IEEE Transactions on Power Apparatus and Systems, Vol. PAS-104, No. 6, June 1985

THE METHOD OF SYMMETRICAL COMPONENTSDERIVED BY REFERENCE FRAME THEORY

Paul C. Krause, Fellow, IEEESchool of Electrical Engineering

Purdue UniversityWest Lafayette, Indiana 47906

Abstract - In 1965 it was shown that all known realtransformations commonly used in the analysis of inductionmachines were contained in one general change of variables. Itis now shown that complex transformations, in particular themethod of symmetrical components, are also contained in thissame general transformation. Hence all change of variables(transformations), real or complex, commonly used in acmachine analysis stem from one general transformation. More-over, even without prior knowledge of the existence of sym-metrical components, this method naturally evolves from thegeneral transformation as the logical technique to solve unbal-anced operation of induction machines. This rigorousapproach to the sOblution of unbalanced stator or rotor condi-tions enables one to appreciate the power and flexibility of themethod of symmetrical components in induction machineanalysis. The general transformation (arbitrary referenceframe) is used to establish the positively and negatively rotat-ing balanced sets for multi-frequency phase variables whichneed only be periodic. The method of symmetrical componentsis then derived from the expressions in the arbitrary referenceframe. -An open circuited stator phase and unbalanced rotorresistors are used as two specific examples to illustrate thefacility of this approach in establishing the symmetrical com-ponent voltage equations necessary to analyze these modes ofoperation.

INTRODUCTION

In the late 1920's, R. H. Park [1] revolutionized electricmachine analysis when he formulated a change of variableswhich transformed the stator variables of a synchronousmachine to a reference frame fixed in the rotor. In the late1930's, H. C. Stanley [2] employed a change of variables whichtransformed the rotor variables of an induction machine to astationary reference frame. He then transformed the statorvariables to the same reference frame by Clarke's [31 transfor-mation. G. Kron 141 introduced a change of variables whichtransformed both the stator and rotor variables of a symmetri-cal induction machine to a reference frame rotating in syn-chronism with the rotating magnetic field (synchronouslyrotating reference frame). D. S. Brereton et.al. [51 first used achange of variables which transformed the stator variables ofan induction machine to a reference frame fixed in the rotor.This is essentially Park's transformation applied to inductionmachines.

The change of variables formulated by Park, Stanley,Clarke, Kron and Brereton et.al. are real transformations eachof which appeared to be unique. Consequently, each wasderived and treated separately in literature [61 until in 1965 [71when it was noted that all known real transformations used in

84 SM 661-5 A paper recommended and approvedby the IEEE Rotating Machinery Committee of theIEEE Power Engineering Society for presentationat the IEEE/PES 1984 Summer Meeting, Seattle,Washington, July 15 - 20, 1984. Manuscript sub-mitted February 1, 1984; made available for print-ing May 4, 1984.

induction machine analysis were contained in one generalchange of variables which transforms the stator and rotor vari-ables to a frame of reference which may rotate at any angularvelocity or remain stationary. All known real transformationsmay then be obtained by simply assigning the appropriatespeed of rotation to this so called "arbitrary reference frame."Later [8] it was noted that the stator variables of a synchro-nous machine could also be'referred to the arbitrary referenceframe.

Eleven years before Park's publication, C. L. Fortescue [91published his classic paper on the method of symmetrical co-ordinates (components). This complex transformation foundimmediate application in the analysis of steady-steady opera-tion of unbalanced induction machines which probably pro-vided the impetus for his work. There has been a host of pub-lications applying symmetrical components to numerous unbal-anced modes of operation of induction machines and othercomplex transformations have evolved in an effort to improveor supplement this method; Lyon's [10] work on instantaneoussymmetrical components and Ku's [11] forward and backwardcomponents are perhaps the best known.

When studying the method of symmetrical componentsthere often appears to be a lack of theoretical basis for theapplication of this concept to the analysis of unbalanced condi-tions of induction machines such as an open circuited phase orunbalanced rotor resistors. If one accepts that an unbalancedset of phasor may be represented by two balanced sets (posi-tive and negative sequence) and a set of zero sequence phasorsit is often difficult to understand the limitations of this tech-nique and to adapt this trigonometric picture to the analysis ofa particular unbalanced mode of operation of an inductionmachine.

Although many have noted a relationship between thesymmetrical component variables and Clarke's steady-statevariables, this has been only an observation and both the com-plex symmetrical component transformation and Clarke's realtransformation are treated as separate, unique transformations[6]. Consequently, the complex transformations (Fortescue,Lyon, Ku, etc.) have been considered as separate from the realtransformations (Parks, Stanley, Clarke, Kron, Brereton et.al.,etc.) and thus separate from the general transformation to thearbitrary reference frame. In this paper it is shown that thissame general transformation also contains the symmetricalcomponent transformation and thus it encompasses complextransformations. Not only is the symmetrical componenttransformation contained within this general transformation itis the natural transformation or mathematical tool whichevolves from this general transformation to analyze steady-state unbalanced operation of an induction machine evenwithout prior knowledge of its existence. This derivationanalytically establishes the theory of symmetrical componentsand it provides a straightforward theoretically based approachto stator and rotor unbalances. Hopefully this information willencourage Fortescue's work to be more readily applied andtherefore more fully appreciated.

In this paper the arbitrary reference frame is used to iden-tify the positively and negatively rotating balanced sets whichoccur due to stator or rotor unbalances. The phase variablesmay each be of different waveform, containing different fre-quencies; they need only be periodic, however, the periodicityof each phase may each be different. This development enables

0018-9510/85/0006-1492$01.00© 1985 IEEE

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one to determine the appropriate reference frame to analyzeeach unbalanced condition. Also, the expressions are given forthe instantaneous electromagnetic torques including all pulsat-ing components. Although some simultaneous stator and rotorunbalances may be handled by symmetrical components, statorand rotor unbalanced conditions are treated separately. Anopen circuited stator phase and unbalanced external rotorresistors are used as two specific examples to illustrate the easeof establishing the symmetrical component voltage equationsappropriate to analyze these modes of operation.

UNBALANCED STATOR CONDITIONS

It is convenient to consider stator and rotor unbalancedconditions separately. Although some types of simultaneousstator and rotor unbalances may be handled quite readily usingsymmetrical components there are other types, such as simul-taneous unsymmetrical stator and rotor circuits, which cannot.Moreover, simultaneous unbalance is quite unlikely in practiceand perhaps a rigorous analysis is only of academic interest.

In the analysis of unbalanced stator conditions, it isassumed that either unbalanced applied stator voltages orunsymmetrical (unbalanced) external stator circuit conditionsmay occur, however, the rotor circuits are assumed to be sym-metrical. If external rotor resistors are present they are bal-anced and may be incorporated with the internal rotor phaseresistance; if the machine is doubly fed the rotor source vol-tages are balanced and contain only one frequency. Althoughthe assumption of symmetrical rotor circuits is necessary inorder to apply the method of symmetrical components, thelatter assumptions are made for notation convenience and, ifnecessary the resulting voltage equations may be extended toinclude unbalanced and/or multi-frequency rotor source vol-tages.

Transformations - Arbitrary Reference FrameIn this analysis it is assumed that the stator variables

(voltages, currents and flux linkages) may be defined by aseries of sinusoidal functions with coefficients which may betime varying.

00

fas= X (faska cos Weskt + fask, sin Weskt) (1)k=O00

fb8= (fbska cos Weskt + fbsk sin weskt) (2)k=O00

fcs = (fcska cos Weskt + fcskfk sin weskt) (3)k=O

In the series expansions, wesk are constant. The a and 6 sub-scripts denote, respectively, the coefficients of the cosine andsine terms. Our primary interest is in the analysis of steady-state operation wherein these coefficients are constants. Ingeneral, however, the coefficients may be time varying toaccount for functions such as exponentially decaying sinusoidalfunctions. Nonsinusoidal components are the coefficients of thecosine terms with k = 0 where, by definition, wesO = 0. Simi-larly, k = 1 is generally used to denote the coefficients associ-ated with the fundamental frequency. The equations are writ-ten to accommodate multi-frequency variables which will occurin electric drive systems where the stator is supplied from aconverter.

Transforming the stator variables to the arbitrary refer-ence frame by (A-1) yields the following relationships.

fqs = X fqskA cos (weskt - 0) + fqskB sin (Weskt - 0)k=O

+t [fqskC COS (weskt + 0) + fqskD sin (Weskt + 0) (4)

1493

00fd= E [fdskA) COS (Weskt - 0) + fd.kB S" (Weskt - e)J

+ S skC cOS (Weskt + 0) + fdskD sin (Weskt + 0) (5)k=O

fos= Y faska + fbska + feska) cos Weskt

+ (faskO + fb8k,6 + fesk,6) sin Weskt = X fOskk=O

where1

fqskA = [faska 2 fbskc - 2fcska + -(fbsk,t fcsk,6)I

(6)

(7)

fqskB = [fask 2 fbsk,6 2 fcskf6 (fbska fcska)J

(8)- fdskA

fqskC = -faska 2 fbska jfcska 2- (fbskg fcsk,6)

(9)

fqskD -fask,8 2 fbsk,6 2f5skf 2-(fbska fcska)

(10)

Regardless of the waveforms of the 3-phase variables, (4) and(5) reveal that the qs- and ds-variables form a series of 2-phasebalanced sets in the arbitrary reference frame [12]. The onlyrestriction is that the 3-phase variables must be of the generalform given by (1) - (3). They may be balanced or unbalancedwith constant or time varying coefficients and each may con-tain the same or different frequency components.

It is noted that (4) and (5) are written with the sinusoidalfunctions of (Weskt - 0) separated from the sinusoidal functionsof (Weskt + 0). This grouping helps to emphasize the fact thatthe qs- and ds-variables consist of balanced sets of differenttime sequence. In particular, for any value of k, four, 2-phasebalanced sets are possible; fqskA cOs (weskt - 9) andfdskB sin (Weskt- 0); fqskB sin (Weskt-0) andfdskA COS (Peskt - 0); fqskC COS (Weskt + 0) and

fdskD sin (Weskt + 0); fqskD sin (weskt + 0) andfdskC cos (Weskt + 0); which of course may be combined intotwo, 2-phase balanced sets. In the case of the balanced setswhich have the argument (Weskt - 0) the qs-variables lag theds-variables by 900 for k > 0 and W < wesk, and lead by 900for k > 0 and W > wesk. Similarly, for the balanced sets withthe argument (Weskt + 0) the qs-variables lead the ds-variablesby 900 for k > 0 and w> Wesk, and lag by 900 for k > 0 andW < Wesk. It is possible to relate these balanced sets to thepositive and negative sequence variables. It is assumed thatthe magnetic axes of the induction machine are arranged sothat a balanced 3-phase set of currents of abc-sequence (posi-tive sequence) will produce an air-gap MMF which rotatescounterclockwise at an angular velocity of We. In this case themagnetic axes of the qs- and ds variables are such that fork > 0 the balanced sets of currents with the argument

= -fdskB

=- fdskD

=: -fdskC

1494

(Weskt - 0) will produce an air-gap MMF which rotates counter-clockwise relative to the stator windings and are referred to aspositively rotating sets. It follows that the balanced sets withthe argument (Weekt + 0) are negatively rotating sets for k >0.

With the assumed constraints, the stator and rotor vari-ables will be in the same form when expressed in the arbitraryreference frame. Hence, (4) - (10) may also be used to expressthe rotor variables in the arbitrary reference frame by, simplyreplacing the s subscript with r, except in the case of Week, andadding a prime to denote rotor variables referred to the statorwindings by a turns ratio. It is useful, however, to express therotor phase variables in terms of fqrIu& f' fI and IqrkD- Ifrk)qrkBi qrkC adfqk Ithe equivalent of (4) - (6) are substituted for the rotor variablesinto the inverse of (A-7) -with 0,0) = 0 the rotor phase vari-ables for unbalanced stator conditions become

00

far= £ rkA COS(Week -Wr)t + fqrkB sin (Week - wr)tk=O

fqrkC cos (Wesk r)t + fqrkD sin (Wesk + +Ir)tk (11)

+ (-- fqk + -ffqkA)sin (WeekWr)t00 1 ' 3

+ (- fqrkC +- fqrkB) COS (Week +Wr)t

2 qrkD j t3rkO) sin (wesk + Wr)t + (12)

fcr = -k 2 fqrkA + 2 fqrkB) cos (Wesk - wr)t

+(- fqrkB- j fqrkA) sin (Week - Wr)t

In the ab

The A and B subscripts are associated with the positivelyrotating balanced sets while the C and D subscripts are associ-ated with the negatively rotating balanced sets.

Analysis of Steady-State OperationIt is convenient to use the stationary reference frame to

analyze steady-state unbalanced stator conditions, since itprovides a direct relationship between the reference frame vari-ables and the stator variables. The equations, which are writ-ten in terms of arbitrary reference frame variables, may bewritten in the stationary reference frame by setting G = 0,whereupon the variables contain only one frequency for eachvalue of k. Using uppercase letters to denote steady-state vari-ables we can write the qs- and ds-variables in phasor form, fork > 0, as

_=v s 1 F 8Fqsk = qs+k + 'qe-k

Fdsk = ds+k + Fd,k

(16)

(17)where (16) comes from (4) and (17) from (5). The phasors withthe + k subscripts come from the first summation on theright-hand-side of (4) and (5); the phasors with the -k sub-scripts come from the second summation. It is clear thatidentical equations may be written for the rotor variables inthe stationary reference frame; we need only replace the s sub-script with r and add the primes. It should be pointed out,however, that the above equations are not valid for constant(dc) quantities where k = 0. This of course is not a seriousrestriction since a dc voltage is generally applied to the statoronly for the purpose of dynamic braking. The analysis for thistype of operation may be handled as a special case.

If we consider (4) and (5) along with (7) - (10) we canwrite

v/X Fqqj+k = FqskA -jFqkBV ds+k = F qskAjFd5kB = J A Fq:+k-~~~~~~Vi Fqs_k = Fqskc - jFqskD/Vf2 = FdskC-FdskD = j d t

(18)(19)(20)(21)

1 f I 4/ Again, it is clear that identical expressions may be written for+( 2 fqrkc - fqrkD) cos (Week + Wr)t the rotor variables expressed in the stationary reference frame.

sin2(Week Substituting (18) and (21) into (17), gives (16) and (17) as

+ I ffqrkD + 2 fqrkc) sin (We1k + w1t 1 foq:+k1(22)ove equations f0rk has been included even though it is FFd5kI [jl ji F+ +k no+wa:+o ;oA+~~~~~~dskqsk __:a+

This equation establishes a complex transformation for a 2-phase system. Actually (22) is the inverse of the well known2-phase symmetrical component transformation which is

Fqss+k 1 -jil qsk

SImiFqrky j j j F (23)Similarly, for the rotor variables

zero aue to tue corstraiuts imposea ou tie rotor circuits.

Electromagnetic TorqueThe electromagnetic torque, positive for motor action,

may be expressed as

Te = ( 2 )( 2 )(Wb) (iqsidr idsiq) (14)

where P is the number of poles. If we use (4) and (5) for iand id., respectively, and equivalent equations for iqr and i4rand if an uppercase index (K) is used in the expressions of iqrand i', since a double summation is necessary, the torque maybe expressed in terms of arbitrary reference frame variables as

P XM 00 00 V .1Te = ()( 2(X ,,) Y (iqskAlqrKB - lqskBiqrKA2 2 Wb k=OK=O

qskCiqrkD +lqskDiqrKC) Cos (Wesk- eK)t-

(iqskAiqrKD iqskCiqrKB + iqskBiqrKC-iqskDiqrYjA)COS(Wesk + WesK)t +

(qskAi'qrKA +iqskBiqrKBiqskCiqrKC-iqskDiqrKfD)sin(Wesk-WesK)t +

(lqskAiqrKC1iqskBlqrKD IqskCqqrKA-lqskDlqrKBr (resk +WesK)t (15)

= 2 1 jl J[Fdrkj

The zero-quantities may be written in phasor form from (6) as

vi Fresk = FOeka jFOek# (25)where Foeka and FOek# are defined by (6).

We can write (A-14) in the stationary reference frame bysetting w = 0 and for steady-state conditions by assuming therotor speed is constant and setting p = jiwsk. If we then sub-stitute the inverses of (23) and (24) for the voltages andcurrents we obtain

(24)

rS+j X5S j XMWb Wb

. Wesk rr ' Wesk iJ-XM - +jW XTrWb Sk Wb

0 O0

0

Vqs+k

Vqr+kSk

qs-k- I,Vqr k

2--~Sk

0

0

.Weekr5 +j

Wb

. WeskJ-XM

0

0

iWeek X

I °b

rr . Wesk i

2S +J (<} rr2-sk WOb

Iqs+k

qr +k (26)

I ,qs-k

I-''qr-k

With Sk = eSk Wr (27)Sk =

Wesk

The Os-voltage equation is

WeSkVOsk = (rs + (L, XlS) 1Osk (28)

Wb

For the assumed conditions of balanced, single frequencyapplied rotor voltages, Vqr+k = Vqr+1 and Vq =k 0 if themachine is doubly fed. We have derived the well known posi-tive and negative sequence voltage equations for unbalancedstator conditions. It is clear from symmet,rical componenttheory that Fqs+i =F S+ F5 = Fa F5 = Far+ and-' I

.qs-qs-1

a--r+1

Fqsi = Fqr where F,s+ and Far+ are the positive sequencephasors associated with a-phase and F, and F are thenegative sequence phasors.

We have yet to derive the 3-phase symmetrical com-

ponent transformation. F s FdS and Fo5k are algebraically

related to Fask, Fbsk and FCek by K. From (A-i)

[Fqsk Fdsk FOekf = Ks Iask Fbsk Fcesk

2 2

1 - l ~- Fas

=31 2 2 Fbsk (29)

1 l FCSk2 2 2where T denotes the transpose. From (23) and (25)

qs+k Fqs-k Foskf = S3qd [Fq:k Fd:k FOskf (30)

_1 ljlIOwith S3qd = j O (31)

O 0 2Substituting (29) into (30) yields

frqs+k Fqsk Fos |T = S3qd Ks |askk Fbsk Fcskf

-= S ~F2ask FbSk FcSk (32)

1495where S is the familiar symmetrical component transformationfor a 3-phase system.

r 12

I a a2= I l 1a2 a

3 1 l l(33)

In (33) "a" is complex, denoting a counterclockwise rotation of22r3 radians. Clearly, (32) may also be used for steady-state

rotor quantities expressed in the stationary reference frame.The phasors representing the stator phase quantities may

be expressed from the +k and -k components by the inverse of(32). However, due to stator unbalance the rotor phase vari-ables will contain two frequencies for each value of k as notedin (11) - (13). Since we cannot use a real transformation totransform phasors from the stationary reference frame to areference frame fixed in the rotor, it is necessary to write thephasors from the expressions for the instantaneous rotor vari-ables. Therefore, from (11) - (13), along with the relationships(3) and (5) written for qr- and dr-quantities, we can have

Fark=~Fqr+kJ ~ + ~ W)+ [k (34)

2bkjFr+J +(aqrJ+IF J (35)(Wesk Wr) (Wesk + Wr) (Wesk-Wr)

Fcrk = laFq:+kJ _ +aqr-k+1 + Fork (3)1(Weekr__) L +Wr) Or iek Wr)

In the above equations the subscripts associated with eachbracket indicate the frequency of the instantaneous quantitybeing represented by the phasors.

The steady-state electromagnetic torque may be expressedin terms of the stationary reference frame component phasorsby appropriate substitution into (15). In particular,

Te P3(XM12 Wb k=1 K=1

j* 5*j'Re (iqs+klqr+K - lqs-klqr-K)] COS (Wesk WesK)t

+ Re [i(-Iqs+kiqr-K + Iqs-kIqr+K)J cos (Wesk + WesK)t

+ Re [Iqs+kiqr+K -Iqekiqr-KJ sin (esk WWesK)t

+ Re 11q:+kIqr-K -Iqs-klqrs+KJ sin (Wesk + Wese)tJ (37)

where the asterisk denotes the conjugate.

Open Circuited Stator PhaseThe analysis of an open circuited stator phase is rather

well known and is not given here for the purpose of new infor-mation. It is instructive, however, to observe the straightfor-ward manner in which the voltage equations necessary toanalyze this unbalanced condition can be established in light ofthe material set forth in this section and in [7]. For this pur-pose let us consider the stator circuits of a 3-phase, wye-connected induction machine as shown in Fig. 1. Phase a isopen circuited at a normal current zero. Since the stator cir-cuit is a 3-wire system ios and vos are zero. With = 0,fq =fa and

1496

las

Fig. 1. Open Circuited Stator Phase

V= Vq = P lbq (38)Wb

With iq = 0,

?qs = XMi'r (39)

and vM = P XMifs (40)Wb

Therefore, if, at the instant ia is zero, the voltage ^ Xmi"s is

applied to phase a, the current i. will be forced to remain atzero [71. From Fig. 1

Vbs egb - Vng (41)

vs = e5c - vng (42)In this system vos is zero, therefore, adding the above equationsand solving the result for vng yields

vn = 2 (es + e..) +I vI (43)

Substituting (43) into (41) and (42) givesvbs =2gegb- egc 2 Vas (44)

Cs = 2 gb 2 gc 2 a(45)The above relationships are valid for transient and steady-stateconditions. A computer can be used to solve the nonlineardifferential equations for the complete electromechanicaldynamics. Constant speed electric transients and steady-stateoperation may be analyzed by the method of multiple referenceframes [121.

For convenience of steady-state unbalanced analysis, letus now assume the source voltages contain only one frequency.Substituting the steady-state phasor equivalent of (40), (44)and (45) into (32) gives

VqS+ j 2 W XM Iqr + E (46)qs+ 2 (b4 q

Vs 1We 47Vqs- =Jj2 -XMlIqrs-E(47)q ~2 w'b- 1with E = 1 (Egb - Egc) (48)

The 1 subscript is dropped for convenience. It is clear that(16) is also valid for I'5s therefore we can write

qr~= is., + (49)Iqr = qr+ +qr- (

There is one other relationship which we can use. lqs may alsobe expressed in terms of its components, from (16), as

Iqs =Iqs+ + Iqss (50)Howeverhi is zero and since 0 and .0 are both zero then I s

L as ~~~~~~~~~~~~~~~~~~~qs7which is Iaw is zero. Thus

voltage equations which can be used to analyze the steady-state operation of an induction machine with phase a open cir-cuited and with the source voltages balanced or unbalanced.

- r+j WeyWee- +Xr 2 b q:+

qr + e r We .

results substituted into (15) to obtain the instantaneous torquewhich of course includes constant and pulsating components.It is noted in (52) that the positive and negative sequence vol-tage equations are coupled. With the rotor circuits symmetri-cal, this occurs whenever the stator circuits become unsym-metrical.

UNBALANCED ROTOR CONDITIONSThe application of reference frame theory to the analysis ofunbalanced rotor conditions parallels that for unbalanced sta-tor conditions but with distinctly different results. Here it isassumed that the stator circuits are symmetrical (balanced)and for convenience of notation the stator applied voltages willbe assumed to be balanced and contain only one frequency.

Transformation- Arbitrary Reference FrameAs in the case of unbalanced stator conditions it is

assumed that the rotor variables may be defined by a series ofsinusoidal functions with coefficients which may be time vary-ing.

II00 + infehtfar = (farha cos Werht + farhd sin Werht)

h=O

f 0r Wbrhct erht + frh' sin Werht)h=O00 ' i W t

cr = (fcrha cos Werht + fcrh erhh=O

(53)

(54)

(55)

As in the case of unbalanced stator conditions, nonsinusoidalcomponents of f fir and fc are the coefficients of the cosineterms with h = 0 since by definition, werO = 0. If (53) - (55)are transformed to the arbitrary reference frame by (A-10)wherein d = 0 - Or with Or = Wrt, we obtain

fqr = [ffrhACOs I(Werh + Wd)t-eJ+ fqrhB sin I(Werh + Wr)t OJ}h=O~

+ S fqrhC COS [Werh WUJr)t + I + fqrhD sin Werh - rWt +0i 5B)

fdr = j fdrhAcos IWerh + Wr)t + idrhs sin |erh + W*t -0100 o dh ~r i i

+ l fdrhC COS I(erh Wr)t + ol + fdirhD sin [(erh Wr)t + 41(57)h=0 I

for= farha + fbrha + ferhcs) cos Werht

+ (farhp + %brh#+ fcrh#) sin Werhtl = f6Orh (58)h=0

= qs + (51) The ABCD-quantities are defined by (7) - (10) with theIf we substitute (49) into (46) and (47) and then substitute the appropriate change in subscripts. The balanced sets whichresults into (26) and if we incorporate (51) we can write the appear in the arbitrary reference frame have the arguments of

(Werh + Wr)t - 0 and (Werh - Wr)t + 0. Generally we - wr isdenoted Weri where We corresponds to the frequency of the bal-anced stator source voltages. In this case,.Werl + Wr = We andWeri - Wr = We - 2Wr. Thus, for h = 1, two rotating air-gapMMF's are produced by the two balanced sets. By letting0 = 0 in (56) and (57) we see that one rotates at we and theother at 2W, - wS. In other words, one balanced set producesan air-gap MMF which rotates positively at We relative to astationary observer. The second balanced set produces an air-gap MMF which rotates negatively for 2wr < We and positivelyfor 2Wr > we relative to a stationary observer. In general,unbalanced rotor conditions give rise to air-gap MiMF's whichrotate at wr + Werh and Wr - Werh with respect to the stationaryobserver.

With the constraint of symmetrical stator conditions thestator variables expressed in the arbitrary reference frame willbe of the same form as the rotor variables expressed in thearbitrary reference frame. Hence, (56) - (58) may be used toexpress the stator variables in the arbitrary reference frame byreplacing all r subscripts with s, except in the case of Werh, andremoving the primes. The stator phase variables may now beexpressed in terms of f9shA) fqshBp f sC and fqshD by substitutingthe stator variable equivalents of(56) - (58) into the inverse of(A-1). Thus, for unbalanced rotor conditions the stator phasevariables may be expressed

fas = fqshA cos (Werh + Wr)t + fqshB sin (Werh + Wjth=O

+ fqshC COS (Werh - Wr)t + fqshD sin (Werh - Wr)t + fosh] (59)

Expressions for fb5 and f,, are not given since they may beascertained by comparing (59) with (11) and then writing thecoefficients of fb5 and f,,s from (12) and (13), respectively.Although fosh is included, it is zero for the constraints assumed.

Electromagnetic TorqueThe expression. for the electromagnetic torque may be

derived by a procedure identical to that used in the case of sta-tor unbalance. The result of this derivation may be obtainedby simply changing k to h, K to H in all subscripts of (15) ands to r in the subscripts of Weak and WeaK. If only the fundamen-tal frequency is present in the rotor currents (h = H = 1),then, during unbalanced conditions, the terms in the equationfor torque with the argument (Werh + WerH)t give rise to asteady-state pulsating torque that varies at twice slip fre-quency or 2(we - Wr).

Analysis of Steady-State OperationIt is assumed, that the stator circuits are symmetrical and

the stator applied voltages are balanced and have only one fre-quency. Since the analysis of steady-state operation duringunbalanced rotor conditions is similar in many respects, to theanalysis for unbalanced stator conditions, the relationships willbe given without lengthy discussion. The principle difference isthe reference frame in which the analysis is carried out. It isconvenient, in the case of rotor unbalanced conditions withsymmetrical stator circuits, to conduct the analysis in the rotorreference frame since therein the variables are of one frequencyfor a given h. From (56) and (57) we can write the followingphasor relationships for h > 0 and with 0 = wrt.

Fqrh = Fqr+h + Fqrh (60)

Fdrh = Fdr+h + Fdr-h (61)In the rotor reference frame the frequency of all variablescorresponds to Werh. Relations identical to (60) and (61) maybe written for the stator variables in the rotor reference frame.From (56), (57) and (7) - (10), with the appropriate change in

1497superscripts and subscripts, we can write phasor relationshipsfor the rotor and stator variables identical in form to (18) -(21). It follows that relationships identical to (22) - (24), withthe appropriate change in notation, may be written for therotor and stator variables. We can write (A-14) in the rotorreference frame by setting w = wr and then by settingP = JW)erh we will obtain the voltage equations for steady-stateconditions. Substituting the inverses of (23) and (24), with theappropriate changes in notation, into the steady-state equa-tions yields

r Werh +rW)Xesrh+r Werh+W Mvq:+h r j0r ~~Wb Wb

We rr WeX

Vqhr Wb Sh WJb0 0

qr-h~ 00

0

0

WJerh-Wrr _ _j( .ehfir)XB

WbWe

j-XMWb

Werhwith Sh - Wr

We

0

0

Werh WrWb

Sh Wb

qs+h

qr+h

qs-h

qr-h*

(62)

(63)

We have derived a set of positive and negative sequence vol-tage equations valid for unbalanced rotor conditions. Sincesymmetrical stator circuits and balanced stator source voltagesare assumed, fOsh is zero. The Or-voltage equation is

VOrh = (r + j -eb X;r) iOrh (64)

It is assumed that the stator is supplied from a balanced, sin-gle frequency source, thus Vq+h = Vqr+1 and V F = 0qs+h qs p~q-h

=

-

With Or = , Kr =K. Thus, (32) may be written

Fqr+h Fqrh FOrhf = S3qd Kr tarh Fbrh Fcrhj

arh F brh Fcrh (65)

The phasors representing the rotor phase quantities maybe expressed in terms of the +h and -h components by theinverse of (65). The stator phase quantities may be expressedin terms of the phasors calculated from (62) by relating thephasors to the coefficients of fa (59), fb8 and fc8

F5h = [Fr(+hJ + + [FqrhJ +) FOshJ (66)

Fb,h = aqr+hJ + [aFq_hJWr) + [FOShI (67)

FFr + F r + LI 8F8h [|atq:+hJ +W ) [qh(erh](i hi(h (68e)The subscripts associated with the brackets denote the fre-quency of the instantaneous quantity represented by the pha-sor. The electromagnetic torque given by (37) may be used toexpress the torque in terms of phasors in the rotor referenceframe if the superscript s is replaced with r, k with h, and Kwith H.

Unbalanced Rotor ResistorsThe analysis of an induction motor with unbalanced rotor

resistors has been of interest for many years [13, 141. Thederivation of the sequence voltage equations valid for a generalunbalance of rotor resistors will be used for the purpose ofillustration. From Fig. 2, the rotor phase voltages may bewritten

var = Vpm larRar (69)

' i'R' ~~~~~~~~~~(70)Vbr = vpm lbrRbr70

ver = vpm- irRcr (71)

V'br+icr

Yar -2Ra~r Rr cR ir

+~~~~~Vbr = 1Rr Rbr br(

R rr R'or

Fig. 2. Unbalanced Rotor Resistors

Since the rotor is assumed to be a 3-wire system, io= 0 and

hence vin= 0. Adding (69) (71) and solving for vpm gives

vpm arRa= +rRb|r i+ CrRr) (72)3

Substituting (72) into (69) (71) yields[

I

var -2R.ar Rbr R cr larI .1

|br - ar =RSr Rcr lbr (73)3

[V R+I R I .(vcr Rar Rbr -2Rcr 'cr~

For the analysis of steady-state operation with h 1 it isconvenient to omit h. and express (73) as

br 4rT=c [if i (74)

where if is defined by (73). Substituting (74) into (65) yields

qr+ qr- Or a4r b~r c~rf(

Substituting the inverse of (65) for the rotor currents yields

r$+ V,' VO'~r SfR( +

r(76)

Since Vorand lor are, zero, the above equationt may be written

qr+ [Rabcr RABi0 bRer 1qr+1

rI II I (77)Iqr- ABCr abcr qr-j

with R bcr (R' + Rbr + Rr)

RABCr = 3 (-Rar + 2Rbr + 1 Rcr)2 2

Rbcr = (Rb Rc)

Substituting (77) into (62) yields

ir

Vqs +

0

V r

0

W +W

rr+j(et r )XWb

Wiej XWb

0

0

(78)

(79)

(80)

W +Wcr r.)XWb

rt +R I

rr 'labor . Se t5s Wb

0

+jRj r-RA'BCr bcr

S

The stator is assumed to be balanced, hence Vqs in the above

equation is zero and Vqs+ is VM,. The electrical angular velo-city We corresponds to the frequency of the stator source vol-

tages and Wer is Werl- Also, Wer +Wr we andWer - Wr = We - 2wr. The above equation may be used todetermine the sequence currents for any unbalance of the rotorresistors and (15), with the appropriate change in notation,may then be used to determine the instantaneous torque.

When both stator and rotor circuits are unsymmetrical,such as unbalanced stator source impedances (or an open cir-cuit) and unbalanced rotor resistors, it becomes necessary totransform unbalanced impedances which gives rise to timevarying impedances. This type of unbalance cannot be han-dled using symmetrical components. However, a stator (rotor)source voltage unbalance and a rotor (stator) with unbalancedexternal impedances may be analyzed using symmetrical com-ponents along with the principle of superposition. For exam-

ple, if the rotor resistors are unbalanced and the stator sourcevoltages are also unbalanced but the stator circuits are sym-

metrical, then a negative sequence voltage would appear in therotor reference frame (Vqr). The frequency of this voltage isWe + WL* Hence, the currents can be determnined by first consid-ering Vqr+ and then V r with the appropriate changes toaccount for the frequency of we + Wr, and the two resultsadded.

CONCLUSIONS

Reference frame theory has been used to derive themethod of symmetrical components and to show that it is thelogical change of variables to analyze steady-state unbalancedoperation of an induction machine. It was previously esta-

blished that all real transformations used in machine analysisstem from one general transformation. This paper establishesthat the complex transformation commonly used in machineanalysis also evolves naturally from this same general transfor-mation. This rigorous approach to the analysis of steady-stateunbalanced operation of induction machines clearly illustratesthe convenience of the method of symmetrical componentswhich is too often not fully appreciated.

APPENDIX A

A change of variables which formulates a transformationof the 3-phase variables of stationary circuit elements to thearbitrary reference frame may be expressed [7]

fqdOs = -sfabcs (A-i)

with (fqdos)T = [fqs Ids fosJ (A-2)

(Tabcs)T = fbs ffs

Cos 6 Cos( _ 2r ) Cos (6 + 2-)3 3

K = 2 sin sin (a_ 2ir) sin (0 + 2r )23

1 1 1

2 2 2

0

0

r jW(ierW )r s

b6

WeeXM

0

er wr)XWb

r +Rfbcr (de9rab +j wX,rS Wb

jr

qs +

r

qr +

(A-3)

(A-4)

(81)

1498

1499

O = f w()de + O(0)0

where ( is a dummy variable of integration. It can bthat for the inverse transformation

cos a sin 0 1

(K.)-l = cos ( - 27) sin ( -27) 1

2 si 27r ICo (6 + -) sin (6 +-) j

3 3

(A-5)

ye shown qs

Vos=( qr(A-6) d

Vdr.

In the above equations, f can represent either voltage, current,flux linkage or electric charge. The superscript T denotes thetranspose of a matrix. The s subscript indicates the variables,parameters and transformation are associated with stationarycircuits. The angular displacement, 6, must be continuous;however, the angular velocity, w, associated with the change ofvariables is unspecified. The frame of reference may rotate atany constant or varying angular velocity or it may remain sta-tionary. The connotation of arbitrary stems from the fact thatthe angular velocity of the transformation is unspecified andcan be selected arbitrarily to expedite the solution of the sys-tem equations or to satisfy system constraints.

A change of variables which formulates a transformationof the 3-phase variables of the rotor circuits of a symmetricalinduction machine to the arbitrary reference frame is

fqdOr = Kr faber (A-7)

with (qdOr)" = [f;r fdr frJ (A-8)

(tabcr) = far fbrJ (A9)

cos #Kr = - sin ,p

1

2

cos(#3- 2i)3

sin(,8- 2ir3

12

cos(P6 + 2i r3

sin(,3 + 2ir3

2

(A-10)

03 O6r (A-11)The angular displacement, 6, is defined by (A-5) and Or is

t

6r = f Wr(W)dC + 6r(0) (A-12)where wr is the electrical angular velocity of the rotor. Theinverse is

cos sin d 1

(rI= cos( 2.y) sin(13 - -7) 1 (A-13)+27r 2ircos(/3+-) sin(/3+-) 1

3 3

The r subscript indicates the variables, parameters andtransformation associated with rotating circuits.

If the currents are selected as independent variables thevoltage equations for a 3-phase symmetrical induction machinemay be expressed [11

rW+P YS Wb Y, 0

- -XSS r, + WX-.. 0Wb Wb

o 0 rs+ w X,.Wb

p XM (-)Xr,M 0Wb Wb

_(WWr )XM p XM 0Wb Wb

o 0 0

-b-XM WbXM--wXMWb

00b0

r, + 2YX~r (-Wr )X;,Wb Wb

-( 'O)X,r rrP+X:r-(Wb r'+Wb

0

0

0

0

0

Wr+ Xl

with XsX= Yjs + XM

iqs

id.

'Os

iqr

(A-14)

(A-15)

Xrr = Xir + XM (A-16)In the above equations X15(X;) is the stator (rotor) leakagereactance, rs(rr) is the stator (rotor) phase resistance, XM is themagnetizing reactance and Wb is the base electrical angularvelocity, generally 377 rad/s.

REFERENCES

[1] R. H. Park, "Two-Reaction Theory of SynchronousMachines - Generalized Method of Analysis - Part I,"AIEE Trans., Vol. 48, July 1929, pp. 716-727.

2] H. C. Stanley, "An Analysis of the Induction Motor,"AIEE Trans., Vol. 57 (Supplement), 1938, pp. 751-755.

[31 E. Clarke, "Circuit Analysis of A-C Power Systems,"Vols. I and II, John Wiley and Sons, Inc., New York,N.Y., 1943 and 1950.

[4] G. Kron, "Equivalent Circuits of Electric Machinery,"John Wiley and Sons, Inc., New York, N.Y., 1951.

[5] D. S. Brereton, D. G. Lewis and C. G. Young,"Representation of Induction Motor Loads During PowerSystem Stability Stud-ies," AIEE Trans., Vol. 76, August1957, pp. 451-461.

[6] D. C. White and H. H. Woodson, "ElectromechanicalEnergy Conversion," John Wiley and Sons, Inc., NewYork, N.Y., 1959.

[7] P. C. Krause and C. H. Thomas, "Simulation of Sym-metrical Induction Machinery," IEEE Trans. PowerApparatus and Systems, Vol. 84, November 1965, pp.1038-1053.

[8] P. C. Krause, F. Nozari, T. L. Skvarenina and D. W.Olive, "The Theory of Neglecting Stator Transients,"IEEE Trans. Power Apparatus and Systems, Vol. 98,January/February 1979, pp. 141-148.

[9] C. L. Fortescue, "Method of Symmetrical Co-ordinatesApplied to the Solution of Polyphase Networks," AIEETrans., Vol. 37, 1918, pp. 1027-1115.

101 W. V. Lyon, "Transient Analysis of Alternating-CurrentMachinery," Technology Press of MIT and John Wileyand Sons, Inc., New York, N.Y., 1954.

[I1] Y. H. Ku, "Electric Energy Conversion," Ronald PressCo., New York, N.Y., 1959.

[121 P. C. Krause, "Method of Multiple Reference FramesApplies to the Analysis of Symmetrical InductionMachinery," IEEE Trans. Power Apparatus and Systems,Vol. 87, January 1968, pp. 218-227.

[13] H. L. Garbarino and E. T. B. Gross, "The GoergesPhenomenon - Induction Motors with Unbalanced RotorImpedances," AIEE Trans., Vol. 69, 1950, pp. 1569-1575.

[14] 0. I. Elgerd, "Nonuniform Torque in Induction MotorsCaused by Unbalanced Rotor Impedances," AIEE Trans.,Vo. 73, December 1954, pp. 1481-1486.

(