II. Polyphase Induction Motor 1. INTRODUCTION 135. The typical induction motor is the polyphase motor. By gradual development from the direct-current shunt motor we arrive at the polyphase induction motor. The magnetic field of any induction motor, whether supplied by polyphase, monocyclic, or single-phase e.m.f., is at normal condition of operation, that is, near synchronism, a polyphase field. Thus to a certain extent all induction motors can be called polyphase machines. When supplied with a polyphase system of e.m.fs. the internal reactions of the induction motor are simplest and only those of a transformer with moving second- ary, while in the single-phase induction motor at the same time a phase transformation occurs, the second or magnetizing phase being produced from the impressed phase of e.m.f. by the rota- tion of the motor, which carries the secondary currents into quadrature position with the primary current. INDUCTION MACHINES 311 The polyphase induction motor of the three-phase or quarter- phase type is the one most commonly used, while single-phase motors have found a more limited application only, and especially for smaller powers. Thus in the following more particularly the polyphase induc- tion machine shall be treated, and the single-phase type discussed only in so far as it differs from the typical polyphase machine. 2. CALCULATION 136. In the polyphase induction motor, Let Y = g — jb = primary exciting admittance, or admit- tance of the primary circuit with open secondary circuit; that is, ge = magnetic power current, be = wattless magnetizing current, where e = counter-generated e.m.f. of the motor; ZQ = r0 + jxQ = primary self -inductive impedance, and Zi = 7*1 + jxi = secondary self-inductive impedance, reduced to the primary by the ratio of turns.1 All these quantities refer to one primary circuit and one corre- sponding secondary circuit. Thus in a three-phase induction motor the total power, etc., is three times that of one circuit, in the quarter-phase motor with three-phase armature 1J^ of the three secondary circuits are to be considered as corresponding to each of the two primary circuits, etc. Let e = primary counter-generated e.m.f., or e.m.f. generated in the primary circuit by the flux interlinked with primary and secondary (mutual induction); s = slip, with the primary fre- quency as unit; that is, s = 0 denoting synchronous rotation, s = l standstill of the motor. We then have 1 — s = speed of the motor secondary as fraction of syn- chronous speed, sf = frequency of the secondary currents, where / = frequency impressed upon the primary; 1 The self -inductive reactance refers to that flux which surrounds one of the electric circuits only, without being interlinked with the other circuits. 312 ELEMENTS OF ELECTRICAL ENGINEERING hence, . se = e.m.f. generated in the secondary. The actual impedance of the secondary circuit at the frequency sf is Zi8 = 7*1 +jsxi; hence, the secondary current is se se where the primary exciting current is /oo =eY = e[g — jb], and the total primary current is /o = e I (ai -f g) — j (a2 + b] where The e.m.f. consumed in the primary circuit by the impedance ZQ is /oZo, the counter-generated e.m.f. is e, hence, the primary terminal voltage is EQ = e + IQZQ = e[l + (bi — j&2) (r0 + jx0)] .= e (ci — jc2), where Ci = 1 + robi + Xobz and c2 = r062 — Xobi. Eliminating complex quantities, we have EQ = e Vci2 + c22, hence, the counter-generated e.m.f. of motor, e = — == , where EQ = impressed e.m.f., absolute value. Substituting this value in the equations of /i, /oo, /o, etc , gives the complex expressions of currents and e.m.fs., and elimi- nating the imaginary quantities we have the primary current, /o = e V&i2 + 622, etc. INDUCTION MACHINES 313 The torque of the polyphase induction motor (or any other motor or generator) is proportional to the product of the mutual magnetic flux and the component of ampere-turns of the sec- ondary, which is in phase with the magnetic flux in time, but in quadrature therewith in direction or space. Since the generated e.m.f. is proportional to the mutual magnetic flux and the num- ber of turns, but in quadrature thereto in time, the torque of the induction motor is proportional also to the product of the gen- erated e.m.f. and the component of secondary current in quadra- ture therewith in time and in space. Since /i = e (a\ — ja2) is the secondary current corresponding to the generated e.m.f. e, the secondary current in the quadrature position thereto in space, that is, corresponding to the e.m.f. je, is jli = e(a2 and die is the component of this current in quadrature in time with the e.m.f. e. Thus the torque is proportional toe X die, or D = ezdi n2 + s*xi* ' (ex2 + c22) (n2 + sV) This value D is in its dimension a power, and it is the power which the torque of the motor would develop at synchronous speed. 137. In induction motors, and in general motors which have a definite limiting speed, it is preferable to give the torque in the form of the power developed at the limiting speed, in this case synchronism, as "synchronous watts," since thereby it is made independent of the individual conditions of the motor, as its number of poles, frequency, etc., and made comparable with the power input, etc. It is obvious that when given in synchronous watts, the maximum possible value of torque which could be reached, if there were no losses in the motor, equals the power input. Thus, in an induction motor with 9000 watts power input, a torque of 7000 synchronous watts means that % of the maximum theoretically possible torque is realized, while the statement, "a torque of 30 pounds at 1-foot radius," would be meaningless without knowing the number of poles and the fre- quency. Thus, the denotation of the torque in synchronous 314 ELEMENTS OF ELECTRICAL ENGINEERING watts is the most general, and preferably used in induction motors. Since the theoretical maximum possible torque equals the power input, the ratio torque in synchronous watts output power input that is, actual torque maximum possible torque* is called the torque efficiency of the motor, analogous to the power efficiency or power output t power input that is, power output maximum possible power output * Analogously torque in synchronous watts volt-amperes input is called the apparent torque efficiency. The definitions of these quantities, which are of importance in judging induction motors, are thus: The "efficiency" or "power efficiency" is the ratio of the true mechanical output of the motor to the output which it would give at the same power input if there were no internal losses in the motor. The "apparent efficiency" or "apparent power efficiency" is the ratio of the mechanical output of the motor to the output which it would give at the same volt-ampere input if there were neither internal losses nor phase displacement in the motor. The "torque efficiency" is the ratio of the torque of. the motor to the torque which it would give at the same power input if there were no internal losses in the motor. The "apparent torque efficiency" is the ratio of the torque of the motor to the torque which it would give at the same volt- ampere input if there were neither internal losses nor phase dis- placement in the motor. The torque efficiencies are of special interest in starting where the power efficiencies are necessarily zero, but it nevertheless INDUCTION MACHINES 315 is of importance to find how much torque per watt or per volt- ampere input is given by the motor. Since D = e2ai is the power developed by the motor torque at synchronism, the power developed at the speed of (1 — s) X synchronism, or the actual power output of the motor, is P = (1 - s) D = e2ai (1 - s) eViS (1 - s) The output P includes friction, windage, etc. ; thus, the net me- chanical output is P — friction, etc. Since, however, friction, etc., depend upon the mechanical construction of the individual motor and its use, it cannot be included in a general formula. P is thus the mechanical output, and D the torque developed at the armature conductors. The primary current IQ = e (bi — jb2) has the quadrature components ebi and ebz. The primary impressed e.m.f. EQ = e (ci - jc2) has the quadrature components eci and ec2. Since the components ebi and ec2, and eb2 and eci, respectively, are in quadrature with each other, and thus represent no power, the power input of the primary circuit is ebl X eci + eb2 X ec2, or P0 = e2 (bid + 62c2). The volt-amperes or apparent input is obviously, Pa + &22) (d2 +c22). 138. These equations can be greatly simplified by neglecting the exciting current of the motors, and approximate values of current, torque, power, etc., derived thereby, which are suffi- ciently accurate for preliminary investigations of the motor at speeds sufficiently below synchronism to make the total motor current large compared with the exciting current. 316 ELEMENTS OF ELECTRICAL ENGINEERING In this case the primary current equals the secondary current, that is, T se IQ = /i == -- = e (oi - ja2), where and = etc. Oi+r)2 + s2zi2 139. Since the counter-generated e.m.f. e (and thus the im- pressed e.m.f. EQ) enters in the equation of current, magnetism, etc., as a simple factor, in the equations of torque, power input and output, and volt-ampere input as square, and cancels in the equation of efficiency, power-factor, etc., it follows that the current, magnetic flux, etc., of an induction motor are propor- tional to the impressed e.m.f., the torque, power output, power input, and volt-ampere input are proportional to the square of the impressed e.m.f., and the torque- and power efficiencies and the power-factor are independent of the impressed voltage. In reality, however, a slight decrease of efficiency and power- factor occurs at higher impressed voltages, due to the increase of resistance caused by the increasing temperature of the motor and due to the approach to magnetic saturation, and a slight decrease of efficiency occurs at lower voltages when including in the efficiency the loss of power by friction, since this is inde- pendent of the output and thus at lower voltage, that is, lesser output, a larger percentage of the output, so that the efficiencies and the power-factor can be considered as independent of the impressed voltage, and the torque and power proportional to the square thereof only approximately, but sufficiently close for many purposes. 3. LOAD AND SPEED CURVES 140. The calculation of the induction motor characteristics is most conveniently carried out in tabulated form by means of above-given equations as follows: Let ZQ = r0 + JXQ = 0.1 -f- 0.3 j = primary self-inductive im- pedance. Zi = TI + jxi = 0.1 + 0.3 j = secondary self-inductive impedance reduced to pri- mary. Y = g — jb = 0.01 — 0.1 j = primary exciting admit- tance. EQ = 110 volts = primary impressed e.m.f. 318 ELEMENTS OF ELECTRICAL ENGINEERING It is then, per phase, „ 1 rO «„ %, X n% [V* H| s ' + -0 *f it § 1 ! • \ h > »*r« :? i* 0 0.0100 0 3 0.010.10 1.031 +0.007 1.031106.6 0.101010.8 0.01 o.oioolo.ioo 3.003 0.11 0.103 1.042 -0.0231.042 105.7 0.1507 15.9 0.02 0.0100 0.200 3.012 0.21 0.112 1.055 -0.052 1.056 104.3 0.238 24.8 0.05 0.0102 0.490 3.073 0.50 0.173 1.102 -0.133 1.110 99.20.522 51.8 0.1 0.0109 0.920 3.276 0.93 0.376 1.206 -0.241 1.230 89.5 1.003 89.7 0.15 0.2 0.0120 0.0136 1.25 1.47 3.563 3.883 1.26 1.48 0.663 0.983 1.325 1.443 -0.308 -0.354 1.360 1.485 80.91.424 74.21.777 115 132 0.3 0.5 1.0 0.01811.66 0.03251.54 O.lOOOjl.OO L.49 2.31 5.00 1.671.50 1.552.41 1.013.10 1.617 1.878 2.031 -0.351 -0.224 +0.007 1.654 1.891 2.031 66.6 58.2 54.1 2.245 2.865 3.261 149 167 176 s e* D = e2oi P = Pa = Eol p = bici + &2C2 Po2 = eff. = P Po app. eff. = P Pa pow.fac. = Po Pa 0 11,360 0 0 1.19 0.011 0.125 0 0 10.5 0.01 11,170 1.117 1. 106 1.75 0.112 1.249 88.5 63.2 71.5 0.02 10,880 2.17C . 2. 133 2.73 0.216 2.350 91.0 78.3 86.2 0.05 9,840 4.82 4. 58 5.70 0.528 5.20 88.3 80.5 91.3 0.1 8,010 7.38 6. 64 9.87 1.030 8.25 80.7 67.3 83.5 0.15 6,540 8.20 6. 97 12.65 1.466 9.60 72.5 55.0 76.0 0.2 5,510 8.10 6. 48 14.52 1.782 9.80 66.0 44.6 67.5 0.3 4,440 7.36 5. 15 16.4 2.154 9.55 53.8 31.5 58.3 0.5 3,390 5.23 2. 61 18.4 2.370 8.04 32.3 14.2 43.8 1.0 2,930 2.93 0 19.4 2.072 6.08 0 0 31.3 Diagrammatically it is most instructive in judging about an induction motor to plot from the preceding calculation — 1st. The load curves, that is, with the load or power output as abscissas, the values of speed (as a fraction of synchronism), of current input, power-factor, efficiency, apparent efficiency, and torque. 2d. The speed curves, that is, with the speed, as a fraction of synchronism, as abscissas, the values of torque, current input, power-factor, torque efficiency, and apparent torque efficiency. The load curves are most instructive for the range of speed near synchronism, that is, the normal operating conditions of the motor, while the speed curves characterize the behavior of the motor at any speed. INDUCTION MACHINES 319 In Fig. 176 are plotted the load curves, and in Fig 177 the speed curves of a typical polyphase induction motor of moderate size, having the folio wing constants: eQ = 110; Y = 0.01 — 0.1 j; Z, = 0.1 + 0.3 j, and Z0 = 0.1 + 0.3 j. As sample of a poor motor of high resistance and high admit- tance or exciting current are plotted in Fig. 178 the load curves of a motor having the following constants: eQ = 110; Y = 0.04 Z0=Zf=0. 1+0.3 j .Y— 0.01 -0.1 j 2000 3000 4000 5000 FIG. 176. — Induction motor load cooo curves. - 0.4 j', Zi = 0.3 + 0.3 j, and ZQ = 0.3 + 0.3 j, showing the overturn of the power-factor curve frequently met in poor motors. 141. The shape of the characteristic motor curves depends entirely on the three complex constants, Y, Zi, and ZQ, but is essentially independent of the impressed voltage. Thus a change of the admittance Y has no effect on the char- acteristic curves, provided that the impedances Z\ and Z0 are 320 ELEMENTS OF ELECTRICAL ENGINEERING changed inversely proportional thereto, such a change merely representing the effect of a change of impressed voltage. A 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 FIG. 177. — Induction motor speed curves. 1000 2000 3000 4000 FIG. 178. — Load curves of poor induction motor. moderate change of one of the impedances has relatively little effect on the motor characteristics, provided that the other impedance changes so that the sum Zi + ZQ remains constant, INDUCTION MACHINES 321 and thus the motor can be characterized by its total internal impedance, that is, Z = Zl + Z0; or r + jx = (ri + r0) + j (xi + XQ). Thus the characteristic behavior of the induction motor de- pends upon two complex imaginary constants, Y and Z, or four real constants, g, 6, r, x, the same terms which characterize the stationary alternating-current transformer on non-inductive load. Instead of conductance g, susceptance 6, resistance r, and react- ance x, as characteristic constants may be chosen: the absolute exciting admittance y = \/g2 -f- &2; the absolute self-inductive impedance z — \/r2-}-x2', the power-factor of admittance 0 = g/y, and the power-factor of impedance a = r/z. 142. If the admittance y is reduced rz-fold and the impedance z increased n-fold, with the e.m.f. \SnEQ impressed upon the motor, the speed, torque, power input and output, volt-ampere input and excitation, power-factor, efficiencies, etc., of the motor, that is, all its characteristic features, remain the same, as seen from above given equations, and since a change of impressed e.m.f. does not change the characteristics, it follows that a change of admittance and of impedance does not change the character- istics of the motor provided the product 7 = yz remains the same. Thus the induction motor is characterized by three constants only: The product of exciting admittance and self-inductive impe- dance 7 = yz, which may be called the characteristic constant of the motor. The power-factor of exciting admittance /? = -• y The power-factor of self-inductive impedance a = -- All these three quantities are absolute numbers. The physical meaning of the characteristic constant or the prod- uct of the exciting admittance and impedance is the following: If IQQ = exciting current and 7i0 = starting current, we have, approximately, E0 z = j-i •MO Jf 00 y = yZ = ~ • 322 ELEMENTS OF ELECTRICAL ENGINEERING The characteristic constant of the induction motor 7 = yz is the ratio of exciting current to starting current or current at standstill. At given impressed e.m.f., the exciting current 7oo is inversely proportional to the mutual inductance of primary and secondary circuit. The starting current Iio is inversely proportional to the sum of the self-inductance of primary and secondary circuit. Thus the characteristic constant 7 = yz is approximately the ratio of total self-inductance to mutual inductance of the motor circuits; that is, the ratio of the flux interlinked with only one circuit, primary or secondary, to the flux interlinked with both circuits, primary and secondary, or the ratio of the waste or leakage flux to the useful flux. The importance of this quantity is evident. 4. EFFECT OF ARMATURE RESISTANCE AND STARTING 143. The secondary or armature resistance TI enters the equa- tion of secondary current thus: S6 I STi . S*Xi -3 \ ) • and the further equations only indirectly in so far as TI is con- tained in ai and a2. Increasing the armature resistance n-fold, to nri, we get at an n-fold increased slip ns, use se 1 " n + jsx that is, the same value, and thus the same values for e, Jo, D, Po, Pa, while the power is decreased from P = (1 — s) D to P = (1 — ns) D, and the efficiency and apparent efficiency are correspondingly reduced. The power-factor is not changed; hence, an increase of armature resistance ri produces a propor- tional increase of slip s, and thereby corresponding decrease of power output, efficiency and apparent efficiency, but does not change the torque, power imput, current, power-factor, and the torque efficiencies. Thus the insertion of resistance in the armature or secondary of the induction motor offers a means of reducing the speed corresponding to a given torque, and thereby the desired torque can be produced at any speed below that corresponding to short- INDUCTION MACHINES 323 circuited armature or secondary without changing the input or current. Hence, given the speed curve of a short-circuited motor, the speed curve with resistance inserted in the armature can be derived therefrom directly by increasing the slip in proportion to the increased resistance. 1.0 0.9 08 SLIP FRACTION SLIP FRACTION C F 8YNC 00 05 04 03 02 0.1 Y » 0.01 - 0. X HO TORQU • WATT« SS& 2000 1.0 0.9 0.8 0.7 0.0 0.5 0.4 0.3 0.2 0.1 0 FIG. 179. — Induction motor speed-torque and -current curves. This is done in Fig. 179, in which are shown the speed curves of the motor Figs. 176 and 177, between standstill and syn- chronism, for — Short-circuited armature, n = 0.1 (same as Fig. 177). 0.15 ohm additional resistance per circuit inserted in armature, r\ = 0.25, that is, 2.5 times increased slip. 324 ELEMENTS OF ELECTRICAL ENGINEERING 0.5 ohm additional resistance inserted in the armature, r\ = 0.6, that is, 6 times increased slip. 1.5 ohm 'additional resistance inserted in the armature, r\ = 1.6, that is, 16 times increased slip. The corresponding current curves are shown on the same sheet. With short-circuited secondary the maximum torque of 8250 synchronous watts is reached at 16 per cent. slip. The starting torque is 2950 synchronous watts, and the starting current 176 amp. With armature resistance TI = 0.25, the same maximum torque is reached at 40 per cent, slip, the starting torque is in- creased to 6050 synchronous watts, and the starting current decreased to 160 amp. With the secondary resistance r\ = 0.6, the maximum torque of 8250 synchronous watts approximately takes place in start- ing, and the starting current is decreased to 124 amp. With armature resistance 7*1 = 1.6, the starting torque is below the maximum, 5620 synchronous watts, and the starting current is only 64 amp. In the two latter cases the lower or unstable branch of the torque curve has altogether disappeared, and the motor speed is stable over the whole range; the motor starts with the maxi- mum torque which it can reach, and with increasing speed, torque and current decrease; that is, the motor has the character- istic of the direct-current series motor, except that its maximum speed is limited by synchronism. 144. It follows herefrom that high secondary resistance, while very objectionable in running near synchronism, is advantageous in starting or running at very low speed, by reducing the current input and increasing the torque. In starting we have s = 1. Substituting this value in the equations of subsection 2 gives the starting torque, starting current, etc., of the polyphase in- duction motor. In Fig. 180 are shown for the motor in Figs. 176, 177 and 179 the values of starting torque, current, power-factor, torque efficiency, and apparent torque efficiency for various values of the secondary motor resistance, from r\ = 0.1, the internal re- sistance of the motor, or R = 0 additional resistance to n = 5.1 INDUCTION MACHINES 325 or R = 5 ohms additional resistance. The best values of torque efficiency are found beyond the maximum torque point. The same Fig. 180 also shows the torque with resistance in- serted into the primary circuit. The insertion of reactance, either in the primary or in the secondary, is just as unsatisfactory as the insertion of resistance in the primary circuit. 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 FIG. 180. — Induction motor starting torque with resistance secondary. in the Capacity inserted in the secondary very greatly increases the torque within the narrow range of capacity corresponding to resonance with the internal reactance of the motor, and the torque which can be produced in this way is far in excess of the maximum torque of the motor when running or when starting with resistance in the secondary. 326 ELEMENTS OF ELECTRICAL ENGINEERING But even at its best value, the torque efficiency available with capacity in the secondary is below that available with resistance. For further discussion of the polyphase inductance motor, see "Theory and Calculation of Alternating-current Phenomena."