motor inductance, and thereby gives fairly constant speed, independent of the load, at various speed values determined by the value of the capacity.
(6) By the use of a resistance of very high negative tempera-
ture coefficient in the armature, so that with increase of load and current the resistance decreases by its increase of temperature, and thus keeps approximately constant speed over a wide range
of load.
Neither of these methods, however, avoids the loss of efficiency incident to the decrease of speed.
9. There is no method of speed variation of the induction motor analogous to field control of the shunt motor, or change of the armature supply voltage by a multivolt supply system.
The field excitation of the induction motor is by what may be
called armature reaction. That is, the same voltage, impressed upon the motor primary, gives the energy current and the field exciting current, and the field excitation thus can not be varied without varying the energy supply voltage, and inversely. Furthermore, the no-load speed of the induction motor does not depend on voltage or field strength, but is determined by
synchronism.
The speed of the induction motor can, however, be changed:
(a) By changing the impressed frequency, or the effective
frequency.
(6) By changing the number of poles of the motor.
Neither of these two methods has any analogy in the directcurrent shunt motor: the direct-current shunt motor has no fre-
quency relation to speed, and its speed is not determined by the number of poles, nor is it feasible, with the usual construction of direct-current motors, to easily change the number of poles.
In the induction motor, a change of impressed frequency correspondingly changes the synchronous speed. The effect of a change of frequency is brought about by concatenation of the

14

ELECTRICAL APPARATUS

motor with a second motor, or by internal concatenation of the motor: hereby the effective frequency, which determines the no-load or synchronous speed, becomes the difference between
primary and secondary frequency. Concatenation of induction motors is more fully discussed in
Chapter III. As the no-load or synchronous speed of the induction motor
depends on the number of poles, a change of the number of poles changes the motor speed. Thus, if in a 60-cycle induction motor, the number of poles is changed from four to six and to eight, the speed is changed from 1800 to 1200 and to 900 revolutions per
minute.
This method of speed variation of the induction motor, by changing the number of poles, is the most convenient, and such "multispeed motors" are extensively used industrially,

A. Pyro-electric Speed Control

10. Speed control by resistance in the armature or secondary has the disadvantage that the speed is not constant, but at a change of load and thus of current, the voltage consumed

by the armature resistance, and therefore the speed changes*

To give constancy of speed over a range of load would require

a resistance, which consumes the same or approximately the

A same voltage at all values of current.

resistance of very

high negative temperature coefficient does this: with increase of

current and thus increase of temperature, the resistance decreases,

and if the decrease of resistance is as large as the increase of

current, the voltage consumed by the resistance, and therefore

the motor speed, remains constant.
"
Some pyro-electric conductors (see Chapter I, of Theory

and

Calculation

of

Electric

77
Circuits )

have

negative

tempera-

ture coefficients sufficiently high for this purpose. Fig. 6 shows

the current-resistance characteristic of a pyro-electric conductor,

consisting of cast silicon (the same of which the characteristic

is given as rod II in Fig. 6 of "Theory and Calculation of Electric

Circuits")' Inserting this resistance, half of it and one and one-

half of it into the secondary of the induction motor of constants :

+ - e = 110; 7o = 0.01

0.1 j;Z* 0.1

0.3 Zl = 0.1 +0.3j

gives the speed-torque curves shown in Fig. 7,
The calculation of these curves is as follows: The speed-
torque curve of the motor with short-circuited secondary, r = 0,

SPEED CONTROL

15

FIG. 6. Variation of resistance of pyro-electric conductor, with current.
PYRO-ELECTRIC RESISTANCE IN SEC9NDARY OF INDUCTION MOTOR, < =110. SPE"E'D CONTROL BY PYRO' ELECTRIC ^CONDUCTOR. SPEED CURVES.

0.1 3> 7.

0.2

0.3

0.4

0.5

0.6

0.7

0O.8a

0.9

1-

Speed control of induction motor by pyro-electric conductor,
speed curves

16

ELECTRICAL APPARATUS

is calculated in the usual way as described on page 318 of

" Theoretical

Elements

of

Electrical

7'
Engineering.

For any

value of slip, s, and corresponding value of torque, T, the secondary

vV + = current is ii e

a2 2.

To this secondary current corre-

sponds, by Fig. 6, the resistance, r, of the pyro-electric conductor, and the insertion of r thus increases the slip in proportion to the

increased secondary resistance: -^r^' where n = 0.1 in the

present instance. This gives, as corresponding to the torque,

!F, the slip:

+ r

TI

where s

=

slip at torque,

Tf with short-circuited ,

armature, or

resistance, r\.
As seen from Fig. 7, very close constant-speed regulation is produced by the use of the pyro-electric resistance, over a wide range of load, and only at light-load the motor speeds up.
Thus, good constant-speed regulation at any speed below synchronism, down to very low speeds, would be produced at a corresponding sacrifice of efficiency, however by the use of suitable pyro-electric conductors in the motor armature.
The only objection to the use of such pyro-electric resistances is the difficulty of producing stable pyro-electric conductors, and
permanent terminal connections on such conductors.

B. Condenser Speed Control
11. The reactance of a condenser is inverse proportional to the frequency, that of an inductance is directly proportional to
the frequency. In the secondary of the induction motor, the frequency varies from zero at synchronism, to full frequency at standstill. If, therefore, a suitable capacity is inserted into the secondary of an induction motor, there is a definite speed, at which inductive reactance and capacity reactance are equal and opposite, that is, balance, and at and near this speed, a large current is taken by the motor and thus large torque developed, while at speeds considerably above or below this resonance speed, the current and thus torque of the motor are small.
The use of a capacity, or an effective capacity (as polarization cell or aluminum cell) in the induction-motor secondary should therefore afford, at least theoretically, a means of speed
control by varying the capacity.

SPEED CONTROL

17

Let, In an induction motor:

YQ = g jb = primary exciting admittance;

+ ZQ = TQ jxQ = primary self-inductive impedance;

+ Zi = TI

jxi = internal self-inductive impedance, at full

frequency ; and let the condenser, C, be inserted into the secondary circuit.
The capacity reactance of C is

at

full

frequency,

and

k-
5

at

the

frequency

of

slip,

s.

The total secondary impedance, at slip, s, thus is :

_ _ _ _ Zi'-ri+^as!-*)

and the secondary current:

___

sE_

_ .
f

$e

(2)

where :

= - jE (ai

ja 2),

a

=

1
m

a2 =

m

(4)

'
(k\SXi -j

The further calculation of the condenser motor, then, is the same as that of the standard motor. 1
12. Neglecting the exciting current:
/oo = EY

the primary current equals the secondary current:
7o = /i

and the primary impressed voltage thus is:

1 " Theoretical Elements of Electrical Engineering/' 4th edition, p. 318.
2

18

ELECTRICAL APPARATUS

and, substituting (3) and rearranging, gives:

or, absolute:

+ + ^i + (ri

sr )

jf

SXQ - -
j

+ + + 2 sro)

(sxi

sx - -

The torque of the motor is :
T = e2ai
and, substituting (4) and (6):

(/c\ + s#i

sa;o

J

As seen, this torque is a maximum in the range of slip, s,
where the second term in the denominator vanishes, while for values of s, materially differing therefrom, the second term in the denominator is large, and the torque thus small.
That is, the motor regulates for approximately constant speed
near the value of s, given by:

+ = SXl

SXQ _ *

o

that is:

/

I

TTX

(8)

and

= so

1,

that

is,

the

motor

gives

maximum

torque

near

standstill, for:

+ k = XQ XL

(9)

13. As instances are shown, in Fig. 8, the speed-torque curves of a motor of the constants:

- Fo - 0.01

0.1 j,

+ Z = Zi = 0.1 0.3 j,

SPEED CONTROL

19

for the values of capacity reactance :

S

= k

0, 0.012, 0,048, 0.096, 0.192, 0.3, 0.6 denoted respectively

by 1, 2, 3, 4, 5, 6, 7.

The impressed voltage of the motor is assumed to be varied

with the change of capacity, so as to give the same maximum

torque for all values of capacity.

The volt-ampere capacity of the condenser is given, at the

frequency of slip, s, by:

substituting (3) and (6), this gives:

= '

+ + (ri

r )>

+ *CO -

(*BI

~)'

SPEED CONTROL OF INDUCTION MOTOR BY CONDENSER IN SECONDARY CAPACITY* REACTANCE k: d'). 0; (2) ".012; (3) .048;
(4-) .096, (5) .192; (6) .3; (7) .6
7
\

\

1.0

.8

xi

FIG. 8. Speed control of induction motor by condenser in secondary circuit. Speed curves.

and, compared with (7), this gives:

At full frequency, with the same voltage impressed upon the condenser, its volt-ampere capacity, and thus its 60-eycle rating, would be:

20

ELECTRICAL APPARATUS

As seen, a very large amount of capacity is required for speed control. This limits its economic usefulness, and makes the use of a cheaper form of effective or equivalent capacity desirable.

C. Multispeed Motors

14. The change of speed by changing the number of poles, in
the multispeed induction motor, involves the use of fractionalpitch windings: a primary turn, which is of full pole pitch for
a given number of motor poles, is fractional pitch for a smaller number of poles, and more than full pitch for a larger number
of poles. The same then applies to the rotor or secondary, if containing a definite winding. The usual and most frequently employed squirrel-cage secondary obviously has no definite number of poles, and thus is equally adapted to any number of

poles.
As an illustration may be considered a three-speed motor

changing between four, six and eight poles.

Assuming that the primary winding is full-pitch for the six-

polar motor, .that is, each primary turn covers one-sixth of the

motor circumference. Then, for the four-polar motor, the

% % primary winding is

pitch, for the eight-polar motor it is

% pitch which latter is effectively the same as

pitch.

Suppose now the primary winding is arranged and connected

as a six-polar three-phase winding. Comparing it with the

same primary turns, arranged as a four-polar three-phase wind-

ing, or eight-polar three-phase winding, the turns of each phase

can be grouped in six sections:
Those which remain in the same phase when changing to a

winding for different number of poles. Those which remain in the same phase, but are reversed when

changing the number of poles. Those which have to be transferred to the second phase.

Those which have to be transferred to the second phase in the

reverse direction.

Those which have to be transferred to the third phase. Those which have to be transferred to the third phase in the

reverse direction.

The problem of multispeed motor design then is, so to arrange
the windings, that the change of connection of the six coil groups
of each phase, in changing from one number of poles to another, is accomplished with the least number of switches.

SPEED CONTROL

21

15. Considering now the change of motor constants when changing speed by changing the number of poles. Assuming that at all speeds, the same primary turns are connected in series,
and are merely grouped differently, it follows, that the selfinductive impedances remain essentially unchanged by a change of the number of poles from n to n'. That is;
ZQ = Z'o, Z l = Z'L
With the same supply voltage impressed upon the same number of series turns, the magnetic flux per pole remains unchanged by the change of the number of poles. The flux density, therefore, changes proportional to the number of poles :
&_ " rf
B n]

therefore, the ampere-turns per pole required for producing the
magnetic flux, also must be proportional to the number of poles:

n

However, with the same total number of turns, the number of

turns per pole are inverse proportional to the number of poles:

N^n

N

n''

In consequence hereof, the exciting currents, at the same

impressed voltage, are proportional to the square of the number

of poles:

_ ZJH)

n^_

ioo

n2 7

and thus the exciting susceptances are proportional to the square of the number of poles :

The magnetic flux per pole remains the same, and thus the
magnetic-flux density, and with it the hysteresis loss in the primary core, remain the same, at a change of the number of poles. The tooth density, however, increases with increasing number of poles, as the number of teeth, which carry the same flux per pole, decreases inverse proportional to the number of

22

ELECTRICAL APPARATUS

poles. Since the tooth densities must be chosen sufficiently low not to reach saturation at the highest number of poles, and their
core loss is usually small compared with that in the primary core itself, it can be assumed approximately, that the core loss of the motor is the same, at the same impressed voltage, regardless of the number of poles. This means, that the exciting conductance, g, does not change with the number of poles.
Thus, if in a motor of n poles, we change to n f poles, or by the

ratio

= ri

a

>

n

the motor constants change, approximately:

from :

= + ZQ

7'

JXQ,

+ = Zi

Tl

JXi,

Fo = g - jb,

to :

+ = ZQ TQ JXQ.

+ Z = 1

Ti

JXL

Fo

=

g

ja*b.

16. However, when changing the number of poles, the pitch of the winding changes, and allowance has to be made herefore in the constants: a fractional-pitch winding, due to the partial neutralization of the turns, obviously has a somewhat higher exciting admittance, and lower self-inductive impedance, than
a full-pitch winding.
As seen, in a multispeed motor, the motor constants at the higher number of poles and thus the lower speed, must be materially inferior than at the higher speed, due to the increase of the exciting susceptance, and the performance of the motor, and especially its power-factor and thus the apparent efficiency,
are inferior at the lower speeds.
When retaining series connection of all turns for all speeds,
and using the same impressed voltage, torque in synchronous watts, and power are essentially the same at all speeds, that is, are decreased for the lower speed and larger number of poles only as far as due to the higher exciting admittance. The actual torque thus would be higher for the lower speeds, and approximately inverse proportional to the speed.
As a rule, no more torque is required at low speed than at high speed, and the usual requirement would be, that the multispeed motor should carry the same torque at all its running speeds, that is, give a power proportional to the speed.
This would be accomplished by lowering the impressed voltage

SPEED CONTROL

23

for the larger number of poles, about inverse proportional to the square root of the number of poles:
= e'

since the output is proportional to the square of the voltage.
The same is accomplished by changing connection from multiple
connection at higher speeds to series connection at lower speeds,
F or from delta connection at higher speeds, to at lowei speeds.
If, then, the voltage per turn is chosen so as to make the actual
torque proportional to the synchronous torque at all speeds, that
MULTISPEED INDUCTION MOTOR] 4 POLES 1800 REV.

FIG. 9. Load curves for nxultispeed induction motor, highest speed, four
poles.

is, approximately equal, then the magnetic flux per pole and the density in the primary core decreases with increasing number of poles, while that in the teeth increases, but less than at constant
impressed voltage.
The change of constants, by changing the number of poles by
the ratio :
n' = a
n
thus is:

from:

6 , YQ,

to

aZo

and the characteristic constant is changed from $ to a?$.
17. As numerical instance may be considered a 60-cycle 100-
volt motor, of the constants:

ELECTRICAL APPARATUS

MULTISPEED INDUCTION 6 POLES 1200 REV.

MULTISPEED INDUCTION MOTORAM/8

FIG. 10. Load curve of multispeed induction motor, middle
speed, six poles.

FIG. 11. Load curves of multispeed induction motor, low speed,
eight poles.

REV.

MULTISPEED INDUCTION MOTOR

1800

4-6-8-POLES. 1800-1200-900 REV.

FIG. 12. Comparison of load curves of three-speed induction motor.

SPEED CONTROL

25

+ + Four poles, 1800 Z rev.: Q = r

jx G = 0.1

0.3 j;

+ Zi = ri+jxi = 0.1

- 0.3 j;

YQ

=

g

jb

=

0.01

0.05 j.

+ + Six poles, 1200 Z rev.: Q = r

j# = 0.15 0.45 j;

+ + Zi = n jxi = 0.15 0.45 j; F = g -jb = 0.0067 ~ 0.0667 j.

+ + Eight poles, 900 Z rev.: Q = r

= jx<>

0.2

0.6 j;

+ + # - - = 7 Zi = ri

ja?!

0.2

0.6 j;

= g

= 0.005 0.1 j.

Figs. 9, 10 and 11 show the load curves of the motor, at the three different speeds. Fig. 12 shows the load curves once more,

MULTISPEED INDUCTION MOTOR 4-6-8 POLES 1800-1200-900 REV

not
100L .KW.
.9. -90.
S._80.

_50.
_30.

100 200 300 400 500 600 700 800 900100011001200130014001500160017001800
FIG, 13. Speed torque curves of three-speed induction motor.
with all three motors plotted on the same sheet, but with the torque in synchronous watts (referred to full speed or fourpolar synchronism) as abscissae, to give a better comparison.
S denotes the speed, I the current, p the power-factor and 7 the
apparent efficiency. Obviously, carrying the same load, that is, giving the same torque at lower speed, represents less power
output, i and in a multispeed motor the maximum power output
should be approximately proportional to the speed, to operate at all speeds at the same part of the motor characteristic. Therefore, a comparison of the different speed curves by the power output does not show the performance as well as a comparison on the basis of torque, as given in Fig. 12.

26

ELECTRICAL APPARATUS

As seen from Fig. 12, at the high speed, the motor performance is excellent, but at the lowest speed, power-factor and apparent
efficiency are already low, especially at light-load.
The three current curves cross : at the lowest speed, the motor takes most current at no-load, as the exciting current is highest ;
at higher values of torque, obviously the current is greatest at
the highest speed, where the torque represents most power. The speed regulation is equally good at all speeds. Fig. 13 then shows the speed curves, with revolutions per
minute as abscissae, for the three numbers of poles. It gives current, torque and power as ordinates, and shows that the
maximum torque is nearly the same at all three speeds, while
current and power drop off with decrease of speed.

CHAPTER II
MULTIPLE SQUIRREL-CAGE INDUCTION MOTOR
18. In an induction motor, a high-resistance low-reactance secondary is produced by the use of an external non-inductive resistance in the secondary, or in a motor with squirrel-cage secondary, by small bars of high-resistance material located close to the periphery of the rotor. Such a motor has a great slip of speed under load, therefore poor efficiency and poor speed regulation, but it has a high starting torque and torque at low and intermediate speed. With a low resistance fairly high-reactance secondary, the slip of speed under load is small, therefore efficiency and speed regulation good, but the starting torque and torque at low and intermediate speeds is low, and the current
in starting and at low speed is large. To combine good start-
ing with good running characteristics, a non-inductive resistance is used in the secondary, which is cut out during acceleration. This, however, involves a complication, which is undesirable
in many cases, such as in ship propulsion, etc. By arranging
then two squirrel cages, one high-resistance low-reactance one,
consisting of high-resistance bars close to the rotor surface,
and one of low-resistance bars, located deeper in the armature iron, that is, inside of the first squirrel cage, and thus of higher reactance, a "double squirrel-cage induction motor' 7 is derived, which to some extent combines the characteristics of the highresistance and the low-resistance secondary. That is, at starting and low speed, the frequency of the magnetic flux in the armature, and therefore the voltage induced in the secondary winding is high, and the high-resistance squirrel cage thus carries considerable current, gives good torque and torque efficiency, while the low-resistance squirrel cage is ineffective, due to its high reactance at the high armature frequency. At speeds near synchronism, the secondary frequency, being that of slip, is low, and the secondary induced voltage correspondingly low. The high-resistance squirrel cage thus carries little current and gives little torque. In the low-resistance squirrel cage, due to its low reactance at the low frequency of slip, in spite of the relatively
27

28

ELECTRICAL APPARATUS

low induced e.m.f., considerable current is produced, which is

effective in producing torque. Such double squirrel-cage induction motor thus gives a torque curve, which to some extent is a
superposition of the torque curve of the high-resistance and that of the low-resistance squirrel cage, has two maxima, one at low

speed, and another near synchronism, therefore gives a fairly good torque and torque efficiency over the entire speed range from standstill to full speed, that is, combines the good features

of both types. Where a very high starting torque requires
locating the first torque maximum, near standstill, and large size
and high efficiency brings the second torque maximum very close to synchronism, the drop of torque between the two maxima may be considerable. This is still more the case, ivhen the motor

is required to reverse at full speed and full power, that is, a very

high torque is required at full speed backward, or at or near

= may slip $

2. In this case, a triple squirrel cage

be used, that

is, three squirrel cages inside of each other: the outermost, of
high resistance and low reactance, gives maximum torque below

standstill, at backward rotation; the second squirrel cage, of
medium resistance and medium reactance, gives its maximum

torque at moderate speed; and the innermost squirrel cage, of

low resistance and high reactance, gives its torque at full speed,

near synchronism.
Mechanically, the rotor iron may be slotted down to the inner-
most squirrel cage, so as to avoid the excessive reactance of a closed magnetic circuit, that is, have the magnetic leakage flux or self-inductive flux pass an air gap.

19. In the calculation of the standard induction motor, it is

usual to start with the mutual magnetic flux, $, or rather with

the voltage induced by this flux, the mutual inductive voltage

$= e,

as

it

is

most

convenient,

with

the

mutual

inductive

voltage, e, as starting point, to pass to the secondary current by

the self-inductive impedance, to the primary current and primary

impressed voltage by the primary self-inductive impedance and
exciting admittance.

In the calculation of multiple squirrel-cage induction motors,

it is preferable to introduce the true induced voltage, that is, the voltage induced by the resultant magnetic flux interlinked with the various circuits, which is the resultant of the mutual

and the self-inductive magnetic flux of the respective circuit. This permits starting with the innermost squirrel cage, and

INDUCTION MOTOR

29

gradually building up to the primary, circuit. The advantage
hereof is, that the current in every secondary circuit is in phase

with

the

true

induced

voltage

of

this

circuit,

and

is

=
ii

>

TI

where TI is the resistance of the circuit. As ei is the voltage

induced by the resultant of the mutual magnetic flux coming

from the primary winding, and the self-inductive flux corre-

sponding

to

the

i&i

of

the

secondary,

the

reactance,

x it

does

not

enter any more in the equation of the current, and e\ is the

voltage due to the magnetic flux which passes beyond the cir-

cuit in which e\ is induced. In the usual induction-motor theory,

the mutual magnetic flux, $, induces a voltage, E, which produces

a current, and this current produces a self-inductive flux, $'j,

giving rise to a counter e.m.f. of self-induction I&i, which sub-

tracts from E. However, the self-inductive flux, <'i, interlinks

with the same conductors, with which the mutual flux, $, inter-

links, and the actual or resultant flux interlinkage thus is $1 =

<

E $'i, and this produces the true induced voltage ei =

I&i, from which the multiple squirrel-cage calculation starts. 1

Double Squirrel-cage Induction Motor

20. Let, in a double squirrel-cage induction motor:

$2 = true induced voltage in inner squirrel cage, reduced

to full frequency,
/2 = current, and
+ Zz r2 jx* = self-inductive impedance at full frequency,

reduced to the primary circuit.
$i = true induced voltage in outer squirrel cage, reduced

to full frequency,
li = current, and
+ Zi TI jxi = self-inductive impedance at full frequency,

reduced to primary circuit.
$ voltage induced in secondary and primary circuits by

mutual magnetic flux,

I$Q = voltage impressed upon primary,

Jo = primary current,

+ ZQ = r

jxo = primary self -inductive impedance, and

= YO

ff

jb = primary exciting admittance.

^ee' "Electric Circuits", Chapter XII. Beactance of Induction Apparatus,

30

ELECTRICAL APPARATUS

The leakage reactance, x*, of the inner squirrel cage is that due to the flux produced by the current in the inner squirrel cage, which passes between the two squirrel cages, and does not include the reactance due to the flux resulting from the current, J2, which passes beyond the outer squirrel cage, as the latter is mutual reactance between the two squirrel cages, and thus meets
the reactance, xi.
It is then, at slip s:

72 = ^-

(1)

=
/i

/TV+ + V = I

T

J

TV

/2 /I -*OV

0)
W/Q^

and:

+ + E $ = 3

jo;i (/i

/)

= TTT

T

j[T/ Q

jT^

i f

yT
^r/ QJt Q

(5)
/\
\ v//

The leakage flux of the outer squirrel cage is produced by the

+ m.m.f. of the currents of both squirrel cages, /i

/2, and the

+ reactance voltage of this squirrel cage, in (5), thus IBJXI (/i /a)-

E As seen, the difference between l and |J2 is the voltage in-

duced by the flux which leaks between the two squirrel cages, in

the path of the reactance, #2, or the reactance voltage, #2/2? the
$ difference between and #1 is the voltage induced by the rotor

flux leaking outside of the outer squirrel cage. This has the
+ m.m.f. /i /a> and the reactance xi, thus is the reactance voltage + E $ #1 (/i /2). The difference between Q and is the voltage
consumed by the primary impedance: Z /o. (4) and (5) .are the

voltages reduced to full frequency; the actual voltages are s

times as high, but since all three terms in these equations are

induced voltages, the 5 cancels.

21. From the equations (1) to (6) follows:

(7)

(9)

INDUCTION MOTOR

where:

a\ = 1

TlT-2,

-- --- -- -= &/, 2

5o !

-J-

J. 2

\ri

r2

r2

thus the exciting current :
/oo = YoE ^ E,(g~- jb) (ai
= ^2 (6i+j62),

where:

and the total primary current is (3) :

where:

0,=- +i -

31
(10)
(ID
(12) (13) (14)

and the primary impressed voltage (6) :
(r

where :
+

(16)

hence, absolute:

-
(17)

io

+ "*0

i

9 2.

/I O\ (lo)

22. The torque of the two squirrel cages is given by the product of current and induced voltage in phase with it, as:

(19)

(20)

32

ELECTRICAL APPARATUS

hence, the total torque :

D

-

D2

+

D l9

(21)

and the power output
P = (i - s) D.

(22)

(Herefrom subtracts the friction loss, to give the net power
output.)
The power input is:

Po = /$o, IQ/'
+ = e^dd, cjd*),

(23)

and the volt-ampere input :

Q=

T>
Herefrom then follows the power-factor 7V7* the torque effi-

ciency

p-,

the

apparent

torque

efficiency -Q,

the

power

efficiency

P

P

~- and the apparent power efficiency pr

-TO

Hf

23. As illustrations are shown, in Pigs. 14 and 15, the speed curves and the load curves of a double squirrel-cage induction motor, of the constants:

e = 110 volts;

+ ZQ = 0.1

0.3 j;

+ Zl = 0.5 0.2 j;

+ 2 = 0.08

0.4 j;

7

- = 0.01

0.1 j;

the speed curves for the range from $ = = to s 2, that is, from
synchronism to backward rotation at synchronous speed. The total torque as well as the two individual torques are shown on the speed curve. These curves are derived by calculating, for
the values of $':

= 5 0, 0.01, 0.02, 0.05, 0.1, 0.15, 0.2, 0.3,
0.4, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6, 1.8, 2.0,

INDUCTION MOTOR

33

DOUBLE SQUIRREL CAGE INDUCTION MOTOR SPEED CURVES

-1.0-.9-8 -.7 -.6 -.5 -.4 -.3 -.2 -.1

.1 .2 .3 .4 .5 ,6 .7 .8 .9 1.0

FIG. 14. Speed curves of double squirrel-cage induction motor.

DOUBLE SQUIRREL CAGE INDUCTION MOTOR
LOAD CURVES

1.0 1.5 20 2.5 30 35 40 45 50 55 6 KW
FIG. 15. Load curves of double squirrel-cage induction motor.

34:
the values:

ELECTRICAL APPARATUS

-

a

I

+

/

r3 V

r2

+ D = Di

Z> 2 ,

P = (1 - s) D,

Po = e2

and:

P_ ,0 P D Po
Pa'Po'C' C' Q*

Triple Squirrel-cage Induction. Motor

24. Let:

E + * = flux, = voltage, / = current, and Z = r

= jx

self-

inductive impedance, at full frequency and reduced to primary

circuit, and let the quantities of the innermost squirrel cage be

denoted by index 3, those of the middle squirrel cage by 2, of

the outer squirrel cage by 1, of the primary circuit by 0, and the

mutual inductive quantities without index.

7 Also let:

== g

jb = primary exciting admittance.

It is then, at slip s:

current in the innermost squirrel cage:

^ - /.- T

S 3-

,

m(D

INDUCTION MOTOR

35

current in the middle squirrel cage:

~ /2 =

2
;

(v 2)y

r2

current in the outer squirrel cage:

=

-1
~J~~>

(3)

primary current:

+ + + /o = /3

/2

/i

Y Q E.

(4)

The voltages are related by:

777 __ 771

I

*

T

777 __ 777

*
|

/T

777

777

* /T

[

+ EQ = E Zo/0,

i^ 7" \

T

j

[

7* \

/C\
/iC\
f7\ (8)

where #3 is the reactance due to the flux leakage between the
third and the second squirrel cage; x% the reactance of the leak-
age flux between second and first squirrel cage; a?i the reactance
of the first squirrel cage and XQ that of the primary circuit, that
+ is, X* XQ corresponds to the total leakage flux between primary
and outer most squirrel cage.
E E$, 2 and $1 are the true induced voltages in the three squirrel E cages, the mutual inductive voltage between primary and secondary, and E Q the primary impressed voltage
25. From equations (1) to (8) then follows:

(9)

(10)

where:

ai = 1

2 i

,

(12)

36

ELECTRICAL APPARATUS

#=

where:

+ = 6 2

&2

^
I
TB

thus the exciting current :

loo = Y E

+ x

J6 2 ) (flf

Jb)

where :

=
C]

and the total primary current, by (4) :

where: where:

4- JL S 2iC 3
,

= + + #3 (di

jdz) fro

ja;o)

thus, the primary impressed voltage, by (8) :

where:

= E3

(14) (15)
(16) (17)
(18) (19) (20) (21) (22) (23)

INDUCTION MOTOR
hence, absolute:
_
^+ 2 (J 2 ,
ei = e3 AoTlz"?.
26. The torque of the innermost squirrel cage thus is;
* = *;
that of the middle squirrel cage:

37
(25) (26) (27) (28)

and that of the outer squirrel cage:
D, = s-;

(30)

the total torque of the triple squirrel-cage motor thus is:
D = D, + D2 + Da,

and the power:

P = (1 - a) D,

(31) (32)

the power input is:

Po = /#,, / /'

= + 2 es (diflri

dtfj),

(33)

and the volt-ampere input :

Q = e io.

(34)
T>

Herefrom then follows the power-factor -j> the torque effi-

ciency

p-,
JT o

apparent

torque

efficiency

Vyj*

power

efficiency *pr

and apparent power efficiency TT

27. As illustrations are shown, in Figs. 16 and 17, the speed and the load curves of a triple squirrel-cage motor with the

constants:

e = 110 volts;

Z = 0.1 +0.3J;

Z1 = 0.8 +0.1J;

Z2 = 0.2 +0.3j;

Z = + 3

0.05

0.8 ,7;

Fo = 0.01 - 0.1 j;

38

ELECTRICAL APPARATUS

TRIPLE SQUIRREL CAGE INDUCTION MOTOR
SPEED CURVES

-1.0-9 -.8"-.7 -.6 -.5 -.4 -.3 -.2 -.1

.1 .2 .3 .4 .5 .6 .7 .8 .9 1.0

FIG. 16. Speed curves of triple squirrel-cage induction motor.

TRIPLE SQUIRREL CAGE INDUCTION MOTOR LOAD CURVES

FIG. 17. Load curves of triple squirrel-cage induction motor.

INDUCTION MOTOR

39

the speed curves are shown from s ~ to s = 2, and on them,

the individual torques of the three squirrel cages are shown in

addition to the total torque.

These numerical values are derived by calculating, for the

values of s:

= s 0, 0.01, 0.02, 0.05, 0.1, 0.15, 0.20, 0.30,

the values:

0.40, 0.60, 0.80, 1.0, 1.2, 1.4, 1.6, 1.8, 20,

=. 1

S*X%Xz
>

2
Co
+2 firs!

= bi

&i -

t>2 == &2 ~

+.I5" ,+5~.+ Cl,

OU 2

O *t- 3

O

j

(

Oi

= 63

1+

= + 2
ei

2

2

ea (fli

a2 2 ),

rz

D = Di + D2 + D3,
P = (1 - s) D,

~ V

60^0?

and
P_ I) P D Po

PO'PO'Q'Q'Q'

CHAPTER III

CONCATENATION

Cascade or Tandem Control of Induction Motors

28. If of two induction motors the secondary of the first motor

is connected to the primary of the second motor, the second
machine operates as a motor with the voltage and frequency impressed upon it by the secondary of the first machine. The first machine acts as general alternating-current transformer
or frequency converter (see Chapter XII), changing a part of the
primary impressed power into secondary electrical power for the supply of the second machine, and a part into mechanical

work.

The frequency of the secondary voltage of the first motor, and thus the frequency impressed upon the second motor, is the frequency of slip below synchronism, s. The frequency of the secondary of the second motor is the difference between its impressed frequency, s, and its speed. Thus, if both motors are connected together mechanically, to turn at the same speed, 1 5, and have the same number of poles, the secondary fre-

quency of the second motor is 2 s

1, hence equal to zero at

= s

0.5.

That is, the second motor reaches its synchronism at

half speed. At this speed, its torque becomes zero, the power component of its primary current, and thus the power component of the secondary current of the first motor, and thus also

the torque of the first motor becomes zero. That is, a system of

two concatenated equal motors, with short-circuited secondary

of the second motor, approaches half synchronism at no-load,

in the same manner as a single induction motor approaches

synchronism. With increasing load, the slip below half syn-

chronism increases.
In reality, at half synchronism, s = 0.5, there is a slight torque

produced by the first motor, as the hysteresis energy current of

the second motor comes from the secondary of the first motor,

and therein, as energy current, produces a small torque.

More generally, any pair of induction motors connected in

concatenation divides the speed so that the sum of their two

40

CONCATENATION

41

respective speeds approaches synchronism at no-load; or, still
more generally, any number of concatenated induction motors run at such speeds that the sum of their speeds approaches

synchronism at no-load. With mechanical connection between the two motors, con-

catenation thus offers a means of operating two equal motors at

full efficiency at half speed in tandem, as well as at full speed, in parallel, and thereby gives the same advantage as does series

parallel control with direct-current motors.
With two motors of different number of poles, rigidly con-

nected together, concatenation allows three speeds : that of the
one motor alone, that of the other motor alone, and the speed of concatenation of both motors. Such concatenation of two motors

of different numbers of poles, has the disadvantage that at the
two highest speeds only one motor is used, the other idle, and the apparatus economy thus inferior. However, with certain ratios of the number of poles, it is possible to wind one and the same motor structure so as to give at the same time two different numbers of poles: For instance, a four-polar and an eightpolar winding; and in this case, one and the same motor structure can be used either as four-polar motor, with the one winding,

or as eight-polar motor, with the other winding, or in concatena-
tion of the two windings, corresponding to a twelve-polar speed. Such "internally concatenated " motors thus give three different
speeds at full apparatus economy. The only limitation is, that only certain speeds and speed ratios can economically be produced

by internal concatenation.

29. At half synchronism, the torque of the concatenated couple

of two equal motors becomes zero. Above half synchronism,

the second motor runs beyond its impressed frequency, that is,

becomes a generator. In this case, due to the reversal of current
in the secondary of the first motor (this current now being out-

flowing or generator current with regards to the second motor)

its torque becomes negative also, that is, the concatenated couple
becomes an induction generator above half synchronism. When

approaching full synchronism, the generator torque of the second motor, at least if its armature is of low resistance, becomes very

small, as this machine is operating very far above its synchronous

speed.

With

regards

to

the

first motqr

(

2

it

thus

begins

to

act

merely as an impedance in the secondary circuit, that is, the first

machine^becomes a motor

dg&m.

'
Thus,

somewhere

between

42

ELECTRICAL APPARATUS

half synchronism and synchronism, the torque of the first motor

becomes zero, while the second motor still has a small negative or

A generator torque.

little above this speed, the torque of the

concatenated couple becomes zero about at two-thirds synchronism with a couple of low-resistance motors and above this, the concatenated couple again gives a positive or motor
torque though the second motor still returns a small negative torque and again approaches zero at full synchronism. Above full synchronism, the concatenated couple once more becomes generator, but practically only the first motor contributes to the generator torque above and the motor torque below full synchronism. Thus, while a concatenated couple of induction motors has two operative motor speeds, half synchronism and full synchronism, the latter is uneconomical, as the second motor

holds back, and in the second or full synchronism speed range, it

is more economical to cut out the second motor altogether, by

short-circuiting the secondary terminals of the first motor.

With resistance in the secondary of the second motor, the

maximum torque point of the second motor above half syn-

chronism is shifted to higher speeds, nearer to full synchronism, and thus the speed between half and full synchronism, at which the concatenated couple loses its generator torque and again becomes motor, is shifted closer to full synchronism, and the motor torque in the second speed range, below full synchronism, is greatly reduced or even disappears. That is, with high resistance in the secondary of the second motor, the concatenated couple becomes generator or brake at half synchronism, and

remains so at all higher speeds, merely loses its braking torque when approaching full synchronism, and regaining it again beyond

full synchronism.
m The speed torque curves of the concatenated couple, shown
Fig. 18, with low-resistance armature, and in Fig. 19, with high
resistance in the armature or secondary of the second motor,

illustrate this.

30. The numerical calculation of a couple of concatenated induction motors (rigidly connected together on the same shaft

or the equivalent) can be carried out as follows :

Let:

n s* number of pairs of poles of the first motor,
nf = number of pairs of poles of the second motor.

CONGA TENA TION

43

a=

= ratio of poles,

(1)

/ = supply frequency.

Full synchronous speed of the first motor then is:

So =

(2)

of the second motor:

= <S'o

(3)

At slip s and thus speed ratio (1 s) of the first motor, its

speed is:

S = (!-*)&- (!-)

(4)

iL

and the frequency of its secondary circuit, and thus the frequency of the primary circuit of the second motor:

synchronous speed of the second motor at this frequency is:

llr

the speed of the second motor, however, is the same as that of

the first motor, 8, hence, the slip of speed of the second motor below its synchronous

speed, is:

n 5

~ t

(1

5)

==

I

-.

n \n

n ) J. I

and the slip of frequency thus is :

+ - = s'

s (1

a)

a.

(5)

This slip of the second motor, s', becomes zero, that is, the couple reaches the synchronism of concatenation, for:

* - iTT

C6)

44

ELECTRICAL APPARATUS

The speed in this case is:

S

=-

(1

so)

(7)

31. If:

0=1,

that is, two equal motors, as for instance two four-polar motors

it is:

n = n' = 4,

= so

0.5,

while at full synchronism:

If: it Is:

~"

"~

n4

a = 2,

n = 4,

= n7

8,

=? ? 3

that is, corresponding to a twelve-polar motor. While:

*-/n4

if: it is:

= a

0.5,

n = 8,

n7 = 4,

CONGA TENA TION

45

that is, corresponding to a twelve-polar motor again. That is, as regards to the speed of the concatenated couple, it is immaterial in which order the two motors are concatenated.
32. It is then, in a concatenated motor couple of pole ratio:

a = n>
n
if:

5 = slip of first motor below full synchronism.

The primary circuit of the first motor is of full frequency.

The secondary circuit of the first motor is of frequency s.

The primary circuit of the second motor is of frequency s.

The

secondary

circuit

of the

second motor is

of

frequency

f
s

=

+ s (1

a)

a.

Synchronism of concatenation is reached at:

Let thus:

_
1+a

CQ = voltage impressed of first motor primary;

YQ

g jb

= exciting admittance of first motor;

F'o = g' jV = exciting admittance of second motor;

+ Zo = TQ

JXQ = self-inductive impedance of first

motor

primary;
+ Z'Q r'o jx'o = self-inductive impedance of second motor

primary;
+ Zi = TI jxi = self-inductive impedance of first motor second-

ary;
+ Z'\ = r\ jx\ = self-inductive impedance of second motor

secondary.
Assuming all these quantities reduced to the same number of turns per circuit, and to full frequency, as usual.

If:
e = counter e.m.f. generated in the second motor by its mutual
magnetic flux, reduced to full frequency. It is then:

secondary current of second motor:

_ _ r,

^e

+ - [s (1

a)

a] e

_- ,

46 where :

-ELECTRICAL APPARATUS

- + rMaq . .

_

o) -a] 1

QF

a

m

+ + m sV = rV

(s (1

-2

a)

a) ;

(10)

exciting current of second motor:

/'oo-eF' = e (/-#'),

(ID

hence, primary current, of second motor, and also secondary current of first motor:

+ = = /'o /i /']

/'oo

= e (61 - #),

(12)

where :

^ &!-! +
+ &2 = a* o,

(13)

the impedance of the circuit comprising the primary of the second, and the secondary of the first motor, is:

+ + + + Z = ZS

Z' 2 = (n

r' )

js (a?,

^o),

(14)

hence, the counter e.m.f., or induced voltage in the secondary of the first motor, of frequency is :
+ sE l - se hZ,

hence, reduced to full frequency:

where:

= + e (ci

jc a),

(15)

-
'o) 61 s

33. The primary exciting current of the first motor is:

/"Ci = V 00

&IJ-

** e(di jds),

where:

j

r - _ i ^ 1

(16)
(17) (18)

CONGA TENA TION

47

thus, the total primary current of the first motor, or supply

current :

+ Jo = Ii /oo
= - e (/i #2),

(19)

where :

+ =

fz

&2

<

and the primary impressed voltage of the first motor, or supply
voltage:

where :
and, absolute: thus:

yz

- mf '-'a I *v(jj i

'OJ2JI

+ e = e VVi2

2
^2 ,

(21) (22) (23)

Tp"

^24)

Substituting now this value of e in the preceding, gives the
values of the currents and voltages in the different circuits. 34. It thus is, supply current:

power input:

Po = /#>, /o/' = - e2 (figi

volt-ampere input:
and herefrom power-factor, etc. The torque of the second motor is:
rpr __. / T It
= a e2 3.
The torque of the first motor is :

48

ELECTRICAL APPARATUS

hence, the total torque of the concatenated couple:

+ + - T = Z"

= Ti

6 2 (a,

Ci/i

C 2 /2),

and herefrom the power output :
P = (I - s) T,

thus the torque and power efficiencies and apparent efficiencies,
etc.
35. As instances are calculated, and shown in Fig. 18, the speed

+ 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0-0.1-0.2-0.3-0.4-0.5-0.0-0.7
FIG. 18. Speed torque curves of concatenated couple with low resistance secondary.

torque curves of the concatenated couple of two equal motors:
= a 1, of the constants: e == 110 volts.

Y = Y'

- = 0.01

0.1 j;

+ Z Q = Z' = 0.1 0.3 j;

+ Z l = Z\ = 0.1 0.3J.

Fig. 18 also shows, separately, the torque of the second motor, and the supply current.
Fig. 19 shows the speed torque curves of the same concate-
nated couple with an additional resistance r = 0.5 inserted into
the secondary of the second motor.
The load curves of the same motor, Fig. 18, for concatenated running, and also separately the load curves of either motor,

CONCATENATION

49

are given on page 358 of " Theoretical Elements of Electrical
Engineering."
36. It is possible in concatenation of two motors of different number of poles, to use one and the same magnetic structure for both motors. Suppose the stator is wound with an n-polar primary, receiving the supply voltage, and at the same time with an nf polar short-circuited secondary winding. The rotor is wound with an n-polar winding as secondary to the n-polar primary winding, but this n-polar secondary winding is not short-circuited, but connected to the terminals of a second

FIG. 19. Speed-torque curves of concatenated couple with resistance in second secondary.
n'-polar winding, also located on the rotor. This latter thus receives the secondary current from the n-polar winding and
acts as n'-polar primary to the short-circuited stator winding as secondary. This gives an n-polar motor concatenated to an n'-polar, and the magnetic structure simultaneously carries an n-polar and an n'-polar magnetic field. With this arrangement of " internal concatenation," it is essential to choose the number of poles, n and n', so that the two rotating fields do not interfere with each other, that is, the n'-polar field does not induce in the n-polar winding, nor the n-polar field in the n'-polar winding.
This is the case if the one field has twice as many poles as the
other, for instance a four-polar and an eight-polar field. If such a fractional-pitch winding is used, that the coil pitch
is suited for an n-polar as well as an n'-polar winding, then the same winding can be used for both sets of poles. In the stator, the equipotential points of a 2p-polar winding are points of opposite polarity of a p-polar winding, and thus, by connecting together the equipotential points of a 2 p-polar primary winding,

50

ELECTRICAL APPARATUS

this winding becomes at the same time a p-polar short-circuited
winding. On the rotor, in some slots, the secondary current of
the tt-polar and the primary current of the n'-polar winding flow
In the same direction, in other slots flow in opposite direction, thus neutralize in the latter, and the turns can be omitted In
concatenation but would be put in for use of the structure as

single motor of n, or of nf poles, where such is desired. Thus, on the rotor one single winding also is sufficient, and this arrange-
ment of internal concatenation with single stator and single rotor

winding thus is more efficient than the use of two separate motors, and gives somewhat better constants, as the self-inductive im-
pedance of the rotor is less, due to the omission of one-third of

the turns in which the currents neutralize (Hunt motor).

The disadvantage of this arrangement of internal concatenation

with single stator and rotor winding is the limitation of the avail-

able speeds, as it is adapted only to 4 -5- 8 -*- 12 poles and

K multiples thereof, thus to speed ratios of 1 *

-* H> tbe last

being the concatenated speed.
Such internally concatenated motors may be used advantage-
ously sometime as constant-speed motors, that is, always running in concatenation, for very slow-speed motors of very large

number of poles.
37. Theoretically, any number of motors may be concatenated.
It is rarely economical, however, to go beyond two motors in concatenation, as with the increasing number of motors, the

constants of the concatenated system rapidly become poorer.

If:

YQ = g

j6,

= + ZQ

7*0

j$Q,

+ = Zi ri fai,

are the constants of a motor, and we denote :

+ + + + Z =* ZQ Zi~ (r

ri)

j Oo

a?i)

= r + jx

then the characteristic constant of this motor which char-

acterizes its performance is:

if now two such motors are concatenated, the exciting admittance
of the concatenated couple is (approximately) :
7' ~ 2 Y f

CONCATENATION

51

as the first motor carries the exciting current of the second motor.
The total self-inductive impedance of the couple is that of both motors in series :
Z" = 2 Z;

thus the characteristic constant of the concatenated couple is:

= *'

y'z'

that is, four times as high as in a single motor; in other words, the performance characteristics, as power-factor, etc., are very
much inferior to those of a single motor. With three motors in concatenation, the constants of the
system of three motors are:
Y" -37,
Z" = 3 Z,
thus the characteristic constant :
= y"z"

or nine times higher than in a single motor. In other words, the characteristic constant increases with the square of the
number of motors in concatenation, and thus concatenation of more than two motors would be permissible only with motors
of very good constants. The calculation of a concatenated system of three or more
motors is carried out in the same manner as that of two motors, by starting with the secondary circuit of the last motor, and building up toward the primary circuit of the first motor,

CHAPTER IV

INDUCTION MOTOR WITH SECONDARY EXCITATION

38. While in the typical synchronous machine and commu-
tating machine the magnetic field is excited by a direct current, characteristic of the induction machine is, that the magnetic
field is excited by an alternating current derived from the alternating supply voltage, just as in the alternating-current transformer. As the alternating magnetizing current is a wattless

reactive current, the result is, that the alternating-current input
Into the induction motor is always lagging, the more so, the
larger a part of the total current is given by the magnetizing current. To secure good power-factor in an induction motor,
the magnetizing current, that is, the current which produces the magnetic field flux, must be kept as small as possible. This means as small an air gap between stator and rotor as mechanic-

ally permissible, and as large a number of primary turns per pole,

that is, as large a pole pitch, as economically permissible.
In motors, in which the speed compared to the motor out-

put is not too low, good constants can be secured. This,

however, is not possible in motors, in which the speed is very

low, that is, the number of poles large compared with the output, and the pole pitch thus must for economical reasons be kept

small as for instance a 100-hp. 60-cycle motor for 90 revolu-

tions, that is, 80 poles or where the requirement of an excessive

momentary overload capacity has to be met, etc. In such motors

of necessity the exciting current or current at no-load which

is practically all magnetizing current is a very large part of

may full-load current, and while fair efficiencies

nevertheless be

secured, power-factor and apparent efficiency necessarily are

very low. As illustration is shown in Fig. 20 the load curve of a typical

100-hp. 60-cycle 80-polar induction motor (90 revolutions per

minute) of the constants:

Impressed voltage: Primary exciting admittance: Primary self-inductive impedance: Secondary self-inductive impedance:

eo = 500.
F = 0.02 0.6 j.
+ Z = 0.1 0.3 j. + = Zi 0,1 0.3 j.

52

INDUCTION MOTOR

53

As seen, at full-load of 75 kw. output, the efficiency is 80 per cent., which is fair for a slow-speed motor.
But the power-factor is 55 per cent., the apparent efficiency only 44 per cent., and the exciting current is 75 per cent, of full-
load current.

This motor-load curve may be compared with that of a typical

induction motor, of exciting admittance:

- 7o = 0.01

0.1 j,

given

on

page

234

of

"
Theory

and

Calculation

of

Alternating-

current Phenomena" 5th edition, and page 319 of "Theoretical

LOW SPEED INDUCTION MOTOR
= 500

FIG. 20. Low-speed induction motor, load curves.
Elements of Electrical Engineering/' 4th edition, to see the
difference.
39. In the synchronous machine usually the stator, in cornmutating machines the rotor is the armature, that is, the element to which electrical power is supplied, and in which electrical power is converted into the mechanical power output of the motor. The rotor of the typical synchronous machine, and the stator of the commutating machine are the field, that is, in them no electric power is consumed by conversion into mechanical work, but their purpose is to produce the magnetic field flux, through which the armature rotates.
In the induction machine, it is usually the stator, which is the

54

ELECTRICAL APPARATUS

primary, that is, which receives electric power and converts it into mechanical power, and the primary or stator of the induction machine thus corresponds to the armature of the synchronous or commutating machine. In the secondary or rotor of the induction machine, low-frequency currents of the frequency of slip---are induced by the primary, but the magnetic field flux is produced by the exciting current which traverses the primary
or armature or stator. Thus the induction machine may be
considered as a machine in which the magnetic field is produced by the armature reaction, and corresponds to a synchronous machine, in which the field coils are short-circuited and the field produced by armature reaction by lagging currents in the
armature.
As the rotor or secondary of the induction machine corresponds structurally to the field of the synchronous or commutating machine, field excitation thus can be given to the induction machine by passing a current through the rotor or secondary and thereby more or less relieving the primary of its function of giv-
ing the field excitation.
Thus in a slow-speed induction motor, of very high exciting current and correspondingly poor constants, by passing an
exciting current of suitable value through the rotor or secondary,
the primary can be made non-inductive, or even leading current
produced, or with a lesser exciting current in the rotor at
least the power-factor increased.
Various such methods of secondary excitation have been pro-
posed, and to some extent used. 1. Passing a direct current through the rotor for excitation. In this case, as the frequency of the secondary currents is the
frequency of slip, with a direct current, the frequency is zero, that is, the motor becomes a synchronous motor.
2. Excitation through commutator, by the alternating supply
current, either in shunt or in series to the armature.
- At the supply frequency, /, and slip, s,the frequency of rotation
and thus of commutation is (1 s)f, and the full frequency cur-
- - rents supplied to the commutator thus give in the rotor the
effective frequency, / (1 s) / tf, that is, the frequency of
slip, thus are suitable as exciting currents.
3. Concatenation with a synchronous motor.
If a low-frequency synchronous machine is mounted on the induction-motor shaft, and its armature connected into the indue-

INDUCTION MOTOR

55

tion-motor secondary, the synchronous machine feeds low-fre-

quency exciting currents into the induction machine, and thereby

permits controlling it by using suitable voltage and phase.
If the induction machine has n times as many poles as the

synchronous machine, the frequency of rotation of the synchro-

nous machine is - that of the induction machine, or

- How-

ti

flj

ever, the frequency generated by the synchronous machine must

be the frequency of the induction-machine secondary currents,

that is, the frequency of slip s.

Hence :

- 1

s

~ S<j

n

'

or:

s

=

_J__
+j
n1

that is, the concatenated couple is synchronous, that is, runs at constant speed at all loads, but not at synchronous speed, but at

+ constant slirp n

r-7' 1

4. Concatenation with a low-frequency commutating machine. If a commutating machine is mounted on the induction-motor shaft, and connected in series into the induction-motor secondary, the commutating machine generates an alternating voltage of the frequency of the currents which excite its field, and if the field is excited in series or shunt with the armature, in the circuit of the induction machine secondary, it generates voltage at the
frequency of slip, whatever the latter may be. That is, the
induction motor remains asynchronous, increases in slip with
increase of load.

5. Excitation by a condenser in the secondary circuit of the
induction motor.

As the magnetizing current required by the induction motor is
a reactive, that is, wattless lagging current, it does not require a generator for its production, but any apparatus consuming leading, that is, generating lagging currents, such as a condenser, can be used to supply the magnetizing current.

40. However, condenser, or synchronous or commutating machine, etc., in the secondary of the induction motor do not merely give the magnetizing current and thereby permit powerfactor control, but they may, depending on their design or application, change the characteristics of the induction machine, as regards to speed and speed regulation, the capacity, etc*

56

ELECTRICAL APPARATUS

If by synchronous or commutating machine a voltage is
inserted into the secondary of the induction machine, this vol-
tage may be constant, or varied with the speed, the load, the slip,
etc., and thereby give various motor characteristics. Further-
more, such voltage may be inserted at any phase relation from
zero to 360. If this voltage is inserted 90 behind the secondary current, it makes this current leading or magnetizing and so increases the power-factor. If, however, the voltage is inserted in phase with the secondary induced voltage of the induction machine, it has no effect on the power-factor, but merely lowers the speed of the motor if in phase, raises it if in opposition to the secondary induced voltage of the induction machine, and hereby permits speed control, if derived from a commutating machine. For instance, by a voltage in phase with and proportional to the secondary current, the drop of speed of the motor can be increased and series-motor characteristics secured, in the same manner as
by the insertion of resistance in the induction-motor secondary. The difference however is, that resistance in the induction-motor
secondary reduces the efficiency in the same proportion as it lowers the speed, and thus is inefficient for speed control. The insertion of an e.m.f., however, while lowering the speed, does not lower the efficiency, as the power corresponding to the lowered
speed is taken up by the inserted voltage and returned as output of the synchronous or commutating machine. Or, by inserting a voltage proportional to the load and in opposition to the induced secondary voltage, the motor speed can be maintained constant,
or increased with the load, etc.
If then a voltage is inserted by a commutating machine in the
induction-motor secondary, which is displaced in phase by angle
a from the secondary induced voltage, a component of this voltage: sin a, acts magnetizing or demagnetizing, the other component : cos a, acts increasing or decreasing the speed, and thus
various effects can be produced.
As the current consumed by a condenser is proportional to the
frequency, while that passing through an inductive reactance is inverse proportional to the frequency, when using a condenser in the secondary circuit of the induction motor, its effective im-
pedance at the varying frequency of slip is:

where #2 is the capacity reactance at full frequency.

INDUCTION MOTOR

57

For s

=

0,

ZS 3

=

oo
,

that

is,

the

motor

has no power at

or

near

synchronism.

For:

or

it is:

and the current taken by the motor is a maximum. The power
output thus is a maximum not when approaching synchronism,
as in the typical induction motor, but at a speed depending on the
slip,
V^' ^ fa2
S \a/i
and by varying the capacity reactance, #2, various values of resonance slip, So, thus can be produced, and thereby speed control of the motor secured. However, for most purposes, this is uneconomical, due to the very large values of capacity required.

Induction Motor Converted to Synchronous

41. If, when an induction motor has reached full speed, a direct
current is sent through its secondary circuit, unless heavily loaded and of high secondary resistance and thus great slip, it drops into synchronism and runs as synchronous motor.
The starting operations of such an induction motor in conversion to synchronous motor thus are (Fig. 21) :

First step: secondary closed through resistance:

A.

Second step: resistance partly cut out:

B.

Third step: resistance all cut out:

C.

Fourth step: direct current passed through the secondary: D.

In this case, for the last or synchronous-motor step, usually the direct-current supply will be connected between one phase and the other two phases, the latter remaining short-circuited to each other, as shown in Fig. 21, D. This arrangement retains
a short-circuit in the rotor now the field in quadrature with the excitation, which acts as damper against hunting (Danielson
motor).

58

ELECTRICAL APPARATUS

In the synchronous motor, Fig. 21, D, produced from the induc-

tion motor, Fig. 21, C, it is:

Let:

7 =g ZQ = r

jb = primary exciting admittance
of the induction machine,
+ JXQ = primary self-inductive impe -

Zl = n + jxi

dance, secondary self-inductive im-

pedance.

-A/W -AATV-
-AAV-
-AAAr-AA/V-
FIG. 21. Starting of induction motor and conversion to synchronous.
The secondary resistance, ri, is that of the field exciting winding, thus does not further come into consideration in calculating the motor curves, except in the efficiency, as iiVi is the loss of power
in the field, if ii == field exciting current. x\ is of little further importance, as the frequency is zero. It represents the magnetic
leakage between the synchronous motor poles. TO is the armature resistance and # the armature self-inductive
reactance of the synchronous machine. However, x is not the synchronous impedance, which enters
the equation of the synchronous machine, but is only the self-
inductive part of it, or the true armature self-inductance. The

INDUCTION MOTOR

59

mutual inductive part of the synchronous impedance, or the effective reactance of armature reaction, a?', is not contained in XQ.
The effective reactance of armature reaction of the synchronous machine, %', represents the field excitation consumed by the armature m.m.f., and is the voltage corresponding to this field excitation, divided by the armature current which consumes this
field excitation.
6, the exciting susceptance, is the magnetizing armature current, divided by the voltage induced by it, thus, &', the effective reactance of synchronous-motor armature reaction, is the reciprocal of the exciting susceptance of the induction machine.
The total or synchronous reactance of the induction machine as synchronous motor thus is;
+ X = XQ Xf

-*o + g-

The exciting conductance, g, represents the loss by hysteresis, etc., in the iron of the machine. As synchronous machine, this loss is supplied by the mechanical power, and not electrically, and the hysteresis loss in the induction machine as synchronous motor thus is: e*g.
We thus have:

The induction motor of the constants, per phase:

Exciting admittance:

YQ = g jb,

Primary self-inductive impedance:

+ Zo r

JXQ,

+ Secondary self-inductive impedance: Z\ 7*1 jxi,

by passing direct current through the secondary or rotor, becomes a synchronous motor of the constants, per phase:

Armature resistance :

r

,

+ Synchronous impedance: x XQ 7-

(1)

Total power consumed in field excitation:

P=

2

2
; n,

(2)

where

= i field exciting current.

Power consumed by hysteresis :

P = 60.

(3)

60

ELECTRICAL APPARATUS

42, Let, in a synchronous motor:

Eo = impressed voltage,
E = counter e.m.f., or nominal induced

voltage,

+ Z = r jx = synchronous impedance,

= J

2*1

= jf2 current,

it is then:

or:
E ~ EQ- ZI

or, reduced to absolute values, and choosing:

= = jB

e

real axis in equation (4),

= = J?

eo real axis in equation (5),

n + (^ = eQ2

(e 4. -j

#4)2 4.

n*

2 )2

e
[

reai axigj^

(g)

= + + e2

(eo

n'i

xi^

(arii

= 2
*2) [eo

real axis].

(7)

Equations (6) and (7) are the two forms of the fundamental equation of the synchronous motor, in the form most convenient for the calculation of load and speed curves.
In (7), ii is the energy component, and 4 the reactive com-
ponent of the current with respect to the impressed voltage, but not with respect to the induced voltage; in (6), i\ is the energy component and i2 the reactive component of the current with respect to the induced voltage, but not with respect to the
impressed voltage.
The condition of motor operation at unity power-factor is :

Thus:

= ;2

in equation (7).

= = at no-load, for ii

0, this gives: e

e^ as was to be expected.

Equation (8) gives the variation of the induced voltage and

thus of the field excitation, required to maintain unity power-

factor at all loads, that is, currents, i\.
From (8) follows:

= e& *

"II,'

tl

Z \^

>

X GQ "'""'

y/A\
(9)

INDUCTION MOTOR

61

Thus, the minimum possible value of the counter e.m.f., e, is given by equating the square root to zero, as :

e

=

x -e

.

For a given value of the counter e.m.f., e, that is, constant field excitation, it is, from (7) :

or, if the synchronous impedance, x, is very large compared with r, and thus, approximately:

(ID

The maximum value, which the energy current, ii, can have,

at

a

given

counter

e.m.f.,

e y

is

given

by equating the

square

root

to zero, as:

- --
t, vU

(12)

For:

= ii

0, or at no-load, it is, by (11):

CQ

e

Equations (9) and (12) give two values of the currents i\ and izf of which one is very large, corresponds to the upper or

unstable part of the synchronous motor-power characteristics

shown on page 325 of "Theory and Calculation of Alternating-

current Phenomena," 5th edition.

43. Denoting, in equation (5) :

E - = e'

je",

(13)

and again choosing $0 = eo, as the real axis, (5) becomes :

>
e

= "

je

( eo

n\

~ - a&"2)

j (ai n*2),

(14)

and the electric power input into the motor then is:
Po = /EQj //' =
e<#i,

(15)

the power output at "the armature conductor is:

62

ELECTRICAL APPARATUS

hence by (14):

= - - + - Pi

ii (e<>

n'i

xi z )

i* (xii

ria),

(16)

expanded, this gives :

- + Pi = e ii

r (if

8
i, )

~ = Po ri*,

(17)

where:

i = total current. That is, the power out-

put at the armature conductors is the power input minus the

i*r loss.

The current in the field is :

=

io

eb,

(18)

hence, the i 2r loss in the field; of resistance, r\.
= *ri e*b*n.

(19)

The hysteresis loss in the induction motor of mutual induced
voltage, e, is : e 2g, or approximately :

= P'

e Q *g,

(20)

in

the

synchronous

motor,

the

nominal

induced

voltage,

e }

does

not correspond to any flux, but may be very much higher, than

corresponds to the magnetic flux, which gives the hysteresis

loss, as it includes the effect of armature reaction, and the hysteresis loss thus is more nearly represented by e^g (20). The

difference, however, is that in the synchronous motor the hys-

teresis loss is supplied by the mechanical power, and not the

electric power, as in the induction motor.
The net mechanical output of the motor thus is:

P = P! - io*ri - P'

- = Po- iar - ioVi

e

2
g

- - - = e ii

i*r

e 2 6Vi

e

z g,

(21)

and herefrom follow efficiency, power-factor and apparent
efficiency.
44, Considering, as instance, a typical good induction motor, of the constants :
= 60 500 volts; Fo = 0.01 - 0.1 j;
+ Z = 0.1 0.3J; + Zi 0.1 0.3 j.

INDUCTION MOTOR

63

The load curves of this motor, as induction motor, calculated
in the customary way, are given in Fig. 22. Converted into a synchronous motor, it gives the constants: Synchronous impedance (1) ;
+ Z = r+jx = 0.1 10.3 j.

Fig. 23 gives the load characteristics of the motor, with the power output as abscissae, with the direct-current excitation, and thereby the counter e.m.f., e, varied with the load, so as to maintain unity power-factor.
The calculation is made in tabular form, by calculating for
various successive values of the energy current (here also the
total current) ii, input, the counter e.m.f., e, by equation (8) :

- + 6 2 = (500

0.1 iiY

100.61

2 ii ,

the power input, which also is the volt-ampere input, the powerfactor being unity, is:

= = PO eoii

500 i\.

From e follow the losses, by (17), (19) and (20):

in armature resistance:
in field resistance: hysteresis loss:

0.1

2
ii

;

0.001 62 ;

2.5 kw.;

and thus the power output :

p - - - = 500 ii

2.5

0.1

2 *i

0.001 e2

and herefrom the efficiency. Fig. 23 gives the total current as i, the nominal induced voltage
as e, and the apparent efficiency which here is the true efficiency,
as y.
As seen, the nominal induced voltage has to be varied very greatly with the load, indeed, almost proportional thereto. That is, to maintain unity power-factor in this motor, the field excitation has to be increased almost proportional to the load.
It is interesting to investigate what load characteristics are given by operating at constant field excitation, that is, constant nominal induced voltage, e, as this would usually represent the
operating conditions.

64

ELECTRICAL APPARATUS

90 100 110 120 130 140 KW.
FIG. 22. Load curves of standard induction motor.

-
,
tion motor converted to synchronous motor.

INDUCTION MOTOR

65

Figs. 24 and 25 thus give the load characteristics of the motor, at constant field excitation, corresponding to:

in Fig. 24: in Fig. 25:

e = 2e

;

= e

5e.

For different values of the energy current, 3 , from zero up to
the maximum value possible under the given field excitation,

INDUCTION MOTOR CONSTANT DIRECT CURRENT EXCITATION
e
=500

= (Z

.1 4-10.3 j)

SYNCHRONOUS

FIG. 24. Load curves Tat constant excitation 2e, of standard induction motor converted to synchronous motor.

as given by equation (12), the reactive current, i%, is calculated

by equation (11):
Fig. 24: Fig. 25:

z*2 = 48.5 - V9410 - i?;

- i2 = 48.5 - V58,800

2
^i .

The total current then is:

the volt-ampere input : the power input:

= Q

eoi;

Po == e ii,

66

ELECTRICAL APPARATUS

the power output given by (21), and herefrom efficiency 77, power-factor p and apparent efficient, 7, calculated and plotted.
Figs. 24 and 25 give, with the power output as abscissae, the total current input, efficiency, power-factor and apparent
efficiency.
As seen from Figs. 24 and 25, the constants of the motor as synchronous motor with constant excitation, are very bad: the no-load current is nearly equal to full-load current, and power-

INDUCTION MOTOR

CONSTANT

DIRECT e

=CU5ReRENT

EXCITATION

__

Yj-.Ol-.1J

Z?.'l -K3J

(Z= .H-10.3J)

SYNCHRONOUS

FIG. 25. Load curves at constant excitation 5 e, of standard induction motor converted to synchronous motor.
factor and apparent efficiency are very low except in a narrow
range just below the maximum output point, at which the
motor drops out of step. Thus this motor, and in general any reasonably good induction
motor, would be spoiled in its characteristics, by converting it into a synchronous motor with constant field excitation.
In Fig. 23 are shown, for comparison, in dotted lines, the apparent efficiency taken from Figs. 24 and 25, and the apparent efficiency of the machine as induction motor, taken from Fig. 22.

INDUCTION MOTOR

67

45, As further instance, consider the conversion into a synchronous motor of a poor induction motor: a slow-speed motor of very high exciting current, of the constants:
60 = 500;
To = 0.02 - 0.6 j;
+ Z Q = 0.1 0.3 j] + Zi 0.1 0.3J.
The load curves of this machine as induction motor are given
in Fig. 20.

FIG. 26. Load curves of low-speed high-excitation induction motor converted to synchronous motor, at unity power-factor excitation.
Converted to a synchronous motor, it lias the constants: Synchronous impedance:
+ Z = 0.1 1.97 j.
Calculated in the same manner, the load curves, when vary-
ing the field excitation with changes of load so as to maintain unity power-factor, are given in Fig. 26, and the load curves for constant field excitation giving a nominal induced voltage:
e == 1.5 eo
are given in Fig. 27.
As seen, the increase of field excitation required to maintain

68

ELECTRICAL APPARATUS

unity power-factor, as shown by curve e in Fig. 26, while still considerable, is very much less in this poor induction motor, than it was in the good induction motor Figs. 22 to 25.
The constant-excitation load curves, Fig. 27, give characteristics, which are very much superior to those of the motor as induction motor. The efficiency is not materially changed, as was
to be expected, but the power-factor, p, is very greatly improved at all loads, is 96 per cent, at full-load, rises to unity above full-

FIG. 27. Load curve of low-speed high-excitation induction motor converted to synchronous motor, at constant field excitation.
load (assumed as 75 kw.) and is given at quarter-load already
higher than the maximum reached by this machine as straight
induction motor.
For comparison, in Fig. 28 are shown the curves of apparent efficiency, with the power output as abscissae, of this slow-speed
motor, as: I as induction motor (from Fig. 20); SQ as synchronous motor with the field excitation varying to maintain unity power-factor (from Fig. 26) ;
S as synchronous motor with constant field excitation (from
Fig. 27).

INDUCTION MOTOR

69

As seen, in the constants at load, constant excitation, S, is practically as good as varying unity power-factor excitation, S , drops below it only at partial load, though even there it is very greatly
superior to the induction-motor characteristic, /. It thus follows:
By converting it into a synchronous motor, by passing a direct
current through the rotor, a good induction motor is spoiled, but a poor induction motor, that is, one with very high exciting current, is greatly improved.

50.

.40.

I INDUCTION MOTOR

S SYNCHRONOUS, UNITY POWER FACTOR 8 SYNCHRONOUS, CONSTANT EXCITATION

-30.

X CSoSYNCHR.CONCAT.INDUCT., UNITY P.P.

A C8 SYNCHR.CONCAT.1NDUCT., CONSTANT EX:CciT_20.

*4'- CO COMMUTAT.MACH.CONCAT.INDUCTION C CONDENSER IN SECONDARY

20 SO 4 50 60 70 80 90 100 1 .0 120 130 140 150 160 170 180 190 200

FIG. 28. Comparison of apparent efficiency and speed curves of highexcitation induction motor with various forms of secondary excitation,

46. The reason for the unsatisfactory behavior of a good induction motor, when operated as synchronous motor, is found in the
excessive value of its synchronous impedance. Exciting admittance in the induction motor, and synchronous
impedance in the synchronous motor, are corresponding quantities, representing the magnetizing action of the armature currents. In the induction motor, in which the magnetic field is produced by the magnetizing action of the armature currents, very high magnetizing action of the armature current is desirable, so as to produce the magnetic field with as little magnetizing current as possible, as this current is lagging, and spoils the powerfactor. In the synchronous motor, where the magnetic field is produced by the direct current in the field coils, the magnetizing action of the armature currents changes the resultant field excitation, and thus requires a corresponding change of the field current to overcome it, and the higher the armature reaction, the more

70

ELECTRICAL APPARATUS

has the field current to be changed with the load, to maintain

proper excitation. That is, low armature reaction is necessary. In other words, in the induction motor, the armature reaction
magnetizes, thus should be large, that is, the synchronous reactance high or the exciting admittance low; in the synchronous motor the armature reaction interferes with the impressed field

excitation, thus should be low, that is, the synchronous imped-

ance low or the exciting admittance high.

Therefore, a good synchronous motor makes a poor induction

motor, and a good induction motor makes a poor synchronous

motor, but a poor induction motor one of high exciting admit-

tance, as Fig. 20 makes a fairly good synchronous motor.

Here a misunderstanding must be guarded against: in the

theory of the synchronous motor, it is explained, that high

synchronous reactance is necessary for good and stable synchro-

nous-motor operation, and for securing good power-factors at all

A loads, at constant field excitation.

synchronous motor of low

synchronous impedance is liable to be unstable, tending to hunt

and give poor power-factors due to excessive reactive currents.

This apparently contradicts the conclusions drawn above in

the comparison of induction and synchronous motor. However, the explanation is found in the meaning of high and
low synchronous reactance, as seen by expressing the synchronous reactance in per cent. : the percentage synchronous reactance is the voltage consumed by full-load current in the synchronous

reactance, as percentage of the terminal voltage*
When discussing synchronous motors, we consider a synchro-
nous reactance of 10 to 20 per cent, as low, and a synchronous

reactance of 50 to 100 per cent, as high.
In the motor, Figs. 22 to 25, full-load current at 75 kw. out-
put is about 180 amp. At a synchronous reactance of x =
10.3, this gives a synchronous reactance voltage at full-load

current, of 1850, or a synchronous reactance of 370 per cent.
In the poor motor, Figs. 20, 26 and 27, full-load current is about
200 amp., the synchronous reactance x = 1.97, thus the react-
ance voltage 394, or 79 per cent., or of the magnitude of good

synchronousrinotor operation.
That is, the motor, which as induction motor would be consid-
ered as of very high exciting admittance, giving a low synchronous impedance when converted into a synchronous motor, would as synchronous motor, and from the viewpoint of synchronous-