THE TEXT IS FLY WITHIN THE BOOK ONLY UNIVERSITY LIBRARIES THEORY AND CALCULATIONS OF ELECTRICAL APPARATUS THEORY AND CALCULATIONS OF ELECTRICAL APPARATUS BY CHARLES PROTEUS STEINMETZ, A. M., PH. D, KDITION SIXTH IMPUEHSION McGRAW-HILL BOOK COMPANY, INC. NEW YORK; 370 SEVENTH AVENUE LONDON: 6 & 8 BOUVEEIE ST., E. C. 4 1917 COPYRIGHT, 1917, BY THE MCGRAW-HILL BOOK COMPANY, INC. PBINTB0 IN THE UNITED HTATEB OF AMBHICA MAPLE PRESS - YORK PREFACE In the twenty years since the first edition of " Theory and Cal- culation of Alternating Current Phenomena" appeared, elec- trical engineering has risen from a small beginning to the world's greatest industry; electricity has found its field, as the means of universal energy transmission, distribution and supply, and our knowledge of electrophysics and electrical engineering has in- creased many fold, so that subjects, which twenty years ago could be dismissed with a few pages discussion, now have expanded and require tin extensive knowledge by every electrical engineer. In the following volume I have discussed the most important characteristics of the numerous electrical apparatus, which have been devised and have found their place in the theory of electrical engineering. While many of them have not yet reached any industrial importance, experience has shown, that not infre- quently apparatus, which had been known for many years but had not found any extensive 4 , practical use, become, with changes of industrial conditions, highly important. It is therefore necessary for the electrical engineer to be familiar, in a general way, with the characteristics of the less frequently used types of apparatus. In some; respects, the following work, and its companion vol- ume, "Theory and Calculation of Electric Circuits," may be considered as continuations, or rather as parts of "Theory and Calculation of Alternating Current Phenomena." With the 4th edition, which appeared nine years ago, "Alternating Current Phenomena" had reached about the largest practical bulk, and who n rewriting it recently for the /)th edition, it became necessary to subdivide it into three volumes, to include at least the most necessary structural elements of our knowledge of electrical engineering. The subject matter thus has been distributed into three volumes: "Alternating Current Phenomena," "Electric Circuits," and "Electrical Apparatus," CHARLES PROTEUS STEINMETZ, CAMP MOHAWK, VIBLK'B CKKKK, July, 1017. CONTENTS PREFACE PAOE CHAPTER T. SPEED CONTROL OP INDUCTION MOTORS. /. Starting and Acceleration 1. The problems of high torque over wide range of speed, and of ....................... constant speed over wide range of load Starting by armature rheostat * 1 2. A, Temperature starting device Temperature rise increasing ....................... secondary resistance with increase of current Calculation of motor 2 3. Calculation of numerical instance Its discussion -Estimation .............. of required temperature rise 4 4. B. Hysteresis starting device Admittance of a closed mag- netic circuit \vith negligible eddy current loss Total secondary ..... impedance of motor with hysteresis starting device 5 5. Calculation of numerical instance Discussion- Similarity of torque curve with that of temperature starting device Close speed regulation Disadvantage of impairment of power factor ........... and apparent efficiency, due to introduction of reactance Re- quired Increase of magnetic density 6 6. (L Eddy current starting device- Admittance of magnetic cir- cuit with high eddy current losses and negligible hysteresis ............... Total secondary impedance of motor with eddy current starting deviceNumerical instance 8 7. Double maximum of torque curve Close speed regulation- High torque efficiency -Poor power factor, requiring increase ............ of magnetic density to get output Relation to double squirrel cage motor and deep bar motor 10 //. Constant Speed Operation H. Speed control by armature resistance Disadvantage of in- ......... eoiwUncy of speed with load Use of condenser in armature or secondary- -Use of pyro-eleetric resistance 12 9, Speed control by variation of the effective frequency: con- ................ catenationBy changing the number of poles: rnultispeed motors ........ 13 10, A. Pyro-electric speed control Characteristic of pyro- .............. olectric conductor Close speed regulation of motor Limita- tion of pyro-eloctrio conductors 14 11, B. Condenser speed control Effect of condenser in secondary, viii CONTENTS PAGE giving high current and torque at resonarxce speed Calcula- tion of motor . 16 12. Equations of motor Equation of torque Speed range of maximum torque . 17 13. Numerical instance Voltampere capacity of required con- denser 18 14. C. Multispeed motors Fractional pitch winding, and switch- ing of six groups of coils in each phase, at a change of the num- ber of poles . . . 20 15. Discussion of the change of motor constants due to a change of the number of poles, with series connection of all primary turns Magnetic density and inferior performance curves at lower speeds . 21 16. Change of constants for approximately constant maximum torque at all speeds Magnetic density and change of coil connection 22 17. Instance of 4 6 -=- -f- 8 pole motor Numerical calculation and discussion 23 CHAPTER II. MULTIPLE SQUIRREL CAGE INDUCTION MOTOR. 18. Superposition of torque curves of high resistance low reactance, and low resistance high reactance squirrel cage to a torque curve with two maxima, at high and at low speed 27 19. Theory of multiple squirrel cage based on the use of the true induced voltage, corresponding to the resultant flux which passes beyond the squirrel cage Double squirrel cage induc- tion motor 28 20. Relations of voltages and currents in the double squirrel cage induction motor 29 21. Equations, and method of calculation 30 22. Continued: torque and power equation 31 23. Calculation of numerical instance of double squirrel cage motor, speed and load curves Triple squirrel cage induction motor 32 24. Equation between the voltages and currents in the triple squirrel cage induction motor 34 .... 25. Calculation of voltages and currents . ... 35 26. Equation of torque and power of the three squirrel cages, and their resultant 37 27. Calculation of numerical instance of triple squirrel cage induc- tion motor Speed and load curves 37 CHAPTER III. CONCATENATION. Cascade or Tandem Control of Induction Motors 28. Synchronizing of concatenated couple at half synchronism The two speeds of a couple of equal motors and the three CONTENTS ix PAGE speeds of a couple of unequal motors Internally concatenated motor . ... 40 29. Generator equation of concatenated couple above half syn- chronism Second range of motor torque near full synchron- ism Generator equation above full synchronism Ineffi- ciency of second motor speed range Its suppression by resistance in the secondary of the second motor 41 30. General equation and calculation of speed and slip of con- catenated couple 42 31. Calculation of numerical instances 44 32. Calculation of general concatenated couple 45 33. Continued 46 34. Calculation of torque and power of the two motors, and of the couple 47 35. Numerical instance 48 36. Internally concatenated motor Continuation of windings into one stator and one rotor winding Fractional pitch No inter- ference of magnetic flux required Limitation of available speed Hunt motor 49 37. Effect of continuation of two or more motors on the character- istic constant and the performance of the motor.' 50 CHAPTER IV. INDUCTION MOTOR WITH SECONDARY EXCITATION. 38. Large exciting current and low power factor of low speed induction motors and motors of high overload capacity Instance 52 39. Induction machine corresponding to synchronous machine ex- cited by armature reaction, induction machine secondary corresponding to synchronous machine field Methods of secondary excitation : direct current, commutator, synchronous machine, commutating machine, condenser 53 40. Discussion of the effect of the various methods of secondary excitation on the speed characteristic of the induction motor . 55 Induction Motor Converted to Synchronous 41. Conversion of induction to synchronous motor Relation of exciting admittance and self-inductive impedance as induction motor, to synchronous impedance and coreloss as synchronous motor Danielson motor -57 42. Fundamental equation of synchronous motor Condition of unity power factor Condition of constant field excitation . . 60 .... 43. Equations of power input and output, and efficiency 61 44. Numerical instance of standard induction motor converted to synchronous Load curves at unity power factor excitation and at constant excitation 62 45. Numerical instance of low speed high excitation induction motor converted to synchronous motor Load curves at unity CONTENTS PAGE power factor and at constant field excitation Comparison with induction motor 67 46. Comparison of induction motor and synchronous motor regard- Ing armature reaction and synchronous impedance Poor induction motor makes good, and good induction motor makes poor synchronous motor 69 Induction Motor Concatenated with Synchronous 47. Synchronous characteristic and synchronizing speed of concatenated couple Division of load between machines The synchronous machine as small exciter . . 71 48. Equation of concatenated couple of synchronous and induction motor Reduction to standard synchronous motor equation . 72 49. Equation of power output and input of concatenated couple . 74 50. Calculation of numerical instance of 56 polar high excitation induction motor concatenated to 4 polar synchronous . . 75 51. Discussion. High power factor at all loads, at constant synchronous motor excitation 76 Induction Motor Concatenated with Commutating Machine 52. Concatenated couple with commutating machine asynchronous Series and shunt excitation Phase relation adjustable Speed control and power factor control Two independent variables with concatenated commutating machine, against one with synchronous machine Therefore greater variety of speed and load curves 78 53. Representation of the commutating machine by an effective impedance, in which both components may be positive or negative, depending on position of commutator brushes ... 80 54. Calculation of numerical instance, with commutating machine series excited for reactive anti-inductive voltage Load curves and their discussion 82 Induction Motor with Condenser in Secondary Circuit 55. Shunted r capacit3 neutralizing lagging current of induction motor Numerical instance Effect of wave shape distortion Condenser in tertiary circuit of single-phase induction motor Condensers in secondary circuit Large amount of capacity required by low frequency 84 56. Numerical instance of low speed high excitation induction motor with capacity in secondary Discussion of load curves and of speed 86 57. Comparison of different methods of secondary excitation, by power factor curves: low at all loads; high at all loads, low at light, high at heavy loads By speed: synchronous or constant speed motors and asynchronous motors in which the speed decreases with increasing load , , , 88 CONTENTS xi Induction Motor with Commutator PAGE 58. Wave shape of commutated full frequency current in induction motor secondary Its low frequency component Full fre- quency reactance for rotor winding The two independent variables: voltage and phase Speed control and power factor correction, depending on brush position 89 59. Squirrel cage winding combined with commutated winding Heyland motor Available only for power factor control Its limitation 91 CHAPTER V. SINGLE-PHASE INDUCTION MOTOK. 60. Quadrature magnetic flux of single-phase induction motor pro- duced by armature currents The torque produced by it The exciting ampere-turns and their change between synchron- ism and standstill 93 61. Relations between constants per circuit, and constants of the total polyphase motor Relation thereto of the constants of the motor on single-phase supply Derivation of the single- phase motor constants from those of the motor as three-phase or quarter-phase motor 94 62. Calculation of performance curves of single-phase induction motor Torque and power 96 63. The different methods of starting single-phase induction motors Phase splitting devices; inductive devices; monocyclic de- vices; phase converter 96 64. Equations of the starting torque, starting torque ratio, volt- ampere ratio and apparent starting torque efficiency of the single-phase induction motor starting device 98 65. The constants of the single-phase induction motor with starting device 100 66. The effective starting impedance of the single-phase induction motor Its approximation Numerical instance 101 67. Phase splitting devices Series impedances with parallel con- nections of the two circuits of a quarter-phase motor Equa- ' tions 103 68. Numerical instance of resistance in one motor circuit, with motor of high and of low resistance armature 104 69. Capacity and inductance as starting device Calculation of values to give true quarter-phase relation 106 70. Numerical instance, applied to motor of low, and of high arma- ture resistance 108 71. Series connection of motor circuits with shunted impedance Equations, calculations of conditions of maximum torque ratio Numerical instance 109 72. Inductive devices External inductive devices Internal in- ductive devices Ill 73. Shading coil Calculations of voltage ratio and phase angle . 112 xil CONTENTS PAGE 74. Calculations of voltages, torque, torque ratio and efficiency . . 114 75. Numerical instance of shading coil of low, medium and high resistances, with motors of low, medium and high armature resistance 116 76. Monocyclic starting device Applied to three-phase motor Equations of voltages, currents, torque, and torque efficiency . 117 77. Instance of resistance inductance starting device, of condenser motor, and of production of balanced three-phase triangle by capacity and inductance 120 78. Numerical instance of motor with low resistance, and with high resistance armature Discussion of acceleration . . .121 CHAPTER VI. INDUCTION MOTOR REGULATION AND STABILITY. 1. Voltage Regulation and Output 79. Effect of the voltage drop in the line and transformer im- pedance on the motor Calculation of motor curves as affected by line impedance, at low, medium and high line impedance . 123 80. Load curves and speed curves Decrease of maximum torque and of power factor by line impedance Increase of exciting current and decrease of starting torque Increase of resistance required for maximum starting torque 126 2. Frequency Pulsation 81. Effect of frequency pulsation Slight decrease of maximum torque Great increase of current at light load 131 3. Load and Stability 82. The two motor speed at constant torque load One unstable and one stable point Instability of motor, on constant torque load, below maximum torque point 132 83. Stability at all speeds, at load requiring torque proportional to square of speed: ship propellor, centrifugal pump Three speeds at load requiring torque proportional to speed Two stable and one unstable speed The two stable and one un- stable branch of the speed curve on torque proportional to speed ... 134 84. Motor stability function of the character of the load General conditions of stability and instability Single-phase motor . , 136 4. Generator Regulation and Stability 85. Effect of the speed of generator regulation on maximum output of induction motor, at constant voltage Stability coefficient of motor Instance 137 CONTENTS xiii PAGE 86. Relation of motor torque curve to voltage regulation of system Regulation coefficient of system Stability coefficient of system 138 87. Effect of momentum on the stability of the motor Regulation of overload capacity Gradual approach to instability . . 141 CHAPTER VII. HIGHER HARMONICS IN INDUCTION MOTORS. 88. Component torque curves due to the higher harmonics of the impressed voltage wave, in a quarter-phase induction motor; their synchronous speed and their direction, and the resultant torque curve . . . 144 89. The component torque curves due to the higher harmonics of the impressed voltage wave, in a three-phase induction motor True three-phase and six-phase winding The single-phase torque curve of the third harmonic 147 90. Component torque curves of normal frequency, but higher number of poles, due to the harmonics of the space distribu- tion of the winding in the air-gap of a quarter-phase motor Their direction and synchronous speeds 150 91. The same in a three-phase motor Discussion of the torque components due to the time harmonics of higher frequency and normal number of poles, and the space harmonics of normal frequency and higher number of poles 154 92. Calculation of the coefficients of the trigonometric series repre- senting the space distribution of quarter-phase, six-phase and three-phase, full pitch and fractional pitch windings 155 M 93. Calculation of numerical values for 0, J, MJ pitch defi- ciency, up to the 21st harmonic 157 CHAPTER VII. SYNCHRONIZING INDUCTION MOTORS. 94. Synchronizing induction motors when using common secondary resistance 159 95. Equation of motor torque, total torque and synchronizing torque of two induction motors with common secondary rheo- stat 160 96. Discussion of equations Stable and unstable position Maxi- mum synchronizing power at 45 phase angle Numerical instance 163 CHAPTER IX. SYNCHRONOUS INDUCTION MOTOR. 97. Tendency to drop into synchronism, of single circuit induction motor secondary Motor or generator action at synchronism Motor acting as periodically varying reactance, that is, as reaction machine Low power factor Pulsating torque below synchronism, due to induction motor and reaction machine torque superposition 166 xlv CONTENTS CHAPTEK X. HYSTERESIS MOTOR. PAGE 98. Rotation of iron disc in rotating magnetic field Equations Motor below, generator above synchronism 168 99. Derivation of equations from hysteresis law Hysteresis torque of standard induction motor, and relation to size 169 100. General discussion of hysteresis motor Hysteresis loop collapsing or expanding 170 CHAPTER XI. ROTARY TERMINAL SINGLE-PHASE INDUCTION MOTORS. 101. Performance and method of operation of rotary terminal single-phase induction Motor Relation of motor speed to brush speed and slip corresponding to the load 172 102. Application of the principle to a self-starting single-phase power motor with high starting and accelerating torque, by auxiliary motor carrying brushes . 173 CHAPTER XII. FREQUENCY CONVERTER OR GENERAL ALTERNATING CURRENT TRANSFORMER. 103. The principle of the frequency converter or general alternating current transformer Induction motor and transformer special cases Simultaneous transformation between primary elec- trical and secondary electrical power, and between electrical and mechanical power Transformation of voltage and of fre- quency The air-gap and its effect 176 104. Relation of e.m.f., frequency, number of turns and exciting current 177 105. Derivation of the general alternating current transformer Transformer equations and induction motor equations, special cases thereof 178 106. Equation of power of general alternating current transformer . 182 107. Discussion: between synchronism and standstill Backward driving Beyond synchronism Relation between primary electrical, secondary electrical and mechanical power . . . .184 108. Calculation of numerical instance 185 109. The characteristic curves: regulation curve, compounding curve Connection of frequency converter with synchronous machine, and compensation for lagging current Derivation of equation and numerical instance 186 110. Over-synchronous operation Two applications, as double synchronous generator, and as induction generator with low frequency exciter 190 111. Use as frequency converter Use of synchronous machine or induction machine as second machine Slip of frequency Advantage of frequency converter over motor generator . . . 191 112. Use of frequency converter Motor converter, its advantages and disadvantages Concatenation for multispeed operation . 192 CONTENTS xv CHAPTER XIII. SYNCHRONOUS INDUCTION GENERATOR. PAGE 113. Induction machine as asynchronous motor and asynchronous generator 194 114. Excitation of induction machine by constant low frequency voltage in secondary Operation below synchronism, and above synchronism 195 115. Frequency and power relation Frequency converter and syn- chronous induction generator 196 1 16. Generation of two different frequencies, by stator and by rotor . 198 117. Power relation of the two frequencies Equality of stator and rotor frequency: double synchronous generator Low rotor frequency: induction generator with low frequency exciter, Stanley induction generator 198 118. Connection of rotor to stator by commutator Relation of fre- quencies and powers to ratio of number of turns of stator and rotor 199 119. Double synchronous alternator General equation Its arma- ture reaction 201 120. Synchronous induction generator with low frequency excita- tion (a) Stator and rotor fields revolving in opposite direc- tion (&) In the same direction Equations 203 121. Calculation of instance, and regulation of synchronous induc- tion generator with oppositely revolving fields 204 122. Synchronous induction generator with stator and rotor fields revolving in the same direction Automatic compounding and over-compounding, on non-inductive load Effect of inductive load 205 123. Equations of synchronous induction generator with fields re- volving in the same direction 207 124. Calculation of numerical instance 209 . . . CHAPTER XIV. PHASE CONVERSION AND SINGLE-PHASE GENERATION. 125. Conversion between single-phase and polyphase requires energy atorage Capacity, inductance and momentum for energy storage Their size and cost per Kva 212 126. Industrial importance of phase conversion from single-phase to polyphase, and from balanced polyphase to single-phase . . . 213 127. Monocyclic devices Definition of monocyclic as a system of polyphase voltages with essentially single-phase flow of energy Relativity of the term The monocyclic triangle for single- phase motor starting 214 128. General equations of the monocyclic square 216 129. Resistance inductance monocyclic square Numerical in- stance on inductive and on non-inductive load Discussion . 218 130. Induction phase converter Reduction of the device to the simplified diagram of a double transformation 220 131. General equation of the induction phase converter 222 xvi CONTENTS PAGE 132. Numerical instance Inductive load Discussion and com- parisons with monocyclic square 223 133. Series connection of induction phase converter in single-phase .... induction motor railway Discussion of its regulation 226 134. Synchronous phase converter and single-phase generation Control of the unbalancing of voltage due to single-phase load, by stationary induction phase balancing with reverse rotation of its polyphase system Synchronous phase balancer. . . . 227 135. Limitation of single-phase generator by heating of armature coils By double frequency pulsation of armature reaction Use of squirrel cage winding in field Its size Its effect on the momentary short circuit current 229 136. Limitation of the phase converter in distributing single-phase load into a balanced polyphase system Solution of the problem by the addition of a synchronous phase balancer to the synchronous phase converter Its construction 230 137. The various methods of taking care of large single-phase loads Comparison of single-phase generator with polyphase generator and phase converter Apparatus economy 232 CHAPTER XV. SYNCHRONOUS RECTIFIERS. 138. Rectifiers for battery charging For arc lighting The arc ma- chine as rectifier Rectifiers for compounding alternators For starting synchronous motors Rectifying commutator Differential current and sparking on inductive load Re- sistance bipass Application to alternator and synchronous motor 234 139. Open circuit and short circuit rectification Sparking with open circuit rectification on inductive load, and shift of brushes 237 140. Short circuit rectification on non-inductive and on inductive load, and shift of brushes Rising differential current and flash- Ing around the commutatorStability limit of brush position, between sparking and flashing Commutating e.m.f . resulting from unsymmetrical short circuit voltage at brush shift Sparkless rectification . 239 141. Short circuit commutation in high inductance, open circuit commutation in low inductance circuits Use of double brush to vary short circuit Effect of loadThomson Houston arc machine Brush arc machine Storage battery charging . , 243 142. Reversing and contact making rectifier Half wave rectifier and its disadvantage by unidirectional magnetization of transformer The two connections full wave contact making recti- fiers Discussion of the two types of full wave rectifiers The mercury arc rectifier 245 143. Rectifier with intermediary segments Polyphase rectifica- tionStar connected, ring connected and independent phase CONTENTS xvii PAGE Y rectifiers connected three-phase rectifier Delta connected three-phase rectifier Star connected quarter-phase rectifier Quarter-phase rectifier with independent phases Ring con- nected quarter-phase rectifier Wave shapes and their discus- sion Six-phase rectifier 250 144. Ring connection or independent phases preferable with a large number of phases Thomson Houston arc machine as con- stant current alternator with three-phase star connected rectifier Brush arc machine as constant current alternator with quarter-phase rectifiers in series connection 254 145. Counter e.m.f. shunt at gaps of polyphase ring connected rectifier Derivation of counter e.m f . from synchronous mo- tor Leblanc's Panchahuteur Increase of rectifier output with increasing number of phases . . 255 146. Discussion: stationary rectifying commutator with revolving brushes Permutator Rectifier with revolving transformer Use of synchronous motor for phase splitting in feeding rectifying commutator: synchronous converter Conclusion . 257 CHAPTER XVI. REACTION MACHINES. 147. Synchronous machines operating without field excitation . . 260 148. Operation of synchronous motor without field excitation de- pending on phase angle between resultant rn.m.f. and magnetic flux, caused by polar field structure Energy component of reactance . . . . ... . 261 149. Magnetic hysteresis as instance giving energy component of reactance, as effective hysteretie resistance . . . 262 150. Make and break of magnetic circuit Types of reaction machines Synchronous induction motor Reaction machine as converter from d.-c. to a.-c 263 151. Wave shape distortion in reaction machine, due to variable reactance, and corresponding hysteresis cycles 264 152. Condition of generator and of motor action of the reactance .... machine, as function of the current phase ... 267 153. Calculation of reaction machine equation Power factor and maximum power 268 154. Current, power and power factor Numerical instance 271 . , . 155. Discussion Structural similarity with inductor machine . 272 CHAPTER XVII. INDUCTOR MACHINES. 156. Description of inductor machine type Induction by pulsating unidirectional magnetic flux 274 157. Advantages and disadvantages of inductor type, with regards to field and to armature 275 158. The magnetic circuit of the inductor machine, calculation of magnetic flux and hysteresis loss 276 xviii CONTENTS PAGE 159. The Stanley type of inductor alternator The Alexanderson high frequency inductor alternator for frequencies of 100,000 cycles and over . 279 160. The Eickemeyer type of inductor machine with bipolar field The converter from direct current to high frequency alternating current of the inductor type . 280 161. Alternating current excitation of inductor machine, and high frequency generation of pulsating amplitude. Its use as amplifier Amplification of telephone currents by high fre- quency inductor in radio communication 281 162. Polyphase excitation of inductor, and the induction motor inductor frequency converter 282 163. Inductor machine with reversing flux, and magneto communi- cation Transformer potential regulator with magnetic com- mutation 284 164. The interlocking pole type of field design in alternators and commutating machines 286 165. Relation of inductor machine to reaction machine Half syn- chronous operation of standard synchronous machine as inductor machine 287 CHAPTER XVIII. SURGING OF SYNCHRONOUS MOTORS. 166. Oscillatory adjustment of synchronous motor to changed condition of load Decrement of oscillation Cumulative oscil- lation by negative decrement 288 167. Calculation of equation of electromechanical resonance . . . 289 168. Special cases and example ... 292 169. Anti-surging devices and pulsation of power 293 170. Cumulative surging Due to the lag of some effect behind its cause Involving a frequency transformation of power 296 . . . CHAPTER XIX. ALTERNATING CURRENT MOTOES IN GENERAL. 171. Types of alternating-current motors 300 172. Equations of coil revolving in an alternating field 302 173. General equations of alteraating-curreat motor 304 174. Polyphase induction motor, equations 307 175. Polyphase induction motor, slip, power, torque 310 176. Polyphase induction motor, characteristic constants . . . .312 177. Polyphase induction motor, example 313 178. Singlephase induction motor, equations 314 179. Singlephase induction motor, continued 316 180. Singlephase induction motor, example 318 181. Polyphase shunt motor, general 319 182. Polyphase shunt motor, equations 320 183. Polyphase shunt motor, adjustable speed motor 321 184. Polyphase shunt motor, synchronous speed motor 323 185. Polyphase shunt motor, phase control by it 324 186. Polyphase shunt motor, short-circuit current under brushes . 327 187. Polyphase series motor, equations 327 188. Polyphase series motor, example 330 CONTENTS xix CHAPTER XX. SINGLE-PHASE COMMUTATOR MOTORS. PAGE 189. General: proportioning of parts of a.-c. commutator motor different from d.-c 331 190. Power factor: low field flux and high armature reaction re- quired Compensating winding necessary to reduce armature self-induction 332 191. The three circuits of the single-phase commutator motor Compensation and over-compensation Inductive compen- sation Possible power factors 336 192. Field winding and compensating winding: massed field winding and distributed compensating winding Under-com- pensation at brushes, due to incomplete distribution of com- pensating winding 338 193. Fractional pitch armature winding to secure complete local compensation Thomson's repulsion motor Eickemeyer in- ductively compensated series motor 339 194. Types of varying speed single-phase commutator motors: con- ductive and inductive compensation; primary and secondary excitation; series and repulsion motors Winter Eichberg Latour motor Motor control by voltage variation and by change of type 341 195. The quadrature magnetic flux and its values and phases in the different motor types .... 345 196. Commutation: e.m.f. of rotation and e.m.f. of alternation Polyphase system of voltages Effect of speed 347 197. Commutation determined by value and phase of short circuit current High brush contact resistance and narrow brushes . 349 198. Commutator leads Advantages and disadvantages of resist- ance leads in running and in starting 351 199. Counter e.m.f. in commutated coil: partial, but not com- plete neutralization possible 354 200. Commutating field Its required intensity and phase rela- tions: quadrature field 356 201. Local commutating pole Neutralizing component and revers- ing component of commutating field Discussion of motor types regarding commutation 358 202. Motor characteristics: calculation of motor Equation of cur- rent, torque, power 361 203. Speed curves and current curves of motor Numerical instance Hysteresis loss increases, short circuit current decreases power factor 364 204. Increase of power factor by lagging field magnetism, by resistance shunt across field 366 205. Compensation for phase displacement and control of power factor by alternating current commutator motor with lagging field flux, as effective capacity Its use in induction motors and other apparatus 370 206. Efficiency and losses: the two kinds of core loss 370 xx CONTENTS PAGE 207. Discussion of motor types: compensated series motors: conductive and inductive compensation Their relative advan- tages and disadvantages 371 208. Repulsion motors: lagging quadrature flux Not adapted to speeds much above synchronism Combination type: series repulsion motor 373 209. Constructive differences Possibility of changing from type to type, with change of speed or load 375 210. Other commutator motors: shunt motor Adjustable speed polyphase induction motor Power factor compensation: Heyland motor Winter-Eichberg motor 377 211. Most general form of single-phase commutator motor, with two stator and two rotor circuits and two brush short circuits . . 381 212. General equation of motor 382 213. Their application to the different types of single-phase motor with series characteristic 383 214. Repulsion motor: Equations 385 215. Continued 388 216. Discussion of commutation current and commutation factor . 391 217. Repulsion motor and repulsion generator 394 218. Numerical instance 395 219. Series repulsion motor: equations 220. Continued 397 398 221. Study of commutation Short circuit current underbrushes . 403 222. Commutation current 404 223. Effect of voltage ratio and phase, on commutation 406 224. Condition of vanishing commutation current 408 225. Numerical example 411 226. Comparison of repulsion motor and various series repulsion motor 414 227. Further example Commutation factors 415 228. Over-compensation Equations 4 IS 229. Limitation of preceding discussion Effect and importance of transient in short circuit current 419 CHAPTER XXI. REGULATING POLE CONVERTER. 230. Change of converter ratio by changing position angle between brushes and magnetic flux, and by change of wave shape . . 422 A. Variable ratio by change of position angle between com- mutator brushes and resultant magnetic flux 422 231. Decrease of a.-c. voltage by shifting the brushes By shifting the magnetic flux Electrical shifting of the magnetic flux by varying the excitation of the several sections of the field pole . 422 232. Armature reaction and commutation Calculation of the re- sultant armature reaction of the converter with shifted mag- netic flux 426 233. The two directions of shift flux, the one spoiling, the other CONTENTS xxi PAGE improving commutation Demagnetizing armature reaction and need of compounding by series field 429 Y B. Variable ratio by change of the wave shape of the voltage 429 234. Increase and decrease of d.-c. voltage by increase or decrease of maximum a.-e. voltage by higher harmonic Illustration by third and fifth harmonic 430 235. Use of the third harmonic in the three-phase system Transformer connection required to limit it to the local converter circuit Calculation of converter wave as function of the pole arc 432 236. Calculation of converter wave resulting from reversal of middle of pole arc . . . 435 237. Discussion 436 238. Armature reaction and commutation Proportionality of resultant armature reaction to deviation of voltage ratio from normal 437 239. Commutating flux of armature reaction of high a.-c. voltage Combination of both converter types, the wave shape distor- tion for raising, the flux shift for lowering the a.-c. voltage Use of two pole section, the main pole and the regulating pole . 437 240. Heating and rating Relation of currents and voltages in standard converter 439 241. Calculation of the voltages and currents in the regulating pole converter 440 242. Calculating of differential current, and of relative heating of armature coil 442 243. Average armature heating of n phase converter 444 244. Armature heating and rating of three-phase and of six-phase regulating pole converter 445 245. Calculation of phase angle giving minimum heating or maxi- mum rating 446 246. Discussion of conditions giving minimum heating Design Numerical instance 448 CHAPTER XXII. UNIPOLAR MACHINES. Homopolar Machines Acyclic Machines 247. Principle of unipolar, homopolar or acyclic machine The problem of high speed current collection Fallacy of unipolar induction in stationary conductor Immaterial whether mag- net standstill or revolves The conception of lines of magnetic force 450 248. Impossibility of the coil wound unipolar machine All electro- magnetic induction in turn must be alternating Illustration of unipolar induction by motion on circular track 452 249. Discussion of unipolar machine design Drum type and disc type Auxiliary air-gap Double structure Series connection of conductors with separate pairs of collector rings . . . 454 xxii CONTENTS PAGE 250. Unipolar machine adapted for low voltage, or for large size high speed machines Theoretical absence of core loss Possibility of large core loss by eddies, in core and in collector rings, by pulsating armature reaction 456 251. Circular magnetization produced by armature reaction Liability to magnetic saturation and poor voltage regulation Compensating winding Most serious problem the high speed collector rings 457 252. Description of unipolar motor meter . . , . 458 CHAPTER XXIII. REVIEW. 253. Alphabetical list of machines: name, definition, principal characteristics, advantages and disadvantages . . ... 459 CHAPTER XXIV. CONCLUSION. 254. Little used and unused types of apparatus Their knowledge important due to the possibility of becoming of great industrial importance Illustration by commutating pole machine . . 472 255. Change of industrial condition may make new machine types important Example of induction generator for collecting numerous small water powers 473 256. Relative importance of standard types and of special types of machines 474 257. Classification of machine types into induction, synchronous, commutating and unipolar machines Machine belonging to two and even three types 474 INDEX 477 THEORY AND CALCULATION OF ELECTRICAL APPARATUS CHAPTER I SPEED CONTROL OF INDUCTION MOTORS I. STARTING AND ACCELERATION 1. Speed control of induction motors deals with two problems: to produce a high torque over a wide range of speed down to standstill, for starting and acceleration; and to produce an approximately constant speed for a wide range of load, for constant-speed operation. In its characteristics, the induction motor is a shunt motor, that is, it runs at approximately constant speed for all loads, and this speed is synchronism at no-load. At speeds below full speed, and at standstill, the torque of the motor is low and the current high, that is, the starting-torque efficiency and especially the apparent starting-torque efficiency are low. Where starting with considerable load, and without excessive current, is necessary, the induction motor thus requires the use of a resistance in the armature or secondary, just as the directcurrent shunt motor, and this resistance must be a rheostat, that is, variable, so as to have maximum resistance in starting, and gradually, or at least in a number of successive steps, cut out the resistance during acceleration. This, however, requires a wound secondary, and the squirrelcage type of rotor, which is the simplest, most reliable and therefore most generally used, is not adapted for the use of a starting rheostat. With the squirrel-cage type of induction motor, starting thus is usually done and always with large motors by lowering the impressed voltage by autotransformer, often in a number of successive steps. This reduces the starting current, but correspondingly reduces the starting torque, as it does not change the apparent starting-torque efficiency. The higher the rotor resistance, the greater is the starting torque, and the less, therefore, the starting current required for 1 2 ELECTRICAL APPARATUS a given torque when starting by autotransformer. However, high rotor resistance means lower efficiency and poorer speed regulation, and this limits the economically permissible resistance in the rotor or secondary. Discussion of the starting of the induction motor by arma- ture rheostat, and of the various speed-torque curves produced by various values of starting resistance in the induction-motor secondary, are given in " Theory and Calculation of Alternating- current Phenomena" and in "Theoretical Elements of Electrical 7 Engineering.' As seen, in the induction motor, the (effective) secondary resistance should be as low as possible at full speed, but should be high at standstill very high compared to the full-speed value and gradually decrease during acceleration, to maintain constant high torque from standstill to speed. To avoid the inconvenience and complication of operating a starting rheostat, various devices have been proposed and to some extent used, to produce a resistance, which automatically increases with increasing slip, and thus is low at full speed, and higher at standstill. A. Temperature Starting Device A 2. resistance material of high positive temperature coeffi- cient of resistance, such as iron and other pure metals, operated at high temperature, gives this effect to a considerable extent: with increasing slip, that is, decreasing speed of the motor, the secondary current increases. If the dimensions of the secondary resistance are chosen so that it rises considerably in temperature, by the increase of secondary current, the temperature and therewith the resistance increases. Approximately, the temperature rise, and thus the resistance rise of the secondary resistance, may be considered as propor- tional to the square of the secondary-current, ii, that is, repre- sented by: = + r r (1 2 aii ). (1) As illustration, consider a typical induction motor, of the constants : e = 110; 7 - - = g jb = 0.01 0.1 j; + ZQ - rQ +jxQ = 0.1 0.3 j; + + Zi = ri jxi = 0.1 0.3 j; A the speed-torque curve of this motor is shown as in Fig. 1, SPEED CONTROL Suppose now a resistance, r, is inserted in series into the secondary circuit, which when cold that is, at light-load equals the internal secondary resistance: but increases so as to double with 100 amp, passing through it. This resistance can then be represented by: + r = r (1 if 10-4) + = 0.1 (1 if 10~ 4 ), FIG. 1. High-starting and acceleration torque of induction motor by posi- tive temperature coefficient of secondary resistance. and the total secondary resistance of the motor then is: n + + = r'i TQ (1 a if) (2) + = 0.2 (1 0.5 if 10- 4 ). To calculate the motor characteristics for this varying resist- ance, r'i, we use the feature, that a change of the secondary resistance of the induction motor changes the slip, s, in proportion to the change of resistance, but leaves the torque, current, powerfactor, torque efficiency, etc., unchanged, as shown on page We 322 of " Theoretical Elements of Electrical Engineering." thus calculate the motor for constant secondary resistance, r 3, but otherwise the same constants, in the manner discussed on page 318 of " Theoretical Elements of Electrical 7 Engineering/ 4 ELECTRICAL APPARATUS A This gives curve of Fig. 1. At any value of torque, T, corre- sponding to slip, s, the secondary current is: + ii - e \/&i2 2 &2 > herefrom follows by (2) the value of r'i, and from this the new value of slip : + = + s' s r'i ri. (3) The torque, T, then is plotted against the value of slip, s', and B B gives curve of Fig. 1. As seen, gives practically constant torque over the entire range from near full speed, to standstill. B Curve has twice the slip at load, as A, as its resistance has been doubled. 3. Assuming, now, that the internal resistance, r x, were made as low as resistance n possible, = 0.05, of high temperature and the rest coefficient: r added as external = 0.05, giving the total resistance : (4) This gives the same resistance as curve A: r\ = 0.1, at light- load, where ii is small and the external part of the resistance cold. But with increasing load the resistance, r'i, increases, and the motor gives the curve shown as C in Fig. 1. As seen, curve C is the same near synchronism as A, but in starting gives twice as much torque as A, due to the increased resistance. C and A thus are directly comparable: both have the same constants and same speed regulation and other performance at speed, but C gives much higher torque at standstill and during acceleration. For comparison, curve A! has been plotted with constant resistance r\ 0.2, so as to compare with B. Instead of inserting an external resistance, it would be pref- erable to use the internal resistance of the squirrel cage, to in- crease in value by temperature rise, and thereby improve the starting torque. Considering in this respect the motor shown as curve C, At = standstill, it is: ii 153; thus r\ = 0.217; while cold, the re- = sistance is: r'i 0.1. This represents a resistance rise of 117 per cent. At a temperature coefficient of the resistance of 0.35, this represents a maximum temperature rise of 335C. As seen, SPEED CONTROL 5 by going to temperature of about 350C. in the rotor conductors which naturally would require fireproof construction it be- A A comes possible to convert curve into C, or f into B, in Fig. 1. Probably, the high temperature would be permissible only in the end connections, or the squirrel-cage end ring, but then, iron could be used as resistance material, which has a materially higher temperature coefficient, and the required temperature rise thus would probably be no higher. B. Hysteresis Starting Device 4. Instead of increasing the secondary resistance with increasing slip, to get high torque at low speeds, the same result can be produced by the use of an effective resistance, such as the effective or equivalent resistance of hysteresis, or of eddy currents. As the frequency of the secondary current varies, a magnetic circuit energized by the secondary current operates at the varying frequency of the slip, s. At a given current, ii, the voltage required to send the current through the magnetic circuit is proportional to the frequency, that is, to s. Hence, the susceptance is inverse proportional to s: y-J- (q The angle of hysteretic advance of phase, a, and the power- factor, in a closed magnetic circuit, are independent of the frequency, and vary relatively little with the magnetic density and thus the current, over a wide 1 range, thus may approxi- mately be assumed as constant. That is, the hysteretic con- ductance is proportional to the susceptance : g' = V tan a. (6) Thus, the exciting admittance, of a closed magnetic circuit of negligible resistance and negligible eddy-current losses, at the frequency of slip, s, is given by: Y' = r g -1 jb = V (tan a j) a .6 -J-,-- 6 ; (, tan-j).x frt\ (7) l " Theory and Calculation of Alternating-current Phenomena," Chapter XII. 6 ELECTRICAL APPARATUS Assuming tan a = 0.6, which is a fair value for a closed mag- netic circuit of high hysteresis loss, it is : r =~ - (0.6 J), the exciting admittance at slip, s. Assume then, that such an admittance, F', is connected in series into the secondary circuit of the induction motor, for the purpose of using the effective resistance of hysteresis, which increases with the frequency, to control the motor torque curve. The total secondary impedance then is : r7f Z/ 1 . *7 JL. l\ -f-yf i+S where : Y = g jb is the admittance of the magnetic circuit at full frequency, and V0 + y = 2 b2. 5. For illustration, assume that in the induction motor of the constants : = e Q 100; - y = o.02 0.2 j; + Z = 0.05 0.15 j; + Z l = 0.05 0.15 j; a closed magnetic circuit is connected into the secondary, of full frequency admittance, and assume: Y = g-jb; g = 0.666 = 4; thus, by (8) : + + Z\ = (0.05 0.11 a) 0.335 js. (9) The characteristic curves of this induction motor with hysteresis starting device can now be calculated in the usual manner, dif- fering from the standard motor only in that Z\ is not constant, m and the proper value of r i; a? 3 and has to be used for every slip, s. Fig. 2 gives the speed-torque curve, and Fig. 3 the load curves of this motor. SPEED CONTROL For comparison is shown, as Tf , in dotted lines, the torque curve of the motor of constant secondary resistance, and of the constants: - 7o = 0.01 0.1 j; Z = + Q 0.01 0.3 j', + Zi = 0.1 0.3 j; As seen, the hysteresis starting device gives higher torque at standstill and low speeds, with less slip at full speed, thus a materially superior torque curve. INDUCTION MOTOR Y =.02-.2j; Z =.05-K15;? ; 6 =100 + Z 1 = (.05 .11s)-K335;7*s SPEED CONTROL BY HYSTERESIS SPEED CURVES FIG. 2. Speed curves of induction motor with hysteresis starting device. p represents the power-factor, T? the efficiency, 7 the apparent efficiency, 77' the torque efficiency and 7' the apparent torqiie efficiency. However, T corresponds to a motor of twice the admittance and half the impedance of Te . That is, to get approximately the same output, with the hysteresis device inserted, as without it, requires a rewinding of the motor for higher magnetic density, the same as would be produced in Tf by increasing the voltage -\/2 times. It is interesting to note in comparing Fig. 2 with Fig. 1, that the change in the torque curve at low and medium speed, pro- duced by the hysteresis starting device, is very similar to that produced by temperature rise of the secondary resistance; at 8 ELECTRICAL APPARATUS speed, however, the hysteresis device reduces the slip, while the temperature device leaves it unchanged. The foremost disadvantage of the use of the hysteresis device is the impairment of the power-factor, as seen in Fig. 3 as p. The introduction of the effective resistance representing the hysteresis of necessity introduces a reactance, which is higher than the resistance, and thereby impairs the motor characteristics. " Comparing Fig. 3 with Fig. 176, page 319 of Theoretical FIG. 3. Load curves of induction motor with hysteresis starting device. Elements of Electrical 7 Engineering/ which gives the load curves of Tf of Fig. 2, it is seen that the hysteresis starting device reduced the maximum power-factor, p, from 91 per cent, to 84 per cent., and the apparent efficiency, 7, correspondingly. This seriously limits the usefulness of the device. C. Eddy-current Starting Device 6. Assuming that, instead of using a well-laminated magnetic circuit, and utilizing hysteresis to give the increase of effective resistance with increasing slip, we use a magnetic circuit having very high eddy-current losses: very thick laminations or solid iron, or we directly provide a closed high-resistance secondary winding around the magnetic circuit, which is inserted into the induction-motor secondary for increasing the starting torque. SPEED CONTROL 9 The susceptance of the magnetic circuit obviously follows the same law as when there are no eddy currents. That is: At a given current, ii, energizing the magnetic circuit, the induced voltage, and thus also the voltage producing the eddy currents, is proportional to the frequency. The currents are proportional to the voltage, and the eddy-current losses, therefore, are proportional to the square of the voltage. The eddy- current conductance, g, thus is independent of the frequency. The admittance of a magnetic circuit consuming energy by eddy currents (and other secondary currents in permanent closed circuits), of negligible hysteresis loss, thus is represented, as function of the slip, by the expression: Y' = g-j~ - (11) s Connecting such an admittance in series to the inductionmotor secondary, gives the total secondary impedance: + Z = ', Z-, + + = Ax + 2-j3\ j /*! * , \' (12) Assuming : = g &. (13) That is, 45 phase angle of the exciting circuit of the magnetic circuit at full frequency which corresponds to complete screening of the center of the magnet core we get: Fig. 4 shows the speed curves, and Fig. 5 the load curves, calculated in the standard manner, of a motor with eddy-current starting device in the secondary, of the constants: 6 = 100; - F = o.03 0.3 j; + Z = 0.033 0.1 jf; + Zi = 0.033 0.1 j; 6-3; 10 thus: ELECTRICAL APPARATUS 7. As seen, the torque curve has a very curious shape: a maximum at 7 per cent, slip, and a second higher maximum at standstill. The torque efficiency is very high at all speeds, and practically constant at 82 per cent, from standstill to fairly close of full speed, when it increases. INDUCTION MOTOR .Sj; 2 ~ -033+ 1j : 100 SPEED CONTROL BY EDDIES .o SPEED CURVES FIG. 4. Speed curves of induction motor with eddy-current starting device. But the power-factor is very poor, reaching a maximum of 78 per cent, only, and to get the output from the motor, required rewinding it to give the equivalent of a \/3 times as high voltage. For comparison, in dotted lines as T1 is shown the torque curves of the standard motor, of same maximum torque. As seen, in the motor with eddy-current starting device, the slip at load is very small, that is, the speed regulation very good. Aside from the poor power-factor, the motor constants would be very satisfactory. The low power-factor seriously limits the usefulness of the device. By differently proportioning the eddy-current device to the secondary circuit, obviously the torque curve can be modified SPEED CONTROL 11 and the starting torque reduced, the depression in the torque curve between full-speed torque and starting torque eliminated, etc. Instead of using an external magnetic circuit, the magnetic circuit of the rotor or induction-motor secondary may be used, and in this case, instead of relying on eddy currents, a definite secondary circuit could be utilized, in the form of a second squirrel cage embedded deeply in the rotor iron, that is, a double squirrel-cage motor. FIG. 5. Load curves of induction motor with eddy-current starting device. In the discussion of the multiple squirrel-cage induction motor, Chapter II, we shall see speed-torque curves of the character as shown in Fig. 4. By the use of the rotor iron as magnetic circuit, the impairment of the power-factor is somewhat reduced, so that the multiple squirrel-cage motor becomes industrially important. A further way of utilizing eddy currents for increasing the effective resistance at low speeds, is by the use of deep rotor bars. By building the rotor with narrow and deep slots filled with solid deep bars, eddy currents in these bars occur at higher frequencies, or unequal current distribution. That is, the current flows practically all through the top of the bars at the high 12 ELECTRICAL APPARATUS frequency of low motor speeds, thus meeting with, a high resistance. With increasing motor speed and thus decreasing secondary frequency, the current penetrates deeper into the bar, until at full speed it passes practically uniformly throughout the entire bar, in a circuit of low resistance but somewhat increased reactance. The deep-bar construction, the eddy-current starting device and the double squirrel-cage construction thus are very similar in the motor-performance curves, and the double squirrel cage, which usually is the most economical arrangement, thus will be discussed more fully in Chapter II. II. CONSTANT-SPEED OPERATION 8. The standard induction motor is essentially a constant-speed motor, that is, its speed is practically constant for all loads, decreasing slightly with increasing load, from synchronism at no-load. It thus has the same speed characteristics as the directcurrent shunt motor, and in principle is a shunt motor. In the direct-current shunt motor, the speed may be changed by: resistance in the armature, resistance in the field, change of the voltage supply to the armature by a multivolt supply circuit, as a three-wire system, etc. In the induction motor, the speed can be reduced by inserting resistance into the armature or secondary, just as in the directcurrent shunt motor, and involving the same disadvantages: the reduction of speed by armature resistance takes place at a sacrifice of efficiency, and at the lower speed produced by armature resistance, the power input is the same as it would be with the same motor torque at full speed, while the power output is reduced by the reduced speed. That is, speed reduction by armature resistance lowers the efficiency in proportion to the lowering of speed. The foremost disadvantage of speed control by armature resistance is, however, that the motor ceases to be a constant-speed motor, and the speed varies with the load: with a given value of armature resistance, if the load and with it the armature current drops to one-half, the speed reduction of the motor, from full speed, also decreases to one-half, that is, the motor speeds up, and if the load comes off, the motor runs up to practically full speed. Inversely, if the load increases, the speed slows down proportional to the load. With considerable resistance in the armature, the induction SPEED CONTROL 13 motor thus has rather series characteristic than shunt character- istic, except that its speed is limited by synchronism. Series resistance in the armature thus is not suitable to produce steady running at low speeds. To a considerable extent, this disadvantage of inconstancy of speed can be overcome: (a) By the use of capacity or effective capacity in the motor secondary, which contracts the range of torque into that of approximate resonance of the capacity with the motor inductance, and thereby gives fairly constant speed, independent of the load, at various speed values determined by the value of the capacity. (6) By the use of a resistance of very high negative tempera- ture coefficient in the armature, so that with increase of load and current the resistance decreases by its increase of temperature, and thus keeps approximately constant speed over a wide range of load. Neither of these methods, however, avoids the loss of efficiency incident to the decrease of speed. 9. There is no method of speed variation of the induction motor analogous to field control of the shunt motor, or change of the armature supply voltage by a multivolt supply system. The field excitation of the induction motor is by what may be called armature reaction. That is, the same voltage, impressed upon the motor primary, gives the energy current and the field exciting current, and the field excitation thus can not be varied without varying the energy supply voltage, and inversely. Furthermore, the no-load speed of the induction motor does not depend on voltage or field strength, but is determined by synchronism. The speed of the induction motor can, however, be changed: (a) By changing the impressed frequency, or the effective frequency. (6) By changing the number of poles of the motor. Neither of these two methods has any analogy in the directcurrent shunt motor: the direct-current shunt motor has no fre- quency relation to speed, and its speed is not determined by the number of poles, nor is it feasible, with the usual construction of direct-current motors, to easily change the number of poles. In the induction motor, a change of impressed frequency correspondingly changes the synchronous speed. The effect of a change of frequency is brought about by concatenation of the 14 ELECTRICAL APPARATUS motor with a second motor, or by internal concatenation of the motor: hereby the effective frequency, which determines the no-load or synchronous speed, becomes the difference between primary and secondary frequency. Concatenation of induction motors is more fully discussed in Chapter III. As the no-load or synchronous speed of the induction motor depends on the number of poles, a change of the number of poles changes the motor speed. Thus, if in a 60-cycle induction motor, the number of poles is changed from four to six and to eight, the speed is changed from 1800 to 1200 and to 900 revolutions per minute. This method of speed variation of the induction motor, by changing the number of poles, is the most convenient, and such "multispeed motors" are extensively used industrially, A. Pyro-electric Speed Control 10. Speed control by resistance in the armature or secondary has the disadvantage that the speed is not constant, but at a change of load and thus of current, the voltage consumed by the armature resistance, and therefore the speed changes* To give constancy of speed over a range of load would require a resistance, which consumes the same or approximately the A same voltage at all values of current. resistance of very high negative temperature coefficient does this: with increase of current and thus increase of temperature, the resistance decreases, and if the decrease of resistance is as large as the increase of current, the voltage consumed by the resistance, and therefore the motor speed, remains constant. " Some pyro-electric conductors (see Chapter I, of Theory and Calculation of Electric 77 Circuits ) have negative tempera- ture coefficients sufficiently high for this purpose. Fig. 6 shows the current-resistance characteristic of a pyro-electric conductor, consisting of cast silicon (the same of which the characteristic is given as rod II in Fig. 6 of "Theory and Calculation of Electric Circuits")' Inserting this resistance, half of it and one and one- half of it into the secondary of the induction motor of constants : + - e = 110; 7o = 0.01 0.1 j;Z* 0.1 0.3 Zl = 0.1 +0.3j gives the speed-torque curves shown in Fig. 7, The calculation of these curves is as follows: The speed- torque curve of the motor with short-circuited secondary, r = 0, SPEED CONTROL 15 FIG. 6. Variation of resistance of pyro-electric conductor, with current. PYRO-ELECTRIC RESISTANCE IN SEC9NDARY OF INDUCTION MOTOR, < =110. SPE"E'D CONTROL BY PYRO' ELECTRIC ^CONDUCTOR. SPEED CURVES. 0.1 3> 7. 0.2 0.3 0.4 0.5 0.6 0.7 0O.8a 0.9 1- Speed control of induction motor by pyro-electric conductor, speed curves 16 ELECTRICAL APPARATUS is calculated in the usual way as described on page 318 of " Theoretical Elements of Electrical 7' Engineering. For any value of slip, s, and corresponding value of torque, T, the secondary vV + = current is ii e a2 2. To this secondary current corre- sponds, by Fig. 6, the resistance, r, of the pyro-electric conductor, and the insertion of r thus increases the slip in proportion to the increased secondary resistance: -^r^' where n = 0.1 in the present instance. This gives, as corresponding to the torque, !F, the slip: + r TI where s = slip at torque, Tf with short-circuited , armature, or resistance, r\. As seen from Fig. 7, very close constant-speed regulation is produced by the use of the pyro-electric resistance, over a wide range of load, and only at light-load the motor speeds up. Thus, good constant-speed regulation at any speed below synchronism, down to very low speeds, would be produced at a corresponding sacrifice of efficiency, however by the use of suitable pyro-electric conductors in the motor armature. The only objection to the use of such pyro-electric resistances is the difficulty of producing stable pyro-electric conductors, and permanent terminal connections on such conductors. B. Condenser Speed Control 11. The reactance of a condenser is inverse proportional to the frequency, that of an inductance is directly proportional to the frequency. In the secondary of the induction motor, the frequency varies from zero at synchronism, to full frequency at standstill. If, therefore, a suitable capacity is inserted into the secondary of an induction motor, there is a definite speed, at which inductive reactance and capacity reactance are equal and opposite, that is, balance, and at and near this speed, a large current is taken by the motor and thus large torque developed, while at speeds considerably above or below this resonance speed, the current and thus torque of the motor are small. The use of a capacity, or an effective capacity (as polarization cell or aluminum cell) in the induction-motor secondary should therefore afford, at least theoretically, a means of speed control by varying the capacity. SPEED CONTROL 17 Let, In an induction motor: YQ = g jb = primary exciting admittance; + ZQ = TQ jxQ = primary self-inductive impedance; + Zi = TI jxi = internal self-inductive impedance, at full frequency ; and let the condenser, C, be inserted into the secondary circuit. The capacity reactance of C is at full frequency, and k- 5 at the frequency of slip, s. The total secondary impedance, at slip, s, thus is : _ _ _ _ Zi'-ri+^as!-*) and the secondary current: ___ sE_ _ . f $e (2) where : = - jE (ai ja 2), a = 1 m a2 = m (4) ' (k\SXi -j The further calculation of the condenser motor, then, is the same as that of the standard motor. 1 12. Neglecting the exciting current: /oo = EY the primary current equals the secondary current: 7o = /i and the primary impressed voltage thus is: 1 " Theoretical Elements of Electrical Engineering/' 4th edition, p. 318. 2 18 ELECTRICAL APPARATUS and, substituting (3) and rearranging, gives: or, absolute: + + ^i + (ri sr ) jf SXQ - - j + + + 2 sro) (sxi sx - - The torque of the motor is : T = e2ai and, substituting (4) and (6): (/c\ + s#i sa;o J As seen, this torque is a maximum in the range of slip, s, where the second term in the denominator vanishes, while for values of s, materially differing therefrom, the second term in the denominator is large, and the torque thus small. That is, the motor regulates for approximately constant speed near the value of s, given by: + = SXl SXQ _ * o that is: / I TTX (8) and = so 1, that is, the motor gives maximum torque near standstill, for: + k = XQ XL (9) 13. As instances are shown, in Fig. 8, the speed-torque curves of a motor of the constants: - Fo - 0.01 0.1 j, + Z = Zi = 0.1 0.3 j, SPEED CONTROL 19 for the values of capacity reactance : S = k 0, 0.012, 0,048, 0.096, 0.192, 0.3, 0.6 denoted respectively by 1, 2, 3, 4, 5, 6, 7. The impressed voltage of the motor is assumed to be varied with the change of capacity, so as to give the same maximum torque for all values of capacity. The volt-ampere capacity of the condenser is given, at the frequency of slip, s, by: substituting (3) and (6), this gives: = ' + + (ri r )> + *CO - (*BI ~)' SPEED CONTROL OF INDUCTION MOTOR BY CONDENSER IN SECONDARY CAPACITY* REACTANCE k: d'). 0; (2) ".012; (3) .048; (4-) .096, (5) .192; (6) .3; (7) .6 7 \ \ 1.0 .8 xi FIG. 8. Speed control of induction motor by condenser in secondary circuit. Speed curves. and, compared with (7), this gives: At full frequency, with the same voltage impressed upon the condenser, its volt-ampere capacity, and thus its 60-eycle rating, would be: 20 ELECTRICAL APPARATUS As seen, a very large amount of capacity is required for speed control. This limits its economic usefulness, and makes the use of a cheaper form of effective or equivalent capacity desirable. C. Multispeed Motors 14. The change of speed by changing the number of poles, in the multispeed induction motor, involves the use of fractionalpitch windings: a primary turn, which is of full pole pitch for a given number of motor poles, is fractional pitch for a smaller number of poles, and more than full pitch for a larger number of poles. The same then applies to the rotor or secondary, if containing a definite winding. The usual and most frequently employed squirrel-cage secondary obviously has no definite number of poles, and thus is equally adapted to any number of poles. As an illustration may be considered a three-speed motor changing between four, six and eight poles. Assuming that the primary winding is full-pitch for the six- polar motor, .that is, each primary turn covers one-sixth of the motor circumference. Then, for the four-polar motor, the % % primary winding is pitch, for the eight-polar motor it is % pitch which latter is effectively the same as pitch. Suppose now the primary winding is arranged and connected as a six-polar three-phase winding. Comparing it with the same primary turns, arranged as a four-polar three-phase wind- ing, or eight-polar three-phase winding, the turns of each phase can be grouped in six sections: Those which remain in the same phase when changing to a winding for different number of poles. Those which remain in the same phase, but are reversed when changing the number of poles. Those which have to be transferred to the second phase. Those which have to be transferred to the second phase in the reverse direction. Those which have to be transferred to the third phase. Those which have to be transferred to the third phase in the reverse direction. The problem of multispeed motor design then is, so to arrange the windings, that the change of connection of the six coil groups of each phase, in changing from one number of poles to another, is accomplished with the least number of switches. SPEED CONTROL 21 15. Considering now the change of motor constants when changing speed by changing the number of poles. Assuming that at all speeds, the same primary turns are connected in series, and are merely grouped differently, it follows, that the selfinductive impedances remain essentially unchanged by a change of the number of poles from n to n'. That is; ZQ = Z'o, Z l = Z'L With the same supply voltage impressed upon the same number of series turns, the magnetic flux per pole remains unchanged by the change of the number of poles. The flux density, therefore, changes proportional to the number of poles : &_ " rf B n] therefore, the ampere-turns per pole required for producing the magnetic flux, also must be proportional to the number of poles: n However, with the same total number of turns, the number of turns per pole are inverse proportional to the number of poles: N^n N n'' In consequence hereof, the exciting currents, at the same impressed voltage, are proportional to the square of the number of poles: _ ZJH) n^_ ioo n2 7 and thus the exciting susceptances are proportional to the square of the number of poles : The magnetic flux per pole remains the same, and thus the magnetic-flux density, and with it the hysteresis loss in the primary core, remain the same, at a change of the number of poles. The tooth density, however, increases with increasing number of poles, as the number of teeth, which carry the same flux per pole, decreases inverse proportional to the number of 22 ELECTRICAL APPARATUS poles. Since the tooth densities must be chosen sufficiently low not to reach saturation at the highest number of poles, and their core loss is usually small compared with that in the primary core itself, it can be assumed approximately, that the core loss of the motor is the same, at the same impressed voltage, regardless of the number of poles. This means, that the exciting conductance, g, does not change with the number of poles. Thus, if in a motor of n poles, we change to n f poles, or by the ratio = ri a > n the motor constants change, approximately: from : = + ZQ 7' JXQ, + = Zi Tl JXi, Fo = g - jb, to : + = ZQ TQ JXQ. + Z = 1 Ti JXL Fo = g ja*b. 16. However, when changing the number of poles, the pitch of the winding changes, and allowance has to be made herefore in the constants: a fractional-pitch winding, due to the partial neutralization of the turns, obviously has a somewhat higher exciting admittance, and lower self-inductive impedance, than a full-pitch winding. As seen, in a multispeed motor, the motor constants at the higher number of poles and thus the lower speed, must be materially inferior than at the higher speed, due to the increase of the exciting susceptance, and the performance of the motor, and especially its power-factor and thus the apparent efficiency, are inferior at the lower speeds. When retaining series connection of all turns for all speeds, and using the same impressed voltage, torque in synchronous watts, and power are essentially the same at all speeds, that is, are decreased for the lower speed and larger number of poles only as far as due to the higher exciting admittance. The actual torque thus would be higher for the lower speeds, and approximately inverse proportional to the speed. As a rule, no more torque is required at low speed than at high speed, and the usual requirement would be, that the multispeed motor should carry the same torque at all its running speeds, that is, give a power proportional to the speed. This would be accomplished by lowering the impressed voltage SPEED CONTROL 23 for the larger number of poles, about inverse proportional to the square root of the number of poles: = e' since the output is proportional to the square of the voltage. The same is accomplished by changing connection from multiple connection at higher speeds to series connection at lower speeds, F or from delta connection at higher speeds, to at lowei speeds. If, then, the voltage per turn is chosen so as to make the actual torque proportional to the synchronous torque at all speeds, that MULTISPEED INDUCTION MOTOR] 4 POLES 1800 REV. FIG. 9. Load curves for nxultispeed induction motor, highest speed, four poles. is, approximately equal, then the magnetic flux per pole and the density in the primary core decreases with increasing number of poles, while that in the teeth increases, but less than at constant impressed voltage. The change of constants, by changing the number of poles by the ratio : n' = a n thus is: from: 6 , YQ, to aZo and the characteristic constant is changed from $ to a?$. 17. As numerical instance may be considered a 60-cycle 100- volt motor, of the constants: ELECTRICAL APPARATUS MULTISPEED INDUCTION 6 POLES 1200 REV. MULTISPEED INDUCTION MOTORAM/8 FIG. 10. Load curve of multispeed induction motor, middle speed, six poles. FIG. 11. Load curves of multispeed induction motor, low speed, eight poles. REV. MULTISPEED INDUCTION MOTOR 1800 4-6-8-POLES. 1800-1200-900 REV. FIG. 12. Comparison of load curves of three-speed induction motor. SPEED CONTROL 25 + + Four poles, 1800 Z rev.: Q = r jx G = 0.1 0.3 j; + Zi = ri+jxi = 0.1 - 0.3 j; YQ = g jb = 0.01 0.05 j. + + Six poles, 1200 Z rev.: Q = r j# = 0.15 0.45 j; + + Zi = n jxi = 0.15 0.45 j; F = g -jb = 0.0067 ~ 0.0667 j. + + Eight poles, 900 Z rev.: Q = r = jx<> 0.2 0.6 j; + + # - - = 7 Zi = ri ja?! 0.2 0.6 j; = g = 0.005 0.1 j. Figs. 9, 10 and 11 show the load curves of the motor, at the three different speeds. Fig. 12 shows the load curves once more, MULTISPEED INDUCTION MOTOR 4-6-8 POLES 1800-1200-900 REV not 100L .KW. .9. -90. S._80. _50. _30. 100 200 300 400 500 600 700 800 900100011001200130014001500160017001800 FIG, 13. Speed torque curves of three-speed induction motor. with all three motors plotted on the same sheet, but with the torque in synchronous watts (referred to full speed or fourpolar synchronism) as abscissae, to give a better comparison. S denotes the speed, I the current, p the power-factor and 7 the apparent efficiency. Obviously, carrying the same load, that is, giving the same torque at lower speed, represents less power output, i and in a multispeed motor the maximum power output should be approximately proportional to the speed, to operate at all speeds at the same part of the motor characteristic. Therefore, a comparison of the different speed curves by the power output does not show the performance as well as a comparison on the basis of torque, as given in Fig. 12. 26 ELECTRICAL APPARATUS As seen from Fig. 12, at the high speed, the motor performance is excellent, but at the lowest speed, power-factor and apparent efficiency are already low, especially at light-load. The three current curves cross : at the lowest speed, the motor takes most current at no-load, as the exciting current is highest ; at higher values of torque, obviously the current is greatest at the highest speed, where the torque represents most power. The speed regulation is equally good at all speeds. Fig. 13 then shows the speed curves, with revolutions per minute as abscissae, for the three numbers of poles. It gives current, torque and power as ordinates, and shows that the maximum torque is nearly the same at all three speeds, while current and power drop off with decrease of speed. CHAPTER II MULTIPLE SQUIRREL-CAGE INDUCTION MOTOR 18. In an induction motor, a high-resistance low-reactance secondary is produced by the use of an external non-inductive resistance in the secondary, or in a motor with squirrel-cage secondary, by small bars of high-resistance material located close to the periphery of the rotor. Such a motor has a great slip of speed under load, therefore poor efficiency and poor speed regulation, but it has a high starting torque and torque at low and intermediate speed. With a low resistance fairly high-reactance secondary, the slip of speed under load is small, therefore efficiency and speed regulation good, but the starting torque and torque at low and intermediate speeds is low, and the current in starting and at low speed is large. To combine good start- ing with good running characteristics, a non-inductive resistance is used in the secondary, which is cut out during acceleration. This, however, involves a complication, which is undesirable in many cases, such as in ship propulsion, etc. By arranging then two squirrel cages, one high-resistance low-reactance one, consisting of high-resistance bars close to the rotor surface, and one of low-resistance bars, located deeper in the armature iron, that is, inside of the first squirrel cage, and thus of higher reactance, a "double squirrel-cage induction motor' 7 is derived, which to some extent combines the characteristics of the highresistance and the low-resistance secondary. That is, at starting and low speed, the frequency of the magnetic flux in the armature, and therefore the voltage induced in the secondary winding is high, and the high-resistance squirrel cage thus carries considerable current, gives good torque and torque efficiency, while the low-resistance squirrel cage is ineffective, due to its high reactance at the high armature frequency. At speeds near synchronism, the secondary frequency, being that of slip, is low, and the secondary induced voltage correspondingly low. The high-resistance squirrel cage thus carries little current and gives little torque. In the low-resistance squirrel cage, due to its low reactance at the low frequency of slip, in spite of the relatively 27 28 ELECTRICAL APPARATUS low induced e.m.f., considerable current is produced, which is effective in producing torque. Such double squirrel-cage induction motor thus gives a torque curve, which to some extent is a superposition of the torque curve of the high-resistance and that of the low-resistance squirrel cage, has two maxima, one at low speed, and another near synchronism, therefore gives a fairly good torque and torque efficiency over the entire speed range from standstill to full speed, that is, combines the good features of both types. Where a very high starting torque requires locating the first torque maximum, near standstill, and large size and high efficiency brings the second torque maximum very close to synchronism, the drop of torque between the two maxima may be considerable. This is still more the case, ivhen the motor is required to reverse at full speed and full power, that is, a very high torque is required at full speed backward, or at or near = may slip $ 2. In this case, a triple squirrel cage be used, that is, three squirrel cages inside of each other: the outermost, of high resistance and low reactance, gives maximum torque below standstill, at backward rotation; the second squirrel cage, of medium resistance and medium reactance, gives its maximum torque at moderate speed; and the innermost squirrel cage, of low resistance and high reactance, gives its torque at full speed, near synchronism. Mechanically, the rotor iron may be slotted down to the inner- most squirrel cage, so as to avoid the excessive reactance of a closed magnetic circuit, that is, have the magnetic leakage flux or self-inductive flux pass an air gap. 19. In the calculation of the standard induction motor, it is usual to start with the mutual magnetic flux, $, or rather with the voltage induced by this flux, the mutual inductive voltage $= e, as it is most convenient, with the mutual inductive voltage, e, as starting point, to pass to the secondary current by the self-inductive impedance, to the primary current and primary impressed voltage by the primary self-inductive impedance and exciting admittance. In the calculation of multiple squirrel-cage induction motors, it is preferable to introduce the true induced voltage, that is, the voltage induced by the resultant magnetic flux interlinked with the various circuits, which is the resultant of the mutual and the self-inductive magnetic flux of the respective circuit. This permits starting with the innermost squirrel cage, and INDUCTION MOTOR 29 gradually building up to the primary, circuit. The advantage hereof is, that the current in every secondary circuit is in phase with the true induced voltage of this circuit, and is = ii > TI where TI is the resistance of the circuit. As ei is the voltage induced by the resultant of the mutual magnetic flux coming from the primary winding, and the self-inductive flux corre- sponding to the i&i of the secondary, the reactance, x it does not enter any more in the equation of the current, and e\ is the voltage due to the magnetic flux which passes beyond the cir- cuit in which e\ is induced. In the usual induction-motor theory, the mutual magnetic flux, $, induces a voltage, E, which produces a current, and this current produces a self-inductive flux, $'j, giving rise to a counter e.m.f. of self-induction I&i, which sub- tracts from E. However, the self-inductive flux, <'i, interlinks with the same conductors, with which the mutual flux, $, inter- links, and the actual or resultant flux interlinkage thus is $1 = < E $'i, and this produces the true induced voltage ei = I&i, from which the multiple squirrel-cage calculation starts. 1 Double Squirrel-cage Induction Motor 20. Let, in a double squirrel-cage induction motor: $2 = true induced voltage in inner squirrel cage, reduced to full frequency, /2 = current, and + Zz r2 jx* = self-inductive impedance at full frequency, reduced to the primary circuit. $i = true induced voltage in outer squirrel cage, reduced to full frequency, li = current, and + Zi TI jxi = self-inductive impedance at full frequency, reduced to primary circuit. $ voltage induced in secondary and primary circuits by mutual magnetic flux, I$Q = voltage impressed upon primary, Jo = primary current, + ZQ = r jxo = primary self -inductive impedance, and = YO ff jb = primary exciting admittance. ^ee' "Electric Circuits", Chapter XII. Beactance of Induction Apparatus, 30 ELECTRICAL APPARATUS The leakage reactance, x*, of the inner squirrel cage is that due to the flux produced by the current in the inner squirrel cage, which passes between the two squirrel cages, and does not include the reactance due to the flux resulting from the current, J2, which passes beyond the outer squirrel cage, as the latter is mutual reactance between the two squirrel cages, and thus meets the reactance, xi. It is then, at slip s: 72 = ^- (1) = /i /TV+ + V = I T J TV /2 /I -*OV 0) W/Q^ and: + + E $ = 3 jo;i (/i /) = TTT T j[T/ Q jT^ i f yT ^r/ QJt Q (5) /\ \ v// The leakage flux of the outer squirrel cage is produced by the + m.m.f. of the currents of both squirrel cages, /i /2, and the + reactance voltage of this squirrel cage, in (5), thus IBJXI (/i /a)- E As seen, the difference between l and |J2 is the voltage in- duced by the flux which leaks between the two squirrel cages, in the path of the reactance, #2, or the reactance voltage, #2/2? the $ difference between and #1 is the voltage induced by the rotor flux leaking outside of the outer squirrel cage. This has the + m.m.f. /i /a> and the reactance xi, thus is the reactance voltage + E $ #1 (/i /2). The difference between Q and is the voltage consumed by the primary impedance: Z /o. (4) and (5) .are the voltages reduced to full frequency; the actual voltages are s times as high, but since all three terms in these equations are induced voltages, the 5 cancels. 21. From the equations (1) to (6) follows: (7) (9) INDUCTION MOTOR where: a\ = 1 TlT-2, -- --- -- -= &/, 2 5o ! -J- J. 2 \ri r2 r2 thus the exciting current : /oo = YoE ^ E,(g~- jb) (ai = ^2 (6i+j62), where: and the total primary current is (3) : where: 0,=- +i - 31 (10) (ID (12) (13) (14) and the primary impressed voltage (6) : (r where : + (16) hence, absolute: - (17) io + "*0 i 9 2. /I O\ (lo) 22. The torque of the two squirrel cages is given by the product of current and induced voltage in phase with it, as: (19) (20) 32 ELECTRICAL APPARATUS hence, the total torque : D - D2 + D l9 (21) and the power output P = (i - s) D. (22) (Herefrom subtracts the friction loss, to give the net power output.) The power input is: Po = /$o, IQ/' + = e^dd, cjd*), (23) and the volt-ampere input : Q= T> Herefrom then follows the power-factor 7V7* the torque effi- ciency p-, the apparent torque efficiency -Q, the power efficiency P P ~- and the apparent power efficiency pr -TO Hf 23. As illustrations are shown, in Pigs. 14 and 15, the speed curves and the load curves of a double squirrel-cage induction motor, of the constants: e = 110 volts; + ZQ = 0.1 0.3 j; + Zl = 0.5 0.2 j; + 2 = 0.08 0.4 j; 7 - = 0.01 0.1 j; the speed curves for the range from $ = = to s 2, that is, from synchronism to backward rotation at synchronous speed. The total torque as well as the two individual torques are shown on the speed curve. These curves are derived by calculating, for the values of $': = 5 0, 0.01, 0.02, 0.05, 0.1, 0.15, 0.2, 0.3, 0.4, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6, 1.8, 2.0, INDUCTION MOTOR 33 DOUBLE SQUIRREL CAGE INDUCTION MOTOR SPEED CURVES -1.0-.9-8 -.7 -.6 -.5 -.4 -.3 -.2 -.1 .1 .2 .3 .4 .5 ,6 .7 .8 .9 1.0 FIG. 14. Speed curves of double squirrel-cage induction motor. DOUBLE SQUIRREL CAGE INDUCTION MOTOR LOAD CURVES 1.0 1.5 20 2.5 30 35 40 45 50 55 6 KW FIG. 15. Load curves of double squirrel-cage induction motor. 34: the values: ELECTRICAL APPARATUS - a I + / r3 V r2 + D = Di Z> 2 , P = (1 - s) D, Po = e2 and: P_ ,0 P D Po Pa'Po'C' C' Q* Triple Squirrel-cage Induction. Motor 24. Let: E + * = flux, = voltage, / = current, and Z = r = jx self- inductive impedance, at full frequency and reduced to primary circuit, and let the quantities of the innermost squirrel cage be denoted by index 3, those of the middle squirrel cage by 2, of the outer squirrel cage by 1, of the primary circuit by 0, and the mutual inductive quantities without index. 7 Also let: == g jb = primary exciting admittance. It is then, at slip s: current in the innermost squirrel cage: ^ - /.- T S 3- , m(D INDUCTION MOTOR 35 current in the middle squirrel cage: ~ /2 = 2 ; (v 2)y r2 current in the outer squirrel cage: = -1 ~J~~> (3) primary current: + + + /o = /3 /2 /i Y Q E. (4) The voltages are related by: 777 __ 771 I * T 777 __ 777 * | /T 777 777 * /T [ + EQ = E Zo/0, i^ 7" \ T j [ 7* \ /C\ /iC\ f7\ (8) where #3 is the reactance due to the flux leakage between the third and the second squirrel cage; x% the reactance of the leak- age flux between second and first squirrel cage; a?i the reactance of the first squirrel cage and XQ that of the primary circuit, that + is, X* XQ corresponds to the total leakage flux between primary and outer most squirrel cage. E E$, 2 and $1 are the true induced voltages in the three squirrel E cages, the mutual inductive voltage between primary and secondary, and E Q the primary impressed voltage 25. From equations (1) to (8) then follows: (9) (10) where: ai = 1 2 i , (12) 36 ELECTRICAL APPARATUS #= where: + = 6 2 &2 ^ I TB thus the exciting current : loo = Y E + x J6 2 ) (flf Jb) where : = C] and the total primary current, by (4) : where: where: 4- JL S 2iC 3 , = + + #3 (di jdz) fro ja;o) thus, the primary impressed voltage, by (8) : where: = E3 (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) INDUCTION MOTOR hence, absolute: _ ^+ 2 (J 2 , ei = e3 AoTlz"?. 26. The torque of the innermost squirrel cage thus is; * = *; that of the middle squirrel cage: 37 (25) (26) (27) (28) and that of the outer squirrel cage: D, = s-; (30) the total torque of the triple squirrel-cage motor thus is: D = D, + D2 + Da, and the power: P = (1 - a) D, (31) (32) the power input is: Po = /#,, / /' = + 2 es (diflri dtfj), (33) and the volt-ampere input : Q = e io. (34) T> Herefrom then follows the power-factor -j> the torque effi- ciency p-, JT o apparent torque efficiency Vyj* power efficiency *pr and apparent power efficiency TT 27. As illustrations are shown, in Figs. 16 and 17, the speed and the load curves of a triple squirrel-cage motor with the constants: e = 110 volts; Z = 0.1 +0.3J; Z1 = 0.8 +0.1J; Z2 = 0.2 +0.3j; Z = + 3 0.05 0.8 ,7; Fo = 0.01 - 0.1 j; 38 ELECTRICAL APPARATUS TRIPLE SQUIRREL CAGE INDUCTION MOTOR SPEED CURVES -1.0-9 -.8"-.7 -.6 -.5 -.4 -.3 -.2 -.1 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0 FIG. 16. Speed curves of triple squirrel-cage induction motor. TRIPLE SQUIRREL CAGE INDUCTION MOTOR LOAD CURVES FIG. 17. Load curves of triple squirrel-cage induction motor. INDUCTION MOTOR 39 the speed curves are shown from s ~ to s = 2, and on them, the individual torques of the three squirrel cages are shown in addition to the total torque. These numerical values are derived by calculating, for the values of s: = s 0, 0.01, 0.02, 0.05, 0.1, 0.15, 0.20, 0.30, the values: 0.40, 0.60, 0.80, 1.0, 1.2, 1.4, 1.6, 1.8, 20, =. 1 S*X%Xz > 2 Co +2 firs! = bi &i - t>2 == &2 ~ +.I5" ,+5~.+ Cl, OU 2 O *t- 3 O j ( Oi = 63 1+ = + 2 ei 2 2 ea (fli a2 2 ), rz D = Di + D2 + D3, P = (1 - s) D, ~ V 60^0? and P_ I) P D Po PO'PO'Q'Q'Q' CHAPTER III CONCATENATION Cascade or Tandem Control of Induction Motors 28. If of two induction motors the secondary of the first motor is connected to the primary of the second motor, the second machine operates as a motor with the voltage and frequency impressed upon it by the secondary of the first machine. The first machine acts as general alternating-current transformer or frequency converter (see Chapter XII), changing a part of the primary impressed power into secondary electrical power for the supply of the second machine, and a part into mechanical work. The frequency of the secondary voltage of the first motor, and thus the frequency impressed upon the second motor, is the frequency of slip below synchronism, s. The frequency of the secondary of the second motor is the difference between its impressed frequency, s, and its speed. Thus, if both motors are connected together mechanically, to turn at the same speed, 1 5, and have the same number of poles, the secondary fre- quency of the second motor is 2 s 1, hence equal to zero at = s 0.5. That is, the second motor reaches its synchronism at half speed. At this speed, its torque becomes zero, the power component of its primary current, and thus the power component of the secondary current of the first motor, and thus also the torque of the first motor becomes zero. That is, a system of two concatenated equal motors, with short-circuited secondary of the second motor, approaches half synchronism at no-load, in the same manner as a single induction motor approaches synchronism. With increasing load, the slip below half syn- chronism increases. In reality, at half synchronism, s = 0.5, there is a slight torque produced by the first motor, as the hysteresis energy current of the second motor comes from the secondary of the first motor, and therein, as energy current, produces a small torque. More generally, any pair of induction motors connected in concatenation divides the speed so that the sum of their two 40 CONCATENATION 41 respective speeds approaches synchronism at no-load; or, still more generally, any number of concatenated induction motors run at such speeds that the sum of their speeds approaches synchronism at no-load. With mechanical connection between the two motors, con- catenation thus offers a means of operating two equal motors at full efficiency at half speed in tandem, as well as at full speed, in parallel, and thereby gives the same advantage as does series parallel control with direct-current motors. With two motors of different number of poles, rigidly con- nected together, concatenation allows three speeds : that of the one motor alone, that of the other motor alone, and the speed of concatenation of both motors. Such concatenation of two motors of different numbers of poles, has the disadvantage that at the two highest speeds only one motor is used, the other idle, and the apparatus economy thus inferior. However, with certain ratios of the number of poles, it is possible to wind one and the same motor structure so as to give at the same time two different numbers of poles: For instance, a four-polar and an eightpolar winding; and in this case, one and the same motor structure can be used either as four-polar motor, with the one winding, or as eight-polar motor, with the other winding, or in concatena- tion of the two windings, corresponding to a twelve-polar speed. Such "internally concatenated " motors thus give three different speeds at full apparatus economy. The only limitation is, that only certain speeds and speed ratios can economically be produced by internal concatenation. 29. At half synchronism, the torque of the concatenated couple of two equal motors becomes zero. Above half synchronism, the second motor runs beyond its impressed frequency, that is, becomes a generator. In this case, due to the reversal of current in the secondary of the first motor (this current now being out- flowing or generator current with regards to the second motor) its torque becomes negative also, that is, the concatenated couple becomes an induction generator above half synchronism. When approaching full synchronism, the generator torque of the second motor, at least if its armature is of low resistance, becomes very small, as this machine is operating very far above its synchronous speed. With regards to the first motqr ( 2 it thus begins to act merely as an impedance in the secondary circuit, that is, the first machine^becomes a motor dg&m. ' Thus, somewhere between 42 ELECTRICAL APPARATUS half synchronism and synchronism, the torque of the first motor becomes zero, while the second motor still has a small negative or A generator torque. little above this speed, the torque of the concatenated couple becomes zero about at two-thirds synchronism with a couple of low-resistance motors and above this, the concatenated couple again gives a positive or motor torque though the second motor still returns a small negative torque and again approaches zero at full synchronism. Above full synchronism, the concatenated couple once more becomes generator, but practically only the first motor contributes to the generator torque above and the motor torque below full synchronism. Thus, while a concatenated couple of induction motors has two operative motor speeds, half synchronism and full synchronism, the latter is uneconomical, as the second motor holds back, and in the second or full synchronism speed range, it is more economical to cut out the second motor altogether, by short-circuiting the secondary terminals of the first motor. With resistance in the secondary of the second motor, the maximum torque point of the second motor above half syn- chronism is shifted to higher speeds, nearer to full synchronism, and thus the speed between half and full synchronism, at which the concatenated couple loses its generator torque and again becomes motor, is shifted closer to full synchronism, and the motor torque in the second speed range, below full synchronism, is greatly reduced or even disappears. That is, with high resistance in the secondary of the second motor, the concatenated couple becomes generator or brake at half synchronism, and remains so at all higher speeds, merely loses its braking torque when approaching full synchronism, and regaining it again beyond full synchronism. m The speed torque curves of the concatenated couple, shown Fig. 18, with low-resistance armature, and in Fig. 19, with high resistance in the armature or secondary of the second motor, illustrate this. 30. The numerical calculation of a couple of concatenated induction motors (rigidly connected together on the same shaft or the equivalent) can be carried out as follows : Let: n s* number of pairs of poles of the first motor, nf = number of pairs of poles of the second motor. CONGA TENA TION 43 a= = ratio of poles, (1) / = supply frequency. Full synchronous speed of the first motor then is: So = (2) of the second motor: = n if: 5 = slip of first motor below full synchronism. The primary circuit of the first motor is of full frequency. The secondary circuit of the first motor is of frequency s. The primary circuit of the second motor is of frequency s. The secondary circuit of the second motor is of frequency f s = + s (1 a) a. Synchronism of concatenation is reached at: Let thus: _ 1+a CQ = voltage impressed of first motor primary; YQ g jb = exciting admittance of first motor; F'o = g' jV = exciting admittance of second motor; + Zo = TQ JXQ = self-inductive impedance of first motor primary; + Z'Q r'o jx'o = self-inductive impedance of second motor primary; + Zi = TI jxi = self-inductive impedance of first motor second- ary; + Z'\ = r\ jx\ = self-inductive impedance of second motor secondary. Assuming all these quantities reduced to the same number of turns per circuit, and to full frequency, as usual. If: e = counter e.m.f. generated in the second motor by its mutual magnetic flux, reduced to full frequency. It is then: secondary current of second motor: _ _ r, ^e + - [s (1 a) a] e _- , 46 where : -ELECTRICAL APPARATUS - + rMaq . . _ o) -a] 1 QF a m + + m sV = rV (s (1 -2 a) a) ; (10) exciting current of second motor: /'oo-eF' = e (/-#'), (ID hence, primary current, of second motor, and also secondary current of first motor: + = = /'o /i /'] /'oo = e (61 - #), (12) where : ^ &!-! + + &2 = a* o, (13) the impedance of the circuit comprising the primary of the second, and the secondary of the first motor, is: + + + + Z = ZS Z' 2 = (n r' ) js (a?, ^o), (14) hence, the counter e.m.f., or induced voltage in the secondary of the first motor, of frequency is : + sE l - se hZ, hence, reduced to full frequency: where: = + e (ci jc a), (15) - 'o) 61 s 33. The primary exciting current of the first motor is: /"Ci = V 00 &IJ- ** e(di jds), where: j r - _ i ^ 1 (16) (17) (18) CONGA TENA TION 47 thus, the total primary current of the first motor, or supply current : + Jo = Ii /oo = - e (/i #2), (19) where : + = fz &2 < and the primary impressed voltage of the first motor, or supply voltage: where : and, absolute: thus: yz - mf '-'a I *v(jj i 'OJ2JI + e = e VVi2 2 ^2 , (21) (22) (23) Tp" ^24) Substituting now this value of e in the preceding, gives the values of the currents and voltages in the different circuits. 34. It thus is, supply current: power input: Po = /#>, /o/' = - e2 (figi volt-ampere input: and herefrom power-factor, etc. The torque of the second motor is: rpr __. / T It = a e2 3. The torque of the first motor is : 48 ELECTRICAL APPARATUS hence, the total torque of the concatenated couple: + + - T = Z" = Ti 6 2 (a, Ci/i C 2 /2), and herefrom the power output : P = (I - s) T, thus the torque and power efficiencies and apparent efficiencies, etc. 35. As instances are calculated, and shown in Fig. 18, the speed + 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0-0.1-0.2-0.3-0.4-0.5-0.0-0.7 FIG. 18. Speed torque curves of concatenated couple with low resistance secondary. torque curves of the concatenated couple of two equal motors: = a 1, of the constants: e == 110 volts. Y = Y' - = 0.01 0.1 j; + Z Q = Z' = 0.1 0.3 j; + Z l = Z\ = 0.1 0.3J. Fig. 18 also shows, separately, the torque of the second motor, and the supply current. Fig. 19 shows the speed torque curves of the same concate- nated couple with an additional resistance r = 0.5 inserted into the secondary of the second motor. The load curves of the same motor, Fig. 18, for concatenated running, and also separately the load curves of either motor, CONCATENATION 49 are given on page 358 of " Theoretical Elements of Electrical Engineering." 36. It is possible in concatenation of two motors of different number of poles, to use one and the same magnetic structure for both motors. Suppose the stator is wound with an n-polar primary, receiving the supply voltage, and at the same time with an nf polar short-circuited secondary winding. The rotor is wound with an n-polar winding as secondary to the n-polar primary winding, but this n-polar secondary winding is not short-circuited, but connected to the terminals of a second FIG. 19. Speed-torque curves of concatenated couple with resistance in second secondary. n'-polar winding, also located on the rotor. This latter thus receives the secondary current from the n-polar winding and acts as n'-polar primary to the short-circuited stator winding as secondary. This gives an n-polar motor concatenated to an n'-polar, and the magnetic structure simultaneously carries an n-polar and an n'-polar magnetic field. With this arrangement of " internal concatenation," it is essential to choose the number of poles, n and n', so that the two rotating fields do not interfere with each other, that is, the n'-polar field does not induce in the n-polar winding, nor the n-polar field in the n'-polar winding. This is the case if the one field has twice as many poles as the other, for instance a four-polar and an eight-polar field. If such a fractional-pitch winding is used, that the coil pitch is suited for an n-polar as well as an n'-polar winding, then the same winding can be used for both sets of poles. In the stator, the equipotential points of a 2p-polar winding are points of opposite polarity of a p-polar winding, and thus, by connecting together the equipotential points of a 2 p-polar primary winding, 50 ELECTRICAL APPARATUS this winding becomes at the same time a p-polar short-circuited winding. On the rotor, in some slots, the secondary current of the tt-polar and the primary current of the n'-polar winding flow In the same direction, in other slots flow in opposite direction, thus neutralize in the latter, and the turns can be omitted In concatenation but would be put in for use of the structure as single motor of n, or of nf poles, where such is desired. Thus, on the rotor one single winding also is sufficient, and this arrange- ment of internal concatenation with single stator and single rotor winding thus is more efficient than the use of two separate motors, and gives somewhat better constants, as the self-inductive im- pedance of the rotor is less, due to the omission of one-third of the turns in which the currents neutralize (Hunt motor). The disadvantage of this arrangement of internal concatenation with single stator and rotor winding is the limitation of the avail- able speeds, as it is adapted only to 4 -5- 8 -*- 12 poles and K multiples thereof, thus to speed ratios of 1 * -* H> tbe last being the concatenated speed. Such internally concatenated motors may be used advantage- ously sometime as constant-speed motors, that is, always running in concatenation, for very slow-speed motors of very large number of poles. 37. Theoretically, any number of motors may be concatenated. It is rarely economical, however, to go beyond two motors in concatenation, as with the increasing number of motors, the constants of the concatenated system rapidly become poorer. If: YQ = g j6, = + ZQ 7*0 j$Q, + = Zi ri fai, are the constants of a motor, and we denote : + + + + Z =* ZQ Zi~ (r ri) j Oo a?i) = r + jx then the characteristic constant of this motor which char- acterizes its performance is: if now two such motors are concatenated, the exciting admittance of the concatenated couple is (approximately) : 7' ~ 2 Y f CONCATENATION 51 as the first motor carries the exciting current of the second motor. The total self-inductive impedance of the couple is that of both motors in series : Z" = 2 Z; thus the characteristic constant of the concatenated couple is: = *' y'z' that is, four times as high as in a single motor; in other words, the performance characteristics, as power-factor, etc., are very much inferior to those of a single motor. With three motors in concatenation, the constants of the system of three motors are: Y" -37, Z" = 3 Z, thus the characteristic constant : = y"z" or nine times higher than in a single motor. In other words, the characteristic constant increases with the square of the number of motors in concatenation, and thus concatenation of more than two motors would be permissible only with motors of very good constants. The calculation of a concatenated system of three or more motors is carried out in the same manner as that of two motors, by starting with the secondary circuit of the last motor, and building up toward the primary circuit of the first motor, CHAPTER IV INDUCTION MOTOR WITH SECONDARY EXCITATION 38. While in the typical synchronous machine and commu- tating machine the magnetic field is excited by a direct current, characteristic of the induction machine is, that the magnetic field is excited by an alternating current derived from the alternating supply voltage, just as in the alternating-current transformer. As the alternating magnetizing current is a wattless reactive current, the result is, that the alternating-current input Into the induction motor is always lagging, the more so, the larger a part of the total current is given by the magnetizing current. To secure good power-factor in an induction motor, the magnetizing current, that is, the current which produces the magnetic field flux, must be kept as small as possible. This means as small an air gap between stator and rotor as mechanic- ally permissible, and as large a number of primary turns per pole, that is, as large a pole pitch, as economically permissible. In motors, in which the speed compared to the motor out- put is not too low, good constants can be secured. This, however, is not possible in motors, in which the speed is very low, that is, the number of poles large compared with the output, and the pole pitch thus must for economical reasons be kept small as for instance a 100-hp. 60-cycle motor for 90 revolu- tions, that is, 80 poles or where the requirement of an excessive momentary overload capacity has to be met, etc. In such motors of necessity the exciting current or current at no-load which is practically all magnetizing current is a very large part of may full-load current, and while fair efficiencies nevertheless be secured, power-factor and apparent efficiency necessarily are very low. As illustration is shown in Fig. 20 the load curve of a typical 100-hp. 60-cycle 80-polar induction motor (90 revolutions per minute) of the constants: Impressed voltage: Primary exciting admittance: Primary self-inductive impedance: Secondary self-inductive impedance: eo = 500. F = 0.02 0.6 j. + Z = 0.1 0.3 j. + = Zi 0,1 0.3 j. 52 INDUCTION MOTOR 53 As seen, at full-load of 75 kw. output, the efficiency is 80 per cent., which is fair for a slow-speed motor. But the power-factor is 55 per cent., the apparent efficiency only 44 per cent., and the exciting current is 75 per cent, of full- load current. This motor-load curve may be compared with that of a typical induction motor, of exciting admittance: - 7o = 0.01 0.1 j, given on page 234 of " Theory and Calculation of Alternating- current Phenomena" 5th edition, and page 319 of "Theoretical LOW SPEED INDUCTION MOTOR = 500 FIG. 20. Low-speed induction motor, load curves. Elements of Electrical Engineering/' 4th edition, to see the difference. 39. In the synchronous machine usually the stator, in cornmutating machines the rotor is the armature, that is, the element to which electrical power is supplied, and in which electrical power is converted into the mechanical power output of the motor. The rotor of the typical synchronous machine, and the stator of the commutating machine are the field, that is, in them no electric power is consumed by conversion into mechanical work, but their purpose is to produce the magnetic field flux, through which the armature rotates. In the induction machine, it is usually the stator, which is the 54 ELECTRICAL APPARATUS primary, that is, which receives electric power and converts it into mechanical power, and the primary or stator of the induction machine thus corresponds to the armature of the synchronous or commutating machine. In the secondary or rotor of the induction machine, low-frequency currents of the frequency of slip---are induced by the primary, but the magnetic field flux is produced by the exciting current which traverses the primary or armature or stator. Thus the induction machine may be considered as a machine in which the magnetic field is produced by the armature reaction, and corresponds to a synchronous machine, in which the field coils are short-circuited and the field produced by armature reaction by lagging currents in the armature. As the rotor or secondary of the induction machine corresponds structurally to the field of the synchronous or commutating machine, field excitation thus can be given to the induction machine by passing a current through the rotor or secondary and thereby more or less relieving the primary of its function of giv- ing the field excitation. Thus in a slow-speed induction motor, of very high exciting current and correspondingly poor constants, by passing an exciting current of suitable value through the rotor or secondary, the primary can be made non-inductive, or even leading current produced, or with a lesser exciting current in the rotor at least the power-factor increased. Various such methods of secondary excitation have been pro- posed, and to some extent used. 1. Passing a direct current through the rotor for excitation. In this case, as the frequency of the secondary currents is the frequency of slip, with a direct current, the frequency is zero, that is, the motor becomes a synchronous motor. 2. Excitation through commutator, by the alternating supply current, either in shunt or in series to the armature. - At the supply frequency, /, and slip, s,the frequency of rotation and thus of commutation is (1 s)f, and the full frequency cur- - - rents supplied to the commutator thus give in the rotor the effective frequency, / (1 s) / tf, that is, the frequency of slip, thus are suitable as exciting currents. 3. Concatenation with a synchronous motor. If a low-frequency synchronous machine is mounted on the induction-motor shaft, and its armature connected into the indue- INDUCTION MOTOR 55 tion-motor secondary, the synchronous machine feeds low-fre- quency exciting currents into the induction machine, and thereby permits controlling it by using suitable voltage and phase. If the induction machine has n times as many poles as the synchronous machine, the frequency of rotation of the synchro- nous machine is - that of the induction machine, or - How- ti flj ever, the frequency generated by the synchronous machine must be the frequency of the induction-machine secondary currents, that is, the frequency of slip s. Hence : - 1 s ~ S X GQ "'""' y/A\ (9) INDUCTION MOTOR 61 Thus, the minimum possible value of the counter e.m.f., e, is given by equating the square root to zero, as : e = x -e . For a given value of the counter e.m.f., e, that is, constant field excitation, it is, from (7) : or, if the synchronous impedance, x, is very large compared with r, and thus, approximately: (ID The maximum value, which the energy current, ii, can have, at a given counter e.m.f., e y is given by equating the square root to zero, as: - -- t, vU (12) For: = ii 0, or at no-load, it is, by (11): CQ e Equations (9) and (12) give two values of the currents i\ and izf of which one is very large, corresponds to the upper or unstable part of the synchronous motor-power characteristics shown on page 325 of "Theory and Calculation of Alternating- current Phenomena," 5th edition. 43. Denoting, in equation (5) : E - = e' je", (13) and again choosing $0 = eo, as the real axis, (5) becomes : > e = " je ( eo n\ ~ - a&"2) j (ai n*2), (14) and the electric power input into the motor then is: Po = /EQj //' = e<#i, (15) the power output at "the armature conductor is: 62 ELECTRICAL APPARATUS hence by (14): = - - + - Pi ii (e<> n'i xi z ) i* (xii ria), (16) expanded, this gives : - + Pi = e ii r (if 8 i, ) ~ = Po ri*, (17) where: i = total current. That is, the power out- put at the armature conductors is the power input minus the i*r loss. The current in the field is : = io eb, (18) hence, the i 2r loss in the field; of resistance, r\. = *ri e*b*n. (19) The hysteresis loss in the induction motor of mutual induced voltage, e, is : e 2g, or approximately : = P' e Q *g, (20) in the synchronous motor, the nominal induced voltage, e } does not correspond to any flux, but may be very much higher, than corresponds to the magnetic flux, which gives the hysteresis loss, as it includes the effect of armature reaction, and the hysteresis loss thus is more nearly represented by e^g (20). The difference, however, is that in the synchronous motor the hys- teresis loss is supplied by the mechanical power, and not the electric power, as in the induction motor. The net mechanical output of the motor thus is: P = P! - io*ri - P' - = Po- iar - ioVi e 2 g - - - = e ii i*r e 2 6Vi e z g, (21) and herefrom follow efficiency, power-factor and apparent efficiency. 44, Considering, as instance, a typical good induction motor, of the constants : = 60 500 volts; Fo = 0.01 - 0.1 j; + Z = 0.1 0.3J; + Zi 0.1 0.3 j. INDUCTION MOTOR 63 The load curves of this motor, as induction motor, calculated in the customary way, are given in Fig. 22. Converted into a synchronous motor, it gives the constants: Synchronous impedance (1) ; + Z = r+jx = 0.1 10.3 j. Fig. 23 gives the load characteristics of the motor, with the power output as abscissae, with the direct-current excitation, and thereby the counter e.m.f., e, varied with the load, so as to maintain unity power-factor. The calculation is made in tabular form, by calculating for various successive values of the energy current (here also the total current) ii, input, the counter e.m.f., e, by equation (8) : - + 6 2 = (500 0.1 iiY 100.61 2 ii , the power input, which also is the volt-ampere input, the powerfactor being unity, is: = = PO eoii 500 i\. From e follow the losses, by (17), (19) and (20): in armature resistance: in field resistance: hysteresis loss: 0.1 2 ii ; 0.001 62 ; 2.5 kw.; and thus the power output : p - - - = 500 ii 2.5 0.1 2 *i 0.001 e2 and herefrom the efficiency. Fig. 23 gives the total current as i, the nominal induced voltage as e, and the apparent efficiency which here is the true efficiency, as y. As seen, the nominal induced voltage has to be varied very greatly with the load, indeed, almost proportional thereto. That is, to maintain unity power-factor in this motor, the field excitation has to be increased almost proportional to the load. It is interesting to investigate what load characteristics are given by operating at constant field excitation, that is, constant nominal induced voltage, e, as this would usually represent the operating conditions. 64 ELECTRICAL APPARATUS 90 100 110 120 130 140 KW. FIG. 22. Load curves of standard induction motor. - , tion motor converted to synchronous motor. INDUCTION MOTOR 65 Figs. 24 and 25 thus give the load characteristics of the motor, at constant field excitation, corresponding to: in Fig. 24: in Fig. 25: e = 2e ; = e 5e. For different values of the energy current, 3 , from zero up to the maximum value possible under the given field excitation, INDUCTION MOTOR CONSTANT DIRECT CURRENT EXCITATION e =500 = (Z .1 4-10.3 j) SYNCHRONOUS FIG. 24. Load curves Tat constant excitation 2e, of standard induction motor converted to synchronous motor. as given by equation (12), the reactive current, i%, is calculated by equation (11): Fig. 24: Fig. 25: z*2 = 48.5 - V9410 - i?; - i2 = 48.5 - V58,800 2 ^i . The total current then is: the volt-ampere input : the power input: = Q eoi; Po == e ii, 66 ELECTRICAL APPARATUS the power output given by (21), and herefrom efficiency 77, power-factor p and apparent efficient, 7, calculated and plotted. Figs. 24 and 25 give, with the power output as abscissae, the total current input, efficiency, power-factor and apparent efficiency. As seen from Figs. 24 and 25, the constants of the motor as synchronous motor with constant excitation, are very bad: the no-load current is nearly equal to full-load current, and power- INDUCTION MOTOR CONSTANT DIRECT e =CU5ReRENT EXCITATION __ Yj-.Ol-.1J Z?.'l -K3J (Z= .H-10.3J) SYNCHRONOUS FIG. 25. Load curves at constant excitation 5 e, of standard induction motor converted to synchronous motor. factor and apparent efficiency are very low except in a narrow range just below the maximum output point, at which the motor drops out of step. Thus this motor, and in general any reasonably good induction motor, would be spoiled in its characteristics, by converting it into a synchronous motor with constant field excitation. In Fig. 23 are shown, for comparison, in dotted lines, the apparent efficiency taken from Figs. 24 and 25, and the apparent efficiency of the machine as induction motor, taken from Fig. 22. INDUCTION MOTOR 67 45, As further instance, consider the conversion into a synchronous motor of a poor induction motor: a slow-speed motor of very high exciting current, of the constants: 60 = 500; To = 0.02 - 0.6 j; + Z Q = 0.1 0.3 j] + Zi 0.1 0.3J. The load curves of this machine as induction motor are given in Fig. 20. FIG. 26. Load curves of low-speed high-excitation induction motor converted to synchronous motor, at unity power-factor excitation. Converted to a synchronous motor, it lias the constants: Synchronous impedance: + Z = 0.1 1.97 j. Calculated in the same manner, the load curves, when vary- ing the field excitation with changes of load so as to maintain unity power-factor, are given in Fig. 26, and the load curves for constant field excitation giving a nominal induced voltage: e == 1.5 eo are given in Fig. 27. As seen, the increase of field excitation required to maintain 68 ELECTRICAL APPARATUS unity power-factor, as shown by curve e in Fig. 26, while still considerable, is very much less in this poor induction motor, than it was in the good induction motor Figs. 22 to 25. The constant-excitation load curves, Fig. 27, give characteristics, which are very much superior to those of the motor as induction motor. The efficiency is not materially changed, as was to be expected, but the power-factor, p, is very greatly improved at all loads, is 96 per cent, at full-load, rises to unity above full- FIG. 27. Load curve of low-speed high-excitation induction motor converted to synchronous motor, at constant field excitation. load (assumed as 75 kw.) and is given at quarter-load already higher than the maximum reached by this machine as straight induction motor. For comparison, in Fig. 28 are shown the curves of apparent efficiency, with the power output as abscissae, of this slow-speed motor, as: I as induction motor (from Fig. 20); SQ as synchronous motor with the field excitation varying to maintain unity power-factor (from Fig. 26) ; S as synchronous motor with constant field excitation (from Fig. 27). INDUCTION MOTOR 69 As seen, in the constants at load, constant excitation, S, is practically as good as varying unity power-factor excitation, S , drops below it only at partial load, though even there it is very greatly superior to the induction-motor characteristic, /. It thus follows: By converting it into a synchronous motor, by passing a direct current through the rotor, a good induction motor is spoiled, but a poor induction motor, that is, one with very high exciting current, is greatly improved. 50. .40. I INDUCTION MOTOR S SYNCHRONOUS, UNITY POWER FACTOR 8 SYNCHRONOUS, CONSTANT EXCITATION -30. X CSoSYNCHR.CONCAT.INDUCT., UNITY P.P. A C8 SYNCHR.CONCAT.1NDUCT., CONSTANT EX:CciT_20. *4'- CO COMMUTAT.MACH.CONCAT.INDUCTION C CONDENSER IN SECONDARY 20 SO 4 50 60 70 80 90 100 1 .0 120 130 140 150 160 170 180 190 200 FIG. 28. Comparison of apparent efficiency and speed curves of highexcitation induction motor with various forms of secondary excitation, 46. The reason for the unsatisfactory behavior of a good induction motor, when operated as synchronous motor, is found in the excessive value of its synchronous impedance. Exciting admittance in the induction motor, and synchronous impedance in the synchronous motor, are corresponding quantities, representing the magnetizing action of the armature currents. In the induction motor, in which the magnetic field is produced by the magnetizing action of the armature currents, very high magnetizing action of the armature current is desirable, so as to produce the magnetic field with as little magnetizing current as possible, as this current is lagging, and spoils the powerfactor. In the synchronous motor, where the magnetic field is produced by the direct current in the field coils, the magnetizing action of the armature currents changes the resultant field excitation, and thus requires a corresponding change of the field current to overcome it, and the higher the armature reaction, the more 70 ELECTRICAL APPARATUS has the field current to be changed with the load, to maintain proper excitation. That is, low armature reaction is necessary. In other words, in the induction motor, the armature reaction magnetizes, thus should be large, that is, the synchronous reactance high or the exciting admittance low; in the synchronous motor the armature reaction interferes with the impressed field excitation, thus should be low, that is, the synchronous imped- ance low or the exciting admittance high. Therefore, a good synchronous motor makes a poor induction motor, and a good induction motor makes a poor synchronous motor, but a poor induction motor one of high exciting admit- tance, as Fig. 20 makes a fairly good synchronous motor. Here a misunderstanding must be guarded against: in the theory of the synchronous motor, it is explained, that high synchronous reactance is necessary for good and stable synchro- nous-motor operation, and for securing good power-factors at all A loads, at constant field excitation. synchronous motor of low synchronous impedance is liable to be unstable, tending to hunt and give poor power-factors due to excessive reactive currents. This apparently contradicts the conclusions drawn above in the comparison of induction and synchronous motor. However, the explanation is found in the meaning of high and low synchronous reactance, as seen by expressing the synchronous reactance in per cent. : the percentage synchronous reactance is the voltage consumed by full-load current in the synchronous reactance, as percentage of the terminal voltage* When discussing synchronous motors, we consider a synchro- nous reactance of 10 to 20 per cent, as low, and a synchronous reactance of 50 to 100 per cent, as high. In the motor, Figs. 22 to 25, full-load current at 75 kw. out- put is about 180 amp. At a synchronous reactance of x = 10.3, this gives a synchronous reactance voltage at full-load current, of 1850, or a synchronous reactance of 370 per cent. In the poor motor, Figs. 20, 26 and 27, full-load current is about 200 amp., the synchronous reactance x = 1.97, thus the react- ance voltage 394, or 79 per cent., or of the magnitude of good synchronousrinotor operation. That is, the motor, which as induction motor would be consid- ered as of very high exciting admittance, giving a low synchronous impedance when converted into a synchronous motor, would as synchronous motor, and from the viewpoint of synchronous-