Revolving-field analysis of asymmetricthree-phase machines and its extension to

single- and two-phase machines

Bhag Singh Guru

Indexing terms: Equivalent circuits, Machine theory, Magnetic flux, Induction motors, Stators

Abstract: The revolving-field theory as applied to single-phase induction motors is extended to develop aconcise yet comprehensive theory for asymmetric three-phase induction motors. Each phase is represented byits equivalent circuit, not only to allow for uneven direct transformer interactions, due to asymmetriclocations of phase windings, but also to account for the different number of turns, wire size, winding factoretc. each phase may have. The accuracy of the theory was confirmed by actual measurements on symmetricand asymmetric three-phase induction motors. Computed and test data on some motors are included forillustration. It is also shown that the general three-phase development can be easily applied to determine thebehaviour of two- and single-phase induction machines, a useful feature for unified computer-aided design, byeliminating one and two phase windings, respectively. Not only the procedural details for determining theperformance of single- and two-phase induction motors are given, but comparisons of numerical and testresults on some output entities are included as well. Though the comparisons among theoretical and testresults are made on fractional horsepower motors, the types being built at Universal Electric, the author iscertain that the development presented here is equally applicable to all sizes of induction motors.

1 Introduction

A symmetric polyphase induction motor, operating on abalanced polyphase power supply, is defined as aninduction motor whose windings are evenly displaced inspace and excited by the currents having the same dis-placement in time phase as the windings have in spacephase. Consequently, the internal interactions of such aninduction motor are simple, and can be easily characterisedby one primary circuit and a corresponding secondarycircuit based on the revolving-field theory. On the otherhand, the simple crossfield analysis of such motors involvesthree times as many circuits as there are phases. Hence thepolyphase induction motors are universally analysed on thebasis of the revolving-field theory. However, quite often thestator core punchings designed for a certain pole configur-ation are also used for another pole configuration, and inthat case, if the number of stator slots per pole pair is nota whole number, the phase windings would have to bearranged at resultant angles other than the conventionalangle for such motors. This results in a spatially unbalancedmotor even though the windings are identical. The actualperformance of an asymmetric motor is bound to differfrom the conventional motor, and therefore necessitates adifferent theoretical approach. As far as the author isaware, no attempt has ever been made in this direction.Therefore, the primary aim of this paper is to develop asystematic method for the analysis of asymmetric three-phase induction motors. In addition, the effects ofunbalanced terminal voltages are also included to make thedevelopment more general. However, note that the wellknown method of symmetrical components1 may be usedto analyse a symmetric (equal windings) three-phaseinduction motor running on an unbalanced power supply.This method has also been extended for the analysis ofasymmetric two-phase induction machines2 by making use

Paper T308 P, first received 9th May and in revised form 24thNovember 1978Dr. Guru is with the Universal Electric Company, 300 East MainStreet, Owosso, Michigan 48867, USA

ELECTRIC POWER APPLICATIONS, FEBRUARY 1979, Vol. 2, No. 1

of suitably chosen components of current and voltage.Further adaptation of this technique to three-phaseinduction machines with unequal windings and arbitraryspatial displacements in addition to unbalanced linevoltages may require a large number of voltage and currenttransformations. At this time, such a development, if thereis any, is not known to the author. Another analyticalmethod would be the direct extension of the revolving-field approach as applied for capacitor motors withwindings not in quadrature.3 Even though this methodrequires only three unknown quantities, the torqueequation by itself is quite involved.

All the foregoing methods, in the author's opinion,satisfy only one particular requirement, that is, if thedevelopment is carried out for a three-phase motor withasymmetric windings, it would require comprehensivechanges to make it useful for an asymmetric two-phasemotor and vice versa. Stated differently, separate computerprograms are needed to analyse different-phase motors.This, in turn, increases the cost of maintaining theseprograms as 'permanent files' on a time-sharing machine. Tocircumvent the situation, we can seek refuge in Kron'sgeneralised theory.4'5 This theory, as Kron states in hisbook,s has to be extended further to encompass the entirerange of asymmetric machines. Moreover, an extension ofKron's theory requires the in depth understanding oftensors, symmetrical components, three-phase/two-phaseand inverse transformations. An average user, however, maynot be able to satisfy some or all of these demands.

In literature, almost, all the time, a symmetric three-phase induction motor is represented by an analyticallysimple yet 'tricky' equivalent circuit. An equivalent circuit,in the author's opinion, should be based upon the physicalpicture as opposed to a 'trick' circuit. In a three-phasemotor, the flux produced by one phase induces voltages bydirect transformer action in the remaining phases and viceversa. An exact representation of this machine, wouldrequire an equivalent circuit for each of its phases and anaccount for the mutual interactions of the fluxes ratherthan a single yet tricky circuit. Although the former may

37

0140-1327/79/010037 + 08 $01-50/0

increase the number of equations needed for an analyticalsolution, it certainly helps the engineer's understanding andthereby enables him to attack more complicated problemsthat he would otherwise hesitate to approach analytically.This paper therefore represents each winding by its equival-ent circuit and then accounts for the direct transformerflux linkages among the windings. Since each phase is rep-resented by its equivalent circuit, the asymmetric three-phase induction motor can be analysed by making use of asimple method published recently.6 By doing so, the finalset of equations are not only useful for analysing symmetricor asymmetric three-phase motors, but also enable theengineer to use them effectively for single- and two-phaseinduction machines, as will be shown later. Therefore, thepresent analytical approach does away with the necessity ofhaving separate computer-aided design programs for single-two- and three-phase machines. This in turn considerablyreduces the cost of maintaining the 'permanent files' forregular use on a time-sharing computing machine.

The present method, however, involves the concept thatany given winding can be made equivalent to two mutuallyorthogonal windings. Although the author employed suchterms as 'd-axis' and 'q-axis' to explain the concept for lackof better names, it is different from the so-called ld- g-axis'technique used by other authors. The latter applies to thetransformation of the voltages and currents while theformer to the windings. Utilising this concept, each windingof a polyphase motor may be looked upon as two separatewindings mutually perpendicular. However, the twowindings have to be assumed to be connected in seriesto have the same effect as the original winding. Thisconcept is explained further in Section 2. In Section 3,some of the basic assumptions generally accepted in motortheory are stated. Section 4 explains the treatment of coreloss. The general theoretical development for three-phaseinduction motors is presented in Section 5. The systematicand efficient implementation of this general developmentfor single- two- and three-phase induction motors isexplained in Section 6. The numerical examples are alsoincluded. Finally, the highlights of this development aresummarised in Section 7.

2 Conceptual background

Since the concept of treating a winding as a set of twomutually perpendicular windings has already beenpublished6 most of the details are omitted here. Themethod of decomposing a winding into two orthogonalwindings is needed here to account for uneven direct trans-former couplings among mutual fluxes of all the windings,since actual windings are not symmetrically arranged. Asummary of the concept is given below.

If two mutually perpendicular axes, direct axis or d-axisfor short, and quadrature axis or g-axis, are oriented in sucha fashion that the axis of one of the three phase-windingscoincides with the d-axis, then the positions of the otherphase windings can be expressed in terms of electricaldegrees circumferentially offset from the reference-phasewinding (r.p.w.). Thus the flux produced by the r.p.w. hasonly a d-axis component while the flux produced by anyother phase-winding can now be resolved into its com-ponents oriented along d- and ^-axes, respectively. Sincethe flux produced by a winding is a measure of its totaleffective conductors, it can likewise be said that thewinding can be resolved into its components whose axescoincide with the d- and q-axes. In other words, a winding

38

displaced at an angle from the r.p.w. can effectively berepresented by two windings connected in series, one inphase with r.p.w., the other in quadrature. The straight-forward extension of this principle states that a given set ofwindings arranged arbitrarily in space can be made fullyequivalent to another arbitrary set of paired quadraturewindings. It is, however, obvious that if two or morewindings are in space phase, that is when they are arrangedso that their magnetic axes coincide, a direct transformertype of coupling exists between their fluxes. In the presentcase, the direct-transformer linkages exist among the fluxesof r.p.w. and tf-axis components of the remaining offsetphase windings on one hand, while such coupling existsamong the fluxes of q-axis components of the offsetwindings, on the other. As usual, the voltages are induced inthe main and crossfield axes of the rotor due to trans-former action and rotation, respectively. Once this conceptis clearly understood, the development of the theoryquantitatively is basically simple and straightforward.

3 Assumptions

The following assumptions are imbedded into the theoreti-cal development that follows in subsequent Sections:

(a) All the windings follow concentric distribution andtherefore, in comparison to the fundamental flux, theeffects of other space-harmonic fluxes are negligible.

(b) Friction, windage, and surface losses are all con-sidered to vary as the square of the speed. These losses areregarded as an extra load on the shaft and are thereforesubtracted from the output.

(c) The magnetic-circuit parameters of the offset phasewindings are all expressed in terms of the magnetic circuitreactances of the reference-phase winding (r.p.w.).

(d) For direct transformer action, the local mutualleakage and magnetising fluxes couple perfectly for thetransformed windings both on the d- and the ^r-axes.

4 Treatment of core loss

Often, while calculating the performance of an inductionmotor, very approximate methods3 are used to account forthe core loss. Since these methods do not interfere withperformance equations, they are often acceptable. To bemore exact, however, the core loss may be handled byintroducing an equivalent resistance, either between rx andxx as suggested by Trickey,9 or in shunt with the magnet-ising branch.4 For fractional horsepower motors the authorfound no appreciable difference between these two equiv-alent representations. In this development, however, theformer method is followed. Even though the equivalent-resistance concept that accounts for the core loss adds tothe complexity of the performance equations, it doesnarrow down the differences between the theoretical andthe test results. Also, for large induction motors, the formerassumption would lead to better correlation between thetheory and tests than the latter.

5 Theory

In a symmetric three-phase induction motor, the number ofturns, wire sizes etc. are the same for each of the threephases and follow identical distribution. Moreover, any twophases are offset circumferentially by 120 electrical degreesin space. For the present development, however, let usconsider that the effective number of conductors, wiresizes etc. are different for each of the phases in addition to

ELECTRIC POWER APPLICATIONS, FEBRUAR Y 1979, Vol. 2, No. 1

the arbitrary displacement of the 2nd- and 3rd-phases by 62

and d3 electrical degrees in space with respect to thelst-phase, as shown in Fig. 1. A and B may be defined asthe ratios of effective number of conductors in the 2nd-and 3rd-phase to the effective number of conductors in thelst-phase, respectively. If kwl and Nlt Kw2 and N2, andkw3 and N3 are the winding factors and the number ofturns for 1st- 2nd- and 3rd-phase, respectively, then A =(kw2N2)l(kwlNx) and B = (kw3N3)/(KwlN1). It is how-ever, obvious that for a symmetric motor, kwx = kw2 = kw3,yV, =N2 = N3, d2 = 120° and 03 =240° . In other words,A=B=l.

To simplify the theoretical development to some extentbut still maintain the generality of the problem, let usconsider that the axis of the lst-phase winding coincideswith the d-axis. This, therefore, is the reference-phasewinding. The 2nd- and 3rd-phase windings can now bedecomposed into their d- and q-axis components asshown in Fig. 2. The ratios of effective number of con-

Fig. 1 Asymmetric three-phase induction motor with 2nd- andSrd-phase windings displaced at d2 and d3 electrical degrees fromthe lst-phase winding

d -axis

ductors for d- and q-ax\s components of the said phasewindings to the effective number of conductors in thelst-phase winding, respectively, are:

d-axis component of 2nd-phase: Ad = Acosd2 (1)

q-axis component of 2nd-phase: Aq = A sin 62 (2)

d-axis component of 3rd -phase: Bd = B cos 03 (3)

g-axis component of 3rd phase: BQ = B sin 03 (4)

For an unbalanced three-phase power supply, the magni-tude of the voltage may be different for each of the phases.Furthermore, each phase may not be 120° apart in time.Assuming the lst-phase voltage as a reference, the 2nd- and3rd-phase voltages may then be given as:

2nd-phase voltage: V2 = C2Vt exp(jt2) (5)

3rd-phase voltage: V3 = C 3 Kjexp( / f 3 ) (6)

where C2, C3 are the constants of proportionality, and t2

and t3 are the phase-shifts for the 2nd- and 3rd-phases,respectively, with reference to the lst-phase supply. For abalanced three-phase supply, it is obvious that C2 = C3 —1, t2 — 120° and t3 = 240° in time. The asymmetric three-phase induction motor of Fig. 2 can now be represented byan equivalent circuit. Such a circuit is shown in Fig. 3. Theequivalent circuit parameters for the 2nd- and 3rd-phase,respectively, are defined in terms of the lst-phase para-meters and the effective winding ratios. The resistances /yel,rfe2 and Ayc3, as shown in Fig. 3, are introduced to accountfor the core loss in each of the three phases. This networkrepresentation involves six unknowns. However, by thestraightforward application of The"venin's theorem,8 theequivalent circuit of Fig. 3 may be transformed intoanother modified equivalent circuit as shown in Fig. 4.Note the terms rm,, rm2 and rm3, and Vml, Vm2 and Vm3.These are the modified resistances and the voltages for the3-phase windings and are given as:

rmi — rpirfei/(rpi

vmi =

(7)

(8)

q -axis

Fig. 2 Equivalent representation of an asymmetric three-phaseinduction motor of Fig. 1

The 2nd- and 3rd-phase windings are shown decomposed into d- and(/-axis components

ELECTRIC POWER APPLICATIONS, FEBRUARY 1979, Vol. 2, No. 1

where the subscript / = 1, 2, or 3 corresponds to 1st- 2nd-or 3rd-phase values of both the resistances and supplyvoltages, respectively. rpl, rp2, and rp3 are the windingresistances for 1st- 2nd- and 3rd-phase, respectively.

As explained earlier, and is now obvious from Fig. 2, inall there are three windings arranged along the d-axis andtwo windings arranged along the q-axis. Since the currents,as shown in Fig. 4, in all the windings oriented either alongthe d-axis or the q-axis flow in a sense in the same direction,the mutual interactions among them are considered to bepositive. Since the d- and 4-axes are in quadrature, the netflux produced by the windings oriented along one axis, saythe d-axis, does not show any direct transformer linkagewith the flux produced by the windings concentric with theother axis, ^-axis, and vice versa.

Thus, the voltage induced in the lst-phase winding by itsown leakage flux and also by direct transformer coupling ofthe leakage fluxes of the d-axis components of the 2nd- and3rd-phase windings is

+AdJ2+BdI3)xl (9)

Similarly, the voltages induced in the d-axis components ofthe 2nd- and 3rd-phase windings by the leakage fluxes are

39

= /(A + AdI2 + BdI3)Adxx

= /(A + A dI2 + BdI3)Bdxx

(10)

00In a similar manner, the voltages induced in the g-axiscomponents of the 2nd- and 3rd-phase windings by theirleakage fluxes are

£», = j(AqI2+BqI3)AqXl (12)

#3<, = j(AqI2+BqI3)BqXl (13)

Voltages induced in the lst-phase by its forward andbackward revolving fields as well as by direct transformerlinkage of the two revolving fields set by the d-axis com-ponents of the 2nd- and 3rd-phase windings are

forward: Efld = (/, +AdI2 + BdI3)Zf (14)

backward: Ebld = (/, + AdI2 + BdI3)Zb (15)

Likewise, the voltages induced in the d-axis components ofthe 2nd- and 3rd-phase windings by the field components,which are set by them as well as by direct transformeraction of the revolving fields set by the lst-phase winding,are:

(a) For the d-axis component of the 2nd-phase:

forward: Ef2d = (h + AdI2 + BdI3)AdZf (16)

backward: Eb2d = (/, + AdI2 + BdI3)AdZb (17)

(b) For the <i-axis component of the 3rd-phase:

forward: End = (/, + AdI2 + BdI3)BdZf (18)

backward: Eb3d = (/t + AdI2 + BdI3)BdZb (19)

Similarly, the voltages induced in the g-axis components ofthe 2nd- and 3rd-phase windings by the forward- andbackward-revolving fields set by them, respectively, are:

(a) For the q-axis component of the 2nd-phase:

forward: Ef2q = (AqI2 + BqI3)AQZf (20)

backward: Eb2q = (AqI2 + BqI3)AqZb (21)

(b) For the g-axis component of the 3rd-phase:

forward: Ef3q = (AqI2 + BqI3)BqZf (22)

backward: Eb3q = (Aqf2 + BqI3)BqZb (23)

We are now in a position to express mathematically thespeed voltages induced in the d-axis components of thewindings by the forward- and backward-revolving fields of

10

Fig. 3 Equivalent circuit of an asymmetric three-phase inductionmotor based upon revolving field theory

40

a.oa

"fid

'm2

rm3

-3d

-f3d

-b3d

Fig. 4 Thevenin 's equivalent of Fig. 3

ELECTRIC POWER APPLICATIONS, FEBRUAR Y 1979, Vol. 2, No. 1

the g-axis components of the said windings and vice versa.After some simplifications, the expressions for thesevoltages are:

A', = -j(AqI2+BqI3)Zf

E* = }(Aqf2+BqI3)Zb

E3 = ~KAqI2 + BqI3)AdZ,

lU = j(AqI2 + BqI3)AdZb

Es = -j(Aqf2 + Bqh)BdZf

£"7 = / ( / , + AdI2 + Bdf3)AqZf

^s = " / (A +^d/2 + BdI3)AqZb

£9 = /(/i + i4d/2 + Bdh)BqZf

ff,o = ~J(h + Adh + Bdh)BqZb

(24)

(25)

(26)

(27)

(28)

(29)

(30)

(31)

(32)

(33)

2q

Simultaneous equations for the voltages in the three circuitsof Fig. 4 can now be written by summing up the voltages ineach of the three stator windings, obtaining

1/ — f. r + / • ' . . -I- F , • + F.. . 4- F\ 4- P- f^&\

' m l •'I'ml ' Lj \d ' '-'fid ' ^b ld ' lj\ 1-'2 W v

Vm2 ~ hrm2 + ^2d + Ef2d + ^b2d + E3 + E4

E-,n + Ef2q + Eb2q + En + E8 (35)i + ^63d + Es + E6

l3q +E9+ E10 (36)

Eqn. 34 is for the referenced-phase winding and can be usedto analyse a single-phase induction motor running on themain winding only, by equating the 2nd- and 3rd-phasecurrents involved implicitly to zero. Adding eqn. 35, we canobtain the performance of two-phase symmetric/asymmetricmotors with balanced/unbalanced terminal voltages at anyspeed of rotation. Remember that a permanent-split capaci-tor motor is only a special case of an unbalanced two-phase induction motor. Further, these equations can also beused to determine the locked-rotor torque and the startingperformance of a single-phase induction motor whoseauxiliary-phase is disconnected at a predetermined speed. Itis, however, obvious that all the above equations are neededto predict the performance of an asymmetric or a sym-metric three-phase induction motor.

By making direct substitutions for the induced voltages,eqns. 9—33, and after some rearranging, eqns. 34—36 canbe rewritten in the general form as

V = I\Z\x +I2Z12 + I3Z\3 (37)

V 2 = hZ2\ +/)Z22 +/3Z23 (38)

Z33 (39)32

where the Z-coefficients are

= r +Zs (40a)

(40b)

Zn - AdZs -jAqZd

Z2x = AdZs+jAqZd

Z72 = rm2 +A2ZS

ELECTRIC POWER APPLICATIONS, FEBRUARY 1979, Vol. 2, No. 1

Zl3

Z23

Z31

Z32

z33

with

zd

Sab

= BdZs

= sabz— BdZs

= SabZ

— rm3

= AdBc

~JBQZd \

s + JCabZd

+ /BqZd

s-jcabzd

\-B2Zs

zb zs

+ AqBq

— Zf ' Zb T jx 1

Cab = AqBd-AdB

(40c)

(41)

Once again, eqn. 40a gives the Z-coefficients for singlephase motors, whereas eqns. 40a and 40b are for two-phasemotors. Needless to say, all the above equations for theZ-parameters are needed for symmetric or asymmetricthree-phase induction machines. Eqns. 37—39 may berewritten in a concise matrix form as

[Z] M = [V] (42)

where [Z] is a complex impedance matrix of dimensions3 x 3 whose elements are given in eqn. 40 in terms ofknown motor-circuit constants. [/] and [V] are thecolumn vectors for the circuit currents and the Thdveninequivalent of the impressed voltages, respectively. Thecircuit currents can then be determined from the followingequation:

[/] = [ Z p t K ] (43)

where [Z]"1 dentoes the inverse of the [Z] matrix.The actual phase currents can now be determined from

the following relation:8

Ipi = (Vt + Iirfei)l(rpi + rfei)

for / = 1,2, and 3, respectively.The total power input in phase / is given by

(44)

(45)

The forward and backward torques can be calculatedindependently for each one of the three phases. This can bedone by calculating the forward and backward voltagesinduced in each of the windings from eqns. 14—33, andmultiplying each voltage by the conjugate of the actualcurrent flowing in the winding of concern and using onlythe real part of the product. Following this procedure, andafter some simplifications, the torques developed insynchronous watts for each of the three phases are:

(a) For the lst-phase:

by forward field: 7>, = Re [IfI?2f] (46)

by backward field: Tbl = Re [IbI*Zb] (47)

(b) For the 2nd-phase:

by forward field: 7>2 = Re{/fI2*Zf (Ad +fAq)} (48)

by backward field: Tb2 = Re {IbI2*Zb (Ad-jAq)} (49)

(c) For the 3rd-phase:

by forward field: 7}3 = Kt{IfI*Zf (Bd + jBq)} (50)

by backward field: Tb3 = Re{IbI3*Zb(Bd-jBq)} (51)

where the symbol * indicates the conjugate of the com-plex quantity involved and

41

If = /, + I2 (Ad -jAq) + I3(Bd -jBq) (52)

Ib = h+h(Ad+jAQ)+I3(Bd+jBq) (53)

The overall forward and backward torques developed, insynchronous watts, including friction and windage lossesare: n

gross forward torque developed = 7} = £ 7},- (54)1 = 1

n

gross backward torque developed = Tb = £ r w (55)1=1

where « specifies the number of phases. For a three-phasemotor, n = 3.

The actual torque developed by the motor, in synchron-ous watts, is

T = Tf-Th (56)

The secondary or the gross output power can be calculatedby multiplying eqn. 56 by the per unit speed of the motor.Subtract the friction and windage losses to obtain theuseful power output.

6 Procedural details

6.1 Three-phase motors

The suggested procedure to predict the performance ofthree-phase induction motors is as follows:

(a) From the known number of turns and the windingfactor for each of the three phases, determine A and B.

(b) From the known locations of the 2nd- and 3rd-phasewindings with respect to the lst-phase winding, calculatethe d- and 4-axis components of these windings, using eqns.1—4. However, for a symmetric motor, set 62 = 120° and03 =240°.

(c) Assuming the lst-phase voltage as a reference, definethe 2nd- and 3rd-phase voltages in terms of their magni-tudes and time-phase displacements. In other words, defineC2 and C3, t2 and t3, respectively. For a balanced three-phase supply, C2 = C3 = 1, and t2 = 120° and t3 = 240°.

(d) Calculate Thdvenin's equivalent of resistances andvoltages by making use of eqns. 7 and 8, respectively.

(e) From the known constants of the motor, computethe Z-parameters as given in eqns. 40a, 40b and 40c.

(/) Use a computer subroutine to invert the compleximpedance [Z] matrix.'

(g) By making use of eqn. 43, calculate the complexvalues of the circuit currents for the modified equivalentcircuit of Fig. 4.

(h) The actual phase currents can now be determinedusing the relation given in eqn. 44.

(/) Calculate the input power and the gross torquesdeveloped by the forward and backward fields from eqns.45—55. The actual torque developed can now be computedfrom eqn. 56.

(/) Determine the forward and backward I2R losses dueto the forward and backward fields by multiplying theforward and backward torques developed by s and 2s,respectively.

(k) Obtain net power developed, total losses, horsepoweretc. by employing standard relations.

Since it is not possible to cite all practical possibilities,theoretical and test data on three different motors are

t A subroutine to solve this matrix is available from the author onrequest

42

studied in the subsequent paragraphs.In the first example, a typical 230V, 60 Hz, 0-4 h.p.,

4-pole symmetric three-phase induction motor rated at1650r/min is considered. The windings in each phaseconsisted of two concentric coils, the inner coil was woundover 6 teeth and had 57 turns, the outer coil was woundover 8 teeth with 76 turns of 24 gauge wire. The per phasevalues of the motor constants obtained from design con-siderations are reproduced below.

r, + /x , = 12-401 +/5-77412

rfe = 822-8 £2

r2 +jx2 = 5-319+/2-653 ft

xm = 88-601 ft

F&W = 8-70W

The laboratory measurements were taken by connecting themotor to a balanced three-phase supply. The test data andtheoretical predictions on various output quantities aregiven in columns 1 and 2, Table 1. Note that these theoreti-cal results in this case can also be obtained by making useof well known equations for symmetric three-phaseinduction motors based on a single equivalent circuit. It is,however, necessary to increase the values of the motorconstants given above by 50%, except for the stator-winding resistances.

A provision was made to add extra windings in serieswith each phase of the above motor to test the motor withstator windings having an unequal number of turns. Thetested and computed results, as given in columns 3 and 4 ofTable 1, are for a motor with 69 and 92, 66 and 88, and 57and 76 turns in the inner and the outer coils, for the lst-2nd- and 3rd- phase windings, respectively.

The last example involves a three-phase induction motorwith unequal turns in each phase and with arbitrary spacingof the phases. The number of turns and coils were same asin the previous example, whereas the 2nd- and 3rd- phasewere displaced at 100 and 200 electrical degress, respect-ively, from the lst-phase. The measured values and theor-etical predictions are given in columns 5 and 6, Table 1.

In all the foregoing cases, as is obvious from Table 1, testdata are in close agreement with theoretical predictions.This, therefore, confirms the accuracy of the presentmethod as applied to both symmetric and asymmetricthree-phase induction machines.

From eqn. 53, it can readily be shown that for a sym-metric three-phase supply, the backward field current Ib isidentically zero. Consequently, there exists no backward-revolving field in the motor, as expected at the outset.

6.2 Two-phase motors (permanent-split capacitor motors)

In the true sense, there are not that many two-phaseinduction motors being built toda>. The reason, of course,is that the two-phase power source needed for the normaloperation of such motors is virtually an extinct species.However, a permanent-split capacitor motor may be treatedas a two-phase motor with unequal windings. The pro-cedure outlined below may be used to determine itsperformance.

(i) Calculate A from the known turns and winding factorfor each of the two windings. Set B equal to zero.

(ii) Use eqns. 1 and 2 to determine the d- and q-

ELECTRICPOWER APPLICATIONS, FEBRUARY 1979, Vol. 2, No. 1

Table 1: Theoretical and test results on three-phase, permanent-split capacitor, and single-phase induction motors

Column

/ p . . A

/PJ,A' P 3 . A

Pint.™Pirn.™Pim.WPout. WT, kgmEfficiency

Three-phase motors

1

Test

1-451-471-45

150 015201515298-9

2

Theory

1-4761-4761-476

156-3156-3156-33063

0-1764 0-18080-66 0-65

3

Test

0-930-672-10

83-780-3

240-2255-5

0-15080-63

4

Theory

0-9020-6782085

80-4477-24

243-1252-5

0-1490-63

5

Test

3-211-123 54

172-584-7

380-1237-9

0-14040-37

6

Theory

322110533-658

177-582-85

384-1235-5

01390-37

Two-phase motors (p.

7

Test

0-730-38—

160-7820

_158-6

009360-65

8

Theory

0-7340-383—

160-781-57—

157-80 09310-65

s.c.)

9

Test

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151-760-3

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008210-66

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Theory

0-7610-258—

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Test

0-97——

181-2-—

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Theory

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112-4) 00664

0-63

components of the 2nd-phase (auxiliary phase for a perma-nent-split capacitor motor).

(iii) Define the magnitude and the phase angle for the2nd-phase voltage reference to 1st-phase voltage. In otherwords, define C2 and t2. For a two-phase motor withbalanced two-phase terminal voltages, C2 = 1 and t2 = 90°(time). However, for a permanent-split capacitor motor(p.s.c.), set C2 = 1 and t2 = 0°.

(iv) Calculate from eqns. 7 and 8, the The'venin's equiv-alent of resistances and voltages. Note that for a p.s.c, thecapacitor impedance should be added to the auxiliary-winding's resistance for all theoretical purposes except forobtaining the primary loss for that winding.

(v) Compute the Z-parameters from eqns. 40a and 40b.Note that eqn. 40c is not needed in this case because thereare only two-phase windings. Therefore, set/3 = 0.

(vi) Steps (vi)-(xi) are essentially the same as those givenin Section 6.2. Compute the power input, the forward andthe backward torques with n = 1.

Two permanent-split capacitor motors were actually builtto verify the accuracy of the present method as extended totwo-phase induction motors. One motor was built with anauxiliary winding displaced in quadrature with respect tothe main winding, while the other with auxiliary arrangedat 60 electrical degrees from the main. Since the samewindings were used for both motors, the following motorconstants are applicable to either of them:

rpx + jxx = 32-255+/ 35-49 ft

rn = 6884-05 ft

r 2+/x2 = 36-453+/21-238 ft

xm = 735-544 ft

A = 1-32107

rp2 = 135-194ft

capacitance = 3 05 mF

These 230 V, 60 Hz, 4-pole permanent-split capacitormotors were tested at 1650r/min. The tested and com-puted results on quadrature and nonquadrature motors aregiven in Table 1, columns 7 and 8, and 9 and 10, respect-ively. It is obvious from these results that the correlationbetween the tested and computed values is excellent andthereby approves the applicability of the general method topermanent-split capacitor motors, and in turn to other suchmotors that use two windings for their operation at therated speed.

6.3 Single-phase motors

The starting performance of single-phase motors can beobtained by using the method explained in Section 6.2,since starting the motor employs both the windings. How-ever, when the motor is running at its rated speed on themain winding only, its performance can be obtained bymodifying the procedure of the previous subsection asfollows:

(a) Determine The'venin's equivalent of resistance andthe voltage from eqns. 7 and 8, respectively, for the mainwinding.

(b) Set I2 = h = 0 and A = B = 0.(c) Compute Zn from eqn. 40a.(d) Calculate the current I\ as follows:

A = Vml/Zn

Other quantities of interest can be obtained by following asimilar procedure given in the previous subsections, ofcourse, overlooking nonessential equations.

One of the p.s.c. motors was selected for the testingpurposes as a single-phase induction motor. At its ratedspeed, 1650r/min, the auxiliary winding was cut off. Thetest data and the theoretical predictions on various outputquantities are given in the last two columns of Table 1. Thefeasibility of using the present development to analysesingle-phase induction motors running on the main windingonly is indicated by the close relationship between themeasured and theoretically predicted values.

Therefore, it can be concluded that a single-analysiscomputer program based on the equations presented in thispaper may be used effectively to analyse single two- andthree-phase induction motors. In addition, the phasewindings may be either equal or unequal, and the spacingamong the phases may be arbitrary.

7 Conclusions

This paper provides, first, a basic theory to predict thebehaviour of an asymmetric three-phase induction motorusing revolving-field theory, secondly, extensions of theabove for single- and two-phase induction motors, andthirdly, detailed step-by-step analysis procedures for suchmotors. A computer program was developed based uponthe present development and is being used extensively bythe Application Department to analyse such inductionmachines as split-phase, permanent-split capacitor witheither quadrature or nonquadrature windings and sym-metric or asymmetric three-phase motors. To demonstratethe effectiveness of the present theoretical approach,however, actual measurements and theoretical predictions

ELECTRIC POWER APPLICATIONS, FEBRUARY 1979, Vol. 2, No. 1 43

on various output quantities are reported. In each casegiven in this paper, and almost all others not included here,a remarkable correlation exists between theory and actualtests. Needless to say there is a reduction in the cost ofmaintaining a single permanent file on a time-sharingsystem instead of three separate files that otherwise wouldbe needed to predict the behaviour of single- two- andthree-phase induction machines.

8 Acknowledgments

The author would like to thank J. Postema for his commentson the manuscript and for several useful discussions duringthe development of this paper. The author is grateful to H.Penhorwood, Director Engineering Division, for grantingpermission to publish this paper. The author also owes hisappreciation to C. Yeiter for expertly typing the manu-script and to D. Austin in drafting the Figures for thispaper.

9 References

1 FORTESCUE, C. L.: 'Method of symmetrical coordinatesapplied to the solution of polyphase networks', Trans. Amer.Inst. Electr. Engrs., 1918, 37, pp. 1027-1115

2 LYON, W. V., and KINGSLEY, C. Jun.: 'Analysis of unsym-metrical machines', ibid., 1936, 55, pp. 471-476

3 PUCHSTEIN, A. F., and LLOYD, T. C: 'Capacitor motors withwindings not in quadrature', ibid., 1935, 54, pp. 1235-1239

4 ALGER, P. L.: 'The nature of induction machines' (Gordon andBreach, 1965), chapters 5 and 11

5 KRON, G.: 'Equivalent circuits of electrical machinery' (Wiley,New York, 1951)

6 GURU, B. S.: 'Revolving-field analysis of capacitor motors withnonquadrature windings', Electr. Mach. & Electromech., (to bepublished)

7 VEINOTT, C. G.: 'Theory and design of small induction motors',(McGraw-Hill, New York, 1959), Chap. 9, pp. 167-175

8 GURU, B. S.: 'Two-equation analysis of a capacitor motor bycross-field theory', Electr. Mach. & Electromech., 1978, 2, pp.147-153

9 TRICKEY, P. H.: 'Iron loss calculations on fractional horse-power induction motors', Trans. Amer. Inst. Electr. Engrs.,1959,pp. 1663-1669

44 ELECTRIC POWER APPLICATIONS, FEBRUARY 1979, Vol. 2, No. 1