induction motor model

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Modeling of Induction Motor

using dq0 Transformations

First Semester 1431/1432

Steady state model developed in previous studies

of induction motor neglects electrical transients

due to load changes and stator frequency

variations. Such variations arise in applications

involving variable-speed drives.

Variable-speed drives are converter-fed from finite

sources, which unlike the utility supply, are

limited by switch ratings and filter sizes, i.e. they

cannot supply large transient power.

Introduction

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Thus, we need to evaluate dynamics of converter-fed variable-speed drives to assess the adequacy of the converter switches and the converters for a given motor and their interaction to determine the excursions of currents and torque in the converter and motor. Thus, the dynamic model considers the instantaneous effects of varying voltages/currents, stator frequency and torque disturbance.

Introduction (cont’d)

Circuit Model of a Three-Phase IM

1. Space mmf and flux waves are considered to be

sinusoidally distributed, thereby neglecting the effect of

teeth and slots.

2. The machine is regarded as group of linear coupled

circuits, permitting superposition to be applied, while

neglecting saturation, hysteresis, and eddy currents.

3. Ls : self inductance per phase of the stator windings.

4. Ms: mutual inductance per phase of the stator windings.

5. rs: resistance per phase of the stator windings.

6. Lr : self inductance per phase of the rotor windings.

7. Mr: mutual inductance per phase of the rotor windings

8. rr: resistance per phase of the rotor windings.

9. Msr: maximum value of mutual inductance between any

stator phase and any rotor phase.

Assumptions and Definitions:

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ias

i bs

ics

iar

i br

icrvas+

+v

bs

+v cs

v ar+

+vbr

vcr +

axis of stator

phase c

axis of stator

phase a

axis of stator

phase b

axis of rotor

phase c

axis of rotor

phase b

axis of rotor

phase a

r

r

r

Circuit Model of a Three-Phase IM

r

Stator Voltage Equations:

as

as as s

dv i r

dt

bsbs bs s

dv i r

dt

cscs cs s

dv i r

dt

Voltage Equations

ias

i bs

ics

iar

i br

icrvas+

+v

bs

+v cs

v ar+

+vbr

vcr +

axis of stator

phase c

axis of stator

phase a

axis of stator

phase b

axis of rotor

phase c

axis of rotor

phase b

axis of rotor

phase a

r

r

r

r

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Rotor Voltage Equations:

ar

ar ar r

dv i r

dt

brbr br r

dv i r

dt

crcr cr r

dv i r

dt

Voltage Equations (cont’d)

ias

i bs

ics

iar

i br

icrvas+

+v

bs

+v cs

v ar+

+vbr

vcr +

axis of stator

phase c

axis of stator

phase a

axis of stator

phase b

axis of rotor

phase c

axis of rotor

phase b

axis of rotor

phase a

r

r

r

r

cos( ) cos( 120 ) cos( 120 )sr sr sar r br r c

as s as s bs s c

r r r

s

M M

L i M i M i

i i M i

cos( 120 ) cos( ) cos( 120 )ar r

bs s as s bs s

br r csr s r rr s

cs

rM M

M i L i M i

Mi i i

Flux Linkage

Equations

ias

i bs

ics

iar

i br

icrvas+

+v

bs

+v cs

v ar+

+vbr

vcr +

axis of stator

phase c

axis of stator

phase a

axis of stator

phase b

axis of rotor

phase c

axis of rotor

phase b

axis of rotor

phase a

r

r

r

r

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(

cos( ) cos( 120 ) cos 20 )

)

( 1sr sar r br r cr

as s as s bs cs

rr srM M

L i M

i i M

i i

i

cos( 120 ) cos( ) c

( )

os( 120 )

bs s as cs s b

sr sr sar r br r cr r

s

r

M i

M M Mi i

i L i

i

Flux Linkage

Equations

0as bs csi i i

In general, we can assume:

ias

i bs

ics

iar

i br

icrvas+

+v

bs

+v cs

v ar+

+vbr

vcr +

axis of stator

phase c

axis of stator

phase a

axis of stator

phase b

axis of rotor

phase c

axis of rotor

phase b

axis of rotor

phase a

r

r

r

r

ss s sL L M

Let:

cos( ) cos( 120 ) cos( 120 )

( )

ar r r r

r ar r br c

as bs cs

r

sr sr sr

L i M

i i M

i i

M M i

Flux Linkage

Equations

0ar br cri i i

In general, we can assume:

ias

i bs

ics

iar

i br

icrvas+

+v

bs

+v cs

v ar+

+vbr

vcr +

axis of stator

phase c

axis of stator

phase a

axis of stator

phase b

axis of rotor

phase c

axis of rotor

phase b

axis of rotor

phase a

r

r

r

r

cos( 120 ) cos( ) cos( 120 )

( )

assr sr sbr r r r

r b

bs cs

r c

r

r ar r

M M M

L i M i

i i i

i

rr r rL L M

Let:

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Flux Linkage Equations

cos( 120 ) cos( 120 ) cos( )cs ar r bss c r r crs sr ri iM iL i

cos( ) cos( 120 ) cos( 120 )as ss sr ar r br r ra crsL i i i iM

cos( 120 ) cos( ) cos( 120 )bs ss bs ar r br r r rsr ci iL M ii

Stator:

Rotor:

cos( ) cos( 120 ) cos( 120 )as bs cssar r r r rrr ariM i ii L

cos( 120 ) cos( ) cos( 120 )as bs cbr r r rs rr brsr i i i L iM

cos( 120 ) cos( 120 ) cos( )ascr r r r rrbs s cr rcs i i i L iM

0 0

0

0 0

0

0

0

0

0

0 0

ar arrr

ar brrr

ar crr

sr

T

s

asssas

bsssb

r

s

ccs

r

sss

iL

iL

i

iL

L

L

L

iL

iL

cos( ) cos( 120 ) cos( 120 )

cos( 120 ) cos( ) cos( 120 )

cos( 120 ) cos( 120 ) cos( )

r r r

r rsr r

r

s

r r

rL M

Flux Linkage Equations

as

as as s

dv i r

dt

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To build up our simulation equations, we

could just differentiate each expression for

, e.g.

But since Lsr depends on position ,

which will generally be a function of time,

the trigonometric terms will lead to a mess!

First raw of the Matrixasas as s

d dv i r

dt dt

Model of Induction Motor

The Park’s transformation is a three-phase to two-

phase transformation for synchronous machine

analysis. It is used to transform the stator variables

of a synchronous machine onto a dq reference

frame that is fixed to the rotor.

The +ve d-axis is aligned with the magnetic axis of

the field winding and the +ve q-axis is defined as

leading the +ve d-axis by /2.

Park’s Transformation

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Park’s Transformation (cont’d)

ias

i bs

ics

iari b

r

icrvas+

+v

bs

+v cs

v ar+

+vbr

vcr +

axis of stator

phase c

axis of stator

phase a

axis of stator

phase b

axis of rotor

phase c

axis of rotor

phase b

axis of rotor

phase a

r

r

r

r

d-axis

q-axis

The result of this transformation is

that all time-varying inductances in

the voltage equations of an induction

machine due to electric circuits in

relative motion can be eliminated.

In induction machine, the

d-axis is assumed to align on

a-axis at t = 0 and rotate

with synchronous speed ()

The Park’s transformation equation is of

the form:

where f can be i, v, or .

0

0

f f

f T f

f f

d a

q dq b

c


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0

2 2cos cos cos

3 3

2 2( ) sin sin sin

3 3

1 1 1

2 2 2

T

d d d

dq d d d dK


where K is a convenient constant. The current id and iq are

proportional to the components of mmf in the direct and quadrature

axes, respectively, produced by the resultant of all three armature

currents, ia, ib, and ic. For balanced phase currents of a given

maximum magnitude, the maximum value of id and iq can be of the

same magnitude. Under balanced conditions, the maximum magnitude

of any one of the phase currents is given by . To

achieve this relationship, a value of 2/3 is assigned to the constant K.

2 2

, , ,a peak b peak c peak d qi i i i i

The inverse transform is given by:

Of course, [T][T]-1=[I]

1

0

cos sin 1

2 2( ) cos sin 1

3 3

2 2cos sin 1

3 3

T

d d

dq d d d

d d


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Thus,

and

0 0

0

d a

q dq b dq abc

c

v v

v T v T v

v v

0 0

0

d a

q dq b dq abc

c

i i

i T i T i

i i


ids

vds+

iqr +vqr

idr

vdr+

iqs +vqs

Induction Motor Model in dq0

d-axis

q-axis

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Lets us define new “dq0” variables.

Our induction motor has two subsystems - the

rotor and the stator - to transform to our

orthogonal coordinates:

So, on the stator,

where [Ts]= [T()], ( = t)

and on the rotor,

where [Tr]= [T()], ( = - r = ( r) t )

0dq s s abcsT

0 [ ]dq r r abcrT

Induction Motor Model in dq0 (cont’d)


0

0 0

1

1

0

"abc": [ ]

"dq0": [ ]

[ ]

STATOR:

[ ]

[ ]

[ ]

abcs ss abcs

dq s s abcs ss s abcs s

dq s ss dq s s

sr

sr

sr

abcr

r r abcr

r dq r

L i

T L T i

L

L

L

T

L

i

T i

i TT

T

i

1

1

0

0 00

"abc": [ ]

"dq0": [ ]

ROTOR:

[ ]

[ ]

[ ][ ]

T

sr

T

abcs

s s abcs

s dq

abcr rr abcr

dq r r abcr r rr r abcr

dq s r rr dq rs

sr

T

sr

L iL

L

i

T T i

TL

T T L T i

T i L i

1 0 0

0 1 0

0 0 1

ss ssL L

1 0 0

0 1 0

0 0 1

rr rrL L

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Now:

Just constants!!

Our double reference frame transformation

eliminates the trigonometric terms found in our

original equations.

1 1

30 0

2

30 0

2

0 0 0

sr

T

sr sr ss s rr rT

M

T L MT T T


Let us look at our new dq0 constitutive law and

work out simulation equations.

0dq s s abcs s abcs s abcs

dv T v T R i T

dt

1 1

0 0s s dq s s s dq s

dT RT i T T

dt


1

0 0dq s s s dq s

dR i T T

dt

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Using the differentiation product rule:

0 0 0

0 0

0 0

0 0 0

dq s dq s dq s

d

dt

d dR i

dt dt


1

0 0 0 0dq s dq s dq s s s dq s

d dv R i T T

dt dt

For the stator this matrix is:

For the rotor the terminal equation is

essentially identical but the matrix is:

0 0

0 0

0 0 0

0 ( ) 0

( ) 0 0

0 0 0

r

r


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Simulation model; Stator Equations:

dsds ds s qs

dv i r

dt

qs

qs qs s ds

dv i r

dt

00 0

ss s s

dv i r

dt


Simulation model; Rotor Equations:

( ) drdr dr r r qr

dv i r

dt

( )qr

qr qr r r dr

dv i r

dt

00 0

rr r r

dv i r

dt


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Zero-sequence equations (v0s and v0r) may be

ignored for balanced operation.

For a squirrel cage rotor machine,

vdr= vqr= 0.


We can also write down the flux linkages:

0

0

0

0

0

0

0 0 3 2 0 0

0 0 0 3 2 0

0 0 0 0 0

3 2 0 0 0 0

0 3 2 0 0 0

0 0 0 0 0

dr rr dr

qr rr qr

r r

ds ss ds

qs ss q

sr

sr

sr

sr

s

r r

s

s s s

L i

L

ML i

M

M

M i

i

L

L i

L

i


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The torque of the motor in qd0 frame is given

by:

where P= # of poles

F=ma, so:

where = load torque

3

2 2qr dr d re r qi i

P

( )re l

dJ

dt

l


induction motor model

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