CHAPTER XI INSTABILITY OF CIRCUITS: INDUCTION AND SYN- CHRONOUS MOTORS C. Instability of Induction Motors 102. Instability of electric circuits may result from causes which are not electrical: thus, mechanical relations between the torque given by a motor and the torque required by its load, may lead to instability. Let D = torque given by a motor at speed, S, and D' = torque required by the load at speed, S. The motor, then, could theoretically operate, that is, run at constant speed, at that speed, S, where Z) = D' (1) However, at this speed and load, the operation may be stable, that is, the motor continue to run indefinitely at constant speed, or the condition may be unstable, that is, the speed change with increasing rapidity, until stability is reached at some other speed, or the motor comes to a standstill, or it destroys itself. In general, the motor torque, D, and the load torque, D', change with the speed, S. If, then, dD' dD the conditions are stable, that is, any change of speed, S, changes the motor torque less than the load torque, and inversely, and thus checks itself. If, however, dD' ^ dD ,,. 'dS^dS ^^^ the operation is unstable, as a change of speed, S, changes the motor torque, Z), more than the load torque, D', and thereby fur- ther increases the change of speed, etc. dD' dD dS dS 201 (4) 202 ELECTRIC CIRCUITS thus ia the expression of the stability limit. For instance, assuming a load requiring a constant torque at all speeds. The load torque thus is given by a horizontal line D' = const. (5) in Fi^. 101. Let then the speed-torque curve of the motor be represented by the curve, D, in Fig. 101. D approximately represents the torque curve of a series motor. At the constant-load torque, D', the motor runs at the speed, S = 0,6, point a of Fig. 101, and the speed is stable, as any tendency to change of speed, checks itself. If "~ •V \ \ a \ \ «. li{ \ \ \ \ D' ■^ d '^ d; ^ Ofl . "-- — f J J 1 Pig. 101. the load torque decreases to D'o, the speed rises to S = 0.865, point ao) if the load torque increases to D'l, the speed drops to S = 0.29, point ai, but the conditions are always stable, until finally with increasing load torque, D', and decreasing speed, standstill is reached at point ag. Let now the speed-torque curve of a motor be represented by^ D in Fig. 102: the curve of a squirrel-cage induction motor witl^^ — moderately high resistance secondary. The horizontal line, D'^ corresponding to a load torque of D' = 10, intersects D at twc^ points, a and b. INSTABILITY OF CIRCUITS 203 At a, jS = 0.905, the speed is st^ible. At 6, however, S = 0.35, the conditions are unstable, and the motor thus can not run at ft, but either — if the speed should drop or the load rise ever so little — the motor begins to slow down, thereby, on curve, D, its torque falls below that of the load, D', thus it slows down still more, and so, with increasing rapidity the motor comes to a standstill. Or, if the motor speed should be a little higher, or the load momen- tarily a little lower, the motor speed rises, until stability ia reached at point a. e ,, / -^ N ° y \ / \ b / ^ i, 0' \ a X \ d. \ ^ \ B - \ Di l"' . ■• With increasing load torque, D', the speed gradually drops, from S — 0.905 at D' = 10, point a, down to point c, at *S = 0-75, D' = 14,3; from there, however, the speed suddenly drops to standstill, that is, it is not possible to operate the motor at speeds less than S = 0.76, at constant load-torque, and the branch of the motor characteristic from the starting point,fir, up to the maximum torque point, c, is unstable on a load requiring con- stant torque. At load torque, D' = 10, the motor can not start the load, can not carry it below ft, S = 0.35; at speeds from 6 to a, iS = 0.35 to 0.905, the motor speeds up; at speeds above a, S = 0.905, the motor slows down, and drops into stable condition at a. 204 ELECTRIC CIRCUITS With a load torque, D'o = 5, the motor starts and runs up to speed tti, S = 0.96. D' = 7.2, point g, thus, is the maximum load torque which the motor can start. 103. Suppose now, while running in stable condition, at point a, with the load torque, D' = 10, the load torque is momentarily- increased. If this increase leaves D' lower than the maximum motor torque. Do = 14.3, the motor speed slows down, but re- mains above c, and thus when the increase of load is taken off, the motor again speeds up to a. If, however, the temporary increase of load torque exceeds the maximum motor torque. Do = 14.3 — ^for instance by starting a line of shafting or other mass of considerable momentimi — then the motor speed continues to drop as long as the excess load exists, and whether the motor will recover when the excess load is taken off, or not, depends on the loss of speed of the motor during the period of overload: if, when the overload is relieved, the motor has dropped to point di in Fig. 102, its speed thus is still above 6, the motor recovers; if, however, its speed has dropped to d2, be- low the speed 6, >S = 0.35, at which the motor torque drops below the load torque, then the motor does not recover, but stops. With a lighter load torque, D'o, which is less than the starting torque, g, obviously the motor will always recover in speed The amount, by which the motor drops in speed at temporary overload, naturally depends on the duration of the overload, and on the momentum of the motor and its moving masses: the higher the momentum of the motor and of the masses driven by it at the moment of overload, the slower is the drop of speed of the motor, and the higher thus the speed retained by it at the moment when the overload is relieved. Thus a motor of low starting torque, that is, high speed regula- tion, may be thrown out of step by picking up a load of high momentum rapidly, while by adding a flywheel to the motor, it would be enabled to pick up this load. Or, it may be troublesome to pick up the first load of high momentum, while the second load of this character may give no trouble, as, due to the momentimi of the load already picked up, the speed would drop less. Thus a motor carrying no load, may be thrown out of step by a load which the same motor, already partly loaded (with a load of considerable momentum), would find no difficulty to pick up. The ability of an induction motor, to carry for a short time INSTABILITY OF CIRCUITS 205 without dropping out of step a temporary excessive overload, naturally also depends on the excess of the maximum motor torque (at c in Fig. 102) over the normal load torque of the motor. A motor, in which the maximum torque is very much higher — several hundred per cent. — ^than the rated torque, thus could momentarily carry overloads which a motor could not carry, in which the maximum torque exceeds the rated torque only by 50 per cent., as was the case with the early motors. However, very high maximum torque means low internal reactance and thus high exciting current, that is, low power-factor at partial loads, and of the two types of motors: (a) High overload torque, but poor power-factor and efficiency at partial loads; (6) Moderate overload torque, but good power-factor and efficiency at partial loads; the type (6) gives far better average operating conditions, except in those rare cases of operation at constant full-load, and is there- fore preferable, though a greater care is necessary to avoid mo- mentary excessive overloads. Gradually the type (a) had more and more come into use, as the customers selected the motor, and the power supply company neglected to pay much attention to power-factor, and it is only in the last few years, that a realization of the harmful effects of low power-factors on the economy of operation of the systems is again directing attention to the need of good power-factors at partial loads, and the industry thus is returning to type (6), especially in view of the increasing tendency toward maximum output rating of apparatus. In distributing transformers, the corresponding situation had been reaUzed by the central stations since the early days, and good partial load efficiencies and power-factors secured. 104. The induction motor speed-torque curve thus has on a constant-torque load a stable branch, from the maximum torque point, c. Fig. 102, to synchronism; and an unstable branch, from standstill to the maximum torque point. However, it would be incorrect to ascribe the stability or in- stability to the induction motor-speed curve; but it is the char- acter of the load, the requirement of constant torque, which makes a part of the speed curve unstable, and on other kinds of load no instability may exist, or a different form of instability. Thus, considering a load requiring a torque proportional to 206 ELECTRIC CIRCUITS the speed, such as would be given, approximately, by an electric generator at constant field excitation and constant resistance as load. The load-torque curves, then, would be straight lines going through the origin, as shown by D'l, D'i,D't, etc., for increasingly larger values of load, in Fig. 103. The motor-torque curve, D, ia the same as in Fig. 102. As seen, all the lines, ly, intersect D at points, Oi, a:, aj . . . , at which the speed is stable, since / / Di ^; M / ^ •A , tj' / V X Y D' / ■> // ' / / y <: / y /. 'V / X y / / r V i. y \ 1 A '<^ / k/^ % ^ --" / . / / / r^ \ \ ■^ ' / y / / y ^ „ / ^ ■' D ^ / / ^ 5^ \ - Do f ^^ " \ / /xx . ^ \ ^ / /^ o ^ \ , V'<^< ^ ^-.ll ,. , . . . , dS - dS" Thus, with this character of load, a torque required propor- tional to the speed, and the motor-torque curve, 2>, no instability exists, but conditions are stable from standstill to synchronism, just as in Fig. 101. That is, with increasing load, the speed de- creases and increases again with decreasing load. If, however, the motor curve is as shown by Do in Fig. 103, that is, low starting torque and a maximum torque point close to synchronism, as corresponds to an induction motor with low resistance secondary, then for a certain range of load, between INSTABILITY OF CIRCUITS 207 D' and D'o, the load-torque line, D'2, intersects the motor curve, Do, in three points 62, ^2, /12. At 62, S = 0.925, and at /12, S = 0.375, conditions are stable; at d2, S = 0.75, instabiUty exists. Thus with this load, D'2, the motor can run at two different speeds in stable conditions: a high speed, above Co, and a low speed, be- low b ; while there is a third, theoretical speed, ^2, which is unstable. In the range below /12, the motor speeds up to A2; in the range between /12 and ^2, the motor slows down to /12; in the range between di and 62, the motor speeds up to 62, and in the range above 62, the motor slows down to 62. There is thus a (fairly narrow) range of loads between D' and X)'o, in which an unstable branch of the induction motor-torque curve exists, at intermediate speeds; at low speed as well as at high speed conditions are stable. For loads less than D', conditions are stable over the entire range of speed; for loads above D'o, the motor can run only at low speeds, A3, A4, but not at high speeds; but there is no load at which the motor would not start and run up to some speed. Obviously, at the lower speeds, the current consumed by the motor is so large, that the operation would be very inefficient. It is interesting to note, that with this kind of load, the "maxi- mum torque point," c, is no characteristic point of the motor- torque curve, but two points, Co and 6, exist, between which the op- eration of the motor is unstable, and the speed either drops down below 6, or rises above Co. 106. With a load requiring a torque proportional to the square of the speed, such as a fan, or a ship propeller, conditions are al- most always stable over the entire range of speed, from standstill to synchronism, and an unstable range of speed may occur only in motors of very low secondary resistance, in which the drop of torque below the maximmn torque point, c, of the motor character- istic is very rapid, that is, the torque of the motor decreases more rapidly than with the square of the speed. This may occur with very large motors, such as used on ship propellers, if the secondary resistance is made too low. More frequently instability with such fan or propeller load or other load of similar character may occur with single-phase motors, as in these the drop of the torque curve below maximum torque is much more rapid, and often a drop of torque with in- creasing speed occurs, especially with the very simple and cheap 208 ELECTRIC CIRCUITS starting devices economically required on very small motors, such as fan motors. Instability and dropping out of step of induction motors also may be the result of the voltage drop in the supply lines, and furthermore may result from the regulation of the generator vol- tage being too slow. Regarding hereto, however, see " Theory and Calculation of Electrical Apparatus, " in the chapter on " Stability of Induction Machines. " D. Hunting of S]rnchronous Machines 106. In induction-motor circuits, instability almost always assumes the form of a steady change, with increasing rapidity, from the unstable condition to a stable condition or to stand- still, etc. Oscillatory instability in induction-motor circuits, as the result of the relation of load to speed and electric supply, is rare. It has been observed, especially in single-phase motors, in cases of considerable oversaturation of the magnetic circuit. Oscillatory instability, however, is typical of the synchronous machine, and the hunting of synchronous machines has probably been the first serious problem of cimiulative oscillations in electric circuits, and for a long time has limited the industrial use of syn- chronous machines, in its different forms: (a) Difficulty and failure of alternating-current generators to operate in parallel. (6) Hunting of synchronous converters. (c) Hunting of synchronous motors. While considerable theoretical work has been done, practically all theoretical study of the hunting of synchronous machines has been limited to the calculation of the frequency of the transi- ent oscillation of the synchronous machine, at a change of load, frequency or voltage, at synchronizing, etc. However, this transient oscillation is harmless, and becomes dangerous only if the oscillation ceases to be transient, but becomes permanent and cumulative, and the most important problem in the study of hunt- ing thus is the determination of the cause, which converts the transient oscillation into a cumulative one, that is, the determina- tion of the source of the energy, and the mechanism of its trans- fer to the oscillating system. To design synchronous machines, so as to have no or very little tendency to hunting, obviously re- INSTABILITY OF CIRCUITS 209 quires a knowledge of those characteristics of design which are instrumental in the energy transfer to the oscillating system, and thereby cause hunting, so as to avoid them and produce the great- est possible inherent stability. If, in an induction motor running loaded, at constant speed, the load is suddenly decreased, the torque of the motor being in ex- cess of the reduced load causes an acceleration, and the speed in- creases. As in an induction motor the torque is a function of the speed, the increase of speed decreases the torque, and thereby de- creases the increase of speed until that speed is reached at which the motor torque has dropped to equality with the load, and thereby acceleration and further increase of speed ceases, and the motor continues operation at the constant higher speed, that is, the induction motor reacts on a decrease of load by an increase of speed, which is gradual and steady without any oscillation. If, in a synchronous motor running loaded, the load is suddenly decreased, the beginning of the phenomenon is the same as in the induction motor, the excess of motor torque causes an ac- celeration, that is, an increase of speed. However, in the synchronous motor the torque is not a function of the speed, but in stationary condition the speed must always be the same, synchronism, and the torque is a function of the relative position of the rotor to the impressed frequency. The increase of speed, due to the excess torque resulting from the decreased load, causes the rotor to run ahead of its previous relative position, and thereby decreases the torque until, by the increased speed, the motor has run ahead from the relative position corresponding to the pre- vious load, to the relative position corresponding to the decreased load. Then the acceleration, and with it the increase of speed, stops. But the speed is higher than in the beginning, that is, is above synchronism, and the rotor continues to run ahead, the torque continues to decrease, is now below that required by the load, and the latter thus exerts a retarding force, decreases the speed and brings it back to synchronism. But when synchron- ous speed is reached again, the rotor is ahead of its proper position, thus can not carry its load, and begins to slow down, until it is brought back into its proper position. At this position, however, the speed is now below synchronism, the rotor thus continues to drop back, and the motor torque increases beyond the load, thereby accelerates again to synchronous speed, etc., and in this manner conditions of synchronous speed, with the rotor position 14 210 ELECTRIC CIRCUITS behind or ahead of the position corresponding to the load, alter- nate with conditions of proper relative position of the rotor, but below or above synchronous speed, that is, an oscillation results which usually dies down at a rate depending on the energy losses resulting from the oscillation. 107. As seen, the characteristic of the synchronous machine is, that readjustment to a change of load requires a change of relative position of the rotor with regard to the impressed fre- quency, without any change of speed, while a change of relative position can be accomplished only by a change of speed, and this results in an over-reaching in position and in speed, that is, in an oscillation. Due to the energy losses caused by the oscillation, the success- ive swings decrease in amplitude, and the oscillation dies down. If, however, the cause which brings the rotor back from the posi- tion ahead or behind its normal position corresponding to the changed load (excess or deficiency of motor torque over the torque required by the load) is greater than the torque which opposes the deviation of the rotor from its normal position, each swing tends to exceed the preceding one in amplitude, and if the energy losses are insufficient, the oscillation thus increases in amplitude and becomes cumulative, that is, hunting. In Fig. 104 is shown diagrammatically as p, the change of the relative position of the rotor, from pi corresponding to the pre- vious load to p2 the position further forward corresponding to the decreased load. V then shows the oscillation of speed corresponding to the oscillation of position. The dotted curve, Wi, then shows the energy losses resulting from the oscillation of speed (hysteresis and eddies in the pole faces, currents in damper windings), that is, the damping power, assumed as proportional to the square of the speed. If there is no lag of the synchronizing force behind the position displacement, the synchronizing force, that is, the force which tends to bring the rotor back from a position behind or ahead of the position corresponding to the load, would be — or may ap- proximately be assumed as — proportional to the position dis- placement, p, but with reverse sign, positive for acceleration when p is negative or behind the normal position, negative or retarding when p is ahead. The synchronizing power, that is, the power exerted by the machine to return to the normal position, then is INSTABILITY OF CIRCUITS 211 derived by multiplying —p with v, and is shown dotted as Wj in Fig. 104. As seen, it has a double-frequency alternation with zero as average. The total resultant power or the resulting damping effect which restores stability, then, is the sum of the synchronizing power ifa and_ the damping power wi, and is shown by the dotted Fio. 104. curve v>. As seen, under the assumption or Pig. 104, in this case a rapid damping occurs. If the damping winding, which consumes a part of all the power, Wi, is inductive — and to a shght extent it always is — the current in the damping winding lags behind the e.m.f. induced in it by the oscillation, that is, lags behind the speed, v. The power, wt, 212 ELECTRIC CIRCUITS or that part of it which is current times voltage, then ceases to be continuously negative or damping, but contains a positive period, and its average is greatly reduced, as shown by the drawn curve, wi, in Fig. 104, that is, inductivity of the damper winding is very harmful, and it is essential to design the damper winding as non- inductive as possible to give efficient damping. With the change of position, p, the current, and thus the ar- mature reaction, and with it the magnetic flux of the machine, changes. A flux change can not be brought about instantly, as it represents energy stored, and as a result the magnetic flux of the machine does not exactly correspond with the position, p, but lags behind it, and with it the synchronizing force, F, as shown in Fig. 104, lags more or less, depending on the design of the machine. The synchronizing power of the machine, Fv, in the case of a lag- ging synchronizing force, F, is shown by the drawn curve, t(?2. As seen, the positive ranges of the oscillation are greater than the negative ones, that is, the average of the oscillating synchronizing power is positive or supplying energy to the oscillating system, which energy tends to increase the amplitude of the oscillation — ^in other words, tends to produce cumulative hunting. The total resulting power, w = Wi + W2, imder these condi- tions is shown by the drawn curve, w, in Fig. 104. As seen, its average is still negative or energy-consuming, that is, the oscilla- tion still dies out, and stability is finally reached, but the average value of w in this case is so much less than in the case above dis- cussed, that the dying out of the oscillation is much slower. If now, the damping power, Wi, were still smaller, or the aver- age synchronizing power, W2j greater, the average w would become positive or supplying energy to the oscillating system. In other words, the oscillation would increase and hunting result. That is: If the average synchronizing power resulting from the lag of the synchronizing force behind the position exceeds the average damping power, hunting results. The condition of stability of the synchronous machine is, that the average damping power ex- ceeds the average synchronizing power, and the more this is the case, the more stable is the machine, that is, the more rapidly the transient oscillation of readjustment to changed circuit con- ditions dies out. INSTABILITY OF CIRCUITS 213 Or, if a — attenuation constant of the oscillating system, a<0 gives cumulative oscillation or hunting. a>0 gives stability. 108. Coimting the time, t, from the moment of maximum back- ward position of the rotor, that is, the moment at which the load on the machine is decreased, and assuming sinusoidal variation, and denoting = 2 ^/t = orf (1) where / = frequency of the oscillation (2) the relative position of the rotor then may be represented by V = — poc*** cos 0, where Po = P2 — Pi = position difference of rotor resulting from change of load, (3) a = attenuation constant of oscillation. (4) The velocity difference from that of uniform rotation then is t; = -^ = 0) -~ = wpo c"*** (sin + ct cos0)- (5) a = tan a; 1 + a^ = A^ (6) Let hence, it is sm a = -j] cos « = -J (7) V = wpoAe-"* sin (0 + a). (8) Let 7 = lag of damping currents behind e.m.f. induced in damper windings (9) the damping power is = -cw2po'^A2e-2''^sin(« + a)sin(« + a-7) (10) where w c = -^ = damping power per unit velocity and vy is v, V lagged by angle 7. (11) 214 ELECTRIC CIRCUITS Let P = lag of synchronizing force behind position displace- ment p (12) and /3 = (joto (13) where ^0 = time lag of synchronizing force. (14) The synchronizing force then is F = bpoe-*'* cos (<^ - /3) (15) where 6 = — = ratio of synchronizing force to po- sition displacement, or specific synchronizing force. (16) The synchronizing power then is W2 = Fv = bcopoAe-^"^ sin (0 + a) cos (<^ - /3). (17) The oscillating mechanical power is d mv^ dv dt e d4 = mco/Spo'^A^e-^ «* sin (0 + a) {cos {4i-\' a) - a sin (<^ + a)} (18) where m = moving mass reduced to the radius, on which p is measured. (19) It is, however, Wi-k-w^-w^O (20) hence, substituting (10), (17), (18) into (20) and canceling, 6 cos (0 — j8) — C(aA sin (0 + a — 7) — moj^Acos (0 + a) + m the equations 6 cos j8 — ccoA sin (a — 7) — ma^A cos a + mw^Aasin a = 6 sin j8 — ccoA cos (a — 7) + mco^A sin a + moj^A cos a = (22) Substituting (6) and (7) and approximating from (13), for iS as a small quantity. cos /3 = 1 ; sin jS = wfc (23) gives 6 — CO? ( a cos 7 — sin 7) — vm^ (1 — a*) = 6 6^0 or, U< cos 7 CCOS7 (28) are the conditions of stability of the synchronous machine. If fc = 7 = it is a = w = \/4m6 — c^ y/\ mh — c^ 2m and, if also, it is c = 0: