CHAPTER VIII SYNCHRONIZING INDUCTION MOTORS 94. Occasionally two or more induction motors are operated in parallel on the same load, as for instance in three-phase rail- roading, or when securing several speeds by concatenation. In this case the secondaries of the induction motors may be connected in multiple and a single rheostat used for starting . and speed control. Thus, when using two motors in concatena- tion for speeds from standstill to half synchronism, from half synchronism to full speed, the motors may also be operated on a single rheostat by connecting their secondaries in parallel. As in parallel connection the frequency of the secondaries must be the same, and the secondary frequency equals the slip, it follows that the motors in this case must operate at the same slip, that is, at the same frequency of rotation, or in synchronism with each other. If the connection of the induction motors to the load is such that they can not operate in exact step with each other, obviously separate resistances must be used in the motor secondaries, so as to allow different slips. When rigidly connect- ing the two motors with each other, it is essential to take care that the motor secondaries have exactly the same relative posi- tion to their primaries so as to be in phase with each other, just as would be necessary when operating two alternators in parallel with each other when rigidly connected to the same shaft or when driven by synchronous motors from the same supply. As in the induction-motor secondary an e.m.f. of definite fre- quency, that of slip, is generated by its rotation through the revolving motor field, the induction-motor secondary is an alternating-current generator, which is short-circuited at speed and loaded by the starting rheostat during acceleration, and the problem of operating two induction motors with their secondaries connected in parallel on the same external resistance is thus the same as that of operating two alternators in parallel. In general, therefore, it is undesirable to rigidly connect induction-motor secondaries mechanically if they are electrically connected in parallel, but it is preferable to have their mechanical connection 159 100 ELECTRICAL APPARATUS sufficiently flexible, as by belting, etc., bo that the motors can drop into exact step with each other and maintain step by their synchronising power. It is of interest, then, to examine the synchronizing power of two induction motors which are connected in multiple with their secondaries on the same rheostat and operated from the same primary impressed voltage. 95. Assume two equal induction motors with their primaries connected to the same voltage, supply and with llieir seeondarioi connected in multiple with each other to a common resistance, r, and neglecting for simplicity the exciting current and the vol- tage drop in the impedance of the motor primaries as not mate- rially affecting the synchronizing power. Let Zi — n + ./j-t = secondary self-inductive impedance at full frequency; s = slip of the two motors, as fraction of syn- chronism; Co = absolute value of impressed voltage and thus, when neglecting the primary impedance, of the voltage generated iu the primary by the rotating field. If then Ihe two motor secondaries are oul of phase with each • if her by angle 2 r, ami the secondary of I he motor 1 is behind in the direction of rotation mid the secondary of the motor 2 ahead of [he average position by angle r. then: #i = sco (cost -+- jsinr) = secondary generated e.m.f. of the first motor, (1) E 3 = scij (cos t — j sin t) = secondary generated e.m.f. of the second motor. (21 And if /i = current coming from the first, 1-. = current coming from the second motor secondary, Hie total current, or currenl in (he external resistance, r, is; / - /, + I,: (8) it is then, in the circuit comprising the first motor secondary and the rheostat, <', {?, - [tZ - fr = 0, (4) in the circuit comprising (lie second motor secondary and the rheostat, r, E, - [,Z - {r = (), where z = n + jsxi; £T3Hmxaenzn& JX3:vr?4xv jip^*r»v«^ t*x « ;jr rff»r**n s. - j-. z - - - i. v^ into equations (6) and substituting ill and v~^ into itf\ give**: /i + /* = 2 ,nyo Y cos r% Ii — I* = + 2j*f*Y\ sin r; V^ hence, /V = sf0 |i*eosr + jY\ sin r| 0^ is the current in the secondary circuit of the motor, and there- fore also the primary load current, that is, the primary current corresponding to the secondary current, ami thus, when neg- lecting the exciting current, also the primary motor current, where the upper sign corresponds to the first, or lagging, the lower sign to the second, or leading, motor. Substituting in (9) for 1" and Y\ gives: /V = se0 { (g cos t ± bi sin r) — j (b cos t "I f/i sin r) | , (1(1) the primary e.m.f. corresponding hereto is: £Y = e0 (cos t J j sin r ) , (II) where again the upper sign corresponds to the first, the lower to the second motor. The power consumed by the current, /2', with the e.m.f., Ij!%\ ii 102 ELECTRICAL APPARATUS is the sum of the products of the horizontal components, and of the vertical components, that is, of the real components and of the imaginary components of these two quantities (as a horizontal component of one does not represent any power with a vertical component of the other quantity, being in quadrature therewith). where the brackets denote that the sum of the product of the corresponding parts of the two quantities is taken. As discussed in the preceding, the torque of an induction motor, in synchronous watts, equals the power consumed by the primary counter e.m.f.; that is: 2V = /Y, and substituting (10) and (11) this gives: D%1 = se02 {cost (g cost ± 6i sin r) + sin t (6 cos r + gr7 sin r)\ (12) , 8eo2 (*+_! _ ?i-^cos 2t ± bl "^sin 2t and herefrom follows the motor output or power, by multiplying with (1 — s). The sum of the torques of both motors, or the total torque, is: 2 Dt = 2>i + D2 = se02 {(gi + g) - (gfi - g) cos 2t}. (13) The difference of the torque of both motors, or the synchroniz- ing torque, is: 2D, = se02 (6i - b) sin 2 t, (14) where, by (7), nil , sxi u sxi } (15) mi nil = ri2 + s2Xi2, In those equations primary exciting current and primary impedance are neglected. The primary impedance can be intro- duced in the equations, by substituting (n + srn) for rl} and (xi + T<>) for X\, in the expression of Mi and m, and in this case only the exciting current is neglected, and the results are suffi- ciently accurate for most purposes, except for values of speeel g = ri-2r m ' 6 = SXi m m = (r, + 2 r)2 + S2Xi2, SYXCHROXIZIXG IXDITTIOX MOTORS 1G3 very close to synchronism, where the motor current is appreciably increased by the exciting current. It is, then: TOi = (ri + r*0)2 + s2 (*i + -To)5, m = (ri + srQ + 2r)J + «* (x, + Jo)5: all the other equations remain the same. From (15) and (16) follows 61 — b _ 2srxi (rx + «r0 + r) 2 mm 1 (16) (17) hence, is always positive. 96. (61 — b) is always positive, that is, the synchronizing torque is positive in the first or lagging motor, and negative in the second or leading motor; that is, the motor which lags in position behind gives more power and thus accelerates, while the motor which is ahead in position gives less power and thus drops back. Hence, the two motor armatures pull each other into step, if thrown together out of phase, just like two alternators. The synchronizing torque (14) is zero if t = 0, as obvious, as for r = 0 both motors are in step with each other. The syn- chronizing torque also is zero if r = 90°, that is, the two motor armatures are in opposition. The position of opposition is unstable, however, and the motors can not operate in opposition, that is, for t = 90°, or with the one motor secondary short- circuiting the other; in this position, any decrease of t below 90° produces a synchronizing torque which pulls the motors together, to r = 0, or in step. Just as with alternators, there thus exist two positions of zero synchronizing power — with the motors in step, that is, their secondaries in parallel and in phase, and with the motors in opposition, that is, their secondaries in opposition — and the former position is stable, the latter unstable, and the motors thus drop into and retain the former position, that is, operate in step with each other, within the limits of their synchronizing power. If the starting rheostat is short-circuited, or r = 0, it is, by (15), 61 = by and the synchronizing power vanishes, as is obvious, since in this case the motor secondaries are short-circuit (id and thus independent of each other in their frequency and speed. With parallel connection of induction-motor armatures a syn- chronizing power thus is exerted between the motors as long as any appreciable resistance exists in the external circuit, and 164 ELECTRICAL APPARATUS the motors thus tend to keep in step until the common starting resistance is short-circuited and the motors thereby become inde- pendent, the synchronizing torque vanishes, and the motors can slip against each other without interference by cross-currents. Since the term — ^ — contains the slip, s, as factor, the syn- chronizing torque decreases with increasing approach to syn- chronous speed. 1 n b, "-- -.. „ t\. - ^ : ID X - -.„ <; \ D ^ > \ N \ \ ^ ^ \ - \ \ v \ \ 1 '. Fig. Fo (12), 9 . — Sy n ch roniz in g r r = 0, or with (15), and (16): induction motors: motor torque and ay torque. the motors in step with each othe s