CHAPTER VI INDUCTION-MOTOR REGULATION AND STABILITY 1. VOLTAGE REGULATION AND OUTPUT 79. Load and speed curves of induction motors are usually calculated and plotted for constant-supply voltage at the motor terminals. In practice, however, this condition usually is only approximately fulfilled, and due to the drop of voltage in the step-down transformers feeding the motor, in the secondary and the primary supply lines, etc., the voltage at the motor terminals drops more or less with increase of load. Thus, if the voltage at the primary terminals of the motor transformer is constant, and such as to give the rated motor voltage at full-load, at no- load the voltage at the motor terminals is higher, but at overload lower by the voltage drop in the internal impedance of the trans- formers. If the voltage is kept constant in the center of distri- bution, the drop of voltage in the line adds itself to the imped- ance drop in the transformers, and the motor supply voltage thus varies still more between no-load and overload. With a drop of voltage in the supply circuit between the point of constant potential and the motor terminals, assuming the cir- cuit such as to give the rated motor voltage at full-load, the voltage at no-load and thus the exciting current is higher, the voltage at overload and thus the maximum output and maximum torque of the motor, and also the motor impedance current, that is, current consumed by the motor at standstill, and thereby the starting torque of the motor, are lower than on a constant-poten- tial supply. Hereby then the margin of overload capacity of the motor is reduced, and the characteristic constant of the motor, or the ratio of exciting current to short-circuit current, is in- creased, that is, the motor characteristic made inferior to that given at constant voltage supply, the more so the higher the voltage drop in the supply circuit. Assuming then a three-phase motor having the following con- stants: primary exciting admittance, Y = 0.01 — 0.1 j; primary self-inductive impedance, Z0 = 0.1 + 0.3 j; secondary self -induc- 123 124 ELECTRICAL APPARATUS tive impedance, Z, = 0.1 + 0.3 j; supply voltage, e0 = 110 volts, and rated output, 5000 waits per phase. Assuming this motor to be operated: 1. By transformers of about 2 per cent, resistance and 4 per cent, reactance voltage, that is, transformers of good regulation, with constant voltage at the transformer terminals. 2. By transformers of ahout 2 per cent, resistance and 15 per cent, reactance voltage, that is, very poorly regulating trans- formers, at constant supply voltage at the transformer primaries. 3. With constant voltage at the generator terminals, and about 8 per cent, resistance, 40 per cent, reactance voltage in line and transformers between generator and motor. This gives, in complex quantities, the impedance between the motor terminals and the constant voltage supply: 1. Z - 0.04 + 0.08 j, 2. Z = 0.04 + 0.3 j", 3. Z = 0.16 + 0.8,/. It is assumed that the constant supply voltage is such u hi give 1 10 volts at the motor terminals at FulHoad. The load and speed curves of the motor, when operating under these conditions, that is, with the impedance, Z, in series between the motor terminals and the constant voltage supply, e., then can be calculated from the motor characteristics at constant termi- nal voltage, eBl as follows: At slip, I, and constant terminal voltage, ea, the current in the motor is i0, its power-factor p = cos 8. The effective or equiva- lent impedance of the motor at this slip then is z" = .-, and, in complex quantities, Z* = ." (cos 0 + i Bin 0), and the total irn- pedance, including that of transformers and line, thus is: Zx = Z° + Z = (?" cos 6 + r) + j(* sin 0 + xj , or, in absolute values: tlm .J(pcos0 4-r)'+ (^sin0+j and, at the supply voltage, e ,, the current thus is INDUCTION-MOTOR REGULATION 125 and the voltage at the motor terminals is: e'o = z°i'i = et. Si If ea is the voltage required at the motor terminals at full-load, and io0 the current, zi° the total impedance at full-load, it is: 1 1 1 1 1 1 1 1 1 1 V -0.01- 0,1 j Z- 0.1 +0.3) TRANSFORMED IMPEDANCE 2(T 0.04+O.OBj CONSTANT PRIMARY POTENTIAL 114.1 VOLTS / 1 / / t. ... T» M .13 / / ■S| m > _n. 7^ / / < V ^ / rf, / / f 31 > 0 X Ji ! si . n a oo 0 o 110 volts at motor hence, the required constant supply voltage is: and the speed and torque curves of the motor under this condi- tion then are derived from those at constant supply voltage, e<,, by multiplying all voltages and currents by the factor "> that is, by the ratio of the actual terminal voltage to the full-load terminal voltage, and the torque and power by multiplying with 126 ELECTRICAL APPARATUS the square of this ratio, while the power-factors and the efficien- cies obviously remain unchanged. In this manner, in the three cases assumed in the preceding, the load curves are calculated, and are plotted in Figs, 43, 44, and 45. 80. It is seen that, even with transformers of good regulation, Fig. 43, the maximum torque and the maximum power are ap- Y-Om-O.ijo Z-0.1 + r 0.5 ohm; and in Fig. 49 for 1.5 ohms addii nserted in the armature. As seen, the line ai mpedance very appreciably lowers the tore at motor it termi- aximum c curves and the ircuited ohm; in tonal re- d trans- ue, and INDUCTION-MOTOR REGULATION 127 v- aci-o.ij i- 0,1 +0Ji CIRCUIT IMPEDANCE. Z,-.L6*.SJ CONSTANT OEHEBATOR POTENTIAL l*4.B VOLTS *- #■>) ' C£ ■" »*r ■T, -■,. / / nn ^ 1D0 " 1 = — / •> 1/ N / *; W f / ^ / m _. y / ST X" i c « 0 1 t«; ^o$* ft-**" 1.0 (1 -it DA** FWOT .,,, ;»■» - 1 .i 128 ELECTRICAL APPARATUS especially the starting torque, which, with short-circuited arma- ture, in the case 3 drops to about one-third the value given at constant supply voltage. 3? i£*" _o »*^- tji-* 1-o^i si* i^ \ \ \ ill ( "■'" n I , ) Fig. 47. — Induction-motor speed torque characteristics with a resistance o 0. 15 onm in secondary circuit. - S2 ^ . "ST -is* £S5i »j_ — ^5 o*i_ - ->* i"i 1*^1 .."■; Fifl. 48. — Induct ion -motor speed turque characterise ch with a resistance of 0.5 ohm in secondary circuit. It is interesting to note that, in Fig. 48, with a secondary resistance giving maximum torque in starting, at constant tcr- INDUCTION -MOTOR REGULATION 129 rainal voltage, with high impedance in the supply, the starting torque drops so much that the maximum torque is shifted to about half synchronism. In induction motors, especially at overloads ami in starting, it therefore is important to have as low impedance as pos- sible between the point of constant voltage and the motor terminals. _>j^_ \^&% °yo^?I L-.K! g I p Friction of Synchroniwn with a resistance of In Table I the numerical values of maximum power, maxi- mum torque, starting torque, exciting current and starting current are given for above motor, at constant terminal voltage and for the three values of impedance in the supply lines, for such supply voltage as to give the rated motor voltage of 110 volts at full load and for 1 10 volts supply, voltage. In the first case, maximum power and torque drop down to their full-load values with the highest line impedance, and far below full-load values in the latter case. 130 ELECTRICAL APPARATUS P & P o o M < J P O W CO o + o II u o o < 1 H -J »-4 o o > o 1 II HH n t** U J 3 H u fc « • 00 • • • • CO CD 00 CO 8I 8 S a i t « ON 00 ■ • • ■ ** ©. ^! «o ** u •S •*» 00 CO © I to n 6 I o I 8 in 3 « 8 * CM 0» m a CO O « 1 3 •« ff w lm O •o ** M M O 1 w 03 ^4 ♦* • 00 o 1 C <0»HH n io no 0050 I- © *" •-• • • • • O f-4 f-4 CO co co o> I* MO hN SSSs 8 8 8 « + ** ao t* co 3*^ CO tO M ssss to co cm §5 TO W rN 8n 0 m * « in So «o 10 CM CO CD N N H 35 5s co cO d» n co *o CM CM »■* iO tO © Q to h S n 10 * cm o tO iO >0 H)hO «o CO © rH v-l rH rH H H H O IO 5 CO «o cm as >* -* go h- CO tO SO O O <<" C? N NOAH 00 CD tO CO CM »h © as •H O) ^4 ■+ 0 t* o> e» «o ^ cm O tO o o 0 2 28 £2 •-• o> 00 CO 0000 O *0 CD O • • ■ ■ © to ^ 00 *M O) t» ^ O CO 0 0 0 CO 00 *• «H »-!•-• CO 110.0; 100.0; 107.5: 102.0 OH OiO • • • • 0 ^1 H *£ ** ** eM -* 0000 • • • • 0000 »H *-l f>4 <-4 ^as .2 31 00 O CO 00 0000 I I I • • • 000 s CO 00 000*0 I I I O © «"« • » • 000 INDUCTION-MOTOR REGULATION 131 3. FREQUENCY PULSATION 81. If the frequency of the voltage supply pulsates with sufficient rapidity that the motor speed can not appreciably follow the pulsations of frequency, the motor current and torque also pulsate; that is, if the frequency pulsates by the fraction, p, above and below the normal, at the average slip, s, the actual slip pulsates between s + p and a — p, and motor current and = =■ - ^pts uw ^ ^^J '^ isV*i ix *!vS \ \ y§ j«fe -r V. ^„ s^ ft \\ Hd \ 5; ^ k V, ** "' ^ \ N y-o.oi-o.ij Zg-o.i*ojj z, -0.0540 ISJ\ \ OR i PERCE HT ^ yp Tin. riF. ;; -ior.o i r,,.» :1,,lM..o'»«M);».«.r/.!l]lr1i.i:1».;l Fig. 50. — Effect of Frequency Pulsation on Induction Motor. torque pulsate between the values corresponding to the slips, s + p and 8 — p. If then the average slip s < p, at minimum frequency, the actual slip, a — p, becomes negative; that is, the motor momentarily generates and returns energy. As instance are shown, in Fig. 50, the values of current and of torque for maximum and minimum frequency, and for the average frequency, for p = 0.025, that is, 2.5 per cent, pulsa- tion of frequency from the average. As seen, the pulsation of current is moderate until synchronism is approached, but be- 132 ELECTRICAL APPARATUS comes very large near synchronism, and from slip, s = 0.025, op to synchronism the average current remains practically con- stant, thus at synchronism is very much higher than the current at constant Frequency. The average torque also drops some- what below the torque corresponding to constant frequency, as shown in the upper pari of Fig. 50. 3. LOAD AND STABILITY 82. At constant voltage and constant frequency the torque of the polyphase induction motor is a maximum at some definite speed and decreases with increase of speed over that correspond- ing to the maximum torque, to zero at synchronism; it also de- ereases with decrease of speed from that at the maximum torque point, to a minimum at standstill, the starting torque. This maximum torque point shifts toward lower speed with increase of the resistance in the secondary circuit, and the starting torque thereby increases. Without additional resistance inserted in the secondary circuit the maximum torque point, however, lies at fairly high speed not very far below synchronism, 10 to 20 per cent, below synchronism with smaller motors of good effi- ciency. Any value of torque between the starting torque and the maximum torque is reached at two different speeds. Thus in a three-phase motor having the following constants: impressed e.m.f., eg = 110 volts: exciting admittance, 1" ~ 0.01 — OAj; primary impedance, Zv = 0.1+ 0.3 j, and secondary impedance, Z\ = 0.1 + 0.3 j, the torque of 5.5 synchronous kw. is reached at. 54 per cent, of synchronism and also at the speed of 94 per cent, of synchronism, as seen in Fig. 51. When connected to a load requiring a constant torque, irre- spective of the speed, as when pumping water against a constant head by reciprocating pumps, the motor thus could carry the load :tl two different speeds, the two points of intersection of the horizontal Hue, L, in Fig. 51. which represents the torque con- sumed by the load, and the motor-torque curve, O. Of these two points, d and r, the lower one, rf, represents unstable con- ditions of operation; that is, the motor can not operate n tln- speed, but either stops or runs up to the higher speed point, C, at which stability is reached. At the lower speed, d, a momen- tary decrease of speed, as by a small pulsation of voltage, load, etc., decreases the motor torque, D, below the torque, L, required by the load, thus causes the motor to slow down, but in doing INDUCTION-MOTOR REGULATION 13,1 so its torque further decreases, and it slows down still more, loses more torque, etc., until it comes to a standstill. Inversely, a momentary increase of speed increases the motor torque, D, beyond the torque, L, consumed by the load, and thereby causes an acceleration, that is, an increase of speed. This increase of speed, however, increases the motor torque and thereby the speed still further, and so on, and the motor increases in speed up to the point, c, where the motor torque, D, again becomes / ^ . 9 l ~/| d/y «\y ^zzzzzzzi 0 HI 02 03 , in three points, 6i, 6i, by, that is, three speeds exist at which the motor gives the torque required by the load: 24 per cent., 00 per cent., and $S per cent, of synchronism. The speeds b, and bs are stable, the speed bi unstable. Thus, with this load the motor starts from standstill, but does not run up to a speed near synchronism, but INDUCTION-MOTOR REGULATION 135 accelerates only to speed bu and keeps revolving at this low speed (and a correspondingly very large current). If, however, the load is taken off and the motor allowed to run up to syn- chronism or near to it, and the load then put on, the motor slows down only to speed b(, and carries the load at this high speed; hence, the motor can revolve continuously at two different speeds, 61 and b%, and either of these speeds is stable; that is, a momen- tary increase of speed decreases the motor torque below that :• », A y. / '• K n-"* T // j the motor thus accelerates up to 61, in the speed range between bt and 61 it slows down to b,. For this character of load, the induction-motor speed curve, D, thus has two stable branches, a lower one, from standstill, t, to the point n, and an upper one, from point m to synchronism, 136 ELECTRICAL APPARATUS where- m and n are the points of contact of the tangents from the required starting torque, p, on to the motor curve, Z>; these two stable branches are separated by the unstable branch, from n to m, on which the motor can not operate. 84. The question of stability of motor speed thus is a func- tion not only of the motor-speed curve but also of the characler of the load in its relation to the motor-speed curve, and if the change of motor torque with the change of speed is less than the change of the torque required by the load, the condition is stable, otherwise it is unstable; that is, it must lie . < ' to give stability, where L is the torque required by the load at speed, S '\ D / / / / / / t 1 0 0 i i 2 0 1 u i ( 1 1 0 7 0 1 ■ J 1 Fin. 5. Occas seated that the speed, phase in to syncl crease in speed c torque-s constant middle c converte .— Spci'il-lurqui' rliiiriirtcriMtii1 nf .-;iin;Ii*-|ilmse induct mally on polyphase induction motors on a loa l Fig. 52 this phenomenon is observed in motor can start the load but can not brin VI ore frequently, however, it is observed duction motors in which the maximum torqu ronism, with some forms of starting devices their effect with increasing speed and thus g laracteristics of forms similar to Fig. 53 iced curve as shown in Fig. 53, even at a loat torque, three speed points may exist of ne is unstable. In polyphase synchronous n rs, when starting by alternating current, t in motor, 1 as icpn- tbe form i it up io >n single- i? is nearer which de- ve motor- With a requiring which the lotors and lat is, as INDUCTION-MOTOR REGULATION 137 induction machines, the phenomenon is frequently observed that the machine starts at moderate voltage, but does not run up to synchronism, but stops at an intermediary speed, in the neighbor- hood of half speed, and a considerable increase of voltage, and thereby of motor torque, is required to bring the machine beyond the dead point, or rather "dead range," of speed and make it run up to synchronism. In this case, however, the phenomenon is complicated by the effects due to varying magnetic reluctance (magnetic locking), inductor machine effect, etc. Instability of such character as here described occurs in elec- tric circuits in many instances, of which the most typical is the electric arc in a constant-potential supply. It occurs whenever the effect produced by any cause increases the cause and thereby becomes cumulative. When dealing with energy, obviously the effect must always be in opposition to the cause (Lenz's Law), as result of the law of conservation of energy. When dealing with other phenomena, however, as the speed-torque relation or the volt-ampere relation, etc., instability due to the effect assisting the cause, intensifying it, and thus becoming cumulative, may exist, and frequently does exist, and causes either indefinite increase or decrease, or surging or hunting, as more fully discussed in Chapters X and XI, of " Theory and Calculation of Electric Circuits/ ' 1 4. GENERATOR REGULATION AND STABILITY 86. If the voltage at the induction-motor terminals decreases with increase of load, the maximum torque and output are de- creased the more the greater the drop of voltage. But even if the voltage at the induction motor terminals is maintained con- stant, the maximum torque and power may bo reduced essen- tially, in a manner depending on the rapidity with which the voltage regulation at changes of load is effected by the generator or potential regulator, which maintains constancy of voltage, and the rapidity with which the motor speed can change, that is, the mechanical momentum of the motor and its load. This instability of the motor, produced by the generator regulation, may be discussed for the case of a load requiring constant torque at all loads, though the corresponding pheno- menon may exist at all classes of load, as discussed under 3, and may occur even with a load proportional to the square of the speed, as ship propellors. 138 ELECTRICAL M'PARA TVS The torque curve of the induction motor at constant terminal voltage consists of two branches, a stable branch, from the maximum torque point to synchronism, and an unstable branch, that is, a branch at which the motor can not operate on a load requiring constant torque, from standstill to maximum torque. With increasing slip, s, the current, i, in the motor increases. If then D = torque of the motor, ,. is positive on the stable, negative on the unstable branch of the motor curve, anil this rate of change of the torque, with change of current, expfnmed as fraction of the current, is: , _ 1 dD * ~ D di' it may be called the stability coefficient of the motor. If k, is positive, an increase of i, caused by an increase of slip, a, that is, by a decrease of speed, increases the torque, D, and thereby checks the decrease of speed, and inversely, that is, the motor is stable. If, however, k, is negative, an increase of i causes a decrease of D, thereby a decrease of speed, and thus further increase of j and decrease of D; that is, the motor slows down with increas- ing rapidity, or inversely, with a decrease of t, accelerates with increasing rapidity, that is, is unstable. For the motor used as illustration in the preceding, of the constants c = 110 volts; Y = 0.01 - 0.1 j; Z0 - 0.1 -f- 0.3 j, Zi = 0.1 + 0.3 j, the stability curve is shown, together with speed, current, and torque, in Fig. 54, as function of the output. As seen, the stability coefficient, k„ is very high for light-load, decreases first rapidly and then slowly, until an output of 7000 watts is approached, and then rapidly drops below zero; that is, the motor becomes unstable and drops out of step, and speed, torque, and current change abruptly, as indicated by the arrows in Fig. 54. The stability coefficient, k„ characterizes the behavior of the motor regarding its load-carrying capacity. Obviously, if the terminal voltage of the motor is not constant, but drops with the load, as discussed in 1, a different stability coefficient results, which intersects the zero line at a different and lower torque. 86. If the induction motor is supplied with constant terminal voltage from a generator of close inherent voltage regulation INDUCTION -MOTOR REGULATION 13!) and of & size very large compared with the motor, over a supply circuit of negligible impedance, so that a sudden change of motor current can not produce even a momentary tendency of change of the terminal voltage of the motor, the stability curve, k„ of Fig. 54 gives the performance of the motor. If, however, II 1 1 1 1 1 1 \ '.Iv Y-0.Q1-O.lj Z-O.l + OJj CONSTANT POTENTIAL HO VOLTS GENERATOR IMPEDANCE Z-0.OZ-0.Sj l> TRANSFORMER IMPEDANCE I-0.0**0.1j 1* 9 STABILITY COEFFICIENT OF MOTOR i,-l STABILITY COEFFICIENT OF SYSTEM k,-k* MAXIMUM OUTPUT POINT • f .<* ±> / rt ja », A E / 1 i // m % .,, v. / \ \ \ J so : ■is* 1 **» / N «. n r / 3D / M """,. '," a oi - >.. , „ Fio. 54. — Induct ion-motor loud curves. at a change of load and thus of motor current the regulation of the supply voltage to constancy at the motor terminals re- quires a finite time, even if this time is very short, the maximum output of the motor is reduced thereby, the more so the more rapidly the motor speed can change. Assuming the voltage control at the motor terminals effected 140 ELECTRICAL APPARA TVS by hand regulation of the generator or the potential regulator in the circuit supplying the motor, or by any other method which is slower than the rate at which the motor speed can adjust itself to a change of load, then, even if the supply voltage at the motor terminals is kept, constant, for a momentary RuctOBtaon of motor speed and current, the supply voltage momentarily varies, and with regard to its stability the motor corresponds not to the condition of constant supply voltage but to a supply voltage which varies with the current, hence the limit of stability is reached at a lower value of motor torque. "At constant slip, s, the motor torque, D, is proportional to the square of the impressed e.m.f., e1. If by a variation of slip caused by a fluctuation of load the motor current, i, varies by di, if the terminal voltage, e, remains constant the motor torque, D, varies by the fraction k, = ,, ..> or the stability coefficient of the motor. If, however, by the variation of current, di, the impressed e.m.f., e, of the motor varies, the motor torque, D, being proportional to e!, still further changes, proportion*! to 1 de* 2 de the change e!, that is, bv the fraction k,= —„ -p- = - ,.• anrl the ' - e* di e d% total change of motor torque resultant from a change, di. of the current, i, thus is k0 = k. + kr. Hence, if a momentary fluctuation of current causes a momen- tary fluctuation of voltage, the stability coefficient of the motor is changed from k, to kn = k, + fc„ and as k, is negative, . the voltage, e, decreases with increase of current, i, the stability coefficient of the system is reduced by the effect of voltage regu- lation of the supply, '.,, and kr thus can be called the regulation coefficient «f the system. kr = ,-. thus represents the change of torque produced by the momentary voltage change resulting from a current change di in the system; hence, is essentially a characteristic of the supply system and its regulation, but depends upon the motor de only in so far as .. depends tijmn the power-factor of the load. In Fig. 54 is shown the regulation coefficient, k,, of the supply- system of the motor, at 110 volts maintained constant at the motor terminals, and an impedance, Z = 0.16 + 0.8 j, between motor terminals and supply e.m.f. As seen, the regulation coefficient of the system drops from a maximum of about 0.03, INDUCTION-MOTOR REGULATION 141 at no-load, down to about 0.01, and remains constant at this latter value, over a very wide range. The resultant stability coefficient, or stability coefficient of the system of motor and supply, A0 = kn + kn as shown in Fig. 54, thus drops from very high values at light-load down to zero at the load at which the curves, k, and fcr, in Fig. 54 intersect, or at 5800 kw., and there become negative; that is, the motor drops out of step, although still far below its maximum torque point, as indicated by the arrows in Fig. 54. Thus, at constant voltage maintained at the motor terminals by some regulating mechanism which is slower in its action than the retardation of a motor-speed change by its mechanical momentum, the motor behaves up to 5800 watts output in exactly the .same manner as if its terminals were connected directly to an unlimited source of constant voltage supply, but at this point, where the slip is only 7 per cent, in the present instance, the motor suddenly drops out of step without previous warning, and comes to a standstill, while at inherently constant terminal voltage the motor would continue to operate up to 7000 watts output, and drop out of step at 8250 synchronous watts torque at 16 per cent. slip. By this phenomenon the maximum torque of the motor thus is reduced from 8250 to 6300 synchronous watts, or by nearly 25 per cent. 87. If the voltage regulation of the supply system is more rapid than the speed change of the motor as retarded by the momentum of motor and load, the regulation coefficient of the system as regards to the motor obviously is zero, and the motor thus gives the normal maximum output and torque. If the regulation of the supply voltage, that is, the recovery of the terminal voltage of the motor with a change of current, occurs at about the same rate as the speed of the motor can change with a change of load, then the maximum output as limited by the stability coefficient of the system is intermediate between the minimum value of 6300 synchronous watts and its normal value of 8250 synchronous watts. The more rapid the recovery of the voltage and the larger the momentum of motor and load, the less is the motor output impaired by this phenomenon of instability. Thus, the loss of stability is greatest with hand regulation, less with automatic control by potential regulator, the more so the more rapidly the regulator works; it is very little 142 ELECTRICAL APPARATUS with compounderl alternators, and absent where the motor terminal voltage remains constant without any control by prac- tically unlimited generator capacity and absence of voltage drop between generator and motor. Comparing the stability coefficient, h„ of the motor load and the stability coefficient, ko, of the entire system under the assumed conditions of operation of Fig. 54, it is seen that the former intersects the zero tine very steeply, that is, the stability remains high until very close to the maximum torque point, and the motor thus can be loaded up close to its maximum torque without impairment of stability. The curve, k0, however, intersects the zero fine under a sharp angle, that is, long before the limit of stability is reached in this case the stability of the system has dropped so close to zero that the motor may drop out of step by some momentary pulsation. Thus, in the case of instability due to the regulation of the system, the maximum output [joint, as found by test, is not definite and sharply defined, but the stability gradually decreases to zero, and during this decrease the motor drops out at some point. Experimentally the difference l>etween the dropping out by approach to the limits of stability of the motor proper and that of the system of supply is very marked by the indefiniteness of the latter. In testing induction motors it thus is necessary to guard against this phenomenon by raising the voltage l>eyond normal before every increase of load, and then gradually decrease the voltages again to normal. A serious reduction of the overload capacity of the motor, due to the regulation of the system, obviously occurs only at very high impedance of the supply circuit; with moderate impedance the curve, It, is much lower, and the intersection between fc, and k, occurs still on the steep part of k„ and the output thus is not materially decreased, but merely the stability somewhat reduced when approaching maximum output. This phenomenon of the impairment of stability of the induc- tion motor by the regulation of the supply voltage is of prac- tical importance, as similar phenomena occur in many instances. Thus, with synchronous motors and converters the regulation of the supply system exerts a similar effect on the overload capacity, and reduces the maximum output so that the motor drops out of step, or starts surging, due to the approach to the stability limit of the entire system. In this case, with syn- INDUCTION-MOTOR REGULATION 143 chronous motors and converters, increase of their field excita- tion frequently restores their steadiness by producing leading currents and thereby increasing the power-carrying capacity of the supply system, while with surging caused by instability of the synchronous motor the leading currents produced by increase of field excitation increase the surging, and lowering the field excitation tends toward steadiness.