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Prediction of the transient performance of induction machines
Electric Power Systems Research, 10 (1986) 241 - 246
241
Prediction o f the Transient Performance of Induction Machines W. C. LIN*, C. E. LIN*, C. L. HUANG*, S. L. CHEN t and Y. T. WANG*
*Department of Electrical Engineering, National Cheng-Kung University, Tainan (Taiwan) t Departrnent of Electrical Engineering, National Tsing-Hua University, Hsinchu (Taiwan) (Received May 29, 1986)
SUMMARY
This paper demonstrates an effective method to predict the transient performance o f induction machines. In the proposed method, a simplified transformation between the conventional steady-state equivalent-circuit model and the phase-variable or d-q models is derived. From the conventional steady-state equivalent-circuit model, data obtained from short-circuit and open-circuit tests can offer sufficient information for the purpose o f transient prediction. The transient response o f induction machines can therefore be predicted by the phase-variable or d-q models through this transformation. From the results obtained, the simulated transient response is shown to be in close agreement with that from experiments. It is concluded that the proposed technique is convenient for data acquisition and can offer sufficient accuracy for transient prediction.
INTRODUCTION
The investigation of the transient behaviour of three-phase induction machines has been an attractive subject [ 1 - 4] for a long time owing to their practical applications in industry. For prediction of the transient phenomena, appropriate modelling of the induction machines is very important. Several models have been well developed, such as the phase-variable and d-q models. In the literature, it was not usually clearly specified which methods were used to obtain the parameters for establishing the models [1, 3, 5]. Some sophisticated computations from the design data or tedious experiments have been adopted. This paper is therefore an attempt to provide a theoretical deriva0378-7796/86/$3.50
tion of the relationship between the transient models and the conventional steady-state equivalent-circuit model. Through the derivations, a direct transformation table is obtained showing the relationship between these two sets of parameters. Short-circuit and open-circuit tests provide some important data with which to establish the conventional equivalent-circuit model of the induction machine. This model has a certain accuracy if the experiments and calculations are properly arranged. By using the transformation, prediction of the transient performance is obtained. In this paper, the theoretical derivation and experiments are described. Comparisons between direct transient measurement and the simulation have resulted in very satisfactory agreement.
FO RMU LA TI O N OF THE T R A N S F O R M A T I O N
The phase-variable and d-q models For transient analysis of three-phase induction motors, the phase-variable model and the two-axis d-q model are discussed as being the two major models. These transient models are mainly based on descriptions of the voltage balance of the machine circuit and the s p e e d - t o r q u e relationship. The voltage balance equations are generally written in matrix form as IV] = [ R ] [ I ] + [LIP[I] + [G][I]PO
(1)
In the phase-variable model, machine equations are written using a pair of magnetically coupled three-phase circuits. Accordingly, eqn. (1) is interpreted as [V] = [V~
V~b V~
V~
V~b V~] T
[I]----lisa [sb Isc Ira Jrrb Irc]T
(2a)
(2b)
© Elsevier Sequoia/Printed in The Netherlands
242 [R] =diag[R s Rs Rs Rr Rr Rr] Msm Ms~]
LSS
[LS]
=
(2c)
~m L~s
MsmJ
(2d)
~mMsm L~
MMCOSOr cos(Or - ~) cos(O~ + a)
[LSR] =
M cos(0, + a)
M cos Or M cos(Or -- ~)
[
L= Mrm Mrm1 [LR] = Mrm Lrt Mr~
M cos(Or - - a ) l M cos(Or + a) M cos Or
(~'f)
:/
(2e)
Lsq
L~.I
M~m Mrm L= J
L L~rl
[LSI [LSRI ]
L~o~
o :o o o
In the d-q model, the actual machine windings are represented by hypothetical windings along a pair of axes in a chosen twodimensional reference frame. The reference frame can be rotated at any speed or be fixed with respect to the stator or the rotor. Equation (1) can be expressed by a fixed reference frame with respect to the stator: [ V] = [ Vsd
Vsq
Vrd
(3a)
Vr q ] W
[I] = [lsd Isq Ird Irq] w
(3b)
[R] =diag[Rs Rs Rr Rr]
(3c)
Rs + LllP I
__q
q
Lm [¢o,L~
--¢°rLm Lm
i R, + L22P i I ¢o~L22
o] o/
/---:L-----:L I LLsr
lLs,
(3e)
J
Derivations of the per-phase equivalent circuit Either the phase-variable representation in eqn. (2) or the d-q axis representation in eqn. (3) can be used in the derivations of the per-phase equivalent circuit. The d-q axis model is chosen for simplicity in the following discussion. Under the circumstances of a uniform air gap and balanced winding configuration on both stator and rotor circuits, the voltage balanced equation for the d-q model can be rearranged in a compact form as
LmP
|
--¢0rL22 / + R~ L2:PJ
Ilsd
1
/22_
(4)
Irq
where L H = Lsd = Lsq, L22 = Lra = Lrq, and Lm = L=. The torque equation is
irq - - ird isq)/ 2 In the preceding derivations, the following definitions are used:
Te = P L m ( isd
L11= L~s + Ms
(5) (6a)
243
L22 = L ~ + Mr
(6b)
Vsq = - - j V s d
Ms = N2M~
(6c)
V',d
(6d)
8
Lm = ~
= M J N = NMr
where L~s, L~r are the stator and the rotor leakage inductances; Ms and Mr are the mutual inductances of the stator and rotor; and N and Lm are the effective turns ratio and the mutual inductance between the stator winding and the rotor winding respectively. By separating the serf-inductances into the leakage and the mutual components, and referring the rotor windings to the stator windings, the voltage relations of eqn. (4) become
(lOb)
-0 _
S I ~ + jcofL~rI'rd + jwfMs(Isd
V,q _ 0 = S
I~)
(10c) (10d)
--j V~d S
'
+
Hence, eqn. (10) can be depicted as in Fig. 1, which is the conventional steady-state equivalent circuit of induction machines. The reactances X~s, XQr, and X m are defined as
X~s = ~ofL~s
Vsd = Rsisd + L~sPisd + Ms(Pisd + Pi 'rd)
(7a)
Vsq = Rsisq + L~sPisq + Ms(Pisq + Pi'rq)
(7b)
X~r = colLar X m = corm s
0 = R'ri'rd + (L~r + Ms)Pi'rd + MsPisd .P
- - vgrMsisq - - C0r(L~r + O
t .t
t
= R ~ rq + (Let +
Ms)Pt
.?
Ms)P/rq
r. +
(7c)
r
x S
X' ir
IS
MsPisq
t
.t
+ ¢o~M~isd + w~(LQr + M~)Pl rd
(7d)
I
I'
sd
The prime notation is used to denote that these equations are referred to the stator windings. By similar substitutions in the torque expression, eqn. (5) yields Te = PMs( isd "lrq
-- i
'rd isq)/2
Re(Vsd) =
R e [ V exp(jco~t)]
= V cos(colt)
(9a)
Vsq = Re[Vsd exp(--90°)] = V sin(cdtt)
(9b)
isd = Re(Isd) = I cos(c0t t -- ~)
(9C)
isq = Re[Isd e x p ( - - 9 0 ° ) ]
x
r'/s m
(8)
Let the three-phase stator windings be supplied with a three-phase, positive-sequence, symmetrical, sine-wave voltage source. The d-q quantities can easily be obtained by Park's transformation. In the steady-state condition, the voltage and current phasors are functions of time. These instantaneous values may be expressed as /)sd =
V
rd
Fig. 1. S t e a d y - s t a t e e q u i v a l e n t circuit o f i n d u c t i o n motors.
Relationships b e t w e e n parameters o f different models According to the relations derived above, the phase-variable and d-q model parameters can be acquired from the steady-state equivalent circuit. Once the parameters in the steady-state equivalent circuit are obtained, by using Table 1, only a direct transformation is required to reach the transient model.
= I sin(colt - - ~b) ( 9 d )
By substituting eqn. (9) into eqn. (8) and letting cor = ( 1 - - S ) c o t , the phasor representation of the voltage equation is V~d = RsIsd + jcofL~I~d + jcofM~(I~d + I ' d )
(10a)
EXPERIMENTS
AND RESULTS
The open-circuit and short-circuit tests on a three-phase, 220 V, 1.5 kW, four-pole induction m o t o r were undertaken to obtain the steady-state equivalent-circuit param-
244 TABLE 1 T r a n s f o r m a t i o n t a b l e : p a r a m e t e r s are r e f e r r e d t o e l e m e n t s o f t h e s t e a d y - s t a t e e q u i v a l e n t c i r c u i t Coefficient
Phase m o d e l
d-q m o d e l
Stator and rotor resistance
Rs, R r
Rs, R r
2 Lss = L~s + ~ M s
S t a t o r w i n d i n g selfinductance
L11 = Lss - - M s m = L~s + M s
_ 1 M s m - - - -sMs
Mutual inductance between stator windings
0
2 L r r = L~r + ~ M s
R o t o r w i n d i n g selfinductance Mutual inductance between rotor windings Mutual inductance between rotor and stator windings
L22 = L r r - - M r m = L~r + M s
1 M r m = - - ~sMs
0
2 M = ~M s
Lm = Ms
TABLE 2 Parameters of the i n d u c t i o n m o t o r u n d e r test Parameter
Rs
Rr
X~s
Xr r
Xm
I m p e d a n c e (~'/)
1.746
0.905
1.825
1.825
37.55
eters. These parameters are calculated and shown in Table 2. Two kinds of experiments were conducted on this induction m o t o r to verify the validity of the proposed method. The first test was the start-up operation under balanced sinusoidal voltage supply. The second test was to feed the m o t o r with a six-step voltage source, which is the usual inverter waveform as shown in Fig. 2. Voltage, current, and angular speed waveforms were recorded in the experiments. Simulated results using the parameters obtained were also plotted. Figures 3 and 4 show the waveforms obtained from the experiments and from the simulation. In Fig. 3, no evident discrepancies are observed in the current waveform. The deviation in the angular speed is due mainly to the deadzone effect of the tachometer. In Fig. 4, the two current waveforms appear to differ in magnitude. During the derivations of eqn. (9), the voltage and current phasors were
assumed to be sine wave by neglecting the harmonic components. Actually, power supply voltage may consist of several harmonic components due to the solid-state switching devices. These voltage harmonics may have a considerable influence on the induced currents, but do not effect the driving torque of the machine. Since only the (1 + 6h)th
E- 12°°--)
-V V
60 °
J
Fig. 2. I n v e r t e r v o l t a g e p o w e r s u p p l y w a v e f o r m .
245
current, k = 1, 2, ..., will contribute to the rotation o f the machine, the variations of the motor speed will not be affected. It can be verified that the start-up t~nes are almost the same both in the experimental and in the simulated results, as shown in Fig. 4.
ON
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(b) 1.0
ON
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(b)
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.
60
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(c)
@ , e 300 36G Time
0.4
30
60
90
120
(c)
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Fig. 3. Waveforms obtained from experiment: (a) s t a t o r voltage, ( b ) s t a t o r c u r r e n t , (c) r o t o r angular v e l o c i t y ; w a v e f o r m s o b t a i n e d b y s i m u l a t i o n : (d) s t a t o r voltage, (e) s t a t o r c u r r e n t , (f) r o t o r angular velocity.
(f)
12o
18o
240
3oo
360 Time
Fig. 4. Waveforms o b t a i n e d f r o m e x p e r i m e n t w h e n f e d b y a n i n v e r t e r v o l t a g e p o w e r s u p p l y : (a) s t a t o r voltage, (b) s t a t o r c u r r e n t , (c) r o t o r angular veloci t y ; w a v e f o r m s o b t a i n e d b y s i m u l a t i o n : (d) s t a t o r voltage, (e) s t a t o r c u r r e n t , (f) r o t o r angular velocity.
246 CONCLUSION
NOMENCLATURE
This paper has presented the well-known phase-variable model, the d-q axis model, and the steady-state equivalent-circuit model for transient performance prediction. The correlations of the parameters in the various models have been clearly noted: Such relationships between the d-q model and the steady-state equivalent-circuit model not only give a clearer insight into the models b u t also provide a simplified approach to parameter acquisition in induction m o t o r simulations. In addition, this simplifies the problem for those control engineers w h o are not familiar with these models. When comparing the simulated results with the experiments, the start-up times of both cases are essentially the same. The current waveforms in Fig. 3 have proved the effectiveness of the method. While supplied with an inverter voltage source, the simulated current waveform appears to differ a little from the experimental result in magnitude. To improve the accuracy, a pure sine-wave voltage supply at different frequencies may be utilized in the open-circuit and short-circuit tests for obtaining the additional information. Generally, it is concluded that the proposed m e t h o d is suitable for modelling transformation. Prediction by the proposed m e t h o d has proved to be a very convenient process, giving reliable results. It is suitable for investigators of different background to use for studies of the transient performance of induction machines.
All the resistances and inductances listed below are per-phase values rotor self-inductances stator self-inductances mutual inductance between stator Lsr and rotor L~, L~ self-inductances in stator and rotor M mutual inductance between stator and rotor Msm, Mrm mutual inductances in stator and rotor rotor resistances in d-q axis Rrd, Rrq R~, Rr resistances in stator and rotor stator resistances in d-q axis Rs(l, Rsq Lrd, Lrq
Lsd, Lsq
Or
angular position of rotor
REFERENCES 1 P. C. Krause, Simulation of symmetrical induction machinery, IEEE Trans., PAS-84 (1965) 1038 - 1053. 2 I. R. Smith and S. Sriharam, Induction motor reswitching transients, Proc. Inst. Electr. Eng., 114 (1967). 3 A. K. Sarka and G. J. Berg, Digital simulation of three-phase induction motors, IEEE Trans., PAS89 (1970) 1031 - 1037. 4 G. Nath and G. J. Berg, Transient analysis of three-phase SCR controlled induction motors, IEEE Trans., IA-17 (1981) 133 - 142. 5 P. K. Kovacs, Transient Phenomena in Electric Machines, Elsevier, New York, 1984.
241
Prediction o f the Transient Performance of Induction Machines W. C. LIN*, C. E. LIN*, C. L. HUANG*, S. L. CHEN t and Y. T. WANG*
*Department of Electrical Engineering, National Cheng-Kung University, Tainan (Taiwan) t Departrnent of Electrical Engineering, National Tsing-Hua University, Hsinchu (Taiwan) (Received May 29, 1986)
SUMMARY
This paper demonstrates an effective method to predict the transient performance o f induction machines. In the proposed method, a simplified transformation between the conventional steady-state equivalent-circuit model and the phase-variable or d-q models is derived. From the conventional steady-state equivalent-circuit model, data obtained from short-circuit and open-circuit tests can offer sufficient information for the purpose o f transient prediction. The transient response o f induction machines can therefore be predicted by the phase-variable or d-q models through this transformation. From the results obtained, the simulated transient response is shown to be in close agreement with that from experiments. It is concluded that the proposed technique is convenient for data acquisition and can offer sufficient accuracy for transient prediction.
INTRODUCTION
The investigation of the transient behaviour of three-phase induction machines has been an attractive subject [ 1 - 4] for a long time owing to their practical applications in industry. For prediction of the transient phenomena, appropriate modelling of the induction machines is very important. Several models have been well developed, such as the phase-variable and d-q models. In the literature, it was not usually clearly specified which methods were used to obtain the parameters for establishing the models [1, 3, 5]. Some sophisticated computations from the design data or tedious experiments have been adopted. This paper is therefore an attempt to provide a theoretical deriva0378-7796/86/$3.50
tion of the relationship between the transient models and the conventional steady-state equivalent-circuit model. Through the derivations, a direct transformation table is obtained showing the relationship between these two sets of parameters. Short-circuit and open-circuit tests provide some important data with which to establish the conventional equivalent-circuit model of the induction machine. This model has a certain accuracy if the experiments and calculations are properly arranged. By using the transformation, prediction of the transient performance is obtained. In this paper, the theoretical derivation and experiments are described. Comparisons between direct transient measurement and the simulation have resulted in very satisfactory agreement.
FO RMU LA TI O N OF THE T R A N S F O R M A T I O N
The phase-variable and d-q models For transient analysis of three-phase induction motors, the phase-variable model and the two-axis d-q model are discussed as being the two major models. These transient models are mainly based on descriptions of the voltage balance of the machine circuit and the s p e e d - t o r q u e relationship. The voltage balance equations are generally written in matrix form as IV] = [ R ] [ I ] + [LIP[I] + [G][I]PO
(1)
In the phase-variable model, machine equations are written using a pair of magnetically coupled three-phase circuits. Accordingly, eqn. (1) is interpreted as [V] = [V~
V~b V~
V~
V~b V~] T
[I]----lisa [sb Isc Ira Jrrb Irc]T
(2a)
(2b)
© Elsevier Sequoia/Printed in The Netherlands
242 [R] =diag[R s Rs Rs Rr Rr Rr] Msm Ms~]
LSS
[LS]
=
(2c)
~m L~s
MsmJ
(2d)
~mMsm L~
MMCOSOr cos(Or - ~) cos(O~ + a)
[LSR] =
M cos(0, + a)
M cos Or M cos(Or -- ~)
[
L= Mrm Mrm1 [LR] = Mrm Lrt Mr~
M cos(Or - - a ) l M cos(Or + a) M cos Or
(~'f)
:/
(2e)
Lsq
L~.I
M~m Mrm L= J
L L~rl
[LSI [LSRI ]
L~o~
o :o o o
In the d-q model, the actual machine windings are represented by hypothetical windings along a pair of axes in a chosen twodimensional reference frame. The reference frame can be rotated at any speed or be fixed with respect to the stator or the rotor. Equation (1) can be expressed by a fixed reference frame with respect to the stator: [ V] = [ Vsd
Vsq
Vrd
(3a)
Vr q ] W
[I] = [lsd Isq Ird Irq] w
(3b)
[R] =diag[Rs Rs Rr Rr]
(3c)
Rs + LllP I
__q
q
Lm [¢o,L~
--¢°rLm Lm
i R, + L22P i I ¢o~L22
o] o/
/---:L-----:L I LLsr
lLs,
(3e)
J
Derivations of the per-phase equivalent circuit Either the phase-variable representation in eqn. (2) or the d-q axis representation in eqn. (3) can be used in the derivations of the per-phase equivalent circuit. The d-q axis model is chosen for simplicity in the following discussion. Under the circumstances of a uniform air gap and balanced winding configuration on both stator and rotor circuits, the voltage balanced equation for the d-q model can be rearranged in a compact form as
LmP
|
--¢0rL22 / + R~ L2:PJ
Ilsd
1
/22_
(4)
Irq
where L H = Lsd = Lsq, L22 = Lra = Lrq, and Lm = L=. The torque equation is
irq - - ird isq)/ 2 In the preceding derivations, the following definitions are used:
Te = P L m ( isd
L11= L~s + Ms
(5) (6a)
243
L22 = L ~ + Mr
(6b)
Vsq = - - j V s d
Ms = N2M~
(6c)
V',d
(6d)
8
Lm = ~
= M J N = NMr
where L~s, L~r are the stator and the rotor leakage inductances; Ms and Mr are the mutual inductances of the stator and rotor; and N and Lm are the effective turns ratio and the mutual inductance between the stator winding and the rotor winding respectively. By separating the serf-inductances into the leakage and the mutual components, and referring the rotor windings to the stator windings, the voltage relations of eqn. (4) become
(lOb)
-0 _
S I ~ + jcofL~rI'rd + jwfMs(Isd
V,q _ 0 = S
I~)
(10c) (10d)
--j V~d S
'
+
Hence, eqn. (10) can be depicted as in Fig. 1, which is the conventional steady-state equivalent circuit of induction machines. The reactances X~s, XQr, and X m are defined as
X~s = ~ofL~s
Vsd = Rsisd + L~sPisd + Ms(Pisd + Pi 'rd)
(7a)
Vsq = Rsisq + L~sPisq + Ms(Pisq + Pi'rq)
(7b)
X~r = colLar X m = corm s
0 = R'ri'rd + (L~r + Ms)Pi'rd + MsPisd .P
- - vgrMsisq - - C0r(L~r + O
t .t
t
= R ~ rq + (Let +
Ms)Pt
.?
Ms)P/rq
r. +
(7c)
r
x S
X' ir
IS
MsPisq
t
.t
+ ¢o~M~isd + w~(LQr + M~)Pl rd
(7d)
I
I'
sd
The prime notation is used to denote that these equations are referred to the stator windings. By similar substitutions in the torque expression, eqn. (5) yields Te = PMs( isd "lrq
-- i
'rd isq)/2
Re(Vsd) =
R e [ V exp(jco~t)]
= V cos(colt)
(9a)
Vsq = Re[Vsd exp(--90°)] = V sin(cdtt)
(9b)
isd = Re(Isd) = I cos(c0t t -- ~)
(9C)
isq = Re[Isd e x p ( - - 9 0 ° ) ]
x
r'/s m
(8)
Let the three-phase stator windings be supplied with a three-phase, positive-sequence, symmetrical, sine-wave voltage source. The d-q quantities can easily be obtained by Park's transformation. In the steady-state condition, the voltage and current phasors are functions of time. These instantaneous values may be expressed as /)sd =
V
rd
Fig. 1. S t e a d y - s t a t e e q u i v a l e n t circuit o f i n d u c t i o n motors.
Relationships b e t w e e n parameters o f different models According to the relations derived above, the phase-variable and d-q model parameters can be acquired from the steady-state equivalent circuit. Once the parameters in the steady-state equivalent circuit are obtained, by using Table 1, only a direct transformation is required to reach the transient model.
= I sin(colt - - ~b) ( 9 d )
By substituting eqn. (9) into eqn. (8) and letting cor = ( 1 - - S ) c o t , the phasor representation of the voltage equation is V~d = RsIsd + jcofL~I~d + jcofM~(I~d + I ' d )
(10a)
EXPERIMENTS
AND RESULTS
The open-circuit and short-circuit tests on a three-phase, 220 V, 1.5 kW, four-pole induction m o t o r were undertaken to obtain the steady-state equivalent-circuit param-
244 TABLE 1 T r a n s f o r m a t i o n t a b l e : p a r a m e t e r s are r e f e r r e d t o e l e m e n t s o f t h e s t e a d y - s t a t e e q u i v a l e n t c i r c u i t Coefficient
Phase m o d e l
d-q m o d e l
Stator and rotor resistance
Rs, R r
Rs, R r
2 Lss = L~s + ~ M s
S t a t o r w i n d i n g selfinductance
L11 = Lss - - M s m = L~s + M s
_ 1 M s m - - - -sMs
Mutual inductance between stator windings
0
2 L r r = L~r + ~ M s
R o t o r w i n d i n g selfinductance Mutual inductance between rotor windings Mutual inductance between rotor and stator windings
L22 = L r r - - M r m = L~r + M s
1 M r m = - - ~sMs
0
2 M = ~M s
Lm = Ms
TABLE 2 Parameters of the i n d u c t i o n m o t o r u n d e r test Parameter
Rs
Rr
X~s
Xr r
Xm
I m p e d a n c e (~'/)
1.746
0.905
1.825
1.825
37.55
eters. These parameters are calculated and shown in Table 2. Two kinds of experiments were conducted on this induction m o t o r to verify the validity of the proposed method. The first test was the start-up operation under balanced sinusoidal voltage supply. The second test was to feed the m o t o r with a six-step voltage source, which is the usual inverter waveform as shown in Fig. 2. Voltage, current, and angular speed waveforms were recorded in the experiments. Simulated results using the parameters obtained were also plotted. Figures 3 and 4 show the waveforms obtained from the experiments and from the simulation. In Fig. 3, no evident discrepancies are observed in the current waveform. The deviation in the angular speed is due mainly to the deadzone effect of the tachometer. In Fig. 4, the two current waveforms appear to differ in magnitude. During the derivations of eqn. (9), the voltage and current phasors were
assumed to be sine wave by neglecting the harmonic components. Actually, power supply voltage may consist of several harmonic components due to the solid-state switching devices. These voltage harmonics may have a considerable influence on the induced currents, but do not effect the driving torque of the machine. Since only the (1 + 6h)th
E- 12°°--)
-V V
60 °
J
Fig. 2. I n v e r t e r v o l t a g e p o w e r s u p p l y w a v e f o r m .
245
current, k = 1, 2, ..., will contribute to the rotation o f the machine, the variations of the motor speed will not be affected. It can be verified that the start-up t~nes are almost the same both in the experimental and in the simulated results, as shown in Fig. 4.
ON
(a) 6.0
O. (a) -6.0
(b) 1.0
ON
.75 .50
.25
(b)
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e 180
i 240
(c)
@ , e 300 36G Time
0.4
30
60
90
120
(c)
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(d)
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90
120
Fig. 3. Waveforms obtained from experiment: (a) s t a t o r voltage, ( b ) s t a t o r c u r r e n t , (c) r o t o r angular v e l o c i t y ; w a v e f o r m s o b t a i n e d b y s i m u l a t i o n : (d) s t a t o r voltage, (e) s t a t o r c u r r e n t , (f) r o t o r angular velocity.
(f)
12o
18o
240
3oo
360 Time
Fig. 4. Waveforms o b t a i n e d f r o m e x p e r i m e n t w h e n f e d b y a n i n v e r t e r v o l t a g e p o w e r s u p p l y : (a) s t a t o r voltage, (b) s t a t o r c u r r e n t , (c) r o t o r angular veloci t y ; w a v e f o r m s o b t a i n e d b y s i m u l a t i o n : (d) s t a t o r voltage, (e) s t a t o r c u r r e n t , (f) r o t o r angular velocity.
246 CONCLUSION
NOMENCLATURE
This paper has presented the well-known phase-variable model, the d-q axis model, and the steady-state equivalent-circuit model for transient performance prediction. The correlations of the parameters in the various models have been clearly noted: Such relationships between the d-q model and the steady-state equivalent-circuit model not only give a clearer insight into the models b u t also provide a simplified approach to parameter acquisition in induction m o t o r simulations. In addition, this simplifies the problem for those control engineers w h o are not familiar with these models. When comparing the simulated results with the experiments, the start-up times of both cases are essentially the same. The current waveforms in Fig. 3 have proved the effectiveness of the method. While supplied with an inverter voltage source, the simulated current waveform appears to differ a little from the experimental result in magnitude. To improve the accuracy, a pure sine-wave voltage supply at different frequencies may be utilized in the open-circuit and short-circuit tests for obtaining the additional information. Generally, it is concluded that the proposed m e t h o d is suitable for modelling transformation. Prediction by the proposed m e t h o d has proved to be a very convenient process, giving reliable results. It is suitable for investigators of different background to use for studies of the transient performance of induction machines.
All the resistances and inductances listed below are per-phase values rotor self-inductances stator self-inductances mutual inductance between stator Lsr and rotor L~, L~ self-inductances in stator and rotor M mutual inductance between stator and rotor Msm, Mrm mutual inductances in stator and rotor rotor resistances in d-q axis Rrd, Rrq R~, Rr resistances in stator and rotor stator resistances in d-q axis Rs(l, Rsq Lrd, Lrq
Lsd, Lsq
Or
angular position of rotor
REFERENCES 1 P. C. Krause, Simulation of symmetrical induction machinery, IEEE Trans., PAS-84 (1965) 1038 - 1053. 2 I. R. Smith and S. Sriharam, Induction motor reswitching transients, Proc. Inst. Electr. Eng., 114 (1967). 3 A. K. Sarka and G. J. Berg, Digital simulation of three-phase induction motors, IEEE Trans., PAS89 (1970) 1031 - 1037. 4 G. Nath and G. J. Berg, Transient analysis of three-phase SCR controlled induction motors, IEEE Trans., IA-17 (1981) 133 - 142. 5 P. K. Kovacs, Transient Phenomena in Electric Machines, Elsevier, New York, 1984.