STATOR TOOTH_ ~~~~~PITCH

COMMON END RING

Fig. 3. A segmented rotor to reduce strayload loss

While equation 12 provides the lower limitof the bridge thickness, its upper limit isprobably 1 or 2 per cent of the depth belowa slot of the stator, depending on the motorsize. Because of the lower flux densities inlarger motors, the ratio of bridge thicknessto depth below stator slot should be smaller.The bridge thickness required to shunt

off the harmonic flux is difficult to deter-mine. However, the following measuresare helpful in reducing this minimum re-quirement:

A. The number of rotor slots being slightlyfewer than that of the stator slots;B. Semiclosed stator slots;C. Sufficient air gap.

3. By segmenting the rotor, the averageharmonic electromotive force for the short-circuit paths is greatly reduced. To thewriter's knowledge, this method has notbeen used previously for the purpose ofreducing stray load loss. Referring to Fig.3, the rotor is divided into two or threesections by thin common end rings. Eachsection has a skew of one stator slot. Asthe common end rings serve only to bringthe currents in phase, their required thick-ness is no more than the average vidth ofthe rotor slot. Their diameters are reducedto avoid eddy-current loss induced by statortooth harmonics. For a rotor with n-sec-tions, the average harmonic electromotiveforce is reduced by a factor of n. If thecondition between the cage bars and lamina-tions remains unchaniged, the stray load lossowing to it should be reduced to 1/n2 of itsprevious value.

Conclusions

The stray load losses are divided intotwo categories; those due to theoreticalimperfection, and those due to industrialimperfection. Among the first groupare rotor currents induced by stator har-

monies, iron losses due to leakage fluxes,and eddy currents loss in stator conduc-tors. Their effects are illustrated inFig. 1.The predominant factor of stray load

loss due to industrial imperfection is theshort circuiting of the cage bars by thelaminations. Measures within a de-signer's control which would minimizethis effect are:

1. Making the rotor slots slightly fewerthan stator slots;2. Using closed-lot rotors;3. Segmenting the rotor.

References

1. EQUIVALENT CIRCUITS AND THEIR APPLICATIONIN DESIGNING SHADED POLE MIOTORS, S. S. L.Chang. AIEE Trais(lctions, vol. 70, pt. I, 1951,pp. 690-99.

2. DiE-CAST ROTORS FOR INDUCTION MOTORS,L. C. Packer. AIEE Transactionis, vol. 68, pt. I,1949, pl). 253-61.

No Discussion

THE adaptability of the inductiongenerator to certain specific types of

applications in power systems prompts acomprehensive discussion of the principleapplication problems. From an over-allengineering standpoint, there are manyapplications where induction generatorsoffer advantages over synchronous ma-chines, resulting in a simplified installa-tion with economy in first cost and inoperating and maintenance expenses.To be sure, induction generators havesome disadvantages when compared withsynchronous generators, but there aremany locations where the advantagesovershadow the disadvantages, and whereinduction machines should be considered.The attractive feature of the induction

generator is simplicity of operation.Placing the machine on the line involvesbringing it to the proper speed and closingthe circuit breaker connecting it to thesystem. All of the features normallyused with synchronous generators forcontrol of speed, excitation, and syn-chronizing are omitted when induction

generators are used. Thus, where auto-matic operation is desired, the controlsystem is greatly simplified.The principle of operation of the induc-

tion generator is described in most a-cmachinery textbooks and in several pub-lished articles and papers. Its use iscontingent upon the system characteris-tics and the requirements of the generat-ing station. The induction machine doesnot have the inherent ability to controlvoltage, frequency, or number of kilovars.The power output is a function of the slipand depends on the prime mover inputand system frequency and voltage.Sufficient synchronous capacity must beavailable on the system to control thesefactors to assure proper operation of theinduction generator.The induction generator generally is

built with a conventional armature wind-ing and a squirrel-cage rotor, which is avery rugged and simple form of construc-tion. However, the fact that the in-duction machine has no field windingsmeans that current to magnetize the ma-

chine must be supplied by the system towhich it is connected. Thus, the systemmust be capable of supplying the laggingkilovars required to establish the air-gapflux in the induction generator. Thesekilovars may be supplied by overexcitedsynchronous machines on the system, orthey may be supplied by shunt capacitors.When shunt capacitors are used, themost desirable location from the systemstandpoint is at the terminals of thegenerator; but when this type of applica-tion is made, the problem of self-excita-tion must be considered.The purpose of this paper is to discuss

the theory of operation of the inductiongenerator, the self-excitation problem,and other application problems thatshould be considered in designing an in-duction generator installation.

Induction Generator Operation

When the stator armature winding ofan induction machine is connected to ana-c system, a synchronously rotating fluxis established in the air gap. Whenoperated as a motor, the rotating flux

Paper 54-26, recommended by the AIEE RotatingMachinery Committee and approved by the AIEECommittee on Technical Operations for presenta-tion at the AIEE Winter General Meeting, NewYork, N. Y., January 18-22, 1954. Manuscriptsubmitted October 21, 1953; made available forprinting November 13, 1953.

J. E. BARKLE and R. W. FERGUSON are with theWestinghouse Electric Corporation, East Pitts-burgh, Pa.

Barkle, Ferguson-Induction Generator Theory and Application

Induction Generator Theory andApplication

J. E. BARKLE R. W. FERGUSONMEMBER AIEE ASSOCIATE MEMBER AIEE

12 FIEBRUARY 1954

of Fig. 1. It will be noticed on thiscurve that there is a definite limit to thepower output of the induction generator.This limiting value of power will becalled the power limit of the generatorand is given by the following equation

B

Fig. 1. Equivalent circuit and phasor diagram of an induction generator

PE2[xm.+xs-x'-2rs I22[rS2+ (xm+xs)(xI) ]

PL = power limit

Xmxr,-! _ v X1---=Xs±m,

M Mr

(5)

induces a current and voltage in the rotorconductors, and the rotor turns at a speedslightly less than that of the flux. Thedifference in speed of rotation betweenthe flux and the rotor, expressed as apercentage of synchronous speed, iscalled the slip. The slip is positive whenthe machine operates as a motor and therotor speed is less than that of the flux.As a generator, the induction machine

is driven by a prime mover; and as thespeed of the rotor is increased to equalsynchronous speed, there is no relativemotion between the rotor conductors andthe flux. Hence, no voltage or current isinduced in the rotor bars. A fiurther in-crease in speed causes a reversal in rela-tive direction of rotation between therotor bars and the flux, and the rotor volt-age and current are correspondingly re-versed. The slip under this condition isconsidered to be negative. Shaft torque,supplied by the prime mover, is trans-ferred across the air gap to the stator,from which it is delivered to the systemas generated power. The net poweroutput is the shaft input less the losseswithin the machine and is a function ofthe slip.-The equivalent circuit and phasor dia-

gram of an induction generator are shownin Fig. 1. For simplicity, core loss isneglected in this equivalent circuit. Theterms of the equivalent circuit are de-fined as follows:

rr, xr, Jr =rotor resistance, reactance andcurrent expressed in terms of statorvoltage and frequency

rs, xs, Is= stator resistance, reactance andcurrent

Xm, Im= magnetizing branch reactance andmagnetizing current

E:= terminal voltageEa = air-gap voltages = slip

The equivalent circuit contains twovariables; the slip s which is a functionof speed, and the magnetizing reactanceXm which is determined by saturationand is a function of the air-gap voltage Ea.The magnetizing reactance can be ob-tained from a saturation curve for themachine in question; but for terminal

voltage variations within the narrowlimits normally encountered on powersystems, it is usually sufficiently accurateto consider the magnetizing reactance acon stant.The power output of the induction

generator can be derived from the equiva-lent circuit of Fig. 1. When the slip sis known or assumed and the magnetiz-ing reactance x,m is known, the equivalentcircuit can be reduced to a single im-pedance.

(r +jx) (JXm)

Z=r,+jx,+ (1)r-'±j(Xr+Xm)s

The stator current is

Is =-E (2)

and the air-gap voltage is

Ea=Et+Is(rs+jxs) (3)

The current in the magnetizing branchis

ImE= (4)J1Xn

The phasor diagram of the voltages andcurrents for a typical design of a slow-speed water-wheel-driven induction gen-erator calculated by this procedure isshown in Fig. 1. A significant feature isthe fact that the stator current, whichrepresents the output current of the gen-erator, leads the- terminal voltage- bythe power factor angle 0. The reactivecomponent of this current, Is (sin 0), isrepresentative of the kilovars that mustbe supplied by overexcited synchronouscapacity or shunt capacitors on thesystem.

Equation 1 shows that a given slipdetermines the value of Z. This in turngives the value of I, and consequentlythe power and var output of the genera-tor EtI8* (Is* is the conjugate of thephasor 18). Thus for a given terminalvoltage there is a fixed relation betweenreal and reactive power. This is illus-trated by curve I in Fig. 3 for the machine

If stator resistance is neglected and thedefinition of x' is used the expression callbe written

PL (XM( Xs)(xm+xr) ( XmXr

2m T Xm+Xr

Typical characteristic curves of an in-duction generator are shown in Fig. 2.The efficiency is good at loadings above 25per cent; in fact, it is comparable to thatof a synchronous generator. Power fac-tor, on the other hand, becomes progres-sively worse as the load and slip are re-duced below rated load values. Thereason for this is that as long as the air-gap voltage is constant, the magnetizingcurrent is constant. Thus, the reactivecurrent required by the generator is sub-stantially constant throughout the entireload range. Consequently, to obtain themost favorable power factor, the ma-chine should be operated as near ratedload as possible at all times.The characteristic curves of Fig. 2 are

for a slow-speed water-wheel-driven ma-chine. The magnetizing requirements ofthe generator are significantly, ffectedby speed and size so that the valuesshown in Fig. 2, and used in all of theexamples of a typical induction genera-tor, may not apply to machines of dif-ferent speed and rating.

Induction generators are particularlysuited to hydroelectric installations ofrelatively small size, and where severalmachines can be installed to comprisethe total capacity. In this way, thenumber of machines in operation can bevaried to suit the water conditions sothat each machine is operated at fullload at all times. The ideal plant loca-tion is where peaking capacity is requiredat a point on the system where sufficientreactive is available to provide theexcitation.

Short-Circuit Characteristics

Since the induction generator dependson the terminal voltage to establish the

Barkle, Ferguson-Induction Generator Theory -and Application

a

FEBRUARY 1954 13

-1.2

-1.1 _

-1.01-_

- .9

- .8z

IL

_

-a

cca

- a.

aL

4

0- .-7Lii

z

- .5

0:

a. - .

-.3

-.2

O -L O 25 50 75

LOAD IN PERC

magnetizing current, it cannot contributeon a sustained basis to a short circuitthat causes a collapse in terminal volt-age. A 3-phase short circuit at the ter-minals of the generator causes the voltageat that point to collapse to zero. At theinstant of fault, certain flux linkagesexist in the rotor circuit. These cannotchange instantaneously, and thereforethe rotor and stator currents increase tomaintain the flux linkages constant.This is similar to the phenomenon thatgives rise to subtransient current in asynchronous machine. In Appendix Ithe fundamental equation

e+ To d =0 (16)dt

is derived where e' is the voltage behindthe transient reactance

Xr+Xm

and is proportional to the rotor flux link-ages. Voltage e is proportional to therotor field current (see equation 13).

Using equations 13, 14, and 15, andneglecting stator resistance r, one ob-tains the following differential equationduring a 3-phase fault where Et = 0

/ +Xr+x dlsXS±XM * wr dt

subject to the initial condition eo-'=eo+'which gives 180+ =eo+'/x'. 0-and O+are used to indicate the instant beforeand instant after the fault. The solutionto this equation is of the form

jO' t/T' (8)xf

CENT

Fig. 2 (left).Characteris ticcurves of an in-duction generator

Fig. 3 (right).P versus 0 curvesfor an inductiongenerator at 100-,120-, and 140-per-cent voltage,100-per-cent fre-

quency

z 2

a-

w 2z

w3:0a.

w

o-44w

0r

letting

T'=(±) (Xir)

where

eo'= the initial value of e' determined bythe load current Is flowing beforethe fault by equation 14

Thus the short-circuit current has a highinitial value but decreases to zero at arate determined by the short-circuit timeconstant T, of the generator. For atypical design, the short-circuit timeconstant is the order of 1 to 3 cycles.To evaluate the effect on circuit breaker

duty and system relaying, it is importantto know the short-circuit characteristicsof the induction generator. In viewof the rapid decay of current, the induc-tion generator has no effect on the inter-rupting duty of 5- or 8-cycle circuitbreakers. The duty on these breakerscan be determined by neglecting theinduction generators.

Induction generators must be con-sidered in computing the momentaryduty of circuit interrupting and protec-tive relaying equipment, and the momen-tary current output of the inductionmachines can be the determining factorin choosing ratings for these protectivedevices. For example, the momentaryrating required of a circuit breaker maybe above that normally supplied in abreaker of sufficient interrupting capacity,and a breaker of the next higher ratingmay be required.The momentarv current contribution

P- POWER IN PER UNIT1 2 3

I-PER UNIT: RATED GENERATOR KW0-SHUNT CAPACITORS AT GENERATOR

TERMINALSA-SYSTEM OF FIG. 4, SHUNT CAPACITOR

KVAR : 2 PER UNIT

of induction generators should not beoverlooked in designing bus work, andin applying current transformers andrelays. The current transformers musthave a mechanical rating sufficient forthe application, and the relays musthave a short-time thermal rating that isadequate.

Relaying of induction generator instal-lations is similar to relaying of syn-chronous machine installations, exceptthat some modification can be made sincethe induction machines do not contrib-ute a sustained current. A form of dif-ferential relaying is normally used, but itcan be supplemented or replaced withsimple overcurrent or directional over-current relavs in some cases.

Self-Excitation of an InductionGenerator

From the previous description of theoperation of an induction generator it canbe seen that the only requirements forobtaining an output are a source of magne-tizing vars and a suitable load below thepower limit. The relationship betweenpower and vars for a typical inductiongenerator is illustrated in Fig. 3. CurveI shows this relation for rated terminalvoltage.The required vars can be supplied by a

synchronous machine or by shunt capaci-tors. Thus an output can be obtained

Barkle, Ferguson-Induction Generator Theory and Application14 FIEBRUARY 1954

from the induction generator when it isconnected to a system consisting only ofshunt capacitors and a load. Naturally,some energy must be introduced into theelectrical system to start the build up ofvoltage to the operating point. Thiscan take the form of an initial charge onthe capacitors or, more effectively, aninitial current in the induction generator.Both of these sources are available whenan induction generator, load, and shuntcapacitors are suddenly separated froma synchronous system.

If the induction generator is separatedfrom a system with a sufficient amountof capacitors connected to its terminals,the voltage rises or falls and the frequencychanges until the net magnetizing varsfrom the capacitors and the load absorbedby the remaining system exactly matchthe requirements of the induction gen-erator. In general, an excess of capaci-tive vars causes the voltage to rise, withthe exception of the case discussed laterwhere the number of vars exceeds thatwhich can be absorbed by the inductiongenerator at the given load when operat-ing at a slip greater than the slip at thepower limit. The frequency of thesystem adjusts itself to a point where thegenerator slip gives the required loadoutput.The frequency of operation of an induc-

tion generator with self-excitation is veryclose to the frequency corresponding tothe speed of the generator since the pull-out slip is normally less thah 10 per cent,and the machine will not operate at aload above the power limit. In thefollowing analysis the frequency of opera-tion was selected first. The slip andconsequent rotor speed is then determinedby the system load characteristics. Thelow value of slip at the power limit allowsper cent frequency and per cent speedvariation to be considered equal.

Analysis of Self-Excitation

A number of methods are available foranalyzing the performance of an induc-tion machine under self-excited condi-tions. Most of these methods deal onlywith the steady-state conditions andconsist of opening the network consistingof the induction generator equivalent andthe system equivalent circuit at somepoint, and of finding a voltage that givesequal and opposite shunt admittanceslooking in either direction from the givenpoint.Another method for analyzing the

induction generator self-excitation voltageis by use of modified synchronous machinetransient theory. This is described in

Appendix I. This method clearly out-lines the region of self-excitation and alsogives an indication of rate of rise of thevoltage. The basic equation for theperformance of the induction generatoris as follows

de'e+To =0 (16)

dt

where

e=per unit voltage that would be inducedin stator in steady-state conditionisfor zero stator current and a rotor currentequal to the instantaneous value of rotorcurrent

e =per unit voltage that would imme-diately appear at the terminals of themachine if the stator current weresuddenly made zero. This quantityis proportional to the rotor flux linkages

TO= open-circuit time constant of the rotor

T0TXwrr

where X is 377 radians per second and theother impedances are defined by theequivalent circuit of the induction generatorin Fig. 1.

Using this equation and relating e and e'by using the assurmed constant systemimpedance, the terminal voltage of thegenerator can be shown to be either ex-ponentially increasing or decaying. Therange of self-excitation is shown in Ap-pendix II to be exactly the shaded regionof Fig. 3 for Et= 100 per cent when thesystem impedances are converted intopower and vars.For fixed constants (constant satura-

tion) the magnitude of the terminal volt-age increases as indicated in the followingequation

Et=Et i.itial (9)

where 1/a is defined in equation 27 andis negative when the voltage is increasing.Thus in 1/10a seconds the magnitudeincreases 10.5 per cent. It should benoted that l/a is always greater than Toand that as a point of stable operation isapproached 1/ca approaches infinity.

Self-Excitation of InductionGenerator Connected to PowerSystem

Since the voltage at the terminals ofthe induction generator will increase onlyif the system P+jQ at the initial voltage

Fig. 4. Systemassumed to beconnected to theinduction gene-

rator

(determined by maintaining constant e'following separation from the synchronoussystem) is within the P and Q curves ofthe induction generator, these curves pro-vide a good means of qualitatively deter-mining the performance of the generator.This requires a knowledge of the varia-tion of the power and var requirements ofthe system as a function of voltage. Inthe analysis of this section the system fre-quency will be assumed to be equal torated frequency.

In the case of capacitors applied at theterminals of the generator where thegenerator capacitor combination can beisolated from the system, the variationof the system var requirement is directlywith the square of the terminal voltage.The P versus Q curves for a typical in-duction generator for 100-, 120-, and140-per-cent voltage are shown in Fig. 3.Assume first that this generator is con-nected to a shunt capacitor whose ratingis 10 per cent greater than the no-loadgenerator var requirements. The pointsindicated by circles give the var outputof the capacitor at 100, 120, and 140volts. The capacitor vars are within theregion of voltage increase for 100- and120- per-cent voltage and just above theregion for 140-per-cent voltage. Thusthe generator voltage would be expectedto rise to slightly less than 140 per centof rated voltage.The system of Fig. 4 corresponds to a

step-up transformer, high-voltage capac-itor bank, step-down transformer, and alightly loaded distribution feeder. Therelative proportions of the elements of thesystem are shown in terms of the capaci-tor bank rating. The var requirementsof such a system are not proportional tothe square of the voltage, as the magnetiz-ing current of transformers increases veryrapidly as the voltage is increased abovethe rated value. The curves for the Pand Q requirements of the system in perunit of the capacitor rating are shown inFig. 5 as a function of generator terminalvoltage.As a second example, assume the induc-

tion generator is connected to the systemof Fig. 4 with the shunt capacitor bankequal to twice the KW rating of the in-duction generator. The triangles inFig. 3 show the P and Q requirements ofthe system for 100-, 120-, and 140-per-cent

STEP UP STEP DOWNTRANSFORMER TRANSFORMERKVA = 70.7% KVA 26.5%

INDUCTIONGENERATORTERMINAL DISTRIBUTION

TRANSFORMERSSHUNT r- T1 KVA - 35.4 %

CAPACI'TOR | LOAD KW: 6.6 %KVA - 100%

Barkle, Ferguson-Induction Generator Theory and Application 15FEBRUARY 1954

I.6

IL0

wo

-auI -

a.

1.4 I-

.2 - _

1.0 - -

I.8

..6

. 2 _

too 110 120 130 140SYSTEM VOLTAGE %

Fig. 5. Variation of P and -Q with voltagefor the system of Fig. 4

voltage and the curve connecting thesetriangles gives the trace of P versus Qfor the system. Again the voltage wouldincrease to approximately 140 per cent.The third example shown in Fig. 6

refers to an induction generator con-nected to the system of Fig. 4 where theshunt capacitor bank is 10 times the rat-ing of the generator. Again the 100-.120-, and 140-per-cent points are indi-cated. If the initial voltage is 100 per centthe voltage will collapse since the systempoint lies outside the induction generatorP versus Q curve for 100-per-cent voltage.However, if an unusual system conditioncauses the initial voltage to be close to140 per cent the system point is withinthe generator 140-per-cent curve and thevoltage will rise to slightly less than 150-per-cent voltage.

In general, if the system power andvars left connected to the inductiongenerator lie inside the area bounded bythe generator power versus var curveand the var axis, the voltage at the ter-minals will increase to a stable operatingpoint. If the system point is above theregion there is a possibility that decreasingvoltage will cause either an increase insystem vars or a fast decrease in generatorvars because of the reduction in saturationthat will move the system point withinthe region and give self-excited operation.This would be the case when starting ateither of the C points in Fig. 3.

Effect of System Frequency onSelf-Excitation Phenomena

If an induction generator is isolatedfrom the synchronous portion of a powersystem, it is quite possible that the re-maining load on the induction generatorwill be less than the prime mover torque.This means that the generator will speedup and consequently the system frequency

will increase. This causes the region ofself-excitation to shrink radially almostinversely proportional to the increase infrequency; however, the var output ofcapacitors connected to the generatorincreases directly with the increase infrequency and consequently the maximumvoltage resulting from self-excitation isincreased. These effects are illustratedin Fig. 7 where the P versus Q curves forthe induction generator used in Fig. 3are shown for 125 per cent of rated fre-quency at 100- and 140-per-cent voltage.The capacitor points A and C have alsobeen repeated and show that the self-excitation voltage under these conditionsis greater than 140-per-cent voltage. Ifthe two cases considered in the previoussection involving the system of Fig. 4are considered at 125-per-cent frequency,the induction generator terminal voltageresulting from self-excitation will be ofthe order of 180 to 190 per cent. Itshould be noted that the voltage at thecapacitor bank is 10 to 20 per cent higherbecause of the heavy var flow through thestep-up transformer bank from the highvoltage capacitor to the induction genera-tors.

Application of Static Capacitorsto Induction Generators

The previous sections have given amethod of analysis of self-excitation ofinduction generators. Also for capacitorslocated at the terminals of the generatorsor for a large bank located on a specificsystem, it has been found that seriousovervoltages can occur, particularly ifthe generator overspeeds.

In the application of static correctionthe voltage possible during self-excitationconditions should not be allowed to goover 10 to 20 per cent above rated volt-age on any part of the system. This canbe accomplished by the following meth-ods

Limiting the amount of correction applied.The allowable correction should be de-termined for the highest system frequencyand the most unfavorable combination ofinduction generators and capacitors thatcould result from a fault or system dis-turbance.Using some protective device such as agap or short-circuiting switch to short-circuit the induction generator in case ofhigh voltage resulting from self-excitation.Such a device would have to be very ac-curately calibrated to differentiate betweennormal voltage and an overvoltage condi-tion. It would also be subject to falsetripping resulting from switching surgesor other transient high voltages. Anysuch device would have to be mechanicallystrong enough to withstand the full momen-tary short-circuit current from the induction

generators when operating at the maximumallowable self-excitation voltage.Distributing the capacitor correction onthe system in such a manner that a danger-ous amount could not be isolated from therest of the system with the inductiongenerators without a substantial load alsobeing present that would prevent damaginghigh voltages.

Summary and Conclusions

1. An explanation of the method of opera-tion of an induction generator has beenpresented.2. A derivation is given for the circlediagram giving the relation between powerand var output of an induction generator.

3. An induction generator does not con-tribute to the interrupting duty of a breaker,but makes substantial contribution to themomentary rating of any devices connectedto the generator. An expression is givenfor calculating the fault current from thegenerator.

4. A method of analysis of self-excitationof induction generators using modifiedsynchronous machine transient theory isgiven.5. This analysis shows that during self-excitation conditions the voltage rises ordecays exponentially and that the coefficientof time in the exponent is less than thereciprocal of the open-circuit time constant.6. The region of system real and reactive

P- POWER IN PER UNITO 1 2 3 4 5 6

0

A__

-2 -

w

5 001- A Et1000

CL Ef~~~C-E2:140/

-12 D - Et:SY 0T/.

8

Fig. 6. P versus 0 curves for an inductiongenerator at 100-, 120-, and 140-per-cent

voltage, 100-per-cent frequency

Barkle, Ferguson-Induction Generator Theory and Application

1.I

16 FEBRUARY 1 90-4

P- POWER IN PER UNIT

I.-z

w -

r.*z

0

uli

1-

4t

lxa

o - SHUNT CAPACITORS AT GENERATORTERMINALS

A - E I100% C-£E 140%

Fig. 7. P versus 0 curves for an inductiongenerator at 100- and 140-per-cent voltage,

125-per-cent frequency

power for self-excitation has been shown tocoincide with the interior of the P versusQ curve of the induction generator describedin conclusion 2.7. A particular system has been analyzed,and voltages of 140 to 150 per cent at ratedfrequency or 180 to 190 per cent at 125-per-cent frequency were shown to bepossible.8. Increasing the frequency of operationduring self-excited conditions increases thepossible overvoltage.9. In applying static correction over-voltages should be avoided by limiting sizeof correction, by use of overvoltage pro-tective devices, and by locating the correc-tion in a manner that will preclude thepossibility of excessive self-excitation volt-ages.

Appendix 1. Self-Excitation ofAn Induction Generator

In an induction generator just as in asynchronous machine there are a certainnumber of flux linkages in the rotor orfield of the machine. The rotor fluxlinkages Nr are given by

(Xr+Xm) XrMXr=Ir 8-Is

Co co(10)

where n = 377 or 2,7 times rated frequency.I, and Ir are phasors referred to an axis

rotating with the rotor. This means thatduring normal operation Ir-=Ir'ejsWt wheres is the rotor slip. All phasors in thisappendix are referred to the rotor axisunless otherwise noted.From the equivalent circuit in Fig. 1

Et+Is(r. +jx8)JlXm

£Et Fr-+j(Xs+xm)1+IsIJXm JXm

substituting in equation 10 gives

XrX+XM[E}I

(rs +jxs+rjxm- )Xr+Xm

Xm [Et+IJ(rs+jx')] (11)COJXm

where

x =Xs+Xm+Xr

In the rotor itself, since there is no excita-tion voltage other than the changing fluxlinkages

dNTrI,r+dXr =0 (12)

dt

Let e be the open circuit voltage for a rotorcurrent Ir

e =hrjxm =Et+Is [rs+j(xs+xm)] (13)

Let e' be the voltage that would exist atthe terminals of the generator if the breakerwere suddenly opened while the generatoris carrying load current.

X, before breaker is opened

Xrr+Xm= JX,[Et +18,(r8+jx')JI

Ar after breaker is opened-Ir (xr+xr)

Et after breaker is opened= Irjxm =eC

Combining this with the fact that Nrbefore= Nr after gives

Ir (after) = [Et+Is (rs +jx') IJXm

e' =E,+Is(rs+±ix')

equation used in analysis of the transientbehavior of a synchronous machine relatinged and ed' as follows

ded,ed+TdO =dt

where e. is the exciter voltage. Naturallythere is no corresponding term to e, inthe induction generator system.

Starting from the equation

Xm+xr de'e+ -rdand for an equivalent circuit of the systemwhere the system is represented by a seriesimpedance Z=Re8+jXe, the equation forthe induction machine voltages can bederived. Both e and e' are phasors withan angle with respect to some referenceaxis of the rotor. They are also movingat synchronous speed with respect to thestator, and they have a fixed angularrelationship with the synchronous field ofthe stator. Thus any change in the angularposition of e or e' will amount to a changein angular position of the rotor with respectto the synchronous field of the stator ofequal magnitude and opposite sign. Therate of change of the angle of e or e' corre-sponds to the slip of the rotor.

Consider the following relations

Et = Is(Re±+jXe) (17)

e' = Ei+Is(rj+-jx')

e =E+1Is [rs+j(xm+x2) I

(18)

(19)1

Using these equations e' and e can beexpressed in terms of Et, the terminalvoltage of the generators, by the followingequations

rrs+Re+j(Xe+Xt)l

e=E fl+ cs±JX

=Etrs+Re+(Xe+XmXs)I=E Re+jXe(14)

Comparing equations 14 and 11, note thate' is proportional to rotor flux linkagescombining 12, 13 and 14 gives

e Xr+xm de'-rr+ . ==0

fxm COxJn dt

thus

Xr±Xnm de'e+ -=0wc7 dt

or

de'e+To- =0

dt

where

To =Xr+xrcorr

(15)

(20)

(21>

Combining equations 20 and 21 with thedifferential equation for e and e' gives thefollowing

Et[r8±+Re±j(7txm±xs)] ±[ Re+X1Xe ]

?rs+Re+j(Xe+x')d,d £tXm+Xd Re+jXeX

rrwo dt

This gives the following

rs+R1+j(Xe+xs+xm) r,co dEtLrs+Re+j(Xe+x') xm±Xr dt

(22)

or

(16) Ety+d=0di

This equation is similar to the fundamental

(23)

where

Barkie, Ferguson-Induction Generator Theory and Application 17F13BRUARY 1954

[Re+rs+j(Xe+xm+xs) r,w _7= R +=ja+(X +_Re+rs+j(Xe+x') Xm+x

ri[(rs+Re)++(Xe+xm±xs)(Xe±x/)]Xrr

Xtn+XT

[(rs+Re)(xm+xs-x')1 cr,L(rs+Re) 2+ (Xe +X')2jXmn +Xr

The solution of this differential equationis of the form Et=EeE7tY where E is de-termined so that e' (t=0)=ee' where ei' isthe value of e' the instant before the induc-tion generators and capacitors are discon-nected from the system. This equationgives

[rs+Re+j(Xe±x')1e'(t = O+) =EL + j =ei

ei'= [Et,+Is(rs+jx')]t=- (24)

or

E=ei'[ Re+jXe ] (25)

By breaking up the exponent the be-havior of the terminal voltage can be moreeasily examined.

Et = Ee- ""e-M

The first portion of the exponent is adamping term and tells whether the voltageis building up or decaying. If a is positivethe voltage decays, while if a is negativethe voltage is increasing. An examinationof the expression for a reveals that its signwill be the same as the sign of (rs+Re)2+(Xe+xm+x )(X,+x'). Thus if the equiva-lent system reactance is inductive, theexpression will be positive and the voltagewill decay.

If the equivalent system reactance isnegative, there are three cases to consider.Let Xe=-Xc

1. If Xc is very large, i.e., Xc>(xxn+Isand Xc>x') then the term will bepositive, and the voltage decays.

2. If Xc is such that X,>x' and X,<Xm+xs then (Xc+xrn+xs)(Xe+x')is negative and if it is greater inmagnitude than (rs+R,)2, a isnegative and the voltage builds up.

3. If Xe is very small, so that X,<x'and Xc<xm+xs is again positive andthe voltage decays.

Case 2 is the most interesting case sinceunder these conditions the voltage buildsup after the machine is disconnected fromthe system. In the actual machine, thereactance xm decreases as the voltageincreases and finally reaches a value suchthat Xc <xm +xs and a = 0. This represenitsa stable operating condition, since if thevoltage decreases a becomes negative,since xm increases, and the voltage againincreases; while if the voltage goes abovethis point, Xm decreases still more and abecomes positive, causing the voltage todecrease. Therefore, when the inductiongenerators are disconnected from thesystem, the voltage will increase until(rs+Re)2-+(Xs+Xm+Xs)(Xe+X')=O and

the region for an increasinig self-excitationvoltage is givenl by

r,±+Re )2+ (Xe+ V,n+Xs)(Xe+x' <0 (26)

Another point of interest is 1/a since themagnitude of the terminal voltage increases10.5 per cent during each 1/lOa seconds.

_ [ (r8±R:)2+(X.X+x')2 )](r,-+-R, + X,)

(.w)t+r)(27)

It starts at a value greater than To, theopen circuit rotor time constalnt, and in-creases to infiniity as (r+R,)2+(Xe+X )(X, +.x +sx,) approaches zero.The quantity ,3 can be related to the slip

of the rotor. The terIm e-jt gives theangular change of the position of et withtime. Since there is always a fixed anglebetween et and e' this also gives the changein the angular position of e'. The rate ofchange of angular position of e' with timeis -f, or the slip of the rotor in per centof the synchronous speed correspondingto the stator frequency is

,a (rs+R,)(x^m+xs-xl)l rr

X' L(r,+Re)2+(XSe+x')2 (x,±+x,)Using the fact that (rs+Re)2+(xm+xs+Xe)(Xe+x')=O the expression for the slips can be reduced to

?r +

R +r,L xm+xr(22)

An examination of the criterion for stableoperation using the method of equatingadmittances described earlier in this sectiongives the same expressions for the relationbetween the X's and R's at the operatingpoint and the same value for the slip atthis point. Thus both methods give thesame results, but the second method givessome idea of the rate of increase of thevoltage and of the range of values of Xethat can cause self-excitation.

Appendix I. Self-ExcitationRegion Expressed in Terms of

Pand Q

Using the values of R, and Xe that satisfythe inequality of Equation 26 the regionof self-excitation can be determined intermns of the powi-er and vars associatedvith the system impedances at a givenvoltage. The boundary of the region isfound by letting the inequality sign becomean equality sign. The region of self-excitation is given by

(rs+R)+ (Xe+x,+ x,)(Xe+x') <0 (28)

This equation can be rewritten by trans-posing terms and adding (Xe+xm*+X')2/4to both sides to the following

(R+r 2+ ((Xe+ Xt+Xm+X' jt

4 -(Xs +X ) (29)

2

Thus in terms of impedance the boundaryfor the region of self-excitation is a circle

with radius x+ and center at Re=2

-rs, X.,-- (X, +Xm +X')

The power and vars absorbed by thesystem for a voltage Et are given by

Re -Xe (30)

The circle determined by equation 28also determines a circle when plotted interms of Pe and Q, from equation 30.This can be seen by substituting in equation28 for R, and X, from equation 30 whichcan be rewritten

= leEt2 QeEt2Repe2*Qe2e ePe+Q2

Thus equation 28, the equation for theregion of self-excitation becomes

[ PeEt2 p2,FQeEt2P2 Q2Pe2Qe'Le+Qei2 5

[QEt2 +x'] =0 (31)

Expanding and collecting terms gives

E±2+ Et22rP + Et2[m+Xs+xl] XP62+ Q,2 P,2+ Q, 2 Pe2+Qe2

Qe+rs2+ (xM+xs)(x')=0 (32)

Multiplying by Pe 2+Qe2 and dividing byrs2(xm+xs)(x') this equation can be written

[t rs2> S)(

Pe ° e +

e 2[r(2+X)(x)+X,(X ][QeEXf +Xm.-X+ ]2 331

4[r22+ (Xm x )(x') ]2This shows the bounidary of the region ofself-excitation in terms of power and varsis a circle with center at

Pe=-

0==-(34)

r,E2+ ({Xm+Xs+)X')Et2 [x.m+xs+x' ]

2 [r,2+ (x,,+x)x') ]and a radius Ke given by

E^2 [xm±xs-x' 3Ke =

2 [rsI2+ (San +x,)(x ) I (35)

Appendix 111. Derivation ofRelation Between Power and Vars

for an Induction Generator

Referring to the equivalent circuit ofFig. 1, the power and vars out of an induc-tion generator are given by the following

PV+jQ9 - Et (36)

where

Et= terminal voltageZ* = conjugate of the impedance seen

Barkle, Ferguson Induction Generator Theory and Application

-Et'r,

is FE-BRUARY 1954

looking into the induction generator

Let -Z=R7+1jXg and -Z*=Ru--JX

then

Krr+jJr (jxm )Rg+jX = -r -jx -

s

an +j(Xr+Xm)

and

Po+jQg = Rg -jXgAdding and subtracting (jxM)2/2j(xr+xXthe right side of 37 and simplifying giv(

R0±jX07= -_rs-[xs+ -xm2(Xr+X,n)

Since x'=-x +±XmxrXm+xr

XM2=XM+Xs-x

xr+Xm

combining this with equation 39

R (xs+xo+Xt)R7+jX,7=- r,-j 2

(37) J(x,±+xs-x')

or

(38) (Rg0r,)+rX) + 2 +)

= -j(x,-+ X,-X') rr .( )-S j(xr+m)

Taking the magnitude of both sides of(39) equation 40 and combining terms results

in

(Rg+rS)2+ (Xg+xS-VM)(Xg+x')=0 (41)

Comparing equations 38 and 30 andequations 41 and 28 shows that the relationbetween P, and Q, (letermined by 41 is thesame as the relation between Pc and Qe.Thus the locus of the power and var output

of the induction generator is a circle withcenter at

rsIf+ (xmn+Xs)(X)Qo, = _E. 2 [X,n +XS±XI]

2 [r 2+ (xrm+-xs)(x') J

and a radius Kg given by

Kg E Xm+X,s-XI ]2 [rs2+ (Xm+Xs)(x' )]

(42)

(43)

References1. CAPACITIVE EXCITATION FOR INDUCTION GENE-RATORS, E. D. Basset, F. M. Potter. AJEETransactions (Electrical Engineering), vol. 54,May 1935, pp. 540-44.

2. SELF-EXCITATION OF INDUCTION MOTORS, C.F. Wagner. AIEE Transactions (Electrical Engi-neering), vol. 58, Feb. 1939, pp. 47-51.

3. THE SQUIRREL-CAGE TNDUCTION GENERATORFOR POWER GENERATION, T. C. Tsao, N. C. Tsang.Electrical Engineering, vol. 70, Sept. 1951, pp.793-95.

4. SrLF-EXCITATION OF INDUCTION MOTORS WITHSERIES CAPACITORS, C. F. Wagner. AIEETransactions, vol. 60, 1941, p. 1241.

No Discussion

P. L. ALGERFELLOW AIEE

Synopsis: This paper presents an analysisof the transients that occur when voltage issuddenly applied across the terminals of aninduction motor, with and without a con-nected capacitor. In wound rotor motors,the transient currents are reduced by con-necting resistance in the rotor circuit, buttransient voltages then appear across theslip rings. Both current and voltage tran-sients are increased when capacitors areconnected across the stator terminals.Calculated curves and test oscillograms aregiven for the transient currents when volt-age was applied to the terminals of a 4-pole3,000-horsepower wound rotor inductionmotor, with an external resistance acrossthe slip rings, with and without a capacitor.The calculations and tests confirm the well-known rule that, when there is an appre-ciable reactance between the powver sourceand the motor, and when voltage is sud-denly applied, the peak transient voltageon the motor capacitor unit will rise abovethe steady-state voltage. If the capacitorkilovolt-amperes (kva) exceed the motormagnetizing kva, the steady-state voltagewill be higher than the source voltage, andthe motor voltage will also rise above itssteady-state value when the motor is dis-connected. It is shown that the capacitor

Y. H. KUFELLOW AIEE

increases the time constant of voltage decayby the factor v/1-Kr2, approximately,where K is the ratio of capacitor to mag-netizing kva, and v is the per unit motorspeed. If Kv2>1, Ei builds up instead ofdecaying. Therefore, a time at least 1/1-Ktimes the normal open-circuit time constantof the motor should be allowed to elapsebefore the line switch is reclosed on thedisconnected motor-capacitor unit, to avoidexcessive motor currents and forces. IfK>l, the additional time required for themotor speed to fall below 1/ NIK shouldelapse.

NDUCTION motors with individualratings of many thousand horsepower

are being used increasingly to drive largepumps, fans, wind tunnels, etc. To pro-vide power factor correction, capacitorsare often connected directly across themotor terminals, and the motor-capacitormay be switched as a unit. It is there-fore of considerable importance to analyzethe transient perfonnance of these largemotor units.

Alger, Ku-Switching Transients in Wound Rotor Indut

How large will the transient currentsbe under full-voltage starting conditions?If different amounts of resistance are

connected across the slip rings, howlarge will the transient voltage across theslip rings be? If the power is inter-rupted, and the circuit is reclosed a fewcycles or seconds later, before the motorvoltage has died away, how large will bethe transients then?To answer these questions, the equiva-

lent circuit of the motor-capacitor unitis set up, and expressions are derived forthe transient currents that flow when a 3-phase voltage is suddenly applied to themotor terminals. Consideration is alsogiven to the transients occurring whenthe motor is disconnected from the lineand allowed to coast with a capacitorconnected across its terminals, and whenthe line is reclosed during this condition.

Analysis

The equivalent circuit' of a polyphasewound rotor induction motor with exter-

Paper 54-23, recommended by the AIEE RotatingMachinery Committee and approved by the AIEECommittee on Technical Operations for presenta-tion at the AIEE Winter General Meeting, NewYork, N. Y., January 18-22, 1954. Manuscriptsubmitted October 20, 1953; made available forprinting November 13, 1953.

P. L. ALGER is with the General Electric Com-pany, Schenectady, N. Y.; Y. H. Ku is with theMoore School of Electrical Engineering, Universityof Pennsylvania, Philadelphia, Pa.

ction Motors 19

Switching Transients in 'WoundRotor Induction Motors

-

FEBRUARY 1954