8/3/2019 New Method to Measure

1/4

New method for measuring Xd and Xgbased on the P-Q diagram of the lossysalient-pole machineA.Ah. Fock. B.E.. and P.M . H art. B.E., M.Eng.Sc.

Indexing terms: Synchronous motors, Reluctance motorsAbstract: The P-Q diagram of the lossy salient-pole synchronous machine is derived from the familiar machinevoltage phasor diagram. The diagram contains a reluctance circle with a centre and diameter which are depen-dent on machine loss resistance. Steady-state stability limits are determined for the lossy salient-pole machine,and it is shown that it is not possible to operate on the extreme motoring point of the reluctance circle. Amethod of measuring Xd, Xq an d R for the lossy salient-pole synchronous machine or lossy reluctance machineis presented which involves measuring three convenient characteristic points on the reluctance circle of the P-Qdiagram. This method is found to have better accuracy than the slip test for smaller machines. The theorypresented in the paper is applied to a 5 kVA test machine and is found to give accurate results.

List of symbolsEoVXdXqR

ZSPQ

= per phase internal voltage of machine= per phase generator terminal voltage= per phase direct-axis reactance= per phase quadrature-axis reactance= per phase machine series resistance

6QT/1

/d= (X \ + R2)112= phase angle between 0 an d V (load angle)= per phase real power at the generator terminals per phase reactive power at the generatorterminals= maximum (generating) power on the reluctancecircle= minimum (motoring) power on the reluctancecircle= maximum reactive power on the reluctance circle= minimum reactive power on the reluctance circle= per phase shaft torque of machine= synchronous speed, rad/s= armature phase currentIntroduction

The authors' interest in testing small (less than 10 kVA)three-phase salient-pole synchronous machines and reluc-tance machines has led to an investigation of the effects ofmachine loss resistance on steady-state performance. Smallmachines are affected proportionately more than largermachines by loss resistance, and it may be necessary toaccount for these effects when analysing steady-state per-formance. The results presented in this paper are applic-able to balanced lossy salient-pole synchronous machinesor reluctance machines for which the losses are representedby a series resistance which is independent of rotor angle.The method of measuring Xd, Xq an d R presented inSection 3 was developed as a result of the authors' dissatis-faction with the accuracy of the slip test when applied tosmall machines.2 P-Q diagram for salient-pole machine allow ingfor machine series resistanceIn this Section expressions for the real power, reactivepower and electromagnetic torque of a polyphase salient-Paper 3460B (PI), received 3rd April 1984The authors are with the Department of Electrical Engineering, Monash University,Clayton, Victoria, Australia 3168

pole synchronous machine with nonnegligible armatureresistance are presented which form the basis of the mea-surement method described in Section 3. Using theseexpressions a modification to the P-Q diagram as given byWalker [1] is presented which accounts for armaturelosses.

The expressions for P an d Q given below can be

Fig. 1 Vector diagram of lossy salient-pole synchronous machineobtained by using trigonometry on the phasor diagram(Fig. 1) for the salient-pole synch ronou s ma chin e:

ZE 0 V sin W + 3) V2RP =V\Xd - Xq) sin 2

8/3/2019 New Method to Measure

2/4

By substituting eqns. 1 and 2 into the expression forshaft electromagnetic torqueP + I2R' (3)

(4)

T =and current

; _ (P 2 + Q 2)112V

the torque is found to beT = {2E0 VZ(XdXq - R2) sin {\jj + 3)- 2E0 VZR(Xd + Xq) cos W +

8/3/2019 New Method to Measure

3/4

and

{(R/Xq)2(X q/X d) (9)Hence the impedances Xd, Xq and R can be determinedonce Pmin, Pma x and gmi n are known. These latter quan-tities can be measured using the arrangement shown inFig. 6. After setting a convenient voltage level (at which

Fig . 6 Measurement connections for application of the circle methodslip current is not excessive) the power input is controlledin both motoring and generating directions until the limit-ing powers Pma x and Pmi n are reached. Because the rate ofchange of power with machine angle approaches zero atthe top and bottom of the reluctance circle, the wattmeterindicates a clear extreme value before their readings col-lapse when slip occurs.The slip test, which is the conventional test for deter-mining XJXq, has serious shortcomings when applied tosmall machines. Fig. 7 shows current and voltage oscillo-

Fig . 7 Slip test oscillograms conducted on the test machine at 150 V(phase)top trace: line current 7.1 A/major divisionbottom trace: l ine voltage 200 V/major divisionslip frequency = 2 Hzgrams which were obtained from a slip test performed onthe test machine at 150 V (phase). It is seen that thecurrent envelope is not sinusoidal, but, rather, has a sharpdip associated with the d-axis alignment position. Thisresults from the relatively small rotor inertia with the con-sequence that the rotor accelerates quickly through thed-axis alignment position, overshoots it and deceleratesmore slowly through the g-axis alignment position.Although it may be possible to improve accuracy byincreasing rotor inertia this is considered to be inconve-nient.The slip test gives inaccurate results because the currentwaveform does not have time to settle to a true minimumand is affected by the transient as well as the steady-statemachine behaviour. Dissatisfaction with the application ofthe slip test to small machines has also been reported byLawrenson and Agu [2].A further criticism of the slip test, although of lesserimportance, is that resistance effects cannot be corrected

for. The effects are accounted for in the proposed newmethod based on the assumption that resistances are notdependent on machine angle. This involves an approx-imation as core losses are likely to be less for the q-axisalignment than for the d-axis alignment. The resistancevalue obtained will be that applicable at the terminalvoltage for which the test is conducted.Fig. 8 gives a comparison of measurements taken on the5048I"

8/3/2019 New Method to Measure

4/4

proved impossible to accurately measure the current (or Q)at slip because of needle overshoot occurring on metersonce instability occurs. The second method requires specialequipment not available to the authors.Machine inductances can also be measured with themachine stationary using fiuxmeter-bridge methods [5, 6]or by measuring time constants of responses to step inputs[7]. These methods allow the variation of self or mutualinductances of coils to be measured as a function of rotorposition. Xd and Xq may then be computed using two-axistheory. Although of definite importance these methodsmust be applied with some caution as '... for accurateresults the inductances must be measured under conditionsapproximating as closely as possible to those of normaloperation' (Reference 5, p. 15).The flux and eddy current distributions and the state ofsaturation of the machine under test will vary according tothe nature of the test applied, and results must be inter-preted accordingly. The circle method presented here isadvantageous in that conditions within the machine arethose applying to steady-state operation for which Xd andXq determine machine behaviour.Practical investigations of small machine inductanceshave been presented by Carter, Leach and Sudworth [8]and Barton and Dunsfield [9]. Deviations from behaviourpredicted by the two-axis theory have been reported inboth papers due to harmonic gap MMF components, thenonsinusoidal variation of inductances with rotor angle,and deviation from assumed equality of various self andmutual inductance coefficients. Two-axis theory remainswidely applied to small machines and in the authors' expe-rience can be applied to small machines with reasonableaccuracy. Deviation from the circular of the reluctancemachine P-Q locus as given in Fig. 5 gives a convenientvisual indication of the likely accuracy of steady-state per-formance as predicted by two-axis theory.4 ConclusionsTh e P-Q diagram of a lossy salient-pole machine has beenpresented (Fig. 2) and is shown to contain a reluctancecircle which has a radius and centre which are affected byloss resistance.The reluctance circle is shifted into the motoring quad-rant when compared with the lossless machine. This shiftmay be significant for smaller machines which have pro-portionately large resistance effects.

The steady-state stability limits are also affected by lossresistance. Although no simple formulas exist to predictthe machine angle at loss of stability for the lossy salient-pole synchronous machine, the lossy reluctance machineloses stability when2(5 = \\i + X 90

A method of measuring Xd, Xq and R for the lossymachine which uses the measured values of three charac-teristic points on the reluctance circle is proposed. Themethod has been successfully applied to a small testmachine and found to have superior accuracy comparedwith the slip test.5 AcknowledgmentThe authors are indebted to Associate Professor W.J.Bonwick for his encouragement and assistance.6 References1 WALKER, J.H.: 'Operating characteristics of salient-pole machines' ,Proc. IEE, 1953,100, Pt. II, pp. 13-242 LAWRENSON, P.J . , and AGU, I .A.: 'Theory and performance ofpolyphase reluctance machine' , ibid., 1964, 111, (8), pp. 1435-14453 STEPHENSON, J .M., and LAWRENSON, P.J . : 'Average asynchro-nous torque of synchronous machines, with particular reference to re-luctance m achines' , ibid, 1969,116, (6), pp. 1049-10514 IEE E test procedures for synchrono us machines, 1965, IEE E pub-lication 115, p. 525 JON ES , C .V.: 'The unified theory of electrical machin es'(Butterworths, 1967)6 PR ESC OTT , J .C., and EL -KHA RAS HI, A.K.: 'A method of measur-ing self inductances applicable to large electrical machines', Proc. IEE,1959,106A, pp. 169-1737 KAMINOSONO, H., and UYEDA, K.: 'New measurement of syn-chronous machine quantities' , IEEE Trans., 1968, PAS-87, pp. 1908-19188 CARTER, G.W., LEACH, W.I , and SUDWORTH, J . : 'The induc-tance coefficients of a salient pole alternator in relation to two-axis

theory', Proc. IEE., 1961,108A, pp. 263-2699 BARTON, T.H, and DUNSFIELD, J .C.: ' Inductances of a pract icalslip-ring primitive, IIan experimental study', IEEE Trans., 1966,PAS-85, pp. 145-1517 AppendixTest machine: 415 V, three phase, 5 kVA, 1500 rev/minsalient-pole alternator having a near sinusoidal waveform:

Xd = 45.5 Q (SC/OC tests) R = 1.83 Q (DC)Made by McColl Electric Works Ltd., Australia.

26 2 IEE PROCEED INGS, Vol. 13J, Pt. B, No. 6, NOVEMB ER 1984