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Precise Method for Measuring Xd andXq Based on S l i p Testo f
Synchronous Machines
Osamu. SUGIURADept. of Elect. and Comp. Science.University of
YamanashiYamanashi, Japan
ABSTRACTAn experimental method for measuring the direct
and quadrature axis synchronous reactances (Xd andxq) of
polyphase synchronous machines with salientpole is newly developed.
It is based on a measuringmethod of slip test. It can be applied to
all thesynchronous machines, but primarily it is developedand
proved for reluctance motors. Precise measuringmethod was derived
by the use of tensor analysismethod, and was introduced to minimize
measurementerrors due to the stator resistance (Rs) as well asthe
machine losses. Detailed results are presentedin case of one
reluctance motor which has a highlyvariable Rs. In fact,
correlation with othermethods is proved to be quite good, The
effects ofRs are observed in the tested motor.
I . INTRODUCTION
The various methods to measure Xd and xa of thesynchronous
machines have been used in many ways inthe past, as in [l ] - [6] .
The slip test haswidely been used to measure Xd and xq. By
thissimple experimental test, he proper measured valuesof Xd and x4
can be obtained. But, this measuringmethod takes no account of R3
when the values of Xdand xq are computed, However, Rs can not be
ignoredin small motors because the resistance is larger insmall
motors than in medium and large motors.
In this paper, the authors made a try to take R3into
consideration. The measuring method of Xd andxq was to make the
theoretical analysis at synchro-nous speed of the reluctance motor.
This method isobtained by applying the analytical results, and
isexperimentally proved.
List of principal symbols
0-7803- 462-x/93$03.OOQ lW3IEEE
Yu i. AKIYAMAYember IEEEDept. of Elect. Engrg.Kanagawa Institute
of TechnologyKanagawa, Japan
R, :resistance of stator winding.LMX IXo
Xd :direct axis synchronous reactance.xq :quadrature axis
synchronous reactance.0
a
o : supply frequency, rad/s.o r :rotating speed, rad/s.p
:differential operator.
t : instantaneous torque.t :mean torque.7 :pulsating torque.
:self inductance of stator winding.:mutual inductance between
the stator windings.:mean value in self reactance of stator
winding.:maximum value of variation in self reactanceof stator
winding.
:angle between stator winding of a-phase and d-axis of
rotor.
:angle between stator winding of a-phase and d-axis of rotor at
time t=o.
Subscripts
s :stator.d :direct axis.q :quadrature axis.a, b, c :symbols of
3-phase.
II. ANALYSIS OF RELUCTANCE YOTOR
Applications of the theory of tensor to analyzerotating machines
were developed by G.Kron, as in
This Krons method is expedient to analyzetwo phase machines,
because the two hypotheticalwindings are assumed to be two phase
windings ofthe machine under consideration. But, for the threephase
machine, it is necessary to transform thethree phase actual
windings into the hypotheticalwindings of the primitive machine
with the aid ofconnection matrix, which will make the
resultantimpedance matrix rather complicated.
To avoid such complicated manipulation, Poly AxisMatrix Yethod
developed by T.Takeuchi as in [8] is
[71 .
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Fig. 1. Winding model of RI.
L
I-*/2 7~ -e
h L
- ? - - - -
Fig. 3. Variation of Y a b with 6 .
I cos20 1
And, the mutual inductance ?dab between the statorwindings
changes sinusoidally as in Fig. 3. Then,the mutual inductance
between the stator windingsis written as
The voltage equations of stator windings are shownas a matrix
formula as in (3 ) .
Fig. 2. Variation of La with 8 . Rs+PLo t $Mob, P n & c
[ ]= [DMbaj R s + P L ,I .Mbc][:! j (3 )used in this paper. This
method analyzes matrix uC PfiIcm PMcb, Rs+PLc
equations on the actual axes of windings and fa-cilitates the
analysis without any confusion. Transforming (3 ) by the three
phase sequence trans-
formation matrix A 3 , we have the voltage equationA. Voltage
Equation of (4) based on the symmetrical co-ordinates axis
Fig. 1 shows the winding model of the reluctancemotor(R.'d) in
which ai, bl and CI are three windingsspaced by 120" each other on
the stator. La, L b andLc are the self inductance of the stator
windings,and M a t , , l b c and M c a are the mutual
inductancebetween the stator windings. The salient pole rotoris
rotating with an angular speed of w in theclockwise direction as
shown by an arrow wt in theFig. 1.
The self inductance La of the stator windingchanges sinusoidally
as in Fig. 2. Then, the selfinductance of the stator windings may
be written as
as
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were B. Stator Current
The zero sequence elements are putted aside from(41, because the
grounding fault do not consider.
Then, the resultant voltage equation of (5 ) iswritten as
The impedance matrix of (5) is a function of timet. To eliminate
8 from the elements of thisimpedance matrix, we define the
commutation matrixK as
which can be chosen by inspecting (51, as in [8] .Transforming
(5 ) by the commutation matrix K
minding that
The impedance matrix of (7) is the stationaryimpedance matrix.
Thus, we can get the inversionof (7) as
where
B I I - . R s + ( L + M ) ( p - j w ) - ( 9 ) p - t j o )
The instantaneous three phase voltages are writtenas
Then, the voltages based on the symmetrical co-ordinates will be
written as
Substituting (9) in (8) obtains (10).
where
The impedance matrix of (7) is the stationaryimpedance matrix
which is no more function of time8 = w t + a . And, the currents
based on the symmetrical co-
ordinates will be written as
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(20) rr =-. xovz s in2(3 =-. (zt%-dV2 s in 6 (25)w x + - x o 2 w
2xdxq
In the right hand side of the above (201, thefirst term is the
power expended in heating theconductors. And the second term is the
rate atwhich energy is being stored in the magnetic field,that do
not contribute mechanical output. Also notethat in the above
(20)
1K=$i] c[L.][i]
is the instantaneous stored energy in the magneticfield.
Therefore, the third term must represent themechanical power
expended by the rotor. And, thepower may be expressed as
On the other hand, z i is an instantaneous torqueand o is an
angular speed of the rotor. Then,o i will be the mechanical power
expended by the
rotor, thus, we have
(21)
This is the general expression of torque obtainedfrom output.
where
~(L0-2Mo)cZ~
L+M3
Equation(25) is the general torque expression ofRM.
IU. DERIVATION OF METHOD FOR MEASURINGxd ANDX g
Fig. 6 shows the vector diagram of I, at the angleof phase
difference d m . In Fig. 6, I m i n is thevector of the minimum
stator current with the angleof phase difference d m .
Fig. 7 shows the variation of W, I and V with thepower angle
6.
The waveforms of input power, stator current andimpressed
voltage are the envelopes of the instan-taneous maximum value. If
RN is rotating nearly atsynchronous speed, it is permitted to
exchange timet for 6. Accordingly, the variation of I and V
isobserved quite the same as experimental data ob-tained by the
slip test.
Then, the impedance 2 i n of RM which is viewedfrom power source
is
I
(23) Fig. 6. Vector diagram of I,.
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W
v
I
I'
4 - - - 4 h Wm ;n I
Fig. 7. Variation of W, I and V with 6.
were
Wm ;
The tangent of (26) is
(27)I V / A - X t X m i n
R s V / A Rs R m i n- - = -a n (6m) =
Thus, comes the following equation as
From Fig. 6, two currents of I m i n and I m a , canbe written
as
Then, two currents of 11 and 1 1 2 may be definedas
I p a x + I m i n
2I tax- I m i n
2
I l l =
I 1 2 =
From (14), we have
I l l - J R s 2 + X i 2 ( V / A >- -r 1 2 x o ( V / A
>
- 4 s 2 + + X 1 *x o
-
(30)
(31)
Substituting (30) in (31) obtains (32)
] (32)I m a x - I m i nI m a x + I m i n
X o = J R s 2 + X i 2
Then, the direct and quadrature axis synchronousreactances Xd
and xq will be
X d = X l + X O , x q = x I - x O (33)
Hence, Xd and xq is derived from three kinds ofmeasurement as
follows.(a) The resistance Rs is measured by a simple d,ctest.(b)
The machine under test is driven steadily atsynchronous speed, and
its impedance Z m i n ismeasured for the rotor angle which gives
the mini-mum current at the terminal (see Fig. 6).
Wm;R m i n = 3 I m i n e 1
Thus, this test gives value of X I as follows.
(c) Being used the slip test, I m a x and I m n whichare a
maximum and a minimum currents, are measuredto get the value of
Xo.
Fig. 8 shows the oscilloscope waveforms of V and
I.From the measured data,
1m a x - I m i n
I m a x f I m i nX D = J R , ~ + X I ~
terminal voltageI I
f stator current I
Fig. 8. Oscilloscope waveforms of V and I.
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'KO -w2Fig. 9. Circuit for measuring Xd and xq.
Then, the value of xd and xq are obtained as
X d = x l + x O
xq = X I - xo
Fig. 9 shows the experimental circuit for measur-ing XI and XC,
thus, Xd and xq.
In Fig. 9, Rg was inserted in the armature circuitfor the
experiment to determine whether the preciseslip test for measuring
Xd and xq is good or not.
IV. EXPERINIENTAL RESULTS
The effect of the stator winding resistance R, onthe computed
values of xd and xq is investigated inthe first place. Then, the
authors computed ap-proximately how large the calculative error is
due
In the second place, he experimental results basedon the precise
slip test are taken into consider-
ation, and a comparison of the conventional sliptest and the
precise slip test with regard to theeffect of R, has been made.
to Rs.
A. Calculative Error in the Slip Test
The authors investigated the effect of Rs on thecomputed values
of xd and Xg by using the equiva-lent circuit in Fig. 10.
It is assumed that xd5 is the direct synchronousreactance which
is computed by the slip test. And,Xd is the true direct synchronous
reactance of RN.
Then, X d s will be
(34)
Changing the above equation into a more suitableform.
Where, K is the error factor, and can be writtenas
K= d 14 - (Rs/Xd)
'(36)
Fig. 11 shows the variation of K with (Rs/Xd).Differentiating K
with respect to (R,/Xd) to in-
vestigate the characteristics of the curve in Fig.11.
d K
d ( R s / X d )K ' =
K " = 1 / ( 1 f ( R s / x ~ )} 3'2> 0
K increases always with (Rs/Xd). WhenK
is dif-ferentiated twice with respect to (Rs/Xd), it canbe
obtained from K " that the curve of Fig. 11 in-creases rapidly with
(R~/xd).
6
Fig. 10 shows the approximate d- axis equivalentcircuit of RI at
power angle 6=0.
4K
Fig. 10.of RM.
Approximate d-axis equivalent circuit
2
Fig. 11. Variation of K with (Rs/xd).
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B. Precise Slip Test 1 A -Precise s l i p t e s tB - s l i p t e
s t
The authors carried out the experiment accordingto the results
of the precise slip test which wasdescribed in Chap. NI. Further,
the authors carriedout the conventional slip test which is based
onthe following assumption; when the rotor i s in thedirect and
also in the quadrature axis position, hecurrents drawn by the
machine are a minimum and amaximum respectively.
Fig. 9 shows the circuit for measuring Xd and xq.The test motor
is rated 1.0 w, 200 volts, 50 Hz.The additional resistance Rq was
connected with thecircuit of stator winding to investigate the
effectof the stator winding resistance R 3,and the experi-ments
were carried out by changing the values of R q ,
Fig. 12 shows these results. Test results of Xd
and Xq obtained by these methods for the test ma-chine are
plotted as function 01 ( R 3 t R s ) in Fig. 12.Fig. 12 shows a
comparison of the calculated valuesof Xd and xq with the test
results obtained by twodifferent tests. An increase in RE I results
in anequivalent increase in Xd and xq in the slip test,and the
values are almost always constant in theprecise slip test.
Therefore, the precise slip test confirms the va-lidity of a
measuring method for Xd and xq.
V . CONCLUSIONS
A new measuring method of x d and Xq which took
account of the stator winding resistance was pro-posed by the
authors. The authors carried out theexperiments according to this
measuring method andexamined the results. The method of
calculatingthe reactances gives results which agree well
withmeasured values of unsaturated state.
The method described in this paper for themeasurement of
salient-pole machine reactances hasmore advantages than the slip
test, and the ex-periments of this method are widely acceptable
andis easier in comparison with other methods.
ACKNOWLEDGMENT
2 0X d
t
Fig. 12. Comparison of the calculated values ofXd and Xq with
the test resultsobtained by two different tests.
his helpful advice and the ldachine Shop Group ofYamanashi
University for producing experimentaldevices.
REFERENCES
P. . Lawrenson, and 1. A. Agu, " Theory and performance
ofpoly-phace reluctance machines, P r m . EE, vol. 3, pp.
1435-1445, Aug 1964.V.B. Honsinger, " The inductance L d and Lg of
reluctancemachines, IEEE, vol. PAS-90, pp. 298-304, Jan/Feb
1971.R.W. Henzies, R.1. Nathur. and H.1. Lee, a Theory and
oper-
ation of reluctance motors with magnetically anistro picrotors,
U -synchronous performance, IEEE, vol. PAS-91, pp.42-45, Jan/Feb
1972.E. A.Klingshirn, " DC standstill torque used to measure Lgof
reluctance and synchronous machines, IEEE, vol. PAS-97,pp.
1862-1869, Sep/Oct 1978.A.A. Fock. and P.M.Hart, " New method for
measuring xd andXg based on the P-Q diagram of the lossy salient
-polemachine," PIEE, Pt.B. vol. 131. pp.259-262. 1984T. Fukao, "
Principles and output characteristics of superhigh-speed reluctance
generator system, IEEE. vol. IA-22.
G. Kron, " The application of tensors to the analysi s
ofrotating electri cal machinery, Gen. Elec. Rev. Aug 1935.May
1938.T. Takeuchi, " Matrix theory of electrical machinery," The
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OHM-SHA LTD., 1967.The authors would like to thank Dr. H. Kazuno
for
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