reference frame

8/9/2019 Reference Frame

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Electric Power Systems Research,

8 (1984/85) 15

-

26

I

t... ...

15

D, Q Reference Frames for the Simulation of Induction Motors

R. J. LEE, P. PILLAY and R. G. HARLEY

Department of Electrical Engineering, University of Natal, King George V Avenue, Durban 4001 South Africa)

(Received April 9, 1984)

SUMMARY

This paper presents the equations of three

preferable reference frames for use in the

simulation of induction machines when using

the d, q 2-axis theory. It uses case studies to

demonstrate that the choice of the reference

frame depends on the problem to be solved

and the type of computer available analog or

digital).

1. INTRODUCTION

reference frames are most frequently used,

the particular reference frame should be

chosen in relation to the problem being inves-

tigated and the type of computer (analog or

digital) that is used. The purpose of this paper

is therefore to provide guidelines (by means

of case studies) in order to choose the most

suitable reference frame. It also shows that

there are instances when the rotor reference

frame, which appears to have been avoided by

most authors, is the best choice.

2. THEORY


2/12

+d axis

rotor phase

A axis

stator phase

A axis

Fig. 1. D, Q axes superimposed onto a three-phase induction motor.

+q axis


3/12


4/12

field of feedback controller design when the

motor equations are linearized around a

steady operating point. It is also possible to

use a larger step length in the digital integ-

ration routine when using this frame, since

the variables are slowly changing DC quanti-

ties during transient conditions.

If there are compensating capacitors con-

nected to the motor, then for the case when

only series capacitors are present (Appendix

B), the terminal voltage eqns. (15) and (16)

become:

Vds =

V

m cos 1

-

Vd

-

RLids

- LLPids

-

0:lsLLiqs

(29)

Vqs =-

Vm sin -y- v~- RLiqs- LUJiqs

+ 0:lsLLids

(30)

The voltage components

Vd

and

v~

across the

series capacitor are found by integrating the

following two differential equations:

PVd

= ids/Cs - wsv~ (31)

pv~ =

iqs/Cs+

0:lsVd

(32)

When there are shunt capacitors connected

across the terminals of the motor (without

series capacitors), eqns. (19)

-

(22) become

Vqr

=

Rriqr + pl/Jqr

(40d)

Following the same procedure as for the

stationary reference frame, the state space

equation for simulation on a digital computer

IS

p[i]

=

[B]{[v]

-

[R][i]

- wr[Gr][i]}

(41)

where the [v] and [i] vectors are the same as

in eqns. (10) and (11), and the other matrices

appear in Appendix A.

Without any capacitors, the terminal volt-

age eqn. (14) becomes

Vds

=

Vm cos S0:lst + 1)

Vqs=- Vm sin(swst + 1)

(42)

(43)

The d,q voltages are therefore of slip fre-

quency and the d-axis rotor current behaves

exactly as the phase A rotor current does.

This means that it is not necessary to com-

pute the phase A rotor current at each step of

the digital integration process through the in-

verse of Park s transform. This saves computer

time and hence is an advantage of the rotor

reference frame when studying rotor quanti-

ties.

If there are compensating capacitors con-

nected to the motor, then for the case when


5/12

Thus far, the voltage equations for three

reference frames have been presented with

their advantages. The electrical torque is given

by the following expression which is indepen-

dent of the reference frame:

Te

=

wbLm idIiqs

-

iqricts)/2

The mechanical motion is described by

PWr =

Te

-

Td J

52)

53)

3. RESULTS

In order to evaluate the usefulness of each

frame of reference, the equations of § 2 are

used to predict the transient behaviour of a

22 kW motor; its parameters appear in

Appendix C.

3.1. Frames of reference

Figure 2 shows the predicted start-up

characteristics against no load, using each one

of the above three reference frames.

The electrical torque Fig. 2 b» exhibits

the familiar initial 50 Hz oscillations, and full

The calculated rotor d-axis variables of this

reference frame are, however, also trans-

formed at 50 Hz frequencies and are therefore

not the same as the actual rotor phase A

variables which are at slip frequency. Figure

3 c) and d) show this difference between

the rotor phase A current and the rotor DR

current.

It would therefore be an advantage when

studying transient occurrences involving the

stator variables to use this reference frame.

3.3. Rotor reference frame

Since in this reference frame the d-axis of

the reference frame is moving at the same

relative speed as the rotor phase A winding

and coincident with its axis, it should be

expected from the considerations of § 3.2

that the behaviour of the d-axis current and

the phase A current would be identical.

Figures 4 c) and d) show how the rotor

phase A current and the rotor d-axis current

are initially at 50 Hz, when the rotor is a

standstill, but gradually change to slip fre-

quency at normal running speed. If this

reference frame is used for the study of rotor


6/12

1.~

TERMINAL VOLTAGE

1... IN

(a)

2SIa. IN

Ia.

...

ffi -5.

0:

0:

::J

U

-11a.

I.B

SIZI.IIIiI

TIME

11Z111.1. I II J

(c)

1sa. I II/ J 2lll1a.RII I

milli-SEC

lSIJ.1N 21i'11 1.B

mi 11 i-SEC

251 1.IN

6.

TORQUE

:::i

n:

4.

t-

2SIa. R1 1a.IN 151 1. 1111 2IIIa. I II/J

(b)

mi 11 i-SEC

:::i

n:

1.

a

w

w

n.

(/)

Ia.

la.1iJIII sa. 111111

TIME

1&1111.N

lSla.1N 21111 1.RII I

(d)

mi

11 i

-SEC

2S1Z1.1 1

1. Ii JIII.

::3

n:

. 5111

Ia...

-.5111

(/)

...

-1.

Nj.J

a

>

-1.

5

V V

la.1N

sa. IN

TIME

DU/\ c /\ '1

1111.

:::i

n:

5.

W Ia.

::J

CJ

0:

a

...

-2.

B.IN

51 1.RII I

TIME

SPEED


7/12

~

0.:

-+

as

51 1.1 11 1

TIME

ll ll l.1 II I l51 1.1 11 1 21 11 1.1 I1 1

a) milli SEC

251 1. I II I

I-

Z 5.

W

It:

It:

::J

U

ar

ll l.

I I.I II I 5111.0111

TIME

11 11 1.1 11 1 l51 1.1 I1 I 21 11 1.1 11 1

c)

milli SEC

251 1.1 I1 1

D-AXIS STATOR CURRENT

~

0.:

11 1.

I I.I II I

51 1.1 I1 1

TIME

ids

ll ll l. I II I l51/J.

21 11 1 . I II I

b) milli SEC

251 1.1 I

11 1.

D-AXIS ROTOR CURRENT

~

0.:

5.

I I.

I-

Z -5.1 I1

W

It:

It:

::J

U

5IZI.1 I1 1

TIME

idr

ll ll l.1 II I l51 1.1 I1 I 21 11 1.1 I1 I

d) milli SEC

251 1.1 1

I-

z 5.

w

It:

It:

::J

U

ll l.

g.1 II I

[

ll l.

0.:

5.


8/12

::5

0:

...

z -5. 1111

UI

0::

0::

:J

U

Sill. 111111

TIME

i

as

1111111.//1//1

a)

15111.//1//1

2//1//1.//1//1

mi 11 i-SEC

25111.//1//1

1//I. //I

PHASE A ROTOR CURRENT

::5

0:

-1//1. //I

13.//1111 5//1.//1//1

TIME

i

ar

1//1111./1//1

c)

15//1.//1111 2111//1.//1//1

mi 11 i -SEC

25//1. //1111

11/1.


+--

::5

0:

...

Z -5. //I

w

0::

0::

:J

U

-1111.//I

13.//1//1 Sill.

TIME

I r

. I

~ds

I

T

i

I

t

I

I

I I

I

15 .111111 2111111.111 //1 25/ /1.111

mi IIi -SEC

1//10.//I .

b)


::5

0:

I I

idr

1 //1.111 15111. 2//1//1.//1111

d) mi 11 i-SEC

25//1.//1


9/12

PHASE A STATOR CURRENT

::5

0:

I-

ffi

-5. IIJI

0::

0::

;:

U

511J 1IJ11J

TIME

i

as

l11J0. IIJIIJ

a)

1511J.00 20 - IIJ0

mi 11 i-SEC

250.0B

1 .

PHASE A ROTOR CURRENT

::5

0:

I-

z -5.

w

0::

0::

;:)

U

50.00

TIME

i

ar


I-

Z

~ -8.11J

0::

;:)

U

ids

l11J0.1IJ11J

c)

15 . 00 20 - DB

mil Ii -SEC

250. BB

-1 -

e. BIIJ 50. 0 10m.IIJ0 15m.IIJ0 200. 00

TIME b) mi 1 Ii-SEC


::5

0:

I-

Z

W

0::

0::

;:)

U

50.011J

TIME

Fig. 5. Predicted starting up current using the synchronously rotating reference frame.

250.00

idr

1011J.00 150. 0 200.00

d milli-SEC

250.00


10/12

studies, since the input to the machine equa-

tions are constant prior to any disturbance.

Also, because the stator and rotor d-axis

variables vary slowly in this frame, a larger

integration step length may be used on a

digital computer, when compared to that used

with the other two reference frames, for the

same accuracy of results.

It is also preferable to use this reference

frame for stability of controller design where

the motor equations must be linearized about

an operating point, since in this frame the

steady state variables are constant and do not

vary sinusoidally with time.

4. CONCLUSIONS

This paper has taken. the three preferred

frames of reference within the d,q two-axis

theory and compared the results obtained

with each frame. From this comparison it is

concluded that:

a) when a single induction motor is being

studied, anyone of the three preferred frames

of reference can be used to predict transient

ration step length may be lengthened without

affecting the accuracy of results;

e) should a multimachine system be

studied then the advantages of the synchro-

nously rotating reference frame appear to out-

weigh those of the other two reference

frames.

KNOWLEDGEMENTS

The authors acknowledge the assistance of

D.C. Levy, R.C.S. Peplow and H.L. Nattrass

in the Digital Processes Laboratory of the

Department of Electronic Engineering and

M.A. Lahoud of the Department of Electrical

Engineering, University of Natal. They are

also grateful for financial support received

from the CSIR and the University of Natal.

NOMEN L TURE

C

shunt capacitor


11/12

R F R N S

1 P. C. Krause and C. H. Thomas, Simulation of

symmetrical induction machinery, IEEE Trans.,

PAS-84 1965 1038 -1053.

2 D. J. N. Limebeer and R. G. Harley, Subsynchro-

nous resonance of single cage induction motors,

Proc. Inst. Electr. Eng., Part B, 128 1981 33 -

42.

3 C. F. Landy, The prediction of transient torques

produced in induction motors due to rapid recon-

nection of the supply, Trans. S. Afr. Inst. Electr.

Eng., 63 1972 178 - 185.

4 P. Pillay and R. G. Harley, Comparison of models

for predicting disturbances caused by induction

motor starting, SAIEE Symposium on Power Sys-

tems Disturbances, Pretoria, September 1983,

SAIEE, Kelvin House, Johannesburg, pp. 11 - 18.

5 B. Adkins and R. G. Harley, The General Theory

of Alternating Current Machines: Application to

Practical Problems, Chapman and Hall, London,

1975, ISBN 0 412 12080 1.

where

Bs

= Lrr/D, Br = Lss/D

Bm= -Lm/D, D =Ls Lrr-Lm 2

APPENDIXA

Bs Bm

0

0

B

Br

0

0

[B

= I m

0 0

Bs

Bm

0 0

Bm Br

0 0 0 0

[Gs] =

I °

0

Lm Lrr

0

0 0 0

-Lm -Lrr

0

0

0 0

Lss Lm

0 0 0

0

[Gr]=1

-Lss -Lm

0 0

0 0 0 0


12/12

26

[UABcJc

= l/C

f[iABcJ dt

B-8

In the Odq reference frame the voltage drop

becomes

[Per1[UABcJc

= 1/C j[Per1[iOdq]

dt

Differentiation gives

[Per1p[UOdq] + p[Per1[UOdq]

=

1/C [Per1[iodq]

Multiplication by

Pe

gives

P[UOdq]

+

[Pe]p[Per1[UOdq]

= l/C [ iOdq]

Therefore

P[UOdq]

= l/C [iodq] - [Pe]p[Per1[UOdq]

Hence

PUd

=

id/C

- WUq

PUq

=

iq/C + WUd

B-9

B-1 0

B-ll

B-12

B-13

B-14

APPENDIX C

22 kW Induction motor parameters

Base time 1 s

Base speed 1 rade S-1

Base power 27.918kVA

Base stator voltage 220 V phase

Base stator current 42.3 A phase

Base stator impedance 5.21 S1

Base torque 177.8 N m

Number of poles 4

Rated output power 22 kW

Stator resistance

Rs

0.021 p.ll.

Rotor resistance

Rr

0.057 p.u.

Stator leakage 0.049 p.u. expressed

reactance Xs

Rotor leakage

reactance Xr

Magnetizing

reactance Xm

0.132 p.u. at 50 Hz

3.038 p.u.

reference frame

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