Static And Dynamic Models Of Im

STATIC AND DYNAMIC MODELS OF INDUCTION MOTORS M.M. Rayan Electrical Engineering Dept. Kuwait University Kuwait

A.M. Sharaf Electrical Engineering Dept. Univereity of New Prunswick Fredericton, NP Canada AP STRACT

The paper deals with the modelling of squirrel cage induction motor8 and deriving steady state and dynamic models that can be utilized in harmonic penetration studies and efficient operation of the induction motor as part of variable voltage variable frequency drive systems. INTRODUCTION Induction motors [l-41 are highly nonlinear highly coupled type load which poses tremendous challenge in control, as well as a nonlinear load in harmonic flow "penetration studies". The paper presents the results of laboratory simulation of motor behavior under variable voltage variable frequency operation. Statistical analysis were performed to assess correlation factors interrelating motor parameters to voltage, stator frequency and speed. Static models were derived for correlating input impedance, power, efficiency, power factor and load torque t o volt/HZ ratio (V /f ) and slip frequency fr. m s Following the statistical correlation, a novel steady state and transient circuit [ 51 was utilized t o investigate the dynamic input impedance characteristics of the induction machine and its dependency on machine parameters and speed. Two sizes of three phase induction motors were tested, the 1 HP and 7 HP motors. Parameters and nominal values are given in the Appendix 1. The derivation of the equivalent apparent dynamic input impedance is given in Appendix 2. Due to space, only results for the 1 HP are included in this paper. RESULTS The three phase induction motor were excited by variable voltage/variable frequency source under constant load conditions for the following cases: Case 1 Figure A1 depicts the test arrangement. 'in

-

30%, 50%, 60%, and 1102 each with

fs = (15, 30, 50, 70 and 90 HZ) and

load power set at: 100, 400, 600, 1000 volts

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Case 2 fS

=

"in

-

15, 30, 50, 70, 90 HZ each with (30%, 50%. 80%, 110%) and

load power set at: 100, 400, 600, 1000 watts Figures 2-12 depicts samples of static model dependency on volt/HZ ratio (Vm/fl) and rotor slip frequency fr for 1 HP 3 phase and single phase motors. Dynamic Apparent Impedance Starting from the classical equivalent T-circuit representation of the induction motor under dynamic condition [SI, a transformed "decoupled" equivalent circuit T-1 as shown in Figure A2 i s utilized to estimate the dynamic "apparent" input impedance (Zin). Appendix 2 depicts the full derivation of (Zin) in terms of machine parameters and speed wm

.

CONCLUSIONS

The paper presents the results of laboratory testing of three phase and single phase induction motors to estimate the static and dynamic models of induction motors. Such models are extremely useful In: 1. Identifying effective control strategies to enhance efficiency of variable voltage/variable frequency motor drives. 2. Assess machine performance with excitation voltage and frequency variations. 3. Identify nonlinear load impedance models of induction motor loads in harmonic penetration programs such as Cyme and Harmflo Software. These models were utilized also in deriving efficient open-loop control strategies based on optimized volt/HZ, (Vm/fs), ratio characteristics for minimum input power, minimum losses and maximum efficiency. REFERENCES Rahman and B . Jeyasurya, "Efficiency improvements in small horsepower single phase electric motors", CEA Report 433 U 492, April 1986. "Techniques for energy conservation in AC motor-drive systems", EPRI Project 1201-13, Prepared by the University of Minnesota, September 1981, pp. 2.1-2.16. S.E. Haque and W. Shepherd, "Thyristor controlled reactive power compensation for electricity supply systems", pp. 504-7, William Clowes & Sons Limited, London. P.J. Tsivitse and E.A. Klingshirn, "Optimal voltage and frequency for polyphase induction motors operating with variable frequency power supplies", IC-WED-2, 1970. "AC motors for high-performance applications", by Sakae Yamamura, Marcel-Dekker, 1986.

M.A.

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1-HP I.M.

frated = 50 Hz,

N r rated = 1400 qxn,

R1 = 1 3 . 3 oh/@. Xll

5 = L2 $:

l1 = l2 = 68 mH (leakage)

41 = 600 mH

ohn/phase (magnetizing)

= 668 mH (self)

'Ibrque base = 8.24

of poles=4

R2=5.81 W p h

= X12 = 21.38 oldphase

5 = 188.5

Pp=E30.

(mgnetizing)

Ifull load = 1 . 8 A = I b s e

Nm

= 240 V "base mtual impauctance between stator and rotm

i

R

Ir

I

> -

Fig.

1 Schematic diagram of the ac va riable frequency va.rtable voltage source.

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APPENDIX 2 Dynamic Equivalent C i r c u i t e q u i v a l e n t c i r c u i t expressed by e q u a t i o n s 151 w i t h

For t h e t r a n s i e n t T-I

a < 1 selected as: 3 . 2 Lm 3

a =

C

2

l2 +

1.0 and

Lm

0

* 2 R2 = a .R2

Lm = a Lm;

l1 = l1

(A2.1)

a i n modifier

3 +7 Lm

(l-a)

We o b t a i n t h e t r a n s i e n t T-1 Equivalent c i r c u i t of Figure e ' = j w ( 3 / 2 ) L ' i -(3 f 2 ) L ' m i ' 2 ) is speed back-emf (speed v o l t a g e ) s m m l

13 where

The g e n e r a l form of transformed dynamic model.

where, wm = r o t o r speed e l e c . r a d f s e c

a i2 = i 2 / a d dt

p =

8

(A2.3)

+ jw

= a

- I

f o r n o n - s i n u s o i d a l e x c i t a t i o n , and

(A2.4)

p = j w f o r s i n u s o i d a l e x c i t a t i o n , w = 2nf

(A2.5)

Where Z v a l u e s f o r n o n s i n u s o i d a l v o l t a g e f c u r r e n t e x c i t a t i o n a r e given by: Z l l = R1

3

+ (I1 + 2 Lm> P

RI + L1p

, L1 3

zl* = 2

(A2.6) =

s e l f s t a t o r inductance

(A2.7)

LmaP

3 ZZ1 = -j- Lm.(P-jWm) A22 = a

2

+a

2

(I2 +

(A2.8) 3 2 Lm)(P-jWm)

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(A2.9)

For sinusoidal excitation we put where w = 2nf

P = jw

Z l l =R1

+ jw(ll +

Z12 = jw z21 =

2

+

Lm)

R1

+

(A2.11)

jwL

. 3

2L

(A2.10)

(A2.12)

a

Lma (jw-jwm) =

3 2

(A2.13)

Lmajwr

and instant power output and torque are given by: Power/ph.

-

Real (e's).Real

Torque/ph.

Po1/(2 Wm/PP)

(1'2)

-

(A2.14)

Pol

, PP-1

of poles

(A2.15)

Impedance dynamic model General Expression for apparent as seen from motor terminals: from dynamic eqns.

(A2.16)

-221

I2 =

(A2.17)

[ X I

v1 = (zll-

z12 z 21I 2 221

(A2.18)

-

v1 z2 1 (zll- 212

so,

(A2.19)

For any given waveform generally (A2.20) then; n

(A2.21)

is a function of machine parameters, R1, R2, l1, 12, L , excitation, and rotor m speed w m

.

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For a given single frequency p = j w 3 ,L2=112+ 2

4’

(A2.23)

(A2.24)

(A2.25) where, w

r

-

(w

- wm)

= rotor s l i p frequency

Correlation matrix f o r a l l v a r i a b l e s : ( 1 HP 3-phase motor)

CO I* r e 1 a t. i on r m t.r i :.:: f cl r a 1 1 v a r i ab 1 es : ’# . ..#.& . .+. ....& f

Var i a t l l e

& :t::+::i:

Vrtt / f s Vrit

I rit

Zin

Fi 17

Po

XEFF

p.f .

Vm / f E Wit I ni

Zin Pin P6 XEFF p.f.

TL Nr

Variable

Writ

fs

fr

Vrii / f s Vm

Im Zin Pin PO XEFF p.f. TL Nr I15

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TL

i::*::i: NI-

Experimental Data for 1-hp 3-Phase I.M. VdfS

4.81 6.8 to40 4.00

4.n 4.00 4.00 1.44

1.40 3.83 t.40

3.83 S.28 3.83 4.28

3.83 S.28 1.72 t.74 3.77 to74

3.n

t.74 3.77

3.n

h,r

Sample Data frfir

?f

72.17 9S.80 W.17 126.00 187.60 130.00 126.60

72.17 120*00 191.70 120.00 191.70

263.80 191.70 263.80 191.70 263.80 126.00 191.70

243.80 ltOo70 213.~0 1T1.30

am

-

1.04 t.48 0.63 0.80

3.03 1.53 2.13 0.4s

0.9 0.70 1.57 1.10 1.65

lo46 1.77 2.40

2.M 035 0.65 Om75 1.00

69.40

106.00 264.00 111*00 1 0 0 e 0 0 15.00 100.00 41.90 460.00 78.40 400.00 s6.30 600.00 111.00 100.00 m.90 100.00 m 9 0 120.0 76.W 400.00 174.30 400.00 159.90 4 0 0 o 0 0 131.30 600.00 149.00 600.00 79.90 1600.00 128.30 1OOO.60 E18.ZO 140.00 095.00 poO.00 351070 P l O e 0 0 rn.30 406.00 400.00 8.9 We00 21T.M 100.00

a.60

o.n m.n 1.48 lop0

W.DO 1.75 ls0.N 1Oooo . 1.33 1200.00 0.H 171.40 140.00 talJ 191.70 0.M 819.so IDo.@o t.T3 ?u,n 0.U 4M.m W.00 t.13 111.70 l o l t 1 7 1 o ) o WoOO z.T3 H3.M .To tT3.10 t-93 t63.80 1.14 t31.0 480.0 t.n u3.m 1.7s 1S.B 1WO .O 4.80 3.17 1.8s 48.90 100.0 C*O 72.17 ..U 106.w I@.@

W O @

FIGURE A2:

337.00 to SlLW

.Do

me00

0.w

46.bO 1.15 445.00 0.U 450.00 47Jt 0.00 m.OO 0.5s 106.00 n.n 0.55 wo.00 N.8 1.13 uS.00 6.3 4.70 745.00 moo 0.M 1410.00 147.U 8.50 1483.00 1SS.SO 0.40 1480.00 153.00 8.0s 129O.ob 135.10 t.2S 1460.00 lS2.W 1.70 1490.00 156.00 3.43 1420.00 1M.10 3.45 1470.00 lS3.N 5.40 1280.00 134.00 5.70 1420.00 14O.R 0,s 2030.00 212.3 0.70 2070.00 216.77 0.70 2070.00 216.77 1.60 2006.00 m.4. 1.60 2010.04 213.0 U 0 1960.00 2OSH . t.M 2020.00 211.so

.ob8

4.H pooo.00 0.M 2soo.00 0.M 2b00.00

UobO

0.41

30.60 7lAO

0.33 0.76 OJb 0.27

31.00 S2v.10 86.70

om

w.)o

0.85 0.81 0.64 00% 0.80

73.80

noto 62.00 P77.00 344.00 265.00 313.00 531.00

0.N 0.31 0.m 0.S 0.a 0.71 0.m 0*60

R3.86 847.40 106.30 151.70 131.70

"bo 13O.W 10.0 165.8 W e 3 4

a1.w su.8

0.41 0.77 0.60

8.n o.n 0.n

o*n 8.71 0, n

u7.w

0.80

bod0

0.4s 0.74

@.so

T-1 Dynamic Equivalent

8T.u

0.61 1.40 1.40 t.10

w.44 2bl.8.

tR.H

M.00 n.4.

13.00 15.00

30.00 Jo.00

30.00 30.00 30.00 Jo.00 s0.M 56.00

.so.00 9.00 50.00 50.00

s.00 so.00 so.00 70.00 70.00 70.00 30.00

R.00 70.00 7o.M 70.00 9o.M w.00

w.m

m . 0 0 tb3.w

w.OO

1680.00 2SM.06 tbT.18 w?. tS30.00 2M.n 1.30 445.06 4b*Y 0.9. n.0

To.@ w.00 9o.W 1s.00 #A0

m.#

m.00

- . . u ~ ~ ~ l Motor on

AMs5 :007

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0. I7 0.M

I .so 8.00 0.00

t.30 3.17

3.00 8.30 0.U 7.00 1d3 0.34 t.67 1.01 7.35 t.67 t.33 1.00 1.00

3.35 t.81 4.44

t.U 30% b.b7

35 t.jr 6.80

3.38 4.M

3.u 0, I7

0.M

1 HP 3-PHASE 1 . M .

1 . 3 4 ClP I-PH4SE I .M,

I:::

3." 3.0

T

FIG 2:

Zin

-vs-

(Vm/f*, fr)

FIG 3:

(3 phase motor)

Zin

-VS-

(Po,fr)

(single phase motor)

1 HP 3-PHASE I . M e

1 HP 3-FkiA.SE- I .M.

. I

IO

FIG 3:

Po

-VS-

(Vm/fss fr)

-' FIG 4:

Efficiency -vs-

(V,/fs,

AMS5:007 I I7

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fr)

m m

GG 1.

.. . ,

I

..

.

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P

P

00

P

S

n*

4

0

~

P

P

m

&

P

a

-

0

0 0

oov

8

c

ro -*

0

f

O

0

0 ,

w

d

N

3

I:

0

I C 0

M

1

\

M

n

c

..

0

P UI

4

r. [P c

..

0

P

P

Iu

*('

EFF

h

.. ..

.P Q,

m M

P CD

... . . . .... ..

P

P

P

Iu

EFF.

-

z

P a

P

wl P. 0

w

la I b

+4

0

C

.'

M

8 h

5


k

.

W

a

3

O 0 O 0

0

m

(0

00

/

\o

zL

\\

c o00 o CO 00,

I I

0

I19

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0

00

0

0