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Copyright, 1917, by the 
McGraw-Hill Book Company, Inc. 

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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 electro physics 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 an 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 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 Pi" Irs- iVeimenilwused types 
of apparatus. 

In some respects, the following work, and its companion vol- 
ume, "Theory and Calculation of. Electric X'irrvits," may be 
considered as continuations, or ratliei as parts o< "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 
when rewriting it recently for the 5th 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, Vtele's Creek. 
July, 1917. 

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Preface v 

Chapter I. — 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 with 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 startin gdeyice — 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. C. 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 
device — Numerical 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 

8. Speed control by armature resistance — Disadvantage of in- 
constancy of speed with load — Use of condenser in armature or 
secondary — Use of pyro-electric resistance 12 

9. Speed control by variation of the effective frequency: con- 
catenation — By changing the number of poles: multispeed 
motors 13 

10. A. Pyro-electric speed control — Characteristic of pyro- 
electric conductor — Close speed regulation of motor — Limita- 
tion of pyro-electric conductors 14 

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

VI 1 


giving high current and torque at resonance 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 Multispecd 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 -5- 6 -5- 8 pole motor — Numerical calculation and 
discussion 23 

Chapter II. Multiple Squikkel 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 



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 in- 
duction motors and motors of high overload capacity — 
Instance 52 

39. Induction machine corresponding to synchronous machine ex- 
cited by armature reaction, induction machine secondary corre- 
sponding 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 

power factor and at constant, field excitation — Comparison 

with induction motor 

4ti. Comparison of induction motor and synchronous motor regard- 
ing armature reaction and synchronous impedance — roor 
induction motor makes kikkI, and good induction motor luaitea 
poor synch roll 01 la motor 

Indliclion Mnlor Canctitenatrd with Sj/iichriimnix 

47. Synchronous characteristic and synchronizing speed of con- 
catenated couple — Division of load between machines — The 
synchronous machine as .small exciter 

48. Equation of concatenated couple of synchronous and induction 
motor — Reduction to standard synchronous motor equation . 

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

50. Calculation of numerical instance of 56 polar high ,- 
induction motor concatenated to 4 polar synchronous . 

51. Discussion. High power factor at all loads, at ( 
synchronous motor excitation 76 

Inthtclion Motor Cimcotaiotcd with i'lxnmiiliitiinj Machine 

52. Concatenated cmiple »il h com tilting machine asynchronous 

— Series and shunt excitation — Phase relation adjustable — 
Speed control and power factor control— Two independent 
variables with concatenated commutaling machine, against one 
with synchronous machine— Therefore greater variety of speed 
and load curves 78 

53. Representation of the commutatiug machine by an effective 
impedance, in which both components may be positive or 
negative, depending on position of commutator brushes ... SO 

54. Calculation of numerical instance, with commutating machine 
series excited for reactive anti-inductive voltage — Load curves 
and their discussion . 82 

Induction Mntnr "-ilk ('iiiiili-iimr in Strouifartj Circuit 

55. Shunted capacity neutralising 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 

50, 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 ami asynchronous motors in which the Bpeed 
decreases with increasing load 8S 


Induction Motor with Commutator 


58. Wave shape of commutatcd 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 80 

59. Squirrel cage winding combined with com mutated winding — 
Hey land motor — Available only for power factor control— Its 
limitation 91 

Chapter V. Single-phase Induction Motor. 

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 maxim urn torque 
ratio — Numerical instance 109 

72. Inductive devices — External inductive devices — Internal in- 
ductive devices Ill 

73. Shading coil — Calculations of voltage ratio and pliane arigh? . 112 



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 



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 — Tl e 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 

93. Calculation of numerical values for 0, %, \i, \i pitch defi- 
ciency, up to the 2l8t 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 




98. Rotation of iron disc in rotating magnetic field — Equations- 
Motor below, generator nhove synchronism 168 

99. Derivation of eqieit imih fin in hysteresis law - I lystcrcsi» (orque 

of standard induction motor, and reJation to size 169 

100. General discussion of hysteresis mot or —Hysteresis loop 

collapsing or expanding . 170 

Chapter XL Rotary Terminal Singli 

I] InIU.'(.'TIO«J MoTOKS. 

101, Performance and method of operation of rotary terminal 
single-phase induction Mot or —Relation of motor speed to 
brush speed and slip corresponding to the loud 172 

102. Applicot inn of tin- principle to n self-starting si iigle-|>h;isc power 
motor with high starting and accelerating torque, by auxiliary 
motor carrying brushes 173 


i Gem 

, Alternating' 

103. The printipli- nf tin- frequency converter nr 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 

IDS. Equation or 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 1S6 

109. The characteristic curves: regulation curve, compounding 
curve— Connection of frequency converter with synchronous 
machine, and eomjieusation 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 M 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 multispced operation . 192 


Chapter XIII. Synchronous Induction Generator. 


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 

116. 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 — (b) 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 
storage — 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 ph&j«e converter . . 222 


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 induciion phase balancing with reverse rotation 

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

135. Limitation of single-phase generator by heating of b 
coils — By double frequency pulsation of annul 
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 uf a synchronous phase balancer lo the 
synchronous phase converter — Its construction 230 

137. The various methods of taking caro 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 i.lifferontial current and flash- 
ing around the commutator — Stability 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 I'sc of double brush 
to vary short circuit — Effect of load — Thomson Houston arc 
machine — Brush arc machine — Storage battery charging . . 243 

142. Reversing and contact making rectifier— Half wave rectifier 
and its disadvantage by unidirectional magnetisation of trans- 
former — 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— Poly phase rectifica- 
tion — Star connected, ripg connected and independent phase 



rectifiers — Y 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 m.m.f. and magnetic 
flux, caused by polar field structure — Energy component of 
reactance 261 

140. Magnetic hysteresis as instance giving energy component of 

reactance, as effective hysteretic 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 Machine*. 

156. Description of inductor machine type —hid actum by pulsating 
unidirectional magnetic flux 274 

157. Advantages and disadvantages of inductor typ*?, with regard* 

to field and to armature 275 

158. The magnetic circuit of the inductor machine, calculation of 
magnetic flux and hysteresis loss 276 


Chapter XX. Single-phase Commutator Motors. 


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 1«gg» n g field magnetism, by 
resistance shunt across field 366 

205. Compensation for phase displacement and control of power 
factor by alternating current commutator motor with t^gf^g 
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 



207. Discussion of motor types: compensated series motors: con- 

ductive and inductive compensation — Their relative advan- 


208. Repulsion motors: lagging quadrature flux — Not adapted to 

speeds much above synchronism — Combination type: series 

repulsion motor 


209, Constructive differences— Possibility of ['hanging from type to 

type, with change of speed or load 


210. Other commutator motors: shunt motor — Adjustable speee: 

polyphase induction motor— Power factor compensation 

Heyland motor — Winter-Eichberg motor 


211. Most general form of single-phase commut at or motor, with two 

stator and two rotor circuits and two brush short circuits . 


212. General equation of motor , 


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

with series characteristic 


214. Repulsion motor; Equations 


215. Continued 


216, Discussion of commutation current and commutation factor 


217. Repulsion motor and repulsion generator , . . 


218, Numerical instance 


219. Series repulsion motor; equations 


220. Continued * 


221. Study of commutation —Short circuit current under brushes . 


222. Commutation current 


223. Effect of voltage ratio and phase, on commutation . . 


224. Conililiui) of vanishing commutation current 


225. Numerical example 


226. Comparison of repulsion motor and various series repulsion 


227. Further example — Commutation factors 


228. Over-compensation — Equations . . , 


229. Limitation of preceding discussion — Effect and importance o: 

transient in short circuit current 


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 . . 


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


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

the magnetic flux — Electrical shifting of the magnetic flux by 

varying the excitation of the several sections of the field pole . 


232. Armature reaction and commutation— Calculation of the re- 

sultant armature reaction of the converter with shifted mag- 

netic flux 


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



improving commutation — Demagnetizing armature reaction 

and need of compounding by series field 429 

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

234. Increase and decrease of d.-c. voltage by increase or decrease 
of maximum a.-c. voltage by higher harmonic — Illustration 

by third and fifth harmonic 430 

235. Use of the third harmonic in the three-phase system — Trans- 
former 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 r . . 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 connec- 
tion of conductors with separate pairs of collector rings . . . 454 



250. Unipolar machine adapted for low voltage, or for large sice 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 magnetizatiftn 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 




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 direct- 
current 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 squirrel- 
cage type of rotor, which is the simplest, most reliable and there- 
fore most generally used, is not adapted for the use of a start- 
ing 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 



a given torque when starting by autotransformor. However, 
high rotor resistance means lower efficiency and poorer speed 
regulation, anil 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- 
ruiTini Phenomena" and in "Theoretical Elements of Electrical 

As Been, in the induction motor, the (effective) secondary re- 
sistance 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 in- 
creasing slip, anil thus is low at full speed, and higher at standstill. 
A. Temperature Starting Device 

2. A 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 extenl : 
with increasing slip, that is, decreasing speed of the motor, the 
secondary current increases. If the dimensions of the secondary 
mfetanoe Me chosen so that it rises considerably in tempera- 
ture, 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 bv: 

r = r° (1 + aii 3 ). (I) 

As illustration, consider a typical inductiou motor, of the 

Co = 110; 

Yt-g-jb" 0.01 - 0.1 j; 
Zo = r„+ j"j: =0.1 +0.3j; 
Z, = r l +jx l = 0.1 -f 0.3j; 
the speed-torque curve of this motor is shown as A in Fig. 1 


Suppose now a resistance, r, i8 inserted in series into the sec- 
ondary 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 + i,« 10-*) 
= 0.1 (1 +»i , 10- 4 ), 





z,=r, + .3i 




































and the total secondary resistance of the motor then is: 

r\ = r, + r<,{l + otV) (2) 

= 0.2 (1 + 0.5 if 10-'). 

To calculate the motor characteristics for this varying resist- 
ance, r'l, we use the feature, that a change of the secondary re- 
sistance of the induction motor changes the slip, s, in proportion 
to the change of resistance, but leaves the torque, current, power- 
factor, torque efficiency, etc., unchanged, as shown on page 
322 of "Theoretical Elements of Electrical Engineering." We 
.thus calculate the motor for constant secondary resistance, n, 
but otherwise the same constants, in the manner discussed on 
page 318 of "Theoretical Elements of Electrical Engineering." 


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

('] = e y/a{ + of, 
herefrom follows by (2) the value of r',, and from this the new 
value of slip: 

e + * - r'i * n. (3) 

The torque, T, then is plotted against the value of slip, .•', and 
gives curve B of Fig. 1. As seen, B gives practically constant 
torque over the entire range from near full speed, to standstill. 

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

3. Assuming, now, that the internal resistance, r lT were made 
as low as possible, t x = 0.05, and the rest added as externa] 
resistance of high temperature coefficient: r" = 0.05, giving the 
total resistance: 

= 0.1 (1 + 0.5 ir 10" 4 ). 


This gives the same resistance as curve A ; r\ = 0.1, at light- 
load, where i L 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 

C and .-1 thus are directly comparable: both have the same 
constants mid same speed regulation and other performance, at 
speed, but C gives much higher torque at standstill and during 

For comparison, curve .4' has heen 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: i, = 153; thus r'i = 0.217; while cold, the re- 
sistnin-c is: r'i = 0.1. Thjs 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 335°C, As seen, 


by going to temperature of about 350°C. in the rotor conductors 
— which naturally would require fireproof construction — it be- 
comes possible to convert curve A into C, or A' 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 increas- 
ing slip, to get high torque at low speeds, the same result can be 
produced by the use of an effective resistance, such as the effect- 
ive 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, i\, the voltage required to send the current 
through the magnetic circuit is proportional to the frequency, 
that is, to 8. Hence, the suaceptance is inverse proportional 
to «: 

V = 6 - (5) 


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 range, 1 thus may approxi- 
mately be assumed as constant. That is, the hysteretic con- 
ductance is proportional to the susceptance : 

g' = V tan a. ((>) 

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

Y' = g' - jb' = V (tan a - j) 

= - J = (tan a - j) (7) 

8 8 8 

1 "Theoiy and Calculation of Al format iri^-rurr^nt Phfjiornwia," 
Chapter XII. 


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

Y' = b g (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 pur- 
pose of using the effective resistance of hysteresis, which in- 
creases with the frequency, to control the motor torque curve. 

The total secondary impedance then is: 


- {» + Q + * (* + J) • « 

Z i — Z\ + v/ 

where: Y = g — jb is the admittance of the magnetic circuit at 
full frequency,, and 

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

6o = 100; 
Y = 0.02 - 0.2 j; 
Zo = 0.05 + 0.15 j; 
Zi = 0.05 + 0. 15 j; 

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

Y = g - jb; 
and assume: 

g = 0.6 b; 

6 = 4; 
thus, by (8) : 

Z\ = (0.05 + 0.11 s) + 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, 
and the proper value of r h %\ and m has to be used for every 
slip, 8. 

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


For comparison is shown, as 7", in dotted lines, the torque 
curve of the motor of constant secondary resistance, and of the 
constants : 

> o.oi - o.i y, 

- 0.01 + 0.3 j; 

> 0.1 + 0.3J; 

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. 




Z,-!.OB + JMI+ 










1 li 









T 1 







- 1-1 















Fia. 2.— Speed c 

a of induction motor with hysteresis starting device. 

p represents the power-factor, tj the efficiency, y the apparent 
efficiency, V the torque efficiency and y' the apparent torque 

However, T corresponds to a motor of twice the admittance 
and half the impedance of 7". 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 7" by increasing the voltage 
y/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 



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. Z 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 


Y 6 = .oa-.9j: Z, -.05+155. e,-100 
Z,-(.05+.ll») + .335 ja 

Mil / 

il ! 1 M /l 



r y~ 

=1 — 




/// i 


// ! 







r c.i i.o 1.5 z.o is i.a s.s <.o is s.» s-s «.e -.s ;s t.s 

Fig. 3.— Load c 

s of induction n 

r with hysteresis starting device. 

Elements of Electrical Engineering." which gives the load curves 
of 7" of Fig. 2, it is seen that the hysteresis starting device reduced 
the maximum power-factor. />. 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 
n=*i-ranr» with increasing slip, we use a magnetic circuit having 
very hizh eddy-current losses: very thick laminations or solid 
iron, or we directly provide a closed high-resistance secondary 
wii-ing around the magnetic circuit, which is inserted into the 
ir.d lotion-motor secondary for increasing the starting torque. 


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

&' = 6 - (10) 


At a given current, i h energizing the magnetic circuit, the in- 
duced 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, there- 
fore, are proportional to the square of the voltage. The eddy- 
current conductance, g f 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) 


Connecting such an admittance in series to the induction- 
motor secondary, gives the total secondary impedance: 

Z J = Z\ + y, 

= Ai + — ^-i-A + 3 /«»i + -™-nr Y (12) 



g = b. (13) 

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

*'• = ( ri + 6TiT>)) + * (*' + 1 ( r+ *>) • < 14) 

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: 

e = 100; 
Y = 0.03 - 0.3 i; 
Z = 0.033 + 0.1 j; 
Z x = 0.033 + 0.1 j; 

6 = 3; 


thus : 

7. As seen, the torque curve has a very curious shape: a 
maximum at 7 per cent, slip, and a second higher maximum at 

The torque efficiency is very high at alt speeds, and prac- 
tically constant at 82 per cent, from standstill to fairly close of 
full speed, when it increases. 


Yir.03-.3j; Z»-.033+ 1j; e -100 









• 7 


























r , 












io. 4. — Speed curves of induction unit or edily-curri'nt starting device. 

But the power-factor is very poor, reaching a maximum of 
8 per cent, only, and to get the output from the motor, required 
ewinding it to give the equivalent of a y/Z times as high voltage. 

For comparison, in dotted lines as 7" is shown the torque curves 
f the standard motor, of same maximum torque. As seen, in 
ic motor with eddy-current starting device, the slip at load is 
ery small, that is, the speed regulation very good. Aside from 
le poor power-factor, the motor constants would be very 
atis factory. 

The low power-factor seriously limits the usefulness of the 

By differently proportioning the eddy-current device to the 
ccondary circuit, obviously the torque curve can be modified 


and the starting torque reduced, the depression in the torque 
curve between full-speed torque and starting torque eliminated, 

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. 


























I I 


■ ft 


t 1 


E 1 


S 6 


i ■ 

a ' 



Fig. S. — Load curves of induction motor with eddy-current atartinR devlra. 

In the discussion of the multiple squirrel-cage induction motor, 
Chapter II, we shall see speed-torque curves of the character us 
shown in Fig. 4. By the use of the rotor iron as magnetic cir- 
cuit, the impairment of the power-factor is somewhat reduced, 
so that the multiple squirrel-cage motor becomes industrially 

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 hIoIh filled 
with solid deep bars, eddy currents in these bars occur at higher 
frequencies, or unequal current distribution. That is, the cur- 
rent flows practically all through the top of the bars at the high 



frequency of low motor speeds, thus meeting with a high resist- 
ance. With increasing motor speed and thus deereMlllg 
secondary frequency, the current penetrates deeper into the bar, 
until at full speed it passes practically uniformly throughout 
the entire bar, in a cireuit 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. 


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 direct- 
current 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 s]>eed can be reduced by inserting 
resistance into the armature or secondary, just as in the direct- 
current 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 arma- 
ture 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, Bpeed 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 D6 
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 conies 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 


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 

The speed of the induction motor can, however, be changed: 

(a) By changing the impressed frequency, or the effective 

(b) By changing the number of poles of the motor. 

Neither of these two methods has any analogy in the direct- 
current 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 corre- 
spondingly changes the synchronous speed. The effect of a 
change of frequency is brought about by concatenation of the 


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 Dumber of poles is changed from four to six and to eight, the 
speed is changed from 1800 to 1200 and to 900 revolutions per 

This method of speed variation of the induction motor, by 
changing the number of poles, is the most convenient, and such 
"multispced 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 
same voltage at all values of current. A 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-etectric conductors (see Chapter I, of "Theory 
and Calculation of Electric 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; >'„ = 0.01 - 0.\j;Z B =0.1 + 0.3 j; Z, = 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, 


1 1 1 1 1 I 1 

























n - 





i - 




li ! 

10 L 

u 1 



Fio. 6. — Variation of resistance of pyro-cleotric conductor, with current. 


Y. = .01-.1; . Zc-A+.3j : Z, = .1 + .3j : r-a,4.6 

























is calculated in the usual way as described on page 318 of 
"Theoretics] Element* of Electrical Engineering." For any 
value of slip, -s, and cor responding value of torque, T, the secondary 
current is *'[ = c y/ac -\- a*-. To this secondary current corre- 
sponds, by Fig. (j, the resistance, r, of the pyro-electric conductor, 
and the insertion of r thus increases the slip in proportion to the 

jni'iea-cil secondary resistance: ■> where ri = 0.1 in the 

present instance. Tliis gives, as corresponding to the torque, 
T, the slip: 

, r + r , 

& = 8, 

where s = slip at torque, T, with short-circuited armature, or 

resistance, r t . 

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. -a peed 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 
permiiiiriit 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 polariza- 
tion cell or aluminum cell) in the induction-motor secondary 
should therefore afford, at least theoretically, a means of speed 
control by varying the capacity. 


Let, in an induction motor: 

Yo = g — jb = primary exciting admittance; 
Z = r + jxo = primary self-inductive impedance; 
Z\ = r\ + jxi = internal self-inductive impedance, at full 

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

k = 2*yc <» 

at full frequency, and - at the frequency of slip, s. 

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

Z{ = n+j («x x - *) (2) 

and the secondary current: 

T sE se 

/l= "w ' k v ti= ~r~^ = ~^ (3) 

r, + j («*t - J ^ + ( 5Xl _ *) 

= E (di - ja 2 ), 

8r Y 

ai m 

s(s Xl - *) 

C2 = " ■ 


m = ri 2 + (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 = $Y 
the primary current equals the secondary current: 

and the primary impressed voltage thus is : 

$Q = # + Zo/o 
1 "Theoretical Elements of Electrical Engineering," 4th edition, p. 318. 



and, substituting (3) and rearranging, gives: 


Eolrt +j(sxi - -)} 

E . _ (g) 

(ri + «r ) + j Isxi + sxu- ■ ) 

or, absolute: 

c 2 = jr — j- (6) 

(ri + sr ) 2 + («Ci + 8X J 

The torque of the motor is : 

T = e 2 a x 
and, substituting (4) and (6) : 

T = srie 2 

(ri + *r ) 2 + \sxi + 8X0 J 


As seen, this torque is a maximum in the range of slip, 8, 
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 : 

that is: 


8X1 + 8X0 = 0, 


* - J— I— (8) 

\£o + x i 

and so = 1, that is, the motor gives maximum torque near 
standstill, for: 

k = xq + xi. (9) 

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

r = 0.01 - 0.1./, 

Z = Zi = 0.1 + 0.3 j, 



for the values of capacity reactance : 

it = 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, a, by : 

«' = ••■* 

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

(n + «r )* + (axi + «ro - -) 



Y =.01-.1i: Z -.l+-3j; Z.-.1+.3) 



























■ s 








V 1 

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-cycle rating, 
would be: 


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 fractional- 
pitch 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 

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 2 .j pitch, for the eight-polar motor it is Jj 
pilch — which latter is effectively the same as ?g pitch. 

Suppose now the primary winding is arranged and connected 
as a six-polar three-phase winding. Comparing it with the 
tune primary burns, 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 wdiich have to be transferred to the third phase in the 
reverse direction. 

The problem of multispeed motor design (hen is, so to arrange 
I he wiiii lings, I hat I he 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. 


16. 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 self- 
inductive impedances remain essentially unchanged by a change 
of the number of poles from n to n'. That is : 

Zn = Z o, 
Z\ = Z i. 

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, there- 
fore, changes proportional to the number of poles: 

& n' 
B n' 

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

F' = n' 
F 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 name 
impressed voltage, are proportional to the square of the number 
of poles: 

t'oo _ n' 2 
too n 2 ' 

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

b n 2 ' 

The magnetic flux per pole remains the same, and thiiM the 
magnetic-flux density, and with it the hystereHin Iohh 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 mimo 
flux per pole, decreases inverse proportional to the number of 


poles. Since the tooth densities must be chosen sufficiently low 
not to reach saturation at the highest number of poles, ami 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 con- 
ductance, y, docs not change with the number of poles. 

Thus, if in a motor of n poles, we change to n' poles, or by the 

the motor constants change, approximately: 
from : to : 

Z a = r + jx , Z a = r„ + j'xo. 

r-jau, Z, = r.+jr,. 

Y» = g- jb, 


- 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 Dumber of poles and thus the lower speed, must be 
materially interior 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 !>e higher for the lower speeds, and approxi- 
mately 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 multi- 
speed 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 


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

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, 
or from delta connection at higher speeds, to Y at lower 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 


1800 HEV 



' IS 














:. i 


5 2 


: : 


I 1 


I s 

o ; 


d e 

t 7 

o • 

iviltispeed induction i 

r, highest speed, four 

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 : 

thus is: 

e 0j Y a , Z„, Zi to e„, aY , aZ a , aZ^ 

and the characteristic constant is changed from d to a*d. 

17. As numerical instance may be considered a 60-cycle 100- 
volt motor, of the constants : 



30 tR ~1~ 



POLES 900 .RE 













/ m 















Sq.5 I 

; 2/i ■:; :; 

36 4 


/o.S 10 1.5 2 

1 II IS E.S«* 

Pto, 18.— Load Btirvofof multi- 

Kiu. 11.— Load curves o( muH 

speed inilui'tion motor, middle speed induction motor, Ion speed 

■peed, six poles. Wgta poles. 




-i ' 












MO 1 3 










mn / 



-— ' 

i ===::::; ? C:; ^- 



' ^ 


















5" o!s 

o lie x'o as o 

n 3 

-, j 

6 E 


t ■ 

r. - 





2. — Comparison of loud turves of three-speed induction 




Four poles, 1800 rev.:Z = n + jx Q = 0.1 + 0.3 j; 

Z x = ri+jxt -0.1 + 0.3j; Y, = g - jb = 0.01 - 0.05 j. 
Six poles, 1200 rev. :Z = U + jx = 0.15 + 0.45 j; 
Z x -= n + jxi = 0.15 + 0.45 j; Y = g -jb = 0.0067 - 0.0667 j. 
Eight poles, 900 rev. : Z„ = r + jx„ = 0.2 + 0.6 j; 

Zi = r, + jx, = 0.2 + 0.6 j; Y = g - jb = 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, 










1 . 































iou a» aoo too wo eoo itx soo 9ooioooiu»i2ooi»oiHoicooi«o(inoai8«) 
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 four- 
polar synchronism) as abscissa), to give a better compariFon. 
5 denotes the speed, / the current, p the power-factor and y the 
apparent efficiency. Obviously, carrying the same load, that 
is, giving the same torque at lower speed, represents less power 
output, 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. There- 
fore, 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. 


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. 



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 clow* 
to the periphery of the rotor. Such a motor has a great slip of 
speed under load, therefore poor efficiency and poor speed regu- 
lation, 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 effi- 
ciency and speed regulation good, but the starting torque arid 
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 clow* 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" in derived, 
which to some extent combines the characteristics of the high- 
resistance and the low-resistance secondary. That is, at start- 
ing and low speed, the frequency of the magnetic flux in the arma- 
ture, and therefore the voltage induced in the secondary winding 
is high, and the high-resistance squirrel cage thus carries con- 
siderable 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 




low induced e.m.f., considerable current is produced, which is 
effective in producing torque. Such double squirrel -cage induc- 
tion 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, Mid 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 ease, when 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 
slip s — 2. In this case, a triple squirrel cage may 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 ami 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 — e, as it is most convenient, with the mutual inductive 
voltage, c, 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 


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 i x = — » 

where ri 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\X\ of the secondary, the reactance, Xi f does not 
enter any more in the equation of the current, and Cj 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, <t>, induces a voltage, E } which produces 
a current, and this current produces a self-inductive flux, <t>'j, 
giving rise to a counter e.m.f. of self-induction I\X\, which sub- 
tracts from E. However, the self -inductive flux, <t>'i, interlinks 
with the same conductors, with which the mutual flux, <t>, inter- 
links, and the actual or resultant flux interlinkage thus is <t>i = 
$ — <t>'i, and this produces the true induced voltage e\ = E — 
I\X\ y 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 vpltage in inner squirrel cage, reduced 

to full frequency, 
It = current, and 
Zi = r 2 + jx 2 = self-inductive impedance at full frequency, 

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

to full frequency, 
/i = current, and 
Z\ = t\ + jxi = self-inductive impedance at full frequency, 

reduced to primary circuit. 
jj? = voltage induced in secondary and primary circuits by 

mutual magnetic flux, 
#o = voltage impressed upon primary, 
/o = primary current, 

Z = r + jx = primary self -inductive impedance, and 
Yo = g — jb = primary exciting admittance. 

*8ee "Electric Circuits", Chapter XII. Reactance of Induction 



The leakage reactance, Xj, of the inner squirrel cage is Hint due 
to the flux produced by the current in the inner squirrel cage, 
which passes between the two squirrel cages, and does not in- 
clude the reactance due to the flux resulting from the current, 
ft, which passes beyond the outer squirrel cage, as the latter is 
mutual reactance between the two squirrel cages, and thus meets 
the reactance, Si, 

It is then, at slip s: 


sE t 

•h + li+Yt 


E, = E t + jx, h- (4) 

E - E,+jx,Ut + h)- (5) 

E = E + Z„f„. (6) 

The leakage flux of the outer squirrel cage is produced by the of the currents of both squirrel cages, /, + ft, and the 
reactance voltage of this squirrel cage, in (5), thus is jxt (f t + /»). 

As seen, the difference between E, and E t is the voltage in- 
duced by the flux which leaks between the two squirrel cages, in 
the path of the reactance, x?, or the reactance voltage, xtft', the 
difference between E and E, is the voltage induced by the rotor 
flux leaking outside of the outer squirrel cage. This has the 
m.m.f. f i + fi, and the reactance X\, thus is the reactance voltage 
xi (fi + /a). The difference between E <, and E is the voltage 
consumed by the primary impedance: Zafn- (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 s cancels. 

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

f .-f(i + if) 

- ft (»J + JOi), 





Oi = 1 ! 

/Xi Xi XjV I 

as = * (- H h — 1 I 

\ri r a r*/ ; 

thus the exciting current : 

= Ei (g - >&) (ai + jot) 
-R(*i+jW, (11) 


6i = <*\9 + Oa6' v 

and the total primary current is (3) : 

'•-*£ + £( 1+i ?) + *- + *} (13) 

= -Fl (Ci + JCi), 



ci = - H 1- 6i 

Ct = — — + bt 


and the primary impressed voltage (6) : 

Eo = Ei{ai + ja t + (r + jx ) (c, + jc,) } 
= & (* + jdi), (15) 


di = a\ + roCi — XoC» 
d* = a, + r^ + XoCi 
hence, absolute: 


C = -=*=• (17) 

to = e,\/ci* + c,*. (18) 

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

D t = /£,, /,/' 


«62 2 

_ sei 2 

<(>+>?)■ «°> 


hence, the total torque: 

D = D 2 + D h (21) 

and the power output: 

P = (1 - s) Z>. (22) 

(Herefrom subtracts the friction loss, to give the net power 

The power input is: 

Po = /#o, Io/' 

= e 2 2 (cidi + c 2 d 2 ), (23) 

and the volt-ampere input: 

Q = eoio. 


Herefrom then follows the power-factor -gr * the torque effi- 
ciency y, , the apparent torque efficiency 7^-, the power efficiency 

P P 

jj- and the apparent power efficiency 7^ 

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

Co = 110 volts; 
Zo = 0.1 + 0.3j; 
Z, = 0.5 + 0.2 j; 
Z 2 = 0.08 + 0.4 j ; 
} r o = 0.01 - 0.1 j; 

the speed curves for the range from s = 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 s: 

s = 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, 






'■ : 




















-8 -.7 -.6 -.6-. 

.1 .2 .3 A .6 .8 .7 .8 .9 1.0 

s o/ douhle squirrel-cage induct iu 






i ••'■ 

i " y~ 

— ^\J\ 





. A, u J. . 

n : 

■■ r 

g j 

■■■ i 

o — 

Fig. 15.— Load c 

s of double squirrel -cage induction motoi 


the values: 

. S 2 X\Xi 

a,\ = 1 > 

rir 2 

lX\ X\ . £ 2 \ 

a% = *( )» 

\r\ r% T\i 

b\ = aig + a 2 6, 
bi = a^g — a\b, 

c i = V + V + b u 

c 2 = h btt 


d\ = a\ + r Ci — XoCj, 

da = a 2 + roC 2 + XqCi, 

. *o* 

e 2 * = 


di 2 + <V 
to = e* Vci 2 + cf, 

Z> = Z>! + Z> 2 , 

P = (1 - s) D, 

Po = e* 1 (cidx + c 2 d 2 ), 

Q = eot'oi 

P D P D Po 

Po'Po'Q'Q' Q* 

Triple Squirrel-cage Induction Motor 
24. Let: 

<*> = flux, E = 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. 

Also let: Yo = g — jb = primary exciting admittance. 

It is then, at slip s: 
current in the innermost squirrel cage: 



current in the middle squirrel cage: 

/* — zr> 
current in the outer squirrel cage: 

* l — 7~> 



primary current: 

/o = U + I* + U + Fo#. (4) 

The voltages are related by: 

& = #» + J*./., (5) 

tf i = #* + jx, (It + /•), (6) 

# = #i + J'x, (/, + /, + /,), (7) 

#o - # + Zo/o, (8) 

where x 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; X\ the reactance 
of the first squirrel cage and x that of the primary circuit, that 
is, Xt + xo corresponds to the total leakage flux between primary 
and outer most squirrel cage. 

# 8 , fit and #i are the true induced voltages in the three squirrel 
cages, $ the mutual inductive voltage between primary and 
secondary, and $o the primary impressed voltage. 

26. From equations (1) to (8) then follows: 

^l-^{l+J- + -(l+J-j+J- 




= #3 (ai + ja 2 )i 


- 8 2 XzXz 

d\ = 1 — - — 

r»r 3 

(x% . 
«*=*(- + 

X2 , xi\ 
r 3 r 3 / 



/, = - #, (O, + jttj), 





E = E s \ ai + jai + r:' («. + jo*) + 3 ^ (l + 3 ?) + 3 ? 

= ^ 3 (6i+j6 2 ), (14) 

where : 

ri r 2 r 8 / J 

hi = aj — 

SXifl2 S #1X3 


r 2 r 3 

fi r 2 r 8 

thus the exciting current: 

/oo = * ov 

= #3 (&i + j6«) (g - jfc) 

= # 3 (Cj +JC 2 ), 


C\ = feigr + 6 2 6, 
c 2 = 6 2 g — 616, 

and the total primary current, by (4) : 
/o = #3 

8 (ai+ja 2 )+^(l+; S f 3 )+J+ Cl +jc, 

r 2 \ T3 / r 8 

where : 

di = ai H 1 h Ci 

ri r 2 r 3 

, s s 2 x s . 

a 2 = - a 2 + ■ — t f2 
n r 2 r 3 

Zo/o = #3 (di + jrf 2 ) (r + ix ) 

= # 3 (/, +# 2 ), 

where : 

/j = r di — Xo^2 

/ 2 = f(K* 2 T" Xo«l 

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


#o = #3 (6i + jbt + /i + jf«) 
= ^3 (j/i + jg 2 ), 

(/j = 6i + /i 
(Jz = b 2 + / 2 










hence, absolute: 

Vgi- + gS 

U = ez Vrfi 8 + _dS, (25) 

e* =e 3y Jl+ ***?> (26) 

e, = e 3 Va, 2 + a**- (27) 

26. The torque of the innermost squirrel cage thus is : 

D, = *?-; (28) 

that of the middle squirrel cage : 

z> 2 = * ea2 ; (29) 

r 2 
and that of the outer squirrel cage: 

0, = s - '*; (30) 

the total torque of the triple squirrel-cage motor thus is: 

D = D, + D 2 + D 3 , (31) 

and the power: 

P = (1 - s) Z>, (32) 

the power input is : 

P Q = /#o, /o/' 

= <?3 2 (dtfi + rf 2 </ 2 ), (33) 

and the volt-ampere input : 

Q = «oio. (34) 


Herefrom then follows the power-factor -~ » the torque effi- 
ciency d", apparent torque efficiency y^ power efficiency -5- 
*o v * o 

and apparent power efficiency ^y 

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

e = 110 volts; 
Z = 0.1 +0.3j; 
Z, = 0.8 + 0.1 j; 
Z 2 = 0.2 + 0.3 j; 
Z 3 = 0.05 + 0.8 j; 
I'o = 0.01 - 0.1 j; 






























, r 




-H f «» 


. i 

. — Speed curves of triple si]iiirre]-(.':igi> induction mo 










_ ,-UXL 




























Fio. 17. — Load curves of triple squirrel-cape induction motor. 



the speed curves are shown from « = to « = 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 = 0, 0.01, 0.02, 0.05, 0.1, 0.15, 0.20, 0.30, 
0.40, 0.60, 0.80, 1.0, 1.2, 1.4, 1.6, 1.8, 20, 
the values: 

ai - 1 - 



/Xt . Xi . Xi\ 

8Xidi 8 2 X\Xt 

6l = Ol - 

n r 2 r 9 

hx\d\ , sxi 


. 8XiG\ . SXi . 8X\ 

bt = «i H h H i 

r\ r 2 r s 

c\ — big + 626, 
ci = btg + bib, 

dx = -- + - + - + c, 
r\ r 2 rz 

j *a 2 * 2 x« 



+ - + c i} 

TiTz rz 

/1 = rodi — Xodj, 

/ 2 = rodt + Xodi, 
9\ = bi + /1, 
gi = b t + ft, 

e 8 * = 



gS + g* 2 

to = e z y/d x 2 + dsS 

** = " 2 ( x + v 1 ) 

ei* = ez 2 (a x 2 + a. 2 ), 

Z>» = 

Z>, = 

S6 8 - 

r 2 

n - 8€l 

D = !>! + D 2 + /) 3 , 
P = (1 - s) D, 
Pa = ez 2 (rfififi + d 2 2 ), 
Q = <?o* 0, 

P D P D P 



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^ part of the 
primary impressed power into secondary electrical power for 
the supply of the second machine, and a part into mechanical 

The frequency of the secondary voltage of the first motor, and 
thus the frequency impressed upon the second motor, is the fre- 
quency of slip below synchronism, s. The frequency of the 
secondary of the second motor is the difference between its im- 
pressed frequency, t, and its speed. Thus, if both motors are 
connected together mechanically, to turn at the same speed, 
1 — s, and have the same number of poles, the secondary fre- 
quency of the second motor is 2h — 1, hence equal to zero at, 
* = u. 5, Thai is, the second motor reaches its synchronism at 
half speed. At this speed, its torque becomes zero, the power 
component of its primary current, ami thus the power bobi- 
poncnl 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 secninbuy 
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 


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 eight- 
polar winding; and in this case, one and the same motor struc- 
ture 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 motor, it thus begins to act 
merely as an impedance in the secondary circuit, that is, the first 
machine becomes a motor again. Thus, somewhere between 



half synchronism and synchronism, the torque of the first motor 
becomes zero, while the second motor still has a small negative or 
generator torque. A little above this speed, the torque of the 
concatenated couple becomes zero— about at two-thirds syn- 
chronism 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 al>ove and the motor torque below full syn- 
chronism. 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 resist- 
ance 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, ami regaining it again beyond 
full synchronism. 

The speed torque curves of the concatenated couple, shown in 
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: 


= number of pairs of poles of the first motor, 
= tiiiuiljer of pairs of poles of the second motor, 


; a = — = ratio of poles, (1) 

m f = supply frequency. 

Full synchronous speed of the first motor then is: 

Se = £ (2) 

of the second motor: 

5'. - £ (3) 

At slip 9 and thus speed ratio (1 — s) of the first motor, its 
speed is: 

S- (1-«)S - U-«)£ (4) 

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: 

sS'o = s *,; 

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

hence, the slip of speed of the second motor below its synchronous 
speed, is: 

.Z_ (1 _.)/.(«,.L=i) /f 

n n \n n / 

and the slip of frequency thus is: 

s' = 8 (1 + a) - a. (5) 

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

« = ^ (6) 


The speed in this case is: 

So = (1 - so) I (7) 

31. If: 

a = 1, 

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

n = W = 4, 
it is: 

while at full synchronism : 


it is: 




i • 




2 71 








— s 

















that is, corresponding to a twelve-polar motor. 


it is: 














5L o 






1.5n 12 


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 = - > 



* = 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 «. 
The secondary circuit of the second motor is of frequency s' = 

* (1 + a) — a. 

Synchronism of concatenation is reached at: 

1 + a 
Let thus: 

eo = voltage impressed of first motor primary; 
Yo = g — jb -= exciting admittance of first motor; 
Y'o = g* — jb' = exciting admittance of second motor; 
Zo = ro + jxo = self-inductive impedance of first motor 

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

Z\ = T\ + jxi = self-inductive impedance of first motor second- 
Z\ = r'i + jx\ = self-inductive impedance of second motor 

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

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/ _ *'* [« (1 + a) - a] e 

1 * "" PT+ fix>\ - ?;+j\MV+ a) - a] x\ = e(fll " Ja '^ (8) 




Ol = 

a» = 

r'x [s (1 + o) - a] 


x', [«.(1 + a) - a]* 


m - r',« + *V (« (1 + a) - a)*; 
exciting current of second motor: 

/' 00 - eY' = e (g' - jb'), 




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


/o s /i = /] + /' 00 

= e (bi - j6a)> 

bi = a x + g', 
bt = a* + 6', 



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

Z = Z/ + ZV - (n + r' ) + js (*, + x' ), 


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

s$i = se + IiZ, 

hence, reduced to full frequency : 


C - 1 + 

*!-« + — 

= e (ci + jc 2 ), 

ri + r -6i + (x 1 + x / )6 a 



c% = (xj + x'o) 6i - 

ri + r' 




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

loo = $]Y 

= e (di - jd 2 ), 

di = C\Q + c»6 

dt = C!& — Cjj/ 




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

/o = /i + /oo 

-«(fi-ifi), (19) 


/i = 61 + di 
ft - 62 + d 2 


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

$0 = -Fi + £0/0 

-«(0i+J0t), (21) 


and, absolute: 

0i = Ci + ro/i + X0/2 i 
02 = Ci + X0/1 — r / 2> 


eo - e VST+'fifi*. (23) 

e = 77-^Y Y (24) 

Vgi 2 + 02* 

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 : 

to - e VP~+'f2 2 = e J fl l *- h ]\ 

power input: 

Po = /#o, /V' 

= e 2 (fig 1 - / 2 2 ) 

01 2 + 02 2 

volt-ampere input: 

Q = Wo, 

and herefrom power-factor, etc. 
The torque of the second motor is : 

r « /«,/,/' 

The torque of the first motor is : 

7\ = /#„ /o/' 

= C 2 (C1/1 - C2/2), 

hence, the total torque of the concatenated couple: 

T = f + Ti - e= (oj + d/. - c/i), 
and herefrom the power output: 

J 3 - (1 -%) T, 
thus the torque and power efficiencies and apparent efficiencies, 

35. As instances are calculated, and shown in Fig. 18, the speed 









^4 i 































a = 



n at 


18.— Sp 

iue cur 





i B . 18 a 

the su 

ig. 19 a 

'd eouj 


Tw loat 

ning, a 

(Wit torque curves of concalenHted couple with low resist 

es of the concatenated couple of two equal mot 
econstants:co = IlOvolts. 

- Y' - 0.01 - 0.1 j; 

- Z', - 0.1+ 0.3 j; 
= Z', = 0.1 + 0.3.). 

ao shows, separately, the torque of the second mc 
j pi j r current. 

jowa the Bpeed torque curves of the - e oofii 

!e with an additional resistance r = 0.5 inserted 
iry of the second motor. 
curves of the same motor, Fig. 18, for concaten 
id also separately 1 he load curves of either mc 






are given on page 358 of "Theoretical Elements of Electrical 

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 n' 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 








i ; 







• « 

* » 

3 «!l 

i •. 

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 e qui potential points of a 2 p-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, 



this winding becomes at the same time a n-polar short-circuited 
winding. On the rotor, in some slots, the secondary current of 
the n-polar and the primary current of the n'-polar winding flow 
in the 3ame direction, in other Blots 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 »' 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-induclive 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 interna) concatenation 
with single stator and rotor winding is the limitation of the avail- 
able speeds, as it is adapted only to 4 -r- 8 + 12 poles and 
multiples thereof, thus to speed ratios of I + % + \i, the last 
being the concatenated speed. 

Such internally concatenated motors may be used advantage- 
ously sometime as constant -speed motors, that is, always run- 
ning in concatenation, for very slow-speed motors of very large 
number of poles. 

37. Theoretically, any numl>er 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. 

Y % - 9 ~ A 
Zo = r + jx a , 
Zi « Tx + 3*u 
are the constants of a motor, and we denote: 

Z = Z„ + Z, = (r + ri) + j (x n + x,) 
= r + jx 
then the characteristic constant of this motor — which char- 
acterizes its performance — is : 

(J = yz; 
if now two such motors are concatenated, the exciting admittance 
of the concatenated couple is (approximately): 
1" = 2 >\ 


as the first motor carries the exciting current of the second 

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' 

= 40, 

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" = 3 7, 
Z" = 3 Z, 

thus the characteristic constant : 

0" = y"z" 
= 9yz 
= 9 0, 

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. 


38. While in the typical synchronous machine and eommu- 
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 alter- 
nating supply voltage, just as in the alternating-current trans- 
former. 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 i«, 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 out- 
put, 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 exutMrVV 
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 
full-load current, and while fair efficiencies may 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: e a = 500. 

Primary exciting admittance: Y a = 0.02 — 0.6 j. 

Primary self-inductive impedance: Zu = 0.1 + 0.3j. 

Secondary self-inductive impedance: Zi = 0.1 + 0.3 j. 



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: 
Y = 0.01 -O.lj, 
given on page 234 of "Theory and Calculation of Alternating- 
current Phenomena" 5th edition, and page 319 of "Theoretical 


LOW 8PEE0 1 




Y.-.02-.SJ Z,-.l+.3j 



— i- 
















1 I 




Fio. 20. — Low-epecd induction motor, load c 

: the 

Elements of Electrical Engineering," 4th edition, 

39. In the synchronous machine usually the stator, in com- 
mutating 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 com mutating machine are the held, 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 



primary, that is, which receives electric power and converts it 
into mechanical power, and the primary or stator of the induc- 
tion machine thus corresponds to the armature of the synchro- 
nous 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 

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 lesaer 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 (I — s) /, and the full frequency cur- 
rents supplied to the commutator thus give in the rotor the 
effective frequency,/ — (1 — s) / = sf, 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- 



tion-inotor 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 thai of the induction machine, or How- 

n n 

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. 

1 -8 

or: 1 

* JT+T 

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

constant slip — ■r^r 

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 scries 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 lead- 
ing, 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 power- 
factor control, but they may, depending on their design or appli- 
cation, change the characteristics of the induction machine, as 
regards to speed and speed regulation, the capacity, etc. 


If by synchronous or com mutating 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 300°. If this voltage is inserted 90° behind the secondary 
current, it makes this current leading or magnetizing and so in- 
creases 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 (he 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 induct ion- 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, docs 
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 vol- 
tage: sin a, acts magnetizing or demagnetizing, the other com- 
ponent: cos a, acts increasing or decreasing the speed, and thus 
various efferts 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: 

Z,' = n+j («i- 7)' 

where x t is the capacity reactance at full frequency. 


For s — 0, Zj* = o° , that is, the motor has no power at or near 

8Xi = 0, 



it is: 

Zf = r h 

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 



and by varying the capacity reactance, x 2 , various values of reson- 
ance slip, So, thus can be produced, and thereby speed control of 
the motor secured. However, for most purposes, this is uneco- 
nomical, 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 conver- 
sion 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 



In the synchronous motor, Fig. 21, D, produced from the induc- 
tion motor, Fig. 21, C, it is: 

l'"» = 8 — jk = primiiry exciting admittance 

of the induction machine, 
Z = r« -f jxn = primary self-inductive impe- 
Z\ = t\ + jxt = secondary self-inductive im- 

Fio. 21. — Sturtiiig of induction motor and 


The secondary resistance, r,, is that of t lie field exciting winding, 
thus does not further come into consideration in calculating the 
motor curves, except in the efficiency, as i|V| is the loss of power 
in the field, if i\ = field exciting current. Xl is of little further 
importance, as the frequency is zero. It represents t he magnetic 
leakage between the synchronous motor poles. 

r is the armature resistance and x a the armature self-inductive 
reactance of the synchronous machine. 

However, x is net the synchronous impedance, which enters 
the equation of the synchronous machine, but is only the self- 
inductive part of il, or the true armature self-induct ancc. The 


mutual inductive part of the synchronous hapedance. or Ik* 
effective reactance of anaatare reaction x\ is not contained in x*. 

The effective reactance of anaaxure reaction of the synchro- 
nous machine, x* f represents the field excitaiiou consumed by the 
armature m.m.f., and is the voltage corresponding to this field 
excitation, divided bj the armature current which consumes this 
field excitation. 

6, the exciting snsoeptance, is the magnetizing armature 
current, divided bj the voltage induced by it, thus, x\ the effect- 
ive reactance of synchronous-motor armature reaction, is the 
reciprocal of the exciting acceptance of the induction machine. 

The total or synchronous reactance of the induction machine 
as synchronous motor thus is: 

* - x« + x' 

= x. + r 

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: 7 = g — jb, 

Primary self-inductive impedance: Z ■= r + jx<>, 
Secondary self-inductive impedance: Z x = r\ + jxi, 

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

Armature resistance: r , 

Synchronous impedance: x = Xo + r* (1) 

Total power consumed in field excitation : 

P = 2 t»r„ (2) 

where i = field exciting current. 

Power consumed by hysteresis: 

P - e*g. (3) 

it is then: 



42. Let, in a synchronous motor: 

E = impressed voltage, 

E = counter e.m.f., or nominal induced 

Z — r + jx = synchronous impedance, 
/ = i\ — 3H = current, 

#o = $ + ZJ 

= # + (n'i + xi 2 ) + j (xt\ - n 2 ), (4) 

$ = $q — Zf 

= #o - (n'i + xz 2 ) ~ j (xii - ri 2 ), (5) 

or, reduced to absolute values, and choosing: 

g = e = r eal axis in equation (4), 
$o = e = real axis in equation (5), 
eo 2 = (e + ri\ + xU) 2 + (xii — ri 2 ) 2 [e = real axis], (6) 
02 = (^o — rii + xi 2 ) 2 + (xii — ri> 2 ) 2 [e = 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), i\ is the energy component, and i 2 the reactive com- 
ponent of the current with respect to the impressed voltage, but 
not with respect to the induced voltage; in (6), t\ is the energy 
component and i 2 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: 

i 2 = in equation (7). 

e 2 = (6o ~ rtf + xW (8) 

at no-load, for i\ = 0, this gives: e = eo, 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 x . 
From (8) follows: 

re ± \/z 2 e 2 — xV , n x 

'* = - - -o - • W 

z l 


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

e = - e<>. 

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

xe , /e* 7. re \* , . 

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

z = x: 

ii = e i ± 4i ~ ^ (11) 

The maximum value, which the energy current, t'i, can have, 
at a given counter e.m.f., e, is given by equating the square root 
to zero, as: 

t, = |- (12) 

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

eo ± e 

ti = ___. 

Equations (9) and (12) give two values of the currents i\ 
and *2, 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) : 

V = «' - je", (13) 

and again choosing J5? = eo, as the real axis, (5) becomes: 

e f — je" = («o — rii — xi 2 ) - j (xii - n 2 ), (14) 

and the electric power input into the motor then is : 

Po = /#o, //' 

= e-oiu (15) 

the power output at the armature conductor is : 

= jttij+e'tt, 


hence by (14): 

" -Pi = U (e — ri\ — xi 2 ) + it (xi t — n 2 ), (16) 

expanded, this gives: 

Pi = cot'i - r (tV + it 2 ) 

= Po - n 2 , (17) 

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

put at the armature conductors is the power input minus the 
t*r loss. 

The current in the field is: 

to - eb, (18) 

hence, the i 2 r loss in the field; of resistance, ri. 

iVn - «Wri. (19) 

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

P' = eoV (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 hys- 
teresis loss thus is more nearly represented by e 2 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 v - toVi - P' 
= Po— i 2 r — t'oV] — e 2 g 
= e ii — i 2 r — 6*ft*ri — e 2 g, (21) 

and herefrom follow efficiency, power-factor and apparent 

44. Considering, as instance, a typical good induction motor, 
of the constants: 

Co = 500 volts; 
7o = 0.01 -O.lj; 
Z = 0.1 + 0.3j; 
Zi = 0.lJ+j0.3j. 


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., c, 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) t'i, input, the counter e.m.f., c, by equation (8): 

e* . (500 - 0.1 t,)« + 100.61 if, 

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

Po = e ii = 500 i\. 
From e follow the losses, by (17), (19) and (20): 

in armature resistance: 0.1 ii 2 ; 
in field resistance: 0.001 e 2 ; 

hysteresis loss: 2.5 kw.; 

and thus the power output: 

P = 500 ii - 2.5 - 0.1 ii 2 - 0.001 e 2 

and herefrom the efficiency. 

Fig. 23 gives the total current as t, 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 excita- 
tion 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. 







Yo-.OI-.U Z,=.l+.3j 



















1 I 



i - 

B f 

> 1 

J 1 

120 t 


1.1 <■■ 

Fm. 22.— Load c 

a of standard induction it 





a - 500 Zo-.1t .3j 

Y -.01-.|J Z|-.1 +.3i 

fZ -.1 + 10. 3j) 













— X 






























1 1 



a so so i 

» 110 ISO I. 


a i 

j i 


u i 



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

in Fig. 24: 
in Fig. 25: 

-- 2e ; 
- 5 e„. 

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


e = 500 . Z -.1 + -3J. 



a - .1 + 10.3;) 





















— - 

— ■— 







Fia. 24.— Load c 

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

by equation (11): 

Fig. 24: u = 48.5 - V9410 - u*; 

Fig. 25: i, = 48.5 - V58,800 - t,». 

The total current then is: 

i = Vfi'T t^i 

the volt-ampere input: 

the power input: 

Q = e i; 
P* = e Q ii, 


the power output given by (21), and herefrom efficiency ij, 
power-factor p and apparent efficient, 7, calculated and plotted. 

Figs. 24 and 25 give, with the power output as al.iscissjc, the 
total current input, efficiency, power-factor and apparent 

As seen from Firs. 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- 

1 1 II 1 1 II 


*= 5e, 

<t„-600 Zo- •« +.3) 
Yo = .01-.1i Zi-.l + .3; 

(I =..14-10.3;) 

















D i 

a 4 



10 *» 

Fib. 25.- 

factor a 
range j 
motor < 

into a s 

In F 

-Load curves at constant excitation 5 e, of standard 
motor converted to synchronous motor. 

nd apparent efficiency are very low except in a 
ust below the maximum output point, at wh 
rops out of step. 

this motor, and in general any reasonably good in 
vould be spoiled in its characteristics, by oonvc 
■nchronous motor with constant field excitation. 
5. 23 are shown, for comparison, in dotted li 
t efficiency taken from Figs. 24 and 25, and the a 
y of the machine as induction motor, taken from 


■ch the 


rting it. 

nes, the 

) parent 
Fig. 22. 


45. As further instance, consider the conversion into a syn- 
chronous motor of a poor induction motor : a slow-speed motor ot 
very high exciting current, of the constants: 

e = 500; 
y„ = 0.02 - 0.6 j; 
Z = 0.1 +0.3j; 
Z, - 0.1 +0.3;'. 

The load curves of this machir 
in Fig. 20. 

as induction motor are given 



e, - BOO Z„-.t + .Si 
Y„ - .0S-.8J Z.-.l + ,3j 
(Z - 1 + 2i) 

































(1 1 

K 1 

'J 1 


n l 

n U 



') i 

» i 

■0 1 

* • 

Converted to a synchronous motor, it has 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 en 
are given in Fig. 27. 

As seen, the increase of field excitation required to maintain 


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 character- 
istics, which are very much superior to those of the motor as in- 
duction 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- 


So-BOO Zo-.1+.3j 

Vo-.02-.6j Z.-.1 + .3; 

(Z-.1 + .2i) 










— ' 










D ! 


S < 


1 G 


a i 


10 1 

» 1 



Flu. 27. — Load curve of luw-ftpnod liifch-pxeirutiiiii induction motor con- 
verted to syiicliriihiiu.s motor, nt cimaUiiil field excitation. 

oad (assumed as 75 kw.) and is given at quarter-load already 
ligher than the maximum reached by this machine as straight 
nduction 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: 

/ as induction motor (from Fig. 20); 

So 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). 


As seen, in the constants at load, constant excitation, S, is prac- 
tically 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, hut 
a poor induction motor, that is, one with very high exciting 
current, is greatly improved. 

Pio. 28. — Comparison of apparent efficiency and speed curves of high- 
excitation induction motor with various forms of secondary excitation. 

46. The reason for the unsatisfactory behavior of a good induc- 
tion 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 quanti- 
ties, representing the magnetizing action of the armature cur- 
rents. 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 cur- 
rent as possible, as this current is lagging, and spoils the power- 
factor. 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 excita- 
tion, and thus requires a corresponding change of the field current 
to overcome it, and the higher the armature reaction, the more 



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 react- 
ance 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 
loads, at constant field excitation. A synchronous motor of low 
synchronous impedance is liable to be unstable, tending to hunt 
and Hive 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 synchro- 
nous reactance in per cent. : the percent age synchronous reaCtMUM 
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 ayDchrOADUl 
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 
synchronous-motor operation. 

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


motor design, be considered as a high synchronous impedance 
motor, while the good induction motor gives as synchronous 
motor a synchronous impedance of several hundred per cent., that 
is far beyond any value which ever would be considered in syn- 
chronous-motor design. 

Induction Motor Concatenated with Synchronous 

47. Let an induction machine have the constants: 

Y = g — jb = primary exciting admittance, 
Zo = r + jxo = primary self-inductive im- 
Z\ = r\ + jxi = secondary self-inductive im- 
pedance at full frequency, 
reduced to primary, 
and let the secondary circuit of this induction machine be con- 
nected to the armature terminals of a synchronous machine 
mounted on the induction-machine shaft, so that the induction- 
motor secondary currents traverse the synchronous-motor arma- 
ture, and let: 

Z% = r s + JX2 = synchronous impedance of 

the synchronous machine, 
at the full frequency im- 
pressed upon the induction 
The frequency of the synchronous machine then is the fre- 
quency of the induction-motor secondary, that is, the frequency 
of the induction-motor slip. The synchronous-motor frequency 

also is the frequency of synchronous-motor rotation, or - times 

the frequency of induction-motor rotation, if the induction motor 
has n times as many poles as the synchronous motor. 

Herefrom follows: 

1 - a 

= *> 



* = ^h <» 

that is, the concatenated couple runs at constant slip, s = — — - - > 

thus constant speed, 

1 — s = — .-— of synchronism. (2) 

n •+- 1 


Thus the machine couple has synchronous- mo tor character- 
istics, and runs at a speed corresponding to synchronous speed 
of a motor having the sum of the induction-motor and syn- 
chronous-motor poles as number of poles. 

If m = 1, that isj the synchronous motor has the same number 
of poles as the induction motor, 

s = 0.5, 
1 - * - 0.5, 
that is, the concatenated couple operates at half synchronous 
speed, and shares approximately equally in the power output. 

If the induction motor has 76 poles, the synchronous motor 
four poles, n = 19, and: 

a = 0.05, 
1 - s = 0.95, 
that is, the couple runs at 95 per cent, of the synchronous speed 
of a 76-polar machine, llius at synchronous speed of an SO-polar 
machine, and thus can be substituted for an 80-polar induction 
motor. In this case, the synchronous motor gives about 5 
per cent., the induction motor 95 per cent, of the output; the 
synchronous motor thus is a small machine, which could be con- 
sidered ms a .synchronous exciter of the induction machine. 
48. Let: 

#n = e'fl + je"o = voltage impressed upon in- 
duction motor. 
Ei m e'i + je"i = voltage induced in induc- 
tion motor, by mutual 
magnetic flux, reduced to 
full frequency. 
#? = e'i + je"? = nominal induced voltage 
of synchronous motor, re- 
duced to full frequency. 
/n ■ i'i — jf'"o • primary current in induc- 
tion motor. 
/l = i'i — ji"i = secondary current of in- 
duction motor and cur- 
rent in synchronous motor. 
Denoting by Z' the impedance, Z, at frequency, s, it is: 
Total impedance of secondary circuit, at frequency, s: 

Z' = Zt + z t - 

= ('-. + r0 + j-{xx + x t ), (3) 



and the equations are: 
in primary circuit: 

in secondary circuit: 
and, current: 

/o = /i + r^i. 

From (6) follows: 

/! = /,- r#„ 

and, substituting (7) into (5) : 

s#i = 8fa + Z'lo - Z'YE h 

sE t + Z'to 

substituting (8) into (4) gives: 

sE 2 + (Z< + sZo + Z'ZoY) /o 



and, transposed: 

8 + Z'Y 


rrzh = ^° ~'(r+ z«f + Zo ) /o - 

i + =-r 


ri + r 2 


= r> 

X\ + Xt = x 7 


= r'+;V = 

_/ i „•_/ r/f | 

it is, substituting into (9; and CIO;: 

£. (1 + ZT; = E t + <Z' + Z» + Z'Z>,Y) U, 

1 + Z'Y ~ E " M + Z'Y + Z ") I* 






E (i + f r) - *, + [* + 2.(1 + * r) ] h, (9) 





E * =V = e '+je" (14) 

1 + ZY 

as a voltage which is proportional to the nominal induced voltage 
of the synchronous motor, and : 

r+^y + Zo = z = r+ix (15) 

and substituting (14) and (15) into (13), gives: 

# = #„ - Z/ . (16) 

This is the standard synchronous-motor equation, with im- 
pressed voltage, #o, current, / , synchronous impedance, Z, and 
nominal induced voltage, $. 

Choosing the impressed voltage, $q = e as base line, and 
substituting into (16), gives: 

e' + je" - (co - ri'o - xi"o) - j (xt'o - ri"o), (17) 

and, absolute: 

e 2 = (e — r i'o — s i"o) 2 + teot'o — r i"o) 2 . (18) 

From this equation (18) the load and speed curves of the 
concatenated couple can now be calculated in the same manner 
as in any synchronous motor. 

That is, the concatenated couple, of induction and synchronous 
motor, can be replaced by an equivalent synchronous motor of 
the constants, e, eo, Z and / . 

49. The power output of the synchronous machine is: 

P 2 = //„ «&/', 

/a+jb, c+jd/' 

denotes the effective component of the double-frequency prod- 
uct: (ac + bd); see "Theory and ^Calculation of Alternating- 
current Phenomena," Chapter XVI, 5th edition. 
The power output of the induction machine is: 

Pi = IK (i - *) £i/', (20) 

thus, the total power output of the concatenated couple: 

P = P -4- P 

= /fi, «Ei + (1 - «)£,/'; (2D 


substituting (7) into (21): 

P = // - r#„ s# 2 + (1 - «)&/'; (22) 

from (8) follows: 

s#2 = #i (« + Z-7) - Z'/o, 
and substituting this into (22), gives: 

P - // - r^i, #, (1 + Z'Y) - Z'lo/'; (23) 

from (4) follows: 

1$\ = #o — Zo/o, 

and substituting this into (23) gives: 

p = fu (i + ZoF) - r#o, #o (i + z*Y) - 

/o (Z- + Z„ + ZoZ'Y)/'. (24) • 

Equation (24) gives the power output, as function of impressed 
voltage, -Fo, and supply current, /o. 

The power input into the concatenated couple is given by: 

Po = /*., /o/', (25) 

or, choosing # = e as base line: 

Po = eoi'o. (26) 

The apparent power, or volt-ampere input ift given by: 

Q = e io, (27) 


to = V'i'o* + i"V 

is the total primary current. 

From P, Po and Q now follow efficiency, power-factor and 
apparent efficiency. 

60. As an instance may be considered the power-factor control 
of the slow-speed 80-polar induction motor of Fig. 20, by a small 
synchronous motor concatenated into its secondary circuit. 

Impressed voltage: 

e« = 500 volts. 

Choosing a four-polar synchronous motor, the induct ion 
machine would have to be redesigned with 70 poles, giving: 

n = 19, 
' * = 0.05, 


With the same rotor diameter of the induction machine, the 
pole pitch would be increased inverse proportional to the number 
of poles, and the exciting susceptance decreased with the square 
thereof, thus giving the constants: 

Y = g -jb = 0.02 - 0.54 j; 
Z = r +jxo = 0.1 + 0.3j; 
Zi = ri+jxt = 0.1 + 0.3./. 

Assuming as synchronous motor synchronous impedance, 
reduced to full frequency: 

Z 2 = r 2 + jx 2 = 0.02 + 0.2 j 

this gives, for s = 0.05: 

Z' = (ri + r 2 ) +js (xi + x 2 ) = 0.12 + 0.025 j, 
and : 

Z' = r' +jx' = Z ' s = 2.4 + 0.5 j, 

Z = r+jx = 0.84 + 1.4 j, 

and from (14): 

E 2 

E = 

1.32- 1.29 j' 


e ~ 1.84 

"339 = (50 ° ~ °' 84 *'° ~ 1-4 *" o) * + (1 ' 4 ? '° ~ ° <84 '"' ,)! ' (28) 
and the power output : 

P = // (0.836 + 0.048 j) - (10 - 270 j), 

(508 - 32 j) - /„ (0.241 + 0.326 j)/'. (29) 

51. Fig. 29 shows the load curves of the concatenated couple, 
under the condition that the synchronous-motor excitation and 
thus its nominal induced voltage, e 2 , is varied so as to maintain 
unity power-factor at all loads, that is: 

t"o = 0; 
this gives from equation (28) : 

~|^ = (500 - 0.84t' ) s + 1.96 tV, 






e„ - 500 z =.i +■ M 
¥,-.02 -.54 j 2i = .l + -3j 
g - .05 Z, = .02 + .2/ 
























I 1 

1 I 





1 ] 





t 1 


'J 1 

:> 1 


e D = 500 Z = .1 +.3) 

8 ■ .OS Z,- .02 + .2J 








„. ( 






u I 


» a 

H 110 1 


] -- 

Fio. 30.— Load n 


P = /(0.8361 i\ - 10) + j (0.048 i' 4- 270), 

(508 = 0.241 1' ) - j (32 + 0.326 *'»/' 

= (0.836 i'» - 10) (508 - 0.241 i' ) - (0.048 j"' d + 270) 

(32 4- 0.326 i\). 

As seen from the curve, e 2 , of the nominal induced voltage, the 
synchronous motor has to be overexcited at all loads. However, 
ei first decreases, reaches a minimum and then increases again, 
thus is fairly constant over a wide range of load, so that with 
this type of motor, constant excitation should give good results. 

Fig. 30 then shows the load curves of the concatenated couple 
for constant excitation, on overexcitation of the synchronous 
motor of 70 per cent., or 

c 2 = 850 volts. 

(It must be kept in mind, that ei is the voltage reduced to full 
frequency and turn ratio 1 :1 in the induction machine: At the 
slip, s = 0.05, the actual voltage of the synchronous motor would 
h« set = 12.5 volts, even if the numher of secondary turns of the 
induction motor equals that of the primary turns, and if, as 
usual, the induction motor is wound for less turns in the secondary 
than in the primary, the actual voltage at the synchronous motor 
terminals is still lower.) 

As seen from Fig. 30: 
the power-factor is practically unity over the entire range of 
load, from less than one-tenth load up to the maximum output 
point, and the current input into the motor thus is practically 
proportional to the load. 

The load curves of this concatenated couple thus arc superior 
to those, which can be produced in a synchronous motor at con- 
stant excitation. 

For comparison, the curve of apparent efficiency, from Fig. 30, 
is plotted as CS in Fig. 28. It merges indisttnguishably into the 
unity power-factor curve, So, except at its maximum output 

Induction Motor Concatenated with Commutating 

62. While the alternating-current commulating machine, MpB* 

cially of the polyphase type, is rather poor at higher frequencies, 

ii becomes better at lower frequencies, and at the extremely low 

' frequency of the induction-motor secondary, it is practically as 


good as the direct-current commutating machine, and thus can 
be used to insert low-frequency voltage into the induction-motor 

With series excitation, the voltage of the commutating machine 
is approximately proportional to the secondary current, and the 
speed characteristic of the induction motor remains essentially 
the same: a speed decreasing from synchronism at no-load, by a 
slip, s f which increases with the load. 

With shunt excitation, the voltage of the commutating machine 
is approximately constant, and the concatenated couple thus 
tends toward a speed differing from synchronism. 

In either case, however, the slip, s, is not constant and independ- 
ent of the load, and the motor couple not synchronous, as when 
using a synchronous machine as second motor, but the motor 
couple is asynchronous, decreasing in speed with increase of load. 

The phase relation of the voltage produced by the commutating 
machine, with regards to the secondary current which traverses 
it, depends on the relation of the commutator brush position 
with regards to the field excitation of the respective phases, and 
thereby can be made anything between and 2t, that is, the 
voltage inserted by the commutating machine can be energy 
voltage in phase — reducing the speed — or in opposition to the 
induction-motor induced voltage — increasing the speed; or it 
may be a reactive voltage, lagging and thereby supplying the 
induction-motor magnetizing current, or leading and thereby 
still further lowering the power-factor. Or the commutating 
machine voltage may be partly in phase — modifying the speed — 
and partly in quadrature — modifying the power-factor. 

Thus the commutating machine in the induction-motor 
secondary can be used for power-factor control or for speed 
control or for both. 

It is interesting to note that the use of the commutating ma- 
chine in the induction- motor secondary gives two independent 
variables: the value of the voltage, and its phase relation to the 
current of its circuit, and the motor couple thus has two degrees 
of freedom. With the use of a synchronous machine in the 
induction-motor secondary this is not the case; only the voltage 
of the synchronous machine can be controlled, but its phase 
adjusts itself to the phase relation of the secondary circuit, and 
the synchronous-motor couple thus has only one degree of free- 
dom. The reason is: with a synchronous motor concatenated to 



the induction machine, the phase of the synchronous machine is 
fixed in space, by the synchronous-motor poles, thus has a fixed 
relation with regards to the induction-motor primary system. 
As, however, the induction-motor secondary has no fixed position 
relation with regards to the primary, but can have any position 
slip, the synchronous-motor voltage has no fixed position with 
regards to the induction-motor secondary voltage and current, 
thus can assume any position, depending on the relation in the 
secondary circuit. Thus if we assume that the synchronous- 
motor field were shifted in space by « position degrees (electrical): 
this would shift the phase of the synchronous-motor voltage by 
a degrees, and the induction-motor secondary would slip in posi- 
tion by the same angle, thus keep the same phase relation with 
regards to the synchronous-motor voltage. In the couple with 
a commutating machine as secondary motor, however, the posi- 
tion of the brushes fixes the relation between commutating- 
machine voltage and secondary current, and thereby imposes I 
definite phase relation in the secondary circuit, irrespective of 
the relations between secondary and primary, and no change of 
relative position between primary and secondary can change this 
phase relation of the commutating machine. 

Thus the commutating machine in the secondary of the induc- 
tion machine permits a far greater variation of condition! of 
operation, and thereby gives a far greater variety of speed and 
load curves of such concatenated couple, than is given by the 
use of a synchronous motor in the induction-motor secondary. 

63. Assuming the polyphase low-frequency commutating 
machine is series-excited, that is, the field coils (and compensat- 
ing coils, where used) in series with the armature. Assuming 
also that magnetic saturation is not reached within the range of 
its use. 

The induced voltage of the commutating machine then is 
proportional to the secondary current and to the speed. 

Thus: e t = pix (1) 

is the commutating-machine voltage at full synchronous speed, 
where tj is the secondary current and p a constant depending 
on the design. 

At the slip, s, and thus the speed (1 — s), the comnmUting 
machine voltage thus is: 

(1 -s)e, = (1 - 

«) pii. 



As this voltage may have any phase relation with regards to 
the current, t'i, we can put: 

£2 = (Pi + jpt)h (3) 


P = Vpi 2 + p 2 2 (4) 


tan w = — (5) 


is the angle of brush shift of the commutating machine. 

(Pi + JPt) ls of the nature and dimension of an impedance, 
and we thus can put : 

Z° = Pi + m (6) 

as the effective impedance representing the commutating machine. 

At the speed (1 — s), 

the commutating machine is represented by the effective 


(1 - s) Z° = (1 - 8) Pl + j (1 - 8) P2. (7) 

It must be understood, however, that in the effective impedance 
of the commutating machine, 

Z° = pi + jp>, 

Pi as well as p% may be negative as well as positive. 

That is, the energy component of the effective impedance, or 
the effective resistance, p lf of the commutating machine, may be 
negative, representing power supply. This simply means, that 
the commutator brushes are set so as to make the commutating 
machine an electric generator, while it is a motor, if pi is positive. 

If pi = 0, the commutating machine is a producer of wattless 
or reactive power, inductive for positive, anti-inductive for 
negative, p 2 . 

The calculation of an induction motor concatenated with a 
commutating machine thus becomes identical with that of the 
straight induction motor with short-circuited secondary, except 
that in place of the secondary inductive impedance of the induc- 
tion motor is substituted the total impedance of the secondary 
circuit, consisting of: 

1. The secondary self-inductive impedance of the induction 


2. The self-inductive impedance of the commutating machine 
comprising resistance and reactance of armature and of field, 
and compensating winding, where such exists. 

3. The effective impedance representing the commutating 

It must be considered, however, that in (1) and (2) the re- 
sistance is constant, the reactance proportional to the slip, s, 
while (3) is proportional to the speed (1 — s). 
64. Let: 

Yo = g— jb = primary exciting admittance 

of the induction motor. 
Z =- r + jxo = primary self-inductive im- 
pedance of the induction 
Z\ = r x + jx\ = secondary self-inductive im- 
pedance of the induction 

motor, reduced to full 


Zi — r2 + jx% = self-inductive impedance of 

the commutating machine, 
reduced to full frequency. 

Z° = pi + jpi = effective impedance repre- 
senting the voltage in- 
duced in the commutating 
machine, reduced to full 

The total secondary impedance, at slip, «, then is: 

Z $ = (r, + jsxi) + (r 2 + jsxi) + (1 - «) (pi + jp 2 ) 

= [r, + r 2 + (1 -» pj + j\s (xi + xt) + (1 - s) p 2 ] (8) 

and, if the mutual inductive voltage of the induction motor is 
chosen as base line, e, in the customary manner, 
the secondary current is: 

h = 7% = (ai-ja 2 )e, (9) 


* [ r\ + r 2 + (1 - s) pi] 
ii\ — 


8[s(xi + X2) + (1 -s)p 2 ] 
a 2 = 




m - [p, + r t + (1 - s) p,]* + [s (Xi + x,) + (1 - 

) Pd*. 

The remaining calculation is the same as on page 318 of 
" Theoretical Elements of Electrical Engineering," 4th edition. 

As an instance, consider the concatenation of a low-frequency 
commutating machine to the low-speed induction motor, Fig. 20. 

The constants then are: 

Impressed voltage: 
Exciting admittance: 

e e = 500; 
Y Q = 0.02 - 0.6 j 
Z* = 0.1 +0.3 j 
Z v = 0.1 +0.3j" 
Z, = 0.02 + 0.3 j" 
Z" = - 0.2 j. 





P,+ Jp.-i-.2j 


Y,,".02-.8i Zi-.l+.3i ASYMCHRONO 


■ -» 


























o - 

1 I 

1 4 


1 t 

i a 


a 1 

o i 


i.l 1 


n l 

n u 

n i 


That is, the commutating machine is adjusted to give only 
reactive lagging voltage, for power-factor compensation. 
It then is: 

Z> = 0.12 + j [0.6 s - 0.2 (1 - s)]. 

The load curves of this motor couple are shown in Fig. 31. As 


Bent, power-factor and apparent efficiency rise to high values, and 
even the efficiency is higher than in the straight induction motor. 
However, at light-load the power-factor and thus the apparent 
efficiency falls off, very much in the same manner as in the con- 
catenation with a synchronous motor. 

It is interesting to note the relatively great drop of speed at 
light-load, while at heavier load the speed remains more nearly 
constant. This is a general characteristic of anti-inductive im- 
pedance in the induction-motor secondary, and shared by the 
use of an electrostatic condenser in the secondary. 

For comparison, on Fig. 28 the curve of apparent efficiency of 
this motor couple is shown as CC. 

Induction Motor with Condenser in Secondary Circuit 

66. As a condenser consumes leading, that is, produces lagging 
reactive current, it can be used to supply the lagging component 
of current of the induction motor and thereby improve the 

Shunted across the motor terminals, the condenser consumes a 
constant current, at constant impressed voltage and frequency, 
Mini as the lagging component of induction-motor current in- 
creases with the load, the characteristics of the combination of 
motor and shunted condenser thus change from leading current 
at no-load, over unity power-factor to lagging current at overload. 
As the condenser is an external apparatus, the characteristics of 
the induction motor proper obviously are not changed by a 
shunted condenser. 

As illustration is shown, in Fig. 32, the slow-speed induction 
motor Fig. 20, shunted by a condenser of 125 kva. per phase. 
Fig. 32 gives efficiency, i?, power-factor, p, and apparent efficiency, 
7, of the combination of motor and condenser, assuming an 
efficiency of the condenser of 99.5 per cent., thai is, 0,5 pet cent 
loss in the condenser, or Z = 0.0025 - 0.5 j, that is, a condenser 

just neutralizing i he magnetising current. 

However, when using a condenser in shunt, it must be realized 
that the current consumed by the condenser is proportional to the 
frequency, and therefore, if the wave of impressed voltage is 
greatly distorted, that is, contains considerable higher harmonics 
— especially harmonics of high order — the condenser may produce 
i uderable higher-frequency currents, and thus by distortion 



of the current wave lower the power-factor, so that in extreme 
cases the shunted condenser may actually lower the power- 
factor. However, with the usual commercial voltage wave 
shapes, this is rarely to be expected. 

In single-phase induction motors, the condenser may be used 
in a tertiary circuit, that is, a circuit located on the same member 
(usually the stator) as the primary circuit, but displaced in posi- 




e„ = 50O Zo = .1 + .3i 
Yo=.02-.6j Z,«.1 + .3J 
Z t -.002S-.SJ 













i i 

I I 



) C 

i) ■ 




Xl 1 

o l 


Fio. 32.— Load c 

tion therefrom, and energized by induction from the secondary. 
By locating the tertiary circuit in mutual induction also with the 
primary, it can be used for starting the single-phase motor, and 
is more fully discussed in Chapter V. 

A condenser may also be used in the secondary of the induction 
motor. That is, the secondary circuit is closed through a con- 
denser in each phase. As the current consumed by a condenser is 
proportional to the frequency, and the frequency in the secondary 
circuit varies, decreasing toward zero at synchronism, the cur- 
rent consumed by the condenser, and thus the secondary current 
of the motor tends toward zero when approaching synchronism, 


and peculiar speed characteristics result herefrom in such a 
motor. At a certain slip, s, the condenser current just balances 
all the reactive lagging currents of the induction motor, resonance 
may thus be said to exist, and a very large current flows into the 
motor, and correspondingly large power is produced. Above this 
"resonance speed," however, the current and thus the power 
rapidly fall off, and so also below the resonance speed. 

It must be realized, however, that the frequency of the sec- 
ondary is the frequency of slip, and is very low at speed, thus a 
very great condenser capacity is required, far greater than would 
be sufficient for compensation by shunting the condenser across 
the primary terminals. In view of the low frequency and low 
voltage of the secondary circuit, the electrostatic condenser 
generally is at a disadvantage for this use, but the electrolytic 
condenser, that is, the polarization cell, appears better adapted. 

56. Let then, in an induction motor, of impressed voltage, e : 

Y a = g — jb — exciting admittance; 

Z» — H + J x <> = primary self-inductive impe- 

Z\ = Ti + jxi = secondary self-inductive im- 
pedance at full frequency; 

and let the secondary circuit be closed through a condenser of 
capacity reactance, at full frequency: 

Z* — U — j*h 

where r%, representing the energy loss in the condenser, usually is 
very small and can lie neglected in the electrostatic condenser, 
so that: 

Zt= - jxj. 

The inductive reactance, Xt, is proportional to the frequency, 
that is, the slip, s, and the capacity reactance, x : , inverse propor- 
tional thereto, and the total impedance of the secondary circuit, 
at slip, j*, thus is: 

Z--r, +;(»!, -»), (I) 

Ihus tlir- secondary current: 

;, - " 

- < <«i - >=), (2) 



Oi = —i 

o 2 = 

(- - ?) 



m = ri 2 + (sxi + yj • 


All the further calculations of the motor characteristics now 
are the same as in the straight induction motor. 

As instance is shown the low-speed motor, Fig. 20, of constants: 

e = 500; 
Y = 0.02 - 0.6 j ; 
Z = 0.1 + 0.3j; 
Zi = 0.1 + 0.3 j; 

with the secondary closed by a condenser of capacity impedance: 

Z, = - 0.012 j, 
thus giving: 


Z' = 0.1 + 0.3j(s-^p) 

Fig. 33 shows the load curves of this motor with condenser 
in the secondary. As seen, power-factor and apparent effi- 
ciency are high at load, but fall off at light-load, being similar 
in character as with a commutating machine concatenated to 
the induction machine, or with the secondary excited by direct 
current, that is, with conversion of the induction into a synchro- 
nous motor. 

Interesting is the speed characteristic: at very light-load the 
speed drops off rapidly, but then remains nearly stationary over 
a wide range of load, at 10 per cent. slip. It may thus be said, 
that the motor tends to run at a nearly constant speed of 90 per 
cent, of synchronous speed. 

The apparent efficiency of this motor combination is plotted 
once more in Fig. 28, for comparison with those of the other 
motors, and marked by C. 

Different values of secondary capacity give different operating 
speeds of the motor: a lower capacity, that is, higher capacity 


•eactance, x t , gives a greater slip, s, that is, lower operating 
peed, and inversely, as was discussed in Chapter I. 

67. It is interesting to compare, in Fig. 28, the various met hods 
>f secondary excitation of the induction motor, in their effect in 
niproving the power-factor and thus the apparent efficiency of 
v motor of high exciting current and thus low power-factor, >mli 
is a slow-speed motor. 

The apparent efficiency characteristics fall into three groups; 




e a = 500 Zo-.t +.3i 

Y„=,02 -.6) Z, = .1 +.3i 

2, = - .012) 







— ~ 














1 7 


■ 1 

n i 

» i 


a i 

B l 

D i 

a i 

o - 

■'i<;. :j:i. — Load curves of liLgh-cxoitiitiou induction motor with condeiisera in 
secondary circuits. 

1. Low apparent efficiency at all loads: the straight slow- 
speed induction motor, marked by /. 

2. High apparent efficiency at all loads: 

The synchronous motor with unity power-factor excitation, So 

Concatenation to synchronous motor with unity power-factor 
excitation, CSq. 

Concatenation to synchronous motor with constant excitation 

These three curves are practically identical, except at great 

3. Low apparent efficiency at iiglit-loads, high apparent 


efficiency at load, that is, curves starting from (1) and rising up 
to (2). 

Hereto belong: The synchronous motor at constant excita- 
tion, marked by S. 

Concatenation to a commutating machine, 

Induction motor with condenser in secondary 
circuit, C. 
These three curves are very similar, the points calculated for 
the three different motor types falling within the narrow range 
between the two limit curves drawn in Fig. 28. 

Regarding the speed characteristics, two types exist : the motors 
So, S, CSo and CS are synchronous, the motors 7, CC and C are 

In their efficiencies, there is little difference between the 
different motors, as is to be expected, and the efficiency curves 
are almost the same up to the overloads where the motor begins 
to drop out of step, and the efficiency thus decreases. 

Induction Motor with Commutator 

58. Let, in an induction motor, the turns of the secondary 
winding be brought out to a commutator. Then by means of 
brushes bearing on this commutator, currents can be sent into 
the secondary winding from an outside source of voltage. 

Let then, in Fig. 34, the full-frequency three-phase currents 
supplied to the three commutator brushes of such a motor be 
shown as A. The current in a secondary coil of the motor, 
supplied from the currents, A, through the commutator, then is 
shown as B. Fig. 34 corresponds to a slip, s = %. As seen from 
Fig. 34, the commutated three-phase current, B, gives a resultant 
effect, which is a low-frequency wave, shown dotted in Fig. 34 
By and which has the frequency of slip, s, or, in other words, the 
commutated current, B, can be resolved into a current of fre- 
quency, $, and a higher harmonic of irregular wave shape. 

Thus, the effect of low-frequency currents, of the frequency 
of slip, can be produced in the induction-motor secondary by 
impressing full frequency upon it through commutator and 

The secondary circuit, through commutator and brushes, can 
be connected to the supply source either in series to the primary, 


or in shunt thereto, and thus given series-motor characteristics, 
or shunt-motor characteristics. 

In either case, two independent variables exist, the value of 
the voltage impressed upon the commutator, and its phase, 
and the phase of the voltage supplied to the secondary Hreuil 
may be varied, either by varying the phase of the impressed 
voltage by a suitable transformer, or by shifting the brushes on 
the commutator and thereby the relative position of the brushes 
with regards to the stator, which has the same effect. 

However, with such a commutator motor, while the resultant 
magnetic effect of the secondary currents is of the low [reqtMQgy 


11 miluction motor 

of slip, the actual current in each secondary coil is of full fre- 
quency, as a section or piece of a full- frequency wave, and thus 
it meets in the secondary the full-frequency reactance. That is, 
the secondary reactance at slip, s, is not: Z" = r, + jsx,, but is: 
Z' = ft + jxi, in other words is very much larger than in the 
motor with short-circuited secoudary. 

Therefore, such motors with commutator always require 
power-factor compensation, by shifting the brushes or choosing 
the impressed voltage so as to be anti-inductive. 

Of the voltage supplied to the secondary through commutator 
and brushes, a component in phase with the induced voltage 
lowers the speed, a component in opposition raises the speed, 
and by varying the commutator supply voltage, speed control 
of such an induction motor can be produced iu the same manner 
and of the same character, as produced in a direct -current motor 


by varying the field excitation. Good constants can be secured, 
if in addition to the energy component of impressed voltage, used 
for speed control, a suitable anti-inductive wattless component 
is used. 

However, this type of motor in reality is not an induction 
motor any more, but a shunt motor or series motor, and is more 
fully discussed in Chapter XIX, on "General Alternating-current 

59. Suppose, however, that in addition to the secondary wind- 
ing connected to commutator and brushes, a short-circuited 
squirrel-cage winding is used on the secondary. Instead of 
this, the commutator segments may be shunted by resistance, 
which gives the same effect, or merely a squirrel-cage winding 
used, and on one side an end ring of very high resistance em- 
ployed, and the brushes bear on this end ring, which thus acts 
as commutator. 

In either case, the motor is an induction motor, and has the 
essential characteristics of the induction motor, that is, a slip, «, 
from synchronism, which increases with the load; however, 
through the commutator an exciting current can be fed into the 
motor from a full-frequency voltage supply, and in this case, the 
current supplied over the commutator does not meet the full- 
frequency reactance, X\, of the secondary, but only the low-fre- 
quency reactance, sxi, especially if the commutated winding is in 
the same slots with the squirrel-cage winding: the short-circuited 
squirrel-cage winding acts as a short-circuited secondary to the 
high-frequency pulsation of the commutated current, and there- 
fore makes the circuit non-inductive for these high-frequency 
pulsations, or practically so. That is, in the short-circuited con- 
ductors, local currents are induced equal and opposite to the 
high-frequency component of the commutated current, and the 
total resultant of the currents in each slot thus is only the low- 
frequency current. 

Such short-circuited squirrel cage in addition to the commu- 
tated winding, makes the use of a commutator practicable for 
power-factor control in the induction motor. It forbids, how- 
ever, the use of the commutator for speed control, as due to the 
short-circuited winding, the motor must run at the slip, s, corre- 
sponding to the load as induction motor. The voltage impressed 
upon the commutator, and its phase relation, or the brush posi- 
tion, thus must be chosen so as to give only magnetizing, but 


no speed changing effects, and this leaves only one degree of 

The foremost disadvantage of this method of secondary excita- 
tion of an induction motor, by a commutated winding in addi- 
tion to the short-circuited squirrel cage, is that secondary excita- 
tion is advantageous for power-factor control especially in 
slow-speed motors of very many poles, and in such, the commuta- 
tor becomes very undesirable, due to the large number of poles. 
With such motors, it therefore is preferable to separate the 
commutator, placing it on a small commutating machine of a 
few poles, and concatenating this with the induction motor. In 
motors of only a small number of poles, in which a commutator 
would be less objectionable, power-factor compensation is rarely 
needed. This is the foremost reason that this type of motor 
(the Heyland motor) has found no greater application. 



60. As more fully discussed in the chapters on the single-phase 
induction motor, in " Theoretical Elements of Electrical Engineer- 
ing" and " Theory and Calculation of Alternating-current 
Phenomena," the single-phase induction motor has inherently, 
no torque at standstill, that is, when used without special device 
to produce such torque by converting the motor into an unsym- 
metrical ployphase motor, etc. The magnetic flux at standstill 
is a single-phase alternating flux of constant direction, and the 
line of polarization of the armature or secondary currents, that 
is, the resultant m.m.f. of the armature currents, coincides with 
the axis of magnetic flux impressed by the primary circuit. 
When revolving, however, even at low speeds, torque appears in 
the single-phase induction motor, due to the axis of armature 
polarization being shifted against the axis of primary impressed 
magnetic flux, by the rotation. That is, the armature currents, 
lagging behind the magnetic flux which induces them, reach 
their maximum later than the magnetic flux, thus at a time when 
their conductors have already moved a distance or an angle 
away from coincidence with the inducing magnetic flux. That is, 

if the armature currents lag ~ = 90° beyond the primary main 

flux, and reach their maximum 90° in time behind the magnetic 
flux, at the slip, s, and thus speed (1 — s), they reach their maxi- 

mum in the position (1 — s) ~ = 90 (1 — s) electrical degrees 

behind the direction of the main magnetic flux. A component 
of the armature currents then magnetizes in the direction at 
right angles (electrically) to the main magnetic flux, and the 
armature currents thus produce a quadrature magnetic flux, 
increasing from zero at standstill, to a maximum at synchronism, 
and approximately proportional to the quadrature component of 
the armature polarization, P: 

P sin (1 — s) • 



The torque of the single-phase motor then is produced by the 
action of the quadrature flux on the energy currents induced by 
the main flux, and thus is proportional to the quadrature flux. 

At synchronism, the quadrature magnetic flux produced by 
the armature currents becomes equal to the main magnetic flux 
produced by the impressed single-phase voltage (approximately, 
in reality it is less by the impedance drop of the exciting current 
in the armature conductors) and the magnetic disposition of the 
single-phase induction motor thus becomes at synchronism iden- 
tical with that of the polyphase induction motor, and approxi- 
mately so near synchronism. 

The magnetic field of the single-phase induction motor thus 
may be said to change from a single-phase alternating field at 
standstill, over an unsymmetrical rotating field at intermediate 
speeds, to a uniformly rotating field at full speed. 

At synchronism, the total volt-ampere excitation of the single- 
phase motor thus is the same as in the polyphase motor at the 
same induced voltage, and decreases to half this value at stand- 
still, where only one of the two quadrature components of 
magnetic flux exists. The primary impedance of the motor is 
that of the circuits used. The secondary impedance varies 
from the joint impedance of all phases, at synchronism, to twice 
this value at standstill, since at synchronism all the secondary 
circuits correspond to the one primary circuit, while at stand- 
still only their component parallel with the primary circuit 
corres ponds. 

61. Hereby the single-phase motor constants are derived from 
the constants of the same motor structure as polyphase motor. 

Let, in a polyphase motor: 

Y = g — jb = primary exciting admittance; 

2o = To + Jin = primary self-inductive im- 

Z\ = fi + jxi = secondary self-inductive im- 
pedance (reduced to the pri- 
mary by the ratio of turns, 
in the usual manner}; 

the characteristic constant of the motor then is: 

& - y (z„ + z x ). (i) 

The total, or resultant admittance respectively impedance of 



the motor, that is, the joint admittance respectively impedance of 
all the phases, then is: 

In a three-phase motor: 

7 a = 3 Y, 

Zo° = H Z,, , (2) 

Zi° = H Zv 

In a quarter-phase motor: 

Y° - 2 Y, ] 

Zo° - H Zo, (3) 

Z,° = M Z,. 1 

In the same motor, as single-phase motor, it is then: at syn- 
chronism: 8 = 0: 

Y' = F°, 
Z' = 2 Z °, | (4) 

Z'! = Zx°, 

hence the characteristic constant : 

t>' - r (z'o + z',) 

- r°(2Z ,, + Z 1 ) ) 


at standstill : * = 1 : 

r = H y«, 

Z'o = 2 Zo°, 
Z'i = 2 Z,«, 


hence, the characteristic constant: 

t>', = Y° (Zo° + Zj ) 


approximately, that is, assuming linear variation of the constants 
with the speed or slip, it is then: at slip, s: 

Y' = F°(l -|), 

Lt o = 2 #0j 

Z'i = Z t » (1 + »). J 
This gives, in a three-phase motor: 

F'=3F(1- *), 

Z'o - % z\ 
Z . x = i + i Zl . 




In 8 quarter-phase motor: 

Y' =27(1-3, 

Z = Zoj 


Thus the characteristic constant, #', of the single-phase motor 
is higher, that is, the motor inferior in its performance than the 
polyphase motor; but the quarter-phase motor makes just as 
good — or poor — a single-phase motor as the three-phase motor. 

62. The calculation of the performance curves of the single- 
phase motor from its constants, then, is the same as that of the 
polyphase motor, except that : 

In the expression of torque and of power, the term (1 — *) 
is added, which results from the decreasing quadrature flux, and 
it thus is: 

T = T(l -*) 

= (1 - *) a*-. (11) 


P* =P(1 -*) 

«(l-*)*aif*. (12) 

However, these expressions are approximate only, as they 
assume a variation of the quadrature flux proportional to the 

63. As the single-phase induction motor is not inherently 
self-starting, starting devices are required. Such are: 

(a) Mechanical starting. 

As in starting a single-phase induction motor it is not neces- 
sary, as in a synchronous motor, to bring it up to full speed, but 
the motor begins to develop appreciable torque already at low 
speed, it is quite feasible to start small induction motors by hand, 
by a pull on the belt, etc.. especially at light-load and if«of high- 
resistance armature. 

(b) By converting the motor in starting into a shunt or series 

This has the great objection of requiring a commutator, and a 
cwuttutating-machine rotor winding instead of the common 
iftd«c*iQ«i-n*otor squirrel-cage winding. Also, as series motor, 
tl* KthiKty exists in the starting connection, of running away; 


as shunt motor, sparking is still more severe. Thus this method 
is used to a limited extent only. 

(c) By shifting the axis of armature or secondary polarization 
against the axis of inducing magnetism. 

This requires a secondary system, which is electrically un- 
symmetrical with regards to the primary system, and thus, since 
the secondary is movable with regards to the primary, requires 
means of changing the secondary circuit, that is, commutator 
brushes short-circuiting secondary coils in the position of effective 
torque, and open-circuiting them in the position of opposing torque. 

Thus this method leads to the various forms of repulsion 
motors, of series and of shunt characteristic. 

It has the serious objection of requiring a commutator and a 
corresponding armature winding; though the limitation is not 
quite as great as with the series or shunt motor, since in the re- 
pulsion motors the armature current is an induced secondary 
current, and the armature thus independent of the primary 
system regards current, voltage and number of turns. 

(d) By shifting the axis of magnetism, that is producing a 
magnetic flux displaced in phase and in position from that in- 
ducing the armature currents, in other words, a quadrature 
magnetic flux, such as at speed is being produced by the rotation. 

This method does not impose any limitation on stator and 
rotor design, requires no commutator and thus is the method 
almost universally employed. 

It thus may be considered somewhat more in detail. 

The infinite variety of arrangements proposed for producing 
a quadrature or starting flux can be grouped into three classes: 

A. Phase-splitting Devices. — The primary system of the single- 
phase induction motor is composed of two or more circuits 
displaced from each other in position around the armature 
circumference, and combined with impedances of different in- 
ductance factors so as to produce a phase displacement between 

The motor circuits may be connected in series, and shunted 
by the impedance, or they may be connected in shunt with each 
other, but in series with their respective impedance, or they 
may be connected with each other by transformation, etc. 

B. Inductive Devices. — The motor is excited by two or more 
circuits which are in inductive relation with each other so as to 
produce a phase displacement. 


This inductive relation may be established outside of the motor 
by an external phase-splitting device, or may take place in the 
motor proper. 

C. Monocyclic Devices. — An essentially reactive quadrature 
voltage is produced outside of the motor, and used to energize 
a cross-magnetic circuit in the motor, either directly through a 
separate motor coil, or after combination with the main voltage 
to a system of voltages of approximate three-phase or quarter- 
phase relation. 

D. Phase Converter. — By a separate external phase converter— 
usually of the induction-machine type — the single-phase supply 
is converted into a polyphase system. 

Such phase converter niay be connected in shunt to the motor, 
or may be connected in series thereto. 

This arrangement requires an auxiliary machine, running idle, 

however. It therefore is less convenient, but has the advantage 

of being capable of giving full polyphase torque and output to 

the motor, and thus would be specially suitable for railroading. 

64. If: 

*o = main magnetic flux of single-phase 
motor, that is, magnetic flux produced 
by the impressed single-phase voltage, 
4> = auxiliary magnetic flux produced by 

starting device, and if 
u> = space angle between the two fluxes, in 

electrical degrees, and 
* = time angle between the two fluxes, 

then the torque of the motor is proportional to: 

T - a** sin u sin tf>) (13) 

in the same motor as polyphase motor, with the magnetic flux, 
#o, the torque is: 

T n = a*, 1 ; (14) 

thus the torque ratio of the starting device is; 

. T * . 

I = y = ^- am w sin <f>, 

or, if: 


= quadrature flux produced by the startiug device, that is, 


component of the auxiliary flux, in quadrature to the main flux, 
$o, in time and in space, it is: 
Single-phase motor starting torque: 

T = afc'So, (16) 

and starting-torque ratio: 

t - £-• (17) 

As the magnetic fluxes are proportional to the impressed vol- 
tages, in coils having the same number of turns, it is: starting 
torque of single-phase induction motor: 

T = be e sin a> sin 
= 6e e', 


and, starting-torque ratio: 

< = -8inw sin 6 



eo = impressed single-phase voltage, 
e = voltage impressed upon the auxiliary or 
starting winding, reduced to the same 
number of turns as the main winding, 
e' = quadrature component, in time and in 
space, of this voltage, e, 
and the comparison is made with the torque of a quarter-phase 
motor of impressed voltage, eo, and the same number of turns. 

Or, if by phase-splitting, monocyclic device, etc., two voltages, 
ei and e 2 , are impressed upon the two windings of a single-phase 
induction motor, it is: 
Starting torque : 

T = be\e% sin a> sin <f> (20) 

and, starting-torque ratio: 

t = -^y sin co sin 0, (21) 


where eo is the voltage impressed upon a quarter-phase motor, 
with which the single-phase motor torque is compared, and all 


these voltages, ej, e^, e , are reduced to the same number of turns 
of the circuits, as customary. 

If then : 

Q = volt-amperes input of the single-phase 

motor with starting device, and 
Qo = volt-amperes input of the same motor 
with polyphase supply, 

, = i (22) 

is the volt-ampere ratio, and thus: 

v = - (23) 


is the ratio of the apparent starting-torque efficiency of the 
single-phase motor with starting device, to that of the same 
motor as polyphase motor, v may thus be called the apparent 
torque efficiency of the single-phase motor-starting device. 

In the same manner the apparent power efficiency of the start- 
ing device would result by using the power input instead of the 
volt-ampere input. 

66. With a starting device producing a quadrature voltage, e', 

t = e ' (24) 

is the ratio of the quadrature voltage to the main voltage, and 
also is the starting-torque ratio. 
The quadrature flux: 

e' = te (25) 

requires an exciting current, equal to t times that of the main 
voltage in the motor without starting device, the exciting current 
at standstill is: 

e i'= 2 

and in the motor with starting device giving voltage ratio, /, 
the total exciting current at standstill thus is: 

'a" U + O 


and thus, the exciting admittance: 

r' = y 2 °(i + 0; (27) 

in the same manner, the secondary impedance at standstill is: 

ZS = W (28) 

and thus: 

in the single-phase induction motor with starting device pro- 
ducing at standstill the ratio of quadrature voltage to main 
voltage : 

t = 


the constants are, at slip, s: 

Z' Q = 2 Zo°, 

Zi __ * ' s 7 o 
1 + 8t 


However, these expressions (29) are approximate only, as they 
assume linear variation with s, and furthermore, they apply only 
under the condition, that the effect of the starting device does 
not vary with the speed of the motor, that is, that the voltage 
ratio, t y does not depend on the effective impedance of the motor. 
This is the case only with a few starting devices, while many 
depend upon the effective impedance of the motor, and thus 
with the great change of the effective impedance of the motor 
with increasing speed, the conditions entirely change, so that no 
general equations can be given for the motor constants. 

66. Equations (18) to (23) permit a simple calculation of the 
starting torque, torque ratio and torque efficiency of the single- 
phase induction motor with starting device, by comparison with 
the same motor as polyphase motor, by means of the calculation 
of the voltages, e' y e h e 2 , etc., and this calculation is simply that 
of a compound alternating-current circuit, containing the induc- 
tion motor as an effective impedance. That is, since the only 
determining factor in the starting torque is the voltage impressed 
upon the motor, the internal reactions of the motor do not come 
into consideration, but the motor merely acts as an effective 
impedance. Or in other words, the consideration of the internal 



reaction of the motor is eliminated by the comparison with the 
polyphase motor. 

In calculating the effective impedance of the motor at stand- 
still, we consider the same as an alternating-current transformer, 
and use the equivalent circuit of the transformer, as discussed 
in Chapter XVII of "Theory and Calculation of Alternating- 
current Phenomena." That is, the induction motor is con- 
sidered as two impedances, Z a and Z ( , connected in series to the 



it of the induction n 

impressed voltage, with a shunt of the admittance, Y a , between 
the two impedances, as shown in Fig. 35. 
The effective impedance then is: 

approximately, this is: 

= Z Q + Zx. 

( :;<n 


This approximation (31), is very close, if Zi is highly inductive, 
as a short-circuited low-resistance squirrel cage, but ceases to be 
a satisfactory approximation if the secondary is of high resistance, 
for instance, contains a starting rheostat. 

As instances are given in the following the correct values of the 
effective impedance, Z, from equation (30), the approximate 
value (31), and their difference, for a three-phase motor without 
starting resistance, with a small resistance, with the resistance 
giving maximum torque at standstill, and a high resistance: 








0.01 - 0.1/ 0.1 + 0.3/ 0.1 + 0.3/ 0.195 + 0.502/ 0.2 + 0.6 / 

0.25 + 0.3 / 0.336 + 0.506/ 0.35 + 0.6/ 

0.6 + 0.3/ 0.661 + 0.620/ 0.7 + 0.6/ 

1.6 +0.3/ 1.552+0.804/ 1.7 +0.6/ 


0.005 - 0.008/ 
0.014 - 0.004/ 
0.030 + 0.020/ 
0.148 +0.204/ 


Parallel Connection 

67. Let the motor contain two primary circuits at right angles 
(electrically) in space with each other, and of equal effective 

Z = r + jx. 
These two motor circuits are connected in parallel with each 

Fio. 36. — Diagram of phase-splitting device with parallel connection of 

motor circuits. 

other between the same single-phase mains of voltage, eo, but 
the first motor circuit contains in series the impedance 

Z\ = ri + jxi, 
the second motor circuit the impedance : 

Z<L = T<l + JX2, 

as shown diagrammatically in Fig. 36. 
The two motor currents then are: 

h = z + z, and h = z~+z\' 

I = h + 1*, 


the two voltages across the two motor coils 


Ei = /jZ and E t = / 2 Z 

= eo z + z\' = eo z + z; (34) 

and the phase angle between #1 and # 2 is given by : 

m (cos <t> + j sin 0) = ^^z" ' ^ 

Denoting the absolute values of the voltages and currents by 
small letters, it is: 

T = beiei sin <f>; (36) 

in the motor as quarter-phase motor, with voltage, e 0} impressed 
per circuit, it is: 

To = 6e 2 , (37) 

hence, the torque ratio: 

t = ei6 ?8m<t>. (38) 

The current per circuit, in the machine as quarter-phase motor, 

to = -> (39) 


hence the volt-amperes: 

Qo = 2e *'o, (40) 

while the volt-amperes of the single-phase motor, inclusive start- 
ing impedances, are: 

Q = eoi, (41) 


and, the apparent torque efficiency of the starting device: 

q CqIZ 
68. As an instance, consider the motor of effective impedance: 

Z = r +jx = 0.1 +0.3J, 

z = 0.316, 


and assume, as the simplest case, a resistance, a = 0.3, inserted 
in series to the one motor circuit. That is : 

Z x = 0, ' (44) 

Zo = a. 

It is then: 
(32):/= <:-. =„, f« - ; h= e ° e " 

r+jx 0.1 + 0.3 j " r + a + jx 0.4 + 0.3 J 
= e (l-3j), = eo (1.6 - 1.2 j); 

(33) : / - « (2.6 - 4.2 j), 

i = 4.94 e ; 

r+jx 0.1 + 0.3 j 

(34): E> = e , U = *o f + a + - = e Q 4 + Q 3 , 

ei = eo, e 2 = 0.632 eo) 

/«^x / , . • • ,\ r+jx 0.1 + 0.3.7 

(35): m (cos * + j sin *) = r + - + . f = Q 4 + Mj 

= 0.52 + 0.36 j, 

tan * = 0.52' 

sin <^> = 0.57; 
(38): t = 0.36; 

(43) : v = 0.46. 

Thus this arrangement gives 46 per cent., or nearly half as 
much starting torque per volt-ampere taken from the supply 
circuit, as the motor would give as polyphase motor. 

However, as polyphase motor with low-resistance secondary, 
the starting torque per volt-ampere input is low. 

With a high-resistance motor armature, which on polyphase 
supply gives a good apparent starting-torque efficiency, v would 
be much lower, due to the lower angle, <f>. In this case, however, 
a reactance, +ja, would give fairly good starting-torque efficiency . 

In the same manner the effect of reactance or capacity inserted 
into one of the two motor coils can be calculated. 

As instances are given, in Fig. 37, the apparent torque efficiency, 
v, of the single-phase induction-motor starting device consisting 
of the insertion, in one of the two parallel motor circuits, of 
various amounts of reactance, inductive or positive, and capacity 


or negative, for a low secondary resistance motor of impedance: 

Z - 0.1 +0.3; 
and a high resistance armature, of the motor impedance: 

Z = 0.3 + 0.1 j 
resistance inserted into the one motor circuit, has the same effect 






1 + 




+ S 4 

i + 

1 + 


| 1 

I * 









E If 








+ i? 



H ( 

+ 1 s 

•E* 1 

+ 1 



in t 



37.— Apparent starting-torque eflutenoei of phase-splitting de 
parallel cumieition uf motor cireuits. 

lie first motor, as positive reactance in the second motor, 

K Higher values of starting-torque efficiency are aecurec 
use of capacity in the one, and inductance in the other m 
nit. It is obvious that by resistance and inductance al 
phase displacement between the two component curre 
thus true quarter-phase relation, can not be reached. 
s resistance consumes energy, the use of resistance is justi 







only due to its simplicity and cheapness, where moderate start- 
ing torques are sufficient, and thus the starting-torque efficiency 
less important. For producing high starting torque with high 
starting-torque efficiency, thus, only capacity and inductance 
would come into consideration. 

Assume, then, that the one impedance is a capacity: 

X2 = — fc, or: Z 2 = — jk, (45) 

while the other, xi, may be an inductance or also a capacity, what- 
ever may be desired: 

Zi = +jx 1} . (46) 

where X\ is negative for a capacity. 
It is, then : 

(35) : m (cos <t> + 3 sin <f>) = 

r + j (xi + x) [r 2 - (xi + x)(k - x)] + jrxik ( ? . 
r -j(k - x) ' r 2 + (fc- x)* " K } 

True quadrature relation of the voltages, e\ and e%, or angle, 

<f> = s' requires: 

cos <t> = 0, 

(xx + x) (k - x) = r 2 (48) 

and the two voltages, e\ and 62, are equal, that is, a true quarter- 
phase system of voltages is produced, if in 

(34): [Z + ZJ = [Z + Z 2 ], 

where the [ ] denote the absolute values. 
This gives: 

r* + (*i + xY = r* + (k - x)\ 

X\ + x = k — x, (49) 

hence, by (48) : 

Xi + x = k — x = r, 

k = r + x >\ (50) 

Xt = r — x. 


Thus, if x > r, or in a low-resistance motor, the second reactance, 
Xi f also must be a capacity. 


70. Thus, let: 
in a low-resistance motor: 

Z = r+jx = 0.1 + 0.3.?, 
k = 0.4, xi = - 0.2, 

Z 2 0.4 i, Z x = -0.2j, 

that is, both reactances are capacities. 

(34) : e x = e 2 = 2.23 e , 

* = 5, 

that is, the torque is five times as great as on true quarter-phase 

41 0.1 + o.i / i2 ai-o-ij' 

/ = 10 e = i, 
that is, non-inductive, or unity power-factor. 

to = y = 3.166o, 

g = 1.58, 
v = 3.16, 

that is, the apparent starting-torque efficiency, or starting torque 
per volt-ampere input, of the single-phase induction motor with 
starting devices consisting of two capacities giving a true quarter- 
phase system, is 3.16 as high as that of the same motor on a 
quarter-phase voltage supply, and the circuit is non-inductive 
in starting, while on quarter-phase supply, it has the power- 
factor 31.6 per cent, in starting. 
In a high-resistance motor: 

Z = 0.3 + 0.1 i, 
it is: 

k = 0.4, xx = 0.2, 

Z 2 = -0.4j, Z 2 = +0.2 j, 

that is, the one reactance is a capacity, the other an inductance. 

ei = e 2 = 0.743 e 0) 

t = 0.555, 

i = 3.33 6o, 

to =3.16 eo, 

q = 0.527, 

v = 1.055, 



that is, the starting-torque efficiency is a little higher than with 
quarter-phase supply. In other words: 

This high-resistance motor gives 5.5 per cent, more torque 
per volt-ampere input, with unity power-factor, on single-phase 
supply, than it gives on quarter-phase supply with 95 per cent, 

The value found for the low-resistance motor, t = 5, is how- 
ever not feasible, as it gives: e x = 6 2 = 2.23 e , and in a quarter- 
phase motor designed for impressed voltage, e , the impressed 
voltage, 2.23 eo, would be far above saturation. Thus the motor 
would have to be operated at lower supply voltage single-phase, 
and then give lower t, though the same value of v = 3.16. At 
e\ = ej = e , the impressed voltage of the single-phase circuit 
would be about 45 per cent, of e , and then it would be: t = 1. 

Thus, in the low-resistance motor, it would be preferable to 
operate the two motor circuits in series, but shunted by the two 
different capacities producing true quarter-phase relation. 

Series Connection 

71. The calculation of the single-phase starting of a motor 
with two coils in quadrature position, shunted by two impedances 

Fia. 38. — Diagram of phase-splitting device with series connection of motor 


of different power-factor, as shown diagrammatically in Fig. 38, 
can be carried out in the same way as that of parallel connection, 
except that it is more convenient in series connection to use the 
term " admittance" instead of impedance. 

That is, let the effective admittance per motor coil equal: 

y = v = ( J - A 


and the two motor coils be shunted respectively by the admit- 

Yi = gi - jb u 

Y 2 = 02 — j&2, 

it is then: 


/ = 1 6 ° =— , (53) 

ir~ + 


Y +Yi ' Y +Y 
the current consumed by the motor, and : 

& = Y~+Ti and ^ 2 = Y + Y 2 ' (54) 

the voltages across the two motor circuits. 

The phase difference between E\ and E 2 thus is given by 

Y + Y* 
m (cos 4> + j sin 4>) = y^Y ' ^ 

and herefrom follows t, q and v. 

As instance consider a motor of effective admittance per cir- 

Y = g-jb = l-3j, 

with the two circuits connected in series between single-phase 
mains of voltage, e<>, and one circuit shunted by a non-inductive 
resistance of conductance, g im 

What value of g\ gives maximum starting torque, and what 
is this torque? 

It is: 

(53} ' ' " 1 , __J_ " 2g + g x - 2j6 ~ (5b} 

ff + flfi — jb - J& 

(54). *-— -—^ *__.___, (57) 

(55) : m (cos * + j sin 0) - *-±-«L^ = [^+_^)^Hl^; 



tan = 

pfa + gO + fc 2 

sin * = , g '_. — - (58) 

VViW + [flf (ff + 91) + 6 s ] 1 


and thus: 

'2» + 

* _ 9*> 

1(2? -f g,)* 

9i)* + *& 

+ 46*] 

and for 


maximum. 1: 



= 2 V» f +6* 
= 2 y = 6.32, 

or, substituting back: 


t = 

,, . „ = 0.18. 




■» \ir ~r yj 

As in single-phase operation, the voltage, e , is impressed upon 
the two quadrature coils in series, each coil receives only about 

—v=. Comparing then the single-phase starting torque with that 
of a quarter-phase motor of impressed voltage, —.-* it is: 

t = 0.36. 

The reader is advised to study the possibilities of capacity 
and reactance (inductive or capacity) shunting the two motor 
coils, the values giving maximum torque, those giving true 
quarter-phase relation, and the torque and apparent torque 
efficiencies secured thereby. 


External Inductive Devices 

72. Inductively divided circuit: in its simplest form, as shown 
diagrammatically in Fig. 39, the motor contains two circuits 
at right angles, of the same admittance. 

The one circuit (1) is in series with the one, the other (2) with 
the other of two coils wound on the same magnetic circuit, M. 
By proportioning the number of turns, n\ and n 2 , of the two coils, 
which thus are interlinked inductively with each other on the 
external magnetic circuit, M, a considerable phase displacement 



between the motor coils, and thus starting torque can be pro- 
duced, especially with a high-resistance armature, that is, a 
motor with starting rheostat. 

A full discussion and calculation of this device is contained in 
the paper on the " Single-phase Induction Motor," page 63, 
A. I. E. E. Transactions, 1898. 


-*&>°* ^M ATURE 

FlO. 39. 

-External inductive 

Fiu. 40. — Diagram of shading coil. 

Internal Inductive Devices 

The exciting system of the motor consists of a stationary pri- 
mary coil and a stationary secondary coil, short-circuited upon 
itself (or closed through an impedance), both acting upon the 
revolving secondary. 

The stationary secondary can either cover a part of the pole 
face excited by the primary coil, and is then called a "shading 
coil," or it has the same pitch as the primary, but is angularly 
displaced therefrom in space, by less than 90° (usually 45° or 60°), 
and then has been called accelerating coil. 

The shading coil, as shown diagrammatically in Fig. 40, is 
the simplest of all the single-phase induction motor-starting 
devices, and therefore very extensively used, though it gives 
only a small starting torque, and that at a low apparent starting- 
torque efficiency. It is almost exclusively used in very small 
motors which require little starting torque, such as fan motors, 
and thus industrially constitutes the most important single- 
phase induction motor-starting device. 

73. Let, all the quantities being reduced to the primary num- 
ber of turns and frequency, as customary in induction machines: 

Z = r<> + jxo = primary self-inductive impedance, 
y = g — jb = primary exciting admittance of unshaded poles 

(assuming total pole unshaded), 


Y' = g' — jb' = primary exciting admittance of shaded poles 

(assuming total pole shaded). 

If the reluctivity of the shaded portion of the pole is the same 
as that of the unshaded, then Y' = Y; in general, if 

b = ratio of reluctivity of shaded to unshaded portion of 

Y' = bY, 

b either = 1, or, sometimes, b > 1, if the air gap under the 
shaded portion of the pole is made larger than that under the 
unshaded portion. 

Yi = gi — jbi = self-inductive admittance of the revolving 

secondary or armature, 

Y* = 02 — jb 2 = self-inductive admittance of the stationary 

secondary or shading coil, inclusive its exter- 
nal circuit, where such exists. 

Z , Yi and Y 2 thus refer to the self-inductive impedances, in 
which the energy component is due to effective resistance, and 
Y and Y' refer to the mutual inductive impedances, in which the 
energy component is due to hysteresis and eddy currents. 

a = shaded portion of pqje, as fraction of total pole; thus 

(1 — a) = unshaded portion of pole. 

eo = impressed single-phase voltage, 

$i = voltage induced by flux in unshaded portion of pole, 
$2 = voltage induced by flux in shaded portion of pole, 
/o = primary current, 
it is then : 

e = #i + #2 + Zo/o. (62) 

The secondary current in the armature under the unshaded 
portion of the pole is: 

/i = #iVY (03) 

The primary exciting current of the unshaded portion of the 
pole : 

/„„ = f l J a , (64) 


h = ft + f«. - & { r, + , } „!• («5) 



The secondary current under the shaded portion of the pole is: 

/'i = frYi. (66) 

The current in the shading coil is: 

h - #2^2. (67) 

The primary exciting current of the shaded portion of the pole 


/ 00 = 



/o = /'i + ho + u = & Yi + -y + Y t 



from (65) and (69) follows: 


y, + - y + Y t 


Fi + 

= m (cos <t> + j sin 0), 


I- a 

and this gives the angle, 4>> of phase displacement between the two 
component voltages, $1 and $2- 

If, as usual, 6=1, and 

if fc = 0.5, that is, half the shaded, it is: 

Ei wB Y l + 2Y+Y* 




74. Assuming now, as first approximation, Z = 0, that is, 
neglecting the impedance drop in the single-phase primary coil — 
which obviously has no influence on the phase difference between 
the component voltages, and the ratio of their values, that is, 
on the approximation of the devices to polyphase relation — then 
it is: 

Pi + Pt = e ] (72) 

thus, from (70) : 

Pi = e 

Yt + ^-Y+Y* 

2Yl + Y (l + v l- a ) + Yi 

Y t + 

E t = e - 

1 - a 

2Y l +Y(l + T ±- ) +Y t ' 
\a 1 — at 




or, for: 

b = 1; a = 0.5; 
JK, + 2F + F, 
** 2 F,"+ 4 F +~F,' 
„ F, + 2 F 

** 2F l + 4F+F*' 


and the primary current, or single-phase supply current is, by 
substituting (73) into (65) : 

Y \ t„ . b 

/o = Co 


or, for: 

b = \;a = 0.5: 

. _ c (F, + 2 YHY, + 2 F - 
/0 " e ° 2 F! + 4 F + F, 

+ F 2 ) 


and herefrom follows, by reducing to absolute values, the torque, 
torque ratio, volt-ampere input, apparent torque efficiency, etc. 
Or, denoting: 

y ' + T--a = r °' 
Y + ^Y+Y t =Y', 


it is: 



= ~- = m (cos <t> + j sin </>) ; 





E, = 




F° + F' 

e F° 

ft + F'» ] 

c F l^_ •' 
F° + F'' 

r = Aetfi sin #, 
Q = e io; 




and for a quarter-phase motor, with voltage —y impressed per 



circuit, neglecting the primary impedance, z , to be comparable 
with the shaded-coil single-phase motor, it is: 

;«= e ° AY+Yt), 

q = 2e f* = eovr + y./, 


T. = A *', 




9 = i\72 

. 2 6i6 2 . . 
*- eo2 sm«, 

t; = • 

75. As instances are given in the following table the compo- 
nent voltages, ei and 62, the phase angle, 4> } between them, the 
primary current, i 0| the torque ratio, t, and the apparent starting- 
torque efficiency, v y for the shaded-pole motor with the constants: 

Impressed voltage: e = 100; 

Primary exciting admittance: Y = 0.001 — 0.01 j. 

6 = 1, that is, uniform air gap. 

a = 0.5, that is, half the pole is shaded. 

And for the three motor armatures : 

Low resistance: Y x = 0.01 — 0.03 j, 

Medium resistance: Y x = 0.02 — 0.02 j, 
High resistance: F, = 0.03 - 0.01 j; 

and for the three kinds of shading coils: 

Low resistance: Y 2 = 0.01 — 0.03 j, 

Medium resistance: V 2 = 0.02 — 0.02 j, 
High resistance: ^2 = 0.03 - 0.01 j. 

As seen from this table, the phase angle, 0, and thus the start- 
ing torque, t } are greatest with the combination of low-resistance 
armature and high-resistance shading coil, and of high-resistance 
armature with low-resistance shading coil; but in the first case 
the torque is in opposite direction — accelerating coil — from what 


it is in the second case — lagging coil. In either case, the torque 
efficiency is low, that is, the device is not suitable to produce 
high starting-torque efficiencies, but its foremost advantage is 
the extreme simplicity. 

The voltage due to the shaded portion of the pole, ۥ, is less 
than that due to the unshaded portion, *i, and thus a somewhat 
higher torque may be produced by shading more than half of 
the pole: a > 0.5. 

A larger air gap: b > 1, under the shaded portion of the pole, 
or an external non-inductive resistance inserted into the shad- 
ing coil, under certain conditions increases the torque somewhat — 
at a sacrifice of power-factor — particularly with high-resistance 
armature and low-resistance shading coil. 

Co = 100 volts; a = 0.5; b = 1; Y - 0.001 - 0.01 j. 

Yii Yti er. e*: <f>: i : /,: v: 

X 10~ 2 X 10~* per cent, per cent. 

. 1 - 3j 1 - Sj 38.3 61.8 +1.9 1.97 + 1.5!) +4.07 

2-2./40.3 60.2 +11.0 2.07 +9.28 +23. (K) 

. 3 - \j 42.0 59.8 +21.5 2.17 +18.36 +43.70 

2 -2>1 -3J 37.2 62.9 -4.3 1.70 -3.52 -.9.65 
2-2J38.5 61.7 +6.2 1.76 +5.12 +13.60 
3 -I; 39.2 62.0 +17.3 1.80 +14.44 +37.40 

. 3 - \j 1 - 3j 37.6 63.0 -11.9 1.66 -9,76 -25.80 

2 - 2j 37.8 62.5 - 0.8 1.66 - 0.66 ' - 1.75 

3 - Ij 37.4 63.0 +10.3 1.64 +8.44 +22.60 

Monocyclic Starting Device 

76. The monocyclic starting device consists in producing ex- 
ternally to the motor a system of polyphase voltages with single- 
phase flow of energy, and impressing it upon the motor, which is 
wound as polyphase motor. 

If across the single-phase mains of voltage, e, two impedances 
of different inductance factors, Z\ and Z 2 , are connected in series, 
as shown diagrammatically in Fig. 41, the two voltages, I$y and 
#2, across these two impedances are displaced in phase from each 
other, thus forming with the main voltage a voltage triangle. 
The altitude of this triangle, or the voltage, # , between the com- 



mon connection of the two impedances, and a point inside of the 
main voltage, e (its middle, if the two impedances are equal), is 
a voltage in quadrature with the main voltage, and is a teazer 
voltage or quadrature voltage of the monocyclic 
system, e, E\, E s , that is, it is of limited energy 
and drops if power is taken off from it. (See 
Chapter XIV.) 

Let then, in a three-phase wound motor, oper- 
ated single-phase with monocyclic starting device, 
and shown diagrammatically in Fig. 42: 

■ — voltage impressed 1 jet. ween single-phase 

/ = current in single-phase lines, 
Y = effective admittance per motor circuit, 
I'i, Ei and I',, and Y 2 , $ 2 and f' 2 = admittance, voltage and 
current respectively, in the two impedances of the mono- 
cyclic starting device, 

Fio. 42. — Three-phase motor with monocyclic starting device. 

/i, /] and /a = currents in the three motor circuits. 

E ,, and /« = voltage and current of the quadrature circuit from 

the common connection of the two impedances, 

to the motor. 



It is then, counting the voltages and currents in the direction 
indicated by the arrows of Fig. 42: 



/o = J'i — I't = /* — h'y 

/'* = EtYt, 
I* = V>Y, 
h = W, 

l$\Yi — jpjl't = (#* ~ jFO F, 
Ai Y t + Y 



Zt Yt+Y 

= m (cos 4> + j sin 0). 


This gives the phase angle, <t>, between the voltages, #1 and #», 
of the monocyclic triangle. Since: 

it is, by (83) : 

Y t +Y 


F, + F, + 2 F 

Ki + r 
Ki+ F, + 2F' 


and the quadrature voltage: 

c Fi - F, 

~2F,+ F, + 2F (86) 

and the total current input into the motor, inclusive starting 

/ = /'. + h + h 
= ViY x + £,F + eY 
,(Y l +Y)(Y i + F) 

= e 

+ Y 

Yi+Y, + 2Y 
= Y\Y t + 2 F(K, +_F 2 ) + 37* 
" "7!+ F S _ +"2F" 


As with the balanced three-phase motor, the quadrature com- 
ponent of voltage numerically is « Vo, it is, when denoting by: 



Ee' the numerical value of the imaginary term of # ; the torque 
ratio is: 

<= 2B ;.'- (^ 

The volt-arnpere ratio is: 

8 = 3i. 


rims I he apparent starting-torque efficiency: 



77. Three eases have become of special importance: 

(a) The resistance-reactance monocyclic starting device; where 
one of the two impedances, Z, and Z t , is a resistance, the other an 
inductance. This is the simplest and cheapest arrangement, 
gives good starting torque, though a fairly high current consump- 
tion and therefore low starting-torque efficiency, and is therefore 
very extensively used for starling single-phase induction motors. 
After starting, the monocyclic device is cut out and the power 
consumption due to the resistance, and depreciation of the power- 
factor due to the inductance, thereby avoided. 

This device is discussed on page 333 of "Theoretical Elements 
of Electrical Engineering" and page 253 of "Theory and Calcu- 
lation of Alternating-current Phenomena." 

(fe) The "condenser in the tertiary circuit," which may be 
considered as a monocyclic starting device, in which one of the 
two impedances is a capacity, the other one is infinity. The 
capacity usually is made so as to approximately balance the mag- 
netizing current of the motor, is left in circuit after starting, as 
it does not interfere with the operation, does not consume power, 
and compensates for the lagging current of the motor, so that 
the motor has practically unity power-factor for all loads. This 
motor gives a moderate starting torque, but with very good start- 
ing-torque efficiency, and therefore is the most satisfactory single- 
phase induction motor, where very high starting torque is not 
needed. It was extensively used some years ago, but went out 
of use due to the trouble with the condensers of these early days, 
and it is therefore again coming into use, with the development 
of the last years, of a satisfactory condenser. 



The condenser motor is discussed on page 249 of " Theory and 
Calculation of Alternating-current Phenomena. ,, 

(c) The condenser-inductance monocyclic starting device. 
By suitable values of capacity and inductance, a balanced three- 
phase triangle can be produced, and thereby a starting torque 
equal to that of the motor on three-phase voltage supply, with 
an apparent starting-torque efficiency superior to that of the 
three-phase motor. 

Assuming thus: 

1\ = +jbi = capacity, 1 . 

} r 2 = — j6 2 = inductance, J 
Y = g - jb. 

If the voltage triangle, e, E u # 2 , is a balanced three-phase tri- 
angle, it is : 

tfi«£(l -jV3), 

Substituting (91) and (92) into (83), and expanding gives: 
(6, - 6i + 26) a/3 - j (6 2 + 6i - 2g y/S) = 0; 

thus : 

6 2 - 6, + 2 b = 0, 

62 + 61 - 2(7 V3 = 0; 
hence : 

bi = gy/S + b, \ 

62 = gy/3 ~ b) 
thus, if: 

b > g V3, 

the second reactance, Z 2 , must be a capacity also; if 

b<g \/3, 

only the first reactance, Zi, is a capacity, but the second is an 

78. Considering, as an instance, a low-resistance motor, and a 
high-resistance motor: 

(«) w 

Y = g - jb = 1 - 3j, Y-g-jb-Z-j, 



it is: 

61 = 4.732, capacity, 61 = 6.196, capacity, 

62 = —1.268, capacity, 6 2 = 4.196, inductance. 
It is, by (86) and (92) 

tfo>= |(#2- E x ) =|V3; 


t = 1, as was to be expected, 
/s = e(g - jb), 
is = e vV + b 2 
= 3.16 6; 

it is, however, by (87) : 

I = e(3g-jb); 

i = 4.243 e, i = 9.06 e, 

and by (89) : 

q = 0.448, g = 0.956, 


v = 2.232, v = 1.046. 

Further discussion of the various single-phase induction motor- 
starting devices, and also a discussion of the acceleration of the 
motor with the starting device, and the interference or non-inter- 
ference of the starting device with the quadrature flux and thus 
torque produced in the motor by the rotation of the armature, is 
given in a paper on the "Single-phase Induction Motor," A. I. 
E. E. Transactions, 1898, page 35, and a supplementary paper on 
"Notes on Single-phase Induction Motors," A. I. E. E. Trans- 
actions, 1900, page 25. 



79. Load and speed curves of induction motors are usually 
calculated and plotted for constant-supply voltage at the motor 
terminals. In practice, however, this condition usually is only 
approximately fulfilled, and due to the drop of voltage in the 
step-down transformers feeding the motor, in the secondary and 
the primary supply lines, etc., the voltage at the motor terminals 
drops more or less with increase of load. Thus, if the voltage 
at the primary terminals of the motor transformer is constant, 
and such as to give the rated motor voltage at full-load, at no- 
load the voltage at the motor terminals is higher, but at overload 
lower by the voltage drop in the internal impedance of the trans- 
formers. If the voltage is kept constant in the center of distri- 
bution, the drop of voltage in the line adds itself to the imped- 
ance drop in the transformers, and the motor supply voltage 
thus varies still more between no-load and overload. 

With a drop of voltage in the supply circuit between the point 
of constant potential and the motor terminals, assuming the cir- 
cuit such as to give the rated motor voltage at full-load, the 
voltage at no-load and thus the exciting current is higher, the 
voltage at overload and thus the maximum output and maximum 
torque of the motor, and also the motor impedance current, that 
is, current consumed by the motor at standstill, and thereby the 
starting torque of the motor, are lower than on a constant-poten- 
tial supply. Hereby then the margin of overload capacity of the 
motor is reduced, and the characteristic constant of the motor, 
or the ratio of exciting current to short-circuit current, is in- 
creased, that is, the motor characteristic made inferior to that 
given at constant voltage supply, the more so the higher the 
voltage drop in the supply circuit. 

Assuming then a three-phase motor having the following con- 
stants: primary exciting admittance, Y = 0.01 — 0.1 j; primary 
self-inductive impedance, Z = 0.1 + 0.3 j; secondary self -induc- 




tive impedance, Z, = 0.1 + 0.3 j; supply voltage, e = 110 volts, 
and rated output, 5000 waits per phase. 
Assuming this motor to be operated: 

1. By transformers of about 2 per cent, resistance and 4 per 
cent, reactance voltage, that is, transformers of good regulation, 
with constant voltage at the transformer terminals. 

2. By transformers of ahout 2 per cent, resistance and 15 per 
cent, reactance voltage, that is, very poorly regulating trans- 
formers, at constant supply voltage at the transformer primaries. 

3. With constant voltage at the generator terminals, and 
about 8 per cent, resistance, 40 per cent, reactance voltage in 
line and transformers between generator and motor. 

This gives, in complex quantities, the impedance between the 
motor terminals and the constant voltage supply: 

1. Z - 0.04 + 0.08 j, 

2. Z = 0.04 + 0.3 j", 

3. Z = 0.16 + 0.8,/. 

It is assumed that the constant supply voltage is such u hi 
give 1 10 volts at the motor terminals at FulHoad. 

The load and speed curves of the motor, when operating under 
these conditions, that is, with the impedance, Z, in series between 
the motor terminals and the constant voltage supply, e., then can 
be calculated from the motor characteristics at constant termi- 
nal voltage, e Bl as follows: 

At slip, I, and constant terminal voltage, e a , the current in the 
motor is i , its power-factor p = cos 8. The effective or equiva- 
lent impedance of the motor at this slip then is z" = .-, and, in 

complex quantities, Z* = ." (cos + i Bin 0), and the total irn- 
pedance, including that of transformers and line, thus is: 
Z x = Z° + Z = (?" cos 6 + r) + j(* sin + xj , 
or, in absolute values: 

tl m .J(pcos0 4-r)'+ (^sin0+j 
and, at the supply voltage, e ,, the current thus is 


and the voltage at the motor terminals is: 

e'o = z°i'i = e t . 


If e a is the voltage required at the motor terminals at full-load, 
and io the current, zi° the total impedance at full-load, it is: 

1 1 1 1 1 1 1 1 1 1 

V -0.01- 0,1 j Z- 0.1 +0.3) 
































Ji ! 

si . n 



o 110 volts at motor 

hence, the required constant supply voltage is: 

and the speed and torque curves of the motor under this condi- 
tion then are derived from those at constant supply voltage, e<,, 
by multiplying all voltages and currents by the factor "> that 

is, by the ratio of the actual terminal voltage to the full-load 
terminal voltage, and the torque and power by multiplying with 


the square of this ratio, while the power-factors and the efficien- 
cies obviously remain unchanged. 

In this manner, in the three cases assumed in the preceding, 
the load curves are calculated, and are plotted in Figs, 43, 44, 
and 45. 

80. It is seen that, even with transformers of good regulation, 
Fig. 43, the maximum torque and the maximum power are ap- 

Y-Om-O.ijo Z-0.1 + <Mj 



































Fio. 44.— 

nal volta 
torque at 
In Fig- 
of the m 
three ca 
Fig. 48 f 
si stance 
former i 

ndnd ion-motor load curves) corresponding to 110 volti 

li'H ii iii;i l.s at 5000 watts load. 

reduced. The values corresponding to consta 
£e are shown, for the part of the curves near rr 

d maximum power, in Figs. 43, 44, and 45. 

. 46, 47, 48, and 49 are given the specd-torqii 
otor, for constant terminal voltage, Z — 0, 

es above discussed; in Fig. 46 for short-< 
es, or running condition; in Fig. 47 for 0.15 
:>r 0.5 ohm; and in Fig. 49 for 1.5 ohms addii 
nserted in the armature. As seen, the line ai 
mpedance very appreciably lowers the tore 

at motor 

it termi- 

c curves 
and the 
ohm; in 
tonal re- 
d trans- 
ue, and 


v- aci-o.ij i- 0,1 +0Ji 
















1 = 

















ST X" 








1.0 (1 







1 .i 


especially the starting torque, which, with short-circuited arma- 
ture, in the case 3 drops to about one-third the value given at 
constant supply voltage. 












ill ( 




, ) 

Fig. 47. — Induction-motor speed torque characteristics with a resistance o 
0. 15 onm in secondary circuit. 

















Fifl. 48. — Induct ion -motor speed turque characterise ch with a resistance of 
0.5 ohm in secondary circuit. 

It is interesting to note that, in Fig. 48, with a secondary 
resistance giving maximum torque in starting, at constant tcr- 



rainal voltage, with high impedance in the supply, the starting 
torque drops so much that the maximum torque is shifted to 
about half synchronism. 

In induction motors, especially at overloads ami in starting, 
it therefore is important to have as low impedance as pos- 
sible between the point of constant voltage and the motor 




L-.K! g 


p Friction of Synchroniwn 

with a resistance of 

In Table I the numerical values of maximum power, maxi- 
mum torque, starting torque, exciting current and starting 
current are given for above motor, at constant terminal voltage 
and for the three values of impedance in the supply lines, for 
such supply voltage as to give the rated motor voltage of 110 
volts at full load and for 1 10 volts supply, voltage. In the first 
case, maximum power and torque drop down to their full-load 
values with the highest line impedance, and far below full-load 
values in the latter case. 




































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81. If the frequency of the voltage supply pulsates with 
sufficient rapidity that the motor speed can not appreciably 
follow the pulsations of frequency, the motor current and torque 
also pulsate; that is, if the frequency pulsates by the fraction, 
p, above and below the normal, at the average slip, s, the actual 
slip pulsates between s + p and a — p, and motor current and 






^ ^^J '^ 

isV*i ix 



\ y§ 


- r 

















y-o.oi-o.ij Zg-o.i*ojj z, -0.0540 











i r,,.» 

:1 ,, l M..o'»«M);».«.r/.!l] l r 1 i.i: 1 ».; l 

Fig. 50. — Effect of Frequency Pulsation on Induction Motor. 

torque pulsate between the values corresponding to the slips, 
s + p and 8 — p. If then the average slip s < p, at minimum 
frequency, the actual slip, a — p, becomes negative; that is, the 
motor momentarily generates and returns energy. 

As instance are shown, in Fig. 50, the values of current and 
of torque for maximum and minimum frequency, and for the 
average frequency, for p = 0.025, that is, 2.5 per cent, pulsa- 
tion of frequency from the average. As seen, the pulsation of 
current is moderate until synchronism is approached, but be- 



comes very large near synchronism, and from slip, s = 0.025, op 
to synchronism the average current remains practically con- 
stant, thus at synchronism is very much higher than the current 
at constant Frequency. The average torque also drops some- 
what below the torque corresponding to constant frequency, 
as shown in the upper pari of Fig. 50. 


82. At constant voltage and constant frequency the torque 
of the polyphase induction motor is a maximum at some definite 
speed and decreases with increase of speed over that correspond- 
ing to the maximum torque, to zero at synchronism; it also de- 
ereases with decrease of speed from that at the maximum torque 
point, to a minimum at standstill, the starting torque. This 
maximum torque point shifts toward lower speed with increase 
of the resistance in the secondary circuit, and the starting torque 
thereby increases. Without additional resistance inserted in 
the secondary circuit the maximum torque point, however, lies 
at fairly high speed not very far below synchronism, 10 to 20 
per cent, below synchronism with smaller motors of good effi- 
ciency. Any value of torque between the starting torque and 
the maximum torque is reached at two different speeds. Thus 
in a three-phase motor having the following constants: impressed 
e.m.f., eg = 110 volts: exciting admittance, 1" ~ 0.01 — OAj; 
primary impedance, Z v = 0.1+ 0.3 j, and secondary impedance, 
Z\ = 0.1 + 0.3 j, the torque of 5.5 synchronous kw. is reached 
at. 54 per cent, of synchronism and also at the speed of 94 per 
cent, of synchronism, as seen in Fig. 51. 

When connected to a load requiring a constant torque, irre- 
spective of the speed, as when pumping water against a constant 
head by reciprocating pumps, the motor thus could carry the 
load :tl two different speeds, the two points of intersection of the 
horizontal Hue, L, in Fig. 51. which represents the torque con- 
sumed by the load, and the motor-torque curve, O. Of these 
two points, d and r, the lower one, rf, represents unstable con- 
ditions of operation; that is, the motor can not operate n tln- 
speed, but either stops or runs up to the higher speed point, C, 
at which stability is reached. At the lower speed, d, a momen- 
tary decrease of speed, as by a small pulsation of voltage, load, 
etc., decreases the motor torque, D, below the torque, L, required 
by the load, thus causes the motor to slow down, but in doing 



so its torque further decreases, and it slows down still more, 
loses more torque, etc., until it comes to a standstill. Inversely, 
a momentary increase of speed increases the motor torque, D, 
beyond the torque, L, consumed by the load, and thereby causes 
an acceleration, that is, an increase of speed. This increase of 
speed, however, increases the motor torque and thereby the 
speed still further, and so on, and the motor increases in speed 
up to the point, c, where the motor torque, D, again becomes 


^ . 9 

l ~/ | d/y «\y 


HI 02 03 <w OK OG 07 OB 09 1.0 

Flu. 61. — Speed -torque elianieteristies of induction motor nnd lorul for 

tic termination of the slnbility point; 

equal to the torque consumed by the load. A momentary in- 
crease of speed beyond c decreases the motor torque, D, and thus 
limits itself, and inversely a momentary decrease of speed below 
c increases the motor torque, D, beyond L, thus accelerates and 
recovers the speed ; that is, at c the motor speed is stable. 

With a load requiring constant torque the induction motor 
thus is unstable at speeds below that of the maximum torque 
point, but stable above it; that is, the motor curve consists of 
two branches, an unstable branch, from standstill, /, to the maxi- 



mum torque point, m, and a stable branch, from the maximum 
torque point., m, to synchronism. 

83. It must be realized, however, that this instability of the 
lower branch of the induction-motor speed curve is a function of 
the nature of the load, and as described above applies only to a 
luad requiring a constant torque, L, Such a load the motor 
could not start (except by increasing the motor torque at low 
speeds by resistance in the secondary), but when brought up to 
a speed above d would carry the load at speed, c, in Fig. 51. 

If, however, the load on the motor is such as to require a 
torque which increases with the square of the speed, as shown 
by curve, C, in Fig. 51, that is, consists of a constant part p 
(friction of bearings, etc.) and a quadratic part, as when driving 
a ship's propeller or driving a centrifugal pump, then the induc- 
tion motor is stable over the entire range of speed, from standstill 
to synchronism. The motor then starts, with the load repre- 
sented by curve C, and runs up to speed, c. At a higher load, 
represented by curve B, the motor runs up to speed, b, and with 
excessive overload, curve A, the motor would run up to low 
speed, point a, only, but no overload of such nature would stop 
the motor, but merely reduce its speed, and inversely, it would 
always start, but at excessive overloads run at low speed only. 
Thus in this case no unstable branch of the motor curve exists, 
hut it is stable over the entire range. 

With a load requiring a torque which increases proportionally 
to the speed, as shown by C in Fig. 52, that is, which consists 
of a constant part., p, and a part proportional to the speed, as 
when driving a direct-current generator at constant excitation, 
connected to a constant resistance as load — -as a lighting sys- 
tem — the motor always starts, regardless of the load — provided 
that the constant, part of the torque, », is less than the starting 
torque. With moderate load, C, the motor runs up to a speed, 
c, near synchronism. With very heavy load, A, the motor starts, 
but runs up to a low speed only. Especially interesting is the 
case of an intermediary load as represented by line B in Fig. 
52. B intersects the motor-torque curve, />, in three points, 
6i, 6i, by, that is, three speeds exist at which the motor gives the 
torque required by the load: 24 per cent., 00 per cent., and $S 
per cent, of synchronism. The speeds b, and b s are stable, the 
speed bi unstable. Thus, with this load the motor starts from 
standstill, but does not run up to a speed near synchronism, but 



accelerates only to speed b u and keeps revolving at this low 
speed (and a correspondingly very large current). If, however, 
the load is taken off and the motor allowed to run up to syn- 
chronism or near to it, and the load then put on, the motor slows 
down only to speed b ( , and carries the load at this high speed; 
hence, the motor can revolve continuously at two different speeds, 
61 and b%, and either of these speeds is stable; that is, a momen- 
tary increase of speed decreases the motor torque below that 



















1 ft 

; a 

.'. u 

1 i 

; < 



'.' 1 

FlO. 52. — Speed torque characteristics of induotioD motor and load for 

determination of the stability point. 

required by the load, and thus limits itself, and inversely a de- 
crease of motor speed increases its torque beyond that correspond- 
ing to the load, and thus restores the speed. At the intermediary 
speed, 6i, the conditions are unstable, and a momentary increase 
of speed causes the motor to accelerate up to speed fej, a momen- 
tary decrease of speed from b\ causes the motor to bIow down to 
speed 61, where it becomes stable again. In the speed range 
between Oj and J>j the motor thus accelerates up to 61, in the 
speed range between b t and 61 it slows down to b,. 

For this character of load, the induction-motor speed curve, 
D, thus has two stable branches, a lower one, from standstill, t, 
to the point n, and an upper one, from point m to synchronism, 


where- m and n are the points of contact of the tangents from the 
required starting torque, p, on to the motor curve, Z>; these two 
stable branches are separated by the unstable branch, from n to 
m, on which the motor can not operate. 

84. The question of stability of motor speed thus is a func- 
tion not only of the motor-speed curve but also of the characler 
of the load in its relation to the motor-speed curve, and if the 
change of motor torque with the change of speed is less than the 
change of the torque required by the load, the condition is stable, 
otherwise it is unstable; that is, it must lie . < ' to give 
stability, where L is the torque required by the load at speed, S 











i i 


1 u 

i ( 




1 ■ 

J 1 

Fin. 5. 

that the 
phase in 
to syncl 
crease in 
speed c 
middle c 

.— Spci'il-lurqui' rliiiriirtcriMtii 1 nf .-;iin;Ii*-|ilmse induct 

mally on polyphase induction motors on a loa 
l Fig. 52 this phenomenon is observed in 

motor can start the load but can not brin 
VI ore frequently, however, it is observed 
duction motors in which the maximum torqu 
ronism, with some forms of starting devices 

their effect with increasing speed and thus g 
laracteristics of forms similar to Fig. 53 
iced curve as shown in Fig. 53, even at a loat 

torque, three speed points may exist of 
ne is unstable. In polyphase synchronous n 
rs, when starting by alternating current, t 

in motor, 

1 as icpn- 
tbe form 
i it up io 
>n single- 
i? is nearer 
which de- 
ve motor- 
With a 
which the 
lotors and 
lat is, as 


induction machines, the phenomenon is frequently observed that 
the machine starts at moderate voltage, but does not run up to 
synchronism, but stops at an intermediary speed, in the neighbor- 
hood of half speed, and a considerable increase of voltage, and 
thereby of motor torque, is required to bring the machine beyond 
the dead point, or rather "dead range," of speed and make it 
run up to synchronism. In this case, however, the phenomenon 
is complicated by the effects due to varying magnetic reluctance 
(magnetic locking), inductor machine effect, etc. 

Instability of such character as here described occurs in elec- 
tric circuits in many instances, of which the most typical is the 
electric arc in a constant-potential supply. It occurs whenever 
the effect produced by any cause increases the cause and thereby 
becomes cumulative. When dealing with energy, obviously 
the effect must always be in opposition to the cause (Lenz's 
Law), as result of the law of conservation of energy. When 
dealing with other phenomena, however, as the speed-torque 
relation or the volt-ampere relation, etc., instability due to the 
effect assisting the cause, intensifying it, and thus becoming 
cumulative, may exist, and frequently does exist, and causes 
either indefinite increase or decrease, or surging or hunting, as 
more fully discussed in Chapters X and XI, of " Theory and 
Calculation of Electric Circuits/ ' 


86. If the voltage at the induction-motor terminals decreases 
with increase of load, the maximum torque and output are de- 
creased the more the greater the drop of voltage. But even if 
the voltage at the induction motor terminals is maintained con- 
stant, the maximum torque and power may bo reduced essen- 
tially, in a manner depending on the rapidity with which the 
voltage regulation at changes of load is effected by the generator 
or potential regulator, which maintains constancy of voltage, and 
the rapidity with which the motor speed can change, that is, 
the mechanical momentum of the motor and its load. 

This instability of the motor, produced by the generator 
regulation, may be discussed for the case of a load requiring 
constant torque at all loads, though the corresponding pheno- 
menon may exist at all classes of load, as discussed under 3, 
and may occur even with a load proportional to the square of 
the speed, as ship propellors. 



The torque curve of the induction motor at constant terminal 
voltage consists of two branches, a stable branch, from the 
maximum torque point to synchronism, and an unstable branch, 
that is, a branch at which the motor can not operate on a load 
requiring constant torque, from standstill to maximum torque. 
With increasing slip, s, the current, i, in the motor increases. If 

then D = torque of the motor, ,. is positive on the stable, 
negative on the unstable branch of the motor curve, anil this 
rate of change of the torque, with change of current, expf nm ed 
as fraction of the current, is: 

, _ 1 dD 

* ~ D di' 

it may be called the stability coefficient of the motor. 

If k, is positive, an increase of i, caused by an increase of 
slip, a, that is, by a decrease of speed, increases the torque, D, and 
thereby checks the decrease of speed, and inversely, that is, the 
motor is stable. 

If, however, k, is negative, an increase of i causes a decrease 
of D, thereby a decrease of speed, and thus further increase of j 
and decrease of D; that is, the motor slows down with increas- 
ing rapidity, or inversely, with a decrease of t, accelerates with 
increasing rapidity, that is, is unstable. 

For the motor used as illustration in the preceding, of the 
constants c = 110 volts; Y = 0.01 - 0.1 j; Z - 0.1 -f- 0.3 j, 
Zi = 0.1 + 0.3 j, the stability curve is shown, together with 
speed, current, and torque, in Fig. 54, as function of the output. 
As seen, the stability coefficient, k„ is very high for light-load, 
decreases first rapidly and then slowly, until an output of 7000 
watts is approached, and then rapidly drops below zero; that is, 
the motor becomes unstable and drops out of step, and speed, 
torque, and current change abruptly, as indicated by the arrows 
in Fig. 54. 

The stability coefficient, k„ characterizes the behavior of the 
motor regarding its load-carrying capacity. Obviously, if the 
terminal voltage of the motor is not constant, but drops with 
the load, as discussed in 1, a different stability coefficient results, 
which intersects the zero line at a different and lower torque. 

86. If the induction motor is supplied with constant terminal 
voltage from a generator of close inherent voltage regulation 



and of & size very large compared with the motor, over a supply 
circuit of negligible impedance, so that a sudden change of 
motor current can not produce even a momentary tendency of 
change of the terminal voltage of the motor, the stability curve, 
k„ of Fig. 54 gives the performance of the motor. If, however, 

II 1 1 1 1 1 1 



Y-0.Q1-O.lj Z-O.l + OJj 















































. — 












>.. , 


Fio. 54. — Induct ion-motor loud curves. 

at a change of load and thus of motor current the regulation 
of the supply voltage to constancy at the motor terminals re- 
quires a finite time, even if this time is very short, the maximum 
output of the motor is reduced thereby, the more so the more 
rapidly the motor speed can change. 

Assuming the voltage control at the motor terminals effected 



by hand regulation of the generator or the potential regulator 
in the circuit supplying the motor, or by any other method which 
is slower than the rate at which the motor speed can adjust itself 
to a change of load, then, even if the supply voltage at the 
motor terminals is kept, constant, for a momentary RuctOBtaon 
of motor speed and current, the supply voltage mom e nta ri ly 
varies, and with regard to its stability the motor corresponds 
not to the condition of constant supply voltage but to a supply 
voltage which varies with the current, hence the limit of stability 
is reached at a lower value of motor torque. 

"At constant slip, s, the motor torque, D, is proportional to the 
square of the impressed e.m.f., e 1 . If by a variation of slip 
caused by a fluctuation of load the motor current, i, varies by di, 
if the terminal voltage, e, remains constant the motor torque, D, 

varies by the fraction k, = ,, ..> or the stability coefficient of 
the motor. If, however, by the variation of current, di, the 
impressed e.m.f., e, of the motor varies, the motor torque, D, 
being proportional to e ! , still further changes, proportion*! to 

1 de* 2 de 
the change e ! , that is, bv the fraction k,= —„ -p- = - ,.• anrl the 
' - e* di e d% 

total change of motor torque resultant from a change, di. of the 
current, i, thus is k = k. + k r . 

Hence, if a momentary fluctuation of current causes a momen- 
tary fluctuation of voltage, the stability coefficient of the motor 
is changed from k, to k n = k, + fc„ and as k, is negative, . the 
voltage, e, decreases with increase of current, i, the stability 
coefficient of the system is reduced by the effect of voltage regu- 
lation of the supply, '.,, and k r thus can be called the regulation 
coefficient «f the system. 

k r = ,-. thus represents the change of torque produced by 
the momentary voltage change resulting from a current change 
di in the system; hence, is essentially a characteristic of the 
supply system and its regulation, but depends upon the motor 

only in so far as .. depends tijmn the power-factor of the load. 

In Fig. 54 is shown the regulation coefficient, k,, of the supply- 
system of the motor, at 110 volts maintained constant at the 
motor terminals, and an impedance, Z = 0.16 + 0.8 j, between 
motor terminals and supply e.m.f. As seen, the regulation 
coefficient of the system drops from a maximum of about 0.03, 


at no-load, down to about 0.01, and remains constant at this 
latter value, over a very wide range. 

The resultant stability coefficient, or stability coefficient of the 
system of motor and supply, A = k n + k n as shown in Fig. 54, 
thus drops from very high values at light-load down to zero at 
the load at which the curves, k, and fc r , in Fig. 54 intersect, or 
at 5800 kw., and there become negative; that is, the motor drops 
out of step, although still far below its maximum torque point, 
as indicated by the arrows in Fig. 54. 

Thus, at constant voltage maintained at the motor terminals 
by some regulating mechanism which is slower in its action than 
the retardation of a motor-speed change by its mechanical 
momentum, the motor behaves up to 5800 watts output in 
exactly the .same manner as if its terminals were connected 
directly to an unlimited source of constant voltage supply, but 
at this point, where the slip is only 7 per cent, in the present 
instance, the motor suddenly drops out of step without previous 
warning, and comes to a standstill, while at inherently constant 
terminal voltage the motor would continue to operate up to 
7000 watts output, and drop out of step at 8250 synchronous 
watts torque at 16 per cent. slip. 

By this phenomenon the maximum torque of the motor thus 
is reduced from 8250 to 6300 synchronous watts, or by nearly 
25 per cent. 

87. If the voltage regulation of the supply system is more 
rapid than the speed change of the motor as retarded by the 
momentum of motor and load, the regulation coefficient of the 
system as regards to the motor obviously is zero, and the motor 
thus gives the normal maximum output and torque. If the 
regulation of the supply voltage, that is, the recovery of the 
terminal voltage of the motor with a change of current, occurs at 
about the same rate as the speed of the motor can change with 
a change of load, then the maximum output as limited by the 
stability coefficient of the system is intermediate between the 
minimum value of 6300 synchronous watts and its normal value 
of 8250 synchronous watts. The more rapid the recovery of 
the voltage and the larger the momentum of motor and load, 
the less is the motor output impaired by this phenomenon of 
instability. Thus, the loss of stability is greatest with hand 
regulation, less with automatic control by potential regulator, 
the more so the more rapidly the regulator works; it is very little 



with compounderl alternators, and absent where the motor 
terminal voltage remains constant without any control by prac- 
tically unlimited generator capacity and absence of voltage drop 
between generator and motor. 

Comparing the stability coefficient, h„ of the motor load and 
the stability coefficient, ko, of the entire system under the assumed 
conditions of operation of Fig. 54, it is seen that the former 
intersects the zero tine very steeply, that is, the stability remains 
high until very close to the maximum torque point, and the motor 
thus can be loaded up close to its maximum torque without 
impairment of stability. The curve, k , however, intersects the 
zero fine under a sharp angle, that is, long before the limit of 
stability is reached in this case the stability of the system has 
dropped so close to zero that the motor may drop out of step by 
some momentary pulsation. Thus, in the case of instability due 
to the regulation of the system, the maximum output [joint, as 
found by test, is not definite and sharply defined, but the stability 
gradually decreases to zero, and during this decrease the motor 
drops out at some point. Experimentally the difference l>etween 
the dropping out by approach to the limits of stability of the 
motor proper and that of the system of supply is very marked 
by the indefiniteness of the latter. 

In testing induction motors it thus is necessary to guard 
against this phenomenon by raising the voltage l>eyond normal 
before every increase of load, and then gradually decrease the 
voltages again to normal. 

A serious reduction of the overload capacity of the motor, due 
to the regulation of the system, obviously occurs only at very 
high impedance of the supply circuit; with moderate impedance 
the curve, It, is much lower, and the intersection between fc, and 
k, occurs still on the steep part of k„ and the output thus is not 
materially decreased, but merely the stability somewhat reduced 
when approaching maximum output. 

This phenomenon of the impairment of stability of the induc- 
tion motor by the regulation of the supply voltage is of prac- 
tical importance, as similar phenomena occur in many instances. 
Thus, with synchronous motors and converters the regulation 
of the supply system exerts a similar effect on the overload 
capacity, and reduces the maximum output so that the motor 
drops out of step, or starts surging, due to the approach to the 
stability limit of the entire system. In this case, with syn- 


chronous motors and converters, increase of their field excita- 
tion frequently restores their steadiness by producing leading 
currents and thereby increasing the power-carrying capacity 
of the supply system, while with surging caused by instability 
of the synchronous motor the leading currents produced by 
increase of field excitation increase the surging, and lowering the 
field excitation tends toward steadiness. 



88. The usual theory and calculation of induction motors, 
.■is discussed in '* Theoretical Elements of Electrical Enginccr- 
ing" and in "Theory and Calculation of Alternating-current 
Phenomena," is based on the assumption of the sine wave. That 
U, it is assumed that the voltage impressed upon the motor 
per phase, and therefore the magnetic flux and the current, KM 
sine waves, and it is further assumed, that the distribution of 
the winding on the circumference of the armature or primary, 
is sinusoidal in space. While in most eases this is sufficicntly 
the ease, it is not always so, and especially the space or air-gap 
distribution of the magnetic flux may sufficiently differ from sine 
shape, to exert an appreciable effect on the torque at lower 
speeds, and require consideration where motor action and 
braking action with considerable power is required throughout 
the entire range of speed. 

Let then: 
r — i j i cos * + e» cos (3 * — a,) + es cos (5 * — at) 4- e? cos 
(7* - a-) + e, cos (9 * - a„) + . . . (1) 

be the voltage impressed u|hjn one phase of the induction motor. 

If the motor is a motor, the voltage of the 

second motor phase, which lags 90° or behind the first motor 
phase, is: 

= e,eos^«- gj + c 3 
+ 8) eos (? 




A 3 * - 

1 . cost 3 <t> - «■( + * } ) + e t coalS <t> - 

+ «odb(7#-«t + 5) + *«*(&* -■•-£) + ■ • ■ W 

The magnetic flux produced by these (wo voltages thus con- 
sists of a series of component fluxes, corresponding respective]] 


to the successive components. The secondary currents induced 
by these component fluxes, and the torque produced by the 
secondary currents, thus show the same components. 

Thus the motor* torque consists of the sum of a series of 

The main or fundamental torque of the motor, given by the 
usual sine-wave theory of the induction motor, and due to the 
fundamental voltage wave: 

ei cos ] 

Iju *\ ( 3 ) 

d cos \<t> ~ 2) 

is shown as T\ in Fig. 55, of the usual shape, increasing from 
standstill, with increasing speed, up j to a maximum torque, and 
then decreasing again to zero at synchronism. 
The third harmonics of the voltage waves are : 

e 3 cos(3 — a 3 ), j 

e 3 cos(3 0- «» + 5) - | (4) 

As seen, these also constitute a quarter-phase system of 
voltage, but the second wave, which is lagging in the funda- 
mental, is 90° leading in the third harmonic, or in other words, 
the third harmonic gives a backward rotation of the poles with 
triple frequency. It thus produces a torque in opposite direc- 
tion to the. fundamental, and would reach its synchronism, that 
is, zero torque, at one-third of synchronism in negative direction, 
or at the speed <S, = — J£, given in fraction of synchronous speed. 
For backward rotation above one-third synchronism, this triple 
harmonic then gives an induction generator torque, and the 
complete torque curve given by the third harmonics thus is as 
shown by curve T* of Fig. 55. 

The fifth harmonics: 

6 6 cos (5 — a 6 ), | 

e b cos ^5 05 - a& - 2) 

give again phase rotation in the same direction as the funda- 
mental, that is, motor torque, and assist the fundamental. But 
synchronism is reached at one-fifth of the synchronous speed of 

the fundamental, or at: S = +}i } and above this speed, the 


fifth harmonic becomes induction, genera tor, due to overayn- 
chronous rotation, and retards. Its torque curve is shown as 
7\ in Fig. 55. 

The seventh harmonic again gives negative torque, due to 
backward phase rotation of the phases, and reaches synchronism 
at S = — J-j, that is, one-seventh speed in backward rotation, 
as shown by curve T-, in Fig. 55. 











, I 



\ f 



T ', 















Flo. 55. — Quarti'r-plmsr imliirtinii riintnr, I'diiipoiii-iil harmonica mid 
resultant torque. 

The ninth harmonic again gives positive motor torque up to 
its synchronism, 5 = %, and above this negative induction 
generator torque, etc. 

We then have the effects of the various harmonics on the 


1 ■( 

+ i -H 

+ll - 



+ '. 




+ '» 


11 is 

-H, + ',. 

- +K. 


Torque positrt 

otherwise n 

peed: S -. . . . 

i'Uj.i tu; .-. = . 




Adding now the torque curves of the various voltage harmonics, 
Tz, 7\, T7, to the fundamental torque curve, 7\, of the induction 
motor, gives the resultant torque curve, T. 

As seen from Fig. 55, if the voltage harmonics are consider- 
able, the torque curve of the motor at lower speeds, forward 
and backward, that is, when used as brake, is rather irregular, 
showing depressions or "dead points." 

89. Assume now, the general voltage wave (1) is one of the 
three-phase voltages, and is impressed upon one of the phases 
of a three-phase induction motor. The second and third 

phase then is lagging by -«- and -«- respectively behind the first 

phase (1): 

e' = ei cos \6 - 3 ~ J + e 8 cos (3 £ - a 8 ) 

+ e h cos (5 - ■-•„* - a b J + e 7 cos \7 <t> - * - a 7 ) 

18 IT 


+ e 9 cos(90- Q ir ~ a »)+ • 
= 61 cos (0 — «- J + e% cos (3 4> — a 8 ) 

+ e b cos (5 - a 5 + q~) + e 7 cos ( 7 <t> - a 7 - ^ J 

+ e 9 cos (9 4> — ag) + . 
e" = 61 cos ( — -s J + e 8 cos (3 — a 8 ) 

+ e& cos (5 — «* + ■ 3*) + e 7 cos ( 7 - a 7 - «- j 

+ e 9 cos (9 4> — ag) + • 

Thus the voltage components of different frequency, impressed 
upon the three motor phases, are : 

ei cos * 

rj COR 

n cos 

e? cos 

r* cos 

(3 4> - a a ) 

(5* - a 4 ) 

(7* - a;) 

(9 * - a») 

ei cos 

/ 2t\ 

ft cos 

es cos 

/ 2t\ 

e 7 cos 

2 f \ 

r» cos 


(3 - oa) 

^♦-« + --j 

17* - ai 


(9 - a.) 

ei cos 


f 3 cos 

(3 - ai) 

f 5 cos 

( 5 *-°' + V) 

ei 00s 

(7* - ai 


n cos 

(9 - a») 

\ / 




\ / 





As spm, in this case of the three-phase motor, the third 
harmonics have no phase rotation, but are in phase with each 
other, or single-phase voltages. The fifth harmonic gives 
backward phase rotation, and thus negative torque, while the 
seventh harmonic has the same phase rotation, as the funda- 
menlal, thus adds its torque up to its synchronous speed, S = 
+ \i, and above this gives negative or generator torque. The 
ninth harmonic again is single-phase. 

Fig. 56 shows the Fundamental torque, 5ft, the higher harmonics 


















T t 

~\ r 

T ; 



Fig. 56. — HtrBC-ph;ini> inilii<-t.inn motor, component harmonica and 
resultant torque. 

of torque, T& ami 5T ; , and the resultant torque, T. As seen, the 
distortion of the torque curve is materially less, due In 1 lie 
absence, in Fig. 50, of the third harmonic torque. 

However, while the third harmonic (and its multiples) in the 
three-phase system of voltages are in phase, thus give no phase 
rotation, they may give torque, as a single-phase induction motor 
has torque, at speed, though al standstill the torque is eero. 

Fig. 57 Ii shows diagrammatical ly, as T, the development of 
the air-gap distribution of a hue three-phase winding, such as 
used in synchronous converters, etc. Each phase 1, 2, 3, coi an 

one-third of the pitch of a pair of poles or -5-, of the upper layer, 



and its return, 1', 2', 3', covers another third of the circumference 
of two poles, in the lower layer of the armature winding, 180° 
away from 1, 2, 3. However, this type of true three-phase wind- 
ing is practically never used in induction or synchronous machines, 
but the type of winding is used, which is shown as S, in Fig. 
57 C. This is in reality a six-phase winding: each of the three 











1o 3' 2' a 






mm a 



'' ^« a 




r 3; 2' 


3 2 

i; 3' 2i 




^ B 




r 3' ft 2' 


& c 

Fig. 57. — Current distribution at air gap of induction motor, fundamental 

and harmonics. 

phases, 1, 2, 3, covers only one-sixth of the pitch of a pair of 

poles, or ~ or 60°, and between the successive phases is placed 

the opposite phase, connected in the reverse direction. Thus 
the return conductors of phases 1,2, 3 of the upper layer, are 
shown in the lower layer as 1', 2', 3'; in the upper layer, above 
1', 2', 3', is placed again the phase 1, 2, 3, but connected in the 
reverse direction, and indicated as 1 , 2o, 3o. As 1 is connected 
in the reverse direction to 1, and 1' is the return of 1, lo is in 


phase with I', and the return of I..: I'o, is in the lower layer, in 

phase with, and beneath I. Tims the phase rotation is: 1,-3, 
2, -1,3, -2, 1, etc. 

For comparison, Fig. 57 .4 shows the usual quarter-phase 
winding, Q, of the same general type as the winding, Fig.' 57 C. 

If then the three third harmonics of 1, 2 and 3 are in phase 
with each other, for these third harmonics the true three-phase 
winding, T, gives the phase diagram shown as 7*» in Fig. 57 D. 
As seen, the current flows in one direction, single-phase, through- 
out the entire upper layer, and in the opposite direction in the 
lower layer, anil thus its magnetizing action neutralizes, that is, 
there can be no third' harmonic flux in the true three-phase 

The third harmonic diagram of the customary six-phase ar- 
rangement of three-phase winding, S, is shown as iS 3 in Fig. 57 
E. As seen, in this case alternately the single-phase third har- 
monic current flows in one direction for 60° or „. and in the 

opposite direction for the next „. In other words, a single-phase 
m.m.f. and single -phase flux exists, of three times as many poles 
:is the fundamental flux. 

Thus, with the usual three-phase induction-motor winding, 
a third harmonic in the voltage wave produces a single-phase 
triple harmonic flux of three times the number of motor poles, 
and this gives a single-phase motor-torque curve, that is, a torque 
which, starting with zero at standstill, increases to a maximum 
in positive direction or assisting, and then decreases again to zero 
at its synchronous speed, and above this, becomes negative as 
single-phase induction-generator torque. Triple frequency with 
three times the number of poles gives a synchronous speed of 
S = +}(>. That is, the third harmonic in a three-phase vol- 
tage may give a single-phase motor torque with a synchronous 
speed of one-ninth that of the fundamental torque, and in cither 
direction, as shown as Tj in dotted lines, in Fig. 56. 

As usually the third harmonic is absent in three-phase vol- 
tages, such a triple harmonic single-phase torque, as shown 
dotted in Fig. 56, is of rare occurrence: it could occur only in a 
four-wire three-phase system, that is, system containing the 
three phase-wires and the neutral. 

90. All the torque components produced by the higher har- 
monics of the voltage wave have the same number of motor poles 



as the fundamental (except the single-phase third harmonic 
above discussed, and its multiples, which have three times as 

/ \_ ! /_ 8INE 

- Q-0 

- S-0 



r £ V is \ 

II^L^,— r JZI s-Vs 

Q-'/i . J. \ 

Q-'A L- 


ZL s-'/t 


Fig. 58. — Current and flux distribution in induction-motor air gap, with 

different types of windings. 

many motor poles), but a lower synchronous speed, due to their 
higher frequency. 

Torque harmonics may also occur, having the fundamental 



frequency, but higher number of pairs of poles than the Funda- 
mental, and thus lower synchronous speeds, doe to the deviation 
of the space distribution of the motor winding from sine, 

The fundamental motor torque, I\, of Figs. 55 and 56, is given 
by ft sine wave of voltage and thus of flux, if the winding of each 
phase is distributed around the circumference of the motor air 
gap in a sinusoidal manner, as shown as F under " Sine," in Fig. 
58, and the flux distribution of each phase around the circum- 
ference of the air gap is sinusoidal also, as shown as * under 
"Sine," in Fig. 58. 

This, however, is never the ease, but the winding is always 
distributed in a non-sinusoidal manner. 

The space distribution of magnetizing force and thus of flux 
of each phase, along the c i re u inference of the motor air gap, 
thus can in tin' general case lie represented by a trigonometw 
series, with to as space angle, in electrical degrees, that is, counting 
a pair of poles as 2jt or 3ti0°. It is then: 

The distribution of the conductors of one phase, in the motor 
air gap: 

= Fa J COS oj -\- flj cos 3 tu + i 

; COS 5 C 


+ Or eos 7 n 
a eos 9 to + . 


hen (he assumption is made, that all the harmonics are in phase, 
that is, the magnetic distribution symmetrical. This is prac- 
tically always the case, and if it were not, it would simply add 
phase angle, a„, to the harmonics, the same as in paragraphs 88 
and 89, but would make no change in the result, as the component 
torque harmonics are independent of the phase relations between 
the harmonic and the fundamental, as seen below. 

In a quarter-phase motor, the second phase is located 90° 
or u) = - displaced in space, from the first phase, and thus 
represented by the expression: 

F'= PojoOfl(« - *) + » s cos(3 w - 3 2 ") + U( .cos(5u - 5 2 T ) 

- a 7 cos(7 w - ■*) + 
" V\ + "' c ° 9 f 3 w + ^ 

+ «;COS(7 

cos 9u - " 




Such a general or non-sinusoidal space distribution of magnetiz- 
ing force and thus of magnetic flux, as represented by F and F', 
can be considered as the superposition of a series of sinusoidal 
magnetizing forces and magnetic fluxes : 

as cos 5co 

m (5 « - D 


cos (• - J) 


a 7 cos 

a 3 cos 3 to 

The first component : 

a 3 cos (3 co + 1 j 
09 cos 9 co 

a 9 cos (9w- J 

cos CO, 

cos (co - p > 

a 5 cos 



gives the fundamental torque of the motor, as calculated in the 
customary manner, and represented by 7\ in F*igs. 55 and 56. 

The second component of space distribution of magnetizing 
force : 

a 3 cos 3 co, j 


a 3 cos 

( 3 »+3 

gives a distribution, which makes three times as many cycles 
in the motor-gap circumference, than (10), that is, corresponds 
to a motor of three times as many poles. This component of 
space distribution of magnetizing force would thus, with the 
fundamental voltage and current wave, give a torque curve 
reaching synchronism as one-third speed; with the third harmonic 
of the voltage wave, (11) would reach synchronism at one-ninth, 
with the fifth harmonic of the voltage wave at one-fifteenth of 
the normal synchronous speed. 

In (11), the sign of the second term is reversed from that in 
(10), that is, in (11), the space rotation is backward from that 
of (10). In other words, (11) gives a synchronous speed of 
S = —% with the fundamental or full-frequency voltage wave. 

The third component of space distribution : 

as cos 5 co, 1 

a 5 cos 



gives a motor of five times as many poles as (10), but with same 
space rotation as (10), and this component thus would give a 
torque, reaching synchronism at S = +^. 



In the same manner, the seventh space harmonic gives 
S «* —%, the ninth apace harmonic S = + }■$, etc. 

91. As seen, the component torque curves of the harmonies 
of the space distribution of magnetizing force and magnetic 
flux in the motor air gap, have the same characteristics as the 
component torque due to the time harmonics of the impioawd 
voltage wave, and thus are represented by the same torque 
diagrams : 

Fig. 55 for a quarter-phase motor, 

Fig. 56 for a three-phase motor. 

Here again, we see that the three-phase motor is less liable 
to irregularities in the torque curve, caused by higher harmonics, 
than the quarter-phase motor is. 

Two classes of harmonics thus may occur in the induction 
motor, and give component torques of lower synchronous speed: 

Time harmonies, that is, harmonics of the voltage wave, 
which are of higher frequency, but the same number of motor 
poles, and 

Space harmonics, that is, harmonics in the air-gap distribu- 
tion, which are of fundamental frequency, but of a higher number 
of motor poles. 

Compound harmonics, that is, higher space harmonics of 
higher time harmonics, theoretically exist, but their torque 
necessarily is already so small, that they can be neglected, except 
where they are intentionally produced in the design. 

We thus get the two elates of harmonics, and their 
characteristics : 

Quarttr- phair motor .- 
PKue tul»tion . 
Synchro noun tpwil 

Tim- II I F *«™™>' 
I No of polr* 

sp.«H(£ et r nr ! 

I No. ol polp» 

Phup rotation .... 



92. The space harmonics usually are more important than the 
time harmonics, as the space distribution of the winding in the 
motor usually materially differs from sinusoidal, while the devia- 
tion of the voltage wave from sine shape in modern electric power- 
supply systems is small, and the time harmonics thus usually 

The space harmonics can easily be calculated from the dis- 
tribution of the winding around the periphery of the motor air 
gap. (See "Engineering Mathematics," the chapter on the 
trigonometric series.) 

A number of the more common winding arrangements are 
shown in Fig. 58, in development. The arrangement of the 
conductors of one phase is shown to the left, under F, and the 
wave shape of the m.m.f. and thus the magnetic flux produced 
by it is shown under <& to the right. The pitch of a turn of the 
winding is indicated under F. 

Fig. 58 shows: 

Full-pitch quarter-phase winding: Q — 0. 

Full-pitch six-phase winding: S — 0. 

This is the three-phase winding almost always used in induction 
and synchronous machines. 

Full-pitch three-phase winding: T — 0. 

This is the true three-phase winding, as used in closed-circuit 
armatures, as synchronous converters, but of little importance 
in induction and synchronous motors. 
%> % and J^-pitch quarter-phase windings: 

Q - H; Q - «; Q - «. 

%l % and J^-pitch six-phase windings: 

S - Ve; S - }£; s-y 2 . 

%-pitch true three-phase windings: T — J^. 

As seen, the pitch deficiency, p, is denoted by the index. 
Denoting the winding, F, on the left side of Fig. 58, by the 
Fourier series: 

F = Fo (cos co + a 3 cos 3 co + a 6 cos 5 co + a 7 cos 7 co + . . . ). (13) 
It is, in general : 


Foa n = - I F cosncodw. 

If, then: p = pitch deficiency, 

q = number of phases 




(four with quarter-phase, Q, six with six-phase, 5, three with 
three-phase, T); 

any fractional pitch winding then consists of the superposition 
of two layers: 


From w = to co = + rt - > 

q 2 

from cj = to co = — jr-t 

q 2 

and the integral (14) become: 

r , Pw v Vt 

4F| f« 2 A" 2 

f ofln = - I <*os wcorfco + I cos ncodco 

?r IJo Jo 

= - ■ ; sin n\ + ~ ) + sin /<( r ~- J :■ 

717T I \(J 2 / \<7 2 / J 

8F . 717T »/?7T 

= — sin — cos ■ > 
q 2 



as for: n = 1; a n = 1, it is, substituted in (15) 


.IT VTT ' 

sin cos r rt 
9 2 

hence, substituting (16) into (15) : 

o„ = 

. nir pur 

sin — cos n 

_ 9_ _. 2 

sin - cos - 
q 2 

For full-pitch winding: 

p = 0. 

It is, from (17): 



«»° = 






and for a fractional-pitch winding of pitch deficiency, p, it thus is : 


cos 2 
a„ = a„° — • (19) 

cos T 

93. By substituting the values: q = 4, 6, 3 and p = 0, %, 
Hj %} i n to equation (17), we get the coefficients a n of the 
trigonometric series: 

F = Fq { cos co + a 3 cos 3 co + a 6 cos 5 co + a 7 cos 7w+ . . . } , 


which represents the current distribution per phase through the 
air gap of the induction machine, shown by the diagrams F of 
Fig. 58. 

The corresponding flux distribution, $, in Fig. 58, expressed by 
a trignometric series: 

<fr = $o {sin o> + 63 sin 3 w + 6 5 sin 5 co + 67 sin 7 co + . . . | 


could be calculated in the same manner, from the constructive 
characteristics of $ in Fig. 58. 

It can, however, be derived immediately from the consideration, 
that $ is the. summation, that is, the integral of F: 


= J>dco (22) 

and herefrom follows: 

b n = * (23) 

and this gives the coefficients, b n , of the series, 4>. 

In the following tables are given the coefficients a n and b n , 
for the winding arrangements of Fig. 58, up to the twenty-first 

As seen, some of the lower harmonics are very considerable 
thus may exert an appreciable effect on the motor torque at low 
speeds, especially in the quarter-phase motor. 






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94. Occasionally two or more induction motors are operated 
in parallel on the same load, as for instance in three-phase rail- 
roading, or when securing several speeds by concatenation. 
In this case the secondaries of the induction motors may be 
connected in multiple and a single rheostat used for starting 
. and speed control. Thus, when using two motors in concatena- 
tion for speeds from standstill to half synchronism, from half 
synchronism to full speed, the motors may also be operated on 
a single rheostat by connecting their secondaries in parallel. 
As in parallel connection the frequency of the secondaries must 
be the same, and the secondary frequency equals the slip, it 
follows that the motors in this case must operate at the same slip, 
that is, at the same frequency of rotation, or in synchronism with 
each other. If the connection of the induction motors to the 
load is such that they can not operate in exact step with each 
other, obviously separate resistances must be used in the motor 
secondaries, so as to allow different slips. When rigidly connect- 
ing the two motors with each other, it is essential to take care 
that the motor secondaries have exactly the same relative posi- 
tion to their primaries so as to be in phase with each other, just 
as would be necessary when operating two alternators in parallel 
with each other when rigidly connected to the same shaft or 
when driven by synchronous motors from the same supply. 
As in the induction-motor secondary an e.m.f. of definite fre- 
quency, that of slip, is generated by its rotation through the 
revolving motor field, the induction-motor secondary is an 
alternating-current generator, which is short-circuited at speed 
and loaded by the starting rheostat during acceleration, and the 
problem of operating two induction motors with their secondaries 
connected in parallel on the same external resistance is thus the 
same as that of operating two alternators in parallel. In general, 
therefore, it is undesirable to rigidly connect induction-motor 
secondaries mechanically if they are electrically connected in 
parallel, but it is preferable to have their mechanical connection 



sufficiently flexible, as by belting, etc., bo that the motors can 
drop into exact step with each other and maintain step by their 
synchronising power. 

It is of interest, then, to examine the synchronizing power of 
two induction motors which are connected in multiple with 
their secondaries on the same rheostat and operated from the 
same primary impressed voltage. 

95. Assume two equal induction motors with their primaries 
connected to the same voltage, supply and with llieir seeondarioi 
connected in multiple with each other to a common resistance, 
r, and neglecting for simplicity the exciting current and the vol- 
tage drop in the impedance of the motor primaries as not mate- 
rially affecting the synchronizing power. 

Let Zi — n + ./j-t = secondary self-inductive impedance at 
full frequency; s = slip of the two motors, as fraction of syn- 
chronism; Co = absolute value of impressed voltage and thus, 
when neglecting the primary impedance, of the voltage generated 
iu the primary by the rotating field. 

If then Ihe two motor secondaries are oul of phase with each 
• if her by angle 2 r, ami the secondary of I he motor 1 is behind in 
the direction of rotation mid the secondary of the motor 2 
ahead of [he average position by angle r. then: 

#i = sco (cost -+- jsinr) = secondary generated 

e.m.f. of the first motor, (1) 

E 3 = scij (cos t — j sin t) = secondary generated 

e.m.f. of the second motor. (21 

And if /i = current coming from the first, 1-. = current coming 
from the second motor secondary, Hie total current, or currenl 
in (he external resistance, r, is; 

/ - /, + I,: (8) 

it is then, in the circuit comprising the first motor secondary 
and the rheostat, <', 

{?, - [ t Z - fr = 0, (4) 

in the circuit comprising (lie second motor secondary and the 
rheostat, r, 

E, - [,Z - {r = (), 

z = n + jsxi; 

£T3Hmxaenzn& JX3:vr? 4 xv jip^*r»v«^ t*x 

« ;jr rff»r**n 

s. - j-. z - - - <v = *. 

fc — i-r — /. Z — - « -A 

£, - (■ i - /-.-/& Z - iV 

'---'«- Z-£ 

and v ^ 

/: - /, = z 

for convenience the abbreviations. 

y = *\ = 0i — ji>i. 


into equations (6) and substituting ill and v~^ into itf\ give**: 

/i + /* = 2 ,nyo Y cos r % 

Ii — I* = + 2j*f*Y\ sin r; V^ 


/V = sf |i*eosr + jY\ sin r| 0^ 

is the current in the secondary circuit of the motor, and there- 
fore also the primary load current, that is, the primary current 
corresponding to the secondary current, ami thus, when neg- 
lecting the exciting current, also the primary motor current, 
where the upper sign corresponds to the first, or lagging, the 
lower sign to the second, or leading, motor. 
Substituting in (9) for 1" and Y\ gives: 

/V = se { (g cos t ± bi sin r) — j (b cos t "I f/i sin r) | , (1(1) 
the primary e.m.f. corresponding hereto is: 

£Y = e (cos t J j sin r ) , (II) 

where again the upper sign corresponds to the first, the lower to 

the second motor. 

The power consumed by the current, / 2 ', with the e.m.f., Ij!%\ 


is the sum of the products of the horizontal components, and 
of the vertical components, that is, of the real components and 
of the imaginary components of these two quantities (as a 
horizontal component of one does not represent any power with 
a vertical component of the other quantity, being in quadrature 

where the brackets denote that the sum of the product of the 
corresponding parts of the two quantities is taken. 

As discussed in the preceding, the torque of an induction motor, 
in synchronous watts, equals the power consumed by the primary 
counter e.m.f.; that is: 

2V = /Y, 

and substituting (10) and (11) this gives: 

D% 1 = se 2 {cost (g cost ± 6i sin r) + sin t (6 cos r + gr 7 sin r)\ 


, 8eo 2 (*+_! _ ?i-^cos 2t ± bl "^sin 2t 

and herefrom follows the motor output or power, by multiplying 
with (1 — s). 
The sum of the torques of both motors, or the total torque, is: 

2 D t = 2>i + D 2 = se 2 {(gi + g) - (gfi - g) cos 2t}. (13) 
The difference of the torque of both motors, or the synchroniz- 
ing torque, is: 

2D, = se 2 (6i - b) sin 2 t, (14) 

where, by (7), 

, sxi u sxi } (15) 

nil = ri 2 + s 2 Xi 2 , 

In those equations primary exciting current and primary 
impedance are neglected. The primary impedance can be intro- 
duced in the equations, by substituting (n + sr n ) for r l} and 
(xi + T<>) for X\, in the expression of Mi and m, and in this case 
only the exciting current is neglected, and the results are suffi- 
ciently accurate for most purposes, except for values of speeel 

g = 



6 = 



m = 

(r, + 2 

r) 2 


S 2 Xi 2 , 


very close to synchronism, where the motor current is appreciably 
increased by the exciting current. It is, then: 

TOi = (ri + r* ) 2 + s 2 (*i + -To) 5 , 
m = ( ri + sr Q + 2r) J + «* (x, + Jo) 5 : 

all the other equations remain the same. 
From (15) and (16) follows 

61 — b _ 2srxi (r x + «r + r) 
2 mm 1 



hence, is always positive. 

96. (61 — b) is always positive, that is, the synchronizing 
torque is positive in the first or lagging motor, and negative in 
the second or leading motor; that is, the motor which lags in 
position behind gives more power and thus accelerates, while the 
motor which is ahead in position gives less power and thus 
drops back. Hence, the two motor armatures pull each other 
into step, if thrown together out of phase, just like two alternators. 

The synchronizing torque (14) is zero if t = 0, as obvious, 
as for r = both motors are in step with each other. The syn- 
chronizing torque also is zero if r = 90°, that is, the two motor 
armatures are in opposition. The position of opposition is 
unstable, however, and the motors can not operate in opposition, 
that is, for t = 90°, or with the one motor secondary short- 
circuiting the other; in this position, any decrease of t below 
90° produces a synchronizing torque which pulls the motors 
together, to r = 0, or in step. Just as with alternators, there 
thus exist two positions of zero synchronizing power — with the 
motors in step, that is, their secondaries in parallel and in phase, 
and with the motors in opposition, that is, their secondaries in 
opposition — and the former position is stable, the latter unstable, 
and the motors thus drop into and retain the former position, 
that is, operate in step with each other, within the limits of their 
synchronizing power. 

If the starting rheostat is short-circuited, or r = 0, it is, by 
(15), 61 = by and the synchronizing power vanishes, as is obvious, 
since in this case the motor secondaries are short-circuit (id and 
thus independent of each other in their frequency and speed. 

With parallel connection of induction-motor armatures a syn- 
chronizing power thus is exerted between the motors as long 
as any appreciable resistance exists in the external circuit, and 


the motors thus tend to keep in step until the common starting 
resistance is short-circuited and the motors thereby become inde- 
pendent, the synchronizing torque vanishes, and the motors can 
slip against each other without interference by cross-currents. 

Since the term — ^ — contains the slip, s, as factor, the syn- 
chronizing torque decreases with increasing approach to syn- 
chronous speed. 





































9 . — Sy n ch roniz in g 

r r = 0, or with 
(15), and (16): 

induction motors: motor torque and ay 

the motors in step with each othe 
s<V (Tl + 2 r) 



r, it is, by 









r i 










(ri + «r + 2D= + .;' (i, +x ) 
lue as found for a single motor. 
jn to both motors, for each moto 

unstable positions of the motors, 

sc *ri 
"' (r, +«-,)*+ 8* ( Xl +JV-' 
lue as the motor would give w 



As the 



circuited armature. This is to be expected, as the two motor 
armatures short-circuit each other. 

The synchronizing torque is a maximum for r = 45°, and is, 
by (14), (15), and (16): 

i), = ^ 6l ~ 6 . (20) 

As instances are shown, in Fig. 59, the motor torque, from 
equation (18), and the maximum synchronizing torque, from 
equation (20), for a motor of 5 per cent, drop of speed at full- 
load and very high overload capacity (a maximum power nearly 
two and a half times and a maximum torque somewhat over 
three times the rated value), that is, of low reactance, as can be 
produced at low frequency, and is desirable for intermittent 
service, hence of the constants : 

Zx = Zo = i+i, 

Y = 0.005 - 0.02 i, 
e = 1000 volts, 

for the values of additional resistance inserted into the armatures: 

r = 0; 0.75; 2; 4.5, 
giving the values: 

1 l + 2r 

p — g — --._, 

Wi m 

, 2« , 8Xi 

Oi = — , b = -, 

Wi m 

mx = (1 + s) 2 + 4 a 2 , m = (1 + s + 2 r) 2 + 4 s\ 

As seen, in this instance the synchronizing torque is higher 
than the motor torque up to half speed, slightly below the motor 
torque between half speed and three-quarters speed, but above 
three-quarters speed rapidly drops, due to the approach to syn- 
chronism, and becomes zero when the last starting resistance 
is cut out. 


97. The typical induction motor consists of one or a number 
df primary circuits acting upon an armature movable thereto, 
which contains a number of closed secondary circuits, displaced 
from each other in space so as to offer a resultant closed secondary 
circuit in any direction and at any position of the armature or 
secondary, with regards to the primary system. In consequence 
thereof the induction motor can be considered as a transformer, 
having to each primary circuit a corresponding secondary cir- 
cuit — a secondary coil, moving out of the field of the primary 
coil,* being replaced by another secondary coil moving into the 

In such a motor the torque is zero a) synchronism, positive 
below, and negative above, synchronism. 

If, however, the movable armature contains one closed cir- 
cuit only, it offers a closed secondary circuit only in the direc- 
tion of the axis of the armature coil, but no secondary circuit at 
right angles therewith. That is, with the rotation of the arma- 
ture the secondary circuit, corresponding to a primary circuit, 
varies from short-circuit at coincidence of the axis of the arma- 
ture coil with the axis of the primary coil, to open-circuit in 
quadrature therewith, with the periodicity of the armature 
speed. That is, the apparent admittance of the primary circuit 
varies periodically from open-circuit admittance to the short- 
circuited transformer admittance. 

At synchronism such a motor represents an electric circuit of 
an admittance varying with twice the periodicity of the primary 
frequency, since twice per period the axis of the armature coil 
and that of the primary coil coincide. A varying admittance 
is obviously identical in effeel with a varying reluctance, which 
will be discussed in the chapter on reaction machines. That 
is, the induction motor with one closed armature circuit is, at 
synchronism, nothing but a reaction machine, and consequently 
gives zero torque at synchronism if the maxima and minima of 
the periodically varying admittance coincide with the 


and zero values of the primary circuit, but gives a definite torque 
if they are displaced therefrom. This torque may be positive 
or negative according to the phase displacement between ad- 
mittance and primary circuit; that is, the lag or lead of the 
maximum admittance with regard to the primary maximum. 
Hence an induction motor with single-armature circuit at syn- 
chronism acts either as motor or as alternating-current generator 
according to the relative position of the armature circuit with 
respect to the primary circuit. Thus it can be called a syn- 
chronous induction motor or synchronous induction generator, 
since it is an induction machine giving torque at synchronism. 

Power-factor and apparent efficiency of the synchronous in- 
duction motor as reaction machine are very low. Hence it is 
of practical application only in cases where a small amount of 
power is required at synchronous rotation, and continuous current 
for field excitation is not available. 

The current produced in the armature of the synchronous 
induction motor is of double the frequency impressed upon the 

Below and above synchronism the ordinary induction motor, 
or induction generator, torque is superimposed upon the syn- 
chronous-induction machine torque. Since with the frequency 
of slip the relative position of primary and of secondary coil 
changes, the synchronous-induction machine torque alternates 
periodically with the frequency of slip. That is, upon the con- 
stant positive or negative torque below or above synchronism 
an alternating torque of the frequency of slip is superimposed, 
and thus the resultant torque pulsating with a positive mean 
value below, a negative mean value above, synchronism. 

When started from rest, a synchronous induction motor will 
accelerate like an ordinary single-phase induction motor, but 
not only approach synchronism, as the latter does, but run up 
to complete synchronism under load. When approaching syn- 
chronism it makes definite beats with the frequency of slip, which 
disappear when synchronism is reached. 



98. In it revolving magnetic field, a circular iron disk, or 
iron cylinder of uniform magnetic reluctance in the direction of 
the revolving field, is set in rotation, even if subdivided so as to 
preclude the production of eddy currents. Thin rotation is due 
to the effect of hysteresis of the revolving disk or cylinder, and 
such a motor may thus be called a hysteresis motor. 

Let / be the iron disk exposed to a rotating magnetic field 
or resultant m.m.f. The axis of resultant magnetization in the 
disk, /, does not coincide with the axis of the rotating field, but 
lags behind the latter, thus producing a couple. That is, the 
component of magnetism in a direction of the rotating disk, /, 
ahead of the axis of rotating m.m.f., is rising, thus below, and 
in a direction behind the axis of rotating m.m.f. decreasing, that 
is, above proportionality with the m.m.f., in consequence of the 
lag of magnetism in the hysteresis loop, and thus the axis of 
resultant magnetism in the iron disk, /, does not coincide with 
the axis of rotating m.m.f., but is shifted backward by an angle, 
«, which is the angle of hysteretic lead. 

The induced magnetism gives with the resultant m.m.f. a 
mechanical couple: 

D = mS'b sin a, 

S = resultant m.m.f., 
4> = resultant magnetism, 
« = angle of hysteretic advance of phase, 
m = a constant. 
The apparent or volt-ampere input of the motor is: 

P ■ wiS*. 
Thus the apparent torque efficiency: 

Q = volt-ampere input, 


and the power of the motor is: 

P = (1 - s) D = (1 - s) m$$ sin a, 

s = slip as fraction of synchronism. 

The apparent efficiency is: 


n = (1 — *) sin a. 

Since in a magnetic circuit containing an air gap the angle, 
a, is small, a few degrees only, it follows that the apparent 
efficiency of the hysteresis motor is low, the motor consequently 
unsuitable for producing large amounts of mechanical power. 

From the equation of torque it follows, however, that at 
constant impressed e.m.f., or current — that is, constant SF — 
the torque is constant and independent of the speed; and there- 
fore such a motor arrangement is suitable, and occasionally used 
as alternating-current meter. 

For s<0, we have a < 0, 

and the apparatus is an hysteresis generator. 

99. The same result can be reached from a different point 
of view. In such a magnetic system, comprising a movable 
iron disk, 7, of uniform magnetic reluctance in a revolving 
field, the magnetic reluctance — and thus the distribution of 
magnetism — is obviously independent of the speed, and conse- 
quently the current and energy expenditure of the impressed 
m.m.f. independent of the speed also. If, now: 

V = volume of iron of the movable part, 
(B = magnetic density, 

rj = coefficient of hysteresis, 

the energy expended by hysteresis in the movable disk, 7, is 
per cycle: 

Wo = V V ® 1 \ 

hence, if / = frequency, the power supplied by the m.m.f. to 
the rotating iron disk in the hysteretic loop of the m.m.f. is: 

p =/Fi ? (B , - e . 

At the slip, sfj that is, the speed (1 — s) f, the power expended 
by hysteresis in the rotating disk is, however: 

Pi = s/FtjCB 1 - 6 . 


Hence, in the transfer from the stationary to the revolving 
member the magnetic power: 

has disappeared, and thus reappears as mechanical work, ami 
the torque is: 

D = 

(1 -«)/ 

. IV 

that is, independent of the speed. 

Since, as seen in " Theory and Calculation of Alternating-cur- 
rent Phenomena," Chapter XII, sin a is the ratio of the energy 
of the hysteretic loop to the total apparent energy of the mag- 
netic cycle, it follows that the apparent efficiency of such a motor 
can never exceed the value (1 — s) sin a, or a fraction of the 
primary hysteretic energy. 

The primary hysteretic energy of an induction motor, as repre- 
sented by its conductance, ij, being a part of the loss in the 
motor, and thus a very small part of its output only, it follows 
that the output of a hysteresis motor is a small fraction only of 
the output which the same magnetic structure could give with 
secondary short-circuited winding, as regular induction motor. 

As secondary effect, however, the rotary effort of the magnet ic 
structure as hysteresis motor appears more or less in all induction 
motors, although usually it. is so small as in be neglected. 

However, with decreasing size of the motor, the torque of the 
hysteresis motor decreases at a lesser rate than that of the in- 
duction motor, so that for extremely small motors, the torque 
as hysteresis motor is comparable with that as induction motor. 

If in the hysteresis motor the rotary iron structure has imi 
uniform reluctance in all directions — but is, for instance, bar- 
shaped or shuttle-shaped — on the hysteresis-motor effect is 
superimposed the effect of varying magnetic reluctance which 
tends to bring the motor to synchronism, and maintain it 
therein, as shall be more fully investigated under "Reaction 
Machine" in Chapter XVI. 

100. In the hysteresis motor, consisting of an iron disk of 
uniform magnetic reluctance, which revolves in a uniformly 
rotating magnetic field, below synchronism, the magnetic mix 
rotates in the armature with the frequency of slip, and the 
resultant line of magnetic induction in the disk thus lags, in 
space, behind the synchronously rotating line of resultant m.m.f 


of the exciting coils, by the angle of hysteretic lead, or, which is 
constant, and so gives, at constant magnetic flux, that is, con- 
stant impressed e.m.f., a constant torque and a power propor- 
tional to the speed. 

Above synchronism, the iron disk revolves faster than the 
rotating field, and the line of resulting magnetization in the disk 
being behind the line of m.m.f. with regard to the direction of 
rotation of the magnetism in the disk, therefore is ahead of it in 
space, that is, the torque and therefore the power reverses at 
synchronism, and above synchronism the apparatus is an 
hysteresis generator, that is, changes at synchronism from motor 
to generator. At synchronism such a disk thus can give me- 
chanical power as motor, with the line of induction lagging, or 
give electric power as generator, with the line of induction 
leading the line of rotation m.m.f. 

Electrically, the power transferred between the electric cir- 
cuit and the rotating disk is represented by the hysteresis loop. 
Below synchronism the hysteresis loop of the electric circuit 
has the normal shape, and of its constant power a part, propor- 
tional to the slip, is consumed in the iron, the other part, pro- 
portional to the speed, appears as mechanical power. At syn- 
chronism the hysteresis loop collapses and reverses, and above 
synchronism the electric supply current so traverses the normal 
hysteresis loop in reverse direction, representing generation of 
electric power. The mechanical power consumed by the 
hysteresis generator then is proportional to the speed, and of 
this power a part, proportional to the slip above synchronism, 
is consumed in the iron, the other part is constant and appears 
as electric power generated by the apparatus in the inverted 
hysteresis loop. 

This apparatus is of interest especially as illustrating the 
difference between hysteresis and molecular magnetic friction: 
the hysteresis is the power represented by the loop between 
magnetic induction and m.m.f. or the electric power in the 
circuit, and so may be positive or negative, or change from the 
one to the other, as in the above instance, while molecular mag- 
netic friction is the power consumed in the magnetic circuit by 
the reversals of magnetism. Hysteresis, therefore, is an electrical 
phenomenon, and is a measure of the molecular magnetic fric- 
tion only if there is no other source or consumption of power in 
the magnetic circuit. 



101. A single-phase induction motor, giving full torque at 
starting and at any intermediate speed, by means of leading the 
supply current into the primary motor winding through brushes 
moving on a segmental commutator connected to the primary 

Diagram of rotary terminal aingle-plia-w induction motor. 

winding, was devised and built by II. Eickemeyer in 1891, and 
further work thereon done later in Germany, but never was 
brought into commercial use. 

Let, in Fig. 60, P denote the primary stator winding of a single- 
phase induction motor, S the revolving squirrel -cage secondary 
winding. The primary winding is arranged as a ring (or drum) 
Winding and connected to a stationary commutator, C. The 
single-phase supply current is led into the primary winding, P, 
through two brushes bearing on the two (electrically) opposite 


points of the commutator, C These brushes, B, are arranged so 
that they can be revolved. 

With the brushes, B, at standstill on the stationary commutator, 
C, the rotor, S, has no torque, and the current in the stator, P, is 
the usual large standstill current of the induction motor. If now 
the brushes, B, are revolved at synchronous speed, /, in the direc- 
tion shown by the arrow, the rotor, S, again has no torque, but 
the stator, P, carries only the small exciting current of the motor, 
and the electrical conditions in the motor are the same, as would 
be with stationary brushes, B, at synchronous speed of the rotor, 
S. If now the brushes, B, are slowed down below synchronism, 
/, to speed, /i, the rotor, S, begins to turn, in reverse direction, as 
shown by the arrow, at a speed, / 2 , and a torque corresponding 
to the slip, 8 = / — (/i + / 2 ). 

Thus, if the load on the motor is such as to require the torque 
given at the slip, s, this load is started and brought up to full 
speed, / — 8 f by speeding the brushes, B, up to or near synchronous 
speed, and then allowing them gradually to come to rest: at brush 
speed, /i = / — s, the rotor starts, and at decreasing, f h accelr- 
ates with the speed / 2 = / — s — /i, until, when the brushes 
come to rest: f\ = 0, the rotor speed is / 2 = / — s. 

As seen," the brushes revolve on the commutator only in start- 
ing and at intermediate speeds, but are stationary at full speed. 
If the brushes, B, are rotated at oversynchronous speed: /i>/, 
the motor torque is reversed, and the rotor turns in the same 
direction as the brushes. In general, it is: 

/i+/ 2 + s=/, 

/i = brush speed, 
/ 2 = motor speed, 

s = slip required to give the desired torque, 
/ = supply frequency. 

102. An application of this type of motor for starting larger 
motors under power, by means of a small auxiliary motor, is 
shown diagrammatically, in section, in Fig. 61. 

Po is the stationary primary or stator, So the revolving squirrel- 
cage secondary of the power motor. The stator coils of P 
connect to the segments of the stationary commutator, Co, 
which receives the single-phase power current through the 
brushes, B . 



These brushes, B v , are carried by the rotating squirrel -cage 
secondary, Si, of a small auxiliary motor. The primary of this. 
Pi, is mounted on thr power shaft, A, of the main motor, and 
carries the commutator, Cj, which receives current from the 
brushes, B,. 

These brushes are speeder! up t<> or near synchronism by some 
means, as hand wheel, H, and gears, G, and then allowed in slow 
down. Assuming the brushes were rotating in 
wise direction, Then, while they are slowing down, the (ex- 
ternal) squirrel-cage rotor. .Si, of the auxiliary motor start* tad 

. (il, — Rotary terminal an nip-phase inriiu-tiuii motor with i trolling 

s[ da up, in clockwise direction, and while the brushes, B,, 

come to rest, .S', comes up to full speed, and thereby brings the 
brushes, B v , of the power motor up to speed in clockwise rotation. 
As soon as Bo has reached sufficient speed, the power motor gets 
torque and its rotor, So, starts, in counter-clockwise rotation. 
As So carries Pi, with increasing speed of So and P, t Bj and with 
il I lie brushes, B„, slow down, until full speed of the power motor, 
So, is reached, the brushes. B B , stand still, anil the brushes, B u 
by their friction on the commutator, <",, revolve together with 
f„ /*, and 8+ 

In whichever direction the brushes, B,. are Btarted, in the 
same direction starts Ihe main motor, So. 


If by overload the main motor, So, drops out of step and slows 
down, the slowing down of Pi starts Si, and with it the brushes, 
Bo, at the proper differential speed, and so carries full torque 
down to standstill, that is, there is no actual dropping out of 
the motor, but merely a slowing down by overload. 

The disadvantage of this motor type is the sparking at the 
commutator, by the short-circuiting of primary coils during the 
passage of the brush from segment to segment. This would 
require the use of methods of controlling the sparking, such as 
used in the single-phase commutator motors of the series type, 
etc. It was the difficulty of controlling the sparking, which 
side-tracked this type of motor in the early days, and later, with 
the extensive introduction of polyphase supply, the single-phase 
motor problem had become less important. 



103. In general, an alternating-current transformer conafete of 
a magnetic circuit, interlinked with two electric circuits or sets 
of electric circuits, the primary circuit, in which power, sup- 
plied by the impressed voltage, is consumed, and the secondary 
circuit, in which a corresponding amount of electric power 
is produced; or in other words, power is transferred through 
space, by magnetic energy, from primary to secondary circuit. 
This power finds its mechanical equivalent in a repulsive llirusi 
acting between primary and secondary conductors. Thus, if 
the secondary is not held rigidly, with regards to the primary, 
it will be repelled and move. This repulsion is used in the 
constant-current transformer for regulating the current for 
constancy independent of the load. In the induction motor, 
this mechanical force is made use of for doing the work: the 
induction motor represents an alternating-current transformer, 
in which the secondary is mounted niovably with regards to 
the primary, in such a manner that, while set in motion, it still 
remains in the primary field of force. This requires, i hat the 
induction motor field is not constant in one direction, but that 
a magnetic field exists in every direction, in other words that 
the magnetic field successively assumes all directions, as a so- 
called rotating field. 

The induction motor and the stationary transformer thus are 
merely two applications of the same structure, the former using 
the mechanical thrust, the latter only the electrical power 
transfer, and both thus are special cases of what may be called 
the "general alternating-current transformer," in which both, 
power and mechanical motion, are utilized. 

The general alternating-current transformer thus consist* of 
a magnetic circuit interlinked with two sets of electric circuits, 
the primary and the secondary, which are mounted rotatably 
with regards to each other. It transforms between primary 
electrical and secondary electrical power, and also between 


electrical and mechanical power. As the frequency of the re- 
volving secondary is the frequency of slip, thus differing from 
the primary, it follows, that the general alternating-current 
transformer changes not only voltages and current, but also 
frequencies, and may therefore be called "frequency converter." 
Obviously, it may also change the number of phases. 

Structurally, frequency converter and induction motor must 
contain an air gap in the magnetic circuit, to permit movability 
between primary and secondary, and thus they require a higher 
magnetizing current than the closed magnetic circuit stationary 
transformer, and this again results in general in a higher self- 
inductive impedance. Thus, the frequency converter and in- 
duction motor magnetically represent transformers of high ex- 
citing admittance and high self-inductive impedance. 

104. The mutual magnetic flux of the transformer is pro- 
duced by the resultant m.m.f. of both electric circuits. It is 
determined by the counter e.m.f., the number of turns, and the 
frequency of the electric circuit, by the equation : 

E = V2 rfnQ 10" 8 , 


E = effective e.m.f., 
/ = frequency, 
n = number of turns, 
$ = maximum magnetic flux. 

The m.m.f. producing this flux, or the resultant m.m.f. of 
primary and secondary circuit, is determined by shape and 
magnetic characteristic of the material composing the magnetic 
circuit, and by the magnetic induction. At open secondary 
circuit, this m.m.f. is the m.m.f. of the primary current, which 
in this case is called the exciting current, and consists of a 
power component, the magnetic power current, and a reactive 
component, the magnetizing current. 

In the general alternating-current transformer, where the 
secondary is movable with regard to the primary, the rate of 
cutting of the secondary electric circuit with the mutual mag- 
netic flux is different from that of the primary. Thus, the fre- 
quencies of both circuits are different, and the generated e.m.fs. 
are not proportional to the number of turns as in the stationary 

transformer, but to the product of number of turns into frequency. 


105. Let, in a general alternating-current transformer: 

.. secondary . tl ,. „ 

« = ratio — : - frequency, or "slip : 

primary n r 

thus, if: 

/ = primary frequency, or frequency of impressed e.m.f., 

sf = secondary frequency; 

and the e.m.f. generated per secondary turn by the mutual flux 
has to the e.m.f. generated per primary turn the ratio, «, 

s = represents synchronous motion of the secondary; 

s < represents motion above synchronism — driven by external 

mechanical power, as will be seen; 
8 = 1 represents standstill; 
s > 1 represents backward motion of the secondary, 

that is, motion against the mechanical force acting between 
primary and secondary (thus representing driving by external 
mechanical power). 

n = number of primary turns in series per circuit; 
n x = number of secondary turns in series per circuit; 

a = = ratio of turns; 


Y = g — jb = primary exciting admittance per circuit; 


g = effective conductance; 
b = susceptance; 
Zq = r + jxo = internal primary self-inductive impedance 
per circuit, 


r = effective resistance of primary circuit; 
Xq = self-inductive reactance of primary circuit; 
Zn = n + jx\ = internal secondary self-inductive im- 
pedance per circuit at standstill, or for « = 1, 


r x = effective resistance of secondary coil; 
Xi = self-inductive reactance of secondary coil at stand- 
still, or full frequency, s = 1. 


Since the reactance is proportional to the frequency, at the 
slip, 8, or the secondary frequency, sf, the secondary impedance 

Zi = ri + jsxi. 

Let the secondary circuit be closed by an external resistance, 
r, and an external reactance, and denote the latter by x at 
frequency, /, then at frequency, «/, or slip, s, it will be = *x, and 

Z = r + jsx = external secondary impedance. 1 

#o = primary impressed e.m.f. per circuit, 
J$' = e.m.f. consumed by primary counter e.m.f., 
#i = secondary terminal e.m.f., 
#\ = secondary generated e.m.f., 
e = e.m.f. generated per turn by the mutual magnetic 
flux, at full frequency, /, 
/o = primary current, 
/oo = primary exciting current, 
/i = secondary current. 

It is then: 

Secondary generated e.m.f. : 

#'i = sriie. 

Total secondary impedance: 

Zi + Z = (n + r) +js(xi + x); 

hence, secondary current: 

T E\ _ snie 

/i ~ v T 7z - 

Zi + Z (n + r) + js (X! + x) 

1 This applies to the case where the secondary contains inductive react- 
ance only; or, rather, that kind of reactance which is proportional to the 
frequency. In a condenser the reactance is inversely proportional to the 
frequency, in a synchronous motor under circumstances independent of the 
frequency. Thus, in general, we have to set, x = x' -f x" + s'", where x' 
is that part of the reactance which is proportional to the frequency, x" that 
part of the reactance independent of the frequency, and x'" that part of the 
reactance which is inversely proportional to the frequency; and have thus, 

at slip, *, or frequency, */, the external secondary reactance, sx' -f x" -f 

x ,n 



Secondary terminal voltage : 

#i = #'i ~ JiZi = fiZ 

= N r t + jgxi I = sntf ( r + jsx) 

1 1 0i + r) + js (xi + x) J \r x + r) + j* (xi + x) 

e.m.f. consumed by primary counter e.m.f. 

$' = n e; 
hence, primary exciting current: 

/oo = #'Fo = no« (flf - jb). 

Component of primary current corresponding to secondary 
current, /\: 

aMCri + O+^Cxi + x)}' 
hence, total primary current: 

/o = /oo + / 

f 1 1 , g - jb 

lfl2 (n + r) + js(xi + x) « 

Primary impressed e.m.f. : 

$o = E' + /oZo 

I a 2 (ri + r) + js (xi + x) J J 

We get thus, as the 
Equations of the General Alternating-current Transformer, of 
ratio of turns, a; and ratio of frequencies, s; with the e.m.f. 
generated per turn at full frequency, e, as parameter, the values: 

Primary impressed e.m.f. : 

ft = «oe { 1 + £ ( - T -^ £L-_ + (r . + ,*„) (, - jb) } . 

Secondary terminal voltage: 
et ft n + jsxi l r + jsx 

1 (n + r) + js (xi + x) J Oi+r) +js fo + x) 

Primary current : 

io = Sttoe ^ - 2 7 , — r 7 — -7 . r + - r 

1 a 2 (ri + r) + js (xi + x) s J 


Secondary current: 

T stiie 

(ri + r) + js (xi + x) 

Therefrom, we get: 
Ratio of currents: 

r - \ I l + t ^ " #> l < ri + r) + i* (atl + *>' r 

Ratio of e.m.fs. : 

* * fl +- 2 ? ■ ?? w° . % + ( r ° + J*>) (g - jb) 
Eo a* a 2 (ri + r) + js (xi + x) 

#1 "~ * - _ r t + jsxi_ 

(ri + r) + js (xi + x) 

Total apparent primary impedance: 

Z« = ^ = ?8 {(ri + r)+i*(ae 1 +*)} 

/ n # 

1 + -it -L ?-^~w v— n + (ro + J*o) (ff - j"6) 
o^ (r, + r) + js (j i + x) > f 

1 + v (9 ~ Jb) l(ri + r) + js (*, + *)] 



, , x" x'" 

x = x' + — + 

8 9 


in the general secondary circuit as discussed in footnote, page 179. 
Substituting in these equations : 

* = 1, 

gives the 

General Equations of the Stationary Alternating-current Transformer 
Substituting in the equations of the general alternating-current 
transformer : 

Z = 0, • 

gives the 

General Equations of the Induction Motor 


(ri + r) 2 + s 2 (Xi + x) 2 = z k \ 


and separating the real and imaginary quantities: 

#o = no6 J [l + -*—; (r (ri + r) + *x (xi + x)) + (rtf + xjb) J 
- 3 \J^i W*i + *) - * (ri + r)) + (rob - Xotf)] J , 

/§ - ^ I Lis^ + a J - 4~iv + iJr 

/i = ^ {(ri + r) - j* (xi + x) 

Neglecting the exciting current, or rather considering it as 
a separate and independent shunt circuit outside of the trans- 
former, as can approximately be done, and assuming the primary 
impedance reduced to the secondary circuit as equal to the 
secondary impedance: 

Yo = 0, - « = Z\. 

Substituting this in the equations of the general transformer 
we get: 

#o = no6 { 1 + \ \t\ (n + r) + sx x (xi + x)] 

- J 2 [sr x (xi + x) - Xi (r! + r)] 


$i = *"- e \[r (n + r) + « 2 x (xi + x)] - js[rxi -xri]}, 



106. The true power is, in symbolic representation: 

P = WY, 


srti 2 e 2 
-. - = w 

Zk 2 


Secondary output of the transformer: 


Internal loss in secondary circuit: 

Total secondary power: 

Pi + Pi 1 = (-"*) % (r + n) =sw(r + r,) ; 

Internal loss in primary circuit: 

Po 1 = to*r = toVid* = ( M ri = 9TiW\ 
Total electrical output, plus loss: 

P l = Pi+ Pi 1 + Po l = ( 8U £) \r + 2 ri ) = 8w(r + 2 r,) ; 
Total electrical input of primary: 

Po = [tfo/o] 1 = s (**) 2 (r + n + «r0 = tp (r + n + Wi); 

Hence, mechanical output of transformer: 

P = Po-P l = w(l -8)(r + ri ); 

me c hanica l out put _- P _ 1—8 _ speed 
total secondary power Pi + Pi 1 « slip 


In a general alternating transformer of ratio of turns, a, and 
ratio of frequencies, «, neglecting exciting current, it is: 
Electrical input in primary: 

p = sni 2 e* (r + r t + r^) . 
(rV+r^ + ^tei+'i) 1 ' 

(r, + r)* + s* (xx + x) 
Mechanical output: 

P _ J!iJ_z *)l | i^!_(L.+i! 1 ) • 

Electrical output of secondary : 

D 8 2 ni 2 e*r 

*i — v 

(n + ry + sHxi + x)*' 

Losses in transformer: 

2 sViiVri 

P i + p t i = pi = 

(ri + r) 2 + 8* (xi + x) 2 


Of these quantities, P 1 and Pi are always positive; P and P 
can be positive or negative, according to the value of s. Thus 
the apparatus can either produce mechanical power, acting U 
a motor, or consume mechanical power; and it can either con- 
sume electrical power or produce electrical power, as a generator, 

107. AI: 

s = 0, synchronism, P a = 0, P - 0, Pi = 0. 

At I) < s < 1, between synchronism and standstill. 

Pi, P and Pa are positive; that is, the apparatus consumes 
electrical power, P„, in the primary, and produces mechanical 
power, P, and electrical power, Pi 4- Pi 1 , in the secondary, which 
is partly, Pi', consumed by the internal secondary resistance, 
partly, Pi, available at the secondary terminals. 

In this case: 

P, + fY _s_ 
P 1 -a' 

that is, of the electrical power consumed in the primary circuit, 
Po, a part Pu l is consumed by the internal primary resistance, 
the remainder transmitted to the secondary, and divides between 
electrical power, Pi + Pi 1 , and mechanical power, P, in the 
proportion of the slip, or drop below synchronism, s, to the 
speed: 1 — s. 

In this range, the apparatus is a motor. 

At 8 > 1; or backward driving, P < 0, or negative; that is, 
the apparatus requires mechanical power for driving. 

Then : 

Po - P,' - Pi' < P,; 
that is, the secondary electrical power is produced partly by 
the primary electrical power, partly by the mechanical power, 
and the apparatus acts simultaneously as transformer nnd as 
alternating-current generator, with the secondary as armature. 

The ralio of mechanical input to electrical input is the ratio 
of speed to synchronism. 

In this case, the secondary frequency is higher than the 
p ill nary. 


a < 0, beyond synchronism, 
P < 0; that is, the apparatus has to be driven by mechanical 


Po < 0; that is, the primary circuit produces electrical power 
from the mechanical input. 

r + ri + sri = 0, or, s = - , 

the electrical power produced in the primary becomes less than 
required to cover the losses of power, and Po becomes positive 

We have thus: 

. r + ri 

8 < 

consumes mechanical and primary electric power; produces 
secondary electric power. 

_ r+ft <S<Q 

consumes mechanical, and produces electrical power in primary 
and in secondary circuit. 

< s < 1 

consumes primary electric power, and produces mechanical and 
secondary electrical power 

1 < s 

consumes mechanical and primary electrical power; produces 
secondary electrical power. 

108. As an example, in Fig. 62 are plotted, with the slip, s, as 
abscissae, the values of: 

Secondary electrical output as Curve I. ; 
total internal loss as Curve II.; 

mechanical output as Curve III.; 

primary electrical output as Curve IV. ; 

for the values : 

nie = 100.0; 

r = 0.4; 

n = 0.1; 

x = 0.3; 

xi = 0.2; 




Pi - 
Pi 1 = 

P - 










+ H= ' 


s (5 
1 + s" 

l + « 













i l 




i> 1 ' 













'" r 








































"*«: : ,' : -;:,-';? 



p s 


:■■ -""<-.\;" 


KNfH«IOR | K 1 MN 




Fio. 6 


2.— Mpeed-power curves of general alternating- current, transfo 
Since the most common practical application ol 
il alternating-current transformer is that of frequ 
rter, that is, to change from one frequency to ano 
with or without change of the number of phases 
ing characteristic curves of this apparatus are of 

'he regulation curve; that is, the change of secon 
lal voltage as function of the load at constant imprt 
ry voltage. 





2. The compounding curve; that is, the change of primary 
impressed voltage required to maintain constant secondary 
terminal voltage. 

In this case the impressed frequency and the speed are con- 
stant, and consequently the secondary frequency is also constant. 
Generally the frequency converter is used to change from a low 
frequency, as 25 cycles, to a higher frequency, as 60 or 62.5 
cycles, and is then driven backward, that is, against its torque, 
by mechanical power. Mostly a synchronous motor is em- 
ployed, connected to the primary mains, which by overexcitation 
compensates also for the lagging current of the frequency 


Y = g — jb = primary exciting admittance per circuit of 
the frequency converter. 

Z\ = fi + j%\ = internal self-inductive impedance per sec- 
ondary circuit, at the secondary frequency. 

Zo = r + jxo = internal self-inductive impedance per primary 
circuit at the primary frequency. 

a = ratio of secondary to primary turns per circuit. 

b = ratio of number of secondary to number of primary 

c = ratio of secondary to primary frequencies. 


e = generated e.m.f. per. secondary circuit at secondary 

Z = r + jx = external impedance per secondary circuit at 
secondary frequency, that is load on secondary system, where 
x — for non-inductive load. 

To calculate the characteristics of the frequency converter, 
we then have: 
the total secondary impedance : 

Z + Zi = (r + r x ) +j(x + x x ); 
the secondary current: 

/i = z + z = e ( fl i -i***); 


r + ri , x + X] 

0l _ — — anc j fl2 = 

(r + nY +(x + x x y m " " 2 (r + r x y + (x + xtf' 


and the secondary terminal voltage: 

*-*'*- -ST* 5 

= e (r + jx) (ai - ja 2 ) = e (61 - j7> 2 ) ; 

61 = (rai + xa 2 ) and 62 = (ra 2 — xa\) : 
primary generated e.m.f. per circuit: 

primary load current per circuit: 

/ l = abji = abe (ai — ja 2 ); 
primary exciting current per circuit: 

loo = -°- = (g — jb) : 
ac ac 

thus, total primary current: 

/o = Z 1 + /oo = e (ci - jc 2 ); 


Ci = abai + and c 2 = aba 2 H 

ac ac 

and the primary terminal voltage: 

= e (di - jd 2 ) 

di = + r Ci + XoC 2 and d 2 = r c 2 — ZoCi; 

or the absolute value is: 

substituting this value of 6 in the preceding equations, gives, 
as function of the primary impressed e.m.f., e : 
secondary current: 

/. = 

eo (<Ji - joj) /o? + a 2 4 

v^ + ^ or ' absolutc ' /l = eo V^ + ^ ; 

secondary terminal voltage: 

„ _ e (6, - jfr 2 ) / 6 7 + 6,2 . 


primary current: 

i _ e 5 ( c ' ~i c ») 

primary impressed e.m.f. : 













1 1 1 1 

fl 30 « M 

Flo. 63.— Regulation c 
secondary output : 

Pi = [AVt]' 
primary electrical input: 

P. = W.W 

u of frequency converter. 

«.' (oioi + o=4e) 
<fi" + oV 

e u s (cidi + c 2 'fs) 

'if + oV ; 

primary apparent input, volt-amperes: 
P., - ejo. 



Substituting thus different values for the secondary external 
impedance, Z, gives the regulation curve of the frequency 

Such a curve, taken from tests of a 200-kw. frequency converter 
changing from 6300 volts, 25 cycles, three-phase, to 2500 volts, 
62.5 cycles, quarter-phase, is given in Fig. 63. 












" T' 1 



Fio. 6*.-— Compounding curve »l frequency converter. 
From the secondary terminal voltage: 
£i = e(b t -jfe), 
it follows, absolute: 

3 Vfcl 1 + 6j* 

e — 

vV + W 

Substituting these values in the above equation gives the 
quantities as functions of the secondary terminal voltage, that 
is, at constant, e-i, or the compounding curve. 

The compounding curve of the frequency converter above 
mentioned is given in Fig. 64, 

110. When running above synchronism: a < 0, the general 
alternating-current transformer consumes mechanical power and 


produces electric power in both circuits, primary and secondary, 
thus can not be called a frequency converter, and the distinc- 
tion between primary and secondary circuits ceases, but both 
circuits are generator circuits. The machine then is a two-fre- 
quency induction generator. As the electric power generated 
at the two frequencies is proportional to the frequencies, this 
gives a limitation to the usefulness of the machine, and it appears 
suitable only in two cases: 

(a) If s = —1, both frequencies are the same, and stator 
and rotor circuits can be connected- together, in parallel or in 
series, giving the "double synchronous-induction generator." 
Such machines have been proposed for steam-turbine alternators 
of small and moderate sizes, as they permit, with bipolar con- 
struction, to operate at twice the maximum speed available for 
the synchronous machine, which is 1500 revolutions for 25 cycles, 
and 3600 revolutions for 60 cycles. 

(b) If 8 is very small, so that the power produced in the low- 
frequency circuit is very small and may be absorbed by a small 
"low-frequency exciter." 

Further discussion of both of these types is given in the 
Chapter XIII on the "Synchronous Induction Generator." 

111. The use of the general alternating-current transformer as 
frequency converter is always accompanied by the production 
of mechanical power when lowering, and by the consumption 
of mechanical power when raising the frequency. Thus a second 
machine, either induction or synchronous, would be placed on the 
frequency converter shaft to supply the mechanical power as 
motor when raising the frequency, or absorb the power as 
generator, when lowering the frequency. This machine may be 
of either of the two frequencies, but would naturally, for eco- 
nomical reasons, be built for the supply frequency, when motor, 
and for the generated or secondary frequency, when generator. 

Such a couple of frequency converter and driving motor and 
auxiliary generator has over a motor-generator set the advan- 
tage, that it requires a total machine capacity only equal to the 
output, while with a motor-generator set the total machine 
capacity equals twice the output. It has, however, the dis- 
advantage not to be as standard as the motor and the generator. 

If a synchronous machine is used, the frequency is constant ; 
if an induction machine is used, there is a slip, increasing with 
the load, that is, the ratio of the two frequencies slightly varies 



with the load, so that the latter arrangement is less suitable when 
tying together two systems of constant frequencies. 

112. Frequency converters may be used: 

(a) For producing a moderate amount of power of a higher or 
a lower frequency, from a large alternating-current system. 

(6) For tying together two alternating-current systems of 
different frequencies, and interchange power between them, so 
that either acts as reserve to the other. In this case, electrical 
power transfer may be either way. 

(c) For local frequency reduction for commutating machines, 
by having the general alternating-current transformer lower the 
frequency, for instance from 60 to 30 cycles, and take up the 
lower frequency, as well as the mechanical power in a commu- 
tating machine on the frequency converter shaft. Such a 
combination has been called a "Motor Converter." 

Thus, instead of a BO-cycle synchronous converter, such a 
6Q/30-cyc!e motor converter would offer the advantage of the 
lower frequency of 30 cycles in the commutating machine. The 
commutating machine then would receive half its input electric- 
ally, as synchronous converter, half mechanically, as direct- 
current generator, and thus would be half converter and half 
generator; the induction machine on the same shaft would change 
half of its (iO-cycle power input into mechanical power, half into 
30-cycle electric power. 

Such motor converter is smaller and more efficient than a 
motor-generator set, but larger and less efficient than a syn- 
chronous converter. 

Where phase control of the direct-current voltage is desired, 
the motor converter as a rule does not require reactors, as the 
induction machine has sufficient internal reactance. 

(rf) For supplying low frequency to a second machine on the 
same shaft, for speed control, as "concatenated motor couple." 
That is, two. induction motors on the same shaft, operating 
in parallel, give full speed, and half speed is produced, 
at full efficiency, by concatenating the two induction ma- 
chines, that is, using the one as frequency converter for feeding 
the other. 

By using two machines of different number of poles, /i, and 
pi, on the same shaft, four different speeds can lie secured, corre- 
sponding respectively to the number of poles: », + p\, » 2 , p h 
pi — p a . That is, concatenation of both machines, opentJoq 


of one machine only, either the one or the other, and differential 

Further discussion hereof see under "Concatenation." 
In some forms of secondary excitation of induction machines, 
as by low-frequency synchronous or commutating machine in 
the secondary, the induction machine may also be considered 
as frequency converter. Regarding hereto see "Induction 
Motors with Secondary Excitation." 



113. If an induction machine is driven above synchronism, 

the power component of the primary current reverses, thai is, 
energy flows outward, and the machine becomes an induction 
generator. The component of current required for magnetiza- 
tion remains, however, the same; that is, the induction generator 
requires the supply of a reactive current for excitation, just as 
the induction motor, and so must be connected to some apparatus 
which gives a lagging, or, what is the same, consumes a leading 

The frequency of the e.m.f. generated by the induction gen- 
erator, /, is lower than the frequency of rotation or speed, / . 
by the frequency, ft, of the secondary currents. Or, inversely, 
the frequency, ft, of the secondary circuit is the frequency of 
slip — that is, the frequency with which the speed of mechaoioal 
rotation slips behind the speed of the rotating field, in the induc- 
tion motor, or the speed of the rotating field slips behind the 
speed of mechanical rotation, in the induction generator. 

A3 in every transformer, so in the induction machine, the 
secondary current must have the same ampere-turns as the 
primary current less the exciting current, that is, the secondary 
current is approximately proportional to the primary current, 
or to the load of the induction generator. 

In an induction generator with short-circuited secondary, 
the secondary currents are proportional, approximately, to the 
e.m.f. generated in the secondary circuit, and this e.m.f. is pro- 
portional to the frequency of the secondary circuit, that is, 
the slip of frequency behind speed. It so follows that the slip 
of frequency in the induction generator with short-circuited 
secondary is approximately proportional to the load, that is, 
such an induction generator does not produce constant syn- 
chronous frequency, but a frequency which decreases slightly 
with increasing load, just as the speed of the induction motor 
decreases slightly with increase of load. 

Induction generator and induction motor so have also l>eeu 


called asynchronous generator and asynchronous motor, but 
these names are wrong, since the induction machine is not 
independent of the frequency, but depends upon it just as much 
as a synchronous machine — the difference being, that the 
synchronous machine runs exactly in synchronism, while the 
induction machine approaches synchronism. The real asyn- 
chronous machine is the commutating machine. 

114. Since the slip of frequency with increasing load on the 
induction generator with short-circuited secondary is due to 
the increase of secondary frequency required to produce the 
secondary e.m.f. and therewith the secondary currents, it follows: 
if these secondary currents are produced by impressing an e.m.f. 
of constant frequency, f lf upon the secondary circuit, the primary 
frequency, /, does not change with the load, but remains con- 
stant and equal to / = / — /i. The machine then is a syn- 
chronous-induction machine — that is, a machine in which the 
speed and frequency are rigid with regard to each other, just as 
in the synchronous machine, except that in the synchronous- 
induction machine, speed and frequency have a constant dif- 
ference, while in the synchronous machine this difference is zero, 
that is, the speed equals the frequency. 

By thus connecting the secondary of the induction machine 
with a 8010*06 of constant low-frequency, f l9 as a synchronous 
machine, or a commutating machine with low-frequency field 
excitation, the primary of the induction machine at constant 
speed, /o, generates electric power at constant frequency, /, 
independent of the load. If the secondary /i = 0, that is, a 
continuous current is supplied to the secondary circuit, the 
primary frequency is the frequency of rotation and the machine 
an ordinary synchronous machine. The synchronous machine so 
appears as a special case of the synchronous-induction machine 
and corresponds to /i = 0. 

In the synchronous-induction generator, or induction machine 
with an e.m.f. of constant low frequency, f h impressed upon the 
secondary circuit, by a synchronous machine, etc., with increas- 
ing load, the primary and so the secondary currents change, and 
the synchronous machine so receives more power as synchronous 
motor, if the rotating field produced in the secondary circuit 
revolves in the same direction as the mechanical rotation — 
that is, if the machine is driven above synchronism of the 
e.m.f. impressed upon the secondary circuit — or the synchronous 


machine generates more power as alternator, if the direction of 
rotation of the secondary revolving field is in opposition to the 
speed. In the former ease, the primary frequency equals speed 
minus secondary impressed frequency: / = fn — j\\ in the latter 
case, the primary frequency equals the sum of speed and sec- 
ondary impressed frequency:/ = f<, + /i, and the machine is a 
frequency converter or general alternating-current transformer, 
with the frequency, /i, as primary, and the frequency, /,as 
secondary, transforming up in frequency to a frequency, /, 
which is very high compared with the impressed frequency, 
so that the mechanical power input into the frequency con- 
verter is very large compared with the electrical power input. 

The synchronous-induction generator, that is, induction gen- 
erator in which the secondary frequency or frequency of slip h 
fixed by an impressed frequency, so can also be considered as a 
frequency converter or general alternating-current transformer. 

116. To transform from a frequency, / (l to a frequency; f t , the 
frequency, f%, is impressed upon the primary of an induction 
machine, and the secondary driven at such a speed, or fre- 
quency of rotation, /«. that the difference between primary 
impressed frequency, /,, and frequency of rotation, / , that, is, 
the frequency of slip, is the desired secondary frequency,/!, 

There are two speeds, /„, which fulfill this condition: one 
below synchronism: / u = f\ —ft, and one above synchronism: 
/« = /i + /=■ That is", the secondary frequency beoomaH f$, 
if the secondary runs slower than the primary revolving field 
of frequency,/,, or if the secondary runs faster than the primary 
field, by the slip, / s . 

In the former case, the speed is below synchronism, that is, 
the machine generates electric power at. the frequency, / = , in the 
secondary, and consumes electric power at the frequency, /,, 
in the primary. If / 3 < f u the speed / = f, — / 5 is between 
standstill and synchronism, and the machine, in addition to 
electric power, generates mechanical power, as induction motor, 
and as has been seen in the chapter on the "General Alternating- 
current Transformer," it is, approximately: 

Electric power input ■*■ electric power output -=- mechanical 
power output ■• f\ ■*■ ft + ft- 

If ft > !'• I hat is, the frequency converter increases the hc- 
quency, the rotation must be in backward direction, against the 
rotating field, so as to give a slip, / ; , greater than the tmpnmd 


frequency, /i, and the speed is /n = f* — f t . In this ease, the 
iiim-tiine consumes mechanical power, since it is driven against 

I the torque given by it as induction motor, and we have: 
Klectric power input ■*■ mechanical power input + electric 
power output - f % -+■ f f + f a . 
That is, the three powers, primary electric, secondary electric, 
and mechanical, are proportional to their respective frequencies. 
As stated, the secondary ■ frequency, St, is also produced by 
driving the machine above synchronism, /,, that is, with a 
negative slip, St, or at a speed, / = /i + Si- In this case, the 
machine is induction generator, that is, the primary circuit 
generates electric power at frequency Si, the secondary circuit 
generates electric power at frequency St- and the machine con- 
sumes mechanical power, and the three powers again arc proper- 
Itional to their respective frequencies: 
Primary electric output + secondary electric output +■ 
mechanical input = /t -s- /» -*- /o- 
Since in this case of oversynchronous rotation, both electric 
circuits of the machine generate, it can not be called a frequency 
converter, but is an electric generator, converting mechanical 
power into electric power at two different frequencies, / L and 
and so is called a synchronous-induction machine, since 
the sum of the two frequencies generated by it equals the fre- 
quency of rotation or speed — that, is, the machine revolves in 
synchronism with the sum of the two frequencies generated 
by it. 

It is obvious that like all induction machines, this synchro- 
nous-induction generator requires a reactive lagging current for 
excitation, which has to be supplied to it by some outside source, 
s a synchronous machine, etc. 

That is, an induction machine driven at speed, /«, when sup- 

ilied with reactive exciting current of the proper frequency, 

;enerates electric power in the stator as well as in the rotor, at 

he two respective frequencies, /■ and/-, which are such that their 

in synchronism with the speed, that is: 

A + />-/.; 

otherwise the Frequencies, /, and /-, are entirely independent, 
i connecting the stator to a circuit of frequency, Si, the 
•otor generates frequency, /» = /o - /i, or connecting the rotor to 



a circuit of frequency, />, the stator generates a frequency 

116. The power generated in the stator, P u and the power 
generated in the rotor, Pi, are proportional to their respective 
frequencies : 

P,:P,:P, -/•:/•:/* 

where P is the mechanical input (approximately, that is, neg- 
lecting losses). 

As seen here the difference between the two circuits, stator 
and rotor, disappears — that is, either can be primary or sec- 
ondary, that is, the reactive lagging current required for excita- 
tion can be supplied to the stator circuit at frequency, ft, or to 
the rotor circuit at frequency, f t , or a part to the stator and a pan 
to the rotor circuit. Since this exciting current is reactive or 
wattless, it can bo derived from a synchronous motor or con- 
verter, as well as from a synchronous generator, or an alter- 
nating comimitating machine. 

As the voltage required by the exciting current is proportional 
to the frequency, it also follows that the reactive power input or 
the volt-amperes excitation, is proportional to the frequency 
of the exciting circuit. Hence, using the low-frequency circuit 
for excitation, the exciting volt-amperes are small. 

Such a synchronous-induction generator therefore is a two- 
frequency generator, producing electric power simultaneously 
at two frequencies, and in amounts proportional to these fre- 
quencies. For instance, driven at 85 cycles, it can connect with 
the stator to :i 25-eycle system, and with the rotor to a 60-cycle 
system, and feed into both systems power in the proportion of 
25 + 60, as is obvious from the equations of the general alter- 
nating-current transformer in the preceding chapter 

117. Since the amounts of electric power at the two fre- 
quencies are always proportional to each other, such a machine 
is hardly of much value for feeding into two different systems, 
but of importance are only the cases where the two frequence 
generated by the machine can he reduced to one. 

This is the case: 

1. If the two frequencies are the same:/! —ft 


In this 

case, stator and rotor can be connected together, in parallel 
or in series, and the induction machine then generates electric 
power at half the frequency of its speed, that is, runs at double 


synchronism of its generated frequency. Such a " double syn- 
chronous alternator" so consists of an induction machine, in 
which the stator and the rotor are connected with each other in 
parallel or in series, supplied with the reactive exciting current 
by a synchronous machine — for instance, by using synchronous 
converters with overexcited field as load — and driven at a speed 
equal to twice the frequency required. This type of machine 
may be useful for prime movers of very high speeds, such as 
steam turbines, as it permits a speed equal to twice that of the 
bipolar synchronous machine (3000 revolutions at 25, and 7200 
revolutions at 60 cycles). 

2. If of the two frequencies, one is chosen so low that the 
amount of power generated at this frequency is very small, and 
can be taken up by a synchronous machine or other low-fre- 
quency machine, the latter then may also be called an exciter. 
For instance, connecting the rotor of an induction machine to a 
synchronous motor of / 2 = 4 cycles, and driving it at a speed 
of /o = 64 cycles, generates in the stator an e.m.f. at f\ = 60 
cycles, and the amount of power generated at 60 cycles is 6 pj[ = 
15 times the power generated by 4 cycles. The machine then 
is an induction generator driven at 15 times its synchronous 
speed. Where the power at frequency, / 2 , is very small, it would 
be no serious objection if this power were not generated, but con- 
sumed. That is, by impressing / 2 = 4 cycles upon the rotor, 
and driving it at / = 56 cycles, in opposite direction to the rotat- 
ing field produced in it by the impressed frequency of 4 cycles,, 
the stator also generates an e.m.f. at f\ = 60 cycles. In this 
case, electric power has to be put into the machine by a generator 
at / 2 = 4 cycles, and mechanical power at a speed of / = 56 
cycles, and electric power is produced as output at /i = 60 cycles. 
The machine thus operated is an ordinary frequency converter, 
which transforms from a very low frequency, / 2 = 4 cycles, to 
frequency /i = 60 cycles or 15 times the impressed frequency, 
and the electric power input so is only one-fifteenth of the electric 
power output, the other fourteen-fifteenths are given by the 
mechanical power input, and the generator supplying the im- 
pressed frequency, / 2 = 4 cycles, accordingly is so small that it 
can be considered as an exciter. 

118. 3. If the rotor of frequency, / 2 , driven at speed, / , is 
connected to the external circuit through a commutator, the 
effective frequency supplied by the commutator brushes to the 



external circuit is/„ — / s ; hence equals/,, or the atator frequency. 
Stator and rotor so give the same effective frequency, /,, and 
irrespective of the frequency, / s generated in the rotor, and the 
frequencies, /[ and / s , accordingly become indefinite, that is, 
jx may lie any frequency, /i then becomes f„ — /,, but. by the 
commutator is transformed to the same frequency, /i. If the 
stator and rotor were used on entirely independent electric 
circuits, the frequency would remain indeterminate. As soon, 
however, as stator and rotor are connected together, a relation 
appears due to the transformer law, that the secondary ampere- 
turns must equal the primary ampere-turns (when neglecting 
the exciting ampere-turns). This makes the frequency dependent 
upon the number of turns of stator and rotor circuit. 

Assuming the rotor circuit is connected in multiple with the 
stator circuit— as it always can be, since by the commutator 
brushes it has been brought to the same frequency. The rotor 
c.m.f. then must be equal to the stator e.m.f. The e.m.f., how- 
ever, is proportional to the frequency times number of turns, 
and it is therefore: 

n-J, - ■»,/,, 

where: /i] = number of effective stator turns, 

>H = number of effective rotor turns, and f\ 

and/* are the respective frequencies. 
Herefrom follows: 

/,+/,-«, + »,; 

that is, the frequencies are inversely proportional to the number 
of effective turns in stator and in rotor. 

Or, since /o = A + /a is the frequency of rotation : 

I +«! 

+ »., 

That is, the frequency, /,, generated by the synchronous- 
induction machine with commutator, is the frequency of imntinn. 
/o, times the ratio of rotor turns, m, to total turns, n, + n». 

Thus, it can lie made anything by properly choosing the 
number of turns in the rotor and in the stator, or, what amounts 
to the same, interposing between rotor and stator a transformer 
of the proper ratio of transformation. 


The powers generated by the stator and by the rotor, how- 
ever, are proportional to their respective frequencies, and so are 
inversely proportional to their respective turns. 


Pi -§-P» =/i +h = n 2 -*- m; 

if n\ and n 2 , and therewith the two frequencies, are very different, 
the two powers, Pj and P2, are very different, that is, one of the 
elements generates very much less power than the other, and 
since both elements, stator and rotor, have the same active 
surface, and so can generate approximately the same power, the 
machine is less economical. 

That is, the commutator permits the generation of any de- 

sired frequency, /1, but with best economy only if f\ = w, or 

half-synchronous frequency, and the greater the deviation from 
this frequency, the less is the economy. If one of the fre- 
quencies is very small, that is, f\ is either nearly equal to syn- 
chronism, /o, or very low, the low-frequency structure generates 
very little power. 

By shifting the commutator brushes, a component of the rotor 
current can be made to magnetize and the machine becomes a 
self-exciting, alternating-current generator. 

The use of a commutator on alternating-current machines is 
in general undesirable, as it imposes limitations on the design, 
for the purpose of eliminating destructive sparking, as discussed 
in the chapter on "Alternating-Current Commutating Machines." 

The synchronous-induction machines have not yet reached a 
sufficient importance to require a detailed investigation, so only 
two examples may be considered. 

119. 1. Double Synchronous Alternator, 

Assume the stator and rotor of an induction machine to be 
wound for the same number of effective turns and phases, and 
connected in multiple or in series with each other, or, if wound 
for different number of turns, connected through transformers 
of such ratios as to give the same effective turns when reduced 
the same circuit by the transformer ratio of turns. . 


Yi — Q — jb — exciting admittance of the stator, 

Z\ = ri + jx\ = self-inductive impedance of the stator, 

Z 2 = r 2 + jx* = self-inductive impedance of the rotor, 



6 = e.m.f. generated in the stator by the mutual inductive 
magnetic field, that is, by the magnetic flux corresponding to 
the exciting admittance, Y\\ 

/ = total current, or current supplied to the external circuit, 
I\ = stator current, 
I2 = rotor current. 

With series connection of stator and rotor: 

/ = /, = h, 
with parallel connection of stator and rotor: 

/ = /i + / 2 . 

Using the equations of the general induction machine, the 
slip of the secondary circuit or rotor is : 

« = -1; 
the exciting admittance of the rotor is: 

Yt = g - jsb = g + jb, 
and the rotor generated e.m.f.: 

E\ = se = — e; 

that is, the rotor must be connected to the stator in the opposite 
direction to that in which it would be connected at standstill, 
or in a stationary transformer. 

That is, magnetically, the power components of stator and 
rotor current neutralize each other. Not so, however, the 
reactive components, since the reactive component of the rotor 

U = i\ + ji\ 

in its reaction on the stator is reversed, by the reversed direction 
of relative rotation, or the slip, s = — 1, and the effect of the 
rotor current, I* } on the stator circuit accordingly corresponds 

i 2 — I 2 — 3 l 2, 

hence, the total magnetic effect is : 

/i-/' 2 = (*\-*'t)+j(i"i + t" t ); 


and since the total effect must be the exciting current: 

i o — to tj 0, 

it follows that : 

i'x — i't = i'o and i"\ + i #/ t = t 


Hence, the stator power current and rotor power current, 
i'x and i\ y are equal to each other (when neglecting the small 
hysteresis power current). The synchronous exciter of the 
machine must supply in addition to the magnetizing current, 
the total reactive current of the load. Or in other words, such 
a machine requires a synchronous exciter of a volt-ampere 
capacity equal to the volt-ampere. excitation plus the reactive 
volt-amperes of the load, that is, with an inductive load, a large 
exciter machine. In this respect, the double-synchronous 
generator is analogous to the induction generator, and is there- 
fore suited mainly to a load with leading current, as over- 
excited converters and synchronous motors, in which the reactive 
component of the load is negative and so compensates for the 
reactive component of excitation, and thereby reduces the size 
of the exciter. 

This means that the double-synchronous alternator has zero 
armature reaction for non-inductive load, but a demagnetizing 
armature reaction for inductive, a magnetizing armature reac- 
tion for anti-inductive load, and the excitation, by alternating- 
reactive current, so has to be varied with the character of the 
load, in general in a far higher degree than with the synchronous 

120. 2. Synchronous-induction Generator with Low-frequency 

Here two cases exist: 

(a) If the magnetic field of excitation revolves in opposite 
direction to the mechanical rotation. 

(6) If it revolves in the same direction. 

In the first case (a) the exciter is a low-frequency generator 
and the machine a frequency converter, calculated by the same 

Its voltage regulation is essentially that of a synchronous 
alternator: with increasing load, at constant voltage impressed 
upon the rotor or exciter circuit, the voltage drops moderately 
at non-inductive load, greatly at inductive load, and rises at 



anti-inductive load. To maintain constant terminal voltage, 
the excitation has to lie changed with ;i change of load and 
character of load. With a low-frequency synchronous machine 
as exciter, this is done by varying the field excitation of the 

At constant field excitation of the synchronous exciter, the 
regulation is that due to the impedance lietween the nominal 
generated e.m.f. of the exciter, and the terminal voltage of the 
stator — that, is, corresponds to: 

Z = Z + Zi + Z„ 
Here 7.t, = synchronous impedance of the exciter, reduced to full 
frequency, /i, 
Zj = self-inductive impedance of the rotor, reduced to full 

frequency. /i, 
Zi = self-inductive impedance of the stator. 
If then E D = nominal generated e.m.f. of the exciter generator, 
that is, corresponding to the field excitation, and, 
/i = i — jti = stator current or output current, the stator 
terminal voltage is: 

E, - E + ZI„ or, E» = E + (r + jx) {i - ji\); 
and, choosing Ei = p| as real axis, and expanding: 

Bo = («i + « + Mf) + j (ri - ri>), 
and the absolute value: 

- (g. + ri 

■ xn) 


nO* — (n + j/i). 

121. As an example is shown, in Fig. 6.5, in dotted lines, with 
the total current, / = y/i- + f"i ! , iis abscissa?, the voltage regu- 
lation of such a machine, or the terminal voltage, t u with a 
four-cycle synchronous generator as exciter of the 60-cycle 
synchronous-induction generator, driven as frequency converter 
at 56 cycles. 

1. For non-inductive load, or I, - i. (Curve I.) 

2. For inductive load of 80 per cent, power-factor, or /i * 
7(0.8 - 0.6 j). (Curve U.J 

3. For anti-inductive load of 80 per cent, power-factor, or 
/, = 7(0.8 + O.Gj). (Curve III.) 


For the constants: 

e„ - 2000 volts, Z, = 1 + 0.5 j, 

Z x = 0.1 + 0.3 j, Z„ = 0.5 + 0.5 j; 


Z = 1.6 + 1.3 j. 

e, = Vi X 10' - (1.3 i - 1.6 ii) 1 - (1.6 i + 1.3 *'i); 
hence, for non-inductive load, ii = 0: 

c, = V4 X 10« - 1.69f* - 1.6 i; 

; «* --'' 

^^ -"" 

^ ^ -- ^ ^ 


"""' ^sz~— --. ~~ ~~ — -_ 

1800 "s^. " ■"•-^ 

S -- 1 "^ iu 

11 ^"i > ^ ^^ 

2 ^*, V 

wo "^-J 

Z ^is 

300 ^ 

Fio. 65. — Synchronous induction generator regulation curves. 

for inductive load of 80 per cent, power-factor j'i = 0.6 /, i = 

e, = Vi X 10' - 00064/* - 2.06/; 

and for anti-inductive load of 80 per cent, power-factor 1 
- 0.6/,* = 0.8/: 


X 10« -4/ 1 - 0.5/. 

As seen, due to the internal impedance, anil especially the 
resistance of this machine, the regulation is very poor, and even 
at the chosen anti-inductive load no rise of voltage occurs. 

122. Of more theoretical interest is the case (b), where the 


exciter is a synchronous motor, and the synchronous-induction 
generator produces power in the stator and in the rotor circuit. 
In this case, the power is produced by the generated e.m.f., E 
(e.m.f. of mutual induction, or of the rotating magnetic field}, 
of the induction machine, and energy flows outward in both 
circuits, in the stator into the receiving circuit, of terminal 
voltage, #i, in the rotor against the impressed e.m.f. of the 
synchronous motor exciter, En. The voltage of one receiving 
circuit, the stator, therefore, is controlled by a voltage impressed 
upon another receiving circuit, the rotor, and this results in some 
interesting effects in voltage regulation. 

Assume the voltage, E a , impressed upon the rotor circuit as 
the nominal generated e.m.f. of the synchronous-motor exciter, 
that is, the field corresponding to the exciter field excitation, 
and assume the field excitation of the exciter, and therewith 
the voltage, E a , to be maintained constant. 

Reducing all the voltages to the stator circuit by the ratio of 
their effective turns and the ratio of their respective frequencies, 
the same e.m.f., E, is generated in the rotor circuit as in the 
stator circuit of the induction machine. 

At no-load, neglecting the exciting current of the induction 
machine, that is, with no current, we have E n = E = E\. 

If a load is put on the stator circuit by taking a current, /, 
from the same, the terminal voltage, E x , drops below I he gene- 
rated e.m.f., E, by the drop of voltage in the impedance, Z u of 
the stator circuit. Corresponding to the stator current, I,, a 
current, /a, then exists in the rotor circuit, giving the same 
ampere-turns as Ii, in opposite direction, and so neutralizing the 
m.m.f. of the stator (as in any transformer). This current, I t , 
exists in the synchronous motor, and the synchronous motor 
e.m.f., Eo, accordingly drops below the generated e.m.f., E, of 
the rotor, or, since Ea is maintained constant, E rises above E a 
with increasing load, by the drop of volt age in the rotor impedance, 
2V, and the synchronous impedance, Z«, of the exciter. 

That is, the stator terminal voltage, E,, drops with increasing 
load, by the stator impedance drop, and rises with increasing 
load by the rotor and exciter impedance drop, since the latter 
causes the generated e.m.f., E, to rise. 

If then the impedance drop in the rotor circuit is greater than 
thai in the stator, with increasing load the terminal voltage, 
Ei, of the machine rises, that is, the machine automatically 


overcompounds, at constant-exciter field excitation, and if the 
stator and the rotor impedance drops are equal, the machine 
compounds for constant voltage. 

In such a machine, by properly choosing the stator and rotor 
impedances, automatic rise, decrease or constancy of the terminal 
voltage with the load can be produced. 

This, however, applies only to non-inductive load. If the 
current, I, differs in phase from the generated e.m.f., E, the 
corresponding current, J 2 , also differs; but a lagging component 
of I\ corresponds to a leading component in It, since the stator 
circuit slips behind, the rotor circuit is driven ahead of the 
rotating magnetic field, and inversely, a leading component of 
7i gives a lagging component of 7 2 . The reactance voltage of 
the lagging current in one circuit is opposite to the reactance 
voltage of the leading current in the other circuit, therefore 
does not neutralize it, but adds, that is, instead of compounding, 
regulates in the wrong direction. 

123. The automatic compounding of the synchronous induc- 
tion generator with low-frequency synchronous-motor excitation 
so fails if the load is not non-inductive. 


Z\ = T\ + jxi = stator self-inductive impedance, 

Z 2 = r 2 + jx<t = rotor self-inductive impedance, reduced to the 

stator circuit by the ratio of the effective turns, t = - , and the 

J x ni' 

ratio of frequencies, a = t) 

Zo = r + jxo = synchronous impedance of the synchronous- 
motor exciter; 
Ei = terminal voltage of the stator, chosen as real axis, = e\\ 
Eo = nominal generated e.m.f. of the synchronous-motor 

exciter, reduced to the stator circuit; 
E = generated e.m.f. of the synchronous-induction generator 
stator circuit, or the rotor circuit reduced to the stator circuit. 
The actual e.m.f. generated in the rotor circuit then is E' = 
taE, and the actual nominal generated e.m.f. of the synchronous 
exciter is E\ = taEo. 

I\ = i ' — jii = current in the stator circuit, or the output 
current of the machine. 


The current in the rotor circuit, in which the direction of 
rotation is opposite, or ahead of the revolving field, then is, 
when neglecting the exciter current : 

li = i + jii. 

. (If Y = exciting admittance, the exciting current is Io — EY f 
and the total rotor current then h + 7 2 .) 
Then in the rotor circuit: 

E = E + (Z + Z 2 ) A, (1) 

and in the stator circuit : 

E = Ex + Zi/i. (2) 

Hence : 

Ex = E + U (Zo + Z 2 ) - 1 X Z U (3) 

or, substituting for 1\ and 7 2 : 

Ex = E + i (Zo + Z 2 - Z x ) + jit (Z + Z 2 + Z,). (4) 
Denoting now: 

Z + Z\ + Z 2 = Z 3 = r 3 + jx 8 , 
Z + Z 2 — Zi = Zj = r 4 + jx 4 , 


and substituting: 

tf, = E + iZ A + jUZ h (6) 

or, since Ex = e\\ 

E = ei — iZ K — }i\Zz 

= (ei - r A i + x 3 t'i) - j (xii + r i i l ) ) (7) 

or the absolute value: 

e 2 = (ex - r A i + x$ix) 2 + (x A i + r 3 /i) 2 . (8) 

Hence : 

ex = vco 2 - (x A i + r 3 ii) 2 + r 4 i - x 3 ii. (9) 

That is, the terminal voltage, ci, decreases due to the decrease 
of the square root, but may increase due to the second term. 
At no-load: 

i = 0, ix = and e\ = <v 

At non-inductive load: 

t"i = and ei = Ve * - z A H 2 + r A i. (10) 

e\ first increases, from its no-load value, e , reaches a maximum, 

and then decreases again. 


r* = r + r 2 - r lt 

X\ = x + Xt — r x , 
at : r 4 = and au = 0, 


ri = r + r 2 , 
Xi = Xo + x 2 , 


ei = ^o, that is, in this case the terminal vol- 
tage is constant at all non-inductive loads, at constant exciter 

In general, or for I\ = i — ji\, 

if ii is positive or inductive load, from equation (9) follows 
that the terminal voltage, e h drops with increasing load; while 

if i\ is negative or anti-inductive load, the terminal voltage, 
6i, rises with increasing load, ultimately reaches a maximum 
and then decreases again. 

From equation (9) follows, that by changing the impedances, 
the amount of compounding can be varied. For instance, at 
non-inductive load, or in equation (10) by increasing the re- 
sistance, r 4 , the voltage, ci, increases faster with the load. 

That is, the overcompounding of the machine can be increased 
by inserting resistance in the rotor circuit. 

124. As an example is shown, in Fig. 65, in full line, with the 
total current, / = y/i 2 + i x 2 , as abscissae, the voltage regulation 
of such a machine, or the terminal voltage, e h with a four- 
cycle synchronous motor as exciter of a 60-cycle synchronous- 
induction generator driven at 64-cycles speed. 

1. For non-inductive load, or Ii = i. (Curve I.) 

2. For inductive load of 80 per cent, power-factor; or 
/, = / (0.8 - 0.6 j). (Curve II.) 

3. For anti-inductive load of 80 per cent, power-factor; or 
Ii - / (0.8+ 0.6 j). (Curve III.) 



For the constants : 

eo = 2000 volts. 
Z t = 0.1 + 0.3j". 
Z t ~ 1 +0.5 j. 


Z = 0.5 + 0.5 j. 
a = 0.067. 

( = 1, that is, the 
same number of 
turns in stator 
and rotor. 

Z, = 1.6 + 1.3 j and Z« = 1.4 + 0.7 j. 
Hence, substituting in equation (9) : 

Cl = V4X 10'- (0.7 1 + 1.6(1)* + 1.4 1 - 1.3 i,; 














■ — 

— it « 





U N 

Fia. 66. — Synchronous induction generator; voltage regulation with power- 
factor of load. 

thus, for non-inductive load, n = 0: 

e, = Vi X 10* - 0.49 i* + 1.4 i; 

for inductive load of 80 per cent. power-factori'i=0.67;i = 0.87: 

«i = V4 X 10» - 2.31 7* + 0.34 7; 

and for anti-inductive load of 80 per cent, power-factor i"i = 
-0.6 1;x -0.8 7: 

e , = V4 X 10« - 0.16 /* + 1.9 7. 

Comparing the curves of this example with those of the same 
machine driven as frequency converter with exciter generator, 


and shown in dotted lines in the same chart (Fig. 65), it is 
seen that the voltage is maintained at load far better, and 
especially at inductive load the machine gives almost perfect 
regulation of voltage, with the constants assumed here. 

To show the variation of voltage with a change of power- 
factor, at the same output in current, in Fig. 66, the terminal 
voltage, 6i, is plotted with the phase angle as abscissae, from 
wattless anti-inductive load, or 90° lead, to wattless inductive 
load, or 90° lag, for constant current output of 400 amp. As 
seen, at wattless load both machines give the same voltage but 
for energy load the type (b) gives with the same excitation a 
higher voltage, or inversely, for the same voltage the type (a) 
requires a higher excitation. It is, however, seen that with the 
same current output, but a change of power-factor, the voltage 
of type (a) is far more constant in the range of inductive load, 
while that of type (b) is more constant on anti-inductive load, 
and on inductive load very greatly varies with a change of 


126. Any polyphase system can, by mean? of two stationary 
transformers, be converted into any other polyphase system, 
and in such conversion, a balanced polyphase system remains 
balanced, while an unbalanced system converts into a polyphase 
system of the same balance factor. 1 

In the conversion between single-phase system and polyphase 
system, a storage of energy thus must take place, as the balance 
factor of the single-phase system is zero or negative, while that 
of the balanced polyphase system is unity. For such energy 
storage may be used capacity, or inductance, or momentum or a 
combination thereof: 

Energy storage by capacity, that is, in the dielectric fu Id, 
required per kilovolt-ampere at 60 cycles about 200O <-.•■. ol 
space, at a cost of about $10. Inductance, that is. energy 
storage by the magnetic field, requires about 1000 c.e. per kilo- 
volt-ampere at 60 cycles, at a cost of $1, while energy storage by 
momentum, as kinetic mechanical energy, assuming iron moving 
at 30 meter-seconds, stores 1 kva. at 60 cycles by about 3 c.c., 
at a cost of 0.2c, thus is by far the cheapest and least bulky 
method of energy storage. Where large amounts of energy have 
to be stored, for a very short time, mechanical momentum thus 
is usually the most efficient and cheapest method. 

However, size and cost of condensers is practically the same 
for large as for small capacities, while the size and cost of induc- 
tance decreases with increasing, and increases with dec ro a n i B g 
kilovolt-ampere capacity. Furthermore, the use of mechanical 
momentum means moving machinery, requiring more or less 
attention, thus becomes less suitable, for smaller values of power. 
Hence, for smaller amounts of stored energy, inductance and 
capacity may become more economical than momentum, and 
for very small amounts of energy, the condenser may lie the 
cheapest device. The above figures thus give only the approxi- 

• "Theorv and Calculation of Alterwi ting-current Phenomena," 
edition, Chapter XXXII. 




mate magnitude for medium values of energy, and then apply 

only to the active energy -storing structure, under the assumption, 
thai during every energy cycle (or half cycle of alternating our- 

^nt and voltage), the entire energy is returned and stored again. 
iile this is the case with capacity and inductance, when using 
momentum for energy storage, as flywheel capacity, the energy 
storage and return is accomplished by a periodic speed variation, 
thus only a part of the energy restored, and furthermore, only 
a part of the structural material (the flywheel, or the rotor of 
the machine) is moving. Thus assuming that only a quarter 
of the mass of the mechanical structure (motor, etc.) is revolving, 
and that the energy storage takes place by a pulsation of speed of 

per cent., then 1 kva. at 60 cycles would require 600 c.e. of 

terial, at 40c, 

Obviously, at the limits of dielectric or magnetic field strength, 
or at the limits of mechanical speeds, very much larger amounts 
of energy per bulk could be stored. Thus for instance, at the 
limits of steam-turbine rotor speeds, about 400 meter-seconds, 
in a very heavy material as tungsten, 1 e.c. of material would 
store about 200 kva. of 60-cycle energy, and the above figures 
thus represent only average values under average conditions. 

126. Phase conversion is of industrial importance in changing 
from single-phase to polyphase, and in changing from polyphase 
to single-phase. 

Conversion from single-phase to polyphase has been of con- 
siderable importance in former times, when alternating-current 
generating systems were single-phase, and alternating-current 
motors required polyphase for their operation. With the prac- 
tically universal introduction of three-phase electric power 
leration, polyphase supply is practically always available for 

itionary electric motors, at least motors of larger size, and 

n version from single-phase to polyphase thus is of importance 


(a) To supply small amounts of polyphase current, for the 
rting of smaller induction motors operated on single-phase 

listribution circuits, 2300 volts primary, or 110/220 volts 
secondary, that is, in those cases, in which the required amount 
of power is not. sufficient to justify bringing the third phase to 
the motor: with larger motors, all the three phases are brought 
to the motor installation, thus polyphase supply used. 

(b) For induction-motor railway installations, to avoid the 



complication and inconvenience incident to the use of two trolley 
wires. In this case, as large amounts of polyphase power arc 
required, and economy in weight is important, momentum is 
generally used for energy storage, that is an induction machine 
is employed as phase converter, and then is used either in series 
or in shunt to the motor. 

For the small amounts of power required by use (a), generally 
inductance or capacity are employed, and even then usually the 
conversion is made not to polyphase, but to monocyclic, as the 
latter is far more economical in apparatus. 

Conversion from polyphase to single-phase obviously means 
the problem of deriving single-phase power from a balanced 
polyphase system. A single-phase load can be taken from any 
phase of a polyphase system, but such a load, when consider- 
able, unbalances the polyphase system, that is, makes the vol- 
tages of the phases unequal and lowers the generator capacity. 
The problem thus is, to balance the voltages and the reaction of 
the load on the generating system. 

This problem has become of considerable importance in the 
last years, for the purpose of taking targe single-phase loads, for 
electric railway, furnace work, etc., from a three-phase supply 
system as a central station or transmission line. For this pur- 
pose, usually synchronous phase converters with synchronous 
phase balancers are used. 

As illustration may thus be considered in the following the 
monocyclic device, the induction phase converter, and the 
synchronous phase converter and balancer. 

Monocyclic Devices 

127. The name "monocyclic" is applied to a polyphase sys- 
tem of voltages (whether symmetrical or unsymmetrical), in 
which the flow of energy is essentially single- phase. 

For instance, if, as shown diagrammatic ally in Fig. 67, we 
connect, between single-phase mains, AB, two pairs of non-in- 
ductive resistances, r, and inductive reactances, x (or in general, 
two pairs of impedances of different inductance factors), such 
that t = x, consuming the voltages E\ and Et respectively, then 
the voltage e» = CD is in quadrature with, and equal to, the 
voltage e = AB, and the two voltages, e and eo, constitute a 
monocyclic system of quarter-phase voltages: e gives the energy 


axis of the monocyclic system, and e the quadrature or wattless 
axis. That is, from the axis, e, power can be drawn, within 
the limits of the power-generating system back of the supply 
voltage. If, however, an attempt is made to draw power from 
the monocyclic quadrature voltage, e > this voltage collapses. 

If then the two voltages, e and eo, are impressed upon a quarter- 
phase induction motor, this motor will not take power equally 
from both phases, e and e , but takes power essentially only from 
phase, e. In starting, and at heavy load, a small amount of 
power is taken also from the quadrature voltage, eo, but at light- 
load, power may be returned into this voltage, so that in general 
the average power of e approximates zero, that is, the voltage, 
eo, is wattless. 

A monocyclic system thus may be defined as a system of poly- 
phase voltages, in which one of the power axis, the main axis 
or energy axis, is constant potential, and the other power axis, 
the auxiliary or quadrature axis, is of dropping characteristic 
and therefore of limited power. Or it may be defined as a poly- 
phase system of voltage, in which the power available in the one 
power axis of the system is practically unlimited compared with 
that of the other power axis. 

A monocyclic system thus is a system of polyphase voltage, 
which at balanced polyphase load becomes unbalanced, that is, 
in which an unbalancing of voltage or phase relation occurs 
when all phases are loaded with equal loads of equal inductance 

In some respect, all methods of conversion from single-phase 
to polyphase might be considered as monocyclic, in so far as the 
quadrature phase produced by the transforming device is limited 
by the capacity of the transforming device, while the main 
phase is limited only by the available power of the generating 
system. However, where the power available in the quadrature 
phase produced by the phase converter is sufficiently large not 
to constitute a limitation of power in the polyphase device sup- 
plied by it, or in other words, where the quadrature phase pro- 
duced by the phase converter gives essentially a constant-poten- 
tial voltage under the condition of the use of the device, then the 
system is not considered as monocyclic, but is essentially 

In the days before the general introduction of three-phase 
power generation, about 20 years ago, monocyclic systems were 



extensively used, and monocyclic generators built. These were 
.^iriulr-pluisi' alternating-eurrenl generators, having a small 
quadrature phase of high inductance, which combined with the 
main phase gives three-phase or quarter- phase voltages. The 
auxiliary phase was of such high reactance as to limit the quadra- 
< i < ti ■ poWCI and thus make the flow of energy essentially single- 
phase, that is, monocyclic. The purpose hereof was to permit 
the use of a small quadrature coil on the generator, and thereby 
to preserve the whole generator capacity for the single-phase 
main voltage, without danger of overloading the quadrature 
phase in case of a high motor load on the system. The genera] 
introduction of the three-phase system superseded the mono- 
cyclic generator, and monocyclic devices are today used only 
for local production of polyphase voltages from single-phase 
supply, for the starting of small siliEle-phiise induction motors, 
clc. The advantage of the monocyclic feature then comaata in 
1 1 Mm him the output and thereby the size of the device, and making 
it (hereby economically feasible with the use of the rather expen- 
sive energy-storing devices of inductance (and capacity) used in 
this case. 

The simplest and most generally used monocyclic device con- 
si sis of I wo impedances, Z, and Z«,ot different inductance factors 
(resistance and inductance, or inductance and capacity), con- 
nected aiTuss the single-phase mains, .4 ami li. The common 
connection, C, between the two impedances, Z, and Z>. then is dis- 
placed in phase from the single-phase supply voltage. A and B, 
and gives with the same a system of out-of-phase voltages, AC, 
Cli and .4 if, or a — more or less unsymmetrical — three-phase 
Iriaiude. Or, between this common connection, C, and the 
middle, D, of an autotransformer connected between the single- 
phase mains, AB, a quadrature voltage, CD, is produced. 

This ■monocyclic triangle" ACB, in its application as singlc- 
tuCtKM) motor-starting device, is discussed in Chapter V. 

Tw.> -mil monoeydic triangles combined give the monocyclic 
square. Fig. (57. 

138. Let then, in the monoeydic square shown diagrammalic- 
ally in Pig. 67: 

1", = g, — j&i = admittance AC and DB; 
)", = j,j — fit = admittance CB and AD: 

Ye = o* — jb* = admittance of the load on 



the monocyclic quadrature voltage, # = CD, and current, /o. 
Denoting then : 

& = e = supply voltage, AB 9 and / = supply current, and 
?it #« = voltages, /i, /i = currents in the two sides of the 
monocyclic square. 

It is then, counting voltages and currents in the direction 
indicated by the arrows in Fig. 67 : 




into (3) gives 

#* + £i = e, 


Ei — $i = #o; 


„ E + Eq 

& = 2 ~' 

w - e ~ Eo - 
** ~ 2 ' 



J =/l+/2, 

/o = /i — /«; 

/o = #0^0, 

h = ^iK„ 

/ s = &y s ; 

/ =#.ri + £*r,, 

^qJ'o = #1^1 — 1$1 







substituting (2) into (5) gives: 

_«(r 1 -_r i ) . 
*° r7+ r7+ 2 iV 


substituting (6) into (2) gives: 

£1 = 

$2 = 

e(K.+ Fo) 




substituting (7) and (6) into (4) and (5) gives the currents: 

eY (Yx - Y t ) 

/o = 

-— z^rl 

Fi + F 2 + 2 Fo 

f = ? (Z?7a_+ y * y * + 2YjY 2 ) 

/l = 

/* = 

Y 1 + Y i + 2Yo 

eY x (y, + r ) 

yi +"y 2 + 2 yV 
ey, (y! + y ) 


y 1 + y 2 + 2 y 

129. For a combination of equal resistance and reactance : 


r- « -7.07 OHMS 
- 100 VOLTS 



Fig. 67. — Resistance-inductance monocyclic square, topographical regula- 
tion characteristic. 

and a load : 

yi = o, 
Y* = -ja; 

Y = a(p-jq); 
equations (6) and (8) give: 

p _ e(l+j) 

1 - j + 2 (p - jj) 

T _ea(p - jq)(l +j) 
/0 l-j+~2(p-j?j 

eo [(p - jg) (1 - j) - 2j ] 
* l-j + 2(p-jg) 


Fig. 67 shows the voltage diagram, and Fig. 68 the regulation, 
that is, the values of e a and i, with i as abscissae, for: 

e = 100 volts, 
a = 0.1 V*2 mho. 






* - 1O0 VOLTS 


















Fia. 68. — Resist ance-inductance monocyclic square, regulation curve. 

For: q = 0, that is, non-inductive load, the voltage diagram 
is a curve shown by circles in Fig. 67, for 0, 2, 4, 6, 8 and 10 amp. 
load, the latter being the maximum or short-circuit value. 

For q = p, or a load of 45° load, the voltage diagram is the 
straight line shown by crosses in Fig. 67. That is, in this case, 
the monocyclic voltage, e B , is in quadrature with the supply voltage, 



f,;it ;ill toads, while For non-inductive load the monocyclic voltage 
c, not only shrinks with increasing load, but also shifts in phase, 
from quadrature position, and the diagram is in the latter case 
shown for 4 amp. load by the clotted lines in Fig. 67. 

In Fig. 68 the drawn tinea correspond to non-inductive bftd 
The regulation for 45° lagging load is shown by dotted lines in 
Fig. 68. 

e'o shows the quadrature component of the monocyclic voltage. 
e ii, at non-inductive load. That is, the component of e«, which is 
in phase with e, and therefore could he neutralised by inserting 
into 6o a part of the voltage, e, by transformation. 

As seen in Fig. 68, the supply current is a maximum of 20 amp. 
at no-load, and decreases with increasing load, to 10 amp. at 
short-circuit load. 

The apparent efficiency of the device, that is, Ihe ratio of the 
volt-ampere output: 

Qa = e n i„ 
to the volt-ampere input: 

Q = ei 
is given by the curve, y, in Fig. 68. 

As seen, the apparent efficiency is very low, reaching a maxi- 
mum of 14 per cent. only. 

If the monocyclic square is produced by capacity and induc- 
tance, the extreme case of dropping of voltage, e,„ witli increase of 
current, i . is reached in that the circuit of the voltage, eo, becomes 
a constant-current circuit, and this case is more fully discussed 
in Chapter XIV of "Theory and Calculation of Electric Circuits " 
as a constant-potential constant-current transforming device. 

Induction Phase Converter 

130. The magnetic field of a single-phase induction motor at 
or near synchronism is a uniform rotating field, or nearly so, 
deviating from uniform intensity and uniform rotation only by 
the impedance drop of the primary winding. Thus, in any coil 
displaced in position from the single-phase primary coil of the 
induction machine, a voltage is induced which is displaced in 
phase from the supply voltage by the same angle as the coil is 
displaced in position from the coil energized by the supply vol- 
tage. An induction machine running at or near synchronism 
thus can be used as phase converter, receiving single-phase sup- 



ply voltage, E , and current, Z , in one coil, and producing a voltage 
of displaced phase, E 2 , and current of displaced phase, J 2 , in 
another coil displaced in position. 

Thus if a quarter-phase motor shown diagrammatically in Fig. 
69A is operated by a single-phase voltage, E , supplied to the one 

\ju y, 


-> E 




Z z, 

















Fig. 69. — Induction phase converter diagram. 

phase, in the other phase a quadrature voltage, E 2 , is produced 
and quadrature current can be derived from this phase. 

The induction machine, Fig. 69 A, is essentially a transformer, 
giving two transformations in series: from the primary supply 
circuit, Eohj to the secondary circuit or rotor, EJx, and from the 
rotor circuit, EJi, as primary circuit, to the other stator circuit 


or second phase, EJ*, as secondary circuit. It thus can be repre- 
sented diagrammatic ally by the double transformer Fig. 69B. 

The only difference between Fig. 69A and 69B is, that in Fig. 
69.4 the synchronous rotation of the circuit, £\Zi, carries the cur- 
rent, 1 1, 90° in space to the second transformer, and thereby pro- 
duces a 90° time displacement. That is, primary current and 
voltage of the second transformer of Fig. 69/* are identical in 
intensity with the secondary currents and voltage of the first, 
transformer, but lag behind them by a quarter period in space 
and thus also in time. The momentum of the rotor takes care 
of the energy storage during this quarter period. 

As the double transformer, Fig. 60S, can be represented by 
the double divided circuit, Fig. 69C, 1 Fig. 69C thus represents 
the induction phase converter, Fig. 69A, in everything except 
that it does not show the quarter- period lag. 

As the equations derived from Fig. 69C are rather complicated, 
the induction converter can, with sufficient approximation for 
most purposes, be represented either by the diagram Fig. 69D, 
or by the diagram Fig. 69i?. Fig. 69Z> gives the exciting current 
of the first transformer too large, but that of the second trans- 
former too small, so that the two errors largely compensate. 
The reverse is the case in Fig. 69E, and the correct value, cor- 
responding to Fig. 69C, thus lies between the limits 69D and 69£. 
The error made by either assumption, 69/> or 69i?, thus man be 
smaller than the difference between these two assumptions. 

131. Let: 

Y = So _ j°o ■ primary exciting admittance of the induc- 
tion machine, 

Zo = r + jxq = primary, and thus also tertiary self-induc- 
tive impedance, 

Zi ™ Ti + jx, = secondary self-inductive impedance, 
all at full frequency, and reduced to the same number of turns. 

Y* = tfi — jbs = admittanceof the load on the second phase; 
denoting further: 

z = z a + z„ 

1 "Theory and Calculation of Al terns ling-curr 
edition, page 204. 

Phenomena," 5th 



it is, then, choosing the diagrammatic representation, Fig. 69D: 

/o — #oFo = /i + $%Yq ■» lit 

$o = #i + 2 Z (/t + #tVo)i 

/« - ft^s; 

substituting (11) into (10) and transposing, gives: 
if the diagram, Fig. 692?, is used, it is: 




jj?2 = 

l + 2Z(Yo + Yt[l+YoZl) 

which differs very little from (12). 
And, substituting (11) and (12) into (9): 

Io = $t(Yo+Y t )+VoYo, 

F Yt + 2Yo + 2Z Y (Yo +Y i ) 
** 1 + 2ZJY.+ Y t ) 



Equations (11), (12) and (13) give for any value of Load, Y%, 
on the quadrature phase, the values of voltage, #*, and current, 
It, of this phase, and the supply current, fo, at supply voltage, ##. 

It must be understood, however, that the actual quadrature 
voltage is not &, but is jjfo carried a quarter phase forward by 
the rotation, as discussed before. 

132. As instance, consider a phase converter operating at con- 
stant supply voltage: 

of the constants: 


and let 

#© * e© «= 100 volts; 

>'o - 0.01 -0.1;, 

Zv = Zi « 0.05 + 0.15;; 

I = 0.1 +0.3;; 

y , = u (j> -jq) 
= a (0.8 - 0.6;;, 

that is, a load of W) per cent, power-factor, winch <x>m*>poudi- 
about to the average power-factor of au inductiou motoi . 


It is, then, substituted into (11) to (13): 

. _ _ioo 

*■ (1.062 + 0.52 a) + j (0.36 a - 0.0 

.r 80j)«L 

= 0, or no-load, this gives: 

e s = 94.1, 
li - 0, 
I, - 19.5; 

= ™, or short-circuit, this gives: 

ei - 0, 
i, - 159, 

The voltage diagram is shown in Fig. 70, and the load char- 
acteristics or regulation curves in Fig. 71. 

As seen: the voltage, e-t, is already at no-load lower than the 
supply voltage, e«, due to the drop of voltage of the exciting cur- 
rent in the self-inductive impedance of the phase converter. 

In Fig. 70 arc marked by circles the values of voltage, en, for 
every 20 per cent, of the short-circuit current. 

Fig. 71 gives the quadrature component of the voltage, e%, as 
e"j, and the apparent efficiency, or ratio of volt-ampere output 
to volt-ampere input: 

and the primary supply current, Jo- 
lt is interesting to compare the voltage diagram and especially 
the load and regulation curves of the induction phase converter, 
Figs. 70 and 71, with those of the monocyclic square, Fige, 89 
and 68. 

As seen, in the phase converter, the supply current at no-load 
is small, is a mere induction-machine exciting current, and in- 
creases with the load and approximately proportional thereto. 

The no-load input of both devices is practically the same, hut 
the voltage regulation of the phase converter is very much better; 
the voltage drops to zero at 150 amp. output, while that of the 


Fie. 70.— Induction phase converter, topographic regulation characteristic. 


Y -.01- .lj, Z -Z,=.05 +.15 j 

Y,= (.6-.6J) 

e - 100 VOLTS 



6 a 









1 1 

■i i 


Fio. 71. — Induction phase converter, regulation c 


monocyclic square reaches zero already at 10 amp. output. 
illustrates the monocyclic character of the latter,' that is, the limi- 
tation of the output of the quadrature voltage. 

As the result hereof, the phase converter reaches fairly good 
apparent efficiencies, 54 per cent., and reaches these already at 
moderate loads. 

The quadrature component, e"g, of the voltage, en, is much 
smaller with the phase converter, and, being in phase with the 
supply voltage, eo, can he eliminated, and rigid quadrature relation 
of e 2 with Bo maintained, by transformation of a voltage — e"j 
from the single-phase supply into the secondary. Furthermore, as 
e"i is approximately proportional to i" u — except at very low loads 
—it could be supplied without regulation, by a series transformer, 
that is, by connecting the primary of a transformer in series with 
the supply circuit, u, the secondary in series with e%. Thereby 
€i would be maintained in almost perfect quadrature relation 
to Co at all important loads. 

Thus the phase converter is an energy- transforming device, 
while the monocyclic square, as the name implies, is a device for 
producing an essentially wattless quadrature voltage. 

133. A very important use of the induction phase converter 
is in series with the polyphase induction motor for which it sup- 
plies the quadrature phase. 

In this case, the phase, e 0l in of the phase converter is connected 
in series to one phase, e'oi'o, of the induction motor driving the 
electric car or polyphase locomotive, into the circuit of the single- 
phase supply voltage, e = e u + e'o, and the second phase of the 
phase converter, e^, ii, is connected to the second phase of the 
induction motor. 

This arrangement still materially improves the polyphase regu- 
lation: the induction motor receives the voltages: 

e t = tt. 

At no-load, e, is a maximum. With increasing load, e t = t 
drops, and hereby also drops the other phase voltage of the ii 
duction motor, e'„. This, however, raises the voltage, e - r - 
e' , on the primary phase of the phase converter, and hereby 
raises the secondary phase voltage, ei = e' t , thus maintains the 


two voltages e' and e' 2 impressed upon the induction motor much 
more nearly equal, than would be the case with the use of the 
phase converter in shunt to the induction motor. 

Series connection of the induction phase converter, to the in- 
duction motor supplied by it, thus automatically tends to regu- 
late for equality of the two-phase voltages, e' and e' 2 , of the induc- 
tion motor. Quadrature position of these two-phase voltages 
can be closely maintained by a series transformer between i and 
t's, as stated above. 

It is thereby possible to secure practically full polyphase motor 
output from an induction motor operated from single-phase sup- 
ply through a series-phase converter, while with parallel connec- 
tion of the phase converter, the dropping quadrature voltage 
more or less decreases the induction motor output. For this 
reason, for uses where maximum output, and especially maximum 
torque at low speed and in acceleration is required, as in rail- 
roading, the use of the phase converter in series connections to 
the motor is indicated. 

Synchronous Phase Converter and Single-phase Generation 

134. While a small amount of single-phase power can be taken 
from a three-phase or in general a polyphase system without dis- 
turbing the system, a large amount of single-phase power results 
in unbalancing of the three-phase voltages and impairment of 
the generator output. 

With balanced load, the impedance voltages, e' = iz, of a three- 
phase system are balanced three-phase voltages, and their effect 
can be eliminated by inserting a three-phase voltage into the 
system by three-phase potential regulator or by increasing the 
generator field excitation. The impedance voltages of a single- 
phase load, however, are single-phase voltages, and thus, com- 
bined with the three-phase system voltage, give an unbalanced 
three-phase system. That is, in general, the loaded phase drops 
in voltage, and one of the unloaded phases rises, the other also 
drops, and this the more, the greater the impedance in the circuit 
between the generated three-phase voltage and the single-phase 
load. Large single-phase load taken from a three-phase trans- 
mission line — as for instance by a supply station of a single-phase 
electric railway — thus may cause an unbalancing of the trans- 
mission-line voltage sufficient to make it useless. 

A single-phase system of voltage, e, may be considered as com- 
bination of two balanced three-phase systems of opposite phase 


rotation: ,,-,,- 2 and v 2 i ^ where t = VI = .-,' 

The unbalancing of voltage caused by a single-phase load of 
impedance voltage, e = iz, thus is the same as that caused by 
two three-phase impedance voltages, e/2, of which the one has 
the same, the other the opposite phase rotation ;iw the three-phase 
supply system. The former can be neutralized by raising the 
supply voltage by e/2, by potential regulator or generator excita- 
tion. This means, regulating the voltage for the average drop. 
It leaves, however, the system unbalanced by the impedance 
voltage, e/2, of rotation. The latter thus can l>c 
compensated, and the unbalancing eliminated, by inserting into 
the three-phase system a set of three-phase voltages, e/2, of re- 
verse-phase rotation. Such a system can be produced by a three- 
phase potential regulator by interchanging two of the phases. 
Thus, if A, B, C are the three three-phase supply voltages, im- 
pressed upon the primary or shunt coils o, b, c of a three-phase 
potential regulator, and 1, 2, 3 are the three secondary or series 
coils of the regulator, then the voltages induced in 1, 3, 2 are 
three-phase of reverse-phase rotation to A, B, C, and can be in- 
serted into the system for balancing the unbalancing due to 
single-phase load, in the resultant voltage: A + 1, B + 3, C + 2. 
It is obviously necessary to have the potential regulator turned 
into such position, that the secondary voltages 1, 3, 2 have the 
proper phase relation. This may require a wider range of turn- 
ing than is provided in the potential regulator for controlling 
balanced voltage drop. 

It thus is possible to restore the voltage balance of a three- 
phase system, which is unbalanced by a single-phase load of im- 
pedance voltage, e', by means of two balanced three-phase poten- 
tial regulators of voltage range, e'/2. connected so that the one 
gives the same, the other the reverse phase rotation of the main 
three-phase system. 

Such an apparatus producing a balanced polyphase system ui 
reversed phase rotation, for inserting in series into a polyphase 
system to restore the balance on single-phase load, is called n 
phase balancer, and in the present case, a stationary inducliun 
photo balancer. 

A synchronous machine of opposite phase rotation to the main 
system voltages, and connected in series thereto, would then be 
a synchronous phase balancer. 



The purpose of the phase balancer, thus, is the elimination of 
the voltage unbalancing due to single-phase load, and its capacity 
must be that of the single-phase impedance volt-amperes. It 
obviously can not equalize the load on the phases, but the flow 
of power of the system remains unbalanced by the single-phase 


136. The capacity of targe .synchronous generators is essentially 
determined by the heating of the armature coils. Increased load 
on one phase, therefore, is not neutralized by lesser load on the 
other phases, in ils limitation of output by heating of the arma- 
ture coils of the generators. 

The most serious effect of unbalanced load on the generator is 
hat due to the pulsating armature reaction. With balanced 
•olyphase load, the armature reaction is constant in intensity 
nd in direction, with regards to the field. With single-phase 
id, however, the armature reaction is pulsating between zero 
ind twice its average value, thus may cause a double-frequency 
pulsation of magnetic flux, which, extending through the field 
circuit, may give rise to losses and heating by eddy currents in 
the iron, etc. With the slow-speed multipolar engine-driven 
alternators of old, due to the large number of poles and low per- 
ipheral speed, the ampere-turns armature reaction per pole 
amounted to a few thousand only, thus were not sufficient to 
cause serious pulsation in the magnetic-field circuit. With the 
large high-speed turbo-alternators of today, of very few poles, 
and to a somewhat lesser extent also with the larger high-speed 
machines driven by high -head waterwheels, the armature reac- 
tion per pole amounts to very many thousands of ampere-turns. 
Section anil length of the field magnetic circuit are very large. 
Even a moderate pulsation of armature reaction, due to the un- 
balancing of the flow of power by single-phase load, then, may 
cause very large losses in the field structure, and by the resultant 
heating seriously reduce the output of the machine. 

It then becomes necessary either to balance the load between 
the phases, and so produce the constant armature reaction of 
balanced polyphase load, or to eliminate the fluctuation of the 
armature reaction. The latter is done by the use of an effective 
squirrel-cage short-circuit winding in the pole faces. The double- 
frequency pulsation of armature reaction induces double-fre- 
! currents in the squirrel cage— just as in the single-phase 
duetion motor— and these induced currents demagnetize, when 



the armature reaction is above, and magnetize when it is below 
the average value, and thereby reduce the fluctuation, that is, 
approximate a constant armature reaction of constant direction 
with regards to the field — that is, a uniformly rotating magnetic 
field with regards to the armature. 

However, for this purpose, the m.m.f. of the currents induced 
in the squirrel-cage winding must equal that of the armature 
winding, that is, the total copper cross-section of the squirrel cage 
must be of the same magnitude as the total copper cross-section 
of the armature winding. A small squirrel cage, such as is suffi- 
cient for starting of synchronous motors and for anti-hunting 
purposes, thus is not sufficient in high armature-reaction machines 
to take care of unbalanced single-phase load. 

A disadvantage of the squirrel -cage field winding, however, 
is, that it increases the momentary short-circuit current of 
the generator, and retards its dying out, therefore increases the 
danger of self-destruction of the machine at short-circuit. In 
the first moment after short-circuit, the field poles still carry full 
magnetic flux — as the field can not die out instantly. No flux 
passes through the armature— except the small flux required to 
produce the resistance drop, ir. Thus practically the total field 
flux must be shunted along the air gap, through the narrow sec- 
tion between field coils and armature coils. As the squirrel-cage 
winding practically bars the flux to cross it, it thereby further 
reduces the available flux section and so increases the Hux density 
and with it the momentary short -circuit current, which gives 
the m.m.f. of this flux. 

It must also be considered that the reduction of generator out- 
put, resulting from unequal heating of the armature coils due to 
unequal load on the phases is not eliminated by a squirrel-cage 
winding, but rather additional heat produced by the currents 
in the squirrel-cage conductors. 

136. A synchronous machine, just as an induction machine, 
may be generator, producing electric power, or motor, receiving 
electric power, or phase converter, receiving electric power in 
some phase, the motor phase, and generating electric power in 
some other phase, the generator phase. In the phase converter, 
the total resultant armature reaction is zero, and the armature 
reaction pulsates with double frequency between equal positive 
and negative Values. Such phase converter thus can be used to 
produce polyphase power from a single-phase supply. The in- 



duction phase converter has been discussed in the preceding, and 
the synchronous phase converter has similar characteristics, but 

a rule a better regulation, that is, gives a lietter constancy of 
voltage, and can be made to operate without producing lagging 
currents, by exciting the fields sufficiently high. 

However, a phase converter alone can not distribute single- 
phase load so as to give a balanced polyphase system. When 
transferring power from the motor phase to the generator phase, 
the terminal voltage of the motor phase equals the induced vol- 
tage plus the impedance drop in the machine, that of the gen- 
erator phase equals induced voltage minus the impedance drop, 
and the voltage of the motor phase thus must be higher than that 
of the generator phase by twice the impedance voltage of the 
phase converter (vectorially combined). 

Therefore, in converting single-phase to polyphase by phase 
converter, the polyphase system produced can not be balanced 
in voitage, but the quadrature phase produced by the converter 

ess than the main phase supplied to it, and drops off the more, 
the greater the load. 

In the reverse conversion, however, distributing a single-phase 
load between phases of a polyphase system, the voltage of the 
generator phase of the converter must be higher, that of the motor 
phase lower than that of the polyphase system, and as the gen- 
erator phase is lower in voltage than the motor phase, it follows, 
that the phase converter transfers energy oidy when the poly- 
phase system has become unbalanced by more than the voltage 
drop in the converter. That iB, while a phase converter may 
reduce the unbalancing due to single-phase load, it can never 
restore complete balance of the polyphase system, in voltage and 
in the flow of power. Even to materially reduce the unbalancing, 
requires large converter capacity and very close voltage regula- 
tion of the converter, and thus makes it an uneconomical machine. 

To balance a polyphase system under single-phase load, there- 
fore, requires the addition of a phase balancer to the phase 
converter. Usually a synchronous phase balancer, would be 
employed in this case, that is, a small synchronous machine of 
opposite phase rotation, on the shaft of the phase converter, 
and connected in series thereto. Usually it is connected into 
the neutral of the phase converter. By the phase balancer, the 
voltage of the motor phase of the phase converter is raised 
above the generator phase so as to give a power transfer sufficient 


to balance the polyphase system, thai is, to shift half of the single 
phase power by a quarter period, and thus produce a uniform 
flow of power. 

Such synchronous phase balancer constructively is a synchro- 
nous machine, having two sets of field poles, A and B, in quad- 
rature with each other. Then by varying or reversing the 
excitation of the two sets of field poles, any phase relation of the 
reversely rotating polyphase system of the halancer to that of the 
converter can be produced, from zero to 360°. 

137. Large single-phase powers, such as are required for single- 
phase railroading, thus can be produced. 

(a) By using single-phase generators and separate .single-phase 
supply circuits. 

(b) By using single-phase generators running in multiple with 
the general three-phase system, and controlling voltage and me- 
chanical power supply so as to absorb the single-phase load by the 
single-phase generators. In this case, however, if the single- 
phase load uses the same transmission line as the three-phase 
load, phase balancing at the receiving circuit may he ncressarv. 

(c) By taking the single-phase load from the three-phase 
system. If the load is considerable, this may require special 
construction of the generators, and phase balancers. 

(d) By taking the power all as balanced three-phase power 
from the generating system, and converting the required amount 
to single-phase, by phase converter and phase balancer. This 
may be done in the generating station, or at the receiving station 
where the single-phase power is required. 

Assuming that in addition to a balanced three-phase load of 
power, Pn, a single-phase load of power, P, is required. Estimating 
roughly, that the single-phase capacity of a machine structure is 
half the three-phase capacity of the structure — which probably 
is not far wrong — then the use of single-phase generators gives 
us /Vkw. three-phase, and P-kw. single-phase generators, and us 
the latter is equal in size to 2 P-kw. three-phase capacity, the 
total machine capacity would lie P<> + 2 P. 

Three-phase generation and phase conversion would require 
Pi + 7* kw. in three-phase generators, and phase converters 
transferring half the single-phase power from the phase which is 
loaded by single-phase, to the quadrature phase. That is, the 
phase converter must have a capacity of P/2 kw. in the motor 
phase, and P/2 kw. capacity in the generator phase, or a total 


capacity of P kw. Thus the total machine capacity required for 
both kinds of load would again be P + 2 P kw. three-phase 

Thus, as regards machine capacity, there is no material differ- 
ence between single-phase generation and three-phase genera- 
tion with phase conversion, and the decision which arrangement is 
preferable will largely depend on questions of construction and 
operation. A more complete discussion on single-phase genera- 
tion and phase conversion is given in A. I. E. E. Transactions, 
November, 1916. 


Self-compounding Alternators— Self-starting Synchro- 
nous Motors — Arc Rectifier — Brush and Thomson 
Houston Arc Machine — Leblanc Panchahuteur — 
Permutator — Synchronous Converter 

138. Rectifiers ffir converting alternating into direct current 
have been designed and built since many years. As mechanical 
rectifiers, mainly single-phase, they have found a limited use for 
small powers since a long time, and during the last years arc 
rectifiers have found extended use for small and moderate powers, 
for storage-battery charging and for series arc lighting by constant 
direct current. For large powers, however, the rectifier does not 
appear applicable, but the synchronous converter takes its place. 
The two most important types of direct-current arc-light ma- 
chines, however, have in reality been mechanical rectifiers, and 
for compounding alternators, and for starting synchronous 
motors, rectifying commutators have been used to a considerable 

Let, in Fig. 72, e be the alternating voltage wave of the supply 
source, and the connections of the receiver circuit with this sup- 
ply source be periodically and synchronously reversed, at the 
zero points of the voltage wave, by a reversing commutator 
driven by a small synchronous motor, shown in Fig. 73. In the 
receiver circuit the voltage wave then is unidirectional but pul- 
sating, as shown by e in Fig. 74. 

If receiver circuit and supply circuit both are non-inductive, 
the current in the receiver circuit is a pulsating unidirectional 
current, shown as i in dotted lines in Fig. 74, and derived from 
the alternating current, i, Fig. 72, in the supply circuit. 

If, however, the receiver circuit is inductive, as a machine field, 
then the current, i«, in Fig. 75, pulsates less than the voltage, e e , 
which produces it, and the current thus does not go down in wo, 
but is continuous, and its pulsation the less, the higher the in- 
ductance. The current, i, in the alternating supply circuit, how- 



Fia. 72. — Alternating sine wave. 

AC or 

Fig. 73. — Rectifying commutator. 

Fig. 74. — Rectified wave on non inductive load. 

Fig. 75. — Rectified wave on-inductive load. 

Fig. 76. — Alternating supply wave to rectifier on inductive load. 


ever, from which the direct current, in, is derived by reversal, must 
go through zero twice during each period, thus must have the 
Bhape shown as i in Fig. 76, that is, must abruptly reverse. If, 
however, the supply circuit contains any self-inductance — and 
every circuit contains some inductance — the current can not 
change instantly, but only gradually, the slower, the higher the 
inductance, and the actual current in the supply circuit ftsamnes 


Fig. 77.— DiiT.TrnM 

itifier on inductive toad. 

a shape like that shown in dotted lines in Fig. 76. Thus the cur- 
rent in the alternating part and that, in the rectified part of the- 
circuit can not he the same, but a difference must exist, as shown 
as i' in Fig. 77. This current, (■', passes between the two parts 

Fid. 78. — Rectifier with AC ami D.C. aliunl resist 

of the circuit, as arc at. the rectifier brushes, and causes I lie recti- 
fying commutator to spark, if there is any appreciable inductance 
in the circuit. The intensity of the sparking current depends 
on the inductance of the rectified circuit , its duration on that of 
the alternating supply circuit. 

By providing a byepath for this differential current, /, ilie 
sparking is mitigated, and thereby the amount of power, which BSD 
Ik 1 rectified, increased. This is done by shunting a non-indaotivc 
resistance across the rectified circuit, r„, or across the alternating 
circuit, r, or both, as shown in Fig. 78. If this resistance is low . 
i considerable power and finally increases sparking 


by the increase of rectified current; if it is high, it has little effect. 
Furthermore, this resistance should vary with the current. 

The belt-driven alternators of former days frequently had a 
compounding series field excited by such a rectifying commutator 
on the machine shaft, and by shunting 40 to 50 per cent, of the 
power through the two resistance shunts, with careful setting of 
brushes as much as 2000 watts have been rectified from single- 
phase 125-cycle supply. 

Single-phase synchronous motors were started by such recti- 
fying commutators through which the field current passed, in 
series with the armature, and the first long-distance power trans- 


Fio. 79. — Open-circuit rectifier. Fig. 80. — Short-circuit rectifier. 

mission in America (Telluride) was originally operated with 
single-phase machines started by rectifying commutator — the 
commutator, however, requiring frequent renewal. 

139. The reversal of connection between the rectified circuit 
and the supply circuit may occur either over open-circuit, or 
over short-circuit. That is, either the rectified circuit is first 
disconnected from the supply circuit — which open-circuits both 
— and then connected in reverse direction, or the rectified circuit 
is connected to the supply circuit in reverse direction, before 
being disconnected in the previous direction — which short-circuits 
both circuits. The former, open-circuit rectification, results if 
the width of the gap between the commutator segments is greater 
than the width of the brushes, Fig. 79, the latter, short-circuit 
rectification, results if the width of the gap is less than the width 
of the brushes, Fig. 80. 

In open-circuit rectification, the alternating and the rectified 
voltage are shown as e and e in Fig. 81. If the circuit is non- 
inductive, the rectified current, t , has the same shape as the vol- 



tage, 60, but the alternating current, t, is as shown in Fig. 81 as t. 
If the circuit is inductive, vicious sparking occurs in this case 
with open-circuit rectification, as the brush when leaving the 



Fio. 81. — Voltage and current waves in open-circuit rectifier on non-induc- 
tive load. 

commutator segment must suddenly interrupt the current. That 
is, the current does not stop suddenly, but continues to flow as 
an arc at the commutator surface, and also, when making con- 


Fiu. 82. — Voltage and current wave in open-circuit rectifier on inductive 

load, showing sparking. 

tact between brush and segment, the current does not instantly 
reach full value, but gradually, and the current wave thus is as 
shoWn as i and to in Fig. 82, where the shaded area is the arcing 
current at the commutator. 
Sparkless rectification may be produced in a circuit of moderate 



inductance, with open-circuit rectification, by shifting the brushes 
so that the brushes open the circuit only at the moment when 
the (inductive) current has reached zero value or nearly so, as 

Fig. 83. — Voltage waves of open-circuit rectifier with shifted brushes. 

shown in Figs. 83 and 84. In this case, the brush maintains con- 
tact until the voltage, e, has not only gone to zero, but reversed 
sufficiently to stop the current, and the rectified voltage then is 
shown by e in Fig. 83, the current by i and to in Fig. 84. 

Fig. 84. — Current waves of open-circuit rectifier with shifted brushes. 

140. With short-circuit commutation the voltage waves are as 
shown by e and e in Fig. 85. With a non-inductive supply and 
non-inductive receiving circuit, the currents would be as shown 
by i and to in Fig. 86. That is, during the period of short-circuit, 




Fig. 85. — Voltage waves of short-circuit rectifier. 


Fig. 80. — Current waves of short-circuit rectifier on non-inductive load. 

Fm. 87. — Current waves of short circuit rectifier on moderately inductive 

load, showing flashing. 



the current in the rectified circuit is zero, and is high, is the short- 
circuit current of the supply voltage, in the supply circuit. 

Xnductance in the rectified circuit retards the dying out of the 
current, but also retards its rise, and so changes the rectified 
c **rrent wave to the shapes shown — for increasing values of in- 
ductance— as to in Figs. 87, 88 and 89. 

Fig. 88. — Current waves of short-circuit rectifier on inductive load at the 

stability limit. 

Inductance in the supply circuit reduces the excess current 
value during the short-circuit period, and finally entirely elimi- 
nates the current rise, but also retards the decrease and reversal 
of the supply current, and the latter thus assumes the shapes 
shown — for successively increasing values of inductance — as i in 
Figs. 87, 88 and 89. 

Fig. 89. — Current waves of short-circuit rectifier on highly inductive load, 

showing sparking but no flashing. 

As seen, in Figs. 86 and 87, the alternating supply current has 
during the short-circuit reversed and reached a value at the end 
of the short-circuit, higher than the rectified current, and at the 
moment when the brush leaves the short-circuit, a considerable 
current has to be broken, that is, sparking occurs. In Figs. 86 , 
and 87, this differential current which passes as arc at the com- 
mutator, is shown by the dotted area. It is increasing with in- 




creasing spark length, that is, the spark or arc at the commutator 
has no tendency to go out — except if the inductance is very small 
— but persists: flashing around the commutator occurs and short- 
circuits the supply permanently. 


Fig. 90. — Voltage wave of short-circuit rectifier with shifted brushes. 

In Fig. 89, the alternating current at the end of the short- 
circuit has not yet reversed, and a considerable differential 
current, shown by the dotted area, d, passes as arc. Vicious 

Fio. 91. — Current waves of short-circuit rectifier with inductive load and the 

brushes shifted to give good rectification. 

sparking thus occurs, but in this case no flashing around the 
commutator, as with increasing spark length the differential 
current decreases and finally dies out. 

In Fig. 88, the alternating current at the end of the short- 
circuit has just reached the same value as the rectified current, 


thus no current change and no sparking occurs. However, if 
the short-circuit should last a moment longer, a rising differential 
current would appear and cause flashing around the commutator. 
Thus, Fig. 88 just represents the stability limit between the 
stable (but badly sparking) condition, Fig. 89, and the unstable 
°* flashing conditions, Figs. 87 and 86. 

By shifting the brushes so as to establish and open the short- 
circuit later, as shown in Fig. 90, the short-circuited alternating 
e -m.f. — shown dotted in Figs. 90 and 85 — ceases to be symmet- 
rical, that is, averaging zero as in Fig. 85, and becomes unsym- 
ttietrical, with an average of the same sign as the next following 
voltage wave. It thus becomes a commiUating e.m.f., causes a 
more rapid reversal of the alternating current during the short- 
circuit period, and the circuit conditions, Fig. 89, then change to 
that of Fig. 91. That is, the current produced by the short- 
circuited alternating voltage has at the end of the short-circuit 
period reached nearly, but not quite the same value as the recti- 
fied current, and a short faint spark occurs due to the differential 
current, d. This Fig. 91 then represents about the best condition 
of stable, and practically sparkless commutation: a greater brush 
shift would reach the stability limit similar as Fig. 88, a lesser 
brush shift leave unnecessarily severe sparking, as Fig. 89. 

141. Within a wide range of current and of inductance — espe- 
cially for highly inductive circuits — practically sparkless and 
stable rectification can be secured by short-circuit commutation 
by varying the duration of the short-circuit, and by shifting the 
brushes, that is, changing the position of the short-circuit during 
the voltage cycle. 

Within a wide range of current and of inductance, in low-in- 
ductance circuits, practically sparkless and stable rectification 
can be secured also by open-circuit rectification, by varying the 
duration of the open-circuit, and by shifting the brushes. 

The duration of open-circuit or short-circuit can be varied by 
the use of two brushes in parallel, which can be shifted against 
each other so as to span a lesser or greater part of the circumfer- 
ence of the commutator, as shown in Fig. 92. 

Short-circuit commutation is more applicable to circuits of 
high, open-circuit commutation to circuits of low inductance. 

But, while either method gives good rectification if overlap and 
brush shift are right, they require a shift of the brushes with every 
change of load or of inductivity of the load, and this limits the 



practical usefulness of rectification, as such readjustment with 
every change of circuit condition is hardly practicable. 

Short-circuit rectification has been used to a large extent on 
constant-current circuits; it is the method by which the Thomson- 

Fig. 92. — Double-brush rectifier. 

Houston (three-phase) and the Brush arc machine (quarter- 
phase) commutates. For more details on this see "Theory and 
Calculations of Transient Phenomena/ ' Section II. 

Ficj. 93. — Volt ago waves of open -circuit rectifier charging storage battery. 

Open-circuit rectification has found a limited use on non-in- 
ductive circuits containing a counter e.m.f., that is, in charging 
ntoragc batteries. 

If, in Fig. 93, e is the rectified voltage, and e x the counter e.m.f. 

p n ^_ p 

of t ho storage battery, the current is i = » where r = ef- 
fective resistance of the battery, and if the counter e.m.f. of the 



battery, e h equals the initial and the final value of e , as in Fig. 
93, eo — e and thus t start and end with zero, that is, no abrupt 
change of current occurs, and moderate inductivity thus gives 
no trouble. The current waves then are: i and i Q in Fig. 94. 





Fio. 94 —Current waves of open-circuit rectifier charging storage battery. 

142. Rectifiers may be divided into reversing rectifiers, like 
those discussed heretofore, and shown, together with its supply 
transformer, in Figs. 95 and 96, and contact-making rectifiers, 
shown in Figs. 97 and 98, or in its simplest form, as half-wave 
rectifier, in Fig. 99. 


Fia. 95. — Reversing rectifier with Fio. 96. — Reversing rectifier 
alternating-current rotor. with direct-current rotor. 

As seen, in Fig. 99, contact is made between the rectified cir- 
cuit and the alternating supply source, T, during one-half wave 
only, but the circuit is open during the reverse half wave, and the 
rectified circuit, B t thus carries a series of separate impulses of cur- 
rent and voltage as shown in Fig. 100 as i\. However, in this 
case the current in the alternating supply circuit is unidirectional 
also, is the same current, i\. This current produces in the trans- 
former, T, a unidirectional magnetization, and, if of appreciable 



magnitude, that is, larger than the exciting current of the trans- 
former, it .saturates the transformer iron. Running at or beyond 
magnetic saturation, the primary exciting current of the trans- 
former then becomes excessive, the hysteresis heating due to the 
unsymmetrical magnetic cycle is greatly increased, and the 
transformer endangered or destroyed. 

Half-wave rectifiers thus are impracticable except for extremely 
small power. 

The full-wave contact-making rectifier, Fig. 97 or 98, does not 
have this objection. In this type of rectifier, the connection be- 
tween rectified receiver circuit and 
alternating supply circuit are not 
synchronously reversed, as in Fig. 95 
or 96, but in Fig. 97 one side of the 
rectified circuit, B, is permanently 
connected to the middle m of the 
alternating supply circuit, T, while the 
other side of the rectified circuit is 
synchronously connected and discon- 
nected with the two sides, a and 
6, of the alternating supply circuit. 
Or we may say: the rectified circuit takes one-half wave from 
the one transformer half coil, ma, the other half wave from 
the other transformer half coil, mb. Thus, while each of the two 
transformer half coils carries unidirectional current, the uni- 
directional currents in the two half coils flow in opposite direc- 
tion, thus give magnetically the same effect as one alternating 



current in one half coil, and no unidirectional magnetization re- 
sults in the transformer. 

In the contact-making rectifier, Fig. 98, the two halves of the 
rectified circuit, or battery, B, alternately receive the two suc- 
cessive half waves of the transformer, T. 

The voltage and current waves of the rectifier, Fig. 97, are 
shown in Fig. 100. e is the voltage wave of the alternating sup- 

Fia. 100. — Voltage and current waves of contact-making rectifier with 

direct-current rotor. 

ply source, from a to b. d and e% then are the voltage waves of 
the two half coils, am and bm, i\ and i 2 the two currents in these 
two half coils, and to the rectified current, and voltage in the 
circuit from m to c. The current, i\ y in the one, and, i% } in the other 
half coil, naturally has magnetically the same effect on the pri- 
mary, as the current, i\ + ii = z'o, in one half coil, or the current, 
io/2 = i, in the whole coil, ab, would have. Thus it may be said: 
in the (full-wave) contact-making rectifier, Fig. 97, the rectified 




voltage, e , is one-half the alternating voltage, e, and the rectified 
current, io, is twice the alternating current, i. However, the i*r 

in the secondary coil, a&, is greater, by y/% 
than it would be with the alternating cur- 
rent, i = io/2. 

Inversely, in the contact-making rectifier, 
Fig. 98, the rectified voltage is twice the 
alternating voltage, the rectified current 
half the alternating current. 

Contact-making rectifiers of the type 
Fig. 97 are extensively used as arc recti- 
fiers, more particularly the mercury-arc 
rectifier shown diagrammatically in Fig. 
Fig. 101.— Mercury- 101. This may be compared with Fig. 
arc rectifier, contact 97. That is, the making of contact during 

one half wave, and opening it during the 
reverse half wave, is accomplished not by mechanical syn- 
chronous rotation, but by the use of the arc as unidirec- 




Fig. 102. — Diagram of mercury-arc rectifier with its reactances. 

tional conductor: 1 with the voltage gradient in one direc- 
tion, the arc conducts; with the reverse voltage gradient 

1 Sec Chapter II of "Theory and Calculation of Electric Circuits/' 



— the other half wave — it does not conduct. A large induc- 
tance is used in the rectified circuit, to reduce the pulsation of 
current, and inductances in the two alternating supply circuits 
— either separate inductances, or the internal reactance of the 
transformer — to prolong and thereby overlap the two half waves, 
and maintain the rectifying mercury arc in the vacuum tube. A 
diagram of a mercury-arc rectifier with its reactances, x x , x 2 , x Q , 


Fio. 103. — Voltage and current waves of mercury-arc rortilier. 

is shown in Fig. 102. The "A.C. reactances" Xi and j* often 
are a part of the supply transformer; the "D.C reactance" x 
is the one which limits the pulsation of the rectified current. The 
waves of currents, i h i 2 and i 0) as overlapped by the inductances, 
Xi, x* and x , are shown in Fig. 103. 

Full description and discussion of the mercury-arc rectifier is 
contained in "Theory and Calculation of Transient Phenomena, 9 ' 
Section II, and in "Radiation, Light and Illumination." 



143. To reduce the sparking at the rectifying commutator, 

the gap between the segments may be divided into a number of 
gaps, by small auxiliary segments, as shown in Fig. 104, and 
these then connected to intermediate points of the shunting re- 

Fio. 104. — Rectifier with intermediate segments. 

sistance, r, which takes the differential current, t — *i or the 
auxiliary segments may be connected to intermediate points of 
the winding of the transformer, T, which feeds the rectifier, 
through resistances, r', and the supply voltage thus succettsnlj 


Via. 105. — Three-phase >'-eonriorted reetifier. 

rectified. Or both arrangements may be combined, that is, the 
intermediate segments connected to intermediate points of the 
resistance, r, and intermediate points of the transformer wind- 
ing, T. 

Polyphase rectification can yield somewhat larger power than 




single-phase rectification. In polyphase rectification, the ■ ,, -- 
ments and circuits may he in star connection, or in ring connAB* 
tion, or independent. 

Thus, Fig. 105 shows the arrangement of a star-connected {<U 
Y -connected) three-phase rectifier. The arrangement of Fig. 103 
is shown again in Fig. 100, in simpler representation, by showing 
the phases of the alternating supply circuit, and their relation 
to each other and to the rectifier segments, by heavy black lines 
inside of the commutator. 

Fig. 107 shows a ring or delta-connected three-phase rectifier. 

Fig. 108 a star-connected quarter-phase rectifier and Fig. 
109 a quarter-phase rectifier with two independent quadra- 

Fio. 112. — Voltage warns of quarter- phase sl!ir-n>uiii.>i'te\) rralificr. 
ture phases, while Fig. 1 10 shows a ring-connected quarter-phase 

The voltage waves of the two coils in Fig. 109 are shown as 
d and e 2 in Fig. 112, in thin lines, and the rectified voltage by the 
heavy black line, e u , in Fig. 112. As seen, in star connection, tin- 
successive phases alternate in feeding the rectified circuit, but 
only one phase is in circuit at a time, except during the Limn of 
the overlap of the brushes when passing tin 1 gap between suc- 
cessive segments. At that time, two sueccssivi- phases arc in 
multiple, and the current changes from the phase of decreasing 
voltage to that of rising voltage. Only a part of the voltage 
wave is thus used. The unused part of the wave, c,. is -hmni 
shaded in Fig. 1 12. 

Fig. 113 shows the voltages of the four phases, ri. fj, cj, f«, in 
ring connection, Fig. 110, and as e the rectified voltage. As 
seen, in this case, all the phases are always in circuit, two phases 
always in series, except during the overlap of the bnwhee »1 the 
gap between the segments, when a phase is short-circuited dur- 
ing commutation. The rectified voltage is higher than that of 
each phase, but twice as many coils are required as BOOroU of 
supply voltage, each carrying half the rectified current. 



By using two commutators in series, as shown in Fig. Ill, the 
two phases can be retained continuously in circuit while using 

Fig. 113. — Voltage waves of water-phase ring-connected rectifier. 

only two coils — but two commutators are required. The voltage 
waves then are shown in Fig. 114. 

Fig. 114. — Voltage waves of quarter-phase rectifier with two commutators. 

A star-connected six-phase rectifier is shown in Fig. 115, with 
the voltage waves in Fig. 117. The unused part of wave e\ is 

Fig. 115. — Six-phase star- 
connected rectifier. 

Fig. 110. — Six-phase ring- 
connected rectifier. 

shown shaded. A six-phase ring-connected rectifier in Fig. 
116, with the voltage waves in Fig. 118. 


144. As seen, with larger number of phases, star connection 
becomes less and less economical, as a lesser part of the alternat- 
ing voltage wave is used in the rectified voltage: in quarter-phase 

Fig. 117— Voltage w 

omieotod rectifier. 

rectification 90° or one-half, in six-phase rectification 60° or 
one-third, etc. In ring connection, however, all the phases are 

Flu. 118.— Voltage 

continuously in circuit, and thus no loss of economy occurs by 
the use of the higher numl>er of phases. 

Fig. 110.— Rectifying 


Therefore, ring connection is generally used in rectification 
of a larger number of phases, and star connection is never used 
beyond quarter-phase, that is, four phases, and where a higher 
number of phases is desired, to increase the output, several 


rectifying commutators are connected in series, as shown in 
Fig. 119. This represents two quarter-phase rectifiers in series 
displaced from each other by 45°, that is, an eight-phase system. 
Three-phase star-connected rectification, Fig. 106, has been 
used in the Thomson-Houston arc machine, and quarter-phase 
rectification, Fig. 108, in the Brush arc machine, and for larger 
powers, several such commutators were connected in series, as 
in Fig. 119. These machines are polyphase (constant-current) 

Fia. 120. — Counter e.m.f. shunting gaps of six-phase rectifier. 

alternators connected to rectifying commutators on the armature 

For a more complete discussion of the rectification of arc 
machine see "Theory and Calculation of Transient Electric 
Phenomena," Section II. 

145. Even with polyphase rectification, the power which can 
be rectified is greatly limited by the sparking caused by the dif- 
ferential current, that is, the difference between the rectified 
current, io, which never reverses, but is practically constant, and 
the alternating supply current. Resistances shunting the gaps 
between adjoining segments, as bye path for this differential cur- 
rent, consume power and mitigate the sparking to a limited extent 
only. A far more effective method of eliminating the sparking 
is by shunting this differential current not through a mere non- 
inductive resistance, but through a non-inductive resistance which 
contains an alternating counter e.m.f. equal to that of the supply 
phase, as shown diagrammatically in Fig. 120. 

In Fig. 120, ei to e* are the six phases of a ring-connected six- 
phase system; e\ to e\ are e.m.fs. of very low self -inductance 



and mock-rate resistance, r, shunted between the rectifier seg- 
ments. Fig. 121 then shows the wave shape of the current, i» — i, 
which passes through these counter e.m.fs.,e' (assuming that the 
circuit of e', t, contains no appreciable self-inductance). 

Such polyphase counter e.m.fs. for shunting the differentia! 
current between the segments, can be derived from the syn- 
chronous motor which drives the rectifying commutator. By 
winding the synchro nous -mo tor armature ring connected and 

shape of differential current. 

of the same number of phases as the rectifying commutator, and 

using a revolving-armature synchronous motor, the synchronous- 
motor armature coils can be connected to the rectifier segments, 
and hyepass the differential current. To carry this current, the 
armature conductor of the synchronous motor has to be increased 
in size, but as the differential current is small, this is relatively 

Fio. 122.— Leb lane's Paiiuliahulciir. 

little. Hereby ihc output which can be derived from a poly- 
phase rectifier can be very largely increased, (he more, (he larger 
the number of phases. This is Leblanc's Panchahuteur, shown 
diagnimmatically in Fig. 122 for six phases. 

Such polyphase rectifier with non-inductive counter 
byepath through the synchronous-motor armature requires as 
'many collector rings as rectifier segments. It can rectify large 
currents, but is limited in the voltage per phase, that is, 
per segment, to 20 to 30 volts at best, and the larger th 



required rectified voltage, the larger thus must be the number of 

146. Any number of phases can be produced in the secondary 
system from a three-phase or quarter-phase primary polyphase 
system by transformation through two or three suitably designed 
stationary transformers, and a large number of phases thus is 
not objectionable regarding its production by transformation. 
The serious objection to the use of a large number of phases 
(24, 81, etc.) is, that each phase requires a collector ring to lead 
the current to the corresponding segment of the rectifying 

This objection is overcome by various means: 

1. The rectifying commutator is made stationary and the 
brushes revolving. The synchronous motor then has revolving 


Fig. 123. — Phase splitting by synchronous-motor armature: synchronous 


field and stationary armature, and the connection from the 
stationary polyphase transformer to the commutator segments 
and the armature coils is by stationary leads. 

Such a machine is called a yermutator. It has been built to a 
limited extent abroad. It offers no material advantage over the 
synchronous converter, but has the serious disadvantage of re- 
volving brushes. This means, that the brushes can not be in- 
spected or adjusted during operation, that if one brush sparks 
by faulty adjustment, etc., it is practically impossible to find out 
which brush is at fault, and that due to the action of centrifugal 
forces on the brushes, the liability to troubles is greatly increased. 



For this reason, the permutator has never been introduced in 

this country, and has practically vanished abroad. 

2. The transformer is mounted on the revolving-motor struc- 
ture, (hereby revolving, permitting direct connection of its 
secondary leads with the commutator segments. In this case 
only the three or four primary phases have to be lead into the 
rotor by collector rings. 

The mechanical design of eucfa structure is difficult, the trans- 
former, not open to inspection during operation, and exposed to 
centrifugal forces, which limit its design, exclude oil and ilm- 
limit the primary voltage, so that with a high-voltage primary- 
supply system, double transformation becomes necessary. 

As this construction offers no material advantage over (3), 
it has never reached beyond experimental design. 

3. A lesser number of collector rings and supply phases is 
used, than the number of commutator segments and synchronous- 
motor armature coils, and the latter are used as autotransformers 
to divide each supply phase into two or more phases feeding suc- 
cessive commutator segments. Fig. 123 shows a 12-phase recti- 
fying commutator connected to a 12-phase synchronous motor 
with six collector rings for a six-phase supply, so that each sup- 
ply phase feeds two motor phases or coils, and thereby two recti- 
fier segments. Usually, more than two segments are used per 
supply phase. The larger the number of commutator segments 
per supply phase, the larger is the differential current in the 
synchronous motor armature coils, and the larger thus must bj| 
I his motor. 

Calculation, however, shows that there is practically no gain 
by the use of more than 12 supply phases, and very little gain 
beyond six supply phases, and that usually the most economical 
design is that using six supply phases and collector rings, qq 
matter how large a number of phases is used on the commutator. 

Fig. 123 is the well-known synchronous converter, which hereby 
appears as the final development, for large powers, of the syn- 
chronous rectifier. 

This is the reason why the synchronous rectifier apparently 
has never been developed for large powers : the development of the 
polyphase synchronous rectifier for high power, by increasing 
the number of phases, byepassing the differential current which 
causes the sparking, by shunting the commutator segments with 
the armature coils of the motor, and finally reducing the number 


of collector rings and supply phases by phase splitting in the 
synchronous-motor armature, leads to the synchronous con- 
verter as the final development of the high-power polyphase 

For " synchronous converter" see "Theoretical Elements of 
Electrical Engineering," Part II, C. For some special types of 
synchronous converter see under "Regulating Pole Converter" 
in the following Chapter XXI. 



147. In the usual treatment of synchronous machines and 
induction machines, the assumption is made that the reactance, 
x, of the machine is a constant. While this is more or less 
approximately the case in many alternators, in others, especially 
in machines of large armature reaction, the reactance, x, is 
variable, and is different in the different positions of the armature 
coils in the magnetic circuit. This variation of the reactance 
causes phenomena which do not find their explanation by the 
theoretical calculations made under the assumption of constant 

It is known that synchronous motors or converters of large 
and variable reactance keep in synchronism, and are able to do 
a considerable amount of work, and even carry under circum- 
stances full load, if the field-exciting circuit is broken, and thereby 
the counter e.m.f., E,, reduced to zero, and sometimes even if 
the field circuit is reversed and the counter e.m.f., £.',. made 

Inversely, under certain conditions of load, the current and 
the e.m.f. of a generator do not disappear if the generator field 
circuit is broken, or even reversed to a small negative value, in 
which tatter case the current is against the e.m.f., E a , of the 

Furthermore, a shuttle armature without any winding (Fig. 
120) will in an alternating magnetic field revolve when once 
brought up to synchronism, and do considerable work as a motor. 

These phenomena are not due to remanent magnetism nor 
to the magnetizing effect of eddy currents, because they exist 
also in machines with laminated fields, and exist if the alternator 
is brought up to synchronism by external means and the rema- 
nent magnetism of the field poles destroyed beforehand by 
application of an alternating current. 

These phenomena can uol be explained under the assump- 
tion of a constant synchronous reactance: because in ilu- oast 
al no-field excitation, the e.m.f. or counter e.m.f. of the machine 





H MVO, ;md the only cm. I', existing in tlic- al tern (it in 1 is the e.m.f. 
of self-induction; that is, the e.m.f. induced by the alternating 
current upon itself. If, however, the synchronous reactance is 
constant, (he counter e.m.f. of self-induction is in quadrature 
with the current and wattless; that is, can neither produce nor 
consume energy. 

In the synchronous motor running without field excitation, 
always a large lag of the current behind the impressed e.m.f. 
exists; and an alternating-current generator will yield an e.m.f. 
without field excitation only when closed by an external circuit 
of large negative reactance; that is, a circuit in which the current 
the e.m.f., as a condenser, or an overexcited synchronous 

iotor, etc. 

14S. The usual explanation of the operation of the synchronous 
machine without field excitation is self-excitation by reactive 
armature currents. In a synchronous motor a lagging, in a 
generator a leading armature current magnetizes the field, and in 
such a case, even without any direct-current field excitation, there 
is a field excitation and thus a magnetic field flux, produced by the 
m.m.f. of the reactive component of the armature current*. In 
the polyphase machine, this is constant in intensity and direc- 
tion, in the single-phase machine constant in direction, hut pul- 
sating in intensity, and the intensity pulsation can be reduced 
by a short-circuit winding around the field structure, as more 
fully discussed under "Synchronous Machines." 

Thus a machine as shown diagram mat ically in Fig. 124, with 
a polyphase (three-phase) current impressed on the rotating 
armature, A, and no winding on the field poles, starts, runs up 
to synchronous and does considerable work as synchronous 
motor, and underload may even give a fairly good (lagging) power- 
factor. With a single-phase current impressed upon the arma- 
ture, A, it does not start, but when brought up to synchronism, 
continues to run as synchronous motor. Driven by mechanical 
power, with a leading current load it is a generator. 

However, the operation of such machines depends on the 
existence of a polar field structure, that is a strucinre having a 
low reluctance in (he direction of the field poles, P — P, and a 
high reluctance in quadrature position thereto. Or, in other 
words, the armature reactance with the coil facing the field poles 

high, and low in the quadrature position thereto. 

In a structure with uniform magnetic reluctance, in which 



therefore the armature reactance does oof vary with the posi- 
tion of the armature in the field, us shown in Fig. 125, such ■ H 
excitation hy reactive armature currents does not occur, and 
direct-current field excitation is always necessary (except in the 
so-called "hysteresis motor"). 

Vectorially this is shown in Figs. 124 and 125 by the relalivc 
position of the magnetic flux, *, the voltage, E, in quadrature to 
*, and the m.m.f. of the current, /. In Fig. 125, where / and 
4> coincide, I and E are in quadrature, that is, the power zero. 
Due In the polar structure in Fig. 124, /and * do not coincide, 

thus / is not in quadrature to E, but contains a positive 01 a 
negative energy component, making the machine motor or 

As the voltage, E, is produced by the current, /, it is an e.m.f. 
of self-induction, and self-excitation of the synchronous machine 
by armature reaction can be explained by the fact that the 
counter e.m.f. of self-induction is not wattless or in quadrature 
with the current, but contains an energy component; that if, 
that the reactance is of the form X = h + jx, where x is the watt- 
less component of reactance and h the energy component of 
reactance, and k is positive if the reactance consumes power — 
in winch case the counter e.m.f. of self-induction lags more than 
90° behind the current — while h is negative if the reactance 
produces power — in which case the counter e.m.f. of self-induction 
lags less than 90° behind the current. 

149. A case of this nature occurs in the effect of hysteresis, 
from a different point of view. In "Theory and Calcuation of Al- 
ternating Current" it was shown, thai -magnetic hysteresis distorts 
the current wave in such a way that the equivalent sine wave, 


that is, the sine wave of equal effective strength and equal power 
with the distorted wave, is in advance of the wave of magnetism 
by what is called the angle of hysteretic advanee of phase a. 
the e.m.f. generated by the magnetism, or counter e.m.f. 
nf self-induction lag* 90° behind the magnetism, it lags 90° + a 
heh ; nd the current ; that is, the self-induction in a circuit contain- 
ing iron is not in quadrature with the current and thereby 
wattless, but lags more than 90° and thereby consumes power, so 
that the reactance has to be represented by X = k + jx, where 
h is what has been called the "effective hysteretic resistance." 

A similar phenomenon takes place in alternators of variable 
reactance, or, what is the same, variable magnetic reluctance. 

Operation of synchronous machines without field excitation 
is most conveniently treated by resolving the synchronous 
reactance, 3"u, in its two components, the armature reaction and the 
true armature reactance, and once more resolving the armature 
reaction into a magnetizing and a distorting component, and 

msidering only the former, in its effect, on the field. The true 
armature self-inductance then is usually assumed as constant. 
Or, both armature reactance and self-inductance, are resolved 
into the two quadrature components, in line and in quadrature 
with the field poles, as shown in Chapters XXI and XXIV of 
"Alternating-Current Phenomena," 5th edition. 

160. However, while a machine comprising a stationary single- 
phase "field coil," A, and a shuttle-shaped rotor, R, shown 
diagrammatically as bipolar in Fig. 120, might still be interpreted 
in this matter, a machine as shown diagrammatically in Fig. 
127, as four-polar machine, hardly allows this interpretation. 
In Fig. 127, during each complete revolution of the rotor, !<', 
it four times closes and opens the magnetic circuit of the single- 
phase alternating coil, A, and twice during the revolution, the 
magnetism in the rotor, n", reverses. 

A machine, in which induction takes place by making and 
breaking (opening and closing) of the magnetic circuit, or in 
general, by the periodic variation of the reluctance of the 
magnetic circuit, is called a reaction machine. 

Typical forms of such reaction machines are shown diagram- 
matically in Figs. 126 and 127. Fig. 126 is a bipolar, Fig. 127 
is a four-polar machine. The rotor is shown to the position of 
closed magnetic circuit, but the position of open magnetic i-in-uit 
is shown dotted. 



Instead of cutting out segments of the rotor, iu Fig. 126, the 
same effect can lie produced, with a cylindrical rotor, by a shorl- 
circuitcd turn, S, as shown in Fig. 128, This gives a periodic 
variation of the effective reluctance, from ft minimum, shown in 
Fig. 128, to a maximum in the position shown in dotted lines in 
Fig. 128. 

This latter structure is the so-called "synchronous-induction 
motor," Chapter VIII, which here appears as a special form of 
I he reaction machine. 

If a direct current is sent through the winding of the machine, 

BIO. 1 2U.— Bipolar n 

Fig. 126 or 127, a pulsating voltage and current is produced in 
this winding. By having two separate windings, and energizing 
the one by a direct current, we get a converter, from direct cur- 
rent in the first, to alternating current in the second winding. 
The maximum voltage in the second winding can not exceed the 
voltage, per turn, in the exciting winding, thus is very limited, 
and so is the current. Higher values are secured by inserting a 
high inductance in series in the direct-current winding. In this 
case, a single winding may be used and the alternating-circuit, 
shunted across the machine terminals, inside of the inductance. 
161. Obviously, if the reactance or reluctance is variable, it 
will perform a complete cycle during the time the armature coil 
moves from one field pole to the next field pole, that is, during 
one-half wave of the main current. That is, in other words, 
the reluctance and reactance vary with twice the frequency of 
the alternating main current. Such a case is shown in Figs. 
129 and 130. The impressed e.m.f., and thus at negligible 
resistance, the counter e.m.f., is represented by the sine MVft, 


E, thus the magnetism produced thereby is a sine wave, $, 90° 
ahead of E. The reactance is represented by the sine wave, x, 



*_j _\ 

/p\ ' y^~/\ /"~1 E 

""N / /*" ' / 7v - ■*" N . 

\ / I / / //\ \ 

•^' V\ / \ Wp^X, J/ ' |\ -"*" \*v 

\ /\ " X/i )\ V^^^ \ ? 

a V ^~ > v/ / 1 — r *\ i V 

v\ /\ Ji. \ A : \\ /\ 

\*-J**T \ p -«-— - p' ^V-— - — * N 

i vSv i i v if / "Y^"^ 

\ \ i Ts/i/ / \ \ 

\ \ / ""■-». ^^ \ ^ 

^-^-V - ^oJ 

i 7 



Fio. 129. — Wave shapes in reaction machine as generator. 




/ ^*r~x* E 

N ~X-v Sl. W^ ^V 

I' fa/ii t- 

\ ■**"'' V, // */J^""Vj \ / 


^ /\ /A ' / \V A 

*» ' \ y~i^ / v / 

\\ / X ' \ // ^N / 

\ \ / / ')/ \\ / I \\/ 

1 ■£*,' > *^ s\' 1 \ /L ' 

» - N ^-^' i \ | "'/■-* 

i / IT / 

R / A I 

IV' vj 

e shape in reaction machine a 

varying with the dpuble frequency of E, and shown in Fig. 129 
to reach the maximiim value during the rise of magnetism, in 


Fig. 130 during the decrease of magnetism. The current, /, 
required to produce the magnetism, *, is found from * and x in 
combination with the cycle of molecular magnetic friction of the 
material, and the power, P, is the product, IE. As seen in Fig. 

__ ] t* _ j/> _ ^ >^ 

Fid. 131.— Hysteresis loop of reaction machine as generator. 

129, the positive part of P is larger than the negative part: 
that is, the machine produces electrical energy as generator, 
In Fig. 130 the negative part of P is larger than the positive: 

/^ '- ""5 

t _z 

: U'-z - 

-I / / 41 


€" J- ; = = 

Fia. 132. — Hysteresis loop of reaction machine as motor. 

that is, the machine consumes, electrical energy and produces 
mechanical energy as synchronous motor. In Figs. 131 and 132 
are given the two hysteretic cycles or looped curves, *, I under 
the two conditions. They show that, due to the variation of 


reactance, x, in the first case, the hysteretic cycle has been over- 
turned so as to represent, not consumption, but production of 
electrical energy, while in the second case the hysteretic cycle has 
been widened, representing not only the electrical energy consumed 
by molecular magnetic friction, but also the mechanical output. 

152. It is evident that the variation of reluctance must be 
symmetrical with regard to the field poles; that is, that the 
two extreme values of reluctance, maximum and minimum, will 
take place at the moment when the armature coil stands in front 
of the field pole, and at the moment when it stands midway 
between the field poles. 

The effect of this periodic variation of reluctance is a distortion 
of the wave of e.m.f., or of the wave of current, or of both. 
Here again, as before, the distorted wave can be replaced by 
the equivalent sine wave, or sine wave of equal effective intensity 
and equal power. 

The instantaneous value of magnetism produced by the 
armature current — which magnetism generates in the arma- 
ture conductor the e.m.f. of self-induction — is proportional to 
the instantaneous value of the current divided by the instan- 
taneous value of the reluctance. Since the extreme values of 
the reluctance coincide with the symmetrical positions of the 
armature with regard to the field poles — that is, with zero and 
maximum value of the generated e.m.f., E , of the machine — 
it follows that, if the current is in phase or in quadrature with 
the generated e.m.f., E , the reluctance wave is symmetrical to 
the current wave, and the wave of magnetism therefore sym- 
metrical to the current wave also. Hence the equivalent sine 
wave of magnetism is of equal phase with the current wave ; that 
is, the e.m.f. of self-induction lags 90° behind the current, or is 

Thus at no-phase displacement, and at 90° phase displace- 
ment, a reaction machine can neither produce electrical power 
nor mechanical power. 

If, however, the current wave differs in phase from the wave 
of e.m.f. by less than 90°, but more than zero degrees, it is un- 
symmetrical with regard to the reluctance wave, and the re- 
luctance will be higher for rising current than for decreasing cur- 
rent, or it will be higher for decreasing than for rising current, 
according to the phase relation of current with regard to generated 
e.m.f., #o. 


In the first, case, if the reluctance is higher for rising, Inner Fat 
decreasing, current, the magnetism, which is proportional to 
current, divided by reluctance, is higher for decreasing than for 
rising current; that is, its equivalent sine wave lugs behind the 
sine wave of current, and the e.m.f. or self-induction will lag 
more than 90° behind the current; that is, it will consume 
electrical power, and thereby deliver mechanical power, and do 
work as a synchronous motor. 

In the second case, if the reluctance is lower for rising, and 
higher for decreasing, current, the magnetism is higher for rising 
than for decreasing current, or the equivalent sine wave of 
magnetism leads the sine wave of the current, and the counter 
e.m.f. of self-induction lags less than 90° behind the current; 
that is, yields electric power as generator, and thereby consumes 
mechanical power. 

In the first ease the reactance will lie represented by X = ft + 
jx, as in the case of hysteresis; while in the second case the 
reactance will be represented by A" = — ft + jx. 

153. The influence of the periodical variation of reactance 
will obviously depend upon the nature of the variation, that is, 
upon the shape of the reactance curve. Since, however, no 
matter what shape the wave has, it can always be resolved in a 
series of sine waves of double frequency, and its higher har- 
monies, in first approximation the assumption can l>e made 
that the reactance or the reluctance varies with double frequency 
of the main current ; that is, is represented in the form: 

x = a + b cos 2 &. 

Let the inductance be represented by: 

L = I + 1' cos 2 ft 
= ((1 +7 cos 2 0); 

■- amplitude of variation of inductance. 

!■ of current behind maximum value 

where 7 

fl = angle of lag of zero vah 
of the inductance, L. 

Then, assuming the current as sine wave, or replacing it by 
the equivalent sine wave of effective intensity, /, current: 

i » / v^sin (tf - 8). 


The magnetism produced by this current is: 



where n = number of turns. 
Hence, substituted: 

= l l^ s in (0.- 0) (1 + 7 cos 2/3), 


or, expanded: 

$ = ^V? /l - A cos sin - (l + *) sin cos /j|, 

when neglecting the term of triple frequency as wattless. 
Thus the e.m.f. generated by this magnetism is: 

e = — n 


hence, expanded: 

e = -2 tt/ZZ V2 I (l - ?) cos cos + (l + |) sin sin 
and the effective value of e.m.f. : 

E = 2wfll yj(l _^ 2 cos 2 0+ (l+|) 2 sin 2 

= 2vfllJ\ + £- 7 cos 20. 
Hence, the apparent power, or the volt-amperes: 

Q = IE = 27r/// 2 ^l+£- 7C os20 

*/*^l + 

J£ J 

1 + * - 7 cos 2 

The instantaneous value of power is: 
p = ei 

= -4tt/// 2 sin (0 - 0) I (l - ^ cos cos + 

(l + J) sin sin /»} ; 


and, expanded: 

V = -2wfll* {(l + l) sin 2 $ sin 2 - (l - J) 

sin 2 0cos 2 + sin 2/3 (cos 2 6 - |) }• 

Integrated, the effective value of power is: 

P = -7r/LT 2 7sin2 0; 

hence, negative, that is, the machine consumes electrical, and 
produces mechanical, power, as synchronous motor, if 6 > 0, 
that is, with lagging current ; positive, that is, the machine pro- 
duces electrical, and consumes mechanical power, as generator, 
if 6 > 0, that is, with leading current. 
The power-factor is: 

P y sin 2 $ 

V = 

Q „ L '. y 2 

2 Jl + -£- - 7 cos 2 6 

hence, a maximum, if: 


or, expanded: 

de =0 > 

2 , 7 

cos 2 = - and = {r 
7 2 

The power, P, is a maximum at given current, /, if: 

sin 2 6 = 1 ; 
that is: 

6 = 45°; 

at given e.m.f., E, the power is: 

p fl 2 7 sin 2 

4wfl(l +^ - 7 cos 2 6) 
hence, a maximum at : 

to ' 

or, expanded: t 


cos 2 e = 

•v 1 
1+ 4- 


154. We have thus, at impressed e.m.f., E, and negligible 
resistance, if we denote the mean value of reactance: 

x - 2 *■//. 

/- E 

x yjl +4*- y cos 20 


xJl + j - 7 cos 2 


E*y sin 2^ 

2x(l + ?- - 7 cos 2 0) 


/et r\ 7 sin 2 
V = cos {E, I) = ,-^ ■ — - 

2 Jl + ^ - 7 cos 2 

Maximum power at : 

cos 2 = y 

7 2 

1+ 4 

Maximum power-factor at: 

2 7 

cos 2 = - and = '-• 
7 2 

> : synchronous motor, with lagging current, 
< 0: generator, with leading current. 

Ah an example is shown in Fig. 133, with angle as abscissa*, 
the values of current, power, and power-factor, for the constants, 
E = 110, x = 3, and 7 = 0.8. 

/ = - -- 41 

Vl.45 - cos 2 6 

p = -2017>in 2 
1.45 - cos 2 

— » 

, „ 7X 0.447 sin 2 

p = cos (E, I) = -/_.:.--=-.--- • 

V 1.45 - cos 2 


As seen from Fig. 133, the power-factor, j>, of such a machine 
is very low — does not exceed 40 per cent, in this instance, 

Very similar to the reaction machine in principle and characiei 
of operation are the synchronous induction motor, Chapter IX, 
and the hysteresis motor, Chapter X, either of which is a gen- 
erator above synchronism, and at synchronism can be motor as 




E -110 
*- 3 








P= vUs-mi* JP 
















































"■ ; . 








- 71 


well a. 

the re 
also d 
it thus 

It h 
in kecj 


Fig. 133. — Load curves of reiirtion machine. 

generator, depending on the relative position b 

ield and rotor. 

The low power-factor and the low weigh! efficSu 
iction machine from extended use for large powe 
K-s the severe wave-shape distortion produced by 

has found a very limited use only in small sizes. 
is, however, the advantage or a high degree of ex 

ing in step, that is, it does not merely keep in synch 
iftfi more or less over a phase angle with respect 

>i ween 

cy bar 
s. Bo 
t, and 



to the 


impressed voltage, but the relative position of the rotor with 
regards to the phase of the impressed voltage is more accurately 
maintained. Where this feature is of importance, as in driving 
a contact-maker, a phase indicator or a rectifying commutator, 
the reaction machine has an advantage, especially in a system 
of fluctuating frequency, and it is used to some extent for such 

This feature of exact step relation is shared also, though to 
a lesser extent, by the synchronous motor with self-excitation 
by lagging currents, and ordinarily small synchronous motors, 
but without field excitation (or with great underexcitation or 
overexcitation) are often used for the same purpose. 

Machines having more or less the characteristics of the reac- 
tion machine have been used to a considerable extent in the 
very early days, for generating constant alternating current for 
series arc lighting by Jablochkoff candles, in the 70's and early 

Structurally, the reaction machine is similar to the inductor 
machine, but the essential difference is, that the former operates 
by making and breaking the magnetic circuit, that is, periodically 
changing the magnetic flux, while the inductor machine operates 
by commutating the magnetic flux, that is, periodically changing 
the flux path, but without varying the total value of the magnetic 




Inductor Alternators, Etc. 

156. Synchronous machines may be built with stationary 
field and revolving armature, as shown diagrammatically in 
Fig. 134, or with revolving field and stationary armature, Fig. 
135, or with stationary field and stationary armature, but 
revolving magnetic circuit. 

The revolving-armature type was the most frequent in the 
early days, but has practically gone out of use except for special 

Fia. 134. — Revolving armature 

Fig. 135.— Revolving field al 

purposes, and for synchronous commutating machines, as the 
revolving-armature type of structure is almost exclusively used 
for commutating machines. The revolving-field type is now 
almost exclusively used, as the standard construction of alter- 
nators, synchronous motors, etc. The inductor type had been 
used to a considerable extent, and had a high reputation in the 
Stanley alternator. It has practically gone out of use for 
standard frequencies, due to its lower economy in the use of 
materials, but has remained a very important type of construc- 
tion, as it is especially adapted for high frequencies and other 
special conditions, and in this field, its use is rapidly increasing. 
A typical inductor alternator is shown in Fig. 136. as eight- 
polar quarter-phase machine. 




Its armature coils, A, are stationary. One stationary field 
coil, F, surrounds the magnetic circuit of the machine, which 
consists of two sections, the stationary external one, B, which 
contains the armature, A, and a movable one, C, which contains 
the inductor, N. The inductor contains as many polar projec- 
tions, N, as there are cycles or pairs of poles. The magnetic flux 
in the air gap and inductor does not reverse or alternate, as in 
the revolving-field type of alternator, Fig. 135, but is constant 
in direction, that is, all the inductor teeth are of the same 
polarity, but the flux density varies or pulsates, between a maxi- 
mum, B\, in front of the inductor teeth, and a minimum, B tl 
though in the same direction, in front of the inductor slots. The 
magnetic flux, *, which interlinks with the armature coils, does 
not alternate between two equal and opposite values, + * and 

Fio. 136. — Inductor alternator. 

— *», as in Fig. 135, but pulsates between a high value, *i, 
when an inductor tooth stands in front of the armature coil, 
and a low value in the same direction, *,, when the armature 
coil faces an inductor slot. 

167. fn the inductor alternator, the voltage induction thus 
is brought about by shifting the magnetic flux produced by a 
stationary field coil, or by what may be called magneto commu- 
tation, by means of the inductor. 

The flux variation, which induces the voltage in the armature 
turns of the inductor alternator, thus is #i — *», while that in 
the revolving-field or revolving-armature type of alternator is 
2 *„. 

The general formula of voltage induction in an alternator is: 


! - y/2 «/«*„, 



where : 

/ = frequency, in hundreds of cycles, 
n = number of armature turns in series, 
* = maximum magnetic flux, alternating 
through the armature turns, in megalines, 
e = effective value of induced voltage. 
*i — * s taking the place of 2 * , in the inductor alternator, 
the equation of voltage induction thus is: 



As seen, *, must be more than twice as large as *o, that is, 
in an inductor alternator, the maximum magnetic flux interlinked 
with the armature coil must be more than twice as large as in the 
standard type of alternator. 

In modern machine design, with (he efficient methods of cool- 
ing now available, economy of materials and usually also effi- 
ciency make it necessary to run the flux density up to near satura- 
tion at the narrowest part of the magnetic circuit — which usually 
is the armature tooth. Thus the flux, *o, is limited merely by 
magnetic saturation, and in the inductor alternator, $,, would be 
limited to nearly the same value as, 4> , in the standard machine, 

*i — *i 

and — i, — thus would be only about one-half or less of the 
permissible value of * - That is, the output of the inductor 
alternator armature is only about one-half that of the standard 
alternator armature. This is obvious, as we would double the 
voltage of the inductor alternator armature, if instead of pulsat- 
ing between 4>, and * 2 or approximately zero, we would alternate 
between *i and — *i. 

On the other hand, the single field-coil construction gives a 
material advantage in the material economy of the field, and 
in machines having very many field poles, that is, high-frequency 
alternators, the economy in the field construction overbalances 
the lesser economy in the use of the armature, especially as at 
higli frequencies it is not feasible any more to push the alter- 
nating flux, $0, up to or near saturation values. Therefore, for 
high-frequency generators, the inductor alternator becomes 
the economically superior types, and is preferred, and for ex- 
tremely high frequencies (20,000 to 100,000 cycles) the inductor 
alternator becomes the only feasible type, mechanically, 

168. In the calculation of the magnetic circuit of the inductor 


alternator, if 3>o is the amplitude of flux pulsation through the 
armature coil, as derived from the required induced voltage by 
equation (1), let: 

p = number of inductor teeth, that is, 
number of pairs of poles (four in 
the eight-polar machine, Fig. 136). 
Pi = magnetic reluctance of air gap in front 
of the inductor tooth, which should 
be as low as possible, 
P2 = magnetic reluctance of leakage path 
through inductor slot into the arma- 
ture coil, which should be as high as 



it is: 
and as: 

$! -f- <f> 2 = — -f- 

Pi P2 

$1 — $2 = 2 $0, 

it follows: 


4*1 — £t 4?o y 
P2 — Pi 

* — 9 * P l 

4^2 — 4 ™0 ) 
P2 — Pi 


and the total flux through the magnetic circuit, C, and out from 
all the p inductor teeth and slots thus is : 

$ = P ($i + $2) 

P2 + Pl 

= 2 p$o 
= 2 p$o 

P2 — Pl 

1 + -^-}- (6) 

P2 — Pl 1 

In the corresponding standard alternator, with 2 p poles, the 
total flux entering the armature is : 

2 p<f>o 

and if pi is the reluctance of the air gap between field pole and 
armature face, p 2 the leakage reluctance between the field poles, 
the ratio of the leakage flux between the field poles, $', to the 
armature flux, $ , is: 

<r> - *' = A + } ; ( 7) 

Pl P2 


$' = $„ -> (8) 



and the flux in the field pole, thus, is 


*„ + 2 *' = 

.(i + ! 

hence the total magnetic flux of the machine, of 2 p pole* 

* = 2p«„(l -J- 2 ")- 

2p L 
As in 16), pi is small compared with p,, — in (6) differs 

As regards to the total magnetic flux required for the induc- 
tion of the same voltage in the same armature, no material 
difference exists between the inductor machine and the standard 
machine ; but in the armature teeth the inductor machine requires 
more than twice the maximum magnetic flux of the standard 

Vu;. |:>7. Siimli'v iti'iiirdir iiltt'Tiinlrtr. 

alternator, and thereby ia at a disadvantage where the limit 
of magnetic density in the armature is set only by magnetic 

As regards to the hysteresis loss in the armature of the in- 
ductor alternator, the magnetic cycle is an unsyrametrieal cycle, 
between two values of the same direction, B x and B%, and the 
loss therefore is materially greater than it would be with a 
symmetrical cycle of the same amplitude. It is given by: 

/B, -Ba 1 ' 6 

' = *°( 2 ) 

n-»P +eB"]. 



Regarding hereto see "Theory and Calculation of Electric 
Circuits," under "Magnetic Constants." 

However, as by the saturation limit, the amplitude of the 
magnetic pulsation in the inductor machine may have to be 
kept very much lower than in the standard type, the core loss 
of the machine may be no larger, or may even be smaller than 
that of the standard type, in spite of the higher hysteresis 
coefficient, 170. 

169. The inductor-machine type, Fig. 136, must have an 


\f\ j\j\/\r\/\j\r ^ 

:f-A J 


! I 


Fig. 138. — Alexanderson high frequency inductor alternator. 

auxiliary air gap in the magnetic circuit, separating the revolving 
from the stationary part, as shown at S. 

It, therefore, is preferable 10 double the structure, Fig. 136, 
by using two armatures and inductors, with the field coil between 
them, as shown in Fig. 137. This type of alternator has been 
extensively built, as the Stanley alternator, mainly for 60 cycles, 
and has been a very good and successful machine, but has been 
superseded by the revolving-field type, due to the smaller size 
and cost of the latter. 

Fig. 137 shows the magnetic return circuit, B, between the two 
armatures, A, and the two inductors N and S as constructed of a 
number of large wrought-iron bolts, while Fig. 136 shows the 
return as a solid cast shell. 


A mollification of this type of inductor machine is the Alex* 
anderson inductor alternator, shown in Fig. 138, which is being 
built for frequencies up to 200,000 cycles per second and over, 
for use in wireless telegraphy and telephony. 

The inductor disc, /, contains many hundred inductor teeth, 
and revolves at many thousands of revolutions between the 
two armatures, A, as shown in the enlarged section, S. It is 
surrounded by the field coil, F, and outside thereof the magnetic 
return, S. The armature winding is a single-turn wave winding 
threaded through the armature faces, as shown in section .S' ami 
face view, Q. It is obvious that in the armature special iron 
of extreme thinness of lamination has to be used, and the rotat- 
ing inductor, 7, built to stand the enormous centrifugal stresses 
nf the great peripheral speed. We must realize that even with 
an armature pitch of less than l fa in. 
per pole, we get at 100,000 cycles per 
second peripheral speeds approaching 
bullet velocities, over 1000 miles per 
hour. For the lower frequencies m 
long distance radio communication, 
20,000 to 30,000 cycles, such' ma- 
chines have been built for large 

160. Fig. 139 shows the Eieke- 
meyer type of inductor alternator. 
In this, the field coil F is not con- 
centric to the shaft, and the inductor 
teeth not all of the same polarity, but 
ductor alternator. the field coil, as seen in Fig. 139, sur- 

rounds the inductor, /, longitudinally, 
and with the magnetic return B thus gives a bipolar magnetic 
field, Half the inductor teeth, the one side of the inductor, thus 
are of the one, the other half of the other polarity, and the 
armature coils, A, are located in the (laminated) pole faces of the 
bipolar magnetic structure. Obviously, in larger machines, a 
multipolar structure could be used instead of the bipolar of Fig. 
139. This type has the advantage of a simpler magnetic struc- 
ture, and the further advantage, that all the magnetic flux 
passes at right angles to the shaft, just as in the revolving field 
or revolving armature alternator. In the types, Figs. 136 and 
137, magnetic flux passes, and the field exciting coil magnetizes 



longitudinally to the shaft, arui thus magnetic stray flux tends 
) pass along the shaft, closing through bearings and supports, 
and causing heating of bearings. Therefore, in the types 136 
,nd 137, magnetic barrier coils have been used where needed, 
that is, coils concentric to the shaft, that ia, parallel to the field 
coil, and outside of the inductor, that is, between inductor and 
bearings, energized in opposite direction lo the field coils. These 
coils then act as counter-magnetizing coils in keeping magnetic 
flux out of the machine bearings. 

The type, Fig. 139, is especially adapted for moderate fre- 
quencies, a few hundreds to thousands of cycles. A modifica- 
ti of it, adopted as converter, is used to a considerable extent: 
he inductor, /, is supplied with a bipolar winding connected to a 
■ommutator, and the machine therefore is a bipolar commutating 
machine in addition to a high-frequency inductor alternator 
(I6-polar in Fig. 130). It thus may be operated as converter, 
receiving power by direct-current supply, as direct-current motor, 
and producing high-frequency alternating power in the inductor 
pole-face winding. 

161. If the inductor alternator, Fig. 139, instead of with direct 
current, is excited with low-frequency alternating current, that 


:o. 140. — Voltage wiive of inductor niter 

nth [ill.- 

, an alternating current, passed through the field coil, F, of a 
requency low compared with that generated by the machine as 
inductor alternator, then the high-frequency current generated 
.• the machine as inductor alternator is not of constant ampli- 
tude, but of a periodically varying amplitude, as shown in Fig. 
140. For instance, with 60-cycle excitation, a 64-polar in- 
ductor (that is, inductor with 32 teeth), and a speed of 1800 
revolutions, we get a frequency of approximately 1000 cycles, 
,nd a voltage and current wave about as shown in Fig. 140. 
The power required for excitation obviously is small compared 
the power which the machine can generate. Suppose, 
icrefore, that the high-frequency voltage of Fig. 140 were 
ftified. It would then give a voltage and current, pulsating 


with the frequency of the exciting current, but of a power, as 
many times greater, as the machine output is greater than the 
exciting power. 

Thus such an inductor alternator with alternating-current 
excitation can be used as amplifier. This obviously applies 
equally much to the other types, as shown in Figs. 13(i. 137 
and 138. 

Suppose now the exciting current is a telephone or micro- 
phone current, the rectified generated current then pulsates with 
the frequencies of the telephone current, and the machine is a 
telephonic amplifier. 

Thus, by exciting the high-frequency alternator in Fig. 138, 
by a telephone current, we get a high-frequency current, of an 
amplitude, pulsating with the telephone current, but of niany 
times greater power than the original telephone current. This 
high-frequency current, being of the frequency suitable for radio 
communication, now is sent into the wireless sending antennae, 
and the current received from the wireless receiving antennae, 
rectified, gives wireless telephonic communications. As seen, 
the power, which hereby is sent out from the wireless antenna?, 
is not the insignificant power of the telephone current, but is the 
high-frequency power generated by the alternator with telephonic 
excitation, and may be many kilowatts, thus permitting long- 
distance radio telephony. 

It is obvious, that the high inductance of the field coil, F, of 
the machine, Fig. 138, would make it impossible to force a tele- 
phone current through it, but the telephonic exciting current 
would be sent through the armature winding, which is of very 
low inductance, and by the use of the capacity the armature 
made self-exciting by leading current. 

Instead of sending the high-frequency machine current, which 
pulsates in amplitude with telephonic frequency, through radio 
transmission and rectifying the receiving current, we can rectify 
directly the generated machine current and so get a current 
pulsating with the telephonic frequency, that is, get a greatly 
amplified telephone current, and send this into telephone circuits 
for long-distance telephony, 

162. Suppose, now, in the inductor alternator, Fig. 139, with 
low-frequency alternating-current excitation, giving a voltage 
wave shown in Fig. 140, we use several alternators excited by 
low-frequency currents of different phases, or instead of II -iimlc- 


phase field, as in Fig. 139, we use a polyphase exciting field. This 
is shown, with three exciting coils or poles energized by three- 
phase currents, in Fig. 141. The high-frequency voltages of 
pulsating amplitude, induced by the three phases, then super- 
pose a high-frequency wave of constant amplitude, and we get, 
in Fig. 141, a high-frequency alternator with polyphase field 

Instead of using definite polar projection for the three-phase 
bipolar exciting winding, as shown in Fig. 141, we could use a 
distributed winding, like that in an induction motor, placed in 
the same slots as the inductor-alternator armature winding. By 

Fig. 141. — Inductor alternator with three-phase excitation. 

placing a bipolar short-circuited winding on the inductor, the 
three-phase exciting winding of the high-frequency (24-polar) 
inductor alternator also becomes a bipolar induction-motor 
primary winding, supplying the power driving the machine. 
That is, the machine is a combination of a bipolar induction 
motor and a 24-polar inductor alternator, or a frequency 

Instead of having a separate high-frequency inductor-alter- 
nator armature winding, and low-frequency induction motor 
winding, we can use the same winding for both purposes, as 
shown diagrammatically in Figs. 142 and 143. The stator 
winding, Fig. 142, bipolar, or four-polar 60-cycle, is a low- 
frequency winding, for instance, has one slot per inductor pole, 
that is, twice as many slots as the inductor has teeth. Successive 
turns then differ from each other by 180° in phase, for the high- 
frequency inductor voltage. Thus grouping the winding in 


two sections, 1 anil 3, and 2 ami 4, the high-frequency voltages 
in the two sections are opposite in phase from each other. Con- 
necting, then, as shown in Fig. 143, 1 and 2 in series, and 4 and 
3 in series into the two phases of the quarter-phase supply cir- 
cuit, no high-frequency induction exists in either phase, but the 
high-frequency voltage is generated between the middle points 

Fio. 142.— Induction type of higb -frequency inductor alternator. 

of the two phases, as shown in Fig. 143, and we thus get another 
form of a frequency converter, changing from low-frequency 
polyphase to high-frequency single-phase. 

• HICH Wrf umJ 1 

Z17" ffmlms\ 1S0 ' 

Fig. 143. — Diagram of connection of induction type of inductor alternator. 

163. A type of inductor machine, very extensively used in 
smalt machines— as ignition dynamos for gasoline engines — is 
shown in Fig. 144. The field, F, and the shuttle-shaped armature, 
A, are stationary, and an inductor, J, revolves between field and 
armature, and so alternately sends the magnetic field flux through 
the armature, first in one, then in the opposite direction. As 
seen, in this type, the magnetic flux in the armature reverses, 
by what may be called magnetic commutation. Usually in these 


small machines the field excitation is not by direct current, but 
by permanent magnets. 
This principle of magnetic commutation, that is, of reversing 

Fig. 144. — Magneto inductor machine. 

the magnetic flux produced by a stationary coil, in another 
stationary coil by means of a moving "magneto commutator" 
or inductor, has been extensively used in single-phase feeder 

Fig. 145.— Magneto com mutation voltage regulator. 

regulators, the so-called " magneto regulator*)." It is illustrated 
in Fig. 145. P is the primary coil (shunt coil connected across 
the alternating supply circuit), 8 the secondary coil (connected 
in series into the circuit which is to be regulated) the magnetic 
inductor, I, in the position shown in drawn lines sends the mag- 



ratio flux produced by the primary coil, through the secondaf] 
coil, in the direction opposite to the direction, in which it would 
send the magnetic flux through the secondary coil when in the 
position /', shown in dotted lines. In vertical position, the 
inductor, /, would pass the magnetic flux through the primary 
coil, without passing it through the secondary coil, that is, with- 
out inducing voltage in the secondary. Thus by moving the 
shuttle or inductor, /, from position I over the vertical position 
to the position /', the voltage induced in the secondary cod. S, 
is varied from maximum boosting over to zero to maximum 

164. l r ig. 146 shows a type of machine, which has been EHri 
still is used to some extent, for alternators as well as for direct- 

i fiTiT]* 

current commutating machines, and which may be called an 

inductor machine, or at least has considerable similarity with flu- 
inductor type. It is shown in Pig. 146 as six-polar machine, 
with internal field and external armature, but can easily be built 
with internal armature and external field. The field contains 
one field coil,/ 1 , concentric to the shaft. The poles overhang the 
field coils, and all poles of one polarity, N, come from the mic 
side, all poles of the other polarity from the other side of the field 
coll. The magnetic structure thus consists of two parts which 
interlock axially, as seen in Fig. 146. 

The disadvantage of this type of field construction is the high 
flux leakage between the field poles, which tends to impair the 
regulation in alternators, and makes commutation more difficult 
for direct-current machines. It offers, however, the advantage 


of simplicity and material economy in machines of small and 
moderate size, of many poles, as for instance in small very low- 
speed synchronous motors, etc. 

166. In its structural appearance, inductor machines often 
have a considerable similarity with reaction machines. The 
characteristic difference between the two types, however, is, 
that in the reaction machine voltage is induced by the pulsation 
of the magnetic flux by pulsating reluctance of the magnetic 
circuit of the machine. The magnetic pulsation in the reaction 
machine thus extends throughout the entire magnetic circuit 
of the machine, and if direct-current excitation were used, the 
voltage would be induced in the exciting circuit also. In the 
inductor machine, however, the total magnetic flux does not 
pulsate, but is constant, and no voltage is induced in the direct- 
current exciting circuit. Induction is produced in the armature 
by shifting the — constant — magnetic flux locally from armature 
coil to armature coil. The important problem of inductor 
alternator design — and in general of the design of magneto com- 
iriutation apparatus — is to have the shifting of the magnetic 
flux from path to path so that the total reluctance and thus the 
total magnetic flux does not vary, otherwise excessive eddy- 
current losses would result in the magnetic structure. 

It is interesting to note, that the number of inductor teeth is 
one-half the number of poles. An inductor with p projections 
thus gives twice as many cycles per revolution, thus as syn- 
chronous motor would run at half the speed of a standard syn- 
chronous machine of p poles. 

As the result hereof, in starting polyphase synchronous 
machines by impressing polyphase voltage on the armature and 
using the hysteresis and the induced currents in the field poles, 
for producing the torque of starting and acceleration, there 
frequently appears at half synchronism a tendency to drop into 
step with the field structure as inductor. This results in an 
increased torque when approaching, and a reduced torque when 
passing beyond half synchronism, thus produces a drop in the 
torque curve and is liable to produce difficulty in passing beyond 
half speed in starting. In extreme cases, it may result even in 
a negative torque when passing half synchronism, and make the 
machine non-self-starting, or at least require a considerable 
increase of voltage to get beyond half synchronism, over that 
required to start from rest. 


166. In the theory of the synchronous motor the assumption 
is made that the mechanical output of the motor equals the power 
developed by it. This is the case only if the motor runs at 
constant speed. If, however, it accelerates, the power input is 
greater; if it decelerates, less than the power output, by the power 
stored in and returned by the momentum. Obviously, the 
motor can neither constantly accelerate nor decelerate, without 
breaking out of synchronism. 

If, for instance, at a certain moment the power prod wed by 
the motor exceeds the mechanical load (as in the moment of 
throwing off a part of the load), the excess power is consumed by 
the momentum as acceleration, causing an increase of speed. 
The result thereof is that the phase of the counter e.m.f., c, 
is not constant, but its vector, e, moves backward to earlier time, 
or counter-clockwise, at a rate depending upon the momentum. 
Thereby the current changes and the power developed changes 
and decreases. As soon as the power produced equals the load, 
the acceleration ceases, but the vector, c, still being in motion, 
due to the increased speed, further reduces the power, causing 
a retardation and thereby a decrease of speed, at a rate depend- 
ing upon the mechanical momentum. In this manner a periodic 
variation of the phase relation between e and to, and correspond- 
ing variation of speed and current occurs, of an amplitude and 
period depending upon the circuit conditions and the mechanical 

If the amplitude of this pulsation has a positive decrement, 
that is, is decreasing, the motor assumes after a while a constant 
position of e regarding e a , that is, its speed becomes uniform. 
If, however, the decrement of Hie pulsation is negative, an 
infinitely small pulsation will continuously increase in amplitude, 
until the motor is thrown out of step, or the decrement becomes 
zero, by the power consumed by forces opposing the pulsation, 
as anti-surging devices, or by the periodic pulsation of the syn- 
chronous reactance, etc. If the decrement is zero, a pulsation 


started once will continue indefinitely at constant amplitude. 
This phenomenon, a surging by what may be called electro- 
mechanical resonance, must be taken into consideration in a 
complete theory of the synchronous motor. 
167. Let: 

E = e = impressed e.m.f. assumed as zero vector. 
E = e (cos P — j sin P) = e.m.f. consumed by counter e.m.f. 
of motor, where: 

P = phase angle between E and E. 

Z = r + jx, 

and z = Vr 2 + x 2 

= impedance of circuit between 
Eo and E, and 

tan a = — 

The current in the system is: 

e — E eo — e cos P + je sin P 

/o = 

r + jx 
= - {[e cos a — e cos (a + P)] 

— j [e sin a — e sin (a + 0)] ) (1) 
The power developed by the synchronous motor is: 

Po = [EI] 1 = - {[cos p [e cos a - e cos (a + 0)] 


+ sin [e sin a — e sin (a + 0)] J 

= {[e cos (a — 0) — e cos a]). (2) 

If, now, a pulsation of the synchronous motor occurs, resulting 
in a change of the phase relation, 0, between the counter e.m.f., e, 
and the impressed e.m.f., e (the latter being of constant fre- 
quency, thus constant phase), by an angle, 5, where 8 is a periodic 
function of time, of a frequency very low compared with the 
impressed frequency, then the phase angle of the counter e.rn.f., 
e, is P + 6; and the counter e.m.f. is: 

E = e {cos (0 + 6) - j sin (p + 6)1, 



hence the current: 

/ = - {[e cos a — e cos (a + + 5)] 


— j [e sin a — e sin (a + + 6)]\ 

= h + ysin* jsin(a + p+ *) + jcos(a + + |) [ (3) 

the power: 


P = {e cos (a — — 5) — e cos a} 

= Po + -— sin ^ sm^a - - 2 J • (4) 

Let now: 

t»o = mean velocity (linear, at radius of gyration) of syn- 
chronous machine; 

a = slip, or decrease of velocity, as fraction of t' , where s is 
a (periodic) function of time; hence 

v = Vq (1 — s) = actual velocity, at time, t. 

During the time element, dt, the position of the synchronous 
motor armature regarding the impressed e.m.f., e , and thereby 
the phase angle, + 6, of e, changes by: 

dd = 2 Tcfsdt 

= sd0, (5) 


= 2 icft, 


/ = frequency of impressed e.m.f., e . 

m = mass of revolving machine elements, and 

M — )i im-'o 2 = mean mechanical momentum, reduced to 

joules or watt-seconds; then the momentum at time, t, and 

velocity v = v (1 — s) is: 

AT = y 2 vivj(\ - s) 2 , 

and the change of momentum during the time element, eft, is: 

dM , .ds. 

svmsixg or srxcHROsors motors 2*1 


for srxihiL 

TATaes Oil 


= — 


™ r di 

" Ml d* 




= 2* 

and from 

5 : 







it is: 



= — 


Since, as discussed, the change of momentum equals the dif- 
ference between produced and consumed power, the excess of 
power being converted into momentum, it is: 

P - Po = d £ • l» 

and. substituting <'4> and (7) into (8) and rearranging: 

C *-° sin * sin (a - fi - *) + 2 t/AT ™ = «• l»> 

'Assuming 5 as a small angle, that is, considering only small 

oscillations, it is: 

. 6 6 

Sln 2 = 2 

sin [a - - ^ J = sin (« - £) ; 

hence, substituted in (18): 

^ 5 sin (« - 0) + 4 ir/Af jj]» - 0, (10) 

and, substituting: 

ce «in(a-/8) m . 

4 rfzMo 

it is: 



This differential equation is integrated by: 

5 = At' 9 , (13) 

which, substituted in (12) gives: 

aAe ce + ACU™ = 0, 
a + C 2 = 0, 

C = ± V- a. 
. 168. 1. If a <0, it is: 

5 = A*+ me + A*- m \ 

/ / ee sin (0 — a) 

\ 4 tt/zA/o 

Since in this case, e* m9 is continually increasing, the syn- 
chronous motor is unstable. That is, without oscillation, the 
synchronous motor drops out of step, if > a. 

2. If a > 0, it is, denoting: 

, ^/- , /ee sin (a - 0) 

\ 4 TT/^Af o 

or, substituting for € +;n * and € + "" 4 * the trigonometric functions: 

6 = (Ai + Ao) cos n0 + j (A x — *4 2 ) sin n$ t 

5 = B cos (n0 + 7). (15) 

That is, the synchronous motor is in stable equilibrium, when 
oscillating with a constant amplitude B, depending upon the 
initial conditions of oscillation, and a period, which for small 
oscillations gives the frequency of oscillation: 

f „ f _ //ee sin (a - 0) 

As instance, let: 

<?o = 2200 volts. Z = 1 + 4 j ohms, or, z = 4.12; a = 76°. 

And let the machine, a 16-polar, 60-cycle, 400-kw., revolving- 
field, synchronous motor, have the radius of gyration of 20 in., 
a weight of the revolving part of 6000 lb. 

The momentum then is Af„ = 850,000 joules. 

Deriving the angles, 0, corresponding to given values of output. 
P, and excitation, r, from the polar diagram, or from the symbolic 


representation, and substituting in (16), gives the frequency of 
oscillation : 

P = 0: 

e = 1600 volts; = - 2°;/ = 2.17 cycles, 

or 130 periods per minute. 

2180 volts 

+ 3° 

2.50 cycles, 

or 150 periods per minute. 

2800 volts 

+ 5° 

2.85 cycles, 

or 169 periods per minute. 

P = 400 kw. 

e = 1600 volts; = 33°; / = 1.90 cycles, 

or 114 periods per minute. 
2180 volts -21° 2.31 cycles, 

or 139 periods per minute. 
2800 volts 22° 2.61 cycles, 

or 154 periods per minute. 

As seen, the frequency of oscillation does not vary much with 
the load and with the excitation. It slightly decreases with 
increase of load, and it increases with increase of excitation. 

In this instance, only the momentum of the motor has been 
considered, as would be the case for instance in a synchronous 

In a direct-connected motor-generator set, assuming the 
momentum of the direct-current-generator armature equal to 
60 per cent, of the momentum of the synchronous motor, the 
total momentum is M = 1,360,000 joules, hence, at no-load: 

P = 0, 
e = 1600 volts ;/ = 1.72 cycles, or 103 periods per minute. 

1.98 cycles, or 119 periods per minute. 
1.23 cycles, or 134 periods per minute. 

169. In the preceding discussion of the surging of synchronous 
machines, the assumption has been made that the mechanical 
power consumed by the load is constant, and that no damping 
or anti-surging devices were used. 

The mechanical power consumed by the load varies, however, 
more or less with the speed, approximately proportional to the 
speed if the motor directly drives mechanical apparatus, as 
pumps, etc., and at a higher power of the speed if driving direct- 
current generators, or as synchronous converter, especially 


when in parallel with other direct-current generators. Assum- 
ing, then, in the general case the mechanical power consumed by 
the load to vary, within the narrow range of speed variation con- 
sidered during the oscillation, at the pth power of the speed, 
in the preceding equation instead of Po is to be substituted, 
Po(l - s)p = P (l - ps). 

If anti-surging devices are used, and even without these in 
machines in which eddy currents can be produced by the oscilla- 
tion of slip, in solid field poles, etc., a torque is produced more 
or less proportional to the deviation of speed from synchronism. 
This power assumes the form, Pi = c 2 s, where c is a function of 
the conductivity of the eddy-current circuit and the intensity 
of the magnetic field of the machine, c 2 is the power which 
would be required to drive the magnetic field of the motor 
through the circuits of the anti-surging device at full frequency, 
if the same relative proportions could be retained at full fre- 
quency as at the frequency of slip, s. That is, Pi is the power 
produced by the motor as induction machine at slip «. In- 
stead of P, the power generated by the motor, in the preced- 
ing equations the value, P + Pi, has to be substituted, then: 

The equation (8) assumes the form : 

P + Pi-PoU-jmO = rf J^ 


(P - Po) - (P. + pPos) = *% • (17) 

or, substituting (7) and (4) : 

2e e 2 ° sin | sin [« - - *] + (c« + pPo) * + 4 ,/Af. ~ = 0; 

and, for small values of 8 : 

4 vfzMo 
b , «! + P p JL. (2 Q) 

Of these two terms b represents the consumption, a the oscilla- 
tion of energy by the pulsation of phase angle, p. b and a thus 


have a similar relation as resistance and reactance in alternating- 
current circuits, or in the discharge of condensers, a is the 
same term as in paragraph 167. 

Differential equation (19) is integrated by: 

5 = At c ', (21) 

which, substituted in (19), gives: 

aAt c * + 2 bCAf + C 2 Ae c * - 0, 

a + 2 bC + C 2 = 0, 

which equation has the two roots: 

Ci - -6 + V b*-a , 

C, = -6 - y/b 2 - a. (22) 

1. If a < 0, or negative, that is > a, C\ is positive and C% 
negative, and the term with C\ is continuously increasing, that 
is, the synchronous motor is unstable, and, without oscillation, 
drifts out of step. 

2. If < a < b 2 , or a positive, and b 2 larger than a (that is, 
the energy-consuming term very large), C\ and C% are both 
negative, and, by substituting, + \/b 2 — a = g } it is: 

Ci= - (6-flf), C, = - (6 + g); 

5 = A l€ " <»-*>• + A 2€ " (» + •)•• (23) 

That is, the motor steadies down to its mean position logarith- 
mically, or without any oscillation. 

b 2 > a, 
hence : 

(c 2 + pPo) 2 eeo sin (a - 0) 
16wfMo > 2 (M) 

is the condition under which no oscillation can occur. 

As seen, the left side of (24) contains only mechanical, the 
right side only electrical terms. 

3. a > b 2 . 

In this case, y/b 2 — a is imaginary, and, substituting: 

g = y/a'-b*, 
it is: 



hence : 

and, substituting the trigonometric for the exponential functions, 
gives ultimately: 

6 = Bt- b °cos(ge + y). (25) 

That is, the motor steadies down with an oscillation of period: 



_ fee sin (a - 0) (c« + pP ) 2 


4 ttzMo 64 T*3f o 2 

and decrement or attenuation constant: 

170. It follows, however, that under the conditions considered, 
a cumulative surging, or an oscillation with continuously increas- 
ing amplitude, can not occur, but that a synchronous motor, 
when displaced in phase from its mean position, returns thereto 
either aperiodically, if b 2 > a, or with an oscillation of vanishing 
amplitude, if b 2 < a. At the worst, it may oscillate with constant 
amplitude, if b = 0. 

Cumulative surging can, therefore, occur only if in the differ- 
ential equation (19): 

«»+ 2 »2 + $- ' (28) 

the coefficient, 6, is negative. 

Since c 2 , representing the induction motor torque of the damp- 
ing device, etc., is positive, and pPo is also positive (p being 
the exponent of power variation with speed), this presupposes 

-A 2 
the existence of a third and negative term, Q rnf , in b: 

O IT J M o 

This negative term represents a power: 

P 2 = -h 2 s; (30) 

that is, a retarding torque during slow speed, or increasing £, and 
accelerating torque during high speed, or decreasing 0. 

The source of this torque may be found external to the motor, 
or internal, in its magnetic circuit. 



External sources of negative, Pi, may be, for instance, the 

magnetic field of a self-exciting, direct -current generator, driven 

r the synchronous motor. With decrease of Speed, this field 

's, due to the decrease of generated voltage, and increases 

vith increase of speed. This change of field strength, however, 

i behind the exciting voltage and thus speed, that is, during 
decrease of speed the output is greater than during increase of 
speed. If this direct-current generator is the exciter of the 
synchronous motor, the effect may l>e intensified. 

The change of power input into the synchronous motor, with 
change of speed, may cause the governor to act on the prime 
mover driving the generator, which supplies power to the motor, 
and the lag of the governor behind the change of output gives a 
pulsation of the generator frequency, of e () , which acts like 
a negative power, Pj. The pulsation of impressed voltage, 
caused by the pulsation of 0, may give rise to a negative, 
/'., also. 

An internal cause of a negative term, / J s , is found in the lag 
of the synchronous motor field behind the resultant m.m.f. In 
the preceding discussion, i- is the "nominal generated e.m.f." 

I of the synchronous machine, corresponding to the field excita- 
tion. The actual magnetic flux of the machine, however, does 
not correspond to e, and thus to the field excitation, but corre- 
sponds to the resultant m.m.f. of field excitation and armature 
reaction, which latter varies in intensity and in phase during the 
oscillation of 0. Hence, while e is constant, the magnetic flux 
is not constant, but pulsates with the oscillations of the machine. 
This pulsation of the magnetic flux lags l>ehind the pulsation of 
m.m.f., and thereby gives rise to a term in 6 in equation (28). 
If P B , &, e, e u , Z are such that a retardation of the motor increases 
the magnetizing, or decreases the demagnetising force of the 
armature reaction, a negative term, P,, appears, otherwise a 
positive term. 
Pi in this case is the energy consumed by the magnetic cycle 
uf the machine at full frequency, assuming the cycle at full fre- 
quency as the same as at frequency of slip, a, 

Or inversely, e may be said to pulsate, due to the pulsation of 
armature reaction, with the same frequency as &, but with a 
phase, which may either lie lagging or leading. Lagging of the 
pulsation of e causes a negative, leading a positive, P t , 

P~, therefore, represents the power due to the pulsation of e 


caused by the pulsation of the armature reaction, as discussed in 
"Theory and Calculation of Alternating-Current Phenomena." 

Any appliance increasing the area of the magnetic cycle of 
pulsation, as short-circuits around the field poles, therefore, 
increases the steadiness of a steady and increases the unsteadi- 
ness of an unsteady synchronous motor. 

In self-exciting synchronous converters, the pulsation of e is 
intensified by the pulsation of direct-current voltage caused 
thereby, and hence of excitation. 

Introducing now the term, P 2 = — h*s, into the differential 
equations of paragraph 169, gives the additional cases: 

b < 0, or negative, that is : 

c 2 + pP Q - h 2 

8./M0 < °- (31) 

Hence, denoting: 

6l 6 . __.. , (32) 


4. If: 6i 2 > a, g = + VV - a, 

6 = ^ l€ + (6l+/ ^ + A 2 € + (6l - /)tf . (33) 

That is, without oscillation, the motor drifts out of step, in 
unstable equilibrium. 

5. If: a > 6i 2 , g = y/a - bS, 

8 = £ € + M cos(00 + 6). (34) 

That is, the motor oscillates, with constantly increasing am- 
plitude, until it drops out of step. This is the typical case of 
cumulative surging by electro-mechanical resonance. 

The problem of surging of synchronous machines, and its 
elimination, thus resolves into the investigation of the coefficient: 



while the frequency of surging, where such exists, is given by: 

f _ jfeeo sin (a - 0) (c 2 + p P - /i 2 ) 2 (Wi 

Case (4), steady drifting out of step, has only rarely l>een 

The avoidance of surging thus requires: 


1. An elimination of the term ft 2 , or reduction as far as possible. 

2. A sufficiently large term, c 2 , or 

3. A sufficiently large term, pP . 

(1) refers to the design of the synchronous machine and the 
system on which it operates. (2) leads to the use of electro- 
magnetic anti-surging devices, as an induction motor winding in 
the field poles, short-circuits between the poles, or around the 
poles, and (3) leads to flexible connection to a load or a mo- 
mentum, as flexible connection with a flywheel, or belt drive of 
the load. 

The conditions of steadiness are : 

and if: 

c 2 + pP - h 2 > 0, 

(c 2 + pP - A 2 ) 2 ^ ee sin (a - 0) 

> — : 

16 t/M 

no oscillation at all occurs, otherwise an oscillation with decreas- 
ing amplitude. 

As seen, cumulative oscillation, that is, hunting or surging, 
can occur only, if there is a source of power supply converting 
into low-frequency pulsating power, and the mechanism of con- 
version is a lag of some effect — in the magnetic field of the 
machine, or external — which causes the forces restoring the 
machine into step, to be greater than the forces which oppose the 
deviation from the position in step corresponding to the load. 
For further discussion of the phenomenon of cumulative surging, 
and of cumulative oscillations in general, see Chapter XI of 
"Theory and Calculation of Electric Circuits. ,, 


171. The starting point of the theory of the polyphase and 
single-phase induction motor usually is the general alternating- 
current transformer. Coining, however, to the commutator 
motors, this method becomes less suitable, and the following 
more general method preferable. 

In its general form the alternating-current motor consists of 
one or more stationary electric circuits magnetically related to 
one or more rotating electric circuits. These circuits can be 
excited by alternating currents, or some by alternating, others 
by direct current, or closed upon themselves, etc., and connec- 
tion can be made to the rotating member either by ooIIesSsi 
rings— that is, to fixed points of the windings — or by commutator 
—that is, to fixed points in space. 

The alternating-current motors can he subdivided into two 
classes — those in which the electric and magnetic relation 
between stationary and moving members do not vary with their 
relative positions, ami those in which they vary with the relatifl 
positions of stator and rotor. In the latter a cycle of rotation 
exists, and therefrom the tendency of the motor results to lock at 
a speed giving a definite ratio between the frequency of rotation 
and the frequency of impressed e.m.f. Such motors, therefore, 
are synchronous motors. 

The main types of synchronous motors are as follows: 

1. One member supplied with alternating and the other with 
direct current — polyphase or single-phase synchronous motors, 

2. One member excited by alternating current, the other 
taining a single circuit closed upon itself — synchronous induction 

3. One member excited by alternating current, the other of 
different magnetic reluctance iii different direction! 
construction) — reaction motors. 

4. One member excited by alternating current, the other by 
altcrnating current of different frequency or different direction 
of rotation- — general alternating-current transformer or fre- 
quency converter and synchronous-induction generator. 


(II is the synchronous motor of the electrical industry. (2) 
and (3) are used occasionally to produce synchronous rotation 
without direct-current excitation, and of very great steadiness 
of the rate of rotation, where weight efficiency and power- 
factor are of secondary importance. (4) is used to some extent 
as frequency converter or alternating-current generator. 

(2) and (3) are occasionally observed in induction machines, 
and in the starting of synchronous motors, as a tendency to 
lock at some intermediate, occasionally low, speed. That is, 
in starting, the motor does not accelerate up to full speed, hut 
the acceleration stops at some intermediate speed, frequently 
half speed, and to carry the motor beyond this speed, the im- 
pressed voltage may have to be raised or even external power 
applied. The appearance of such "dead points" in the speed 
curve is due to a mechanical defect — as eccentricity of the 
rotor — or faulty electrical design: an improper distribution of 
primary and secondary windings causes a periodic variation of 
the mutual inductive reactance and so of the effective primary 
inductive reactance, (2) or the use of sharply defined and im- 
properly arranged teeth in both elements causes a periodic 
magnetic lock (opening and closing of the magnetic circuit, (3) 
and so a tendency to synchronize at the speed corresponding to 
this cycle. 

Synchronous machines have been discussed elsewhere. Here 
shall be considered only that type of motor in which the electric 
and magnetic relations between the slator and rotor do not vary 
with their relative positions, and the torque is, therefore, not 
limited to a definite synchronous speed. This requires that the 
rotor when connected to the outside circuit l>e connected through 
a commutator, and when closed upon itself, several closed cir- 
cuits exist, displaced in position from each other so as to offer a 
resultant closed circuit in any direction. 

The main types of these motors are: 

1. One member supplied with polyphase or single-phase alter- 
nating voltage, the other containing several circuits closed upon 
themselves — polyphase and single-phase induction machines. 

2. One member supplied with polyphase or single-phase alter- 
nating voltage, the other connected by a commutator to an 
alternating vol I age — compensated induction motors, commutator 

lotors with shunt-motor characteristic. 
:?. lioth members connected, through a commutator, directly 



or inductively, in series with each other, to an alternating vol- 
tage — alternating-current motors with series-motor characteristic. 

Herefrom then follow three main classes of alternating-current 
motors ; 

Synchronous motors. 

Induction motors. 

Commutator motors. 

There are, however, numerous intermediate forms, which 
belong in several classes, as the synchronous-induction motor, 
the c o oipe n sat ed-in due lion motor, etc. 

172. An alternating current, /, in an electric circuit produces 
a magnetic flux, 4 1 , interlinked with this circuit. Considering 
equivalent sine waves of / and *, 4> lags behind / by the angle 
of hysteretic lag, a. This magnetic flux, $, generates an e.m.f., 
5 = 2 tt/;i<I>, where / = frequency, n = number of turns of 
electric circuit. This generated e.m.f., E, lags 90° behind the 
magnetic flux, *, hence consumes an e.m.f. 90° ahead of ♦, 
or 90—ci degrees ahead of /. This may be resolved in a reactive 
component: E = 2x/ft* eos a = 2 t/LI = xl, the o.m.f, con- 
sumed by self-induction, and power component: E" = 2r/n* 
sin a = 2irfHI = r"I = e.m.f. consumed by hysteresis (eddj 
currents, etc.), and is, therefore, in vector representation denoted 

E' = jxf and E" = f>% 

x = 2 irfL — reactance, 

L = inductance, 
r" = effective hysteretic resistance. 

The ohmic resistance of the circuit, r', consumes an e.n 
r'(, in phase with the current, and the total or effective resistance 
of the circuit is, therefore, r = r' + r", and the total e.m.f. 
consumed by the circuit, or the impressed e.m.f.. is: 

E = (r+jx)I = Z{, 
.where : 

Z = r + jx = impedance, in vector denotation, 
z = Vr* + i* = impedance, in absolute terms. 

If an electric circuit is in inductive relation to another electa 
circuit, it is advisable to separate the inductance, L, of the cir- 


cuit in two parts — the self-inductance, S, which refers to that 
part of the magnetic flux produced by the current in one circuit 
which is interlinked only with this circuit but not with the other 
circuit, and the mutual inductance, M , which refers to that part 
of the magnetic flux interlinked also with the second circuit. 
The desirability of this separation results from the different char- 
acter of the two components: The self-inductive reactance gen- 
erates a reactive e.m.f. and thereby causes a lag of the current, 
while the mutual inductive reactance transfers power into the 
second circuit, hence generally does the useful work of the ap- 
paratus. This" leads to the distinction between the self-inductive 
impedance, Z = r + jx , and the mutual inductive impedance, 
Z = r + jx. 

The same separation of the total inductive reactance into self- 
inductive reactance and mutual inductive reactance, represented 
respectively by the self-inductive or "leakage" impedance, and 
the mutual inductive or "exciting" impedance has been made 
in the theory of the transformer and the induction machine. In 
those, the mutual inductive reactance has been represented, not 
by the mutual inductive impedance, Z, but by its reciprocal 

value, the exciting admittance: Y = ■=• It is then: 

r is the coefficient of power consumption by ohmic resistance, 
hysteresis and eddy currents of the self-inductive flux — effective 

x is the coefficient of e.m.f. consumed by the self-inductive or 
leakage flux — self-inductive reactance. 

r is the coefficient of powfer consumption by hysteresis and 
eddy currents due to the mutual magnetic flux (hence contains 
no ohmic resistance component). 

x is the coefficient of e.m.f. consumed by the mutual magnetic 

The e.m.f. consumed by the circuit is then: 

# = Zol + Zh l (1) 

If one of the circuits rotates relatively to the other, then in 
addition to the e.m.f. of self-inductive impedance : Z /, and the 
e.m.f. of mutual-inductive impedance or e.m.f. of alternation: 
ZJ y an e.m.f. is consumed by rotation. This e.m.f. is in phase 
with the flux through which the coil rotates — that is, the flux 
parallel to the plane of the coil — and proportional to the speed — 



that, is, the frequency of rotation — while the e.m.f. of alternation 
is 90° ahead of the flux alternating through the coil— thai is, Uw 

flux parallel to the axis of the coil— and proportional to the fre- 
quency. If, therefore, Z' is the impedance corresponding to the 
former flux, the e.m.f. of rotation is —jSZ'J, where S is the 
ratio of frequency of rotation to frequency of alternation, or the 
speed expressed in fractions of synchronous speed. The total 
e.m.f. consumed in the circuit is thus: 

g = z i + XI - jSZ'l. 
Applying now these considerations to the alternating-current 
motor, we assume all circuits reduced to the same number of 
turns— that is, selecting one circuit, of n effective turns, as start- 
ing point, if n, = number of effective turns of any other circuit, 
all the e.m.fs. of the latter circuit arc divided, the currents multi- 
plied with the ratio, -> the impedances divided, the admittances 

multiplied with I -) . This reduction of the constants of all 
circuits to the same number of effective turns is convenient by 
eliminating constant factors from the equations, and so permit- 

ting a direct comparison. 

When speaking, therefore, in (he fol- 
lowing of the impedance, etc., of the 
different circuits, we always refer to 
their reduced values, as it is cus- 
tomary in induction-motor designing 
practice, and has been done in pre- 
ceding theoretical investigations. 
173. Let, then, in Fig. 147: 
Pn, f«, Zn = impressed voltage, 
current and self-inductive impedance 
respectively of a stationary circuit, 
F, c . 147. Pu h, Z> = impressed voltage, 

current and self-inductive impedance 
respectively of a rotating circuit, 

r = space angle between the axes of the two circuits, 
Z = mutual inductive, or exciting impedance in the direction 
mI the axis (if the stationary coil, 

Z' = mutual inductive, or exciting impedance in the direction 
of the axis of the rotating coil, 

Z" - mutual inductive or exciting impedance in the direction 
at right angles to the axis of (he rotating coil, 


S = speed, as fraction of synchronism, that is, ratio of fre- 
quency of rotation to frequency of alternation. 

It is then : 

E.m.f. consumed by self-inductive impedance, Z /o. 

E.m.f. consumed by mutual-inductive impedance, Z (/ + J\ 
cos r) since the m.m.f. acting in the direction of the axis of the 
stationary coil is the resultant of both currents. Hence: 

$o - Zo/o + Z (/o + /i COS r). (3) 

In the rotating circuit, it is: 

E.m.f. consumed by self-inductive impedance, Zi/i. 

E.m.f. consumed by mutual-inductive impedance or " e.m.f. of 
alternation": Z' (/i + / cos r). (4) 

' E.m.f. of rotation, — jSZ"lo sin t. (5) 

Hence the impressed e.m.f. : 

#i = ZJi + Z' (/! + U cos r) - jSZ"/o sin r. (6) 

In a structure with uniformly distributed winding, as used in 
induction motors, etc., Z' = Z" = Z, that is, the exciting im- 
pedance is the same in all directions. 

Z is the reciprocal of the "exciting admittance," Y of the in- 
duction-motor theory. 

In the most general case, of a motor containing n circuits, of 
which some are revolving, some stationary, if: 

l$k, hy Zk = impressed e.m.f., current and self-inductive im- 
pedance respectively of any circuit, fc. 

Z\ and Z" = exciting impedance parallel and at right angles 
respectively to the axis of a circuit, i, 

t*» = space angle between the axes of coils k and i, and 

S = speed, as fraction of synchronism, or "frequency of 

It is then, in a coil, i: 

$ { = ZJi + Z i $* /* cos Tu 1 - jSZ" >* h sin r k \ (7) 

i i 


Ziji = e.m.f. of self-inductive impedance; (8) 


Z*^ /* cos r* 1 = e.m.f. of alternation; (9) 


E'i = - jSZ iiS jkJ k sin tV = e.m.f. of rotation; (10) 


which latter = in a stationary coil, in which 5 = 0. 



The power output of the motor is the sum of the powers of all 
the e.m.fs. of rotation, hence, in vector denotation: 


- - S £ tfZ«J* /* sin r*S /J 1 , (11) 


and herefrom the torque, in synchronous watts: 

D - ^ - - J? ljZ u i h sin r*S UK (12) 

o i i 

The power input, in vector denotation, is: 


Po = F [E i9 h] 

= £ [E it hy + J? [ft, /j/ 
= Po 1 + jPo>; 

and therefore: 

Po 1 = true power input; 

P</ = wattless volt-ampere input; 

Q = VPo 1 + Po* = apparent, or volt-ampere 

D . = efficiency; 

n = apparent efficiency; 
iTi = torque efficiency; 
-~ = apparent torque efficiency; 


-Jr = power-factor. 

From the n circuits, i = 1, 2 . . . n, thus result n linear 
equations, with 2 n complex variables, /< and #». 

Hence n further conditions must be given to determine the 
variables. These obviously are the conditions of operation of 
the n circuits. 

Impressed e.m.fs. /? t may be given. 

Or circuits closed upon themselves #» = 0. 

Or circuits connected in parallel c^i = c*#*, where c, and c* 


are the reduction (actors of the circuits to equal number of 
effective turns, as discussed before. 

Or circuits connected in series: -■* = --» etc. 

Ci c k 

When a rotating circuit is connected through a commutator, 
the frequency of the current in this circuit obviously is the same 
as the impressed frequency. Where, however, a rotating circuit 
is permanently closed upon itself, its frequency may differ from 
the impressed frequency, as, for instance, in the polyphase in- 
duction motor it is the frequency of slip, s = 1 — *S, and the 
self -inductive reactance of the circuit, therefore, is sx; though in 
its reaction upon the stationary system the rotating system nec- 
essarily is always of full frequency. 

As an illustration of this method, its application to the theory 
of some motor types shall be considered, especially such motors 
as have either found an extended industrial application, or have 
at least been seriously considered. 


174. In the polyphase induction motor a number of primary 
circuits, displaced in position from each other, are excited by 
polyphase e.m.fs. displaced in phase from each other by a phase 
angle equal to the position angle of the coils. A number of sec- 
ondary circuits are closed upon themselves. The primary usu- 
ally is the stator, the secondary the rotor. 

In this case the secondary system always offers a resultant 
closed circuit in the direction of the axis of each primary coil, 
irrespective of its position. 

Let us assume two primary circuits in quadrature as simplest 
form, and the secondary system reduced to the same number of 
phases and the same number of turns per phase as the primary 
system. With three or more primary phases the method of 
procedure and the resultant equations are essentially the same. 

Let, in the motor shown diagrammatically in Fig. 148: 

#o and — j$o, /o and — j'/o, Z = impressed e.m.f., currents 
and self-inductive impedance respectively of the primary system. 

0, /i and —jl\ 9 Z\ = impressed e.m.f., currents and self-in- 
ductive impedance respectively of the secondary system, reduced 
to the primary. Z = mutual-inductive impedance between 
primary and secondary, constant in all directions. 



S = speed; s = 1 — S — slip, as fraction of synchronism. 
The equation of the primary circuit is then, by (7) : 

E* = ZoJ + Z (/, - /,). (14) 

The equation of the secondary circuit: 

- ZJi + Z (/i - /.) + jSZ (jf x - jh), (15) 

from (15) follows: 

Zo (1 - S) 

= /< 


Z(l-S) + Z l * V Z& + Zi 




Fio. 148. 




and, substituted in (14): 
Primary current: 

/o = Eo 
Secondary current: 

Za + Zi 

ZZoS + ZZi + ZoZi 

r - w - _ Zs _ 

/ 1 *° ZZos + ZZ X + Z^Zi 


Exciting current: 

/oo = h - h = Eo zzliTzzT+zifi 

E.m.f. of rotation: 

E' = jSZ (jh - jh) = sz (/. - /t). 


- SEi 

ZZoS + ZZi + ZoZi 

= (l-8)E0-r ? yT- j-' 


ZZoS + ZZ X + ZoZi 






It is, 

at synchronism ; s 

- 0: 


~ Z'+'Zo 


= 0; 


" fa 


2?oZ £10 
Z + Zo t , Zo 

1 + z 

At standstill: 

8 • 

- i; 


£» (Z + zo . 

ZZo H~ ZZi -f- ZoZi 



_ 2?oZ 

ZZo "h ZZi + ZoZi 


ZZo + ZZi + ZoZi 

#' = 0. 

Introducing as parameter the counter e.m.f ., or e.m.f. of mutual 
induction : 

# = #o — Zo/o, (21) 


#o = # + Zo/o, (22) 

it is, substituted : 
Counter e.m.f. : 

v = ^° zZoT+zz\ + ZoZV (23) 


Primary impressed e.m.f.: 

« „ ZZos + Zi + ZZoZi , .v 

#o = # 22 ' '**' 

E.m.f. of rotation: 

#' = #S = # (1 - s). (25) 

Secondary current: 

h = Jj- (26) 

Primary current: 

/o " ^~zzT' z; + z (27) 


Exciting current: 

/oo = § = $Y. (28) 

These are the equations from which the transformer theory of 
the polyphase induction motor starts. 

176. Since the frequency of the secondary currents is the fre- 
quency of slip, hence varies with the speed, S = 1 — 8, the sec- 
ondary self-inductive reactance also varies with the speed, and 
so the impedance: 

Zi = n + J8 Xl . (29) 

The power output of the motor, per circuit, is: 

P = [£', /i] 

ri ""v'z" r - <7^-i2 ( r i ~ J**i)» ( 3 °) 

[ZZos + ZZl + ZoZx] 2 

where the brackets [ ] denote the absolute value of the term in- 
cluded by it, and the small letters, c , z, etc., the absolute values 
of the vectors, #o, Z, etc. 

Since the imaginary term of power seems to have no physical 
meaning, it is: 
Mechanical power output: 

p _c Vs(l - s)r x ( . 

[ZZos + ZZx + ZoZtf K } 

This is the power output at the armature conductors, hence in- 
cludes friction and windage. 
The torque of the motor is: 

D = 

1 - 8 

eJ&iS • _ eoV sis* _ ,o 2 \ 

[ZZos + ZZi + ZoZif 2 J [ZZos + ZZ X + ZoZtf ^ ; 

The imaginary component of torque seems to represent the 
radial force or thrust acting between stator and rotor. Omitting 
this we have: 

T\ _ ^o z __^} s (1*1} 

~ [ZZos + ZZx + z&W* 



The power input of the motor per circuit is: 

Po = [#o, /o] 

= *° 2 L 1 ' ZZ08 +*ZZ l + ZoZj (34) 

= P'o - jPoj 

P'o = true power, 

PJ = reactive or "wattless power," 

Q = a/PV + iV* = volt-ampere input. 

Herefrom follows power-factor, efficiency, etc. 
Introducing the parameter: #, or absolute e, we have: 
Power output: 

- [* a 

= -— * - jii'Szi. (35) 

Power input: 
Po = [#o, /o] 

t [ZZ os + Z Z t + ZoZi Za_+ Z t 

" c L zzi ' zii . 

Z (Z« + Zi) . , Za + Z, 

t rZo(Z8 + ZQ Z« + Z,1 

~ C L ZZ t " + *' ZZ, ~J 


. rZs + Zn* / , . . e*s , . . , e 1 . . , 

- e \rzzv\ l ( ° " J o) + ^» (ri ~ • , * Cl) + 2« (r " Ja:) 

= *V (r, - jxo) + ii* (~ - j*,) + too* (r - j*). (36) 

And since: 

r, 5 + s Su , 

= — n - - + r„ 



it is: 

P o = (*o 2 r« + i, 2 r! + iooV + P) - j (tV*o + *V*i + ioo*x). (37) 

2*o 2 r = primary resistance loss, 

i\ 2 ri = secondary resistance loss, 

tooV = core loss (and eddy-current loss), 

P = output, 

io 2 Xo = primary reactive volt-amperes, 

ii 2 X\ = secondary reactive volt-amperes, 

ioo 2 x = magnetizing volt-amperes. 

176. Introducing into the equations, (16), (17), (18), (19), (23) 
the terms: 

z - *°' 


Where Xo and Xi are small quantities, and X = Xo + Xi is the 
"characteristic constant" of the induction motor theory, it is: 
Primary current: 

j = &± « + Xt _ _ E s + Xi . 

/0 Z sX + Xi +"X X 1 2«Xo + X * J 

Secondary current: 

/. = E ° * = Eo ! (40) 

41 Z sXo + Xi + XoXi Z 5X0 + Xi v ' 

Exciting current: 

r _ -Bo Xi __ Eo Xi . 

/0 ° " Z sXo + X~i + XoXi ~~ Z 5X0 + Xi' l l) 

E.m.f . of rotation : 

W = QoS .— ,-^ r .-. = VoS -\ - - (42) 

sXo + Xi + X0X1 5X0 + Xi ' 

Counter e.m.f.: 

sXo + Xi + X0X1 sXo + Xi v ' 



177. As an example are shown, in Fig. 149, with the speed 
as abscissae, the curves of a polyphase induction motor of the 

e = 320 volts, 

Z = 1 + 10j ohms, 

Z = Z l = 0.1 + 0.3 j ohms; 
hence : 

X = Xi = 0.0307 - 0.0069 j. 


320 VOLTS 



rr \i\ . 




1 WW 


^ s 




\ i 







i i 
\ i 
\ | 






> \i 




.— — 5^ 



•_ _,- — ^» 


V4|— 150- 

\ 11 1AA 


— "»? 





M ^ft 

*1 WJ 


—I 10 


0.2 0.8 

0.4 0.5 0.6 
Fi«. 149. 


0.8 0.9 


It is: 
/o = 

320{ 10.30 s - (« + 0.1)i| 
(103 + 1.63*) - j(0.11 - 5.99«) amp * 

D = (1.03 + 1.63^ + (0.11 - 5.99 s)' *y™ h ™<>™ kw - 

P = (1 - s) D 

0.11 - 5.99 s 

tan 6" = 

1.03 + 1.63 s 

tan*'= * + 01 ; 
10.3 s ' 

cos (0' — 6") = power-factor. 

Fig. 149 gives, with the speed S as abscissae: the current, J; 
the power output, P; the torque, D; the power-factor, p; the 
efficiency, rj. 



The curves show the well-known characteristics of the poly- 
phase induction motor: approximate constancy of speed at all 

loads, and good efficiency and power- fact or within this narrow- 
speed range, but poor constants at all other speeds. 

178. In the single-phase induction motor one primary circuit 
acts upon a system of closed secondary circuits which are dis- 
placed from each other in position on the secondary member. 

Let the secondary be assumed as two-phase, that is, containing 
or reduced to two circuits closed upon themselves at right angles 

Fio. 150. — SiiiKle-phosp induction n 

to each other. While it then offers a resultant closed secondary 
circuit to the primary circuit in any position, the electrical dis- 
position of the secondary is not symmetrical, but the directions 
parallel with the primary circuit and at right angles thereto are 
to be distinguished. The former may be called the secondary 
energy circuit, the latter the secondary magnetizing circuit, since 
in the former direction power is transferred from the primary to 
the secondary circuit, while in the latter direction the secondary 
circuit can act magnetizing only. 

Let, in the diagram Fig. 150: 

E a , Ja, Z n = impressed e.m.f., current and self-inductive im- 
pedance, respectively, of the primary circuit, 

l\, Z\ = current and self-inductive impedance, respectively, 
of the secondary energy circuit, 

/), Zi = current and self-inductive impedance, respectively, 
of the secondary magnetizing circuit, 

Z = mutual -inductive impedance, 

S m speed, 
and let s = 1 - S 2 (where s is not the slip). 

It is then, by equation (7) : 


Primary circuit: 

E = Zolo + Z(h~ /i). ' (44) 

Secondary energy circuit: 

= Z X U + Z (/i - /„) - jSZU- (45) 

Secondary magnetizing circuit: 

= ZJ, + ZU ~ )SZ (/. - /,) ; (46) 

hence, from (45) and (46) : 

/l ~ /o zuT+2Zz; + zs' (47) 

h = + jS/. ^^2 iz, + Z? ' (48) 

and, substituted in (44) : 
Primary current: 

h = #o ^ (49) 

Secondary energy current: 

U = *. *<*» + *> . (50) 

Secondary magnetizing current: 

/. = + jSE ^ (51) 

E.m.f. of rotation of secondary energy circuit: 

#i = - jSZh = S'# Z ^- (52) 

E.m.f. of rotation of secondary magnetizing circuit: 

E' t = - jsz (/. - /o = - is^o ZZl ( ^ + Zl) ; (53) 

where : 

X = Z„ (Z*So + 2 ZZ, + Z,») + ZZ l (Z + Z,). (54) 

It is, at synchronism, S = 1, s = 0: 

, _ f 2Z + Z t . 

*° *' Z«(2Z + Zi)+Z(Z+Zi)' 

Jl = ^° Z (2 Z + ZO +~Z(Z +' Zi) ' 

/s = + j#o z7(2Z + Z,y + zlz + Z",")" 


Hence, at synchronism, the secondary current of the single- 
phase induction motor does not become zero, as in the polyphase 
motor, but both components of secondary current become equal. 

At standstill, S = 0, s = 1, it is: 

/0 ^° ZZo + ZZ X + ZoZi' 

? l = ^° zz7+~ zzV+lz'oZi' 
U = 0. 

That is, primary and secondary current corresponding thereto 
' have the same values as in the polyphase induction motor, as 
was to be expected. 

179. Introducing as parameter the counter e.m.f., or e.m.f. of 
mutual induction: 

and substituting for / from (49), it is: 
Primary impressed e.m.f.: 

_ „ Z (Z 2 s + 2 ZZ X + Z?) + ZZ X (Z + Z,) , 

*°-* zz^z + zo ' (o5) 

Primary current: 

r „ Z'so + 2 ZZ, + Z, ! riw v 

h ~v zzaz + ZiT " m 

Secondary energy circuit: 

_ p ZsojH Z, _ s E S s £_ , _> 

71 " * z7(z + z.) z, ~ l ~ z + z, v " 

£'. = S*V g-Z z (58) 

Secondary magnetizing circuit : 

/.-+J Z + z ; (59) 

Vt-JW-' ^ (60) 


/o - /i = ^ (61) 

These equations differ from the equations of the polyphase 

induction motor by containing the term s = (1 — S 2 ), instead 

of s = (1 — S), and by the appearance of the terms, y~r^~ and 

S 2 E 
~-, «■» of frequency (1 + S), in the secondary circuit. 



The power output of the motor is: 

P - [Eu /J + [Et, h] 

= ^^-{[ZZu Zs + ZJ - [Z, (Z + Z,), ZJ} 


and the torque, in synchronous watts: 

D "s~~~~m • m 

From these equations it follows that at synchronism tor- 
que and power of the single-phase induction motor are already 

Torque and power become zero for: 

SoZ 2 — Zi 2 = 0, 



that is, very slightly below synchronism. 

Let z = 10, Zi = 0.316, it is, S = 0.9995. 

In the single-phase induction motor, the torque contains the 
speed S as factor, and thus becomes zero at standstill. 

Neglecting quantities of secondary order, it is, approximately : 

h = £o zJZ^ + Zl )+2Zai (65) 

/. - + jSE, z (z^-zj+YzX (67) 

^ = S ^ Z(Z„ So + Z 1 )+2Z.Z l ' (68) 


. ^ = " jS ** Z (Zoso + zi)+2 ZoZi (69) 

P = SWz^sp , 

[z TZoso + zo + 2 ZoZj*' uu; 

n = _Se Vri8 / 71 x 

[Z (ZoSo + Z\j + 2 ZoZi] 2 ' u i} 

This theory of the single-phase induction motor differs from 
that based on the transformer feature of the motor, in that it 
represents more exactly the phenomena taking place at inter- 



mediate speeds, which are only approximated by the transformer 
theory of the single-phase induction motor. 

For studying the action of the motor at intermediate and at 
low speed, as for instance, when investigating the performance 
of a starting device, in bringing the motor up to speed, that is, 
during acceleration, this method so is more suited. An applica- 
tion to the "condenser motor," that is, a single-phase induction 
motor using a condenser in a stationary tertiary circuit (under 
an angle, usually 60°, with the primary circuit) is given in the 
paper on "Alternating-Current Motors," A. I. E. E. Transac- 
tions, 1904. 


Fig. 151. 

180. As example are shown, in Fig. 151, with the speed as 
abscissae, the curves of a single-phase induction motor, having 
the constants: 

e = 400 volts, 

Z = 1 + 10 j ohms, 



Z = Zi = 0.1 + 0.3 j ohms; 


Io = 400 j* amp. ; 

AT = (s + 0.2) + j(10s + 0.6 - 0.6 S); 

K = (0.1+0.3j)AT+(l + 10j)(0.1+j)(0.3-0.3S); 

D = 

1616 Ss 


synchronous kw. 



Fig. 151 gives, with the speed, S, as abscissa*: the current, 7«, 
the power output, P, the torque, D, the power-factor, p, the 
efficiency, y. 


18L Since the characteristics of the polyphase motor do not 
depend upon the number of phases, here, as in the preceding, a 
two-phase system may be assumed: a two-phase stator winding 
acting upon a two-phase rotor winding, that is, a closed-coil 
rotor winding connected to the commutator in the same manner 
as in direct-current machines, but with two sets of brushes in 
quadrature position excited by a two-phase system of the same 
frequency. Mechanically the three-phase system here has the 
advantage of requiring only three sets of brushes instead of four 

\ jl<* 

Fio. 152. 

as with the two-phase system, but otherwise the general form 
of the equations and conclusions are not different. 

Let #o and — j# = e.m.fs. impressed upon the stator, #i and 
— jfli = e.m.fs. impressed upon the rotor, O « phase angle be- 
tween e.m.f., #o and #i, and 0i ■» position angle lwtween the 
stator and rotor circuits. The e.m.fH., #o and — j# , produce the 
same rotating e.m.f. as two e.m.fH. of equal intensity, but dis- 
placed in phase and in position by angle O from #», and jf/l,,, 
and instead of considering a displacement of phase, 0, h arid a dis- 
placement of position, 0i, between stator and rotor circuits, we 
can, therefore, assume zero-phase displacement and diMplacemeut 
in position by angle O + 0i = 0. Phase diMplaecmcnf l*etween 
stator and rotor e.m.fH. is, therefore, equivalent to n fluff of 
brushes, hence gives no additional feature beyond those pro- 
duced by a shift of the commutator bru«he*. 


Without losing in generality of the problem, we can, therefore, 
assume the stator e.m.fs. in phase with the rotor e.m.fs., and the 
polyphase shunt motor can thus be represented diagrammatically 
by Fig. 152. 

182. Let, in the polyphase shunt motor, shown two-phase in 
diagram, Fig. 152: 

#o and — j#o, /o and — j/o, Z = impressed e.m.fs., currents 
and self-inductive impedance respectively of the stator circuits, 

c$o and — jc# , /i and — j/i, Z\ = impressed e.m.fs., currents 
and self-inductive impedance respectively of the rotor circuits, 
reduced to the stator circuits by the ratio of effective turns, c, 

Z = mutual-inductive impedance, 

S = speed; hence s = 1 — S = slip, 

= position angle between stator and rotor circuits, or 
"brush angle." 

It is then : 

#o = Zo/o + Z(h- /i cos - jh sin 0). (72) 


cGo = ZJ X + Z (/i - /o cos 6 + jlo sin 6) - 

jSZ ( - j/i + h sin 6 + j[o cos 0). (73) 

a = cos 6 — j sin 0, 
b = cos + j sin 0, 
it is: 

ad = 1, (75) 

and : 

£o = Zo/o + Z (/o - 5/0, (76) 

c#o = Z,U + Z(f x - alo) + jSZ 07i " Wo) 

= Z,/i + sZ (/i - cr/o). (77) 

Herefrom follows: 

(« + «c) Z + Z, 

' ° _ *"> 7zz»~+ zzT+ ZoZi (78) 

' ' " * izz7+"zzr+~z^' ( ' 9) 

for c = o, this gives: 

, _ „ *Z_+ Z\ 

/0 " *" sZZ* + ZZ\ + ZoZi 

j - v sZ ' 

* l -' r « szz -+zzl + z z i ' 



that is, the polyphase induction-motor equations, a = cos + 

j sin = 1» representing the displacement of position between 
stator and rotor currents. 

This shows the polyphase induction motor as a special case of 
the polyphase shunt motor, for c = o. 

The e.m.fs. of rotation are: 

£'i = -jSZ (- jh + h sin + j/o cos 0) 

- SZ (*h- I i)i 
hence : 

& l ^'iZZl+ZZx + ZtZt' (80) 

The power output of the motor is: 
P - [£., 7.1 

= m . + zz l + za# l( * Zl " cZo) z > (ff8 + c) z + cZ ° ] > (81) 

which, suppressing terms of secondary order, gives: 

p _ <Se V { g(r i + c (x sin — r cos 0) ) + c (r i c os + x t sin — cr ) } 

~ [sZZo + ZZi + ZoZfr ' 


for Sc = o, this gives: 

p Seohhri 

[sZZo + ZZ! + ZoZJ' 1 ' 

the same value as for the polyphase induction motor. 

In general, the power output, as given by equation (82), be- 
comes zero: 

p = °' 

for the slip: 

r x cos + Xi sin - cr , QON 

fi + c (x sin — r cos 0) 

183. It follows herefrom, that the speed of the polyphase 
shunt motor is limited to a definite value, just as that of a direct- 
current shunt motor, or alternating-current induction motor. 
In other words, the polyphase shunt motor is a constant-speed 
motor, approaching with decreasing load, and reaching at no- 
load a definite speed : 

So = 1 - so. (84) 

The no-load speed, S , of the polyphase shunt motor is, how- 
ever, in general not synchronous speed, as that of the induction 



motor, but depends upon the brush angle, 0, and the ratio, c, of 
rotor -J- stator impressed voltage. 

At this no-load speed, So, the armature current, i\, of the 
polyphase shunt motor is in general not equal to zero, as it is 
in the polyphase induction motor. 

Two cases are therefore of special interest: , 

1. Armature current, I\ = o, at no-load, that is, at slip, *o. 

2. No-load speed equals synchronism, s = o 
1. The armature or rotor current (79): 

T _ F vsZ + c (Z + Zi) 
41 ** sZZo + ZZi + ZoZi 
becomes zero, if: 


c = — as 

Z + Z x 

or, since Z x is small compared with Z, approximately: 

c = — as = — s (cos — j sin 6); 

hence, resolved: 

c = — s cos 6, 
o = s sin 6; 

.' : -.. ) « 

That is, the rotor current can become zero only if the brushes 
are set in line with the stator circuit or without shift, and in this 
case the rotor current, and therewith the output of the motor, 
becomes zero at the slip, s = — c. 

Hence such a motor gives a characteristic curve very similar 
to that of the polyphase induction motor, except that the stator 
tends not toward synchronism but toward a definite speed equal 
to (1 + c) times synchronism. 

The speed of such a polyphase motor with commutator can, 
therefore, be varied from synchronism by the insertion of an 
e.m.f. in the rotor circuit, and the percentage of variation is the 
same as the ratio of the impressed rotor e.m.f. to the impressed 
stator e.m.f. A rotor e.m.f., in opposition to the stator e.m.f. 
reduces, in phase with the stator e.m.f., increases the free-run- 
ning speed of the motor. In the former case the rotor impressed 
e.m.f. is in opposition to the rotor current, that is, the rotor 
returns power to the system in the proportion in which the speed 


is reduced, and the speed variation, therefore, occurs without 
loss of efficiency, and is similar in its character to the speed con- 
trol of a direct-current shunt motor by varying the ratio between 
the e.m.f . impressed upon the armature and that impressed upon 
the field. 

Substituting in the equations: 

6 = 0, 

8 + C = S\ 


it is: 

h ~ V° sZZ Q + ZZ l + ZoZ/ (87) 

h = ®» sZZo + ZZ X + ZoZ/ (88) 

p _ Se<?z*8i (ri - cr ) ( 

r [sZZo + ZZ X + ZoZi] 2 ' K ] 

These equations of 7 and I\ are the same as the polyphase 
induction-motor equations, except that the slip from synchron- 
ism, s, of the induction motor, is, in the numerator, replaced by 
the slip from the no-load speed, «i. 

Insertion of voltages into the armature of an induction motor 
in phase with the primary impressed voltages, and by a com- 
mutator, so gives a speed control of the induction motor without 
sacrifice of efficiency, with a sacrifice, however, of the power- 
factor, as can be shown from equation (87). 

184. 2. The no-load speed of the polyphase shunt motor is in 
synchronism, that is, the no-load slip, s = o, or the motor out- 
put becomes zero at synchronism, just as the ordinary induction 
motor, if, in equation (83) : 

t\ cos 6 + Xi sin 6 — cr = o; 
hence : 

e = n«" !±* "Lf; (90) 

or, substituting: 

= tan «i, (91) 

* where ai is the phase angle of the rotor impedance, it is: 

c = - 1 cos (c*i — 0), 
r " 



cos ( ttl - 6) = - c, (92) 


c = !i_^_(«j_^). (93) 


Since r is usually very much smaller than z X) if c is not very 
large, it is: 

cos (c*i — 6) = o; 
hence : 

= 90° - «i. (94) 

That is, if the brush angle, 0, is complementary to the phase 
angle of the self-inductive rotor impedance, a\, the motor tends 
toward approximate synchronism at no-load. 


At given brush angle, 0, a value of secondary impressed e.m.f., 
c#o, exists, which makes the motor tend to synchronize at no- 
load (93), and, 

At given rotor-impressed e.m.f., c# , a brush angle, 0, exists, 
which makes the motor synchronize at no-load (92). 

185. 3. In the general equations of the polyphase shunt motor, 
the stator current, equation (78) : 

sZ + Zi +^bcZ 
/o " *° sZZo + ZZ\ + ZoZi 

can be resolved into a component: 

/ "° = ^ IZZ* + ZZ, + ZoZy (95) 

which does not contain c, and is the same value as the primary 
current of the polyphase induction motor, and a component: 

r ° = $* 'aZzT+zzx'+ZoZ'i (96) 

Resolving /"o, it assumes the form: 

/"o«£o*c(j1, -jA 2 ) 

= c { A\ cos + A 2 sin 0) + j (Ai sin — A 2 cos 0) }. (97) 

This second component of primary current, 7" , which is pro- 
duced by the insertion of the voltage, c#, into the secondary cir- 
cuit, so contains a power component: 

z'o = c (Ai cos + A 2 sin 0), (98) 


and a wattless or reactive component: 

t"o = +jc (A i sin - A 2 cos 0); (99) 

where : 

/"o = t'o - j*"o. (100) 

The reactive component, i"o, is zero, if: 

Ai sin — At cos = o; (101) 


tan 0! = + ^ 2 - (102) 

In this case, that is, with brush angle, 0i, the secondary im- 
pressed voltage, c#, does not change the reactive current, but 
adds or subtracts, depending on the sign of c, energy, and so 
raises or lowers the speed of the motor: case (1). 

The power component, t'o, is zero, if: 

Ax cos + A 2 sin = o, (103) 


tan 2 = - 4 1 ' (104) 

In this case, that is, with brush angle, 2 , the secondary im- 
pressed voltage, cE, does not change power or speed, but pro- 
duces wattless lagging or leading current. That is, with the 
brush position, 2 , the polyphase shunt motor can be made to 
produce lagging or leading currents, by varying the voltage im- 
pressed upon the secondary, c$, just* as a synchronous motor 
can be made to produce lagging or leading currents by varying 
its field excitation, and plotting the stator current, /o, of such a 
polyphase shunt motor, gives the same V-shaped phase charac- 
teristics as known for the synchronous motor. 

These two phase angles or brush positions, 0i and 2 , are in 
quadrature with each other. 

There result then two distinct phenomena from the insertion 
of a voltage by commutator, into an induction-motor armature : 
a change of speed, in the brush position, 0i, and a change of phase 
angle, in the brush position, 2 , at right angles to 6\. 

For any intermediate brush position, 0, a change of speed so 
results corresponding to a voltage: 

c$ cos (0i - 0) ; 


and a change of phase angle corresponding to a voltage: 

c$ cos (0 2 - 0), 
= c& sin (0i - 0), 

and by choosing then such a position, 0, that the wattless current 
produced by the component in phase with 0*, is equal and op- 
posite to the wattless lagging current of the motor proper, /'o, 
the polyphase shunt motor can be made to operate at unity 
power-factor at all speeds (except very low speeds) and loads. 
This, however, requires shifting the brushes with every change 
of load or speed. 

When using the polyphase shunt motor as generator of watt- 
less current, that is, at no-load and with brush position, 2 , it is: 

s = 0; 
hence, from (78) : 

'• - e-zrtrxr (105) 

''• - zfz. < l06 > 

or, approximately: 

7'o = 



that is, primary exciting current: 


'"• - * znrho)' (107) 

or, approximately, neglecting Z against Z: 

„ Eo&c 
1 ° - "Z' x 

_ EpC (cos + j sin 0) 

~" ri + jxi 

= -° ■ { (ri cos + Xi sin 0) — j (Xi cos — n sin 0) ) , 


and, since the power component vanishes: 

r x cos + X\ sin = 0, 

tan 2 = - r -- (109) 



Substituting (109) in (108) gives: 

/"o = ■-!- (^1 cos 2 — t\ sin 2 ) 


= -J 


. EoC, 




T Eo . EoC 

I'- T~' I,' 

- * (i - 1 c - f.) ) 


186. In the exact predetermination of the characteristics of 
such a motor, the effect of the short-circuit current under the 
brushes has to be taken into consideration, however. When a 
commutator is used, by the passage of the brushes from segment 
to segment coils are short-circuited. Therefore, in addition to 
the circuits considered above, a closed circuit on the rotor has 
to be introduced in the equations for every set of brushes. Re- 
duced to the stator circuit by the ratio of turns, the self-inductive 
impedance of the short-circuit under the brushes is very high, 
the current, therefore, small, but still sufficient to noticeably af- 
fect the motor characteristics, at least at certain speeds. Since, 
however, this phenomenon will be considered in the chapters on 
the single-phase motors, it may be omitted here. 


187. If in a polyphase commutator motor the rotor circuits 
are connected in series to the stator circuits, entirely different 

Fig. 153. 

characteristics result, and the motor no more tends to synchronize 
nor approaches a definite speed at no-load, as a shunt motor, but 
with decreasing load the speed increases indefinitely. In short, 


the motor has similar characteristics as the direct-current series 

In this case we may assume the stator reduced to the rotor by 
the ratio of effective turns. 

Let then, in the motor shown diagrammatically in Fig. 153: 

#o and —j$o, lo and — j/o, Z = impressed e.m.fs., currents 
and self-inductive impedance of stator circuits, assumed as two- 
phase, and reduced to the rotor circuits by the ratio of effective 
turns, c, 

#i and — j$\, /i, and — jfi, Z\ = impressed e.m.fs. currents 
and self-inductive impedance of rotor circuits, 

Z = mutual-inductance impedance, 

5 = speed; and, s = 1 — S = slip, 

6 = brush angle, 

c = ratio of effective stator turns to rotor turns. 

If, then : 
P and — j$ = impressed e.m.fs., / and — jj = currents of 
motor, it is: 

/i = /, (112) 

h = c/, (113) 

c#o + #i = E; (114) 

and, stator, by equation (7) : 

#o = Zoh + Z(f - h cos 6 - jlx sin 6); (115) 


#, = Zi/i + Z (A - U cos 6 + jfo sin 6) - jSZ (- jf l + / 

sin0 + j/ o cos0); (116) 

and, e.m.f. of rotation: 

Q\ = - jSZ (- jfi + /o sin 6 + jfi cos 0). (117) 

Substituting (112), (113) in (115), (116), (117), and (115), (116) 
in (114) gives: 

(c 2 Zo + Z,j + Z(1 + c 2 -2ccos0) + SZ(cc - i)' vllo; 

where : 

a = cos0 - jsin 0, (119) 


-, _ _ SZE(ce-l) . 

* ' " (c*Z ~+ Zx) = Z (1 + c* - 2 c cos 0) + SZ (c<r - 1)] ' 




and the power output: 

P = U?\ hY 

Se 2 { c (r cos + x sin 0) — r\ 

[{c*Z + Zi) + Z (1 + c 2 - 2 c cos 0) + SZ (or - 1)]* 


The characteristics of this motor entirely vary with a change 

Se 2 r(x— 1) 
of the brush angle, 0. It is, for = 0: P = — rj^i > hence 



110 S* 









6^0 V 









S i 

\ In 


" y^' 













^r « 

* ^ 



— ^ - 

\ 1 * 




i\ 1 









240 %: 
220 110 

200 100 

180 90 

100 80 

140 70 

120 60 

100 60 

80 40 

60 80 

40 20 

20 10 

.2 .4 .6 

.8 1.0 1.2 1.4 

Fig. 154. 


1.8 2.0 

Ss 2 (xc — * t) 
very small, while for = 90°: P = r~p — , hence consider- 
able. Some brush angles give positive P: motor, others negative, 
P, generator. 

In such a motor, by choosing and c appropriately, unity 
power-factor or leading current as well as lagging current can be 

That is, by varying c and 0, the power output and therefore 
the speed, as well, as the phase angle of the supply current or 
the power-factor can be varied, and the machine used to produce 
lagging as well as leading current, similarly as the polyphase 
shunt motor or the synchronous motor. Or, the motor can be 
operated at constant unity power-factor at all loads and speeds 
(except very low speeds), but in this case requires changing the 


brush angle, 0, and the ratio, c, with the change of load and speed. 
Such a change of the ratio, c, of rotor -f- stator turns can be pro- 
duced by feeding the rotor (or stator) through a transformer of 
variable ratio of transformation, connected with its primary cir- 
cuit in series to the stator (or. rotor). 

188. As example is shown in Fig. 154, with the speed as 
abscissae, and values from standstill to over double synchronous 
speed, the characteristic curves of a polyphase series motor of 
the constants: 

e = 640 volts, 
Z = 1 + 10 j ohms, 
Z = Zx = 0.1 + 0.3 j ohms, 

c = l, 

$ = 37 ; ( a in 6 = 0.6; cos 6 = 0.8); 


. 640 

(0.6 + 5.8 S) + j (4.6 - 2.6 S) amp '' 

P = 4 _673 S kw 

(0.6 + 5.8 S) 2 + (4.6 - 2.6 S)* 

As seen, the motor characteristics are similar to those of the 
direct-current series motor: very high torque in starting and at 
low speed, and a speed which increases indefinitely with the de- 
crease of load. That is, the curves are entirely different from 
those of the induction motors shown in the preceding. The 
power-factor is very high, much higher than in induction motors, 
and becomes unity at the speed S = 1.77, or about one and three- 
quarter synchronous speed. 


I. General 

189. Alternating-current commutating machines have so far 
become ef industrial importance mainly as motors of the series 
or varying-speed type, for single-phase railroading, and as con- 
stant-speed motors or adjustable-speed motors, where efficient 
acceleration under heavy torque is necessary. As generators, 
they would be of advantage for the generation of very low fre- 
quency, since in this case synchronous machines are uneconom- 
ical, due to their very low speed, resultant from the low frequency. 

The direction of rotation of a direct-current motor, whether 
shunt or series motor, remains the same at a reversal of the im- 
pressed e.m.f., as in this case the current in the armature circuit 
and the current in the field circuit and so the field magnetism 
both reverse. Theoretically, a direct-current motor therefore 
could be operated on an alternating impressed e.m.f. provided 
that the magnetic circuit of the motor is laminated, so as to fol- 
low the alternations of magnetism without serious loss of power, 
and that precautions are taken to have the field reverse simul- 
taneously with the armature. If the reversal of field magnetism 
should occur later than the reversal of armature current, during 
the time after the armature current has reversed, but before the 
field has reversed, the motor torque would be in opposite direc- 
tion and thus subtract; that is, the field magnetism of the alter- 
nating-current motor must be in phase with the armature cur- 
rent, or nearly so. This is inherently the case with the series 
type of motor, in which the same current traverses field coils 
and armature windings. 

Since in the alternating-current transformer the primary and 
secondary currents and the primary voltage and the secondary 
voltage are proportional to each other, the different circuits of 
the alternating-current commutator motor may be connected 
with each other directly (in shunt or in series, according to the 
type of the motor) or inductively, with the interposition of a 




transformer, and for this purpose either a separate transformer 
may be used or the transformer feature embodied in the motor, 
as in the so-called repulsion type of motors. This gives to the 
alternating-current commutator motor a far greater variety of 
connections than possessed by the direct-current motor. 

While in its general principle of operation the alternating- 
current commutator motor is identical with the direct-cums! 
motor, in the relative proportioning of the parts a great differ- 
ence exists. In the direct-current motor, voltage is consumed 
by the counter e.m.f. of rotation, which represents the power 
output of the motor, and by the resistance, which represents 
the power loss. In addition thereto, in the alternating-cur rent 
motor voltage is consumed by the inductance, which is wattless 
or reactive and therefore causes a lag of current behind the vol- 
tage, that is, a lowering of the power-factor. While in the direct- 
current motor good design requires the combination of a strong 
field and a relatively weak armature, so as to reduce the armature 
reaction on the field to a minimum, in the design of the alter- 
iiatiiig-current motor considerations of power-factor predominate; 
that is, to secure low self-inductance and therewith a high power- 
factor, the combination of a strong armature and a weak field is 
required, and necessitates the use of methods to eliminate the 
harmful effects of high armature reaction. 

As the varying-speed single-phase commutator motor has 
found an extensive use as railway motor, this type of motor 
will as an instance be treated in the following, and the other 
types discussed in the concluding paragraphs. 

II. Power-factor 

190. In the commutating machine the magnetic field flux gen- 
erics the in the revolving armature conductors, which 
gives the motor output; the armature reaction, that is, the mag- 
net k Mux produced by the armature current, distorts and weakens 
the field, and requires a shifting of the brushes to avoid Bparldag 
due to the short-circuit current under the commutator brushes, 
and where the brushes can not l>e shifted, as in a reversible motor. 
this necessitates the use of a strong field and weak armature to 
keep down the magnetic flux at the brushes. In the alternating- 
current motor the magnetic field flux generates in the armature 
conductors by their rotation the e.m.f. which does the work of 
the motor, but, as the field flux is alternating, it also generates 


in the field conductors an e.m.f. of self-inductance, which is not 
useful but wattless, and therefore harmful in lowering the power- 
factor, hence must be kept as low as possible. 

This e.m.f. of self-inductance of the field, e , is proportional 
to the field strength, $, to the number of field turns, n , and to 
the frequency, /, of the impressed e.m.f. : 

eo = 2 ir/no* 10" 8 , (1) 

while the useful e.m.f. generated by the field in the armature 
conductors, or "e.m.f. of rotation," e, is proportional to the field 
strength, $, to the number of armature turns, n h and to the fre- 
quency of rotation of the armature, /<>: 

e = 2ir/on 1 <i> 10" 8 . (2) 

This later e.m.f., e, is in phase with the magnetic flux, $, and 
so with the current, i, in the series motor, that is, is a power e.m.f., 
while the e.m.f. of self-inductance, e , is wattless, or in quadrature 
with the current, and the angle of lag of the motor current thus 
is given by: 

tan 6 = -^ (3) 

6 -r it 

where ir = voltage consumed by the motor resistance. Or ap- 
proximately, since ir is small compared with e (except at very 
low speed) : 

tan 6 = -> (4) 


and, substituting herein (1) and (2): 

tan 6 - { -°- (5) 

Small angle of lag and therewith good power-factor therefore 
require high values of / and n\ and low values of / and n . 

High /o requires high motor speeds and as large number of 
poles as possible. Low / means low impressed frequency; there- 
fore 25 cycles is generally the highest frequency considered for 
large commutating motors. 

High ni and low n means high armature reaction and low 
field excitation, that is, just the opposite conditions from that 
required for good commutator-motor design. 

Assuming synchronism, /o = /, as average motor speed — 750 
revolutions with a four-pole 25-cyclc motor — an armature reac- 



tion, n,, equal to the field excitation, n , would then give tan 
6 = 1, 9 = 45°, or 70.7 per cent, power-factor; that is, with an 
armature reaction beyond the limits of good motor design, the 
power-factor is still too low for use. 

The armature, however, also has a 
self -inductance; that is, the magnetic 
flux produced by the armature cur- 
rent as shown diagrammatically in 
Fig. 155 generates a reactive e.m.f. in 
the armature conductors, which again 
lowers the power-factor. While this 
armature self-inductance is low with 
small number of armature turns, it 
becomes considerable when the num- 
ber of armature turns, rti, is large 
compared with the field turns, n , 

Let fflo = field reluctance, that 
is, reluctance of the magnetic 
field circuit, and <Ri =- -r- = the armature reluctance, that is, 

6 = =- = ratio of reluctances of the armature and the field mag- 
netic circuit; then, neglecting magnetic saturation, the field flux 

Fig. 155. — Distribution of 

iiain field and field of a 
;ure reaction. 

the armature flux i; 

and the e.m.f. of self-inductance of the armature circuit h 
e x = 2»/ni*,10- s 

hence, the total e.m.f. of self -inductance of the motor, or wattless 
e.m.f., by (1) and (7) is: 

«, + ex = 2 J* 10- C'*,, 6 ' 1 '' )' (8) 


and the angle of lag, 0, is given by: 

eo + ei 

tan = 

f n 2 + bni 2 . 

/o notii 
or, denoting the ratio of armature turns to field turns by: 

q = -> 

tan - J ' 


=mJ + ^ < w > 

and this is a minimum; that is, the power-factor a maximum, for: 

^-{tanfl} = 0, 

*o = ^ (ID 

and the maximum power-factor of the motor is then given by: 

tan 0o = / -> • (12) 

h \/b 

Therefore the greater b is the higher the power-factor that 
can be reached by proportioning field and armature so that 

Tli 1 

no * y/b 

Since b is the ratio of armature reluctance to field reluctance, 
good power-factor thus requires as high an armature reluctance 
and as low a field reluctance as possible; that is, as good a mag- 
netic field circuit and poor magnetic armature circuit as feasible. 
This leads to the use of the smallest air gaps between field and 
armature which are mechanically permissible. With an air gap 
of 0.10 to 0.15 in. as the smallest safe value in railway work, b 
can not well be made larger than about 4. 

Assuming, then, 6 = 4, gives q = 2, that is, twice as many 
armature turns as field turns; rti = 2 n . 

The angle of lag in this case is, by (12), at synchronism:/© = /, 

tan O = 1, 

giving a power-factor of 70.7 per cent. 

It follows herefrom that it is not possible, with a mechanically 


safe construction, at 25 cycles to get a good power-factor 
moderate speed, from a straight series motor, even if such a 
design as discussed above were not inoperative, due to rate 
distortion and therefore destructive sparking. 

Thus it becomes necessary in the single-phase com mutator 
motor to reduce the magnetic flux of armature reaction, thai is, 
increase the effective magnetic reluctance of the armature fur 
beyond the value of the true magnetic reluctance. This is m- 
complished by the compensating winding devised by Eirkemeyer, 
by surrounding the armature with a stationary winding chist-ly 
adjacent and parallel to the armature winding, and energized by 
a current in opposite direction to the armature currem. ;imi ti 
the same m.m.f., that is, the same number of ampere-turns, 
the armature winding. 




( / 



\ 1 

f » 



/ > 

C v 





commutator n 

191. Every single-phase commutator motor thus comprises a 
field winding, F, an armature winding, A, and a compensating 
winding, C, usually located in the pole faces of the field, as shown 
in Figs. 156 and 157. 

The compensating winding, 0, is either connected in aeriea - Imt 
in reversed direction) with the armature winding, and then has 
the same number of effective turns, or it is short-circuited upon 
itself, thus acting as a short-circuited secondary with the arma- 
ture winding as primary, or the compensating winding i| ener- 
gized by the supply current, and the armature short-circuited as 



secondary. The first rase Rives the eonduetively compensated 
series motor, the second case the inductively compensated series 
motor, the third case the repulsion motor. 

In the first case, by giving the compensating winding more 
turns than the armature, overcompensation, by giving it lesB 
turns, undercompensation, is produced. In the second case 
always complete (or practically complete) compensation results, 
irrespective of the number of turns of the winding, as primary 
and secondary currents of a transformer always are opposite in 
direction, and of the same m.m.f. (approximately), and in the 
third case a somewhat less complete compensation. 

With a compensating winding, C, of equal and opposite m.m.f. 
to the armature winding, A, the resultant armature reaction is 
zero, and the field distortion, therefore, disappears; that is, the 
ratio of the armature turns to field turns has no direct effect on 
the commutation, but high armature turns and low field turns 
can be used. The armature self-inductance is reduced from that 
corresponding to the armature magnetic flux, *i, in Fig. 155 to 
that corresponding to the magnetic leakage flux, that is, the 
magnetic flux passing between armature turns and compensating 
turns, or the "slot inductance," which is small, especially if rela- 
tively shallow armature slots and compensating slots are used. 

The compensating winding, or the "cross field," thus fulfils 
the twofold purpose of reducing the armature self-inductance to 
that of the leakage flux, and of neutralizing the armature reac- 
tion and thereby permitting the use of very high armature 

The main purpose of the compensating winding thus is to de- 
crease the armature self-inductance; that is, increase the effect- 
ive armature reluctance and thereby its ratio to the field reluc- 
tance, b, and thus permit the use of a much higher ratio, q = ', 
before maximum power-factor is reached, and thereby a higher 

Even with compensating winding, with increasing q, ultimately 
a point is reached where the armature self-inductance equals 
the field self-inductance, and beyond this the power-factor again 
decreases. It becomes possible, however, by the use of the com- 
pensating winding, to reach, with a mechanically good design, 
values of 6 as high as 16 to 20. 

Assuming b = 16 gives, substituted in (11) and (12): 



that is, four times as many armature turns as field turns, i*j 
4 no and : 

tan ,. - fe 
hence, at synchronism: 

fa = / : tan O = 0.5, or 89 per cent, power-factor. 

At double synchronism, which about represents maximum motor 
speed at 25 cycles: 

/o = 2/ : tan 8 = 0.25, or 98 per cent, power-factor; 

that is, very good power-factors can be reached in the single- 
phase commutator motor by the use of a compensating winding, 
far higher than are possible with the same air gap in polyphase 
induction motors. 

III. Field Winding and Compensating Winding 

192. The purpose of the field winding is to produce the maxi- 
mum magnetic flux, $, with the minimum number of turns, n,. 
This requires as large a magnetic section, especially at the air 
gap, as possible. Hence, a massed field winding with definite 
polar projections of as great pole arc as feasible, as shown in Fig. 
157, gives a better power-factor than a distributed field winding. 

The compensating winding must be as closely adjacent, to the 
armature winding as possible, so as to give minimum teoksfj 
flux between armature conductors and compensating conductors, 
and therefore is a distributed winding, located in the field poll 
faces, as shown in Fig. 1,57. 

The armature winding is distributed over the whole timuO; 
feretice of the armature, but the compensating winding only in 
the field pole faces. With the same ampere-turns in armature 
and compensating winding, their resultant ampere-turns are 
equal and opposite, and therefore neutralize, but locally the two 
windings do not neutralize, due to the difference in the distribu- 
tion curves of their m.m.fs. The m.m.f. of the field winding is 
constant over the pole faces, and from one pole corner to the next 
pole corner reverses in direction, as shown diagninniui i. ■:! . 
by F in Fig. 158, which is the development of Fig. 157. The 
m.m.f. of the armature is a maximum at the brushes, midway 
between the field poles, as shown by A in Fig. 158, and from there 
decreases to zero in the center of the field pole. The m.m.f. of 


the compensating winding, however, is constant in the space 
from pole corner to pole corner, as shown by C in Fig. 158, and 
since the total m.m.f. of the compensating winding equals that 
of the armature, the armature m.m.f. is higher at the brushes, 
the compensating m.m.f. higher in front of the field poles, as 
shown by curve R in Fig. 158, which is the difference between 
A and C; that is, with complete compensation of the resultant 
armature and compensating winding, locally undercompensation 
exists at the brushes, overcompensation in front of the field 

Fio. 158. — Distribution of m.m.f. in compensated motor. 

poles. The local undercompensated armature reaction at the 
brushes generates an e.m.f. in the coil short-circuited under the 
brush, and therewith a short-circuit current of commutation 
and sparking. In the conductively compensated motor, this can 
be avoided by overcompensation, that is, raising the flat top of 
the compensating m.m.f. to the maximum armature m.m.f., but 
this results in a lowering of the power-factor, due to the self- 
inductive flux of overcompensation, and therefore is undesirable. 
193. To get complete -compensation even locally requires the 
compensating winding to give the same distribution curve as the 
armature winding, or inversely. The former is accomplished by 
distributing the compensating winding around the entire cir- 
cumference of the armature, as shown in Fig. 159. This, how- 
ever, results in bringing the field coils further away from the 
armature surface, aftd so increases the magnetic stray flux of the 
field winding, that is, the magnetic flux, which passes through 
the field coils, and there produces a reactive voltage of self-in- 



ductance, but does not pass through the armature conductor? 
and so does no work; that is, it lowers the power factor, just 
overcompensation would do. The distribution curve of the 
armature winding can, however, W 
made equal to that of the compen- 
sating winding, and therewith local 
complete compensation secured, by 
using a fractional pitch armature 
winding of a pitch equal to the pole 
arc. In this case, in the space be- 
tween the pole corners, the current* 
are in opposite direction in the 
upper and the lower layer of con- 
ductors in each armature slot, 
shown in Fig. 160, ami thus DeutmUlB 
magnetically; that is, the armature 
reaction extends only over the spacr 
of the armature circumference covered 
by the pole arc, where it is neutralized 
by the compensating winding in the pole face. 

To produce complete compensation even locally, without im- 
pairing tbe~power-factor, therefore, requires a fractional-pitch 

Fio. 159.— Completely 
distributed compensating 


armature winding, of a pitch equal to the field pole arc, or s< 
equivalent arrangement. 

Historically] the first compensated single-phase commutfttov 
motors, built about 20 years ago, were Prof. Elihu Thomeea^ 
repulsion motors. In these the field winding and coin pen sating 


winding were massed together in a single coil, as shown diagram- 
matically in Fig. 161. Repulsion motors are still occasionally 
built in which field and compensating coils are combined in a 
single distributed winding, as shown in Fig. 162. Soon after the 
first repulsion motor, conductively and inductively compensated 
series motors were built by Eickemeyer, with a massed field 
winding and a separate compensating winding, or cross coil, 
either as single coil or turn or distributed in a number of coils or 
turns, as shown diagrammatically in Fig. 163, and by W. Stanley. 



Fig. 162. — Repulsion motor with Fig. 163. — Eickemeyer inductively 
distributed winding. compensated series motor. 

For reversible motors, separate field coils and compensating 
coils are always used, the former as massed, the latter as dis- 
tributed winding, since in reversing the direction of rotation 
either the field winding alone must be reversed or armature and 
compensating winding are reversed while the field winding re- 
mains unchanged. 

IV. Types of Varying-speed Single-phase Commutator Motors 

194. The armature and compensating windings are in induc- 
tive relations to each other. In the single-phase commutator 
motor with series characteristic, armature and compensating 
windings therefore can be connected in series with each other, or 
the supply voltage impressed upon the one, the other closed upon 
itself as secondary circuit, or a part of the supply voltage im- 
pressed upon the one, and another part upon the other circuit, 
and in either of these cases the field winding may be connected 
in series either to the compensating winding or to the armature 
winding. This gives the motor types, denoting the armature by 










Fio. 164. — Types of alternating-current coramutating motors. 


* • 

A, the compensating winding by C, and the field winding by F, 
shown in Fig. 164. 

Primary Secondary 


• • • 

Series motor. 

A + C + F 

• • • 

Conductively compensated 
series motor. (1) 

A +F 


Inductively compensated 
series motor. (2) 


C + F 

Inductively compensated 
series motor with second- 
ary excitation, or inverted 
repulsion motor. (3) 

C + F 


Repulsion motor. (4) 


A +F 

Repulsion motor with sec- 
ondary excitation. (5) 



• • • j 

Series repulsion motors. 

A, C + F 

(6) (7) 

Since in all these motor types all three circuits are connected 
directly or inductively in series with each other, they all have 
the same general characteristics as the direct-current series 
motor; that is, a speed which increases with a decrease of load, 
and a torque per ampere input which increases with increase of 
current, and therefore with decrease of speed, and the different 
motor types differ from each other only by their commutation 
as affected by the presence or absence of a magnetic flux at the 
brushes, and indirectly thereby in their efficiency as affected by 
commutation losses. 

In the conductively compensated series motor, by the choice 
of the ratio of armature and compensating turns, overcompensa- 
tion, complete compensation, or undercompensation can be pro- 
duced. In all the other types, armature and compensating 
windings are in inductive relation, and the compensation there- 
fore approximately complete. 

A second series of motors of the same varying speed charac- 
teristics results by replacing the stationary field coils by arma- 
ture excitation, that is, introducing the current, either directly 
or by transformer, into the armature by means of a second set 
of brushes at right angles to the main brushes. Such motors 
are used to some extent abroad. They have the disadvantage of 



Fig. 1(55.- 

req Hiring two sots of brushes, but the advantage that their 
power-factor can be controlled and above synchronism even 
lending current produced. Fig. ll>5 shows diagrammatical!}- surli 
a motor, as designed by Winter- Eichberg-Latour, the so-called 
compensated repulsion motor. In this case componsatei! meant: 
compensated for power-factor. 

The voltage which can be used in the motor armature is limited 
by the commutator: the voltage per commutator segment is 
limited by the problem of sparkless commutation, the number 
of commutator segments Frew 
brush to brush is limited 
mechanical consideration of 
commutator speed and width 
of segments. In those motet 
types in which the supply cur- 
rent traverses the armature, the 
supply voltage is thus limited 
to values even lower than in 
the direct-current motor, while 
in the repulsion motor (4 and 
5), in which the armature is the 
secondary circuit, the armature voltage is independent of the 
supply voltage, so can be chosen to suit the requirement! i ■: 
commutation, while the motor can be built for any supply 
voltage for which the stator can economically 1m? insulated. 

Alternating-current motors as well as direct-current scries 
motors can be controlled by series parallel connection of two or 
more motors. Further control, as in starting, with direct -current 
motors is carried out by rheostat, while with alternating-current 
motors potential conlrol, that is, a change of supply voltage by 
transformer or autotransformer, offers a more efficient method 
of control. By changing from one motor type to another motor 
type, potential control can bo, used in alternating-current motors 
without any change of supply voltage, by appropriately choosing 
the ratio of turns of primary and secondary circuit. For in- 
stance, with an armature wound for half the voltage and thus 

twice the current as the compensating winding (ratio of turns 

- = 2) , a change of connection from tvpc 3 to type 2, or from 

type 5 to type 4, results in doubling the field current and there- 


with the field strength. A change of distribution of voltage be- 
tween the two circuits, in types 6 and 7, with A and C wound 
for different voltages, gives the same effect as a change of supply 
voltage, and therefore is used for motor control. 

196. In those motor types in which a transformation of power 
occurs between compensating winding, C, and armature winding, 
A, a transformer flux exists in the direction of the brushes, that 
is, at right angles to the field flux. In general, therefore, the 
single-phase commutator motor contains two magnetic fluxes in 
quadrature position with each other, the main flux or field flux, 
A', in the direction of the axis of the field coils, or at right angles 
to the armature brushes, and the quadrature flux, or transformer 
flux, or commu taring flux, *j, in line with the armature brushes, 
or in the direction of the axis of the compensating winding, that 
is, at right angles (electrical) with the field flux. 

The field flux, *, depends upon and is in phase with the field 
current, except as far as it is modified by the magnetic action of 
the short-circuit current in the armature coil under the commu- 
tator brushes. 

In the conductively compensated series motor, 1, the quad- 
rature flux is zero at complete compensation, and in the direc- 
tion of the armature reaction with undercompensation, in oppo- 
sition to the armature reaction at overcompensation, but in 
either ease in phase with the current and so approximately with 
the field. 

In the other motor types, whatever quadrature flux exists is 
not in phase with the main flux, but as transformer flux is due 
to the resultant m.ui.f. of primary and secondary circuit. 

In a transformer with non-inductive or nearly non-inductive 
secondary circuit, the magnetic flux is nearly 90° in time phase 
behind the primary current, a little over 90° ahead of the sec- 
ondary current, as shown in transformer diagram, Tig. 166. 

In a transformer with inductive secondary, the magnetic flux 
is less than 90" liehind the primary current, more than 90° ahead 
of the secondary current, the more so the higher is the inductivity 
of the secondary circuit, as shown by the transformer diagram, 
Fig. 166. 

Herefrom it follows that: 

In the inductively compensated series motor, 2, the quad- 
rature flux is very small and practically negligible, as very little 
voltage is consumed in the low impedance of the secondary cir- 
cuit, C; whatever flux there is, lags behind the main flux. 



In the inductively compensated series ipotor with secondary 
excitation, or inverted repulsion motor, 3, the quadrature flux, 
$1, is quite large, as a considerable voltage is required for the 
field excitation, especially at moderate speeds and therefore high 
currents, and this flux, $i, lags behind the field flux, $, but this 
lag is very much less than 90°, since the secondary circuit is 


Fig. 166. — Transformer diagram, inductive and non-inductive load. 

highly inductive; the motor field thus corresponding to the con- 
ditions of the transformer diagram, Fig. 166. As result hereof, 
the commutation of this type of motor is very good, flux, $i, 
having the proper phase and intensity required for a commu- 
tating flux, as will be seen later, but the power-factor is poor. 

In the repulsion motor, 4, the quadrature flux is very consid- 
erable, since all the voltage consumed by the rotation of the 
armature is induced in it by transformation from the compen- 



sating winding, and this quadrature flux, *i, laps nearly 90° be- 
hind the main flux, *, since the secondary circuit is nearly non- 
inductive, especially at speed. 

In the repulsion motor with secondary excitation, 5, the quad- 
rature flux, *i, is also very large, and practically constant, corre- 
sponding to the impressed e.m.f., but lags considerably less than 
90° behind the main flux, $, the secondary circuit being induct- 
ive, since it contains the field coil, F. The lag of the flux, *i, 
increases with increasing speed, since with increasing speed the 
e.m.f. of rotation of the armature increases, the e.m.f. of self- 
inductance of the field decreases, due to the decrease of current, 
and the circuit thus becomes less inductive. 

The series repulsion motors 6 and 7, give the same phase rela- 
tion of the quadrature flux, $i, as the repulsion motors, 5 and 6, 
but the intensity of the quadrature flux, $i, is the less the smaller 
the part of the supply voltage which is impressed upon the com- 
pensating winding. 

V. Commutation 

196. In the commutator motor, the current in each armature 
coil or turn reverses during its passage under the brush. In the 
armature coil, while short-circuited by the commutator brush, 
the current must die out to zero and then increase again to its 
original value in opposite direction. The resistance of the arma- 
ture coil and brush contact accelerates, the self-inductance re- 
tards the dying out of the current, and the former thus assists, 
tin 1 latter impairs commutation. If an e.m.f. is generated in 
the armature coil by its rotation while short-circuited by the 
commutator brush, this e.m.f. opposes commutation, that is, 
retards the dying out of the current, if due to the magnetic flux 
of armature reaction, and assists commutation by reversing the 
armature current, if due to the magnetic flux of overcompensa- 
tion, that is, a magnetic flux in opposition to the armature 

Therefore, in the direct-current commutator motor with high 
field strength and low armature reaction, that is, of negligible 
magnetic flux of armature reaction, fair commutation is produced 
with the brushes set midway between the field poles — -that is, 
in the position where the armature coil which is being commu- 
tated encloses the full field flux and therefore cuts no flux and 
has no generated e.m.f. — by using high-resistance carbon brushes, 



as the resistance of the brush contact, increasing when the arma- 
ture coil begins to leave the brush, tends to reverse the current. 
Such "resistance commutation" obviously can not be perfect; 
perfect commutation, however, is produced by impressing upon 
the motor armature at right angles to the main field, thai is, UD 
the position of the commutator brushes, a magnetic field oppo- 
site to that of the armature reaction and proportional to the 
armature current. Such a field is produced by overcompensa- 
tion or by the use of a commutating pole or interpole. 

As seen in the foregoing, in the direct-current motor t he counter 
e.m.f. of self-inductance of commutation opposes the reversal of 
current in the armature coil under the commutator brush, and 
this can be mitigated in its effect by the use of high-resistance 
brushes, and overcome by the commutating field of overcompen- 
sation. In addition hereto, however, in the alternating-current 
commutator motor an e.m.f. is generated in the coil short-cir- 
cuited under the brush, by the alternation of the magnetic flux, 
and this e.m.f., which does not exist in the direct-current motor, 
makes the problem of commutation of the alternating-current 
motor far more difficult. In the position of commutation no 
e.m.f. is generated in the armature coil by its rotation through 
the magnetic field, as in this position the coil encloses the maxi- 
mum field flux; but as this magnetic flux is alternating, in this 
position the e.m.f. generated by the alternation of the flux en- 
closed by the coil is a maximum. This "e.m.f. of alternation* 1 
lags in time 90° behind the magnetic flux which generates it, h 
proportional to the magnetic flux and to the frequency, but is 
independent of the speed, hence exists also at standstill, while 
the "e.m.f. of rotation" — which is a maximum in the position 
of the armature coil midway between the brushes, or parallel to 
the field flux — is in phase with the field flux and proportional 
thereto and to the speed, but independent of the frequency. In 
the alternating-current commutator motor, no position therefore 
exists in which the armature coil is free from a generated e.m.f., 
but in the position parallel to the field, or midway between tin 
brushes, the e.m.f. of rotation, in phase with the field flux, is a 
maximum, while the e.m.f. of alternation is zero, and in the posi- 
tion under the commutator brush, or enclosing the total field 
flux, the e.m.f. of alternation, in electrical space quadrature with 
the field flux, is a maximum, the e.m.f. of rotation absent, while 
in any other position of the armature coil its generated e.m.f. has 


a component due to the rotation — a power e.m.f. — and a com- 
ponent due to the alternation — a reactive e.m.f. The armature 
coils of an alternating-current commutator motor, therefore, are 
the seat of a system of polyphase e.m.f s., and at synchronism 
the polyphase e.m.fs. generated in all armature coils are equal, 
above synchronism the e.m.f. of rotation is greater, while 
below synchronism the e.m.f. of alternation is greater, and in 
the latter case the brushes thus stand at that point of the com- 
mutator where the voltage between commutator segments is a 
maximum. This e.m.f. of alternation, short-circuited by the 
armature coil in the position of commutation, if not controlled, 
causes a short-circuit current of excessive value, and therewith 
destructive sparking; hence, in the alternating-current commuta- 
tor motor it is necessary to provide means to control the short- 
circuit current under the commutator brushes, which results from 
the alternating character of the magnetic flux, and which docs 
not exist in the direct-current motor; that is, in the alternating- 
current motor the armature coil under the brush is in the posi- 
tion of a short-circuited secondary, with the field coil as primary 
of a transformer; and as in a transformer primary and secondary 
ampere-turns are approximately equal, if n = number of field 
turns per pole and i = field current, the current in a single arma- 
ture turn, when short-circuited by the commutator brush, tends 
to become io = n i, that is, many times full-load current; and 
as this current is in opposition, approximately, to the field cur- 
rent, it would demagnetize the field; that is, the motor field 
vanishes, or drops far down, and the motor thus loses its torque. 
Especially is this the case at the moment of starting; at speed, 
the short-circuit current is somewhat reduced by the self-induc- 
tance of the armature turn. That is, during the short time 
during which the armature turn or coil is short-circuited by the 
brush the short-circuit current can not rise to its full value, if 
the speed is considerable, but it is still sufficient to cause destruc- 
tive sparking. 

197. The character of the commutation of the motor, and 
therefore its operativeness, thus essentially depends upon the 
value and the phase of the short-circuit currents under the com- 
mutator brushes. An excessive short-eimiit current, gives de- 
structive sparking by high-current density under the brushes 
and arcing at the edge of the brushes due to the great and sud- 
den change of current in the armature coil when leaving the 



brush. But even with a moderate short-circuit current, the 
sparking at the commutator may be destructive and the motor 
therefore inoperative, if the phase of the short-circuit current 
greatly differs from that of the current in the armature coil after 
it leaves the brush, and so a considerable and sudden change of 


n - 



I ' 


n ' 



a i 


Fin. 167.— E.m.f. consumed at contact of copper brush. 

current must take place at the moment when the armature coil 
leaves the brush. That is, perfect commutation occurs, if the 
short-circuit current in the armature coil under the commutator 
brush at the moment when the coil leaves the brush has the 
same value and the same phase as the main-armature current in 



I 2 










B 1 



. 168.— E.m.f. consumed ft 

t of higli-i 

earhon brush. 

the coil after leaving the brush. The commutation of such a 
motor therefore is essentially characterized by the difference 
between the main-armature current after, and the short-circuit 
current before leaving the brush. The investigation of the short- 
circuit current under the commutator brushes therefore is of 


fundamental importance in the study of the alternating-current 
commutator motor, and the control of this short-circuit current 
the main problem of alternating-current commutator motor 

Various means have been proposed and tried to mitigate or 
eliminate the harmful effect of this short-circuit current, as high 
resistance or high reactance introduced into the armature coil 
during commutation, or an opposing e.m.f . either from the out- 
side, or by a commutating field. 

High-resistance brush contact, produced by the use of very 
narrow carbon brushes of high resistivity, while greatly improv- 
ing the commutation and limiting the short-circuit current so 
that it does not seriously demagnetize the field and thus cause 
the motor to lose its torque, is not sufficient, for the reason that 
the resistance of the brush contact is not high enough and also is 
not constant. The brush contact resistance is not of the nature 
of an ohmic resistance, but more of the nature of a counter 
e.m.f.; that is, for large currents the potential drop at the brushes 
becomes approximately constant, as seen from the volt-ampere 
characteristics of different brushes given in Figs. 167 and 168. 
Fig. 167 gives the voltage consumed by the brush contact of a 
copper brush, with the current density as abscissae, while Fig. 
168 gives the voltage consumed by a high-resistance carbon 
brush, with the current density in the brush as absciss®. It is 
seen that such a resistance, which decreases approximately in- 
versely proportional to the increase of current, fails in limiting 
the current just at the moment where it is most required, that 
s, at high currents. 

Commutator Leads 

198. Good results have been reached by the use of metallic 
resistances in the leads between the armature and the commuta- 
tor. As shown diagrammatically in Fig. 169, each commutator 
segment connects to the armature, A , by a high non-inductive 
resistance, CB y and thus two such resistances are always in the 
circuit of the armature coil short-circuited under the brush, but 
also one or two in series with the armature main circuit, from 
brush to brush. While considerable power may therefore l>c 
consumed in these high-resistance leads, neverthelebs the effi- 
ciency of the motor is greatly increased by their use; that is, the 
reduction in the loss of power at the commutator by the reduction 


of the short-circuit current, usually is far greater than the mfltt 
of power in the resistance leads. To have any apprecial <[•■ i-ffoci , 
the resistance of the commutator lead must lie far higher thau 
that of the armature coil to which it connects. Of the e.m.f. 
of rotation, that is, the useful generated e.m.f., the armature re- 
sistance consumes only a very small part, a few per cent. only. 
The e.m.f. of alternation is of the same magnitude as the 
of rotation — higher below, lower above synchronism. With B 
short-circuit current equal to full-load current, the resistance of 

Fee. 160, — Commutation with resistance leads. 

the short-circuit coil would consume only a small part of the 
e.m.f. of alternation, and to consume the total e.m.f. the short- 
circuit current therefore would have to lie about as many times 
larger than the normal armature current as the useful generated 
e.m.f. of the motor is larger than the resistance drop in the arma- 
ture. Long before this value of short-circuit current is reached 
the magnetic field would have disappeared by the demagnetuui| 
force of the short-circuit current, that is, the motor would have 
lost its torque. 

To limit the short-circuit current under the brush to a value 
not very greatly exceeding full-load current, thus requires a re- 
sistance of the lead, many times greater than that of (he animt un- 
coil. The i-r in the lead, and thus the heat produced in it, then, 
is many times greater than that in the armature coil. The space 
available for the resistance lead is, however, less than that avail- 
able for the armature coil. 

It is obvious herefrom that it is not feasible to build these 
resistance leads so that each lead can dissipate continuously, or 
even for any appreciable time, without rapid si -If-drst ruction, 
the heat produced in it while in circuit. 

When the motor is revolving, even very slowly, thiaiaatH DM 
essary, since each resistance lead is only a very short tmn- in 


circuit, during the moment when the armature c 
to it are short-circuited by the brushes; that is, if t 

i connecting 

■ number of 

armature turns from brush to brush, the lead is only - of the 

time in circuit, and though excessive current densities in mate* 
rials of high resistivity are used, the heating is moderate. In 
starting the motor, however, if it does not start instantly, the 
current continues to flow through the same resistance leads, and 
thus they are overheated and destroyed if the motor does not 
start promptly. Hence care has to be taken not to have such 
motors stalled for any appreciable time with voltage on. 

The most serious objection to the use of high- re si stance leads, 
therefore, is their liability to self-destruction by heating if the 
motor fails to start immediately, as for instance in a railway 
motor when putting the voltage on the motor before the brakes 
are released, as is done when starting on a steep up-grade to 
keep the train from starting to run back. 

Thus the advantages of resistance commutator leads are the 
improvement in commutation resulting from the reduced short- 
circuit current, and the ahsence o fa serious demagnetizing effect 
on the field at the moment of starting, which would result from 
an excessive short-circuit current under the brush, and such 
leads are therefore extensively used ; their disadvantage, however, 
is that when they are used the motor must be sure to start im- 
mediately by the application of voltage, otherwise they are liable 
to l>e destroyed. 

It is obvious that even with high -resistance commutator leads 
the commutation of the motor can not be as good as that of the 
motor on direct-current supply; that is, such an alternating- 
current motor inherently is more or less inferior in commutation 
to the direct-current motor, and to compensate for this effect 
far more favorable constants must be chosen in the ruotui design 
than permissible with a direct-current motor, that is, a lower 
voltage per commutator segment and lower magnetic flux per 
pole, hence a lower supply voltage on the armature, and thus B 
larger armature current and therewith a larger commutator, etc. 

The insertion of reactance instead of resistance in the leads 
connecting the commutator segments with the armature nib of 
the single-phase motor also has Ix-i-ri proposed And UWd f'ir 
limiting the short-circuit current, under I lie commutator brush. 

Reactance has the advantage over resistance, that the voltage 


consumed by it is wattless and therefore produces no scrum- 
heating and reactive leads of low resistance thus are not liable 
to self-destruction by heating if the motor fails to start im- 

On account of the limited apace available in the railway motor 
considerable difficulty, however, is found in designing sufficiently 
high reactances which du not saturate and thus decrease nt 
larger currents. 

At speed, reactance in the armature coils is very objectionable 
in retarding the reversal of current, and indeed one of the most 
important problems in the design of commu.tating machines 
give the armature coils the lowest possible reactance. There- 
fore, the insertion of reactance in the motor leads tnterfffW 
seriously with the commutation of the motor at speed, and ihn- 
requires the use of a suitable commutating or reversing flux, thai 
is, a magnetic field at the commutator brushes of sufficient 
strength to reverse the current, against the self-inductance of the 
armature coil, by means of an e.m.f. generated in the armature 
coil by its rotation. This commutating flux thus must he m 
phase with the main current, that is, a flux of overcompensation. 
Reactive leads require the use of a commutating flux of over- 
compensation to give fair commutation at speed. 

Counter E.m.fs. in Commutated Coil 

199. Theoretically, the correct way of eliminating the de- 
structive effect of the short-circuit current under the 
tutor brush resulting from the e.m.f. of alternation of the main 
flux would be to neutralize the e.m.f. of alternation by an equal 
but opposite e.m.f. inserted into the armature coil or generated 
therein. Practically, however, at least with most motor typos, 
considerable difficulty is met in producing such a neutralizing 
e.m.f. of the proper intensity as well as phase. Since the alter- 
nating current has not only an intensity but also a phase displace- 
ment, with an alternating-current motor the production of com 
mutating flux or commutating voltage is more difficult than with 
direct-current motors in which the intensity is the only v;n t.iM. 

By introducing an external e.m.f. into the short-circuited 
under the brush it is rml possible entirely to neutralise itfl BJB ' 
of alternation, hut simply to reduce it to one-half. Several such 
arrangements were developed in the early days by Ekkettoyar, 


for instance the arrangement shown in Fig. 170, which represents 
the development of a commutator. The commutator consists 
of alternate live segments, S, and dead segments, S', that is, seg- 
ments not connected to armature coils, and shown shaded in 
Fig. 170. Two sets of brushes on the commutator, the one, B t , 


Fig. 170. — Commutation with external e.m.f, 

ahead in position from the other, B t , by one commutator seg- 
ment, and connected to the first by a coil, N, containing an e.m.f. 
equal in phase, but half in intensity, and opposite, to the e.m.f. 
of alternation of the armature coil; that is, if the armature coil 
contains a single turn, coil A' is a half turn located in the main 

Fio. 171. — Commutation by external e 
field space; if the armature coil, A, contains m turns, '„ turns in 
the main field space are used in coil, N. The dead segments, S', 
are cut between the brushes, B x and /Jj, so as not to short-circuit 
between the brushes. 

In this manner, during the motion of the brush over the com- 


mutator, as shown by Fig. 171 in its successive steps, in position: 

1. There is current through brush, B\\ 

2. There is current through both brushes, Si and B«, and the 

armature coil, A, is closed by the counter e.m.f. of coil, 
.V, that is, the difference, A — JV, is short-circuited; 

3. There is current through brush B 3 ; 

4. There is current through both brushes, B, and B t , and the 

coil, JV, is short-circuited; 

5. The current enters again by brush B i ; 

thus alternately the coil, JV, of half the voltage of the armature 
coil, A, or the difference between A and JV is short-circuited, 
that is, the short-circuit current reduced to one-half. 

Complete elimination of the short-circuit current can be pro- 
duced by generating in the armature coil an opposing e.m.f. 
This e.m.f. of neutralization, however, can not be generated by 
the alternation of the magnetic flux through the coil, as this would 
require a flux equal but opposite to the full field flux travers- 
ing the coil, and thus destroy the main field of the motor. The 
neutralizing e.m.f., therefore, must be generated by the rotetifin 
of the armature through the commutating field, and thus can 
occur only at speed; that is, neutralization of the short-circuit 
current is possible only when the motor is revolving, but not while 
at rest. 

200. The e.m.f. of alternation in the armature coil short-cir- 
cuited under the commutator brush is proportional to the main 
field, *, to the frequency, /, and is in quadrature with the main 
field, being generated by its rate of change; hence, it can be rep- 
resented by 

eo -2r/*10-*/. (17) 

The e.m.f., e,, generated by the rotation of the armature coil 
through a commutating field, *', is, however, in phase with the 
field which produces it; and since d must be equal and in phase 
with e to neutralize it, the commutating field, *', therefore, must 
be in phase with e , hence in quadrature with *; that is, the com- 
mutating field, *', of the motor must be in quadrature witfa tin 
main'field, *, to generate a neutralizing voltage, e,, of the proper 
phase to oppose the e.m.f. of alternation in the short-circuited 
coil. This e.m.f., ei, is proportional to its generating field. *', 
and to the speed, or frequency of rotation, f„, hence is: 

ei =2t/„*'10- s , ,lv. 

nil Si = p„ it then follow* that: 

*' = j-t 






that is, the commutating field of the single-phase motor must 
be in quadrature behind and proportional to the main field, pro- 
portional to the frequency and inversely proportional to the 
speed; hence, at synchronism, /» = /, the commutation field 
equals the main field in intensity, and, being displaced therefrom 
1 quadrature both in time and in space, the motor thus must 
have a uniform rotating field, just as the induction motor. 

Above synchronism, fa > f, the commutating field, *', is less 
than the main field; below synchronism, however, /& < /, the 
commutating field must be greater than the main field to give 
complete compensation. It obviously is not feasible to increase 
the commutating field much beyond the main field, u this would 
require an increase of the iron section of the motor beyond that 
required to do the work, that is, to carry the main field flux. At 
standstill *' should be infinitely large, that is, compensation is 
not possible. 

Hence, by the use of a commutating field in time and space 
quadrature, in the single-phase motor the short-circuit current 
under the commutator brushes resulting from the e.m.f. of alter- 
nation can be entirely eliminated at and above synchronism, 
and more or less reduced below synchronism, the more the nearer 
the speed is to synchronism, but no effect can be produced at 
standstill. In such a motor either some further method, as re- 
sistance leads, must, be used to take care of the short-circuit cur- 
rent at standstill, or the motor designed so that its commutator 
can carry the short-circuit current for the small fraction of time 
when the motor is at. standstill or running at very low speed. 

The main field, *,of the series motor is approximately inversely 
proportional to the speed, / , since the product of speed and field 
strength, / *, is proportional to the e.m.f. of rotation, or useful 
e.m.f. of the motor, hence, neglecting losses anil phase displace- 
ments, to the impressed e.m.f., that is, constant. Substituting 
. — a. n>tior» &.. — main field at synchronism, into 



that is, the commutating field is inversely proportional to the 
square of the speed; for instance, at double synchronism it should 
be one-quarter as high as at synchronism, etc. 

201. Of the quadrature field, <!>', only that part is needed for 
commutation which enters and leaves the armature at the posi- 
tion of the brushes; that is, instead of producing a quadrature 
field, <!>', in accordance with equation (20), and distributed around 
the armature periphery in the same manner as the main field, ♦, 
but in quadrature position thereto, a local commutating field 
may be used at the brushes, and produced by a commutating 
pole or commutating coil, as shown diagrammatically in Fig. 172 

Fig. 172. — Commutation with commutating poles. 

as K\ and K. The excitation of this commutating coil, A', then 
would have to be such as to give a magnetic air-gap density <B' 
relative to that of the main field, (B, by the same equations (19) 
and (20) : 

(B' = j(B { 

= *<) 2 


As the alternating flux of a magnetic circuit is proportional to 
the voltage which it consumes, that is, to the voltage impressed 
upon the magnetizing coil, and lags nearly 90° l>ehind it, the mag- 
netic flux of the commutating poles, K, can be produced by ener- 
gizing these poles by an e.m.f. e, which is varied with the speed 
of the motor, by equation: 

e = * (£) -, 
whore e„ is its proper value at synchronism. 



Since (B' lags 90° behind its supply voltage, e, and also lags 90° 
behind (B, by equation (2), and so behind the supply current 
and, approximately, the supply e.m.f. of the motor, the voltage, 
e, required for the excitation of the commutating poles is approxi- 
mately in phase with the supply voltage of the motor; that is, 
a part thereof can be used, and is varied with the speed of the 

Perfect commutation, however, requires not merely the elimi-' 
nation of the short-circuit current under the brush, but requires 
a reversal of the load current in the armature coil during its 
passage under the commutator brush. To reverse the current, 
an e.m.f. is required proportional but opposite to the current and 
therefore with the main field; hence, to produce a reversing e.m.f. 
in the armature coil under the commutator brush a second com- 
mutating field is required, in phase with the main field and ap- 
proximately proportional thereto. 

The commutating field required by a single-phase commutator 
motor to give perfect commutation thus consists of a component 
in quadrature with the main field, or the neutralizing component, 
which eliminates the short-circuit current under the brush, and 
a component in phase with the main field, or the reversing com- 
ponent, which reverses the main current in the armature coil 
under the brush; and the resultant commutating field thus must 
lag behind the main field, and so approximately behind the sup- 
ply voltage, by somewhat less than 90°, and have an intensity 
varying approximately inversely proportional to the square of 
the speed of the motor. 

Of the different motor types discussed under IV, the series 
motors, 1 and 2, have no quadrature field, and therefore can be 
made to commutate satisfactorily only by the use of commutator 
leads, or by the addition of separate commutating poles. The 
inverted repulsion motor, 3, has a quadrature field, which de- 
creases with increase of speed, and therefore gives a better com- 
mutation than the series motors, though not perfect, as the quad- 
rature field does not have quite the right intensity. 

The repulsion motors, 4 and 5, have a quadrature field, lag- 
ging nearly 90° behind the main field, and thus give good com- 
mutation at those speeds at which the quadrature field has the 
right intensity for commutation. However, in the repulsion 
motor with secondary excitation, 5, the quadrature field is con- 
stant and independent of the speed, as constant supply voltage 


is impressed upim the commutating winding, C, which produces 
the quadrature field, and in the direct repulsion motor, 4, the 
quadrature field increases with the speed, as the voltage consumed 
by the main field F decreases, and that left for the compensating 
winding, C, thus increases with the speed, while to give proper 
commutating flux it should decrease with the square of the speed. 
It thus follows that the commutation of the repulsion motors 
improves with increase of speed, up to that speed where the 
quadrature field is just right for commutating field — which is 
about at synchronism — but above this speed the commutation 
rapily becomes poorer, due to the quadrature field being far in 
excess of that required for commutating. 

In the series repulsion motors, 6 and 7, a quadrature field also 
exfsts, just as in the repulsion motors, but this quadrature field 
depends upon that part of the total voltage which is impressed 
upon the commutating winding, C, and thus can be varied by 
varying the distribution of supply voltage between the two cir- 
cuits; hence, in this type of motor, the commutating flux can be 
maintained through all (higher) speeds by impressing the total 
voltage upon the compensating circuit and short-circuiting the 
armature circuit for all speeds up to that at which the required 
commutating flux has decreased to the quadrature, flux given by 
the motor, and from this speed upward only a part of the supply 
voltage, inversely proportional (approximately) to the square of 
the speed, is impressed upon the compensating circuit, the rest 
shifted over to the armature circuit. The difference between 
6 and 7 is that in 6 the armature circuit is more inductive, and 
the quadrature flux therefore lags less behind the main flux than 
in 7, and by thus using more or less of the field coil in the arma- 
ture circuit its inductivity can be varied, and therewith the 
phase displacement of the quadrature flux against, the main flux 
adjusted from nearly 90° lag to considerably less lag, hence not 
only the proper intensity but also the exact phase of the required 
commutating flux produced. 

As seen herefrom, the difference between the different motor 
types of IV is essentially found in their different actions regarding 

It follows herefrom that by the selection of the motor-type 
quadrature fluxes, *i, can be impressed upon the motor, as com- 
mutating flux, of intensities and phase displacements against 
the main flux, *, varying over a considerable range. The main 


advantage of the series-repulsion motor type is the possibility 
which this type affords, of securing the proper commutating 
field at all speeds down to that where the speed is too low to 
induce sufficient voltage of neutralization at the highest available 
commutating flux. 

VI. Motor Characteristics 

202. The single-phase commutator motor of varying speed or 
series characteristic comprises three circuits, the armature, the 
compensating winding, and the field winding, which are connected 
in series with each other, directly or indirectly. 

The impressed e.m.f. or supply voltage of the motor then con- 
sists of the components: 

1. The e.m.f. of rotation, e h or voltage generated in the arma- 
ture conductors by their rotation through the magnetic field, $. 
This voltage is in phase with the field, $>, and therefore approxi- 
mately with the current, i, that is, is power e.m.f., and is the 
voltage which does the useful work of the motor. It is propor- 
tional to the speed or frequency of rotation,/o, to the field strength, 
$, and to the number of effective armature turns, tii. 

«i = 2ir/ n 1 <i> 10" 8 . (23) 

The number of effective armature turns, n if with a distributed 
winding, is the projection of all the turns on their resultant direc- 
tion. With a full-pitch winding of n series turns from brush to 
brush, the effective number of turns thus is: 

♦* 2 

fii = m [avg cos] \ « m. (24) 

With a fractional-pitch winding of the pitch of r degrees, the 
effective number of turns is: 

fix = m [avg cos] / « m sin * (2/5) 

2. The e.m.f. of alternation of the field, e«, tliat is, the voltage 
generated in the field turns by the alternation of the magnetic 
flux, 4>, produced by them and thus enclosed by them. This vol- 
tage is in quadrature with the field flux, 4>, and thus approxi- 
mately with the current [, is proportional to tin* frequency of the 


impressed voltage, /, to the field strength, 4>, and to the number of 
field turns, n„. 

«o = 2jirfn * 10~ 8 . (26) 

3. The impedance voltage of the motor: 

e' = IZ (27) 

and: Z = r + jx, 

where r = total effective resistance of field coils, armature with 
commutator and brushes, and compensating winding, x = total 
self-inductive reactance, that is, reactance of the leakage flux of 
armature and compensating winding — or the stray flux passing 
locally between the armature and the compensating conductors 
— plus the self-inductive reactance of the field, that is, the reac- 
tance due to the stray field or flux passing between field coils 
and armature. 

In addition hereto, x comprises the reactance due to the quad- 
rature magnetic flux of incomplete compensation or overcom- 
pensation, that is, the voltage generated by the quadrature flux, 
$', in the difference between armature and compensating con- 
ductors, ni — n 2 or n% — n\. 

Therefore the total supply voltage, E y of the motor is: 

E = ei + e + e' 

= 2 irforii* 10- 8 + 2jirfn x * lO" 8 + (r + jx) /. (28) 

Let, then, R = magnetic reluctance of field circuit, thus 

$ = ~tr = the magnetic field flux, when assuming this flux as in 
phase with the excitation /, and denoting: 

as the effective reactance of field inductance, corresponding to 
the e.m.f. of alternation: 

S = y = ratio of speed to frequency, or speed 
f as fraction of synchronism, 


c = = ratio of effective armature turns to 
n ° field turns; 



substituting (30) and (31) in (28): 

# = cSxol + jxol + (r + jx) I 

= [(r + cSxo) + j(x + x )] I; (32) 


1 = (r + cSxo) + 7 (x + *o)' (33) 

and, in absolute values: 

t = , -_ — . • (34) 

V(r + cSxo) 2 + (x + x ) 2 

The power-factor is given by: 

tan * - 7T& (35) 

The useful work of the motor is done by the e.m.f. of rotation: 

#i = cSxof, 

and, since this e.m.f., #i, is in phase with the current, /, the 
useful work, or the motor output (inclusive friction, etc.), is: 

p = EJ = cSxoi 2 

cSxoe 2 

(r + cSx Q ) 2 + (x + xo) 2 

and the torque of the motor is : 


D = -~ = cxoi 2 


cxtf 2 

(r + cSxo) 2 + (x + xo) 2 
For instance, let: 

e = 200 volts, c = — = 4, 




Z =± r+jx = 0.02 + 0.06 j, x = 0.08; 
. 10,000 

1 " vu + i^ 2 +49 amp -' 

♦ a 1 + 16 S 
cot = — =- > 

p = 32^0005 , 

(1 + 16 Sj 2 "+ 49 ' 

n _ 32,0 00 , 

(1 + 16 S) 2 + 49 Syn * kW * 


203. The behavior of the motor at different speeds is l*^t 
shown by plotting i, p = cos 8, P and D as ordinates with the 
speed, 8, as abscissae, as shown in Fig. 173. 

In railway practice, by a survival of the practice of former 
times, usually the constants are plotted with the current, /, as 
abscissae, as shown in Fig. 174, though obviously this arrange- 
ment does not as well illustrate the behavior of the motor. 

Graphically, by starting with the current, /, as zero axis,0/, the 
motor diagram is plotted in Fig. 175. 


















































1 l'i 


1 L5 1 






Fia. 173. — Bingle-phase commutator-motor spoori characteristics. 

he voltage consumed by the resistance, r, is OE, — ir. in pi 
l 01; the voltage consumed by the reactance, x, is OE, = 
90° ahead of 01. OE, and OE, combine to the voltage c 
ed by the motor impedance, OE' — iz. 
ombining OE' = iz, OE\ — e it and OE = c thus gives 
linal voltage, OE = e, of the motor, and the phase an 

= e. 

l this diagram, and in the preceding approximate calculat 
magnetic flux, *, has been assumed in phase with the curren 
l reality, however, the equivalent sine wave of magn 
$, lags behind the equivalent sine wave of exciting curren 
he angle of hysteresis lag, and still further by the po 







consumed by eddy currents, and, especially in the commutator 
motor, by the power consumed in the short-circuit current under 
the brushes, and the vector,0*,therefore is behind the current 
vector, 01, by an angle a, which is small in a motor in which the 
short-circuit current under the brushes is eliminated and the 
eddy currents are negligible, but may reach considerable values 
in the motor of poor commutation. 


" r 

" I 


" g / .r. 

" \ ■£/«•' 

u -^ -, ». f- 

" V? 7 

d.. ~~"S_^ -l X m 

Z i C 5 ^ ' j H: 

-/ V ^ \--T 

7 K , V \ 

04 D Z ^ \A-»- 

7 s \"Z 

. ^ i- \ ■ 

M 30 3OT aio MOWI ^Twu W KOI* 1500 i: MlN 

Fio. 174. — Single-phase commutator-motor current characteristics. 

Assuming then, in Fig. 176, 0* lagging behind 01 by angle 
a, OEi is in phase with 0*, hence lagging behind 01; that is, 
the e.m.f. of rotation is not entirely a power e.m.f., but contains 
a wattless lagging component. The e.m.f. of alternation, OE , 
is 90° ahead of O*, hence less than 90° ahead of OI, and therefore 
contains a power component representing the power consumed 
by hysteresis, eddy currents, and the short-circuit current under 
the brushes. 

Completing now the diagram, it is seen that Hie phase angle, 9, 
is reduced, that is, the power-factor of the motor increased by 



the increased loss of power, but is far greater than corresponding 
thereto. It is the result of the lag of the e.m.f. of rotation, which 
produces a lagging e.m.f. component partially compensating for 
the leading e.m.f. consumed by self -inductance, a lag of the e.m.f. 
being equivalent to a lead of the current. 

Fig. 175. — Single-phase commutator-motor vector diagram. 

As the result of this feature of a lag of the magnetic flux, $, 
by producing a lagging e.m.f. of rotation and thus compensating 
for the lag of current by self-inductance, single-phase motors 
having poor commutation usually have better power-factors, and 

Fig. 176. — Single-phase commutator-motor diagram with phase displace- 
ment between flux and current. 

improvement in commutation, by eliminating or reducing the 
short-circuit current under the brush, usually causes a slight de- 
crease in the power-factor, by bringing the magnetic flux, <f>, more 
nearly in phase with the current, /. 

204. Inversely, by increasing the lag of the magnetic flux, <t>, 
the phase angle can bo decrejised .and the power-factor improved. 
Such a shift of the magnetic flux, 4>, behind the supply current, ?, 
can be produced by dividing the current, i, into components, i' 


and i", and using the lagging component for field excitation. 
This is done most conveniently by shunting the field by a non- 
inductive resistance. Let r be the non-inductive resistance in 
shunt with the field winding, of reactance, x Q + x\ f where X\ is 

Fig. 177. — Single-phase commutator-motor improvement of power-factor 

by introduction of lagging e.m.f. of rotation. 

that part of the self-inductive reactance, x, due to the field coils. 

The current, i', in the field is lagging 90° behind the current, i", 

in a non-inductive resistance, and the two currents have the 

ratio .,-, = — -7- — ; hence, dividing the total current, 01 , in this 

* Xo T ^1 

proportion into the two quadrature components, OF and 01" > 

Fig. 178. — Single-phase commutator motor. Unity power-factor produced 

by lagging e.m.f. of rotation. 

in Fig. 177, gives the magnetic flux, 0$, in phase with 01', and 
so lagging behind 01, and then the e.m.f. of rotation is OE h the 
e.m.f. of alternation OE 0) and combining 0E h OE 0) and OE' 


gives the impressed e.ra.f., OE, nearer in phase to 01 than with 
0* in phase with 01. 

In this manner, if the e.m.fa, of self-inductance arc not CM 
large, unity power-factor can be produced, as shown in Fig. 178. 

Let 01 = total current, OE' = impedance voltage of the 
motor, OE = impressed e.m.f. or supply voltage, and assumed 
in phase with 01. OE then must be the resultant of OE' and of 
OEi, the voltage of rotation plus that of alternation, and resolv- 
ing therefore 0E% into two components, 0E% and 0E a , in quadra- 

ture with each other, and proportional respectively to the e.m.f. 
of rotation and the e.m.f. of alternation, gives the magnetic flux, 
0*, in phase with the e.m.f. of rotation, 0E U and the component 
of current in the field, 01', and in the non-inductive resistance-, 
01", in phase and in quadrature respectively with 04>, which 
combined make up the total current. The projection of the 
e.m.f. of rotation 0E\ on 01 then is the power component of 
the e.m.f., which does the work of the motor, and the quadra- 
ture projection of, 0E t , is the compensating component of the 
e.m.f. of rotation, which neutralizes the wattless component of 
the e.m.f. of self-inductance. 

Obviously such a compensation involves some loss of power 
in the non-inductive resistance, r a , shunting the field coils, and as 
the power-factor of the motor usually is sufficiently high, such 
compensation is rarely needed. 

In motors in which some of the circuits are connected induct ively 
in series with the others the diagram is essentially thesame, except 


that a phase displacement exists between the secondary and the 
primary current. The secondary current, Ii, of the transformer 
lags behind the primary current, Jo, slightly less than 180° ; that is, 
considered in opposite direction, the secondary current leads the 
primary by a small angle, 0o, and in the motors with secondary 
excitation the field flux, 4>, being in phase with the field current, 
1 1 (or lagging by angle a behind it), thus leads the primary 
current, Jo, by angle O (or angle do — a). As a lag of the mag- 
netic flux $ increases, and a lead thus decreases the power-factor, 
motors with secondary field excitation usually have a slightly 

Fig. 180. — Single-phase commutator motor with secondary excitation 
power-factor improved by shunting field winding with non-inductive 

lower power-factor than motors with primary field excitation, 
and therefore, where desired, the power-factor may be improved 
by shunting the field with a non-inductive resistance, r . Thus 

for instance, if, in Fig. 179, 01 = primary current, 01 \ = sec- 
ondary current, OEi, in phase with 01 1, is the e.m.f. of rotation, 
in the case of the secondary field excitation, and OEo, in quadra- 
ture ahead of 01 1, is the e.m.f. of alternation, while OE' is the 
impedance voltage, and OEi, OEo and OE' combined give the 

supply voltage, OE, and EOI = the angle of lag. 

Shunting the field by a non-inductive resistance, r , and thus 

resolving the secondary current OI\ into the components OI\ in 

the field and 01" \ in the non-inductive resistance, gives the dia- 
gram Fig. 180, where a = I'iO$ = angle of lag of magnetic 



205. The action of the commutator in an alternating-eurreri 
motor, in permitting compensation for phase displacement iwl 

thus allowing a control of the power-factor, is very imn. ■-m-.- 
and important, and can also be used in other types of machines, 
as induction motors am! alternators, by supplying these machines 
with a commutator for phase control. 

A lag of the current is the same as a lead of the e.m.f., and in- 
versely a leading current inserted into a circuit has the same ef- 
fect as a lagging e.m.f. inserted. The commutator, however 
produces an e.m.f. in phase with the current. Inciting the field 
l>y a lagging current in the field, a lagging e.m.f. of rotation is 
produced which is equivalent to a leading current. As it is easy 
to produce a lagging current by self-inductance, the commutator 
thus affords an easy means of producing the equivalent of a 
leading current. Therefore, the alternating-current commutator 
is one of the important methods of compensating for lagging: 
currents. Other methods are the use of electrostatic or electro- 
lytic condensers and of overexcited synchronous machines. 

Based on this principle, a number of designs of induction 
motors and other apparatus have been developed, using Qm 
commutator for neutralizing the lagging magnetizing current 
and the lag caused by self-inductance, and thereby produdng 
unity power-factor or even leading currents. So far, however, 
none of them has come into extended use. 

This feature, however, explains the very high power-factor* 
feasible in single-phase commutator motors even with COQndtf 
able air gaps, far larger than feasible in induction motors. 

VII. Efficiency and Losses 

206. The losses in single-phase commutator motors ate BOB ' 
tially the same as in other types of machines: 

(a) Friction losses— air friction or windage, lwaring friction 
and commutator brush friction, and also ■ :■ . 
mechanical transmission losses. 

(6) Core losses, as hysteresis and eddy currents. These an 1 
of two classes — the alternating core hiss, due to the alternation 
of the magnetic flux in the main field, quadrature field, and arma- 
ture and the rotating core loss, due to the rotation of the arma- 
ture; through the magnetic field. The former depends upon the 
frequency, the latter upon the speed. 

(cj Commutation losses, as the power consumed by the slum- 


circuit current under the brush, by arcing and sparking, where 
such exists. 

(d) ihr losses in the motor circuits — the field coils, the compen- 
sating winding, the armature and the brush contact resistance. 

(e) Load losses, mainly represented by an effective resistance, 
that is, an increase of the total effective resistance of the motor 
beyond the ohmic resistance. 

Driving the motor by mechanical power and with no voltage 
on the motor gives the friction and the windage losses, exclusive 
of commutator friction, if the brushes are lifted off the commu- 
tator, inclusive, if the brushes are on the commutator. Ener- 
gizing now the field by an alternating current of the rated fre- 
quency, with the commutator brushes off, adds the core losses 
to the friction losses; the increase of the driving power theto 
measures the rotating core loss, while a wattmeter in the field 
exciting circuit measures the alternating core loss. 

Thus the alternating core loss is supplied by the impressed 
electric power, the rotating core loss by the mechanical driving 

Putting now the brushes down on the commutator adds the 
commutation losses. 

The ohmic resistance gives the i 2 r losses, and the difference 
between the ohmic resistance and the effective resistance, calcu- 
lated from wattmeter readings with alternating current in the 
motor circuits at rest and with the field unexcited, represents 
the load losses. 

However, the different losses so derived have to be corrected 
for their mutual effect. For instance, the commutation losses 
are increased by the current in the armature; the load losses are 
less with the field excited than without, etc. ; so that this method 
of separately determining the losses can give only an estimate of 
their general magnitude, but the exact determination of the effi- 
ciency is best carried out by measuring electric input and me- 
chanical output. 

VIII. Discussion of Motor Types 

207. Varying-speed single-phase commutator motors can be 
divided into two classes, namely, compensated series motors and 
repulsion motors. In the former, the main supply current is 
through the armature, while in the latter the armature is closed 
upon itself as secondary circuit, with the compensating winding 



as primary or supply circuit. As the result hereof the repulsion 
motors contain a transformer flux, in quadrature position to the 
main flux, and lagging behind it, while in the series motors no 
such lagging quadrature flux exists, but in quadrature position 
to the main flux, the flux either is zero — complete compensation 
■ — or in phase with the main flux — over- or undercompensation. 

A. Compensated Series Motors 

Series motors give the best power-factors, with the exreptioii 
of those motors in which by increasing the lag of the field flux 
a compensation for power-factor is produced, as discussed in V. 
The commutation of the series motor, however, is equally poor 
at all speeds, due to the absence of any eommutating flux, and 
with the exception of very small sizes such motors therefore are 
inoperative without the use of either resistance leads or eom- 
mutating poles. With high-resistance leads, however, fair opera- 
tion is secured, though obviously not of the same class with 
that of the direct-current motor; with eommutating poles or coils 
producing a local quadrature flux at the brushes good results 
have been produced abroad. 

Of the two types of compensation, conductive compensation. 
1, with the compensating winding connected in series with the 
armature, and inductive compensation, 2, with the compensated 
winding short-circuited upon itself, inductive compensation nec- 
essarily is always complete or practically complete compensa- 
tion, while with conductive compensation a reversing flux can 
be produced at the brushes by overcompensation, and the com- 
mutation thus somewhat improved, especially at speed, at (fat 
sacrifice, however, of the power-factor, which is lowered by the 
increased self-inductance of the compensating winding. On the 
short-circuit current under the brushes, due to the e.m.f. of alter- 
nation, such overcompensation obviously has no helpful effect. 
Inductive compensation has the advantage that the compen- 
sating winding is not connected with the supply circuit, can I* 
made of very low voltage, or even of individually short-circuited 
turns, and therefore larger conductors and less insulation used, 
which results in an economy of spaee, and therewith an infix Hfld 
output for the same size of motor. Therefore inductive compttf- 
satiou is preferable where it can be used. It is not permissible, 
however, in motors which are required to operate also on direct 
current, since with direct-current supply no induction takes place 



and therefore the compensation fails, and with the high ratio of 
armature turns to field turns, without compensation, the field 
distortion is altogether too large to give satisfactory commutation, 
except in small motors. 

The inductively compensated series motor with secondary ex- 
citation, or inverted repulsion motor, 3, takes an intermediary 
position between the series motors and the repulsion motors; it 
is a series motor in so far as the armature is in the main supply 
circuit, but magnetically it has repulsion-motor characteristics, 
that is, contains a lagging quadrature flux. As the field exci- 
tation consumes considerable voltage, when supplied from the 
compensating winding as secondary circuit, considerable voltage 
must he generated in this winding, thus giving a corresponding 
transformer flux. With increasing speed and therewith decreas- 
ing current, the voltage consumed by the field coils decreases, 
and therewith the transformer flux which generates this voltage. 
Therefore, the inverted repulsion motor contains a transformer 
flux which has approximately the intensity and the phase re- 
quired for commutation; it lags behind the main flux, but less 
than 90°, thus contains a component in phase with the main 
flux, as reversing flux, and decreases with increase of speed. 
Therefore, the commutation of the inverted repulsion motor is 
very good, far superior to the ordinary series motor, and it can 
be operated without resistance leads; it has, however, the serious 
objection of a poor power-factor, resulting from the lead of the 
field flux against the armature current, due to the secondary ex- 
citation, as discussed in V. To make such a motor satisfactory 
in power-factor requires a non-inductive shunt across the field, 
and thereby a waste of power. For this reason it has not come 
into commercial use. 

B. Repulsion Motors 
208. Repulsion motors are characterized by a lagging quadra- 
ture flux, which transfers the power from the compensating wind- 
ing to the armature. At standstill, and at very low speeds, re- 
pulsion motors and series motors are equally unsatisfactory in 
commutation; while, however, in the series motors the commu- 
tation remains bad (except when using commutating devices), 
in the repulsion motors with increasing speed the commutation 
rapidly improves, and becomes perfect near synchronism. As 
the result hereof, under average conditions a much inferior com- 


mutation can be allowed in repulsion motors at. very low speeds 
than in series motors, since in the former the period of poor 
commutation lasts only a very short time. While, therefore, 
series motors can not be satisfactorily operated Hit hoot resist a m.-e 
leads (or commutating poles), in repulsion motors peristanofl 
leads are not necessary and not used, and the excessive current 
density under the brushes in the moment of starting permitted, 
as it lasts too short a time to cause damage to the commutator. 

As the transformer field of the repulsion motor is approximately 
constant, while the proper commutating field should decrease 
with the square of the speed, above synchronism the transformer 
field is too large for commutation, and at speeds considerably 
above synchronism — 50 per cent, and more — the repulsion motor 
becomes inoperative because of excessive sparking. At syn- 
chronism, the magnetic field of the repulsion motor is a rotating 
field, like that of the polyphase induction motor. 

Where, therefore, speeds far above synchronism are required, 
the repulsion motor can not be used; but where synchronous 
speed is not much exceeded the repulsion motor is preferred be- 
cause of its superior commutation. Thus when using a commu- 
tator as auxiliary device for starting single- phase induction 
motors the repulsion-motor type is used. For high frequencies. 
as 60 cycles, where peripheral speed forbids synchronism being 
greatly exceeded, the repulsion motor is the type to be considers! 

Repulsion motors also may be built with primary and str- 
ondary excitation. The latter usually gives a lietter commuta- 
tion, because of the lesser lag of the transformer flux, and i here- 
with a greater in-phase component, that is, greater reversing flux, 
especially at high speeds. Secondary excitation, however, gives 
a slightly lower power-factor. 

A combination of the repulsion-motor and series-motor types 
is the series repulsion motor, 6 and 7. In this only a part of 
the supply voltage is impressed upon the conqx-nsating winding 
and thus transformed to the armature, while the rest of the sup- 
ply voltage is impressed directly upon the armature, just as in 
the series motor. As result thereof the transformer flux of tin 1 
series repulsion motor is less than that of the repulsion motor. 
in the same proportion in which the voltage impressed upon the 
compensating winding is less than the total supply voltage. 
Such a motor, therefore, reaches equality of the transformer flux 
with the commutating flux, and gives perfect commutation at a 


higher speed than the repulsion motor, that is, above synchron- 
ism. With-the total supply voltage impressed upon the compen- 
sating winding, the transformer flux equals the commutating 
flux at synchronism. At n times synchronous speed the com- 
mutating flux should be — 2 of what it is at synchronism, and by 


impressing —^ of the supply voltage upon the compensating wind- 


ing, the rest on the armature, the transformer flux is reduced 
to —j of its value, that is, made equal to the required commuta- 

ting flux at n times synchronism. 

In the series repulsion motor, by thus gradually shifting the 
supply voltage from the compensating winding to the armature 
and thereby reducing the transformer flux, it can be maintained 
equal to the required commutating flux at all speeds from syn- 
chronism upward; that is, the series repulsion motor arrange- 
ment permits maintaining the perfect commutation, which the 
repulsion motor has near synchronism, for all higher speeds. 

With regard to construction, no essential difference exists be- 
tween the different motor types, and any of the types can be 
operated equally well on direct current by connecting all three 
circuits in series. In general, the motor types having primary 
and secondary circuits, as the repulsion and the series repulsion 
motors, give a greater flexibility, as they permit winding the 
circuits for different voltages, that is, introducing a ratio of trans- 
formation between primary and secondary circuit. Shifting one 
motor element from primary to secondary, or inversely, then 
gives the equivalent of a change of voltage or change of turns, 
Thus a repulsion motor in which the stator is wound for a higher 
voltage, that is, with more turns, than the rotor or armature, 
when connecting all the circuits in series for direct-current opera- 
tion, gives a direct-current motor having a greater field excita- 
tion compared with the armature reaction, that is, the stronger 
field which is desirable for direct-current operating but not per- 
missible with alternating current. 

209. In general, tthe constructve differences between motor 
types are mainly differences in connection of the three circuits. 
For instacne, let F = field circuit, A = armature circuit, C = 
compensating circuit, T = supply transformer, R = resistance 
used in starting and at very low speeds. Connecting, in Fig. 181, 
the armature, A, between field F and compensate" Ending, C. 



With switch open the starting resistance is in circuit . dofBO| 
switch short-circuits the starting resistance and gives the run- 
ning conditions of the motor. 

With all the other switches open the motor is a conductively 

compensated series motor. 

Fia. ] 

raged to operate 

Closing 1 gives the inductively compensated series motor. 
Closing 2 gives the repulsion motor with primary excitation. 
Closing 3 gives the repulsion motor with secondary excitation. 
Closing 4 or 5 or 6 or 7 gives the successive speed steps of the 
scries repulsion motor with armature excitation. 


Connecting, in Fig. 182, the field, F, between armature, .1 , ud 
compensating winding, C, the resistance, R, is again controlled by 

switch 0. 

All other switches open gives the conductively compensated 
series motor. 



Switch 1 closed gives the inductively compensated series 

Snitch 2 closed gives the inductively compensated series 
motor with secondary excitation, or inverted repulsion motor. 

Switch 3 closed gives the repulsion motor with primary 

Switches 4 to 7 give the different speed steps of the series re- 
pulsion motor with primary excitation. 

Opening the connection at x and closing at y (as shown in 
dotted tine), the steps 3 to 7 give respectively the repulsion motor 
with secondary excitation and the successive steps of the series 
repulsion motor with armature excitation. 

Still further combinations can be produced in this manner, as 
for instance, in Fig. 181, by closing 2 and 4, but leaving open, 
the field, F, is connected across a constant- potential supply, in 
series with resistance, R, while the armature also receives con- 
stant voltage, and the motor then approaches a finite speed, that 
is, has shunt motor characteristic, and in starting, the main 
field, F, and the quadrature field, AC, are displaced in phase, so 
give a rotating or polyphase field (unsymmetrical). 

To discuss all these motor types with their in some instances 
very interesting characteristics obviously is not feasible. In 
general, they can all be classified under series motor, repulsion 
motor, shunt motor, and polyphase induction motor, and com- 
binations thereof. 

IX. Other Commutator Motors 

210. Single-phase commutator motors have been developed as 
varying-spced motors for railway service. In other directions 
commutators have been applied to alternating-current motors 
and such motors developed : 

(a) For limited speed, or of the shunt-motor type, that is, 
motors of similar characteristic as the single-phase railway 
motor, except that the speed does not indefinitely increase with 
decreasing load but approaches a finite no-load value. Several 
types of such motors have been developed, as stationary motors 
for elevators, variable-speed machinery, etc., usually of the 
single-phase type. 

By impressing constant voltage upon the field the magnetic 
field flux is constant, and the speed thus reaches a finite limiting 
value at which the of rotation of the armature through 



the constant field flux consumes the impressed voltagi 
armature. By changing the voltage supply to the field different 
speeds can be produced, that is, an adjustable-speed motor. 
The main problem in the design of such motors is to get the 
field excitation in phase with the armature current and thus pro- 
duce a good power-factor. 

(b) Adjustable-speed polyphase induction motors. In the 
secondary of the polyphase induction motor an e.m.f. is gener- 
ated which, at constant impressed e.m.f. and therefore apprffld- 
mately constant flux, is proportional to the slip from synchron- 
ism. With short-circuited secondary the motor closely ap- 
proaches synchronism. Inserting resistance into the secondary 
reduces the speed by the voltage consumed in the secondary. 
As this is proportional to the current and thus to the load, the 
speed control of the polyphase induction motor by resistance in 
the secondary gives a speed which varies with the load, just M 
the speed control of a direct-current motor by resistance in tin- 
armature circuit ; hence, the speed is not constant, and the opera- 
tion at lower speeds inefficient. Inserting, however, a con&taitf 
voltage into the secondary of the induction motor the speed is 
decreased if this voltage is in opposition, and is increased if this 
voltage is in the same direction as the secondary generated e.m.f., 
and in this manner a speed control can be produced. If c = 
voltage inserted into the secondary, as fraction of the voltage 
which would be induced in it at full frequency by the rotating 
field, then the polyphase induction motor approaches at no-load 
and runs at load near to the speed (1 — c) or (1 + c) times syn- 
chronism, depending upon the direction of the inserted voltage. 

Such a voltage inserted into the induction-motor secondary 
must, however, have the frequency of the motor secondary cur- 
rents, that is, of slip, and therefore can be derived from the full- 
frequency supply circuit only by a commutator revolving with 
the secondary, If cf is the frequency of slip, then (1 — c)f is 
the frequency of rotation, and thus the frequency of commuter 
tion, and at frequency, /, impressed upon the commutator the 
effective frequency of the eommutated current is/ — (1 — c)/ = 
cf, or the frequency of slip, as required. 

Thus the commutator affords a means of inserting voltage 
into the secondary of induction motors and thus varying its 

However, while these eommutated currents in their resultant 


give the effect of the frequency of slip, they actually consist of 
sections of waves of full frequency, that is, meet the full station- 
ary impedance in the rotor secondary, and not the very much 
lower impedance of the low-frequency currents in the ordinary 
induction motor. 

If, therefore, the brushes on the commutator are set so that 
the inserted voltage is in phase with the voltage generated in the 
secondary, the power-factor of the motor is very poor. Shifting 
the brushes, by a phase displacement between the generated and 
the inserted voltage, the secondary currents can be made to lead, 
and thereby compensate for the lag due to self-inductance and 
unity power-factor produced. This, however, is the case only 
at one definite load, and at all other loads either overcompensa- 
tion or undercompensation takes place, resulting in poor power- 
factor, either lagging or leading. Such a polyphase adjustable- 
speed motor thus requires shifting of the brushes with the load 
or other adjustment, to maintain reasonable power-factor, and 
for this reason has not been used. 

(c) Power-factor compensation. The production of an alter- 
nating magnetic flux requires wattless or reactive volt-amperes, 
which are proportional to the frequency. Exciting an induction 
motor not by the stationary primary but by the revolving sec- 
ondary, which has the much lower frequency of slip, reduces the 
volt-amperes excitation in the proportion of full frequency to 
frequency of slip, that is, to practically nothing. This can be done 
by feeding the exciting current into the secondary by commuta- 
tor. If the secondary contains no other winding but that con- 
nected to the commutator, the motor gives a poor power-factor. 
If, however, in addition to the exciting winding, fed by the com- 
mutator, a permanently short-circuited winding is used, as a 
squirrel-cage winding, the exciting impedance of the former is 
reduced to practically nothing by the short-circuit winding coin- 
cident with it, and so by overexcitation unity power-factor or 
even leading current can be produced. The presence of the short- 
circuited winding, however, excludes this method from speed 
control, and such a motor (Heyland motor) runs near synchron- 
ism just as the ordinary induction motor, differing merely by the 
power-factor. Regarding hereto see Chapter on "Induction 
Motors with Secondary Excitation." 

This method of excitation by feeding the alternating current 
through a commutator into the rotor has been used very success- 


fully abroad in the so-called "compensated repulsion motor" of 
Winter-Eichberg. This motor differs from the ordinary repul- 
sion motor merely by the field coil. F, in Fig. 183 being replaced 
by a set of exciting brushes, G, in Fig. 184, at right angles to the 
main brushes of the armature, that is, located so that the m_mJ. 
of the current between the brushes, G, magnetizes in the same 



Fig. 1*3. — Plain repulsion motor. 

direction as the field coils. F, in Fig. 1S3. Usually the exciting 
brushes are supplied by a transformer or autotransfonner. so as 
to vary the excitation and thereby the speed. 

This arrangement then lowers the e.mJ. of self-inductance of 
field excitation of the motor from that corresponding to full fre- 


qoency in the ordinary repulsion motor to that cc the frequency 
of sop. hence to a negative value above syririmrJsn: so that 
hereby a compensation for lagging current ;*r he produced 
above synchronism, and unity power-dacKc cc even leading 
currents produced. 


211. Theoretical Investigation. — In its most general form, the 
single-phase commutator motor, as represented by Fig. 185, 
comprises: two armature or rotor circuits in quadrature with 
each other, the main, or energy, and the exciting circuit of the 
armature where such exists, which by a multisegmental commu- 
tator are connected to two sets of brushes in quadrature position 
with each other. These give rise to two short-circuits, also in 
quadrature position with each other and caused respectively by 
the main and by the exciting brushes. Two stator circuits, the 

field, or exciting, and the cross, or compensating circuit, also in 

quadrature with each other, and in line respectively with, the 

exciting and, the main armature circuit. 

These circuits may be separate, or may be parts or components 

of the same circuit. They may be massed together in a single 

slot of the magnetic structure, or may be distributed over the 

whole periphery, as frequently done with the armature windings, 

and then as their effective number of turns must be considered 

their vector resultant, that is: 

2 , 

n = -n 


where n' = actual number of turns in series between the arma- 
ture brushes, and distributed over the whole periphery, that is, 
an arc of 180° electrical. Or the windings of the circuit may be 
distributed only over an arc of the periphery of angle, w, as 
frequently the case with the compensating winding distributed 
in the pole face of pole arc, w ; or with fractional-pitch armature 
windings of pitch, w. In this case, the effective number of turns 


2 . . a) 
n = - n sin « 


where n' with a fractional-pitch armature winding i: 
of series turns in the pitch angle, w, that is: 

n" being the number of turns in series between the brushes, dun 
in the spaed (*■ — w) outside of the pitch angle the armature 
conductors neutralize each other, that is, conductors curryine 
current in opposite direction arc superposed upon each other. 
See fractional-pitch windings, chapter "Commutating Machine," 
"Theoretical Elements of Electrical Engineering." 

212. Let: 

Bo, /o, Z a = impressed voltage, current, and self-inductive 
impedance of the magnetizing or exciter circuit of stator (field 
coils), reduced to the rotor energy circuit by the ratio of effective 
turns, Cn, 

Ei, I,, Zi = impressed voltage, current and self-inductive im- 
pedance of the rotor energy circuit (or circuit at right angles 
to /„), 

Et, It, Z t = impressed voltage, current and self-inductive im- 
pedance of the stator compensating circuit (or circuit parallel to 
/l) reduced to the rotor circuit by the ratio of effective turn*, - ■.. 

fia, t», Z\ = impressed voltage, current'and self-inductive im- 
pedance of the exciting circuit of the rotor, or circuit parafld 

It, Z t = current and self-inductive impedance of the short- 
circuit under the brushes, /,, reduced to the rotor cireuit, 

h, /... = current and self-inductive impedance of the short- 
circuit under the brushes, / B , reduced to the rotor circuit. 

Z = mutual impedance of field excitation, that is, in the direc- 
tion of h, /,, /,, 

Z' = mutual impedance of armature reaction, that, is, in the 
direction of /,, I,, /&. 

Z' usually either equals Z, or is smaller than Z. 

Ii and I a are very small, Z, and Z& very large quantities. 

Let S = speed, as fraction of synchronism. 

Using then the general equations "Chapter XIX, which ftpplji 
to any alternating-current circuit revolving with speed, S, bhnmgjb 
a magnetic field energized by alternating-current circuits, gives 
for the six circuits of the general single-phase commutator motor 
the six equations: 


G o = ZJ b + Z (/ 
E , = Z,/, + Z' (/ 
& = Z-/- + Z' (/ 

£* - Zl/, + Z (/; 

o = ZJ 4 + Z (/ 
o = Z>h + Z' (/. 

+ /i - W, (1) 

+ A - W - J'SZ (/• + /. - /«>, (2) 

- /i - W, (3) 
+ /o - A> - jSZ v /« - /» - h\ (4) 

- /o - /,) - jSZ tf. + /* - f,\ (5) 
+ /i - /») - JSZ (h + /. - M. 16) 

These six equations contain ten variables: 

/o, /it /•* lit /i, Is, £o, £l, £2, £Y 

and so leave four independent variables, that is, four conditions, 
which may be chosen. < 

Properly choosing these four conditions, and substituting them 
into the six equations (1) to (6), so determines all ten variables. 
That is, the equations of practically all single-phase commutator 
motors are contained as special cases in above equations, and 
derived therefrom, by substituting the four conditions, which 
characterize the motor. 

Let then, in the following, the reduction factors to the arma- 
ture circuit, or the ratio of effective turns of a circuit, t , to the 
effective turns of the armature circuit, be represented by 0*. 
That is, 

number of effective turns of circuit, i 
number of effective turns of armature circuit* 

and if #,, /,, Z» are voltage, current and impedance of circuit, i, 

reduced to the armature circuit, then the actual voltage, current 

and impedance of circuit, i, are: 

d$i } * cc Zi. 

213. The different forms of single-phase commutator motors, 
of series characteristic are, as shown diagrammatically in Fig. 

1. Series motor: 

e = c #o + -Pi; h = Colx] h = 0; / 3 = 0. 

2. Conduetively compensated series motor (Eickcmeyer 
motor) : 

e = c #o + #i + c 2 # 2 ; h = co/i; / 2 = c 2 /i; h = 0. 

3. Inductively compensated series motor (Eickcmeyer motor) : 

e = coEq + Pi; # 2 = 0; /o = c /i; / 3 = 0. 



4. Inverted repulsion motor, or series motor with secondary 
excitation : 

e = #1; cqE + c 2 E 2 = 0; c 2 /o = c / 2 ; It = 0. 

5. Repulsion motor (Thomson motor) : 

e = c #o + c 2 # 2 ; #1 = 0; C2I0 = coA; It = 0. 

6. Repulsion motor with secondary excitation: 

c = c 2 # 2 ; co#> + #1 = 0; lo = co/i; It = 0. 

Fig. 186. 

7. Series repulsion motor with secondary excitation : 

ei = co#o + #i;.e 2 = # 2 ; h = c /i; /s = 0. 

8. Series repulsion motor with primary excitation (Alexander- 
sen motor) : 

ei = #1; e 2 = c #o + c 2 # 2 ; c 2 / = c / 2 ; Js = 0. 

9. Compensated repulsion motor (Winter and Eichberg 
motor) : 

e = C2E2 + cz$ z ; gi = 0; / = 0; C3/2 = c 2 / 3 . 



Fig. 187. 

10. Rotor-excited series motor with conductive compensation : 

e = & + cj# 2 + c 8 # 3 ; U = c 2 /i; h = ci/ij /o = 0. 

11. Rotor-excited series motor with inductive compensation: 

. e = & + c,# 3 ; ft - 0; /o = 0; /, - c 3 /i. 

Numerous other combinations can be made and have been 

All of these motors have series characteristics, that is, a speed 
increasing with decrease of load. 

(1) to (8) contain only one set of 
brushes on the armature; (9) to (11) 
two sets of brushes in quadrature. 

Motors with shunt characteristic, 
that is, a speed which does not vary 
greatly with the load, and reaches such 
a definite limiting value at no-load 
that the motor can be considered a constant-speed motor, can 
also be derived from the above equations. For instance: 

Compensated shunt motor (Fig. 187) : 

#1 = 0; c a ft = c 8 #3 = e; /o = 0. 

In general, a series characteristic results, if the field-exciting 
circuit and the armature energy circuit are connected in series 
with each other directly or inductively, or related to each other 
so that the currents in the two circuits are more or less propor- 
tional to each other. Shunt characteristic results, if the voltage 
impressed upon the armature energy circuit, and the field excita- 
tion, or rather the magnetic field flux, whether produced or in- 
duced by the internal reactions of the- motor, are constant, or, 
more generally, proportional to each other. 

ReptUtsion Motor 

As illustration of the application of these general equations, 
paragraph 212, may be considered the theory of the repulsion 
motor (5), in Fig. 180. 

214. Assuming in the following the armature of the repulsion 
motor as short-circuited upon itself, and applying to the motor 
the equations (1) to ((>), the four conditions characteristic of the 
repulsion motor are: 



1. Armature short-circuited upon itself. Hence: 

2. Field circuit and cross-circuit in series with each other con- 
nected to a source of impressed voltage, e. Hence, assuming 
the compensating circuit or cross-circuit of the same number of 
effective turns as the rotor circuit, or, c% = 1 : 

Cq$0 + #2 = e. 

Herefrom follows: 

3. io == CqI 2. 

4. No armature excitation used, but only one set of commu- 
tator brushes; hence: 

/•-0 f 
and therefore: 

/ 6 = 0. 

Substituting these four conditions in the six equations (1) to 
(6), gives the three repulsion motor equations: 
Primary circuit: 

Z 2 / 2 + Z' (h - /1) + Co 2 Zo/ 2 + CoZ (co/ 2 - h) = e; (7) 
Secondary circuit: 

Zxh + Z' (h - h) - jSZ (co/ 2 - I a) = 0; (8) 

Brush short-circuit: 

Z4/4 + Z(h- coh) - jSZ'ih - /,) = 0; (9) 

Substituting now the abbreviations: 

Z 2 + co 2 (Z + Z) = Z 8 , (10) 





z x 


= K 



, + z 

= X4; 


where Xi and X«, especially the former, are small quantities. 
From (9) then follows: 

h = X 4 1/, (c, - jSA) + jShA } ; (14) 

from (8) follows, by substituting (14) and rearranging: 

' ' - '• 1 + x, - x,s< 

and, substituting (15) in (14), gives: 

t _x r (co-i5A)(l + X,-X 4 -S*)+jSA-S»co-X i j,S(-SA + jiCo) 

/« - Wi ! q: Xi - ^ » 

or, canceling terms of secondary order in the numerator: 

'• - k '''TTT^?> < I6) 

Equation (7) gives, substituting (10) and rearranging: 

h (Z, + Z') - hZ' - 7 4 coZ = 6. (17) 

Substituting (15) and (16) herein, and rearranging, gives: 
Primary Current: 

_e(l+X, -_S%) , 

#2 = ^ > (18) 


# = (i43-iSco)+X 1 (A3+4)-X4(S J i4 3 -S 2 Co+co 2 -jSco), (19) 


a, = | 8 ; (20) 

or, since approximately: 

A z = Co 2 , (21) 

it is: 

X = (4, - jSco) + Xi (c« 2 + A) - X4C0 (Co - j/S). (22) 

Substituting (18), (19), (20) in (15) and (16), gives: 
Secondary Current: 

.{.+*»- **(* + fl } 
'' ZK 

Brush Short-circuit Current: 


/4 = x««oU-.s*). (24) 


As seen, for S = 1, or at synchronism, J \ = 0, that is, the 
short-circuit current under the commutator brushes of the re- 
pulsion motor disappears at synchronism, as was to be expected, 
since the armature coils revolve synchronously in a rotating field. 

215. The e.m.f. of rotation f that is, the e.m.f. generated in the 
rotor by its rotation through the magnetic field, which e.m.f., 
with the current in the respective circuit, produces the torque 
and so gives the power developed by the motor, is: 

Main circuit: 

Q\ = jSZ (coh ~ M. (25) 

Brush short-circuit: 

V< = jSZ' (/i - /,). (26) 

Substituting (18), (23), (24) into (25) and (26), and rearrang- 
ing, gives: 

Main Circuit E.m.f. of Rotation: 

£'. =^ 0f ll+Xi-M- (27) 

Brush Short-circuit E.m.f. of Rotation: 

$\ = ™{Sco+ j\iA - c X 4 } ; (28) 

or, neglecting smaller terms: 

£\ = -£*• (29) 

The Power produced by the main armature circuit is: 

Pi = [#'., /.is 

hence, substituting (22) and (27) : 

i% e |i + x, - M, J ±-gg4 4li] • 




m = [ZK] (31) 

be the absolute value of the complex product, ZK, and: 


A = cl + ja" 

X, = X', - j\'\ 
X 4 = X'« + jX" 4 



it is, substituting (31), (32) in (30), and expanding: 

Pi = ^jr U«U - Scoa") - rSc*a'] + (1 - Sc*r") (x(X\ - X' 4 ) 

- r(X"i + X" 4 )] - Scoa'[r(\\ - X'„) + jr(X", + X" 4 )| 

- x (X'«iS* - X'iScoa" + X" 4 »Sc a') -f r (k\ScW 
-X" 4< S* + X" 4 Sc,a")}, (33) 

after canceling terms of secondary order. 
As first approximation follows herefrom: 

Pl = Sc ^ x U _ Sc oa " - ?Sc* a >) 
m 2 \ x / 



S#X jl - Sc»(«" + ^a') 
= C (l +S l ) 

hence a maximum for the speed S, given by: 

dS u ' 



So = ^l+ Co 2 (a" + £ a') 7 - c« (a" + ^ «'). 


and equal to: 


Pi = 
1 2 

{^l+a«(^ + ^) , -^(^ / + ^« # )|. (36) 

The complete expression of the power of the main circuit ia, 
from (33) : 

Pi - ***£-{ [l - &* (</' + V)]'-ft* - M - M*;, (37) 

where fc» T b Xf k* are functions of X'i, X"i, X' 4 , X"*, as derived by 
rearranging (33). 
The Power produced in the bru*h short-circuit is: 

P 4 - [#',, /M«; 


hence, substituting (24) and (28) : 

p pS*c e X 4 ec (l - S*)y 
4 IK' ZK J 

SWe* ( 1 - S«) r . . , .. „ "I 

5'coV (1 - 8*) . . , , . „ , 
= ±r (rX 4 + xX" 4 ); 

TO 2 


hence positive, or assisting, below synchronism, retarding above 

The total Power, or Output of the motor then is: 

P = Pi + P< 

Power Output: 

p = -- 


m 2 

{ [l -&. («" + \ «') ] - 6o+S [c„ (x" 4 + ^X' 4 ) - 6t] 

- S'6, - S'co (x" 4 + r x X' 4 ) } ; (39) 

or, approximately: 

1 - 8* (a" + T - a') ] (40) 

,s^i- sco («"+;;«')} (41) 

co(i + s 2 ) 


given in synchronous watts. 

The power input into the motor, and the volt-ampere input, 
are, if: 

2 = I 2 — Jl 2, 

and : (42) 

*2 — V 2 T * 2 , J 

given by: 
Power Input: 

Po = ei\ (43) 

Volt-ampere Input: 


Pa = «,, (44) 

V - j\ (45) 


Apparent Efficiency: 

V = p , (46) 

PV = p -, (47) 


216. While excessive values of the short-circuit current under 
the commutator brushes, / 4 , give bad commutation, due to ex- 
cessive current densities under the brushes, the best commuta- 
tion corresponds not to the minimum value of I a — as the zero 
value at synchronism in the repulsion motor — but to that value 
of Ii for which the sudden change of current in the armature 
coil is a minimum, at the moment where the coil leaves the com- 
mutator brush. 

J 4 is the short-current in the armature coil during commuta- 
tion, reduced to the armature circuit, /i, by the ratio of effective 

__ short-circuited turns under brushes ,.~. 

4 total effective armature turns 

The actual current in the short-circuited coils during commuta- 
tion then, is: 

// = y, (49) 

or, if we denote: 

£ = A h (50) 

where A\ is a fairly large quantity, and substitute (24), it is: 

/'< = dWi_z_s') (51) 

Before an armature coil passes under the commutator brushes, 
it carries the current, — /i; while under the brushes, it carries 
the current, J'\\ and after leaving the brushes, it carries the cur- 
rent, +/i. 


While passing under the commutator brushes, the current in 
the armature coils must change from, — f,, to f',, or bjr: 

v, - r. + u- 

In the moment of leaving the commutator brushes, the rur- 
rent in the armature coils must change from, f, to -+- f x , or hy: 

I» = h~ /'«- (53) 

The value, /' 8 , or the current change in the armature coils 
while entering commutation, is of less importance, since during 
this change the armature coils are short-circuited by the brushes. 

Of fundamental importance for the commutation is the value, 
/„, of Ihe current change in the armature coils while leaving the 
commutator brushes, since this change has to be brought about 
hy the 'resistance of the brush contact while the coil approaches 
the edge of the brush, and if considerable, can not be comptdtad 
thereby, but the current, / B , passes as arc beyond the edge of the 

Essential for good commutation, therefore, is thai the current, 
/„, should be zero or a minimum, and the study of the commu- 
tation of the single-phase commutator thus resolves itself largely 
into an investigation of the commutation current, /„, or its abso- 
lute value, if. 

The ratio of the commutation current, i t , to the main 
current, >\, can be called the commutation constant: 

k - 

For good commutation, this ratio should l>e small or zero. 

The product of the commutation current, t„ and the speed, S, 
is proportional to the voltage induced by the break of ih<- mt- 
rent, or the voltage which maintains the arc at the edge of the 
commutator brushes, if sufficiently high, and may lie called the 
commutation vettage: 

C = Si,. (55) 

In the repulsion motor, it is, substituting (23) and [51 En 
and dropping the term with X«, as of secondary order: 

Commutation Current: 


A,c (l - S») 


Commutation Constant: 

I 1 + j ~° - ami - S 1 ) 


1 + 


= 1 - 

A<co(l - S*) 

"*" A 


Or, denoting: 

A A = a\+ ja" t ; 

substituting (32) and expanding: 

*' ZD 

I, e { 1 -c, [Sa" + (1 - S 8 ) a'J -jc [(1 - S 1 ) a", - Sa'} ) 
/i (r-5coa")+"iScoa' 

and, absolute: 



*' = m^ ( * ~ c ° (Sa " + ( l ~ ,S2) a ' 4] I * + C »M ( 1 " 5*) «"4 - Sa'\ • 



= # 

- C [Sa" + (1 - fi 1 ) a' 4 ]) 1 + C«« {(1 - .S*) a" 4 - Sa'} 4 

(1 - Scoa") 1 + S'c V 2 


Perfect commutation, or /„ = 0, would require from equation 

(58) : 

1 - c [Sa" + (1-S s )a' 4 ] =0, 
(1 - S*) a" 4 - Sa' = 0; 

1 - CoSa" 


« 4 = 

c (l-S 8 )* 

,» _ Sa ' _ i ' 

' - = = 1 — a 4. 

« 4 = 

1 - s* 


This condition can usually not be fulfilled. 
The commutation is best for that speed, S, when the commu- 
tation current, ?'„, is a minimum, that is: 



= 0; 


~{(l-c*[Sa"+(l-S*)a'<))*+Co\(l-S i )a"<-Sa') i \=0 



This gives a cubic equation in S, of which one root, < Si < 1, 
represents a minimum. 

The relative commutation, that is, relative to the current con- 
sumed by the motor, is best for the value of speed, St, where the 
commutation factor, fc, is a minimum, that is: 

£-«• (65) 

217. The power output of the repulsion motor becomes aero 
at the approximate speed given by substituting P = in the 
approximate equation (40), as: 

s.= ! 

eo(a"+ r -a') (66) 


and above this speed, the power, P, is negative, that is, the 
repulsion motor consumes power, acting as brake. 

This value, So, however, is considerably reduced by using the 
complete equations (39), that is, considering the effect of the 
short-circuit current under the brushes, etc. 

For S < 0, P < 0; that is, the power is negative, and the 
machine a generator, when driven backward, or, what amounts 
to the same electrically, when reversing either the field-circuit, 
/o, or the primary energy circuit, /j. In this case, the machine 
then is a repulsion generator. 

The equations of the repulsion generator are derived from those 
of the repulsion motor, given heretofore, by reversing the sign 
of S. 

The power, P 4 , of the short-circuit current under the brushes 
reverses at synchronism, and becomes negative above synchron- 
ism. The explanation is: This short-circuit current, /«, and a 
corresponding component of the main current, /i, are two cur- 
rents produced in quadrature in an armature or secondary, short- 
circuited in two directions at right angles with each other, and 
so offering a short-circuited secondary to the single-phase pri- 
mary, in any direction, that is, constituting a single-phase in- 
duction motor. The short-circuit current under the brushes so 
superimposes in the repulsion motor, upon the repulsion-motor 
torque, a single-phase induction-motor torque, which is positive 
Mow synchronism, zero at synchronism, and negative above 
synchronism, as induction-generator torque. It thereby lowers 



the speed, S<>, at which the total torque vanishes, and reduces 
the power-factor and efficiency. 

218. As an example are shown in Fig. 188 the characteristic 
curves of a repulsion motor, with the speed, S, as abscissa?, for 
the constants: 

Impressed voltage: e = 500 volts. 
Exciting impedance, main field: Z = 0.25 + 3 j ohms, 
cross field: Z' = 0.25 + 2.5 j ohms. 

i i i 

e -5oo volts 

L25+3J Z,-aolS*0.075i 

as+i» z, -0.02s + ot07w 












■ ; 














■; / 




Self-inductive impedance, main field: Z = 0.1 + 0.3 j ohms, 
cross field: Z» = 0.025 + 0.075johms. 
armature: Z x = 0.025 4- 0.075 j ohms, 
brush short-circuit: Z* = 7.5 + 10 j ohms. 
Reduction factor, main field: c — 0.4. 
brush short-circuit: c A = 0.04, 


Z, = 0.08 + 0.60 j ohms. 
A =0.835 - 0.014 j. 

j - a'+ja"- 1.20 + 0.02 j. 

Xi = 0.031 - 0.007;. 
X, = 0.179 + 0.087 j. 
A t = 4.475 + 2.175 j. 
A» = 0.202 - 0.010 j. 


Then, substituting in the preceding equations: 

K = (0.204 - 0.035 S) - j (0.031 + 0.328 S), 
ZK = (0.144 + 0.975 S) + j (0.604 - 0.187 S). 

Primary or Supply Current: 

. _ 500 { (1.031 - 0.179 S*) - j (0.007 + 0.087 S*) \ 
/2 ~ " ZK 

Secondary or Armature Current : 

T 500 {(1 + 0.048 5 -0.1 79 S*) +j 0.4 S - 0.087 S*)) 

II --------- 2K — : 

Brush Short-circuit Current: 

500 (1 - S s ) (0.072 - 0.035 j) 
U= . ZK ' 

and absolute: 

40 (1 - S*) 

lx = 


Commutation Factor: 

. = /(L508 S l - 0.673) 2 + (0.718 - 6.4 S - 0.704S*)* 
V (0.697 + 0.4 S- 0.014 )* 

Main E.m.f. of Rotation: 

500 5 (4.052 + 0.792 j) 
El = z _ 

Commutation E.m.f. of Rotation : 

„, 500 S* (0.4 - 4.8 j) 
E<= - ZK - • 

Power of Main Armature Circuit: 

P = 250 S (4 052 _ Q 122 s _ Q 65? S2) Jn kw 
m 2 

Power of Brush Short-circuit: 

n 49.2 S 2 (1 - S 2 ) . . 
P 4 = zlt~ — y in kw. 

Total Power Output: 

m 2 

p = ?™ ? (4.052 + 0.075 S - 0.657 S 2 - 0.197 S 8 ). 
m 2 

Torque : 

D = -°~ (4.052 + 0.075 S - 0.657 S 2 - 0.197 S 8 ), 
m 2 



These curves are derived by calculating numerical values in 
tabular form, for S = 0, 0.2, 0.4, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6, 1.8, 
2.0, 2.2, 2.4. 

As seen from Fig. 188, the power-factor, p, rises rapidly, reach- 
ing fairly high values at comparatively low speeds, and remains 
near its maximum of 90 per cent, over a wide range of speed. 
The efficiency, 17, follows a similar curve, with 90 per cent, maxi- 
mum near synchronism. The power, P, reaches a maximum of 
192 kw. at 60 per cent, of synchronism — 450 revolutions with a 
four-pole 25-cycle motor — is 143 kw. at synchronism, and van- 
ishes, together with the torque, D, at double synchronism. The 
torque at synchronism corresponds to 143 kw., the starting 
torque to 657 synchronous kw. 

The commutation factor, fc, starts with 1.18 at standstill, the 
same value which the same motor would have as series motor, 
but rapidly decreases, and reaches a minimum of 0.23 at 70 per 
cent, of synchronism, and then rises again to 1.00 at synchron- 
ism, and very high values above synchronism. That is, the 
commutation of the repulsion is fair already at very low speeds, 
becomes very good somewhat below synchronism, but poor at 
speeds considerably above synchronism : this agrees with the ex- 
perience on such motors. 

In the study of the commutation, the short-circuit current 
under the commutator brushes has been assumed as secondary 
alternating current. This is completely the case only at stand- 
still, but at speed, due to the limited duration of the short-circuit 
current in each armature coil — the time of passage of the coil 
under the brush — an exponential term superimposes upon the 
alternating, and so modifies the short-circuit current and thereby 
the commutation factor, the more, the higher the speed, and 
greater thereby the exponential term is. The determination of 
this exponential term is beyond the scope of the present work, 
but requires the methods of evaluation of transient or momentary 
electric phenomena, as discussed in "Theory and Calculation of 
Transient Electric Phenomena and Oscillations." 

B. Series Repulsion Motor 

219. As fuither illustration of the application of these funda- 
mental equations of the single-phase commutator motor, (1) to 
(6), a motor may be investigated, in which the four independent 
constants are chosen as follows: 


1. Armature and field connected in series with each other. 
That is: 

#1 + Co#o = # = «ii (67) 


Co = reduction factor of field winding to armature; that is, 

. „ field turns 

ratio of effective r — — : 

armature turns 

It follows herefrom: 

/o = coli. (68) 

2. The e.m.f. impressed upon the compensating winding is 
given, and is in phase with the e.m.f., ei, which is impressed upon 
field plus armature: 

#2 = e 2 . (69) 

That is, #2 is supplied by the same transformer or compensator 
as 6i, in series or in shunt therewith. 

3. No rotor-exciting circuit is used: 

h = 0, (70) 

and therefore: 

4. No rotor-exciting brushes, or brushes in quadrature posi- 
tion with the main-armature brushes, are used, and so: 

U - 0, (71) 

that is, the armature carries only one set of brushes, which give 
the short-circuit current, J\. 

Since the compensating circuit, e 2 , is an independent circuit, 
it can be assumed as of the same number of effective turns as 
the armature, that is, e 2 is the e.m.f. impressed upon the com- 
pensating circuit, reduced to the armature circuit. (The actual 
e.m.f. impressed upon the compensating circuit thus would be: 

«. , • compensating turns \ 
c 2Cs , where c, = ratio effective — mature turn8 ) 

220. Substituting (68) into (1), (2), (3), and (5), and (1) and 
(2) into (67), gives the three motor equations: 


e, = Zih + Z' (/, - /,) - jSZ (co/, - /«) 

+ c *Zo /i + CoZ (c /o — h), 
ei = Z t J t + Z' (Jt - /,), (73) 

= Z4i + Z(I<- Co/,) - jSZ' (/, - /,). (74) 


Substituting now: 

Y* = ~ = qua dratur e, or transformer exciting | 

it — Xa — X • — jX 3 

z + z< 


= X 4 = X' i +jV' 4 


~ = A = a' — jV = impedance ratio of the 
two quadrature (tuxes, 

Zi + c<r <Z» — Zi = Z* 

« = «l + «2, 



Adding (72? and ' 73 r and rearrangim^, ajv#* : 
< = ZJ« + [ l (Zi - jikj - IJL 

_ ,«• 


^ = ***/* + U 'A* - ;.v % - /. '- - # .! 
From (73/ follows: 


/; = /; 1 - X : - tt)\ 

(. = f 1 - x. - *>' 

From '74 foBow« 

= /< Z ~ Z. - / tJ? ~ ,,>/', - ^Z'/, 

> > 


/ 7* 


in its eratafcT^oi:. That !•?>, wh*n *>uU*ututiiAg '7^ jij '70,, // 
can be drr>pj^o : 






Hence, (80) substituted in (79) gives: 

= U (Z + Z*) ~ CoZh + jSet, 




Hence : 

and actual value of short-circuit current: 

r. -».{/.-£). 



b = "°, a fairly large quantity, and 

C\ = reduction factor of brush short-circuit 
to armature circuit. 

The commutation current then is: 

/. - /l - /'4 

»/l(l -6X4) + 

jSetb \< 
c Q Z 

Substituting (81) and (80) into (77), gives: 

/i = 

or, denoting: 

e 1 ^JSt\i (c - j-S)_— jf_X»_ 

Z A 8 - j«Sco ~ XiCo (c - jS) + \ 2 A 

it is: 

K = A* - jSco - X 4 c (c - j/S) + X 2 A, 

. _ ejl -jSt\i(co -jS) -t\i\ 
(1 - Zk 

It is, approximately : 

A 3 = 


— /._2 

hence : 

z = 
x, = 0, 

7C = c (l — C0X4) (c -j<S), 
= efl — JSOn (c - jS) 1 
'' c„Z(l -c„X 4 )(co - j»S)" 

c Z(l — CoX<) i Co — /<S 

' 1 ) 










Substituting now (85) respectively (87), (88) into (78), (81) (84), 
and into: 

r,=i,SZ(co/i-/4), 1 

#\ - jSZ' (/, - /,), 
gives the 

Equation* of the Series Repulsion Motor: 

K = A z - jSc - \iC (c — -jS) + \sA, 

K = Co (1 - c X 4 ) (Co - jS). 

Inducing, or Compensator Current: 

_ e { 1 - i»Mc - jS) - (1 +_0_X 2 | , eta - X 2 ) 
/2 Z# " """ " Z' 



j»X 4 « 



c Z (1 - C0X4) (co - jS) c Z (1 - C0X4) Z' 

Armatore, or Secondary Current: 

. e\l -jSikifa -jS ) -<\» ) 
tl ~ — ZK~ ' 


j __ _ e f 1_ .« I 

' l " 60Z (f- C0X4) 1 Co - jS ^ 4 / ' 

Brush Short-circuit Current: 

T cX 4 

#4 — wr\ : 

f 1 

Z(l - C0X4) lco -jS 

cX 4 j 1 

- jSt(l +X4- c X 4 )!» 

/4 = 

Z(l -c-oX 4 ) I c - jS 


Commutation Current: 






/. -- 

1 - \ t b 

c Z (1 — c X<) I Co — jS 

!: — ,^+is<x 4 [(6- 1) 

+ 6X4(1 - Co)]}' 


h = 

e (1 ■ 

c Z(l — CoX<)lco 

^ + jSt\ i b j 





Main E.m.f. of Rotation: 



f 1 -X 4 



^ + jS<X«*(l -co)} 

Wl = JSe (1 





(1 - coX*) (c - jS) 
Quadrature E.m.f. of Rotation: 

E\ = + jSte. 

Power Output: 

P - P, + P 4 

- [#'1, /J 1 + [^'4, /d l . 
Power Input: 

Po = [«i, /1] 1 + [62, /2] 1 . 
Volt-ampere Input: 

P = ea'i + e*i2 

= e{(l - 0*i + ^'2}, 

where the small letters, i\ and it, denote the absolute values of 
the currents, /1 and / 2 . 

When t'i and i 2 are derived from the same compensator or 
transformer (or are in shunt with each other, as branches of the 
same circuit, if e\ = 62), as usually the case, in the primary cir- 
cuit the current corresponds not to the sum, {(1 — i x + tit] of 
the secondary currents, but to their resultant, [(1 — /1 + f/2] 1 , 
and if the currents, J\ and /2, are out of phase with each other, 
as is more or less the case, the absolute value of their resultant 
is less than the sum of the absolute values of the components. 
The volt-ampere input, reduced to the primary source of power, 
then is: 

P ao = e[(l - /1 + th\\ (99) 


*■ <iq ^s. 1 a» 


From these equations then follows the torque: D = ■», the 


power-factor, p = p , etc. 

These equations (90) to (99) contain two terms, one with, and 
one without t = - , and so, for the purpose of investigating the 


effect of the distribution of voltage, e, between the circuits, e\ 
and 62, they can be arranged in the form: F = K\ + tK*. 


t = 0, 

that is, all the voltage impressed upon the armature circuit, and 
the compensating circuit short-circuited, these equations are 
those of the inductively compensated series motor. 


that is, all the voltage impressed upon the compensating or in- 
ducing circuit, and the armature circuit closed in short-circuit, 
that is, the armature energizing the field, the equations are those 
of the repulsion motor with secondary excitation. 


a reverse voltage is impressed upon the armature circuit. 

Study of Commutation 

221. The commutation of the alternating-current commutator 
motor mainly depends upon : 

(a) The short-circuit current under the commutator brush, 

which has the actual value: J\ = — * High short-circuit current 


causes arcing under the brushes, and glowing, by high current 

(6) The commutation current, that is, the current change in 
the armature coil in the moment of leaving the brush short-cir- 
cuit, J* = /1 — l\. This current, and the e.m.f. produced by 
it, SI ot produce sparking at the edge of the commutator brushes, 
and is destructive, if considerable. 

(a) Short-circuit Current under Brushes 

Using the approximate equation (93), the actual value of the 
short-circuit current under the brushes is: 

*'« - Jo"- coxo I trhs - ** } ■' ( 100 > 


Co 1 

b = — , or t = reduction factor of short-circuit under brushes, 

C4 ' 


to field circuit, that is: 

, _ number of fi e ld turns 

number of effective short-circuit turns' 
hence a large quantity. 

The absolute value of the short-circuit current, therefore, h: 

call - cXJ (c* + 8*) 

hence a minimum for that value of I, where: 
/ = co J + SMI - ( (c,, 1 + S 2 ))* = mi 
= 1 - ( (c 1 + S s ) = 0, hence, 

t -- 


" a,' + S>' 

8 " >f" * 

That is, t = — = j,- v" — j gives minimum short-circuit cur- 
rent at speed, S, and inversely, speed 5 = »/- — e#*, gives 

minimum short-circuit current at voltage ratio, (. 

For ( =■ 1, or the repulsion motor with secondary excitation, 
the short-circuit current is minimum at speed, S = y/\ — c a '. or 
somewhat below synchronism, and is j'» = - , while in ihe re- 
pulsion motor with primary excitation, the short-circuit current 
is a minimum, and equals zero, at synchronism S = 1. 

The lower the voltage ratio, t = ! , the higher is the speed, S, 
at which the short-circuit current reaches a minimum. 

The short-circuit current, f\, however, is of far less importance 
than the commutation current, /,. 

(6) Commutation Current 

222. While the value, J'„ = [\ + f t , or the current change in 
the armature coils while entering commutation, is of niiuor im- 
portance, of foremost importance for good commutation is that 
the current change in the armature coils, when leaving the short- 
circuit under the brushes: 

h = h- /'. (103) 

is zero or a minimum. 


Using the approximate equation of the commutation current 
(94), it is: 

/. - 

c Z(l — c X 4 ) I Co — jS 

[ l " X ?* + jStkJb 

c Z(l - c X 4 )(co -jS) 

and, denoting: 

X 4 = X\ + j\"<, 
it is, expanded : 


; -r { 1 - X 4 6 + jS (co - jS) t\ A b] ; (104) 


-m HI - X'lfr + Stb(S\\ - C X" 4 )] 

CoZ(l-c X 4 )(co+iS) 

- j [\"<b - S» (c X' 4 + SX" 4 )] ) ; (105) 
hence, absolute: 

h = 

C<>z[l - C X 4 ]-y/ Co 2 + fi[2 

\[1 - X' 4 fc + Sft(flX # 4 - c X" 4 )] 2 + [X" 4 fc - Stb(c \\ + SX" 4 )] 2 , 

where [1 — c X 4 ] denotes the absolute value of (1 — c X 4 ). 

The commutation current is zero, if either S = 00 } that is, 
infinite speed, which is obvious but of no practical interest, or 
the parenthesis in (105) vanishes. 

Since this parenthesis is complex, it vanishes when both of 
its terms vanish. This gives the two equations: 

1 - X' 4 fc + Stb (SX' 4 - c X" 4 ) = 0, 
X" 4 6 - Stb (c X' 4 + SX" 4 ) = 0. 


From these two equations are calculated the two values, the 
speed, S, and the voltage ratio, t, as: 


on — 

h = 

Soto = 

C» (6X 4 * - X' 4 ) 



c W(6X 4 * - X' 4 ) ' 
X" 4 



For instance, if: 

Z =0.25 + 3j, 
Z< = 5 + 2.5 j: 




u ' ' 


= 0.4, 


- 0.04; 



- 10; 

and herefrom 


- 2.02, 

- 0.197 

- 0.248 j = X', + j\"„ 

that is, at about double synchronism, for e s = te = 0.197 e, or 
about 20 per cent, of e, the commutation current vanishes. 

In general, there is thus in the series repulsion motor only one 
speed, Su, at which, if the voltage ratio has the proper value, (o, 
the commutation current, i„, vanishes, and the commutation i.- 
perfect. At any other speed some commutation current is left, 
regardless of the value of the voltage ratio, (. 

With the two voltages, e\ and e t , in phase with each other, t lie 
commutation current can not be made to vanish at any desired 
speed, S. 

223. It remains to be seen, therefore, whether by a phase dis- 
placement between e t and e», that is, if ei is chosen out of phase 
with the total voltage, e, the commutation current can be made 
to vanish at any speed, S, by properly choosing the value of the 
voltage ratio, and the phase difference. 

Assuming, then, ej out of phase with the total voltage, e, hence 
denoting it. by: 

# s = et (cos 0s - j sin (?,), (109) 

the voltage ratio, (, now also is a complex quantity, and expressed 

T = ^ = t (cob 0i - j sin $i) = t' - jt". (110) 

Substituting (110) in (105), and rearranging, gives: 

'• " cza-cx.Xc-.M I[1 - v ' fc + m w ' - **"•> 

+ Xt"b ta,X' ( + SX",)| - j|K",6 - St'b (c„X', + SX",i 
+ S!"c.(SX'. - e„X",)]]; (HI) 

and this expression vanishes, if: 

1 - X',6 + Sfb (SX\ - c,X",) + .Sf'o ta,X', + .SX",) - 0, I 
X'Vi - Sl'b (e.X', + SX",) + Sf'b (SX', - oX",) - 0:| 


and herefrom follows: 
, = Sb A 2 - S\\ + c X" 4 

S6X 4 2 (S 2 + Co 2 ) Co 2 + S 2 

1 - 

S\ 4 — CqX 4 

„ = Co6X 4 2 - CqX / 4-iSX // 4 _ 1 

S6X 4 2 0S 2 +co 2 ) 
or approximately: 

V = 
t" = 

SfcX 4 2 
Co Cq\\ + SX" 4 ] 
Co 2 + S 2 \ S Sb\ A 2 



Co 2 + S 2 ' 



S (co 2 + S 2 ) 

t" = substituted in equation (112) gives S = So, the value 
recorded in equation (108). 

It follows herefrom, that with increasing speed, S, t f and still 
more t", decrease rapidly. For S = 0, V and t" become infinite. 
That is, at standstill, it is not possible by this method to produce 
zero commutation current. 

The phase angle, 2 , of the voltage ratio, T = t' — jt", is given 


, „ _ ^ _ C06X4 2 - Cp\' A - SX" 4 . 
tan 2 - t , - Sfex ^ 2 _ sx ,^ + CqVV 

rearranged, this gives: 

Co sin 2 + S cos 2 &X4 2 — W . 


and, denoting: 

Co sin 62 — S sin 2 

S f 

— = tan a, 

Co ' 




where a may be called the "speed angle," it is, substituted in 

6X4 2 - X\ 

tan (02 + <r) = 

hence : 


X" 4 
= constant; 

02 + o- = 7, 
2 — y __ ffm 
— r-77 is a large quantity, hence 7 near 90°. 

A 4 



<r is also near 90° for all speeds, S, except very slow speeds, since 
in (116) Co is a small quantity. 



Hence S-i is near zcru for all except very low speeds. 

For very low, a is small, and $2 thus large and positive. 

That is, the voltage, £ 2 , impressed upon the compensating 
circuit to get negligible commutation current, must l>e approxi- 
mately in phase with e for all except low speeds. At low speeds, 
it must lag, the more, the lower the speed. Its absolute value 
is very large at low speeds, but decreases rapidly with increasing 
speed, to very low values. 

For instance, let, as before: 

X, - 0.304 - 0.248 j, 
c„ = 0.4, 

fl, = for <t = 79°; hence, by (116), S„ = 2.02, or double syn- 
chronism. Above this speed, 2 is leading, but very small, since 
the maximum leading value, for infinite speed, S = <* . is given 
by d = 90°, as, : = - 11°. Below the speed, So, 0i is positive, 
or lagging; 

for S = 1, it isff = 68°, S = +11°, hence still approximately 
in phase; 

for S = 0.4, it is o- = 45", S = 34°; hence Et is still nearer in 
phase than in quadrature to e. 

The corresponding values of T = t' + t"- are, from (112): 

S - 2.02, e 5 = 0, T = 0.197, t = 0.197, 

B - \, 81 = -+-11°, T = 0.747 +0.140 j, I = 0.760, 
S = 0.4, 2 = 34°, T = 3.00 -2.00 j, ( = 3.61. 

224. The introduction of a phase displacement between tW 
compensating voltage, E 5 , and the total voltage, e, in general is 
more complicated, and since for all but the lowest epeedfl tbt 
required phase displacement, 0;, is small, it is usually sufficient 
to employ a compensating voltage, e., in phase with e. 

In this case, no value of / exists, which makes the commutation 
current vanish entirely, except, at the speed, So. 

The problem then is, to determine for any speed, S, that value 


of the voltage ratio, t, which makes the commutation current, i 99 
a minimum. This value is given by: 

~dt = °» (120) 

where i is given by equation (106). 

Since equation (106) contains t only under the square root, 
the minimum value of i u is given also by: 

where : 

K = [1 - b\' A + Sib {S\' 4 - c X" 4 )] 2 + [fcX" 4 - Sib (c \'i + S\"<)]\ 

Carrying out this differentiation, and expanding, gives: 

/ = S6\ 4 2 ~ S\'< + Co*"* = 1 L _ SX< - c X'^l mn 
1 Sb\J (co 2 + S 2 ) Jco 2 + S 2 1 " S6X 4 2 " f u ; 

This is the same value as the real component, t f , of the complex 
voltage ratio, T h which caused the commutation current to 
vanish entirely, and was given by equation (112). 

It is, approximately: 

' - cTW (122) 

Substituting (121) into (105) gives the value of the minimum 
commutation current, i 0% . 

Since the expression is somewhat complicated, it is preferable 
to introduce trigonometric functions, that is, substitute: 

tan 6 - *" 4 > (123) 

A 4 

where 6 is the phase angle of X4, and therefore: 

J/ = J 4 " !' I (124) 

X 4 =* X 4 cos 6, I 

and also to introduce, as before, the speed angle (116): 

tan a = ■> 



v - \/c« 2 »; , 


,S - 7*111*, I (126) 

Ct, «■ 7 com 0, J 



Substituting these trigonometric values into the expression 
(121) of the voltage ratio for minimum commutation current, 
it is: 

1_ __ sin (a — 6) 

r Sbq x< < 127) 

Substituting (117) into (106) and expanding gives a relatively 
simple value, since most terms eliminate: 

Jg = e {[cos 2 (a — 6) + b\ (sin a sin (a — 6) — cos 6)] 

+ j [ sin (a — 5) cos (a — 6) — 6X4 (sin a cos (cr — 6) — sin 6)]} 

**o = 




CoZ(l - C0X4, (c +JS) 

and the absolute value: 

e ( cos (<r — 6) — 6X 4 cos <r) . 

coz [ 1 - C0X4] Vc? + S* ' 
or, resubstituting for a and 5: 

. eJSVV^Co (X 4 2 fe - X/) } 

lf ° coz [1 - CoX" 4 ] (co 2 + S 2 ) ' 
From (129) and (130) follows, that i g = 0, or the commutation 

current vanishes, if: 

cos (a — 6) — 6X 4 cos a = 0, (131) 


SX"4 - co (X 4 2 fe - X' 4 ) = 0. 
This gives, substituting, X" 4 = y/\f — XV, and expanding: 

X 4 

X , t = 

Co 2 + S 2 

cos (a — 6) = 
From (131) follows: 

{6X4C0 2 ± SVS 2 -"co 2 "(fe 2 X4 2 - 1)), 


Vco 2 + S 2 


cos (<r — 6) = 6X4 cos <r. 

Since cos (c — 6) must be less than one, this means: 

6X4 cos<r < 1, 


X 4 < 

b cos 


or, inversely: 

x ^VcV+S*' 
A4 < y 

S > Co V6 2 X4 2 - 1. 



That is: 

The commutation current, i 0i can be made to vanish at any 
speed, S, at given impedance factor, X 4 , by choosing the phase 
angle of the impedance of the short-circuited coil, 5, or the resist- 
ance component, X', provided that X 4 is sufficiently small, or the 
speed, S, sufficiently high, to conform with equations (133). 

From (132) follows as the minimum value of speed, S, at which 
the commutation current can be made to vanish, at given X 4 : 

Si = Co Vfe 2 x 4 2 - 1, 


v - 1 - 


X". = ^ - I • 

For high values of speed, S, it is, approximately: 

cos (<r — 5) = 0, 
a - 6 = 90°, 

tan a = — ; 


hence: <r = 90° 

6 = ' 

X 4 = X 4 . 

That is, the short-circuited coil under the brush contains no 
inductive reactance, hence: 

At low and medium speeds, some inductive reactance in the 
short-circuited coils is advantageous, but for high speeds it is 
objectionable for good commutation. 

225. As an example are shown, in Figs. 189 and 192, the char- 
acteristic curves of series-repulsion motors, for the constants: 

Impressed voltage: e = 500 volts, 

Exciting impedance, main field: Z = 0.25 + 3j ohms, 

Exciting impedance, cross field: Z' = 0.25 +2.5 j ohms, 
Self-inductive impedance, main 

field : Z = 0. 1 + 0.3 j ohms, 
Self-inductive impedance, cross 

field : Z % = 0.025 + 0.075 j ohms, 















e-500 VOLTS -f»-o 

z- o.;5* si Zi- 002s + o.07sj 

j!=0.IS+2.Sj ZJ-0.O2S+ 0,075) 





-0.4 C •- 0.04 















Z= 0.25+3 j Zi=0.025+0.075j 

Z' 0.;S+2_5J Z,-0.O:5 4-0.O75j 
ZrO.1 + 0.3J Z,= 7.S + 1D) 



l\" I - 









H ; 




... a 














C- BOO VOLTS -f = 0.5 



0.25»3j Z, -0 025+0075. 






















;t -- 








Fig. 191. — Scries repulsion motor. 


Self-inductive impedance arma- 
ture: Zi= 0.025 + 0.075 j ohms, 
Self-inductive impedance, brush 

short-circuit: Z A = 7.5 + lOjohms, 

Reduction factor, main field: c„ = 0.4, 

brush short-circuit c* = 0.04; 
that is, the same constants as used in the repulsion motor, 
Fig. 188. 

Curves are plotted for the voltage ratios; 
t = 0: inductively compensated series motor, Fig. 189. 
t = 0.2: series repulsion motor, high-speed, Fig. 190. 
( = 0.5: series repulsion motor, medium-speed, Fig. 191. 
( = 1.0: repulsion motor with secondary excitation, low-speed, 
Fig. 192. 




Z'"0.!i*2.Sj Z,-0-025t0.07Sj 



s v 

C. - O.J C, 0.04 













— . 

























Fig. 192— Rcpulsio 

secondary excitation. 

is, from above constants 

Z 3 -Zi + c»(Z»+Z) 

- 0.08 + 0.60 j. 


- 0.202 - 0.010}. 


- 0.835 - 0.014 j. 


- 0.031 - 0.007 j. 

x, - - z - 
' Q + Z, 

- 0.179 + 0.087 j. 


- 10. 



Hence, substituting into the preceding equations: 

(90) ZK = Z, - jSeoZ - X,CoZ (e, - jS) + X 3 Z' 

= (0,160 + 0.975 S) +j {0.590 - 0.187 S), 

(92) /, = £, - ^ US^ic-jS) + U\ 

= IR + ^iC-°- 031 + 0035 ' s - |>i: '' s 

- j" ( - 0.007 + 0.072 S + 0.087 #') | , 

(91) /, = /, (0.969 + 0.007 j) + et (0.010 - 0.096 j), 

a (0.072 +0.035;)+ 3d [ (0.016 - 0.072 6')-j0.045+ 0.035g) | 

h = 


226. Ah seen, these four curves are very similar to each other 
and to those of the repulsion motor, with the exception of the 
commutation current, i,, and commutation factor, k = -?- 

The commutation factor of the compensated series motor, 
that is, the ratio of current change in the armature coil while 
leaving the brushes, to total armature current, is constant in the 
series motor, at all speeds. In the series repulsion motors, the 
commutation factor, h, starts with the same value at standstill, 
as the series motor, but decreases with increasing speed, thus 
giving a superior commutation to that of the series motor, reaches 
a minimum, and then increases again. Beyond the minimum 
commutation factor, the efficiency, power-factor, torque and out- 
put of the motor first slowly, then rapidly decrease, due to the 
rapid increase of the commutation losses. These higher values, 
however, are of little practical value, since the commutation is 

The higher the voltage ratio, (, that is, the more voltage is 
impressed upon the compensating circuit, and the less upon the 
armature circuit, the lower is the speed at which the commuta- 
tion factor is a minimum, and the commutation so good or perfect. 
That is, with ( = 1 , or the repulsion motor with secondary ex- 
citation, the commutation is best at 70 per cent, of synchronism, 
and gets poor above synchronism. With t = 0.5, or a series 
repulsion motor with half the voltage on the compensating, half 
on the armature circuit, the commutation is best just above syn- 
chronism, with the motor constants chosen in this instance, and 


gets poor at speeds above 150 per cent, of synchronism. With 
t = 0.2, or only 20 per cent, of the voltage on the compensating 
circuit, the commutation gets perfect at double synchronism. 

Best commutation thus is secured by shifting the supply vol- 
tage with increasing speed from the compensating to the arma- 
ture circuit. 

t > 1, or a reverse voltage, — ei, impressed upon the armature 
circuit, so still further improves the commutation at very low 

For high values of t, however, the power-factor of the motor 
falls off somewhat. 

The impedance of the short-circuited armature coils, chosen 
in the preceding example: 

Z K = 7.5 + 10 j, 

corresponds to fairly high resistance and inductive reactance in 
the commutator leads, as frequently used in such motors. 

227. As a further example are shown in Fig. 193 and Fig. 194 
curves of a motor with low-resistance and low-reactance com- 
mutator leads, and high number of armature turns, that is, low 
reduction factor of field to armature circuit, of the constants: 

Z 4 = 4 + 2j; 
hence : 

X 4 = 0.373 + 0.267 j, 

Co = 0.3, 

d = 0.03, 

the other constants being the same as before. 

Fig. 193 shows, with the speed as abscissae, the current, torque, 
power output, power-factor, efficiency and commutation current, 
i 0f under such a condition of operation, that at low speeds t = 1.0, 
that is, the motor is a repulsion motor with secondary excita- 
tion, and above the speed at which t = 1.0 gives best commuta- 
tion (90 per cent, of synchronism in this example), t is gradually 
decreased, so as to maintain i g a minimum, that is, to maintain 
best commutation. 

As seen, at 10 per cent, above synchronism, i g drops below t, 
that is, the commutation of the motor becomes superior to that 
of a good direct-current motor. 

Fig. 194 then shows the commutation factors, fc = -?> of the 



i i 1 1 i i i i i i 1 1 


Z =0.25 +3 j OHM Z,-0.O26 + 0.079 J0"»t 

\ ,- 




c. 0.3 







. i 




















1 1 






























a. 00 


\ V 














sj /" 


s . 










— ■ 









O.ffl 0.1O 0.60 0.80 l.Of 


different motors, all under the assumption of the same constants: 

Z = 0.25 + Sj, 
Z' = 0.25 +2.5j, 
Z = 0. 1 4- 0.3 j, 
Z 2 = 0.025 + 0.075 j, 
Zi = 0.025 + 0.075 j, 
Z 4 = 4 + 2 j, 

Co = 0.3, 

c 4 = 0.03. 

Curve I gives the commutation factor of the motor as induct- 
ively compensated series motor (t = 0), as constant, k = 3.82, 
that is, the current change at leaving the brushes is 3.82 times 
the main current. Such condition, under continued operation, 
would give destructive sparking. 

Curve II shows the series repulsion motor, with 20 per cent, of 
the voltage on the compensating winding, t = 0.2; and 

Curve III with half the voltage on the compensating winding, 
t = 0.5. 

Curve IV corresponds to t = 1, or all the voltage on the com- 
pensating winding, and the armature circuit closed upon itself: 
repulsion motor with secondary excitation. 

Curve V corresponds to t = 2, or full voltage in reverse direction 
impressed upon the armature, double voltage on the compen- 
sating winding. 

Curve VI gives the minimum commutation factor, as derived 
by varying t with the speed, in the manner discussed before. 

For further comparison are given, for the same motor 

Curve VII, the plain repulsion motor, showing its good com- 
mutation below synchronism, and poor commutation above 
synchronism; and 

Curve VIII, an overcompensated series motor, that is, con- 
ductively compensated series motor, in which the compensating 
winding contains 20 per cent, more ampere-turns than the arma- 
ture, so giving 20 per cent, overcompensation. 

As seen, overcompensation does not appreciably improve 
commutation at low speeds, and spoils it at higher speeds. 

Fig. 194 also gives the two components of the compensating 
e.m.f., E 2 , which are required to give perfect commutation, or 
zero commutation current: 




Substituting these trigonometric values into the expression 
(121) of the voltage ratio for minimum commutation current, 
it is: 

# _ _1 __ sin (a — b) 

1 ~ .2 

01 x ( 127 ) 

q* Sbq\t v 

Substituting (117) into (106) and expanding gives a relatively 
simple value, since most terms eliminate: 

Jg = e {[cos 2 (a — 6) + b\ (sin a sin (a — 6) — cos 6)] 

+ j [ sin (a — 6) cos (a — 5) — 6X4 (sin <r cos (a — 6) — sin 6)]} 




CoZ (1 - C0X4, (c + jS) 

and the absolute value: 

e ( cos (q- — 6) — 6X4 cos <r) . 

l '° = Coz[l - CoXjv^ +"£*"' 
or, resubstituting for <r and 5: 

. e{5X // 4-c ( X4 2 fe-X4 / )} 

*'° Coz [1 - CoXJ (co 2 + S 2 ) " 

From (129) and (130) follows, that i gQ = 0, or the commutation 

current vanishes, if: 

cos (<r — 6) — 6X 4 cos a = 0, (131) 


SX" 4 - Co (X 4 2 fc - X\) = 0. 
This gives, substituting, X" 4 = VX4 2 — XV, and expanding: 


X\ - ^rj-gi 1&W ± S VS 2 - Co 2 (6 2 XV - 1)}, 

6 X4C0 
cos (, - I) - ^qrii' 

From (131) follows: 

cos (a — 6) = 6X4 cos cr. 

Since cos (c — 8) must be less than one, this means: 

6X 4 cos<r < 1, 


or K 

X 4 < 

b cos 


or, inversely: 

x ^Vc 2 + s 2 ' 

A4 < 1 > 

c b 

S > c V& 2 X 4 2 - 1. 



That is: 

The commutation current, i 0} can be made to vanish at any 
speed, S y at given impedance factor, \ 4 , by choosing the phase 
angle of the impedance of the short-circuited coil, 5, or the resist- 
ance component, X', provided that X 4 is sufficiently small, or the 
speed, S f sufficiently high, to conform with equations (133). 

From (132) follows as the minimum value of speed, S, at which 
the commutation current can be made to vanish, at given X 4 : 

Si = Co V6 2 X 4 * - 1, 

x « ~b' 


x "< - V x *' - I- 

For high values of speed, S, it is, approximately: 

cos (<r — 8) = 0, 
<r - 6 = 90°, 

tan <r = - ; 

hence: a- = 90° 

X 4 = X4. 

That is, the short-circuited coil under the brush contains no 
inductive reactance, hence: 

At low and medium speeds, some inductive reactance in the 
short-circuited coils is advantageous, but for high speeds it is 
objectionable for good commutation. 

225. As an example are shown, in Figs. 189 and 192, the char- 
acteristic curves of series-repulsion motors, for the constants: 

Impressed voltage: e = 500 volts, 

Exciting impedance, main field : Z = 0.25 + 3j ohms, 

Exciting impedance, cross field: Z'= 0.25 +2.5 j ohms, 
Self-inductive impedance, main 

field : Z = 0. 1 + 0.3 j ohms, 
Self-inductive impedance, cross 

field: Z 2 = 0.025 + 0.075 j ohms, 


• I-- .11 . r .tii.i* * s 

* -* • . »" " .»- -.I*.". ^f-n**r:irr-i n •> 
• - ... -•... r-'STfiiiT 'irr^r^ i: ;. 

-.-- * * - ..'•"• *:;<- iiicnc-r * 4 \t- -nft--:. ■- 

. i j- !"!.> ••XDi»nt*:i w :^i --rni : 

* • . ■ . - .• --irr'UTt**: v ■ r:t* iTnn:' 

- • -- :- v. : ; i> r: :i^-rr::t";n^- 

'■-r '!i;i»»*!>:iTii«r:. ■ r.u* - 

- r • -. !•.«*.. 1* .'4. -J.* 1 ! .« 

■ .". ^ if: : "»-:inn"i *•■ ••::*r . 
:•■• r* »i "rrtiiMi-!'.! i!:^r:.»:; 

v^ %%il 

- :<:■:»■*".«•!] »i ■•iiiiiim:;!":. 
• ■ •-.'iir' 1 '* i 'fi:i!ni;'; 
• -:.' i: :: -i::u-" "■• - -je 
" >• t:**:::i: I2*»- ■•irr-rr 

■-■\" .>*:•?- i.;rr'-x::iiar.-»i 

. • . " ■ rrv i. ::: 'hi- t 
•■-- ■ " ..- • '::..'■ ■ r " ■i.'i-ri; 

* " .' ""•" ! i i iT -»i .]"- ■■'.r 

■ • • 


Choosing the e.m.f., E 2f impressed upon the compensating 
winding in phase with, and its magnetic flux, therefore in quad- 
rature (approximately), behind the main field, gives a com- 
mutation in the repulsion and the series repulsion motor which 
is better than that calculated from paragraphs 221 to 224, for all 
speeds up to the speed of best commutation, but becomes in- 
ferior for speeds above this. Hence the commutation of the 
repulsion motor and of the series repulsion motor, when con- 
sidering the self-induction of commutation, is superior to the 
calculated values below, inferior above the critical speed, that 
is, the speed of minimum commutation current. The com- 
mutation of the overcompensated series motor is superior to the 
values calculated in the preceding, though not of the same 
magnitude as in the motors with quadrature commutating flux. 

It also follows that an increase of the inductive reactance of the 
armature coil increases the exponential and decreases the alter- 
nating term of e.m.f. and therewith the current in the short- 
circuited coil, and therefore requires a commutating flux earlier 
in phase than that required by an armature coil of lower reac- 
tance, hence improves the commutation of the series repulsion 
and the repulsion motor at low speeds, and spoils it at high 
speeds, as seen from the phase angles of the commutating flux 
calculated in paragraphs 221 to 224. 

Causing the armature current to lag, by inserting external 
inductive reactance into the armature circuit, has the same 
effect as leading commutating flux: it improves commutation at 
low, impairs it at high speeds. In consequence hereof the com- 
mutation of the repulsion motor with secondary excitation — 
in which the inductive reactance of the main field circuit is in 
the armature circuit — is usually superior, at moderate speeds, 
to that of the repulsion motor with primary excitation, except 
at very low speeds, where the angle of lag of the armature cur- 
rent is very large. 



change of current in the armature coil when passing under the 
brush, superimposes upon the e.m.f. generated in the short- 
circuited coil, and so on the short-circuit current under the 
brush, and modifies it. the more, the higher the speed, that is, 
I lie [juicier the current change. Tiiis exponential term of e.m.f. 
generated in the armature coil short-circuited by the commutator 
brush, is the so-called "e.m.f. of self-induction of commutation." 
It exists in direct-current motors as well as in alternating-current 
motors, and is controlled by overcompensation, that is, hj i 
oommutating field in phase with the main field, and approxi- 
mately proportional to the armature current. 

The investigation of the exponential term of generated e.m.f. 
and of short-circuit current, the change of the commutation 
current and commutation factor brought about thereby IBd 
the study of the conimutating field required to control this 
exponential term leads into the theory of transient phenomena. 
that is, phenomena temporarily occurring during and immedi- 
ately after a change of circuit condition.' 

The general conclusions are: 

The control of the e.m.f. of self-induction of commutsti I 

the single-phase commutator motor requires a COmmutatlDf 
field, that is, a field in quadrature position in space to the mam 
field, approximately proportional to the armature current Ittd 
in phase with the armature current, hence approximately in 
phase with the main field. 

Since the conimutating field required to control, in the arma- 
ture coil under the commutator brush, the e.m.f. of alternation 
of the main field, is approximately in quadrature behind I he 
main field — and usually larger than the field controlling the 
e.m.f. of self-induction of commutation — it follows thai Mm 
total conimutating field, or the quadrature flux required to give 
best commutation, must be ahead of the values derived in 
paragraphs 221 to 224. 

As the field required by the e.m.f. of alternation in the -)i"ii- 
circuited coil was found to lag for speeds below the speed of brsl 
commutation, and to lead above this speed, from the poatMl 
in quadrature behind the main field, the total GOmmutatiag 
field must lead this field controlling the e.m.f. of alternation, 
and it follows: 

'See "Theory ami Calculations of Transient Electric Phenomena and 
Oscillations," Sections I and II, 


Choosing the e.m.f., E 2 , impressed upon the compensating 
winding in phase with, and its magnetic flux, therefore in quad- 
rature (approximately), behind the main field, gives a com- 
mutation in the repulsion and the series repulsion motor which 
is better than that calculated from paragraphs 221 to 224, for all 
speeds up to the speed of best commutation, but becomes in- 
ferior for speeds above this. Hence the commutation of the 
repulsion motor and of the series repulsion motor, when con- 
sidering the self-induction of commutation, is superior to the 
calculated values below, inferior above the critical speed, that 
is, the speed of minimum commutation current. The com- 
mutation of the overcompensated series motor is superior to the 
values calculated in the preceding, though not of the same 
magnitude as in the motors with quadrature commutating flux. 

It also follows that an increase of the inductive reactance of the 
armature coil increases the exponential and decreases the alter- 
nating term of e.m.f. and therewith the current in the short- 
circuited coil, and therefore requires a commutating flux earlier 
in phase than that required by an armature coil of lower reac- 
tance, hence improves the commutation of the series repulsion 
and the repulsion motor at low speeds, and spoils it at high 
speeds, as seen from the phase angles of the commutating flux 
calculated in paragraphs 221 to 224. 

Causing the armature current to lag, by inserting external 
inductive reactance into the armature circuit, has the same 
effect as leading commutating flux: it improves commutation at 
low, impairs it at high speeds. In consequence hereof the com- 
mutation of the repulsion motor with secondary excitation — 
in which the inductive reactance of the main field circuit is in 
the armature circuit — is usually superior, at moderate speeds, 
to that of the repulsion motor with primary excitation, except 
at very low speeds, where the angle of lag of the armature cur- 
rent is very large. 


230. With a sine wave of alternating voltage, and the com- 
mutator brushes set at the magnetic neutral, that is, at right 
angles to the resultant magnetic flux, the direct voltage of a syn- 
chronous converter is constant at constant impressed alternating 
voltage. It equals the maximum value of the alternating voltaga 
between two diametrically opposite points of the commutator, 
or "diametrical voltage," and the diametrical voltage is twice 
the voltage between alternating lead and neutral, or star or J 
voltage of the polyphase system. 

A change of the direct voltage, at constant, impressed alter- 
nating voltage (or inversely), can be produced: 

Either by changing the position angle between the eiuimjuia- 
tor brushes and the resultant magnetic flux, so that the direct 
voltage between the brushes is not the maximum diametrical 
alternating voltage but only a part thereof. 

Or by changing the maximum diametrical alternating voltage, 
at constant effective impressed voltage, by wave-shape distortion 
by the superposition of liigher harmonics. 

In the former case, only a reduction of the direct voltage lx*- 
low the normal value can lie produced, while in the latter case 
an increase as well as a reduction can be produced, an increase 
if the higher harmonies are in phase, and a reduction if the higher 
harmonics are in opposition to the fundamental wave of the dia- 
metrical or Y voltage. 

A. Variable Ratio by a Change of the Position Angle between 
Commutator Brushes and Resultant Magnetic Flux 

231. Let, in the commutating maclane shown diagrammatic- 
ally in Fig. 195, the potential difference, or alternating voltage 
between one point, a, of the armature winding and the neutral, 
(that is, the 1' voltage, or half the diametrical voltage) be repre- 
sented by the sine wave, Fig. 197. This potential difference is 
a maximum, e, when a stands at the magnetic neutral, at A or Ji. 




If, therefore, the brushes are located at the magnetic neutral, 
A and B, the voltage between the brushes is the potential differ- 
ence between A and B, or twice the maximum Y voltage, 2 c, 
as indicated in Fig. 197. If now the brushes are shifted by an 
angle, r, to position C and D, Fig. 196, the direct voltage between 

s = 

r- -N 

Fig. 195. — Diagram of Fig. 196. — E.m.f. variation 

com mutating machine by shifting the brushes, 
with brushes in the mag- 
netic neutral. 

the brushes is the potential difference between C and D f or 2 e 
cos f with a sine wave. Thus, by shifting the brushes from the 
position A, B, at right angles with the magnetic flux, to the posi- 
tion E, F, in line with the magnetic flux, any direct voltage be- 

Fig. 197. — Sine wave of e.m.f. 

tween 2 e and can be produced, with the same wave of alter- 
nating volage, a. 

As seen, this variation of direct voltage between its maximum 
value and zero, at constant impressed alternating voltage, is in- 



dependent of the wave shape, and thus run be produced whether 
the alternating voltage is a sine wave or any other wave. 

It is obvious that, instead of shifting the brushes on the com- 
mutator, the magnetic field poles may \k< shifted, in the opposite 
direction, by the same angle, as shown in Fig. 198, A, B, C. 

Instead of mechanically shifting the field poles, they can bt 
shifted electrically, by having each field pole consist of a numUr 
of sections, and successively reversing the polarity of these sec- 
tions, as shown in Fig. 199, A, B, C, D. 

by mechanically shifting llie poles. 

Instead of having a large number of field pole sections, obvi- 
ously two sections are sufficient, and the same gradual change 
can be brought about by not merely reversing the sections but 
reducing the excitation down to zero and bringing it up again In 
opposite direction, as shown in Fig. 200, A, B, C, D, E. 

Fin. 11)9.— 

by electrically shifting the polos. 

In this case, when reducing one section in polarity, the othtf 
section must be increased by approximately the same amount] 
to maintain the same alternating voltage. 

When changing the direct voltage by mechanically shifting 
the brushes, as soon as the brushes come under the field pole 
faces, self-inductive sparking on the commutator would result 
if the iron of the field poles were not kepi away from the brush 


position by having a slot in the field poles, as indicated in dotted 
line in Fig. 196 and Fig. 198, B. With the arrangement in Figs. 
196 and 198, this is not feasible mechanically, and these arrange- 

,f. variation by shifting-flux distribution. 

ments are, therefore, unsuitable. It is feasible, however, as 
shown in Figs. 199 and 200, that is, when shifting the resultant 
magnetic flux electrically, to leave a commutating space between 

Fio. 201. — Variable ratio or split-pole converter. 

the polar projections of the field at the brushes, as shown in Fig. 
200, and thus secure as good commutation as in any other com- 
mutating machine. 


Such a variable-ratio converter, then, comprises an armature 
A, Fig. 201, with the brushes, H, B', in fixed position and field 
poles, P,P', separated by inter polar spaces, C, C, of such width as 
required for commutation. Each field pole consists of two parts, 
P and Pi, usually of different relative size, separated by a narrow 
space, DD', and provided with independent windings. By vary- 
ing, then, the relative excitation of the two polar sections, Pand 
Pi, an effective shift of the resultant field flux and a corresponding 
change of the direct voltage is produced. 

As this method of voltage variation does not depend upon the 
wave shape, by the design of the field pole faces and the pitch 
of the armature winding the alternating voltage wave can 1* 
made as near a sine wave as desired. Usually not much atten- 
tion is paid hereto, as experience shows that the usual distributed 
winding of the commutating machine gives a sufficiently close 
approach to sine shape. 

Armature Reaction and Commutation 

232. With the brushes in quadrature position to the resultant 
magnetic flux, and at normal voltage ratio, the direct -current 
generator armature reaction of the converter equals the syn- 
chronous-motor armature reaction of the power component of 
the alternating current, and at unity power-faetor the converter 
thus has no resultant armature reaction, while with a lagging 
or leading current it has the magnetizing or demagnetizing re- 
action of the wattless component of the current. 

If by a sliift of the resultant flux from quadrature position 
with the brushes, by angle, t, the direct voltage is reduced by 
factor cos r, the direct current and therewith the direct-current 
armature reaction are increased, by factor, -. as by the law 

of conservation of energy the direct-current output must equal 
the alternating-current input (neglecting losses). The dueet- 
current armature reaction, ff, therefore ceases to be equal to the 
armature reaction of the alternating energy current, 5F», but is 
greater by factor, '■ 

The alternating-current armature reaction, S u , at no | 
placement, is in quadrature position with the magnetic flux. 


The direct-current armature reaction, £, however, appears in the 
position of the brushes, or shifted against quadrature position 
by angle t; that is, the direct-current armature reaction is not in 
opposition to the alternating-current Armature reaction, but 
differs therefrom by angle t, and so can be resolved into two 
components, a component in opposition to the alternating-cur- 
rent armature reaction, £0, that is, in quadrature position with 
the resultant magnetic flux: 

£" = $ cos T = $0, 

that is, equal and opposite to the alternating-current armature 
reaction, and thus neutralizing the same; and a component in 
quadrature position with the alternating-current armature reac- 
tion, $0, or in phase with the resultant magnetic flux, that is, 
magnetizing or demagnetizing: 

$' = $ sin t = $0 tan r; 

that is, in the variable-ratio converter the alternating-current 
armature reaction at unity power-factor is neutralized by a 
component of the direct-current armature reaction, but a result- 
ant armature reaction, 5', remains, in the direction of the resultant 
magnetic field, that is, shifted by angle (90 — r) against the 
position of brushes. This armature reaction is magnetizing or 
demagnetizing, depending on the direction of the shift of the 
field, t. 

It can be resolved into two components, one at right angles 
with the brushes : 

5'i = & cos t = $0 sin r, 

and one, in line with the brushes: 

$'2 = $' sin t = £ sin 2 r = $0 sin t tan t, 

as shown diagrammatically in Figs. 202 and 203. 

There exists thus a resultant armature reaction in the direc- 
tion of the brushes, and thus harmful for commutation, just as 
in the direct-current generator, except that this armature reac- 
tion in the direction of the brushes is only $' 2 = & sin 2 t, that is, 
sin 2 t of the value of that of a direct-current generator. 

The value of 5' 2 can also be derived directly, as the difference 
between the direct-current armature reaction, (F, and the com- 


Fig. 203. — Diagram of minis, in split-pole converter. 


IHiuciii of i hi- alternating-current armature reaction, in the direc- 
tion of the brushes, 5 cos r, that is: 

ff'j ■ 

= ff (1 ■ 

COS ! t) ■- 

-- Jo sin r ton r 

233. The shift of the resultant magnetic flux, by angle r, gives 
ii component of the m.m.f. of field excitation, 5"/ = S/sinr, 
(where ;T, = m.m.f. of field excitation), in the direction of the 
commutator brushes, and either in the direction of armature 
reaction, thus interfering with commutation, or in opposition to 
the armature reaction, thus improving commutation. 

If the magnetic flux is slutted in the direction of armature 
rotation, that is, that section of the field pole weakened toward 
which the armature moves, as in Fig. 202, the component 5"/ 
of field excitation at the brushes is in the same direction as the 
armature reaction, 3'j, thus adds itself thereto and impairs the 
commutation, and such a converter is hardly operative. In this 
case the component of armature reaction, 5', in the direction of 
the field flux is magnetizing. 

If the magnetic flux is shifted in opposite direction to the 
armature reaction, that is, that section of the field pole weakened 
which the armature conductor leaves, as in Fig. 203, the Com- 
ponent, it",, of field excitation at the brushes is in opposite direc- 
tum to the armature reaction, J'i, therefore reverses it, if suffi- 
ciently large, and gives a commutating or reversing flux, $„ that 

, improves commutation so that this arrangement is used in 
such converters. In this case, however, the component of arma- 
ture reaction, $', in the direction of the field flux is demagnet- 
izing, and with increasing load the field excitation has to be in- 
creased by ff* to maintain constant flux. Such a converter thus 
requires compounding, as by a series field, to take care of the 
demagnetizing armature reaction. 

If the alternating current is not in phase with the field, but 
lags or leads, the armature reaction of the lagging or leading 
component of current superimposes upon the resultant armature 
reaction, 5', and increases it — with lagging current in Fig. 202, 
leading current in Fig. 203 — or decreases it — with lagging cur- 
rent in Fig. 203, leading current in Fig. 202 — anil with lag of the 
alternating current, by phase angle, 6 = t, under the conditions 
of Fig. 203, the total resultant armature reaction vanishes, that is, 
the lagging component of synchronous-motor armature reaction 
compensates for the component of the direct -current reaction, 



which is not compensated by the armature reaction of the power 
component of the alternating current. It is interesting t<> note 
that in this case, in regard to heating, output based (hereon, etc., 
the converter equals that of one of normal voltage ratio. 

B. Variable Ratio by Change of Wave Shape of the Y Voltage 
234. If in the converter shown diagranimatieally in Fig. 204 
the magnetic flux disposition and the pitch of the armature 
winding are such that the potential difference between the point, 
a, of the armature and the neutral 0, 
or the 1" voltage, is a sine wave, Fig, 
205 A, then the voltage ratio is 
normal. Assume, however, thai 
the voltage curve, a, differs from 
sine shape by the superposition of 
some higher harmonics: the third 
harmonic in Figs. 205 B and C. 
the fifth harmonic in Figs. 20.5 D 
and E. If, then, these higher 
harmonics are in phase with the 
fundamental, that is, their maxima 
coincide, as in Figs. 205 B and D, 
they increase the maximum of the 
—Variable ratio con- alternating voltage, and thereby the 
s shape direct voltagc;andiflhescharmonics 
are in opposition to the funda- 
mental, as in Figs. 205 C and E, they decrease the maximum 
alternating and thereby the direct voltage, without Appreciably 
affecting the effective value of the alternating voltage. For in- 
stance, a higher harmonic of 30 per cent, of the fundamental 
increases or decreases the direct voltage by 30 per cent . bill 
varies the effective alternating voltage only by -y/i + 0.3' = 
1.044, or 4.4 per cent. 

The superposition of higher harmonics thus offers a DMUM '•> 
increasing as well as decreasing the direct voltage, at i-urist.mi 
alternating voltage, and without shifting the angle between the 
brush position and resultant magnetic flux. 

Since, however, the terminal voltage of the converter does not 
only depend on the generated e.m.f. of the converter, but also 
on that of the generator, and is a resultant of the two e.m.fs. in 
approximately inverse proportion to the impedances from the 
converter terminals to the two respective generated e.mJs., hi 



varying the converter ratio only such higher harmonics can be 
used which may exist in the Y voltage without appearing in the 
converter terminal voltage or supply voltage. 

In general, in an n-phase system an nth harmonic existing in 
the star or Y voltage does not appear in the ring or delta voltage, 

Fig. 205. — Superposition of harmonics to change the e.m.f. ratio. 

as the ring voltage is the combination of two star voltages dis- 

placed in phase by — degrees for the fundamental, and thus by 


180°, or in opposition, for the nth harmonic. 

Thus, in a three-phase system, the third harmonic can be in- 
troduced into the Y voltage of the converter, as in Figs. 205 B 
and C, without affecting or appearing in the delta voltage, so 
can be used for varying the direct-current voltage, while the fifth 
harmonic can not be used in this way, but would reappear and 



cause a short-circuit current in the supply voltage, hence should 
be made sufficiently small to be harmless. 

235. The third harmonic thus can be used for varying the 
direct voltage in the three-phase converter diagrammatically 
shown in Fig. 206 A, and also in the six-phase converter with 

Fio. 206. — Transformer connections for varying the e.m.f. ratio by super- 
position of the third harmonic. 

double-delta connection, as shown in Fig. 206 B, or double-}' 
connection, as shown in Fig. 206 C, since this consists of two sepa- 
rate three-phase triangles of voltage supply, and neither of them 
contains the third harmonic. In such a six-phase converter 
with double- Y connection, Fig. 206 C, the two neutrals, however, 



must not be connected together, as the third harmonic voltage 
exists between the neutrals. In the six-phase converter with 
diametrical connections, the third harmonic of the Y voltage ap- 
pears in the terminal voltage, as the diametrical voltage is twice 
the Y voltage. In such a converter, if the primaries of the sup- 

Fig. 207. — Shell-type transformers. 

ply transformers are connected in delta, as in Fig. 206 D, the 
third harmonic is short-circuited in the primary voltage triangle, 
and thus produces excessive currents, which cause heating and 
interfere with the voltage regulation, therefore, this arrangement 


> f 





Fig. 208. — Core-type transformer. 

is not permissible. If, however, the primaries are connected in 
Y, as in Fig. 206 E, and either three separate single-phase trans- 
formers, or a three-phase transformer with three independent 
magnetic circuits, is used, as in Fig. 207, the triple-frequency 
voltages in the primary are in phase with each other between 




the line and the neutral, and thus, with isolated neutral, can not 
produce any current. With a three-phase transformer as shown 
in Fig. 208, that is, in which the magnetic circuit of the third 
harmonic is open, triple- frequency currents can exist in the sec- 
ondary and this arrangement therefore is not satisfactory. 

In two-phase converters, lugher harmonics can he used for 
regulation only if the transformers are connected in such a man- 
ner that the regulating harmonic, which appears in the converter 
terminal voltage, does not appear in the transformer terminals, 
that is, by the connection analogous to Figs. 206 E and 207. 

Since the direct-voltage regulation of a three-phase or sis- 
phase converter of this type is produced by the third harmonic, 

Fig. 209.— V e.m.f. wa' 

the problem is to design the magnetic circuit of the converter 
so as to produce the maximum third harmonic, the minimum 
fifth and seventh harmonics. 

If q = interpolar space, thus (1 — q) = pole arc, as fraction 
of pitch, the wave shape of the voltage generated between the 
point, a, of a full-pitch distributed winding — as generally used 
for commutating machines — and the neutral, or the induced Y 
voltage of the system is a triangle with the top cut off for dis- 
tance q, as shown in Fig. 209, when neglecting magnetic spread 
at the pole corners. 

If then Co = voltage generated per armature turn while in 
front of the field pole (which is proportional to the magnetic den- 
sity in the air gap), m = series turns from brush to brush, the 
maximum voltage of the-wave shown in Fig. 209 is: 

E tt = ?nc (l - g); 
developed into a Fourier series, tliis gives, as the equation of the 
voltage wave a, Fig. 188: 

(2»- 1)^ 

*F ^ COS 2 

(1 -?)t 5 i (2n - 1)' 


or, substituting for 2? , and denoting: 

A 8 tneo 
A = — ^-i 

2n - 1 
- cos 2" -& 

e = A r (2W-D' cos(2ri " x) • 

f ir 1 ir 1 i* 

= il | cos q s cos + q cos 3 g _ c <> s 3 + j- cos 5 q = cos 5 

1 ir 

+ T« cos 7 ^ ^ cos 7 + 

Thus the third harmonic is a positive maximum for q = 0, or 
100 per cent, pole arc, and a negative maximum for q = % y or 
33.3 per cent, pole arc. 

For maximum direct voltage, q should therefore be made as 
small, that is, the pole arc as large, as commutation permits. 
In general, the minimum permissible value of q is about 0.15 to 

The fifth harmonic vanishes for q = 0.20 and q = 0.60, and 
the seventh harmonic for q = 0.143, 0.429, and 0.714. 

For small values of q, the sum of the fifth and seventh har- 
monics is a minimum for about q = 0.18, or 82 per cent, pole arc. 
Then for q = 0.18, or 82 per cent, pole arc: 

ei = A {0.960 cos + 0.0736 cos 3 6 + 0.0062 cos 5 6 

- 0.0081 cos 7 d + . . . } 

= 0.960 A {cos 6 + 0.0766 cos 3 + 0.0065 cos 5 

- 0.0084 cos 7 + . . . } ; 

that is, the third harmonic is less than 8 per cent., so that not 
much voltage rise can be produced in this manner, while the 
fifth and seventh harmonics together are only 1.3 per cent., thus 

236. Better results are given by reversing or at least lowering 
the flux in the center of the field pole. Thus, dividing the pole 
face into three equal sections, the middle section, of 27 per cent, 
pole arc, gives the voltage curve, q = 0.73, thus: 

e 2 = A {0.411 cos - 0.1062 cos 3 + 0.0342 cos 5 

-0.0035 cos 7 . . .} 
= 0.411 A {cos - 0.258 cos 3 + 0.083 cos 5 

-0.0085 cos 7 . . .}• 

The voltage curves given by reducing the pole center to one- 


hall intensity, to aero, reversing it to half intensity, to full in- 
tensity, anil to Btich intensity that the fundamental disappear*, 
then are given 1 >y : 

(1) full, e = e, = O.96OA|cos0+O.O77cos30 

+0.0065 cos 5 0-0.0084 cos 7 0. . ,| 

(2) 0.5, «■ = <>, -0.5c 2 = O.755A(cos0+O.168cos30 

-0.0144 cos 5 0-0.0085 cos 7 0. . . | 

(3) 0, e = e,-e, =0.549 ,4 {cos 0+0.328 cos 3 

-0.053 cos 5 0-0.084 cos 7 0. . .| 

(4) -0.5 e=«i-1.5e, =0.344 A (cos 6 +0.680 cos 3 B 

-0.131 cos 5 0-0.0084 cos 7 0. . \ 

(5) - full, c = e,-2<> 2 =0.138 A | cos 0+2.07 cos 30 

-0.45 cos 5 0-0.008 cos 7 0. . | 

(6) -1.17, e = c,-2.34e 2 = 0.322^jcos 3 0-0.227 cos 5 0. | 
It is interesling tn note that in the last case the fundamental 

frequency disappears and the machine is a generator of triple 
frequency, that is, produces or consumes a frequency equal to 
three times synchronous frequency. In this ease the sevmUl 
harmonic also disappears, and only the fifth is appreciable. Iiut 
could be greatly reduced by a different kind of pole inc. From 
above table follows: 

(1) (2) (3) (4) (5) (6) normal 

MilMIIIHII: fuuiln- 

rocntal alter- 0.960 0.755 0,549 0.344 0.138 0.960 

n&ting voile . . 

Direct volte 1.033 0.883 0.743 0.578 0.423 0.322 0.960 

237. It is seen that a considerable increase of direel voltage 

beyond the normal ratio involves a sacrifice of output, due to 
the decrease or reversal of a part of the magnetic flux, whereby 
the air-gap section is not fully utilized. Thus it is not advisable 
to go too far in tliis direction. 

By the superposition of the third harmonic upon the funda- 
mental wave of the Y voltage, in a converter with three seetwni 
per pole, thus an increase of direct voltage over its norma! 
voltage can be produced by lowering the excitation of the middk 
section and raising that of the outside sections of the field pole, 
and also inversely a decrease of the direct voltage l>e!ow its 
normal value by raising the excitation of the middl 


and decreasing that of the outside sections of the field poles; 
that is, in the latter case making the magnetic flux distribution 
at the armature periphery peaked, in the former case by making 
the flux distribution flat-topped or even double-peaked. 

Armature Reaction and Commutation 

238. In such a split-pole converter let p equal ratio of direct 
voltage to that voltage which it would have, with the same 
alternating impressed voltage, at normal voltage ratio, where 
p > 1 represents an overnormal, p < 1 a subnormal direct 
voltage. The direct current, and thereby the direct-current 
armature reaction, then is changed from the value which it 

would have at normal voltage ratio, by the factor — , as the 

product of direct volts and amperes must be the same as at 
normal voltage ratio, being equal to the alternating power 
input minus losses. 

With unity power-factor, the direct-current armature reac- 
tion, $, in a converter of normal voltage ratio is equal and opposite, 
and thus neutralized by the alternating-current armature reac- 
tion, $0, and at a change of voltage ratio from normal, by factor 

p, and thus change of direct current by factor — The direct- 

current armature reaction thus is: 

* = *-• 


hence, leaves an uncompensated resultant. 

As the alternating-current armature reaction at unity power- 
factor is in quadrature with the magnetic flux, and the direct- 
current armature reaction in line with the brushes, and with 
this type of converter the brushes stand at the magnetic neutral, 
that is, at right angles to the magnetic flux, the two armature 
reactions are in the same direction in opposition with each other, 
and thus leave the resultant, in the direction of the commutator 

5' = $ - So 


The converter thus has an armature reaction proportional to 
the deviation of the voltage ratio from normal. 
239. If p > 1, or overnormal direct voltage, the armature 


reaction is negative, or motor reaction, and the magnetic Hux 
produced by it at the commutator brushes thus a commutfttWfl, 
flux. If p < I, or subnormal direct voltage, the armature 

reaction is positive, that is, the same as in a direct-cum-rii gen- 
erator, but less in intensity, and thus the magnetic flux of arma- 
ture reaction tends to impair commutation. In a direct-current 
generator, by shifting the brushes to the edge of the field poles, 
the field flux is used as reversing flux to give commutation. In 
this converter, however, decrease of direct voltage is produced by 
lowering the outside sections of the field poles, and the edge of 
the field may not have a sufficient flux density to give commuta- 

d . LHJ . UHLJ . L 

Fio. 210. — Three-section pole tor variable -ratio 

tion, with a considerable decrease of voltage l)elow normal, and 
thus a separate commutating pole is required. Preferably this 
type of converter should be used only for raising the voltage, 
for lowering the voltage the other type, which operates by a 
shift of the resultant flux, and so gives a component of the main 
field flux as commutating flux, should be used, or a combination 
of both types. 

With a polar construction consisting of three sections, thia 
can be done by having the middle section at low, the nul.sicfe 
sections at high excitation for maximum voltage, and, to de- 
crease the voltage, raise the excitation of the center section, but 
instead of lowering both outside sections, leave the section in the 
direction of the armature rotation unchanged, while lowering 
the other outside section twice as much, and thus produce, in 
addition to the change of wave shape, a shift of the flux, as 
represented by the scheme Fig, 210. 

Pole section . . 
Max. voltage , 

Min. voltage . 

Magnetic Density 
3 1' 

.+<B +<B 

+&M +*:' -3+-S 
+63 +<B 

+ > s<B - 



Where the required voltage range above normal is not greater 
than can be produced by the third harmonic of a large pole arc 
with uniform density, this combination of voltage regulation 
by both methods can be carried out with two sections of the 
field poles, of which the one (toward which the armature moves) 
is greater than the other, as shown in Fig. 211, and the variation 
then is as follows: 

Magnetic Density 

Pole section 1 2 1' 2' 

Mat. voltage + <B+ (B — (B — (B 

+ M« + i^(B - y 2 & -1J4® 

1^(B -1^(B 

Min. voltage - M® + 1?£(B + V 2 (R -1?£(B 

Fio. 211. — Two-section pole for variable-ratio converter. 

Heating and Rating 

240. The distribution of current in the armature conductors 
of the variable-ratio converter, the wave form of the actual 
or differential current in the conductors, and the effect of the 
wattless current thereon, are determined in the same manner as 
in the standard converter, and from them are calculated the local 
heating in the individual armature turns and the mean armature 

In an n-phase converter of normal voltage ratio, let E = 
direct voltage; /<> = direct current; E° = alternating voltage 
between adjacent collector rings (ring voltage), and J° = alter- 
nating current between adjacent collector rings- (ring current); 
then, as seen in the preceding: 

£ sin 

E° = 




and as by the law of conservation of energy, the output must 
equal the input, when neglecting losses: 


1° = 

n sm- 




where I* is the power component of the current corresponding 
In the duvet -current output. 

The voltage ratio of a converter can be varied: 

(a) By the superposition of a third harmonic upon the 
tar voltage, or diametrical voltage, which does not appear in 

"he ring voltage, or voltage between the collector rings of lbs 

(6) By shifting the direction of the magnetic flux. 

(ii) can be used for raising the direct voltage as well as for 
lowering it, but is used almost, always for the former purpose, 
since when using this method for lowering the direct voltage 
Commutation is impaired. 

(b) can Ik* used only for lowering (he direct voltage. 

It is possible, by proportioning (he relative amounts by which 
the two methods contribute to the regulation of the voltage, 
to maintain a proper commutating field at the brushes for all 
loads and voltages. Where, however, this is not done, the 
brushes are shifted to the edge of the next field pole, and into 
the fringe of its field, thus deriving the commutating field. 

241. In such a variable-ratio converter let, then, ( = intensity 
of the third harmonic, or rather of that component of it which 
is in line with the direct-current brushes, and thus (hies the 
voltage regulation, as fraction of the fundamental wave. / fa 
chosen as positive if the third harmonic increases the maximum 
of the fundamental wave (wide pole arc) and thus raises the 
direct voltage, and negative when lowering the maximum of the 
fundamental and therewith the direct voltage (narrow pole arc). 

pi = loss of power in the converter, which is supplied by the 
current (friction and core loss) as fraction of the alternating 
input (assumed as 4 per cent, in the numerical example). 

T,, = angle of brush shift on the commutator, counted positive 
in the direction of rotation. 

0i = angle of time lag of the alternating current (thus negative 
for lead). 

r„ = angle of shift of the resultant field from the position :>t 
right angles to the mechanical neutral (or middle between the 
pole corners of main poles and auxiliary poles), counted positive 
in the direction opposite to the direction of armature rotation. 
that is, positive in that direction in which the field flux has been 
shifted to get good commutation, as discussed in the preceding 


Due to the third harmonic, f, and the angle of shift of the field 
flux, r a , the voltage ratio differs from the normal by the factor: 

(1 + t) COST«, 

and the ring voltage of the converter thus is: 

E = r—- r ~ — : (3) 

(1 + /) COST* 

hence, by (1): 

E = 

£ sin 

n (4) 

V2(l +/)eosr 

and the power component of the ring current corresponding to 
the direct -current output thus is, when neglecting losses, from 


J' = Jo(! +t ) COSTa 

= IoV2(l+t)cos T a . (5) 


n sin - 

Due to the loss, pi, in the converter, this current is increased by 
(1 + pi) in a direct converter, or decreased by the factor 
(1 — pi) in an inverted converter. 

The power camponent of the alternating current thus is: 

/, = /'(!+ Vl ) 

T \/2(l+0 (1+P /) COS T a 

= '0 t 


n sin - 

where pi may be considered as negative in an inverted converter. 
With the angle of lag 0i, the reactive component of the current 

J 2 = I\ tan 0i, 
and the total alternating ring current is: 

z = _i v 

cos 0t 
_ JoV2(l+0 (l+p ( )cosT a (?) 

n sin - cos 0i 



or, introducing for simplicity the abbreviation: 

t - (1 + 00 + PJW T. 


242. Let, in Fig. 212, Il'OA represent the center line of the 
magnetic field structure. 

The resultant magnetic field flux, 0*, then leads OA by angle 
*Oi = r a . 

The resultant m.m.f.of the alternating power current,/], isO/i, 


f \ \ 

1 v\ 

/ \1 


* is 




Fio. 212.— Diagram of variable ratio converter. 

at right angles to 0$, and the resultant m.m.f . of the alternating 
reactive current, h, is Olt, in opposition to 0* f while the total 
alternating current, I, is 01, lagging by angle 6\ behind <)/,. 

The m.m.f. of direct-current armature reaction is in the direc- 
tion of the brushes, thus lagging by angle r» behind the position 
OB, where BOA = 90°, and given by 0~lo- 

The angle by which the direct-current m.m.f., O/ , lags in space 
behind the total alternating m.m.f., 01, thus is, by Pig. 212: 

r„ = Si - r„ - r h . (10) 

If the alternating m.m.f. in a converter coincides with the 
direct-current m.m.f., the alternating current and the direct cur- 
rent are in phase with each other in the armature coil midway 



between adjacent collector rings, and the current heating thus 
a minimum in this coil. 

Due to the lag in space, by angle t , of the direct-current 
m.m.f. behind the alternating current m.m.f., the reversal of the 
direct current is reached in time before the reversal of the alter- 
nating current in the armature coil; that is, the alternating 
current lags behind the direct current by angle, 6 — t , in the 

Fig. 213. — Alternating and direct current in a coil midway between 

adjacent collector leads. 

armature coil midway between adjacent collector leads, as 
shown by Fig. 213, and in an armature coil displaced by angle, t, 
from the middle position between adjacent collector leads the 
alternating current thus lags behind the direct current by angle 
(r + O ), where t is counted positive in the direction of armature 
rotation (Fig. 214). 

Fig. 214. — Alternating and direct current in a coil at the angle t from the 

middle position. 

The alternating current in armature coil, t, thus can be ex- 
pressed by: 

i = JV2sin(0 - r - 0«); (11) 

hence, substituting (9): 

i = sin(0-r-0o), (12) 


and as the direct current in this armature coil is -~>and opposite 



to the alternating current, i, the resultant current in the arma- 
ture coil, r, is: 


to = l — 


- sin (0 - t - O ) - 1 


n sin - 


and the ratio of heating, of the resultant current, io, compared 
with the current, ^, of the same machine as direct-current gen- 
erator of the same output, thus is: 

io 2 

h\ 2 



w sin 

sin(0 - t - O ) - 1 




sin (0 - r - O ) - 1 1 d$. (15) 

n sin 

Averaging (14) over one half wave gives the relative heating 
of the armature coil, r, as: 

Integrated, this gives: 

8fc 2 

n 2 sin 2 - 


16 k cos (r + O ) 


wn sm 


243. Herefrom follows the local heating in any armature 

coil, t, in the coils adjacent to the leads by substituting t = ± - , 

and also follows the average armature heating by averaging 

7 T fromr = — - to t = H — • 

n n 

The average armature heating of the n-phase converter there- 
fore is : 

+ ~ 

r - V f > 


or, integrated: 

r = - 


+ 1 - 

16 k cos 0o 

n 2 sin 2 - 




This is the same expression as found for the average armature 
heating of a converter of normal voltage ratio, when operating 
with an angle of lag, O , of the alternating current, where k denotes 
the ratio of the total alternating current to the alternating 
power current corresponding to the direct-current output. 

In an n-phase variable ratio converter (split-pole converter), 
the average armature heating thus is given by: 

8fc 2 . . 16fccos0 o 

r = 

n 2 sin 2 - 

+ 1 - 



h _ (1 + (1 + yi) cos t„ 

cos 0i ' 


0o = 0i - t« - r 6 ; (10) 

and / = ratio of third harmonic to fundamental alternating 
voltage wave; p t = ratio of loss to output; 0i = angle of lag of 
alternating current; r = angle of shift of the resultant mag- 
netic field in opposition to the armature rotation, and n = angle 
of shift of the brushes in the direction of the armature rotation. 
244. For a three-phase converter, equation (18) gives (n = 3): 

qo k 2 1 

T = 27 + 1 - 1-621 k cos 0„ 

= 1.185 A* + 1 - 1.621 k cos O . J 
For a six-phase converter, equation (18) gives (n = 6): 

8fc 2 

r = 


+ 1 - 1.621 k cos O 

= 0.889 k 2 + 1 - 1.621 k cos 0„. 
For a converter of normal voltage ratio: 

t = 0, r = 0, 

using no brush shift : 


n = 0; 

when neglecting the losses: 

Pi = 0, 

it is: 


:- y 

COS 0i 

00 — 01, 


and equations (19) and (20) assume the form: 

Six-phase : 

r = i^f -0.621. 

COS 2 0i 

r = **» - 0.621. 

COS 2 0i 

The equation (18) is the most general equation of the relative 
heating of the synchronous converter, including phase displace- 
ment, 0i, losses, pi } shift of brushes, n y shift of the resultant mag- 
netic flux, t , and the third harmonic, t. 

While in a converter of standard or normal ratio the armature 
heating is a minimum for unity power-factor, this is not in gen- 
eral the case, but the heating may be considerably less at same 
lagging current, more at leading current, than at unity power- 
factor, and inversely. 

245. It is interesting therefore to determine under which con- 
ditions of phase displacement the armature heating is a minimum 
so as to use these conditions as far as possible and avoid con- 
ditions differing very greatly therefrom, as in the latter case 
the armature heating may become excessive. 

Substituting for A; and O from equations (8) and (10) into 
equation (18) gives: 

_ t ,8(1 +0 2 (1 + ? ><) 2 cos 2 t 

n 2 sin 2 - cos 2 0j 

16 (1 + (1 + pi) cos r tt cos (0i - T a - n) . . 
__«>s_ (19) 


- sin - = m t (20) 

which is a constant of the converter type, and is for a three- 
phase converter, w 3 = 0.744; for a six-phase converter, w 6 = 
0.955; and rearranging, gives: 

8(1+ 2 (1 + Pi) 2 COS 2 T a 

r = i + 

TT 2 W 2 

1 P 

- - 2 (1 + t) (1 + p t ) COS T a COS (t + T b ) 


8 (1 + !)■ (1 + p,)» COB' r. tanl 
it 2 ra 2 

2 (1 + (1 + pi) cos r sin (t + r 6 ) tan B\. (21) 

r is a minimum for the value, 0i, of the phase displacement 
given by: 

dr =0 

d tan 0i ' 
and this gives, differentiated: 

*« fb = ^-J™ ( J ^-^ (22) 

(1 + (1 +Pl) C0ST o 

Equation (22) gives the phase angle, 2 , for which, at given 
r 0> T6, J and pi, the armature heating becomes a minimum. 

Neglecting the losses, p/, if the brushes are not shifted, n = 0, 
and no third harmonic exists, t = 0: 

tan 0' 2 = ra 2 tan t 


where m 2 = 0.544 for a three-phase, 0.912 for a six-phase 

For a six-phase converter it thus is approximately B\ — r a , 
that is, the heating of the armature is a minimum if the alter- 
nating current lags by the same angle (or nearly the same angle) 
as the magnetic flux is shifted for voltage regulation. 

From equation (22) it follows that energy losses in the con- 
verter reduce the lag, 2 , required for minimum heating; brush 
shift increases the required lag; a third harmonic, t y decreases 
the required lag if additional, and increases it if subtractive. 

Substituting (22) into (21) gives the minimum armature heat- 
ing of the converter, which can be produced by choosing the 
proper phase angle, 2 , for the alternating current. It is then, 
after some transpositions: 

r. - i + £ { [^- + ° (1 + p,) c ° 8 T °]'- 2(1 + 0(1 + pj • 

= 1- 

cos r a cos (r + n) — m 2 sin 2 (r« + n) 
8m 2 /, r(l +0 (1 + pi) cost, 

7T 2 

nl + t) (1 + pi) COS T a , , ,1 s ) /rtrtX 

1 " L m 2 C0S (r ° + T6) J | (23) 

The term To contains the constants /, p/, t , n only in the 
square under the bracket and thus becomes a minimum if this 



square vanishes, that is, if between the quantities (, p h i 

relations exist thai : 


246. Of the quantities I, p,, r a , r b ; p, and t„ are determined 
by the machine design. ( and r„, however, are equivalent lo 
each other, that is, the voltage regulation can be accomplished 
either by the flux shift, r„, or by the third harmonic, (, or by both, 
and in the latter ease can be divided between t u and / so as to 
give any desired relations between them. 

Equation (24) gives: 

!■- cos |r„ + n.) 

J -■ 


(1 + p t COS T„) 

and by choosing the third harmonic, (, as function of the angle of 
flux shift r a , by equation (25), the converter heating becomes a 

minimum, and is: 

■ 1 

8 m* 




Tu" — 0.551 for a three-phase converter, 
IV = 0.261 for a six-phase converter. 
Substituting (25) into (22) gives: 

tan 02 = tan (r„ + n); 

fl 2 = u + n; (29) 

or, in other words, the converter gives minimum heating IV if 
the angle of lag, 2 , equals the sum of the angle of flux shift, r„ and 
of brush shift, r t . 

It follows herefrom that, regardless of the losses, /tj, of the 
brush shift, t,„ and of the amount of voltage regulation required, 
that is, at normal voltage ratio as well as any other ratio, the 
same minimum converter heating IV can be secured by dividing 
the voltage regulation between the angle of flux shift, r # , and 
the third harmonic, 1, in the manner as given by equation [jMQi 
and operating at a phase angle between alternating current and 
voltage equal to the sum of the angles of flux shift, r„ and of bru-h 
shift, n; that, is, the heating of the split-pole Converter OSS !■<■ 
made the same as that of the standard converter of normal 
voltage ratio. 



Choosing p t = 0.04, or 4 per cent, loss of current, equation 
(25) gives, for the three-phase and for the six-phase converter: 
(a) no brush shift (n = 0) : 

/ 3 ° = 0.467, 1 (30) 

/ 6 ° = 0.123; J 

that is, in the three-phase converter this would require a third 
harmonic of 46.7 per cent., which is hardly feasible; in the six- 
phase converter it requires a third harmonic of 12.3 per cent., 
which is quite feasible. 

(6) 20° brush shift (r 6 = 20) : 

#0 1 ft-QQ COS ( T « + T *) 

cos T a 

< 6 <»=l-0.877 CO8(T,, -+ Tfc) ; 

COS T a 

for r a = 0, or no flux shift, this gives: 

*3 00 = 0.500, ) ( 

h 00 = 0.176./ K ' 

Since " — -- < 1 for brush shift in the direction of 

cos r a 

armature rotation, it follows that shifting the brushes increases 
the third harmonic required to carry out the voltage regulation 
without increase of converter heating, and thus is undesirable. 

It is seen that the third harmonic, t, does not change much 
with the flux shift, r , but remains approximately constant, and 
positive, that is, voltage xaising. 

It follows herefrom that the most economical arrangement 
regarding converter heating is to use in the six-phase converter 
a third harmonic of about 17 to 18 per cent, for raising the vol- 
tage (that is, a very large pole arc), and then do the regulation 
by shifting the flux, by the angle, r a , without greatly reducing the 
third harmonic, that is, keep a wide pole arc excited. 

As in a three-phase converter the required third harmonic is 
impracticably high, it follows that for variable voltage ratio the 
six-phase converter is preferable, because its armature heating 
can be maintained nearer the theoretical minimum by propor- 
tioning t and r a . 




Homopolar or Acyclic Machines 

247.. If a conductor, C, revolves around, one pole of a stationary 
magnet shown as NS in Fig. 215, a continuous voltage is induced 
in the conductor by its cutting of the lines of magnetic force of 
the pole, N, and this voltage can be supplied to an external cir- 
cuit, D, by stationary brushes, Bi and B 2) bearing on the ends 
of the revolving conductor, C. 

The voltage is: 

e = /$ 10- 8 , 

where / is the number of revolutions per second, $ the magnetic 
flux of the magnet, cut by the conductor, C. 


Fig. 215. — Diagrammatic illustration of unipolar machine with two high- 
speed collectors. 

Such a machine is called a unipolar machine, as the conductor 
during its rotation traverses the same polarity, in distinction of 
bipolar or multipolar machines, in which the conductor during 
each revolution passes two or many poles. A more correct name 
is homopolar machine,* signifying uniformity of polarity, or 
acyclic machine, signifying absence of any cyclic change: in all 
other electromagnetic machines, the voltage induced in a con- 
ductor changes cyclically, and the voltage in each turn is alter- 
nating, thus having a frequency, even if the terminal voltage 
and current at the corjimutator are continuous. 



By bringing the conductor, C, over the end of the magnet close 
to the shaft, as shown in Fig. 216, the peripheral speed of motion 
of brush, J3 2 , on its collector ring can be reduced. However, at 
least one brush, J5i, in Fig. 216, must bear on a collector ring 
(not shown in Figs. 215 and 216) at full conductor speed, because 
the total magnetic flux cut by the conductor, C, must pass through 
this collector ring on which Bi bears. Thus an essential char- 
acteristic of the unipolar machine is collection of the current from 
the periphery of the revolving conductor, at its maximum speed. 
It is the unsolved problem of satisfactory current collection from 
high-speed collector rings, at speeds of two or more miles per 

Fia. 216. — Diagrammatic illustration of unipolar machine with one high- 
speed collector. 

minute, which has stood in the way of the commercial intro- 
duction of unipolar machines. 

Electromagnetic induction is due to the relative motion of con- 
ductor and magnetic field, and every electromagnetic device is 
thus reversible with regards to stationary and rotary elements. 
Howeyer, the hope of eliminating high-speed collector rings in 
the unipolar machine, by having the conductor standstill and 
the magnet revolve, is a fallacy: in Figs. 215 and 216, the con- 
ductor, C, revolves, and the magnet, NS, and the external circuit, 
D, stands still. The mechanical reversal thus would be, to have 
the conductor, C, stand still, and the magnet, NS, and the external 
circuit revolve, and this would leave high-speed current collection. 

Whether the magnet, NS, stands still or revolves, is immaterial 
in any case, and the question, whether the lines of force of the 
magnet are stationary or revolve, if the magnet revolves around 
its axis, is meaningless. If, with revolving conductor, C, and 
stationary external circuit, D, the lines of force of the magnet 
are assumed as stationary, the induction is in C, and the return 
circuit in D; if the lines of force are assumed as revolving, the 



Induction is in D, and C is the return, but the voltage in the m- 
ouit, CD, is the same. If, (hen, V and D both stand still, 
there is no induction in either, or, assuming the lines of magnetic 
force lo revolve, equal and opposite voltages are induced in ( 
and D, and the voltage in circuit, CD, is zero just the wum. 
However, the question whether the lines of force of a revolving 
magnet rotate or not, is meaningless for this reason: the lines <>i 
force are a pictorial representation of the magnetic field in space. 
The magnetic field at any point is characterized by an intensity 
and a direction, and as long as intensity and direction at ;inv point 
arc constant or stationary, the magnetic field is constant or sta- 
tionary. This is the case in Figs. 215 and 210, regardJesa vht&ht i 
the magnet revolves around its axis or not, and the rotation o| 
the magnet thus has no effect whatsoever on the induction phe- 
nomena. The magnetic field is stationary at any point of space 
outside of the magnet, and it is also stationary at any point dJ 
space inside of the magnet, even if the niagncl revolves, and a4 
the same time it is stationary also with regards to any efemetd 
of the revolving magnet. lising then the pictorial representation 
of the lines of magnetic force, we can assume these lines of force- 
as stationary in space, or as revolving with the rotating magnet, 
whatever best suits the convenience of the problem at hand: but 
whichever assumption we make, makes no difference on 
tion of the problem, if we reason correctly from the assumption. 

248. As in the unipolar machine each conductor (cot 
ing to a half turn of the bipolar or multipolar machine) requires 
a separate high-speed collector ring, many attempts have 
been made fund arc still Ix'ing made) to design a coil-wound 
unipolar machine, that is, a machine connecting a number of 
peripheral conductors in series, without going through collector 
rings. This is an impossibility, and unipolar induction, that is, 

continues induction of a unidirectional voltage, is possible | 

in mi open conductor, but not in a coil or turn, as the voltage 
electro magnetically induced in a coil or turn must alwi 
alternating voltage, 

The fundamental law of elect romagnetic induction i- 
indnced voltage is proportional to the rate of cutting of the con- 
ductor through the lines of force of the magnetic field. Applying 
this to a closed circuit or turn; every line of magnetic fun,- bbJ 

by a turn must either go limn il utside to the inside, or from 

the inside to the outside of the turn. This mean- : 



induced in a turn is proportional (or equal, in absolute units) to 
the rate of change of the number of lines of magnetic force en- 
closed by the turn, and a decrease of the lines of force enclosed 
by the turn, induces a voltage opposite to that induced by an 
increase. As the number of lines of force enclosed by a turn can 
not perpetually increase (or decrease), it follows, that a voltage 
can not be induced perpetually in the same direction in a turn. 
Every increase of lines of force enclosed by the turn, inducing 


Fig. 217. — Mechanical an- 
alogy of bipolar induction. 

Fig. 218. — Mechanical analogy of 
unipolar induction. 

a voltage in it, must sometime later be followed by an equal 
decrease of the lines of force enclosed by the turn, which induces 
an equal voltage in opposite direction. Thus, averaged over a 
sufficiently long time, the total voltage induced in a turn must 
always be zero, that is, the voltage, if periodical, must be alter- 
nating, regardless how the electromagnetic induction takes place, 
whether the turn is stationary or moving, as a part of a machine, 
transformer, reactor or any other electromagnetic induction 
device. Thus continuous-voltage induction in a closed turn 
is impossible, and the coil-wound unipolar machine thus a 
fallacy. Continuous induction in the unipolar machine is pos- 
sible only because the circuit is not a closed one, but consists of a 
conductor or half turn, sliding over the other half turn. Mechan- 
ically the relation can be illustrated by Figs. 217 and 218. If 
in Fig* 217 the carriage, C, moves along the straight track of 
finite length — a closed turn of finite area — the area, A, in front of 
C decreases, that B behind the carriage, C, increases, but this 
decrease and increase can not go on indefinitely, but at some time 
C reaches the end of the track, A has decreased to zero, B is a 



Fig. 219. — Drum type of 
unipolar machine with sta- 
tionary magnet core, section. 

maximum, and any further change can only be an increase ol I 
and decrease of H, by a motion of (' in opposite direction, repre- 
senting induction of a reverse voltage. On the endless circular 
D track, Fig. 218, however, the carriage, 

C, can continuously move in the same 
direction, continuously reduce the 
area, A, in front and increase that of 
H behind C, corresponding to con- 
tinuous induction in the same direc- 
tion, in the unipolar machine. 

249. In the industrial design of a 
unipolar machine, naturally a closed 
magnetic circuit would be used, and 
the form, Fig. 216, would be exe- 
cuted as shown in length section in Fig. 219. N is the same 
pole as in Fig. 216, but the magnetic return circuit is shown 
by S, concentrically surrounding N. C is the cylindrical con- 
ductor, revolving in the cylindrical gap be- 
tween N and 8. B, and B% are the two sets 
of brushes bearing on the collector rings at 
the end of the conductor, C, and F is the 
field exciting winding. 

The construction, Fig. 219, has the me- 
chanical disadvantage of a relatively light 
structure, (", revolving at high speed between 
two stationary structures, N and S. As it is 
immaterial whether the magnet is stationary 
Of revolving, usually the inner core, iV, is re- 
volved with the conductor, as shown in 
Figs. 221 and 222. This shortens the gap 
between N and S, but introduces an aux- 
iliary gap, G. Fig. 221 has the disadvantage 
of a magnetic end thrust, and thus the con- 
struction, Fig. 222, is generally used, or its 
duplication, shown in Fig. 223. 

The disk type of unipolar machine, shown 
in section in Fig. 220, has been frequently proposed in fon&0 
times, but is economically inferior to the construction of Figs. 
221, 222 and 223. The limitation of the unipolar machine is the 
high collector speed. In Fig. 220, the average conductor speed 
is less than the collector speed, and the latter thus relatively 


higher than in Figs. 221 to 223, where it equals the conductor 

Higher voltages then can be given by a single conductor, are 

Fig. 221. — Drum type of unipolar Fio. 222.— Drum type o/ unipolar 
machine with revolving magnet core machine with revolving magnet core 
and auxiliary end gap, section. and auxiliary cylinder gap, section. 

derived in the unipolar machine by connecting a number of con- 
ductors in series. In this case, every series conductor obviously 

• ;^-.,^,:,& L, 

q * n ro 

Fio. 223. — Double drum type of unipolar machine, section, 
requires a separate pair of collector rings. This is shown in Figs. 

224 and 225, the cross-section and length section of the rotor of 

Fig. 224. — Multi- Fia. 225.— Mult i-«inductor unipolar machine, 
conductor unipolar length section. 

machine, ■ 

a four-circuit unipolar. As seen in Fig. 224, the cylindrical con- 
ductor is slotted into eight sections, and diametrically opposite 



sections, 1 and 1', 2 and 2', 3 and 3', 4 and 1', are connected in 
multiple (to equalize the flux distribution) between four pairs of 
collector rings, shown in Fig. 225 as 1 and 1|, 2 anil 2j, 3 and 3i, 
4 and 4i. The latter are connected- in series. This machine. 
Figs. 224 and 225, thus could also be used as a three-wire or 
five-wire machine, or as a direct-current converter, bj 
nut intermediary connections, from the collector rings 2, 3. 4. 

250, As each conductor of the unipolar machine requires a 
separate pair of collector rings, with a reasonably moderate 
number of collector rings, unipolar machines of medium capacity 
are suited for low voltages only, such as for electrolytic machines, 
and have been built for this purpose to a limited extent, but in 
general it has been found more economical by series connection 
of the electrolytic cells to permit the use of higher voltages, and 
then employ standard machines. 

For commercial voltages, 250 or f>00, to keep the number of 
collector rings reasonably moderate, unipolar machines m|iun 
very large magnetic fluxes — that is, large units of capadl j and 
very high peripheral speeds. The latter requirement made tin- 
machine type unsuitable during the days of theslow-spe 
connected steam engine, but when the high-speed steam turbttM 
arrived, the study of the design of high-powered steam-turbine- 
driven unipolars was undertaken, and a number of such machines 
built and installed. 

In the huge turbo-alternators of today, the largest lo— i- the 

core loss: hysteresis and eddies in the iron, which often is K 

than all the other losses together. Theoretically, the Uni point 
machine has no core loss, as the magnetic flux does not change 
anywhere, and solid steel thus is used throughout — and has to 
be used, due to the shape of the magnetic circuit. However. 
with the enormous magnetic fluxes of these maclunes, in suinl 
iron, the least variation of the magnetic circuit, such as caused 
by small unequalities of the air gap, by the reaction of the :ir ma- 
ture currents, etc., causes enormous core losses, mostly addfaa, 
and while theoretically the unipolar has no cove loss, designing 
experience has shown, that it is a very difficult problem to keep 
the core loss in such machines down to reasonable values. Fur- 
thermore, in and at the collector rings, the magnetic n 
the armature currents is alternating or pulsating. Thus in Vfgt. 
224 and 225, the point of entrance of the current from the arma- 
ture conductors into the collector rings revolves with the rotation 


of the machine, anil from this point flows through the collector 
ring, distributing between the next brushes. While this circular 
flow of current in the collector ring represents effectively a frac- 
tion of a turn only, with thousands of amperes of current it 
represents thousands of ampere-turns m.m.f., causing high losses, 
which in spite of careful distribution of the brushes to equalize 
the current flow in the collector rings, can not be entirely 

251. The unipolar machine is not free of armature reaction, as 
often believed. The current in all the armature conductors 
(Fig. 224) flows in the same direction, and thereby produces a 
circular magnetization in the magnetic return circuit, S, shown 
by the arrow in Fig. 224. While the armature conductor mag- 
netically represents one turn only, in the large machines it repre- t 
sents many thousand ampere-turns. As an instance, assume a 
peripheral speed of a steam-turbine-driven unipolar machine, of 
12,000 ft. per minute, at 1800 revolutions per minute. This 
gives an armature circumference of 80 in. At }'i in. thickness 
of the conductor, and 2500 amp. per 
square inch, this gives 100,000 ampere- 
turns m.m.f. of armature reaction, 
which probablyis sufficient to magnetic- 
ally saturate the iron in the pole faces, in 
the direction of the arrow in Fig. 224. 
At the greatly lowered permeability at 
saturation, with constant field excita- 
tion the voltage of the machine greatly 
drops, or, to maintain constant voltage, f, 220. — Multi-con- 
a considerable increase of field excita- ductor unipolar machine 
.... -it wit" compensating pole 

turn under load is required. Large f ftce winding, erow-wection. 
unipolar machines thus are liable to 
give poor voltage regulation and to require high compounding. 

To overcome the circular armature reaction, a counter m.m.f. 
may be arranged in the pole faces, by returning the current of 
each collector ring 1,, 2i, 3i, 4,, of Fig. 225, to the collector rings 
on the other end of the machine, 2, 3, 4 in Kg. 225, not through 
an external circuit, but through conductors imbedded in the pole 
face, as shown in Fig. 226 as 1', 2', 3', 4'. 

The most serious problem of the unipolar machine, however, 
is that of the high-speed collector rings, and this has not yet been 
solved. Collecting very large currents by numerous collector 



rings at Bpeeda of 10,000 bo 15,000 ft. per minute, leads to high 
losses and correspondingly low machine efficiency, high tempero- 
ture rise, and rapid wear of the brushes and collector rings, and 
this has probably been the main cause of abandoning the develop- 
ment of the unipolar machine for steam-turbine drive. 

A contributing cause was that, when the unipolar steam-tur- 
bine generator was being developed, the days of the huge direct- 
current generator were over, and its place had been taken by 
turbo-alternator and converter, and the unipolar machine offered 

no advantage in reliability, or efficiency, but the disadvantage 

of lesser flexibility, as it requires a greater concentration of direct- 
current generation in one place, than usually needed. 

262. The unipolar machine may be used :i^ motor as well 
as generator, and has found some application as motor meter. 
The general principle of a unipolar meter may be illustrated by 
Fig. 227. 

The meter shaft, A , with counter, F, is pivoted at P, anil carries 
the brake disk and conductor, a copper or aluminum disk. D, be- 
tween the two poles, N and S, of a circular magnet. The shaft, .4, 
dips into a mercury cup, C, which is insulated and contains tbc 
one terminal, while tiie other terminal goes to a circular mercury 
trough, 67. An iron pin, B, projects from the disk, D, into this 
mercury trough and completes the circuit. 



263. In reviewing the numerous types of apparatus, methods 
of construction and of operation, discussed in the preceding, 
an alphabetical list of them is given in the following, comprising 
name, definition, principal characteristics, advantages and dis- 
advantages, and the paragraph in which they are discussed. 

Alexanderson High-frequency Inductor Alternator. — 159. 
Comprises an inductor disk of very many teeth, revolving at very 
high speed between two radial armatures. Used for producing 
very high frequencies, from 20,000 to 200,000 cycles per second. 

Amortisseur. — Squirrel-cage winding in the pole faces of the 
synchronous machine, proposed by Leblanc to oppose the hunt- 
ing tendency, and extensively used. 

Amplifier. — 161. An apparatus to intensify telephone and 
radio telephone currents. High-frequency inductor alternator 
excited by the telephone current, usually by armature reaction 
through capacity. The generated current is then rectified, be- 
fore transmission in long-distance telephony, after transmission 
in radio telephony. 

Arc Machines. — 138. Constant-current generators, usually 
direct-current, with rectifying commutators. The last and most 
extensively used arc machines were: 

Brush Arc Machine. — 141-144. A quarter-phase constant- 
current alternator with rectifying commutators. 

Thomson-Houston Arc Machine. — 141-144. A three-phase 
F-connected constant-current alternator with rectifying commu- 

The development of alternating-current series arc lighting by 
constant-current transformers greatly reduced the importance 
of the arc machine, and when in the magnetite lamp arc 
lighting returned to direct current, the development of the 
mercury-arc rectifier superseded the arc machine. 

Asynchronous Motor. — Name used for all those types of 
alternating-current (single-phase or polyphase) motors or motor 
couples, which approach a definite synchronous speed at no-load, 
and slip below this speed with increasing load. 




Brush Arc Machine. — (Sec 1 "Are Machines.' 1 } 
Compound Alternator. — 138. Alternator with rectifying com- 
mutator, connected in Beriea to the armature, either con- 
ductive!}-, or inductively through transformer, and exciting a 
scries field winding by the rectified current. The limitation of 
l he power, which can be rectified, and the need of readjusting the 
brushes with a change of the inductivity of the load, hasmade njGfl 
compounding unsuitahie for the modern high-power altcrnu- 


Condenser Motor. — 77. Single-phase induction motor with 
condenser in tertiary circuit on stator, for producing shirting 
torque and high power-factor. The space angle between pri- 
mary and tertiary stator circuit usually is 45° to 60°, and often a 
three-phase motor is used, with single-phase supply on one phase. 
and condenser on a Becond phase. With the small amount of 
capacity, sufficient for power-factor compensation, usually the 
starting torque is small, unless a starting resistance is used, Imi 
the torque efficiency is high. 

Concatenation. — III, 28. Chain connection, tandem connec- 
tion, cascade connection. Is the connection o the secondary nl 
an induction machine with a second machine. The Bttt&d 
machine may be: 

1. An Induction Machine. — The couple then is asynchronous. 
Hereto belong: 

The induction frequency converter or genera] aUernai\ 
transformer, XII, 103. It transforms between alternating-ear- 
rent systems of different frequency, and has over the indoetiOB- 
motor generator set the advantage of higher efficiency and lesser 
capacity, but the disadvantage of not being standard. 

The' concatenated couple of induction motors, 9, 28, 111. It 
permits multispeed operation. It has the disadvantage against 
the multispeed motor, that, two motors are required; but where 
two or more motors are used, as in induction-motor railroading, 
it has the advantage of greater simplicity. 

The internally concatenated motor {Hunt mtttt>r), 36. I' H 
more efficient than the concatenated couple or the multispeed 
motor, but limited in design to certain speeds and speed ratios. 

2. A Synchronous Machine. — The couple then is synchronous. 
Hereto belong: 

The synchronous frequency converter, XII, 103. It has a defi- 
nite frequency ratio, while that of the induction frequi 


verter slightly changes with the load, by the slip of the induction 

Induction Motor with Low-frequency Synchronous Exciter. — 47. 
The synchronous exciter in this case is of small capacity, and 
gives speed control and power-factor compensation. 

I nductionGenerator with Low-frequency Exciter. — 110, 121. Syn- 
chronous induction generator. Stanley induction generator. In 
this case, the low-frequency exciter may be a synchronous or a 
commutating machine or any other source of low frequency. 
The phase rotation of the exciter may be in the reverse direc- 
tion of the main machine, or in the same direction. In the first 
case, the couple may be considered as a frequency converter 
driven backward at many times synchronous speed, the exciter 
is motor, and the generated frequency less than the speed. In 
the case of the same phase rotation of exciter and main machine, 
the generated frequency is higher than the speed, and the 
exciter also is generator. This synchronous induction generator 
has peculiar regulation characteristics, as the armature reaction 
of non-inductive load is absent. 

3. A Synchronous Commutating Machine. — 112. The couple 
is synchronous, and called motor converter. It has the advantage 
of lower frequency commutation, and permits phase control by 
the internal reactance of the induction machine. It has higher 
efficiency and smaller size than a motor-generator set, but is 
larger and less efficient than the synchronous converter, and 
therefore has not been able to compete with the latter. 

4. A direct-current commutating machine, as exciter, 41. This 
converts the induction machine into a synchronous machine 
(Danielson motor). A good induction motor gives a poor syn- 
chronous motor, but a bad induction motor, of very low power- 
factor, gives a good synchronous motor, of good power-factor, 

5. An alternating-current commutating machine, as low-fre- 
quency exciter, 52. The couple then is asynchronous. This 
permits a wide range of power-factor and speed control as motor. 
As generator it is one form of the Stanley induction generator 
discussed under (2). 

6. A Condenser. — This permits power-factor compensation, 
55, and speed control, 11. The power-factor compensation 
gives good values with very bad induction motors, of low power- 
factor, but is uneconomical with good motors. Speed control 



usually requires excessive amounts of capacity, and given rather 
poor constants. The machine is asynchronous. 

Danielson Motor. — 11. An induction motor converted to a 
synchronous motor by direct-current excitation. (8ee "COB- 
catenation (4).") 

Deep-bar Induction Motor. — 7. Induction motor with deep 
and narrow rotor bars. At the low frequency near synchronism, 
the secondary current traverses the entire rotor conductor, and 
the secondary resistance thus is low. At high slips, u 
ing, unequal current distribution in the rotor bars concentrates 
the current in the top of the bars, thus gives a greatly increased 
effective resistance, and thereby higher torque. However, the 
high reactance of the deep bar somewhat impairs the power- 
factor. The effect is very closely the same as in the double 
squirrel cage. (See "Double Squirrel-cage Induction Motor. "I 

Double Squirrel-cage Induction Motor.— II, 18. Induction 
motor having a high-resistance low-reactance squirrel cage, plan 
to the rotor surface, and a low-resistance high-reactance squirrel 
cage, embedded in the core. The latter gives torque at good 
speed regulation near synchronism, but carries little current at 
lower speeds, due to itst high reactance. The surface squirrel 
cage gives high torque and good torque efficiency at Ion SpMdfl 
and standstill, due to its high resistance, but little torque near 
synchronism. The combination thus gives a uniformly high 
torque over a wide speed range, but at some sacrifice of power- 
factor, due to the high reactance of the lower squirrel i 
get close speed regulation near synchronism, together with high 
torque over a very wide speed range, for instance, down to full 
speed in reverse direction (motor brake), a triple sgt 
may be used, one high resistance low reactance, one medium 
resistance and reactance, and one very low resistance and high 
reactance (24). 

Double Synchronous Machine. — 110, 119. An induction ma- 
chine, in which the rotor, running at double synchronism, is 
connected with the stator, either in series or in parallel, but with 
reverse phase rotation of the rotor, so that the two rotating fields 
coincide and drop into step at double synchronism. The machine 
requires a supply of lagging current for excitation, just tike ;itr. 
induction machine. It may be used as synchronous induction 
generator, or as synchronous motor. As generator, the armature 
reaction neutralizes at non-inductive, but not at inductive load, 


and thus gives peculiar regulation characteristics, similar as the 
Stanley induction generator. It has been proposed for steam- 
turbine alternators, as it would permit higher turbine speed 
(3000 revolutions at 25 cycles) but has not yet been used. As 
motor it has the disadvantage that it is not self-starting. 

Eickemeyer Inductively Compensated Single-phase Series 
Motor. — 193. Single-phase commutating machine with series 
field and inductive compensating winding. 

Eickemeyer Inductor Alternator. — 160. Inductor alternator 
with field coils parallel to shaft, so that the magnetic flux disposi- 
tion is that of a bipolar or multipolar machine, in which the 
multitooth inductor takes the place of the armature of the stand- 
ard machine. Voltage induction then takes place in armature 
coils in the pole faces, and the magnetic flux in the inductor re- 
verses, with a frequency much lower than that of the induced 
voltage. This type of inductor machine is specially adopted for 
moderately high frequencies, 300 to 2000 cycles, and used in in- 
ductor alternators and inductor converters. In the latter, the in- 
ductor carries a low-frequency closed circuit armature winding 
connected to a commutator to receive direct current as motor. 

Eickemeyer Rotary Terminal Induction Motor. — XI, 101. 
Single-phase induction motor with closed circuit primary winding 
connected to commutator. The brushes leading the supply cur- 
rent into the commutator stand still at full speed, but revolve 
at lower speeds and in starting. This machine can give full maxi- 
mum torque at any speed down to standstill, depending on the 
speed of the brushes, but its disadvantage is sparking at the com- 
mutator, which requires special consideration. 

Frequency Converter or General Alternating-current Trans- 
former. — XII, 103. Transforms a polyphase system into another 
polyphase system of different frequency and where desired of differ 
ent voltage and different number of phases. Consists of an induc- 
tion machine concatenated to a second machine, which may be 
an induction machine or a synchronous machine, thus giving the 
induction frequency converter and the synchronous frequency con- 
verter. (See "Concatenation.") In the synchronous frequency 
converter the frequency ratio is rigidly constant, in the induction 
frequency converter it varies slightly with the load, by the slip 
of the induction machine. When increasing the frequency, the 
second machine is motor, when decreasing the frequency, it is 
generator. Above synchronism, both machines are generators 



and the machine thus a synchronous induction generator. En 
concatenation, the first machine always nets an Frequency con- 
verter. The frequency converter has the advantage "I [MM 
machine capacity than the motor generator, but the disadvantage 
of not being standard yet. 

Heyland Motor. — 59, 210. Squirrel-cage induction motor with 
commutator for power-factor compensation. 

Hunt Motor. — 30. Internally concatenated induction motor. 
(See "Concatenation (l).") 

Hysteresis Motor. — X, 98. Motor with polyphase stain ud 
laminated rotor of uniform reluctance in all directions, without 
winding. Gives constant torque at all speeds, by the hystereBM 
of the rotor, as motor below and as generatoi above HynchroDBSB, 
while at synchronism it may be either. Poor power-factor and 
small output make it feasible only in very small BU6S, BUCt H 
motor meters. 

Inductor Machines. — XVII, 150. Synchronous machine, gen- 
erator or motor, in which field and armature coils stand still ami 
the magnetic field flux is constant, and the voltage is induced b) 
changing the flux path, that is, admitting and withdrawing the 
flux from the armature coils by means of a revolving inductor. 
The inducing Hux in the armature coils thus does not alternate, 
but pulsates without reversal. For standard freqiirnrM-. tin- 
inductor machine is less economical and little used, but it offers 

great constructive advantages at high frequencies and is ll nlv 

feasible type at extremely high frequencies. Excited by alter- 
naling currents, the inductor machine may be Used U amplilic: 
(see "Amplifier"); excited by polyphase currents, ii a 
(ton inductor frequency converter, 102; with a direct-current wind- 
ing on the inductor, it is a direct-current kigh-frequi <>> . 
(See "Eickemeyer Inductor Alternator.") 

Leading current, power-factor compensation and phase eontn I 
can be produced by: 


Polarisation cell. 

Overexcited synchronous motor or synchronous col 

Induction machine concatenated to condenser, to sj I 
motor or to low-frequency commutating machine. 

Alternating-current commutating machine with lagging field 

Leblanc'sPanchahuteur. — 145. Synchronous rectifier of many 


phases, fed by polyphase transformer increasing the number of 
phases, and driven by a synchronous motor having as many cir- 
cuits as the rectifier has phases, each synchronous motor circuit 
being connected in shunt to the corresponding rectifier phase to 
byepass the differential current and thereby reduce inductive 
sparking. Can rectify materially more power than the standard 
rectifier, but is inferior to the converter. 

Magneto Commutation. — 163. Apparatus in which the induc- 
tion is varied, with stationary inducing (exciting) and induced 
coils, by shifting or reversing the magnetic flux path by means 
of a movable part of the magnetic circuit, the inductor. Applied 
to stationary induction apparatus, as voltage regulators, and to 
synchronous machines, as inductor alternator. 

Monocyclic. — 127. A system of polyphase voltages with essen- 
tially single-phase flow of power. A system of polyphase vol- 
tages, in which one phase regulates for constant voltage, that is, 
a voltage which does not materially drop within the range of 
power considered, while the voltage in quadrature phase thereto 
is of limited power, that is, rapidly drops with increase of load. 
Monocyclic systems, as the square or the triangle, are derived 
from single-phase supply by limited energy storage in inductance 
or capacity, and used in those cases, as single-phase induction 
motor starting, where the use of a phase converter would be 

Motor Converter. — 112. An induction machine concatenated 
with a synchronous commutating machine. (See " Concatenation 
(3).") The latter thus receives part of the power mechanically, 
part electrically, at lower frequency, and thereby offers the ad- 
vantages incident to a lower frequency in a commutating machine. 
It permits phase control by the internal reactance of the induc- 
tion machine. Smaller than a motor-generator set, but larger 
than a synchronous converter, and the latter therefore preferable 
where it can be used. 

Multiple Squirrel-cage Induction Motor. — (See " Double 
Squirrel-cage Induction Motor.") 

Multispeed Induction Motor. — 14. Polyphase Induction 
Motor with the primary windings arranged so that by the opera- 
tion of a switch, the number of poles of the motor, and thereby 
its speed can be changed. It is the most convenient method of 
producing several economical speeds in an induction motor, and 
therefore is extensively used. At the lower speed, the power- 
factor necessarily is lower. 



Permutator. — 146. Machine to convert polyphase alternating 
to direct current, consisting of a stationary polyphase tean.-- 
former with many secondary phases connected to a stationary 
commutator, with a set of revolving brushes driven by a syn- 
chronous motor. Thus essentially a synchronous converter 
with stationary armature and revolving field, but with two 
armature windings, primary and secondary. The t of MCTl 
objection is the use of revolving brushes, which do not permit 
individual observation and adjustment during operation, and 
thus are liable to sparking. 

Phase Balancer.— 134. An apparatus producing a. polyphase 
system of opposite phase rotation for insertion in series to I 
polyphase system, to restore the voltage balance disturbed by a 
single-phase load. It may be: 

A stationary induction-phase balancer, consisting of an induc- 
tion regulator with reversed phase rotation of the series winding. 

A synchronous-phase balancer, consisting of a synchronous 
machine of reversed phase rotation, having two sets of field wind- 
ings in quadrature. By varying, or reversing the excitation of the 
latter, any phase relation of the balancer voltage with those of 
the main polyphase system can be produced. The synchronous 
phase balancer is mainly used, connected into the neutral of n 
synchronous phase converter, to control the latter so as to make 
the latter balance the load and voltage of a polypoM 
with considerable single-phase load, such as that of a single- 
phase railway system. 

Polyphase Commutator Motor. — Such motors may be shunt, 
181, or series type, 187, for mtiltispeed, adjustable-speed and 
varying-speed service. In commutation, they tend to be inferior 
to single-phase commutator motors, as their rotating field does 
not leave any neutral direction, in which a commutating field 
could be produced, such as is used in single-phase oommotttw 
motors. Therefore, polyphase commutator motors have been 
built with separate phases and neutral spaces between the phases, 
for commutating fields: Scherbius motor.. 

Reaction Machines. — XVI, 147. .Synchronous machine, motor 
or generator, in which the voltage is induced by pulsation of the 
magnetic reluctance, that is, by make and break of the magnetic 
circuit. It thus differs from the inductor machine, in that in 
the latter the total field flux is constant, but is shifted with re- 
gards to the armature coils, while in the reaction machine the 


total field flux pulsates. The reaction machine has low output 
and low power-factor, but the type is useful in small synchronous 
motors, due to the simplicity resulting from the absence of direct- 
current field excitation. 

Rectifiers. — XV, 138. Apparatus to convert alternating into 
direct current by synchronously changing connections. Rec- 
tification may occur either by synchronously reversing connec- 
tions between alternating-current and direct-current circuit: 
reversing rectifier, or by alternately making contact between the 
direct-current circuit and the alternating-current circuit, when 
the latter is of the right direction, and opening contact, when 
of the reverse direction: contact-making rectifier. Mechanical 
rectifiers may be of either type. Arc rectifiers, such as the mer- 
cury-arc rectifier, which use the unidirectional conduction of the 
arc, necessarily are contact-making rectifiers. 

Full-wave rectifiers are those in which the direct-current cir- 
cuit receives both half waves of alternating current; half -wave 
rectifiers those in which only alternate half waves are rectified, 
the intermediate or reverse half waves suppressed. The latter 
type is permissible only in small sizes, as the interrupted pul- 
sating current traverses both circuits, and produces in the alter- 
nating-current circuit a unidirectional magnetization, which 
may give excessive losses and heating in induction apparatus. 
The foremost objection to the mechanical rectifier is, that the 
power which can be rectified without injurious inductive spark- 
ing, is limited, especially in single-phase rectifiers, but for small 
amounts of power, as for battery charging and constant-current 
arc lighting they are useful. However, even there the arc recti- 
fier is usually preferable. The brush arc machine and the 
Thomson Houston arc machine were polyphase alternators with 
rectifying commutators. 

Regulating Pole Converter. — Variable-ratio converter. Split- 
pole converter, XXI, 230. A synchronous converter, in which 
the ratio between direct-current voltage and alternating-current 
voltage can be varied at will, over a considerable range, by shift- 
ing the direction of the resultant magnetic field flux so that the 
voltage between the commutator brushes is less than maximum 
alternating-current voltage, and by changing, at constant im- 
pressed effective alternating voltage, the maximum alternating- 
current voltage and with it the direct-current voltage, by the 
superposition of a third harmonic produced in the converter in 


such a manner, that this harmonic exists only in the local COB- 
verier circuit. This is done by separating the field pole into 
two parts, a larger main pole, which has constant excitation, 
anil a smaller regulating pole, in which the excitation is varied 

and reversed. A resultant armature reaction exists in the 
regulating pole converter, proportional to the deviation of the 
voltage ratio from standard, and requires the use of a series Beld 
Regulating pole converters are extensively used for adjltataHo 
voltage service, as direct-current distribution, storaja 
charging, etc., due to their simplicity and wide voltage range at 
practically unity power-factor, white for automatic pottage 
control under fluctuating load, as railway service, phase control 
of the standard converter is usually preferred. 

Repulsion Generator. — 217. Repulsion motor operated 

Repulsion Motor.— 194, 208, 214. Single-phase commutator 
motor in which the armature is short-circuited and energised 
fay induction from a stationary con pensa ting winding as primary. 
Usually of varying speed or series characteristic. Gives betta 
commutation than the series motor at moderate speeds, 

Rotary Terminal Single-phase Induction Motor.— XI, 101 
(See "Eickemeyer Rotary Terminal Induction Motor.") 

Shading Coil. — 73. A short-circuited turn surrounding i part 
of the pole face of a single-phase induction motor with definite 
poles, for the purpose of giving a phase displacement of I he 
flux, and therehy a starting torque. It is the simplest ;iml cheap- 
est single-phase motor-starting device, but gives only low start- 
ing torque and low torque efficiency, thus is not well suited for 
larger motors. It thus is very extensively used in small motors. 
almost exclusively in alternating-current fan motors. 

Single-phase Commutator Motor. —XX, 189. Commutator 
motor with alternating-current field excitation, and such modi- 
fications of design, as result therefrom. Thai is. lamination of 
the magnetic structure, high ratio of armature reaction to Geld 
excitation, and compensation for armature reaction and self- 
induction, etc. Such motor thus comprises three circuits: the 
armature circuit, the field circuit, and the compensating circuit 
in quadrature, on the stator, to the field circuit. These cir- 
cuits may be energized by conduction, from the main current, 
or by induction, as secondaries with the main current as pri- 
mary. If the armature receives the main current, the motor is 


a series or shunt motor; if it is closed upon itself, directly or 
through another circuit, the motor is called a repulsion motor. 
A combination of both gives the series repulsion motor. 

Single-phase commutator motors of series characteristic are 
used for alternating-current railroading, of shunt characteristic 
as stationary motors, as for instance the induction repuhion 
motor, either as constant-speed high-starting-torque motors, or 
as adjustable-speed motors. 

Lagging the field magnetism, as by shunted resistance, pro- 
duces a lead of the armature current. This can be used for 
power-factor compensation, and single-phase commutator motors 
thereby built with very high power-factors. Or the machine, 
with lagging quadrature field excitation, can be used as effective 
capacity. The single-phase commutator motor is the only type 
which, with series field excitation, gives a varying-speed motor 
of series-motor characteristics, and with shunt excitation or its 
equivalent, give speed variation and adjustment like that of the 
direct-current motor with field control,, and is therefore exten- 
sively used. Its disadvantage, however, is the difficulty and 
limitation in design, resulting from the e.m.f . induced in the short- 
circuited coils under the brush, by the alternation of the main 
field, which tends toward sparking at the commutator. 

Single-phase Generation. — 135. 

Speed Control of Polyphase Induction Motor. — 

By resistance in the secondary, 8. Gives a speed varying 
with the load. 

By pyro-electric resista7ice in the secondary, 10. (lives good 
speed regulation at any speed, but such pyro-electric conductors 
tend toward instability. 

By condenser in the secondary, 11. (lives good speed regula- 
tion, but rather poor power-factor, and usually requires an un- 
economically large amount of capacity. 

By commutator, 58. Gives good speed regulation and per- 
mits power-factor control, but has the disadvantage and com- 
plication of an alternating-current commutator. 

By concatenation with a low-frequency commutating machine 
as exciter, 52. Has the disadvantage of complication. 

Stanley Induction Generator. — 117. Induction machine with 
low-frequency exciter. (See "Concatenation (2).") 

Stanley Inductor Alternator. — 150. Inductor machine with 
two armatures and inductors, and a concentric field coil between 
the same. (See "Inductor Machine. ,, ) 



Starting Devices. — Polyphase induction motor: 

Remittance of high temperature coefficient, 2. Gives good torque 
curve at low speed and good regulation at speed, but requires 
high temperature in the resistance. 

Hysteresis device, 4. Gives good speed regulation and good 
torque at low speed and in starting, but somewhat impairs the 

Eddy-current device, 5; double and triple squirrel-cage, 18. 20, 
24; and deep-bar rotor, 7. Give good speed regulation combined 
with good torque at low speed and in starting, but somewhat 
impairs I lie power-factor. (See " Double Squirrel-cage Induction 
Motor" and "Deep-bar Induction Motor."} 

Single-phase induction motor: 

Phase-splitting devices, 67. Resistance in one phase, 68. In- 
ductive devices, 72. Shading coil, 73. (See "Shading CoiL") 
Monocyclic devices, 76. Resistance-reactance device or mono- 
cyclic triangle. Condenser motor, 77. (See " Condenser Motor.") 

Repulsion-motor starting. 

Series-motor starting. 

Synchronous-induction Generator.— XIII, 113. Induction 
machine, in which the secondary is connected so as to Ba a definite 
speed. This may be done: 

1. By connecting the secondary, in reverse phase rotation, in 
shunt or in series to the primary: double gynehrotUrUi 

(See "Double Synchronous Machine.") 

2. By connecting the secondary in shunt to the primary 
through a commutator. In this ease, the resultant frequency is 
fixed by speed antl ratio of primary to secondary turns. 

3. By connecting the secondary to a source of constant low 
frequency; Stanley induction generator. In this case, the low- 
frequency phase rotation impressed upon the secondary may l>c 
in the same or in opposite direction to the speed. (See "Con* 
catenation (2).") 

Synchronous-induction Motor. — IX, 97. An induction motor 
with single-phase secondary. Tends to {hop into step as syn- 
chronoue motor, and then becomes generator when driven by 
power. Its low power-factor makes it unsuitable except fur 
small sixes, where the simplicity due to the absence of djreet 
current excitation may make it convenient as self-starling syn- 
chronous motor. As reaction machine, 150. 

Thomson-Houston Arc Machine. — 141-144. Three-phase V- 


connected constant-current alternator with rectifying commu- 

Thomson Repulsion Motor. — 193. Single-phase compensated 
commutating machine with armature energized by secondary 
current, and field coil and compensating coil combined in one coil. 

Unipolar Machines. — Unipolar or acyclic machine, XXII, 247. 
Machine in which a continuous voltage is induced by the rotation 
of a conductor through a constant and uniform magnetic field. 
Such machines must have as many pairs of collector rings as there 
are conductors, and the main magnetic flux of the machine must 
pass through the collector rings, hence current collection occurs 
from high-speed collector rings. Coil windings are impossible 
in unipolar machines. Such machines either are of low voltage, 
or of large size and high speed, thus had no application before 
the development of the high-speed steam turbine, and now three- 
phase generation with conversion by synchronous converter has 
eliminated the demand for very large direct-current generating 
units. The foremost disadvantage is the high-speed current 
collection, which is still unsolved, and the liability to excessive 
losses by eddy currents due to any asymmetry of the magnetic 

Winter-Eichbery-Latour Motor. — 194. Single-phase compen- 
sated series-type motor with armature excitation, that is, the 
exciting current, instead of through the field, passes through the 
armature by a set of auxiliary brushes in quadrature with the 
main brushes. Its advantage is the higher power-factor, due to 
the elimination of the field inductance, but its disadvantage the 
complication of an additional set of alternating-current commu- 
tator brushes. 



254. Numerous apparatus, structural features and principles 
have been invented and more or less developed, but have fOQMJ 

a limited industrial application only, or arc not used at all, l>e- 
cause there is no industrial demand for them. Nevertheless B 
knowledge of these apparatus is of (treat importance to the elec- 
trical engineer. They may bo considered as filling the storehouse 
of electrical engineer inn, waiting until they are needed. Wry 
often, in the development of the industry, a demand arises for 
certain types of apparatus, which have been known for many 
years, but not used, because they offered no material advan- 
tage, unlil with the change of the industrial conditions their 
use became very advantageous and this led to their extrusive 

Thus for instance the com mutating pole ("interpole") in 
direct-current machines has been known since very many years, 
has been discussed and recommended, but used very little, in 
short was of practically no industrial importance, while now 
practically all larger direct-current machines and synchronous 
converters use commutating poles. For many years, with tin- 
types of direct-current machines in use, the advantage of tin 
commutating pole did not appear sufficient to compensate to* 
the disadvantage of the complication and resuliunt increase o4 
size and cost. But when with the general introduction of tin- 
steam-turbine high-speed machinery became popular, and lij(ilii-i - 
speed designs were introduced in direct-current machinery also, 
with correspondingly higher armature reaction and greater Deed 
of commutation control, the use of the commutating pole became 
of material advantage in reducing size and cost of apparatus, 
and its general introduction followed. 

Similarly we have seen the three-phase transformer find gen- 
eral introduction, after it had been unused for many years; so 
also the alternating-current commutator motor, etc. 

Thus for a progressive engineer, it is dangerous not to be fjuuil- 

iar with the characteristics ^iiit! possibilities of the known but 



unused types of apparatus, since at any time circumstances may 
arise which lead to their extensive introduction. 

255. With many of these known but unused or little used ap- 
paratus, we can see and anticipate the industrial condition which 
will make their use economical or even necessary, and so lead to 
their general introduction. 

Thus, for instance, the induction generator is hardly used at all 
today. However, we are only in the beginning of the water- 
power development, and thus far have considered only the largest 
and most concentrated powers, and for these, as best adapted, 
has been developed a certain type of generating station, compris- 
ing synchronous generators, with direct-current exciting circuits, 
switches, circuit-breakers, transformers and protective devices, 
etc., and requiring continuous attendance of expert operating 
engineers. This type of generating station is feasible only with 
large water powers. As soon, however, as the large water powers 
will be developed, the industry will be forced to proceed to the 
development of the numerous scattered small powers. That is, 
the problem will be, to collect from a large number of small 
water powers the power into one large electric system, similar 
as now we distribute the power of one large system into numer- 
ous small consumption places. 

The new condition, of collecting numerous small powers — 
from a few kilowatts to a few hundred kilowatts — into one sys- 
tem, will require the development of an entirely different type of 
generating station: induction generators driven by small and 
cheap waterwheels, at low voltage, and permanently connected 
through step-up transformers to a collecting line, which is con- 
trolled from some central synchronous station. A cheap hy- 
draulic development, no regulation of waterwheel speed or gen- 
erator voltage, no attendance in the station beyond an occasional 
inspection, in short an automatically operating induction gen- 
erator station controlled from the central receiving station. 

In many cases, we can not anticipate what application an 
unused type of apparatus may find, and when its use may be 
economically demanded, or we can only in general realize, that 
with the increasing use of electric power, and with the intro- 
duction of electricity as the general energy supply of modern 
civilization, the operating requirements will become more diver- 
sified, and where today one single type of machine suffices — as 
the squirrel-cage induction motor — various modifications thereof 



will become necessary, to suit the conditions of service, such :,s 
the double squirrel-cage induction' motor in ship propulsion sad 
similar uses, the various types of concatenation of induction 
machines with synchronous and commutating machines, etc 

256. In general, a new design or new type of machine or 
apparatus has economically no right of existence, if it. is only 
jnst aa good as the existing one. 

A new type, which offers only a slight advantage in efficiency, 
size, coat of production or operation, etc., over the existing type, 
is economically preferable only, if it can entirely supersede tfw 
existing type; but if its advantage is limited to certain applica- 
tions, very often, even usually, the new type is economically 
inferior, since the disadvantage of producing and operating two 
different types of apparatus may !»■ greater than the advantage 
of i he new type. Tims a standard type is economically superior 
ami preferable to a special one, even if the latter has some small 
superiority, unless, and until, the industry lias extended so far, 
that both types can find such extensive application as in ju:-tif>- 
the existence of two standard types. This, for instance, was the 
reason which retarded the introduction of the three-phase trans- 
former: its advantage was not sufficient to justify the dupli- 
cation of standards, until three-phase systems had bet <■ my 

numerous and widespread, 

In other words, the advantage offered by a new type of appara- 
tus over existing standard types, must be very material, to 
economically justify its industrial development. 

The error most frequently made in modern engineering is m>t 
the undue adherence to standards, but is the reverse. The 
undue preference of special apparatus, sizes, methods, eic. 
where standards would be almost a3 good in their characteristics, 
and therefore would be economically preferable. It is the most 
serious economic mistake, to use anything special, when' standard 
can l>e made to serve satisfactorily, and this mistake i> the BitxA 
frequent in modern electrical engineering, due In the innate. 
individualism of the engineers. 

267. However, while existing standard types of apparatus are 
economically preferable wherever they can be used, it is ohvious 
that with the rapid expansion of the industry, new types of 
apparatus will be developed, introduced and become standard, 
to meet new conditions, and for this reason, aa Btated above, I 
knowledge of the entire known field of apparatus is 
to the engineer. 


Most of the less-known and less-used types of apparatus have 
been discussed in the preceding, and a comprehensive list of 
them is given in Chapter XXIII, together with their definitions 
and short characterization. 

While electric machines are generally divided into induction 
machines, synchronous machines and commutating machines, 
this classification becomes difficult in considering all known 
apparatus, as many of them fall in two or even all three classes, 
or are intermediate, or their inclusion in one class depends on the 
particular definition of this class. 

Induction machines consist of a magnetic circuit inductively 
related, that is, interlinked with two sets of electric circuits, 
which are movable with regards to each other. 

They thus differ from transformers or in general stationary 
induction apparatus, in that the electric circuits of the latter are 
stationary with regards to each other and to the magnetic circuit. 

In the induction machines, the mechanical work thus is pro- 
duced — or consumed, in generators — by a disappearance or 
appearance of electrical energy in the transformation between 
the two sets of electric circuits, which are movable with regards 
to each other, and of which one may be called the primary cir- 
cuit, the other the secondary circuit. The magnetic field of the 
induction machine inherently must be an alternating field 
(usually a polyphase rotating field) excited by alternating 

Synchronous machines are machines in which the frequency of 
rotation has a fixed and rigid relation to the frequency of the 
supply voltage. 

Usually the frequency of rotation is the same as the frequency 
of the. supply voltage: in the standard synchronous machine, 
with direct-current field excitation. 

The two frequencies, however, may be different: in the double 
synchronous generator, the frequency of rotation is twice the 
frequency of alternation; in the synchronous-induction machine, 
it is a definite percentage thereof; so also it is in the induction 
machine concatenated to a synchronous machine, etc. 

Commutating machines are machines having a distributed 
armature winding connected to a segmental commutator. 

They may be direct-current or alternating-current machines. 

Unipolar machines are machines in which the induction is 
produced by the constant rotation of the conductor through a 
constant and continuous magnetic field. 


The list of machine types and their definitions, given in 
Chapter XXIII, shows numerous instances of machines belong- 
ing into several classes. 

The most common of these double types is the converter, or 
synchronous commutating machine. 

Numerous also are the machines which combine induction- 
machine and synchronous-machine characteristics, as the double 
synchronous generator, the synchronous-induction motor and 
generator, etc. * 

The synchronous-induction machine comprising a polyphase 
stator and polyphase rotor connected in parallel with the stator 
through a commutator, is an induction machine, as stator and 
rotor are inductively related through one alternating magnetic 
circuit; it is a synchronous machine, as its frequency is definitely 
fixed by the speed (and ratio of turns of stator and rotor), and 
it also is a commutating machine. 

Thus it is an illustration of the impossibility of a rigid classi- 
fication of all the machine types. 


Also see alphabetical list of apparatus in Chapter XXIII. 

Acyclic, see Unipolar. 

Adjustable speed polyphase motor, 

321, 378 
Alcxanderson very high frequency 

inductor alternator, 279 
Amplifier, 281 
Arc rectifier, 248 
Armature reaction of regulating 

pole converter, 426, 437 
of unipolar machine, 457 


Balancer, phase, 228 
Battery charging rectifier, 244 
Brush arc machine as quarterphase 
rectifier, 244, 254 

Capacity storing energy in phase 

conversion, 212 

Cascade control, see Concatenation. 

Coil distribution giving harmonic 

torque in induction motor, 


Commutating e.m.f. in rectifier, 239 

field, singlephase commutator 

motor, 355, 359 
machine, concatenation with in- 
duction motor, 55, 78 
pole machine, 472 
poles, singlephase commutator 
motor, 358 
Commutation current, repulsion 
motor, 392 
series repulsion motor, 400, 
factor, repulsion motor, 392 

Commutation factor of series repul- 
sion motor, 415 
of regulating pole converter, 

426, 437 
of series repulsion motor, 403 
of singlephase commutator 
motor, 347 
Commutator excitation of induction 
motor, 54, 89 
induction generator, 200 
leads, singlephase commutator 

motor, 351 
motors, singlephase, 331 
Compensated series motor, 372 
Compensating winding, singlephase 
commutator motor, 336, 
Concatenation of induction motors, 

14, 40 
Condenser excitation of induction 
motor secondary, 55, 84 
singlephase induction motor, 

speed control of induction 
motor, 13, 16 
Contact making rectifier, 245 
Cumulative oscillation of synchro- 
nous machine, 299 


Deep bar rotor of induction motor, 

Delta connected roctifier, 251 

Direct current in induction motor 
secondary, 54, 57 

Disc type of unipolar machine, 454 

Double squirrel cage induction 
motor, 29 

Double synchronous induction gen- 
erator, 191, 199, 201 

Drum type of unipolar machine, 454 





Eddy current starting device of in- 
duction motor, 8 
in unipolar machine, 456 

Eickemeyer high frequency inductor 
alternator, 280 


Flashing of rectifier, 249 
Frequency converter, 176 

pulsation, effect in induction 
motor, 131 
Full wave rectifier, 245 


General alternating current motor, 

Generator regulation affecting induc- 
tion motor stability, 137 


Half wave rectifier, 245 

Harmonic torque of induction motor, 

Heyland motor, 92 

Higher harmonic torques in induc- 
tion motor, 144 

Homopolar, see Unipolar. 

Hunt motor, 49 

Hunting, see Surging. 

Hysteresis generator, 169 
motor, 168 

starting device of induction 
motor, 5 


Independent phase rectifier, 251 
Inductance storing energy in phase 

conversion, 212 
Inductive compensation of single- 
phase commutator motor, 
devices starting singlephase in- 
duction motor, 97, 111 

Inductive excitation of singlephase 

commutator motor, 343 
•Induction frequency converter, 191 
generator, 473 

motor inductor frequency con- 
verter, 284 
phase balancer stationary, 228 
phase converter, 220 
Inductor machines, 274 
Interlocking pole type of machine, 

Internally concatenated induction 
motor, 41, 49 

Lead of current produced by lagging 
field of singlephase com- 
mutator motor, 366 

Leblanc's rectifier, 256 

Load and stability of induction 
motor, 132 

Low frequency exciter of induction 
generator, 199, 203 


Magneto commutation, 285 
inductor machine, 285 
Mechanical starting of singlephase 

induction motor, 96 
Mercury arc rectifier, 247 
Meter, unipolar, 458 
Momentum storing energy in phase 

conversion, 212 
Monocyclic devices, 214 

starting singlephase induction 

motor, 98, 117 
Motor converter, 192 
Multiple speed induction motor, 14, 

Multiple squirrel cage induction 

motor, 11, 27 


Open circuit rectifier, 237 
Over compensation, singlephase com- 
mutator motor, 418 




Permutator, 257 
Phase balancer, 228 

control by polyphase shunt 

motor, 324 

by commutating machine 

with lagging field flux, 370 

conversion, 212 

converter starting singlephase 

induction motor, 98 
splitting devices starting single- 
phase induction motor, 97, 
Polyphase excitation of inductor 
alternator, 283 
induction motor, 307 
rectifier, 250 
series motor, 327 
shunt motor, 319 
Position angle of brushes affecting 

converter ratio, 422 
Power factor compensation by com- 
mutator motor, 379 
of frequency converter, 178, 184 
Pyroelectric speed control of induc- 
tion motor, 14 


Quart erphase rectifier, 251 


Reaction converter, 264 
machine, 260 

Rectifier, synchronous, 234 

Regulating pole converter, 422 

Regulation coefficient of system and 
induction motor stability, 
of induction motor, 123 

Regulator, voltage-, magneto com- 
mutation, 285 

Repulsion motor, 343, 373, 385 

starting of singlephase induc- 
tion motor, 97 

Resistance speed control of induc- 
tion motor, 12 

Reversing rectifier, 245 
Ring connected rectifier, 251 
Rotary terminal singlephase induc- 
tion motor, 172 


Secondary excitation of induction 

* motor, 52 

Self induction of commutation, 420 
Semi -inductor type of machine, 286 
Series repulsion motor, 343, 374, 397 
Shading coil starting device, 112 
Short circuit rectifier, 237 
Shunt resistance of rectifier, 235 
and series motor starting of 
singlephase induction 
motor, 96 
Singlephase commutator motor, 331 
generation, 212, 229 
induction motor, 93, 314 
self starting by rotary ter- 
minals, 172 
Six-phase rectifier, 253 

regulating pole converter, 446 
Split pole converter, see Regulating 

pole converter. 
Square, monocycle, 216 
Stability coefficient of induction 
motor, 138 
of system containing induc- 
tion motor, 141 
Stability of induction motor and 
generator regulation, 137 
limit of rectifier, 249 
and load of induction motor, 132 
Stanley inductor alternator, 275 
Star connected rectifier, 251 
Surging of synchronous machine, 288 
Synchronizing induction motor on 

common rheostat, 159 
Synchronous exciter of induction 
motor, 72 
frequency converter, 191 
induction generator, 191, 194 
induction generator with low 
frequency exciter, 199, 203 
induction motor, 166 
as reaction machine, 264 



Synchronous machines, surging, 288 
motor, concatenation with in- 
duction motor, 54, 71 
phase balancer, 228 
phase converter, 227 
rectifier, 234 


Unipolar induction, 452 
machines, 400 
motor meter, 458 

Tandem control, see Concatenation. 
Temperature starting device of 

induction motor, 2 
Third harmonic wave controlling 

converter ratio, 432 
Thomson-Houston arc machine as 

three-phase rectifier, 244, 

Three-phase rectifier, 251 

regulating pole converter, 445 
transformer, 472 
Transformer, general alternating, 176 
Triangle, monocyclic, 216 
Triple squirrel cage induction motor, 


Variable ratio converter, see Regu- 
lating pole converter. 


Wave shape affecting converter 

ratio, 430 
harmonics giving induction 

motor torque, 145 
Winter-Eichbcrg motor, 380 

V connected rectifier, 251 

t • 



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