621.313.333.2 The Institution of Electrical EngineersPaper No. 2757 U

Feb. 1959

A NEW METHOD OF DETERMINING ROTOR LEAKAGE REACTANCE ANDRESISTANCE OF A THREE-PHASE INDUCTION MOTOR

By Prof. M. V. DESHPANDE, B.E., M.S., Member, and T. R. SUBRAHMANYAN, B.E., M.E., Student.(The paper was first received 11 th March, and in revised form 23rd July, 1958.)

tion losses, while the locked-rotor test gives the total impedanceof the induction motor. This is separated into the total leakagereactance and total resistance. In the case of a slip-ring machine,the total resistance obtained from the locked-rotor test is dividedinto stator resistance and rotor resistance in the ratio of theirrespective d.c. resistances. In the case of a squirrel-cagemachine, the stator a.c. resistance is obtained by multiplying thed.c. resistance by a factor which varies from 1 -2 to 1 -6 to allowfor skin effect. The rotor resistance is obtained by subtractingthis stator resistance from the total resistance. In the case oftotal leakage reactance, the division is made half to the statorand half to the rotor; alternatively it is divided in the ratio oftheir respective resistances. The paper develops a more accuratemethod of dividing the equivalent leakage reactance andresistance.

SUMMARYThe paper develops a new method of determining the rotor leakage

reactance and resistance of a 3-phase induction motor. Stationaryflux distribution of constant magnitude is produced in the air-gap ofthe machine by supplying direct current to the stator windings. Therotor is driven at various slip speeds simulating the conditions of a3-phase induction motor. The power input to the rotor is measured.The flux distribution in the stator teeth per pole pitch is measured byuse of search coils and a ballistic galvanometer when d.c. excitation toall three phases of the stator is suddenly removed. The voltage inducedper phase in the stator due to this change of flux linkage is calculated.An expression is developed for rotor leakage reactance and resistance.

The theory is proved by experimental work. The performance iscalculated using the rotor constants measured by this method and iscompared with that obtained by no-load and locked-rotor tests andby an actual load test. A comparison of results justifies the deter-mination of rotor constants by this new method.

LIST OF SYMBOLSb — Susceptance.B = Flux density.

Cn = Maximum value of nth harmonic.Vph = Phase voltage.

g = Conductance.Idcb> hec — Equivalent direct currents for phases a, b and c.

Io = R.M.S. value of the no-load phase current.hn max — Maximum value of the nth harmonic of the

no-load phase current.1 = Length of conductor.

w( = Number of stator phases.n — Rotor speed, r.p.m.

ns = Synchronous speed, r.p.m.P — Rotor input, watts.r'2 = Rotor resistance per phase referred to stator.s = Fractional slip.v = Velocity of conductor.

x£ = Rotor standstill leakage reactance referred tostator.

ocn = Angle of lag of the nth harmonic with respectto the fundamental.

(1) INTRODUCTIONThe general method of obtaining the performance charac-

teristics of a polyphase induction motor is to perform a loadtest. Otherwise, the standard method is to determine theequivalent circuit from the no-load and locked-rotor tests on jdthe motor and then to calculate the performance. For anaccurate determination of the equivalent circuit, it is necessaryto determine accurately the stator and rotor constants referred tothe stator. The no-load test gives the core, windage and fric-

Writtsn contributions on papers published without being read at meetings areinvited for consideration with a view to publication.

Prof. Deshpande is Professor of Electrical Engineering, L.D. College of Engineering,Abmedabad, India. At present, he is at the University of Illinois, on a visiting appoint-ment. Mr. T. R. Subrahmanyan was Assistant Lecturer in Electrical Engineering,L.D. College of Engineering, Abmedabad, India, when this work was done in 1956-57.

[46]

(2) PRINCIPLE OF MEASUREMENT OF ROTOR LEAKAGEREACTANCE AND RESISTANCE

The method is based on the measurement of power input tothe rotor when driven in a stationary flux of the same magnitudeas the rotating flux, thus simulating the conditions of running ofan induction motor, and on the measurement of flux per pole inthe air-gap by the use of a search coil under each tooth.

(2.1) Production of Stationary Flux Distribution in Air-GapThe stationary flux distribution in the air-gap is obtained by

connecting two of the stator terminals together and connectingthese and the other stator terminal across a d.c. supply. Adirect current Idc — {y/2)l0 is supplied when the stator is con-nected in star, and Idc - 1 • 5(-\/2)/o when the stator is delta con-nected, where 70 is the no-load phase current. If the waveshapeof the current is purely sinusoidal, this will give the correct station-ary flux distribution to simulate the a.c. condition when the instan-taneous current in one phase is a maximum and the currents inthe two other phases are negative half-maximum.

If the waveshape of the no-load phase current with a.c. opera-tion is not sinusoidal, the waveshape of the phase current isobtained and the various harmonics are calculated. The equiva-lent direct current is calculated for each harmonic, and the totalequivalent direct current is calculated. The effective directcurrent required in each phase to produce the same flux distribu-tion will be:Jdca = Cj + C3 cos a3 -f- C5 cos a5 + . . . + Cn cos <xn + . . .

. . . . (1)

heb = I Q COS y + C7 COS ( y + <X7)

+ C13 cos ( y + a13) + . . .1 + I C5 cos ( y + a5)

+ C n c o s ( y + a n ) + .

+ (C3 cos a3 + C9 cos a9 + Cl5 cos a15 + . . .) . (2)

DESHPANDE AND SUBRAHMANYAN: ROTOR LEAKAGE REACTANCE AND RESISTANCE OF INDUCTION MOTOR 47

hcc = Ci cos y + C7 cos ^ y + a7 \ a- = 2p _szp\ - 4 (6)

C13 COS ( y where

+ (C3 cos a3 -t- C9 cos a9 -|- Ci5 cos a15 + . . .)

These equations are derived in Appendix 10.1.

(3)

(2.2) Simulation of Conditions of Running as an InductionMotor

If the rotor is stationary, there is no relative motion betweenthe rotor conductors and the stationary air-gap flux produced asdescribed in Section 2.1. Therefore a stationary rotor with astationary flux distribution of constant magnitude in the air-gapcorresponds to a synchronously-running rotor with synchro-nously-rotating flux of constant magnitude, and a rotor drivenin the stationary flux by an external motor at a speed n is equi-valent to a rotor running at slip s with a rotating flux of thesame magnitude, where

s = nlns (4)

The main difference between normal operation and this methodis that here no torque which can be used as a mechanical outputis developed in the rotor.

To simulate the actual conditions, the magnitude and wave-shape of the alternating current taken by the motor under normaloperation with a slip s must be known. From this the equivalentd.c. excitation can be calculated and the stator phases supplied•with the appropriate currents.

(2.3) Measurement of Flux per PoleA search coil is introduced at the root of each tooth, the ends

of the coils are connected in turn to a ballistic galvanometer,and d.c. excitation to the three phases is suddenly removed.The ballistic galvanometer having been calibrated with the helpof the Hibbert magnetic standard, the magnitude of the changeof flux linkages, and hence the change in flux, can be calculated.Neglecting the residual flux in the teeth, this will be the fluxexisting in the teeth with d.c. excitation on. The flux in all theteeth is measured in this way. Then assuming that the flux isuniformly distributed in each tooth, and neglecting the fluxpassing through the bottom of the slot, the distribution at theroot of the teeth can be obtained. This will be the same as theair-gap flux distribution, if the slot leakage is neglected. Fromthis distribution, the voltage induced in the stator per phase canbe calculated.

(2.4) Theory of the New MethodThe stator is supplied with equivalent direct currents, and by

rotating the rotor at a speed n = sns, the working conditions atslip s are simulated. Knowing the losses in the driving motorand the friction and windage loss of the induction motor atspeed n, the input to the induction-motor rotor is calculated.

The input power to the rotor is made up of the rotor I2R lossand the rotor hysteresis and eddy-current losses. In the normaloperation of induction motors, the slip is low (less than 10%)and at the slip frequency iron losses can be neglected.

The rotor I2R losses, Pi at a speed corresponding to slip sltand P2 at a speed corresponding to slip s2, are related as follows:

(5)

wheremia\s\

5f)

(8)

(9)

(r'lV2

The voltage induced in the stator per phase, F,, can becalculated knowing the flux distribution per pole, and the valueof r2 and x2 can then be calculated.

(3) DETAILS OF INDUCTION MOTORThe machine is a 230-volt 3-phase squirrel-cage induction

motor, normally delta connected, and designed to develop 3h.p.as a 4-pole machine with an efficiency of 80% and a powerfactor of 85 %. All 36 coils in the stator are brought out to aterminal board. Search coils (5 turns of No. 25 s.w.g.) are intro-duced at the root of each tooth, and their terminals are broughtout to a separate board. A d.c. motor with dynamometer(220 volts, lOh.p., 1500r.p.m., compound wound) was used todrive the rotor of the induction motor.

(4) TESTS AND CALCULATIONSThe following tests and calculations were carried out:(a) Core and windage loss of the d.c. machine for the various

speeds and excitations.(b) Friction and windage loss of the induction motor at various

speeds.(c) Variable-voltage test (no-load) on the induction motor.(d) Locked-rotor test on the induction motor.(e?) Measurement of stator resistance per phase of the induc-

tion motor.(/) Load test on the induction motor. The load was adjusted

by varying the excitation of the d.c. machine so that it workedas a generator connected back to the d.c. mains.

(g) Determination of the waveshape of the phase current ofthe induction motor at various speeds.

(h) Harmonic analysis of the wave. The 12-ordinate methodwas used for the analysis.

(0 Determination of equivalent excitation: Using eqns. (1),(2) and (3), for 1450r.p.m. or 3-3% slip, the equivalent d.c.excitations are: Idca = 2-91 amp; Idcb = — 1-379amp; J.dcc= — 1 • 378 amp. For 1400 r.p.m. or 6 • 67 % slip, the equivalentd.c. excitations are: Idca — 4 0175amp; Idcb = — 1-88amp;Idcc = - 2 08 amp.

0") Calibration of ballistic galvanometer with a Hibbertmagnetic standard.

(k) Determination of rotor input and flux distribution by thenew method:

The direct currents flowing through phases a, b and c of thestator of the induction motor are adjusted to the values Idca,ldcb and Idcc calculated for the slip. The speed of the d.c.machine running as a motor is adjusted to the required slipspeed. Using the ballistic galvanometer connected to all thesearch coils in turn, the complete flux distribution is obtained.The experiment is repeated for various slips. The rotor input ismeasured at different slips. At ^i = 0 • 066 7, P{ =91-40 watts;at s2 — 0-0333, P2 = 22-85 watts.

48 DESHPANDE AND SUBRAHMANYAN: A NEW METHOD OF DETERMINING ROTOR

(/) Determination of voltage induced per phase in the stator:The ballistic galvanometer throws obtained in (k) are expressed

in webers. Assuming that the flux is distributed uniformly ineach tooth, and knowing the length of each tooth, the flux perunit length of the tooth, and hence the flux distribution over the

Fig. 1.—Equivalent circuit.

16

14

12

V-

1

LJ

o 8

t

6

4

2

0

'inI

/f

•4/

20

15

pole pitch is determined. The flux will be rotating at synchro*nous speed when the motor is working normally on a.c. mains.

Using the relation, e = Blv, the voltage induced in a conductorcutting the flux at a velocity corresponding to the synchronousspeed is calculated. Considering the distribution of the winding,the voltage induced in one coil, in a coil group, and in each phaseis calculated. V%h is plotted and its r.m.s. value is calculated.This turned out to be 213 volts.

(m) Determination of r'2 and x2':Knowing the values of P{ and P2 at slips s{ and s2, from

eqn. (6) the value of a was obtained as 0-943. From eqn. (9),b' = 1 -542 x 10"4. From eqn. (8),

and

x'2 = b'V\ = 1-542 X 10~4 x 2132 = 7 ohms

,-2' = flA-2' = 0-943 x 700 = 6-6 ohms

o

/

o

oO J

o / /

/ /

/ /

5 10 15 20SLIP, •/.

(a)

1 2 3 4OUTPUT, h.p.

(6)

I 5<

/

c

/

/

2 3OUTPUT, h.p.

(O

1 0

0-6

o 0 6

0 - 4

<

100

. 80

yt 60u

4 0

1

0 O O n

0 1 2 3 4 0 1OUTPUT, h.p.

id)Fig. 2.—Motor characteristics.

O Load test readings. No-load and locked-rotor test, x^" x\.(a) Torque/slip.(b) Slip/output.(c) Stator-phase-current/output.(d) Power factor/output.(«) Efficiency/output.

OUTPUT, h.p.

New method, x{

LEAKAGE REACTANCE AND RESISTANCE OF A THREE-PHASE INDUCTION MOTOR 49

(5) RESULTSThe equivalent circuit is shown in Fig. 1. The equivalent-

circuit constants obtained by the new method and by no-loadand locked-rotor tests are shown in Table 1.

Table 1EQUIVALENT-CIRCUIT CONSTANTS BY VARIOUS METHODS

Reactance of stator slotDifferential reactance .Magnetizing reactance .

ohms0-9252 090109

xi, ohmsr\, ohmsxi, ohmsri, ohmsg, mhosb, mhos

No-load andlocked-rotor tests

61153-361155-820001 2900081

New method

5-232-57 06-600012900081

The performance of the induction motor is calculated by thefollowing methods:

(a) Equivalent circuit using constants obtained by new method.(b) Equivalent circuit by no-load and locked-rotor tests.(c) Load test.A comparison of performance obtained by all three methods

is shown in Pig. 2.

(6) CONCLUSIONSIt can be seen that the performance characteristics calculated

by the new method lie very near to the actual load-test results.The performance calculated by no-load and locked-rotor testsgive optimistic results. It is obvious that the method of dividingthe total reactance equally between the stator and rotor is notvery accurate because of the difference in winding structure of thestator and rotor, especially in the case of squirrel-cage machines.The method developed here, however, serves to allocate thereactances between stator and rotor more accurately.

In the new method the flux distribution measured is at theroot of the stator teeth. This flux distribution is used in cal-culating the voltage induced in the stator. This is possiblebecause of the following assumptions:

(a) The stator slot leakage flux and zigzag leakage flux aresmall in comparison with the mutual flux, and hence can beneglected.

(b) The effects of distortion on the air-gap flux waveshape dueto fringing, zigzag and belt leakage can be neglected.

(c) The fundamental current in an induction motor producesa fundamental m.m.f. rotating at synchronous speed and severalharmonic m.m.f.'s rotating at —|, \, etc., of the synchronousspeed. Also the nth harmonic current produces fields rotatingat n, —n/5, n\l, etc., times synchronous speed. When the statorwindings are supplied with d.c, all these harmonic m.m.f.'s existbut are stationary with respect to the stator.

In an induction motor running at slip s, a field rotating atspeed kns cuts the rotor at a speed (k — 1 + s)ns. If directcurrent is supplied to the stator, and the rotor turns at sns,then the rotor cuts all the harmonic components of the m.m.f.wave at the same speed, sns. Hence conditions correspondingto that of an induction motor running at slip s are simulatedcorrectly only for the fundamental m.m.f. with a speed ns. Itis expected that the error introduced due to this will be smalland similar to that of differential leakage.

In normal well-designed induction motors the stator-slotleakage is about 1-2 % and the differential leakage is about 2-3 %of the mutual flux. In the machine designed and tested thevalues obtained by calculation are:

so that the above assumptions are justified. Further, the actualair-gap flux is less than the flux behind the stator teeth by theamount of the various stator leakages. This is to some extentoffset by the fact that the flux measured by the new method doesnot include the radial flux passing through the depth of thestator slots. By calculating the permeance of the teeth andslots across their depth, it is found that the flux passing radiallyalong the depth of a slot is 5-919 X 10~5Wb, which is about1 • 5 % of the flux passing through the teeth (349 • 35 x 10~5 Wb).

This new method of finding the reactance division betweenstator and rotor is laborious and at present of academic interest.It is hoped that such a study may lead to a more accurate andreasonable division of the total reactance between stator androtor.

(7) ACKNOWLEDGMENTSThe induction motor was designed and built by the authors

at the L.D. College of Engineering, Ahmedabad, India, and theinvestigation was carried out in the Electrical EngineeringDepartment in connection with the dissertation of Mr. T. R.Subrahmanyan for M.E. (Electrical Machine Design) degree ofGujarat University.

(8) REFERENCES(1) CARTER, F. W.: 'The Magnetic Field of the Dynamo-

Electric Machine', Journal LEE., 1926, 64, p. 1115.(2) DOHERTY, R. E., and NICKLE, C. A.: 'Synchronous Machines

Part I—An Extension of Blondel's Two Reaction Theory',Transactions of the American I.E.E., 1926, 45, p. 912.

(3) PARK, R. H., and ROBERTSON, B. L.: 'The Reactances ofSynchronous Machines', ibid., 1928, 47, p. 514.

(4) ALGER, P. L.: 'The Calculation of Armature Reactance ofSynchronous Machines', ibid., 1928, 47, p. 493.

(5) HELLMUND, R. E., and VBINOTT, C. G.: 'Transformer Ratioand Differential Leakage', ibid., 1930, 49, p. 1043.

(6) KILGORB, L. A.: 'Calculation of Synchronous MachineConstants', ibid,, 1931, 50, p. 1201.

(7) LIWSCHITZ, M. M.: 'Field Harmonics in Induction Motors',ibid., 1942, 61, p. 797.

(8) LLOYD, T. C.: 'Some Aspects of Electric Motor Design—Polyphase Induction Motor Design to Meet FixedSpecifications', ibid., 1944, 63, p. 19.

(9) LIWSCHITZ, M. M.: 'Differential Leakage with Respect tothe Fundamental Wave and to the Harmonics', ibid.,1944, 63, p. 1139.

(10) LIWSCHITZ, M. M.: 'Differential Leakage of a FractionalSlot Winding', ibid, 1946, 65, p. 314.

(11) ALGBR, P. L., and WEST, H. R.: 'The Airgap Reactance ofPolyphase Machines', ibid., 1947, 66, p. 1331.

Books.(12) LANGLOIS-BARTHELOT, R.: 'Electro-Magnetic Machines'

(MacDonald, London, 1953).(13) SAY, M. G.: 'Performance and Design of Alternating

Current Machines' (Isaac Pitman and Co., England).(14) BEWLEY, L. V.: 'Alternating Current Machinery' (The

Macmillan Company, New York, 1949).(15) ALGER, P. L.: 'The Nature of Polyphase Induction

Machines' (John Wiley and Sons, New York, 1951).(16) WEBER, E.: 'Electro-Magnetic Fields' (John Wiley and Sons,

New York, 1954).(17) KARAPETOFF, V., and DENISSON: 'Experimental Electrical

Engineering' (John Wiley and Sons, New York).

50 DESHPANDE AND SUBRAHMANYAN: ROTOR LEAKAGE REACTANCE AND RESISTANCE OF INDUCTION MOTOR

The r.m.s. value of the no-load phase current will be(18) WYLIE, C. R.: 'Advanced Engineering Mathematics'(McGraw-Hill, New York).

(19) GRAY, A.: 'Electrical Machine Design' (McGraw-Hill BookCompany, New York).

(20) KUHLMAN, J. H.: 'Design of Electric Apparatus' (JohnWUey and Sons, New York).

(21) STILL, A.: 'Elements of Electrical Machine Design' (JohnWiley and Sons, New York).

(9) APPENDICES(9.1) Calculation of Equivalent D.C. Excitation

No-load phase current waveshape:

y = m (10)In most cases of electromagnetic induction without a com-mutator, there is no d.c. component and even harmonics areabsent.

Therefore y = A{ sin / + A3 sin 3/ + A5 sin 5/ + . . .

+ B{ cos t 4- B3 cos 3/ + ^5 cos 5/ + . . .

where the peak value of the «th harmonic is

Cn=V(A2n+B*) (11)

and the angle by which the nth harmonic lags the fundamentalis given by

Per-unit content of the wth harmonic:

V2V + p2

(14)

from which the maximum value of the fundamental componentwill be

C, = (V2)/0/(l +PZ+1% + . . .)ll2 . . . (15)

Knowing Cu the maximum values of all the harmonics and theirphase angles with respect to the fundamental can be determined.The effective direct currents required in each phase to producethe same flux distribution will be those given in eqns. (l)-(3).

(9.2) Details of Induction Motor

230 volts, 3-phase, 50c/s. Line current, 8-8amp. Phasecurrent, 5 -09 amp. When delta connected for 4-pole operation,output is 3h.p. All winding terminals are brought out to aterminal board for multipole operation if required.

Design Calculations

(13)

Total resistance per phaseTotal reactance per phaseShort-circuit power factorFriction and windage lossShort-circuit currentNo-load stator I2R lossMagnetizing current

Active component of no-load currentNo-load currentNo-load power factorFull-load currentFull-load slipFull-load torque

8-26 ohms11-55 ohms0-65845 watts17 amp49-2 watts2-19amp« 44% of full-

load current0-415 amp2-23 amp01868 • 8 amp13-75%13 lb-ft