IEEE TRANSACTIONS ON POWER APPARATUS AND SYSTEMS, VOL. PAS-88, NO. 11, NOVEMBER 1969

The Quadrature-Axis Equivalent Circuit of theSynchronous Machine with a Grill

STANIMIR B. JOVANOVSKI

Abstract-The complete equivalent circuits of a synchronousmachine with a grill (open end rings) are analyzed. Because thedirect-axis equivalent circuits for both the squirrel-cage and thegrill machines are the same and are well known, the paper developsonly the quadrature-axis equivalent circuit for a machine with agrill.

INTRODUCTIONTHE equivalent circuits of a synchronous machine with a

squirrel cage proposed by Rankin [1] have been consideredthe most complete. In certain cases, however, synchronousmachines are designed with a grill (open end rings; i.e., withoutconnections between the adjacent poles). The structure of therotor circuits along the d-axis is the same for both machines.Therefore, the direct-axis equivalent circuit is the same for bothtypes of machines. On the other hand, the grill circuits along theq-axis are different from those defined for a squirrel cage.In this paper the grill q-axis circuits are defined, and the

formulas for all parameters needed to represent the completeequivalent circuit are given. Thus, this paper may be con-sidered as an extension of [1].

GRILI-CIRCUIT DEFINITIONOriginally [2], one circuit of the squirrel cage was defined as

formed by two bars symmetrically located with respect to thecenterline of d- and q-axes. These two bars are normally made ofthe same material, and the air gap as well as the whole electro-magnetic structure of the machine is symmetrical with respectto these bars.

In Fig. 1, a simplified scheme of a two-pole synchronousmachine with its q-axis circuits is presented. The squirrel-cageq-axis electrical circuits, and the induced current that flowsthrough them during the transient asynchronous operation, areindicated. The linkage flux c?gnnq, created by any circuit of thecage has the same direction.

In the case of a grill, the end rings are open. The current E = 1I^ will now flow through the outermost bars (Fig. 2). Thedirection of the current in the two outermost bars and in anyother bar of a pole tip are opposite. Thus, the linkage flux

-f11,created by the first circuit opposes the linkage flux 4'1,nnQ, of thenth circuit.The grill q-axis electrical circuits may be replaced by two

identical systems, symmetrically disposed with respect to thecenterline of the quadrature axis (Fig. 3). One circuit of the grill

Paper 69 TP 41-PWR, recommended and approved by the Rotat-ing Machinery Committee of the IEEE Power Group for presenta-tion at the IEEE Winter Power Meeting, New York, N. Y., January26-31, 1969. Manuscript submitted September 9, 1968; madeavailable for printing November 7, 1968; revised February 7, 1969.The author was with the Department of Electrical Engineering,

Clarkson College of Technology, Potsdam, N. Y., on leave from theUniversity of Skopje. He is now with the Electromechanical Faculty,University of Skopje, Yugoslavia.

Fig. 1. Quadrature-axis circuits of squirrel-cage synchronousmachine.

d

Fig. 2. Quadrature-axis circuits of synchronous machine with grill.

is now composed of the outermost bars located at the edges of thepole tips and any other bar of the corresponding pole halves.Thus, the first grill circuit is composed of the bars 1 and 2, thesecond circuit of the bars 1 and 3, etc.

PARAMATERS FOR QUADRATURE-AXIS GRILL CIRCUITSThe grill q-axis circuit per-unit parameters may be determined

from the corresponding parameters of the squirrel cage [1] asshown in the following.

vth Grill-Circuit ParametersThe vth grill-circuit resistance and reactance are

l= qb+Rq (1)X" = Xqb1 +x Qev + Xq v (2)

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JOVANOVSKI: QUADRATURE-AXIS EQUIVALENT CIRCUIT OF SYNCHRONOUS MACHINE

Fig. 4. Armature quadrature-axis MMF and flux waves.

Mutual Reactance Between Armature and vth Grill CircuitThe mutual reactance between the armature circuit and Pth

grill circuits is (Fig. 4)Xqa =Xanq - Xalq, v = n - 1. (16)

Fig. 3. Quadrature-axis grill circuits and air-gap flux wave.

where

DEVELOPMENT OF QUADRATURE-AXIS EQUIVALENT CIRCUITFor a machine with five bars per pole, the flux per-unit equa-

tions written for the q-axis armature flux, and for the q-axisgrill-circuit fluxes (for both halves of the two adjacent pole tips,Fig. 3), are

RqeP-rq

Xqep = 2xqe"and (Fig. 3)

Xqgy = Xgnnq - Xgllq, v = n - 1.

(3)(4)

(5)If all bars are identical and made of the same material the bar

parameters are

Rq = 2Rbnnqxqby, = 2Xbnnql5

(6)(7)

For the circuit formed by the bar lying directly on the polaraxis and the outermost bar (i.e., for the last grill circuit), we have

Rgb = 3RbnmQ (8)XQbVp = 3Xbnnfl (9)

If the bars are different, their parameters are determined byRqb = 2(rbl + rbnq) (10)Xqbp = 2(Xi1i + Xbnq) (11)

and, for the last grill circuitRq = 2(rblq + 2rbnq) (12)

X = 2(Xblq + 2XInq). (13)Mutual Parameters Between vth and kth Grill Circuits (k > v)The mutual resistance and reactance of pth-kth grill circuit

(k > v) areRqk = Rqepp (14)

Xq = Xqe"' + Xq p (15)

(17)*e = XqIq + XqalIql + Xq2Iq2*4I = xqlaIq, + XqllIql + Xql2Iq2I,f2 = X2aIq + Xq2lIql + Xq22Iq2.

(18)(19)

If the armature resistance is neglected, and because the grillcircuit voltages are zero, the electromotive force (EMF) per-unitequations for the motor operation are given by the followingexpressions:

Vq = jXqIq + jX5alIql + jXqa2JI (20)0 = jxq1l"q + (ft + jxqil) I1l + (ft + jXql2) I02 (21)

0 = jX2I5 + + 1X5221) I + (s-+ jX522) Iq2. (22)

Since the mutual parameters of two circuits have the samevalues, substitution of the R14" and XqTM' from (1) and (2) into(21) and (22), together with the conditions determined by (14)and (15) results in the following EMF equations:

/fbi0 = jX/'alI, + 'K + jXqbll) Iqg+ (B;1 + ;X:C11 + jXqgll)(Iq' + Iq2) (23)

0 = jX5a2II +( 2 Xs2) Iq' +( + jX52b22B5e22\

+ q + jXqt22 + jXq522 152. (24)+ /

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IEEE TRANSACTIONS ON POWER APPARATUS AND SYSTEMS, NOVEMBER 1969

By adding and subtracting the term

(121 + jXq21) Iq2 = (R; + jXell +j Iq2in (24), we obtain

0 = jX a2 +(Rq + jX6ell + jXqgll) (Iql + Iq2)-Rb22 R ~~e22- ell

+ jX b22 + Rq + j(Xqe22 Xqell)S S

+ j(X022 Xq11) Iq2 (25)According to the q-axis electrical circuit definition of the syn-

chronous machine with a grill (Figs. 2-4), the armature windingwill be represented on the q-axis equivalent circuit by six series-connected reactances (for the considered machine with five barsper pole). Because both halves of the grill q-axis circuits areidentical (Fig. 3), the corresponding mutual reactances betweenarmature and grill circuits on both halves of the equivalentcircuit are the same (Fig. 5).At two points v = 2 of each half of the armature equivalent

circuit the corresponding grill circuits are connected. TheEMF equation of the armature equivalent circuit (Fig. 5) isVe = j(Xa4 + Xa3 + 2X02 + 2Xal)Iq + 2j(Xa2 + Xal)Iq2

+ 2jXalIqI' (26)By comparing (20) and (26), and considering (16), one finds

Xa2q - Xalq2

Xa3q - Xa2qAo2 = -2Xa3 = XalqXa4= Xq - Xa31. (27)

In Fig. 6 one half of the equivalent circuit, with grill circuitsadded, is represented. The EMF equation of grill-circuit I is

0 = jXalIq + ZlIq' + (Z' + jXql)(Iql + Iq2).Comparing (28) with (23) (divided by 2), we have

Z Rb22q + jXs

if all bars are identical [(6) and 7)]; orZ

=

rblq + rb2q j(X,iq + Xb2q)s

(28)

(29)

(30)

if all bars are not identical [(10) and (11) ].Considering (3)-(5), and the first term of (27), the impedance

Z' is

Zi = rel+ jxqe11 + iXg22q - Xgllq + j Xa2q - Xalq2 2

The EMF equation of grill circuit II (Fig. 6) isO j(Xal + Xa2)Iq + (Z' + iXal)(IqI + Iq2)

+ (Z2 + Z" + jXa2)Iq2. (32)Comparing (32) with (25) (divided by 2), we have

Fig. 5. Armature quadrature-axis equivalent circuit.

Fig. 6. Half of armature quadrature-axis equivalentcircuit with grill added.

Z2= R3 + jXb33a)if all bars are identical [(8) and (9) 1; or

Z2 = + + I(Xblq + 2Xb3q)s

(33)

(34)

if all bars are not identical [(12) and (13) 1.r.e22 _r ell

z r - + j(Xqe22 - Xqell)

+ XQ33q - Xg22q X_ a2q (35)2 av2 (5

In Fig. 7, the complete q-axis equivalent circuit of a machinewith a grill, with five bars per pole, is represented.

CONCLUSIONThe solution of synchronous machine problems involving a

number of simultaneous equations (for instance the asynchro-nous torque-speed characteristic, the damper-bar currents, etc.),is usually made by digital computers today. However, thecomplete equivalent circuits, in which the armature circuit, thefield-winding circuit and the multiple damper-winding circuits

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JOVANOVSKI: QUADRATURE-AXIS EQUIVALENT CIRCUIT OF SYNCHRONOUS MACHINE

-2q

FR 1(0 F~~~~r22 ellI13 b33q qjX q - re q

*I x33q- Xg22q)

-lq L- jfXa3q qb22qs '~+ IXb22q )J

L rbl,

rell |q I,

q rb2q).i(xbIq4xb2q) (2 e Iq

I j(X22q Xg21

2 9-i(X02q - jIIqFig. 7. Complete quadrature-axis equivalent circuit of machine

with grill for five bars per pole: (1)-bars identical; (2)-barsdifferent.

RI2kv mutual resistance Pth-kth grill circuit (k > v)rbnq = resistance of one damper barrqelvlv resistance of end ring, corresponding to the

length Lqe^P (two rings)s slipVq armature q-axis voltagexq armature q-axis reactanceXbnnqf, Xn reactance components of nth q-axis squirrel-

cage circuit due to bar-slot flux and air-gapflux, respectively

Xqbv, Xq'ep, reactance components of vth q-axis grill circuitXqgl'v due to bar-slot flux, end-ring flux, and air-

gap flux, respectivelyXanq, Xqap mutual reactances-armature to nth squirrel-

cage circuit, and armature to vth grill circuit,respectively

Xqkv mutual reactance P,th-kth grill circuits (k > v)XbnnqXbn=q reactance of one damper bar due to slot flux2

Xqev reactance of end ring, due to ring flux corre-sponding to length Lqe,,, (two rings)

P, k grill circuits -v is general term, k refers to grillcircuit external to v (k > v)

Ognnq nth squirrel-cage circuit average flux in air gapt*q, tTfqV quadrature-axis linkages-armature anld vth

grill circuit, respectively.REFERENCES

[1] A. W. Rankin, "The direct- and quadrature-axis equivalentcircuits of the synchronous machine," AIEE Trans., vol. 64,pp. 861-868, December 1945.[2] T. M. Linville, "Starting performance of salient-pole synchro-nous motors," AIEE Trans., vol. 49, pp. 531-547, April 1930.

are individually represented, remain a useful tool in analyzingsynchronous machine transient phenomena.The synchronous machine equivalent circuits have been

analyzed in several papers, the most complete to date beingthose developed in [1]. However, all these treatments haveconsidered only the squirrel-cage machine, even though thesynchronous machine with a grill is occasionally manufactured.

In this paper, a synchronous machine with a grill was analyzed.The rotor d-axis circuits being similar to those of a machinewith a squirrel cage, the emphasis was on the rotor q-axis cir-cuits.The grill q-axis electrical circuits were defined and the formulas

for all parameters necessary when presenting the equivalentcircuit were assembled. From the established flux and EMFequations, the complete q-axis equivalent circuit for a machinewith a grill was developed.

NOMENCLATUREThe symbols used in the paper are defined below. All quantities

are per-unit values unless otherwise specified. Vectors areindicated by boldface type.

Iq, Ig( quadrature-axis currents--armature and Ath grillcircuit, respectively

Laevv length of end ring between two bars forming agrill circuit (see Fig. 3) (mm)

n squirrel-cage circuit (general term)Rbnnq bar resistance of nth squirrel-cage circuitRqbnP, RR"' resistances of vth grill circuit-bar and end ring,

respectively

Discussion

Philip L. Alger (Schenectady, N. Y.): This problem of how tocalculate the performance of a synchronous machine with open endrings has been of interest to me for many years. In my opinion, thetwo most important papers in this area have been those by I. Giaever[3] and W. A. Lewis [4]. I regret that the author has not referred toeither of these, since they both record material advances over theearly papers, [ 1] and [2].

In my opinion, the transient performance of a synchronousmachine with a grill can be materially improved either by placingthe end bars in the lower portions of the pole tips with their slotsopening into the inner pole-tip surface, or by adding extra barslocated in the pole body just below the pole tips, so that all theflux in the q-axis will be linked by the squirrel cage. Such a bararrangement is readily possible with cast aluminum windings. Itappears to me that because of the disadvantages of outside end ringswith bolted joints for high-speed machines, and the alternativedisadvantages of the usual grill construction due to the high q-axissubtransient reactance, it should be worthwhile to provide squirrelcages with end bars below the pole tips.

Since the currents in the various bars are so widely different,their temperature rises may also be very different. If so, failure islikely to occur on brazed windings after frequent starting due to theunequal expansion of the bars. This difficulty is overcome by makingthe pole face winding of cast aluminum.The great objection to cast aluminum pole face windings for

synchronous machines has been the difficulty of making good endring connections between poles. The location of the end bars below

Manuscript received February 3, 1969.

-2

F.1[j(Xq - Xa3q)]Fi 12 (Xa3q Xc2q)j

I L .1

2*i(Xa2q XGaiq)i

Vq fiXaiqj

[2 i(Xa3q - Xa2q)J

lzj(xc2q - X)l)2 a2q

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IEEE TRANSACTIONS ON POWER APPARATUS AND SYSTEMS, VOL. PAS-88, NO. 11, NOVEMBER 1969

the pole tips, which obviates the need for end rings between poles,should make the cast alurninum construction eminently satisfactory.

REFERENCES[3] I. Giaever, "A complete equivalent circuit of a synchronous

machine," AIEE Trans. (Power Apparatus and Systems), vol.77, pp. 204-209, June 1958.

[4] W. A. Lewis, "A basic analysis of synchronous machines-pt.I," AIEE Trans. (Power Apparatus and Systems) vol. 77, pp.436-456, August 1958.

Stanimir B. Jovanovski: The author is indebted to Mr. Alger forhis interest in the paper, and for the additional references concernedwith the equivalent circuits of synchronous machines.Mr. Alger's suggestion of how to improve the transient per-

formance of a synchronous machine with a grill is very interestingone, and deserves further consideration.

Manuscript received March 19, 1969.

A Digital Model for Three-Phase Induction MachinesSTUART D. T. ROBERTSON, MEMBER, IEEE, AND KATTINGERI M. HEBBAR, MEMBER, IEEE

Abstract-A digital model for a three-phase induction machineis developed, which is particularly adapted for studying its dynamicperformance when fed from an inverter. Conventionally, the in-duction motor is analyzed in terms of its d-q variables, while theoperation of the inverter generally needs continuous monitoring ofthe state of its phase quantities. Thus in a study of the compositeinverter-induction-machine system, one is faced with the problemof matching the two sets of variables. The proposed model over-comes this problem by describing the machine behavior directly interms of the stator phase variables, at the same time retaining acomputational simplicity comparable to that of the d-q equations.Furthermore, it is shown that the machine equations, expressed interms of the stator terminal variables, can conveniently handlethe various terminal conditions, like open phases, that may arisewhen the machine is fed from an inverter. Detailed equations in aform suitable for digital solution are also presented for possibleterminal constraints of this nature, with or without the machineneutral connected to supply.

NOMENCLATUREDfi

ia ) ib,8* 8IC

?, rp i r* 8

iTY8

damping coefficientfundamental source frequencygeneral current vector, instantaneousactual rotor phase currents, instantaneous

actual stator phase current, instantaneous

transformed rotor currents, instantaneoustransformed stator currents, instantaneous

Paper 69 TP 6-PWR, recommended and approved by the RotatingMachinery Committee of the IEEE Power Group for presentationat the IEEE Winter Power Meeting, New York, N.Y., January26-31, 1969. Manuscript submitted February 12, 1968; made avail-able for printing October 15, 1968.The authors are with the University of Toronto, Toronto, Ont.,

Canada.

i7' transpose of ilar, lJT transformed rotor current phasors, rmsIa8, Ib,8 actual stator current phasors, rms

IcsIo , Ie', transformed stator current phasors, rms

'I JR

Jkrks[LILrLs1O17

angular moment of inertia of the rotating systemcoefficient of coupling between two rotor coilscoefficient of coupling between two stator coilsgeneral inductance matrixself-inductance of one rotor phaseself-inductance of one stator phaseeffective stator self-inductance for zero-sequence

currentsLp,,T effective rotor self-inductance of one phase for

positive or negative sequence currentsLp, effective stator self-inductance of one phase for

positive or negative sequence currentM maximum of the mutual inductance between a stator

phase and a rotor phasen number of pole pairsp differential operator d/dtPi electrical input power into the machine, instantaneousPm mechanical power developed by the machine, in-

stantaneous[R]RrRS

T

v

Vc

ScrVuct,X

vc8

general resistance matrixrotor resistance per phasestator resistance per phasesliptorque developed by the machine, instantaneousload torquegeneral voltage vector, instantaneousactual rotor phase to neutral voltages, instantaneous

actual stator phase to neutral voltages, instantaneous

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