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 direct- current shunt motor, and this resistance must be a rheostat, that is, variable, so as to have maximum resistance in starting, and gradually, or at least in a number of successive steps, cut out the resistance during acceleration. This, however, requires a wound secondary, and the squirrel- cage type of rotor, which is the simplest, most reliable and there- fore most generally used, is not adapted for the use of a start- ing rheostat. With the squirrel-cage type of induction motor, starting thus is usually done — and always with large motors — by lowering the impressed voltage by autotransformer, often in a number of successive steps. This reduces the starting current, but correspondingly reduces the starting torque, as it does not change the apparent starting-torque efficiency. The higher the rotor resistance, the greater is the starting torque, and the less, therefore, the starting current required for 1 2 ELECTRICAL APPARATUS a given torque when starting by autotransformor. However, high rotor resistance means lower efficiency and poorer speed regulation, anil this limits the economically permissible resistance in the rotor or secondary. Discussion of the starting of the induction motor by arma- ture rheostat, and of the various speed-torque curves produced by various values of starting resistance in the induction-motor secondary, are given in "Theory and Calculation of Alternating- ruiTini Phenomena" and in "Theoretical Elements of Electrical Engineering." As Been, in the induction motor, the (effective) secondary re- sistance should be as low as possible at full speed, but should be high at standstill — very high compared to the full-speed value— and gradually decrease during acceleration, to maintain constant high torque from standstill to speed. To avoid the inconvenience and complication of operating a starting rheostat, various devices have been proposed and to some extent used, to produce a resistance, which automatically increases with in- creasing slip, anil thus is low at full speed, and higher at standstill. A. Temperature Starting Device 2. A resistance material of high positive temperature coeffi- cient of resistance, such as iron and other pure metals, operated at high temperature, gives this effect to a considerable extenl : with increasing slip, that is, decreasing speed of the motor, the secondary current increases. If the dimensions of the secondary mfetanoe Me chosen so that it rises considerably in tempera- ture, by the increase of secondary current, the temperature and therewith the resistance increases. Approximately, the temperature rise, and thus the resistance rise of the secondary resistance, may be considered as propor- tional to the square of the secondary-current, ii, that is, repre- sented bv: r = r° (1 + aii3). (I) As illustration, consider a typical inductiou motor, of the oonatants: Co = 110; Yt-g-jb" 0.01 - 0.1 j; Zo = r„+ j"j:0 =0.1 +0.3j; Z, = rl+jxl = 0.1 -f 0.3j; the speed-torque curve of this motor is shown as A in Fig. 1 SPEED CONTROL 3 Suppose now a resistance, r, i8 inserted in series into the sec- ondary circuit, which when cold — that is, at light-load — equals the internal secondary resistance: but increases so as to double with 100 amp. passing through it. This resistance can then be represented by: r = r° (1 + i,« 10-*) = 0.1 (1 +»i,10-4), NDUCTION MOTOR -110 I ^ z,=r, + .3i SPEED CONTROL BY POSITIVE TEMPERATURE COEFFICIENT r, SPEED CURVES i 1 £ ..■„.. "' TC ,«6*.B'.*?«f*) e ... vo" ^_ A . "if S£ ** ^ G.U B u v _u -'.u 0 i 0 3 I) , (j , 0 * fa 7 0 S 0 5 and the total secondary resistance of the motor then is: r\ = r, + r<,{l + otV) (2) = 0.2 (1 + 0.5 if 10-'). To calculate the motor characteristics for this varying resist- ance, r'l, we use the feature, that a change of the secondary re- sistance of the induction motor changes the slip, s, in proportion to the change of resistance, but leaves the torque, current, power- factor, torque efficiency, etc., unchanged, as shown on page 322 of "Theoretical Elements of Electrical Engineering." We .thus calculate the motor for constant secondary resistance, n, but otherwise the same constants, in the manner discussed on page 318 of "Theoretical Elements of Electrical Engineering." 4 ELECTRICAL APPARATUS This gives curve A of Fig. 1. At any value of torque, T, corre- sponding to slip, s, the secondary current is: ('] = e y/a{ + of, herefrom follows by (2) the value of r',, and from this the new value of slip: e + * - r'i * n. (3) The torque, T, then is plotted against the value of slip, .•', and gives curve B of Fig. 1. As seen, B gives practically constant torque over the entire range from near full speed, to standstill. Curve B has twice the slip at load, as A, as its resistance has heen doubled. 3. Assuming, now, that the internal resistance, rlT were made as low as possible, tx = 0.05, and the rest added as externa] resistance of high temperature coefficient: r" = 0.05, giving the total resistance: = 0.1 (1 + 0.5 ir 10"4). (4) This gives the same resistance as curve A ; r\ = 0.1, at light- load, where iL 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 .-1 thus are directly comparable: both have the same constants mid same speed regulation and other performance, at speed, but C gives much higher torque at standstill and during acceleration. For comparison, curve .4' has heen plotted with constant resistance r, = 0.2, so as to compare with B. Instead of inserting an external resistance, it would be pref- erable to use the internal resistance of the squirrel cage, to in- crease in value by temperature rise, and thereby improve the starting torque. Considering in this respect the motor shown as curve C. At standstill, it is: i, = 153; thus r'i = 0.217; while cold, the re- sistnin-c is: r'i = 0.1. Thjs represents a resistance rise of 117 per cent. At a temperature coefficient of the resistance of 0.35, this represents a maximum temperature rise of 335°C, As seen, SPEED CONTROL 5 by going to temperature of about 350°C. in the rotor conductors — which naturally would require fireproof construction — it be- comes possible to convert curve A into C, or A' into B} in Fig. 1. Probably, the high temperature would be permissible only in the end connections, or the squirrel-cage end ring, but then, iron could be used as resistance material, which has a materially higher temperature coefficient, and the required temperature rise thus would probably be no higher. B. Hysteresis Starting Device 4. Instead of increasing the secondary resistance with increas- ing slip, to get high torque at low speeds, the same result can be produced by the use of an effective resistance, such as the effect- ive or equivalent resistance of hysteresis, or of eddy currents. As the frequency of the secondary current varies, a magnetic circuit energized by the secondary current operates at the varying frequency of the slip, s. At a given current, i\, the voltage required to send the current through the magnetic circuit is proportional to the frequency, that is, to 8. Hence, the suaceptance is inverse proportional to «: V = 6- (5) 8 The angle of hysteretic advance of phase, a, and the power- factor, in a closed magnetic circuit, are independent of the frequency, and vary relatively little with the magnetic density and thus the current, over a wide range,1 thus may approxi- mately be assumed as constant. That is, the hysteretic con- ductance is proportional to the susceptance : g' = V tan a. ((>) Thus, the exciting admittance, of a closed magnetic circuit of negligible resistance and negligible eddy-current losses, at the frequency of slip, «, is given by: Y' = g' - jb' = V (tan a - j) = - J = (tan a - j) (7) 8 8 8 1 "Theoiy and Calculation of Al format iri^-rurr^nt Phfjiornwia," Chapter XII. 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: Y' = bg (0.6 - j), the exciting admittance at slip, s. Assume then, that such an admittance, F', is connected in series into the secondary circuit of the induction motor,* for the pur- pose of using the effective resistance of hysteresis, which in- creases with the frequency, to control the motor torque curve. The total secondary impedance then is: 1 Y - {» + Q + * (* + J) • « Z i — Z\ + v/ where: Y = g — jb is the admittance of the magnetic circuit at full frequency,, and 5. For illustration, assume that in the induction motor of the constants: 6o = 100; Y0 = 0.02 - 0.2 j; Zo = 0.05 + 0.15 j; Zi = 0.05 + 0. 15 j; a closed magnetic circuit is connected into the secondary, of full frequency admittance, Y = g - jb; and assume: g = 0.6 b; 6 = 4; thus, by (8) : Z\ = (0.05 + 0.11 s) + 0.335 js. (9) The characteristic curves of this induction motor with hysteresis starting device can now be calculated in the usual manner, dif- fering from the standard motor only in that Z\ is not constant, and the proper value of rh %\ and m has to be used for every slip, 8. Fig. 2 gives the speed-torque curve, and Fig. 3 the load curves of this motor. SPEED CONTROL 7 For comparison is shown, as 7", in dotted lines, the torque curve of the motor of constant secondary resistance, and of the constants : > o.oi - o.i y, - 0.01 + 0.3 j; > 0.1 + 0.3J; As seen, the hysteresis starting device gives higher torque at standstill and low speeds, with less slip at full speed, thus a materially superior torque curve. INDUCTION MOTOR 5 p Z,-!.OB + JMI+ .335.fi -J. ■ 9 SPEED CURV "fj ao 5 ? :■;■ 1 li 7n" r A CO T -- r \ T1 "* m . — ■— "" - 1-1 J . from 91 per cent, to 84 per cent., and the apparent efficiency, 7. correspondingly. This seriously limits the usefulness of the device. C. Eddy-current Starting Device 6. Assuming that, instead of using a well-laminated magnetic circuit, and utilizing hysteresis to give the increase of effective n=*i-ranr» with increasing slip, we use a magnetic circuit having very hizh eddy-current losses: very thick laminations or solid iron, or we directly provide a closed high-resistance secondary wii-ing around the magnetic circuit, which is inserted into the ir.d lotion-motor secondary for increasing the starting torque. SPEED CONTROL 9 The susceptance of the magnetic circuit obviously follows the same law as when there are no eddy currents. That is: &' = 6- (10) s At a given current, ih energizing the magnetic circuit, the in- duced voltage, and thus also the voltage producing the eddy currents, is proportional to the frequency. The currents are proportional to the voltage, and the eddy-current losses, there- fore, are proportional to the square of the voltage. The eddy- current conductance, gf thus is independent of the frequency. The admittance of a magnetic circuit consuming energy by eddy currents (and other secondary currents in permanent closed circuits), of negligible hysteresis loss, thus is represented, as function of the slip, by the expression: Y'-g-j-- (11) © Connecting such an admittance in series to the induction- motor secondary, gives the total secondary impedance: Z J = Z\ + y, = Ai + — ^-i-A + 3 /«»i + -™-nr Y (12) r^t) Assuming: g = b. (13) That is, 45° phase angle of the exciting circuit of the magnetic circuit at full frequency — which corresponds to complete screen- ing of the center of the magnet core — we get: *'• = (ri + 6TiT>)) + * (*' + 1 ( r+ *>) • <14) Fig. 4 shows the speed curves, and Fig. 5 the load curves, calculated in the standard manner, of a motor with eddy-current starting device in the secondary, of the constants: e0 = 100; Y0 = 0.03 - 0.3 i; Z0 = 0.033 + 0.1 j; Zx = 0.033 + 0.1 j; 6 = 3; 10 ELECTRICAL APPARATUS thus : 7. As seen, the torque curve has a very curious shape: a maximum at 7 per cent, slip, and a second higher maximum at standstill. The torque efficiency is very high at alt speeds, and prac- tically constant at 82 per cent, from standstill to fairly close of full speed, when it increases. i 1NOUCTION MOTOR Yir.03-.3j; Z»-.033+ 1j; e0-100 SPEED CONTROL BY EDDIES SPEED CURVES ? i~~ ;- s ? ... p • 7 1 it n >™. t ""■ ue 1 ^ "' \ m V^ Y ^' ,- t> _- — -" ' _ 0 i 0 Z 0 1 0 r, 0 t 0 7 0 s a 9 1 ' t c t i t s ( a io. 4. — Speed curves of induction unit or wil.li edily-curri'nt starting device. But the power-factor is very poor, reaching a maximum of 8 per cent, only, and to get the output from the motor, required ewinding it to give the equivalent of a y/Z times as high voltage. For comparison, in dotted lines as 7" is shown the torque curves f the standard motor, of same maximum torque. As seen, in ic motor with eddy-current starting device, the slip at load is ery small, that is, the speed regulation very good. Aside from le poor power-factor, the motor constants would be very atis factory. The low power-factor seriously limits the usefulness of the evice. By differently proportioning the eddy-current device to the ccondary 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. IN AUCTION MOTOR JN \ SPEED CONTROL BY EDDIES LOAD CURVES s \ / / / f ■=> • I / v y / -< / r. U .. I h I I 0 1 ■ ft 0 : t 1 0 1 E 1 0. S 6 0 D i ■ a ' 0 i 0 T 0 Fig. S. — Load curves of induction motor with eddy-current atartinR devlra. In the discussion of the multiple squirrel-cage induction motor, Chapter II, we shall see speed-torque curves of the character us shown in Fig. 4. By the use of the rotor iron as magnetic cir- cuit, the impairment of the power-factor is somewhat reduced, so that the multiple squirrel-cage motor becomes industrially 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 hIoIh filled with solid deep bars, eddy currents in these bars occur at higher frequencies, or unequal current distribution. That is, the cur- rent flows practically all through the top of the bars at the high 12 ELECTRICAL APPARATUS frequency of low motor speeds, thus meeting with a high resist- ance. With increasing motor speed and thus deereMlllg secondary frequency, the current penetrates deeper into the bar, until at full speed it passes practically uniformly throughout the entire bar, in a cireuit of low resistance— but somewhat increased reactance. The deep-bar construction, the eddy-current starting device and the double squirrel-cage construction thus are very similar in the motor-performance curves, and the double squirrel cage, which usually is the most economical arrangement, thus will be discussed more fully in Chapter II. 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 direct- current shunt motor, and in principle is a shunt motor. In the direct-current shunt motor, the speed may be changed by: resistance in the armature, resistance in the field, change of the voltage supply to the armature by a multivolt supply circuit, as a three-wire system, etc. In the induction motor, the s]>eed can be reduced by inserting resistance into the armature or secondary, just as in the direct- current shunt motor, and involving the same disadvantages: the reduction of speed by armature resistance takes place at a sacrifice of efficiency, and at the lower speed produced by arma- ture resistance, the power input is the same as it. would be with the same motor torque at full speed, while the power output is reduced by the reduced speed. That is, Bpeed reduction by armature resistance lowers the efficiency in proportion to the lowering of speed. The foremost disadvantage of speed control by armature resistance is, however, that, the motor ceases to D6 a constant -speed motor, and the speed varies with the load: with a given value of armature resistance, if the load and with it the armature current drops to one-half, the speed reduction of the motor, from full speed, also decreases to one-half, that is, the motor speeds up, and if the load conies off, the motor runs up to practically full speed. Inversely, if the load increases, the speed slows down proportional to the load. With considerable resistance in the armature, the induction 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. (b) By changing the number of poles of the motor. Neither of these two methods has any analogy in the direct- current shunt motor: the direct-current shunt motor has no fre- quency relation to speed, and its speed is not determined by the number of poles, nor is it feasible, with the usual construction of direct-current motors, to easily change the number of poles. In the induction motor, a change of impressed frequency corre- spondingly changes the synchronous speed. The effect of a change of frequency is brought about by concatenation of the 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 Dumber of poles is changed from four to six and to eight, the speed is changed from 1800 to 1200 and to 900 revolutions per minute. This method of speed variation of the induction motor, by changing the number of poles, is the most convenient, and such "multispced motors" are extensively used industrially. A. Pyro-electric Speed Control 10. Speed control by resistance in the armature or secondary has the disadvantage that the speed is not constant, but at a change of load and thus of current, the voltage consumed by the armature resistance, and therefore the speed changes. To give constancy of speed over a range of load would require a resistance, which consumes the same or approximately the same voltage at all values of current. A resistance of very high negative temperature coefficient does this: with increase of current and thus increase of temperature, the resistance decreases, and if the decrease of resistance is as large as the increase of current, the voltage consumed by the resistance, and therefore the motor speed, remains constant. Some pyro-etectric conductors (see Chapter I, of "Theory and Calculation of Electric Circuits") have negative tempera- ture coefficients sufficiently high for this purpose. Fig. 6 shows the current-resistance characteristic of a pyro-electric conductor, consisting of cast silicon {the same of which the characteristic is given as rod II in Fig. 6 of " Theory and Calculation of Electric Circuits"). Inserting this resistance, half of it and one and one- half of it into the secondary of the induction motor of constants: e„ = 110; >'„ = 0.01 - 0.\j;ZB =0.1 + 0.3 j; Z, = 0.1 +0.3J gives the speed-torque curves shown in Fig. 7. The calculation of these curves is as follows: The speed- torque curve of the motor with short-circuited secondary, r = 0, SPEED CONTROL 1 1 1 1 1 I 1 l.B 1.7 RESISTANCE OF PVRO ELECTRIC CONDUCTOR [SILICON ROD NO.ll. FIQ.fl ■'ELECTRIC CIRCUITS' ) \ 1.1 1.3 1.2 I.I 1.0 0.'.' M 0.8 «.« \ \ \ \ 3 LI S \ v II \ v \ n - D 0 ■ i 0 5 i - i 1) 0 0 1 li ! 10 L u 1 0 ■ -:. Fio. 6. — Variation of resistance of pyro-cleotric conductor, with current. PYRO-ELECTRIC RESISTANCE IN SECONDARY OF INDUCTION MOTOR. «o-1!0 Y. = .01-.1; . Zc-A+.3j : Z, = .1 + .3j : r-a,4.6 4 SPEED CONTROL BY PYRO ELECTRIC CONDUCTOR. p SPEED CURVES. L 3 WJ. 1 o ■■■"■' "X s \ V •\ rn J T °\ '0 n \ \ \ V \ LI 16 ELECTRICAL APPARATUS is calculated in the usual way as described on page 318 of "Theoretics] Element* of Electrical Engineering." For any value of slip, -s, and cor responding value of torque, T, the secondary current is *'[ = c y/ac -\- a*-. To this secondary current corre- sponds, by Fig. (j, the resistance, r, of the pyro-electric conductor, and the insertion of r thus increases the slip in proportion to the jni'iea-cil secondary resistance: ■> where ri = 0.1 in the present instance. Tliis gives, as corresponding to the torque, T, the slip: , r + r, & = 8, where s = slip at torque, T, with short-circuited armature, or resistance, rt. As seen from Fig. 7, very close constant-speed regulation is produced by the use of the pyro-electric resistance, over a wide range of load, and only at light-load the motor speeds up. Thus, good constant. -a peed regulation at any speed below synchronism, down to very low speeds, would be produced— at a corresponding sacrifice of efficiency, however — by the use of suitable pyro-electric conductors in the motor armature. The only objection to the use of such pyro-electric resistances is the difficulty of producing stable pyro-electric conductors, and permiiiiriit terminal connections on such conductors. B. Condenser Speed Control 11. The reactance of a condenser is inverse proportional to the frequency, that of an inductance is directly proportional to the frequency. In the secondary of the induction motor, the Frequency varies from zero at synchronism, to full frequency at standstill. If, therefore, a suitable capacity is inserted into the Secondary of an induction motor, there is a definite speed, at which inductive reactance and capacity reactance are equal and Opposite, that is, balance, and at and near this speed, a large current is taken by the motor and thus large torque developed, while at speeds considerably above or below this resonance speed, the current and thus torque of the motor are small. The use of a capacity, or an effective capacity (as polariza- tion cell or aluminum cell) in the induction-motor secondary should therefore afford, at least theoretically, a means of speed control by varying the capacity. SPEED CONTROL 17 Let, in an induction motor: Yo = g — jb = primary exciting admittance; Z0 = r0 + jxo = primary self-inductive impedance; Z\ = r\ + 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 k = 2*yc <» k at full frequency, and - at the frequency of slip, s. The total secondary impedance, at slip, «, thus is: Z{ = n+j («xx - *) (2) and the secondary current: T sE se /l= "w ' kvti=~r~^=~^ (3) r, + j («*t - J ^ + (5Xl _ *) = E (di - ja2), where: 8rY ai m s(sXl - *) C2 = " ■ W m = ri2 + (sxi — j (4) The further calculation of the condenser motor, then, is the same as that of the standard motor.1 12. Neglecting the exciting current: /oo = $Y the primary current equals the secondary current: and the primary impressed voltage thus is : $Q = # + Zo/o 1 "Theoretical Elements of Electrical Engineering," 4th edition, p. 318. 2 18 ELECTRICAL APPARATUS and, substituting (3) and rearranging, gives: b Eolrt +j(sxi - -)} E . _ (g) (ri + «r0) + j Isxi + sxu- ■ ) or, absolute: c2 = jr — j- (6) (ri + sr0)2 + («Ci + 8X0 J The torque of the motor is : T = e2ax and, substituting (4) and (6) : T = srie02 (ri + *r0)2 + \sxi + 8X0 J (7) As seen, this torque is a maximum in the range of slip, 8, where the second term in the denominator vanishes, while for values of s, materially differing therefrom, the second term in the denominator is large, and the torque thus small. That is, the motor regulates for approximately constant speed near the value of s, given by : that is: k 8X1 + 8X0 = 0, s * - J— I— (8) \£o + xi and so = 1, that is, the motor gives maximum torque near standstill, for: k = xq + xi. (9) 13. As instances are shown, in Fig. 8, the speed-torque curves of a motor of the constants: r0 = 0.01 - 0.1./, Z0 = Zi = 0.1 + 0.3 j, SPEED CONTROL 19 for the values of capacity reactance : it = 0, 0.012, 0.048, 0.096, 0.192, 0.3, 0.6— denoted respectively by 1, 2, 3, 4, 5, 6, 7. The impressed voltage of the motor is assumed to be varied with the change of capacity, so as to give the same maximum torque for all values of capacity. The volt-ampere capacity of the condenser is given, at the frequency of slip, a, by : «' = ••■* substituting (3) and (6), this gives: (n + «r0)* + (axi + «ro - -) II 1 II II 1 II II - SPEED CONTROL OF INDUCTION MOTOR BY CONDENSER IN SECONDARY Y0=.01-.1i: Z0-.l+-3j; Z.-.1+.3) 5, ,A A ^ K\^ \ ^ X s '?■ s lv £ i. B \ I ■i ■ \ \ ^. V \ -> V ■s \ \ \ 10 . . \ V1 and, compared with (7), this gives T. At full frequency, with the same voltage impressed upon the condenser, its volt-ampere capacity, and thus its 60-cycle rating, would be: 20 ELECTRICAL APPARA TVS As seen, a very large amount of capacity is required for speed control. This limits its economic usefulness, and makes the use of a cheaper form of effective or equivalent capacity desirable. C. Multispeed Motors 14. The change of speed by changing the number of poles, in the multispeed induction motor, involves the use of fractional- pitch windings: a primary turn, which is of full pole pitch for a given number of motor poles, is fractional pitch for a smaller number of poles, and more than full pitch for a larger number of poles. The same then applies to the rotor or secondary, if containing a definite winding. The usual and most frequently employed squirrel-cage secondary obviously has no definite number of poles, and thus is equally adapted to any number of 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 2.j pitch, for the eight-polar motor it is Jj pilch — which latter is effectively the same as ?g pitch. Suppose now the primary winding is arranged and connected as a six-polar three-phase winding. Comparing it with the tune primary burns, arranged as a four-polar three-phase wind- ing, or eight-polar three-phase winding, the turns of each phase can be grouped in six sections: Those which remain in the same phase when changing to a winding for different number of poles. Those which remain in the same phase, but are reversed when changing the number of poles. Those which have to be transferred to the second phase. Those which have to be transferred to the second phase in the reverse direction. Those which have to be transferred to the third phase. Those wdiich have to be transferred to the third phase in the reverse direction. The problem of multispeed motor design (hen is, so to arrange I he wiiii lings, I hat I he change of connection of the six coil groups of each phase, in changing from one number of poles to another, is accomplished with the least number of switches. SPEED CONTROL 21 16. Considering now the change of motor constants when changing speed by changing the number of poles. Assuming that at all speeds, the same primary turns are connected in series, and are merely grouped differently, it follows, that the self- inductive impedances remain essentially unchanged by a change of the number of poles from n to n'. That is : Zn = Z o, Z\ = Z i. With the same supply voltage impressed upon the same number of series turns, the magnetic flux per pole remains unchanged by the change of the number of poles. The flux density, there- fore, changes proportional to the number of poles: & n' B n' therefore, the ampere-turns per pole required for producing the magnetic flux, also must be proportional to the number of poles: F' = n' F n However, with the same total number of turns, the number of turns per pole are inverse proportional to the number of poles : N' n N n' In consequence hereof, the exciting currents, at the name impressed voltage, are proportional to the square of the number of poles: t'oo _ n'2 too n2 ' and thus the exciting susceptances are proportional to the square of the number of poles : b n2' The magnetic flux per pole remains the same, and thiiM the magnetic-flux density, and with it the hystereHin Iohh in the primary core, remain the same, at a change of the number of poles. The tooth density, however, increases with increasing number of poles, as the number of teeth, which carry the mimo flux per pole, decreases inverse proportional to the number of 22 ELECTRICAL APPARATUS poles. Since the tooth densities must be chosen sufficiently low not to reach saturation at the highest number of poles, ami their core loss is usually small compared with that in the primary core itself, it can be assumed approximately, that the core loss of the motor is the same, at the same impressed voltage, regardless of the number of poles. This means, that the exciting con- ductance, y, docs not change with the number of poles. Thus, if in a motor of n poles, we change to n' poles, or by the ratio the motor constants change, approximately: from : to : Za = r0 + jx0, Za = r„ + j'xo. r-jau, Z, = r.+jr,. Y» = g- jb, Y0 - ja''b. 16. However, when changing the number of poles, the pitch of the winding changes, and allowance has to be made herefore in the constants: a fractional-pitch winding, due to the partial neutralization of the turns, obviously has a somewhat higher exciting admittance, and lower self-inductive impedance, than a full-pitch winding. As seen, in a multispeed motor, the motor constants at the higher Dumber of poles and thus the lower speed, must be materially interior than at the higher speed, due to the increase of the exciting susceptance, and the performance of the motor, and especially its power-factor and thus the apparent efficiency, are inferior at the lower speeds. When retaining series connection of all turns for all speeds, and using the same impressed voltage, torque in synchronous watts, and power are essentially the same at all speeds, that is, are decreased for the lower speed and larger number of poles only as far as due to the higher exciting admittance. The actual torque thus would !>e higher for the lower speeds, and approxi- mately inverse proportional to the speed. As a rule, no more torque is required at low speed than at high speed, and the usual requirement would be, that the multi- speed motor should carry the same torque at all its running speeds, that is, give a power proportional to the speed. This would be accomplished by lowering the impressed voltage SPEED CONTROL 23 for the larger number of poles, about inverse proportional to the square root of the number of poles : since the output is proportional to the square of the voltage. The same is accomplished by changing connection from multiple connection at higher speeds to series connection at lower speeds, or from delta connection at higher speeds, to Y at lower speeds. If, then, the voltage per turn is chosen so as to make the actual torque proportional to the synchronous torque at all speeds, that M^,LTIJ^ElS_ NDUCTION MOTOR 1800 HEV , s ' IS ». * f, f \ll \ j f 1 'T I V P— :. i 0 I 5 2 OS : : 0 3 I 1 0 1 I s o ; It d e t 7 o • iviltispeed induction i poles. r, highest speed, four is, approximately equal, then the magnetic flux per pole and the density in the primary core decreases with increasing number of poles, while that in the teeth increases, but less than at constant impressed voltage. The change of constants, by changing the number of poles by the ratio : thus is: from: e0j Ya, Z„, Zi to e„, aY0, aZa, aZ^ and the characteristic constant is changed from d to a*d. 17. As numerical instance may be considered a 60-cycle 100- volt motor, of the constants : 24 ELECTRICAL APPARATUS 5 .POLES IS 30 tR~1~ — H POLES 900 .RE v 1 f / 3 1 y < / "/ I / / m n / y // T // T i // *i p 10 L p Sq.5 I ; 2/i ■:; :; 0 36 4 fi-w /o.S 10 1.5 2 1 II IS E.S«* Pto, 18.— Load Btirvofof multi- Kiu. 11.— Load curves o( muH speed inilui'tion motor, middle speed induction motor, Ion speed ■peed, six poles. Wgta poles. -f- -Ill IK -i ' — ~ s, r i: li r » ml n — *M MO1 3 .,- — .._ .-_ Si ... „J ^=£^™™iL ; mn / '/I „,- -— ' i===::::;?C:;^- mf/7 V ' ^ '' tJL , I) / nt Ml (/ -" ij_ ^Jr 11 *y~\ y 1M *z>— in 5" o!s o lie x'o as o n 3 -, j 0 6 E 0 I t ■ 0 0 r. - (li> m S o. 2. — Comparison of loud turves of three-speed induction nolor. SPEED CONTROL 25 Four poles, 1800 rev.:Z0 = n + jxQ = 0.1 + 0.3 j; Zx = ri+jxt -0.1 + 0.3j; Y, = g - jb = 0.01 - 0.05 j. Six poles, 1200 rev. :Z0 = U + jx0 = 0.15 + 0.45 j; Zx -= n + jxi = 0.15 + 0.45 j; Y0 = g -jb = 0.0067 - 0.0667 j. Eight poles, 900 rev. : Z„ = r0 + jx„ = 0.2 + 0.6 j; Zi = r, + jx, = 0.2 + 0.6 j; Y0 = g - jb = 0.005 - 0.1 j. Figs. 9, 10 and 11 show the load curves of the motor, at the three different speeds. Fig. 12 shows the load curves once more, .. [SPEED INDUCTION MOT --,-■ 1, \ \ ':,;: \ 11(1 1 . \ **. inn " "* •*> 1. f> \ X ty^ \ / y % r y t "p, / y- \ 7 ^ s „ p, 1 t ,c. iou a» aoo too wo eoo itx soo 9ooioooiu»i2ooi»oiHoicooi«o(inoai8«) Fig. 13. — Speed torque curves of three-speed induction motor, with all three motors plotted on the same sheet, but with the torque in synchronous watts (referred to full speed or four- polar synchronism) as abscissa), to give a better compariFon. 5 denotes the speed, / the current, p the power-factor and y the apparent efficiency. Obviously, carrying the same load, that is, giving the same torque at lower speed, represents less power output, and in a multispeed motor the maximum power output should be approximately proportional to the speed, to operate at all speeds at the same part of the motor characteristic. There- fore, a comparison of the different speed curves by the power output does not show the performance as well as a comparison on the basis of torque, as given in Fig. 12. 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.