Inductionmotor Dyn 2

7/29/2019 Inductionmotor Dyn 2

1/41

Dynamic Model of

Induction Machine

MEP 1522ELECTRIC DRIVES


2/41

WHY NEED DYNAMIC MODEL?

In an electric drive system, the machine is partof the control system elements

To be able to control the dynamics of the drive

system, dynamic behavior of the machine needto be considered

Dynamic behavior of of IM can be described

using dynamic model of IM


3/41

WHY NEED DYNAMIC MODEL?

Dynamic model complex due to magneticcoupling between stator phases and rotor phases

Coupling coefficients vary with rotor position

rotor position vary with time

Dynamic behavior of IM can be described by

differential equations with time varying coefficients


4/41

a

b

bc

c

aSimplified

equivalent statorwinding

ias

Magnetic axis of

phase A

Magnetic axis of

phase B

Magnetic axis of

phase C

ics

ibs

DYNAMIC MODEL, 3-PHASE MODEL


5/41

stator, b

stator, c

stator, a

rotor, b

rotor, a

rotor, c

r

DYNAMIC MODEL 3-phase model


6/41

Lets look at phase a

Flux that links phase a is caused by:

Flux produced by winding a

Flux produced by winding b

Flux produced by winding c



7/41

Flux produced by winding b

Flux produced by winding c


The relation between the currents in other phasesand the flux produced by these currents that linked

phase a are related by mutual inductances



8/41


r,ass,asas

csacsbsabsasas iLiLiL crcr,asbrbr,asarar,as iLiLiL

Mutual inductance

between phase a and

phase b of stator

Mutual inductance

between phase a and

phase c of stator

Mutual inductance

between phase a of stator

and phase a of rotor

Mutual inductance


and phase b of rotor

Mutual inductance


and phase c of rotor



9/41

vabcs = Rsiabcs + d(abcs)/dt - stator voltage equationvabcr = Rrriabcr + d(abcr)/dt -rotor voltage equation

cs

bs

as

abcs

cs

bs

as

abcs

cs

bs

as

abcs

i

i

i

i

v

v

v

v

cr

br

ar

abcr

cr

br

ar

abcr

cr

br

ar

abcr

i

i

i

i

v

v

v

v

abcs flux (caused by stator and rotor currents) that

links stator windings

abcr flux (caused by stator and rotor currents) that

links rotor windings



10/41

r,abcss,abcsabcs

s,abcrr,abcrabcr

cs

bs

as

csbcsacs

bcsbsabs

acsabsas

s,abcs

i

i

i

LLL

LLL

LLL

cr

br

ar

cr,csbr,csar,cs

cr,bsbr,bsar,bs

cr,asbr,asar,as

r,abcs

i

i

i

LLL

LLL

LLL

Flux linking stator winding due to stator current

Flux linking stator winding due to rotor current



11/41


cr

br

ar

crbcracr

bcrbrabr

acrabrar

r,abcr

i

i

i

LLL

LLL

LLL

cs

bs

as

cs,crbs,cras,cr

cs,brbs,bras,br

cs,arbs,aras,ar

s,abcr

i

i

i

LLL

LLL

LLL

Flux linking rotor winding due to rotor current

Flux linking rotor winding due to stator current

Similarly we can write flux linking rotor windings caused by rotor and stator currents:


12/41


The self inductances consist of magnetising andleakage inductancesLas = Lms + Lls Lbs = Lms + Lls Lcs = Lms + Lls

The magnetizing inductance Lms, accounts for the fluxproduce by the respective phases, crosses the airgap and

links other windings

The leakage inductance Lls, accounts for the flux produceby the respective phases, but does not cross the airgap

and links only itself


13/41


It can be shown that the magnetizing inductance is given by

4g

rlNL 2soms

It can be shown that the mutual inductance between stator

phases is given by:

o2soacsbcsabs 120cos

4g

rlNLLL

2

L

8g

rlNLLL ms2soacsbcsabs


14/41


The mutual inductances between stator phases (androtor phases) can be written in terms of magnetising

inductances

cs

bs

as

lsmsmsms

mslsms

ms

msmslsms

s,abcs

i

i

i

LL2

L

2

L2

LLL

2

L2

L2

LLL


15/41


The mutual inductances between the stator and rotorwindings depends on rotor position

cr

br

ar

rrr

rrr

rrr

ms

s

rr,abcs

i

i

i

cos3

2cos3

2cos

32coscos

32cos

32cos

32coscos

LNN

cs

bs

as

rrr

rrr

rrr

ms

s

rs,abcr

i

i

i

cos3

2cos3

2cos

32coscos

32cos

32cos

32coscos

LN

N


16/41


cr

br

ar

rrr

rrr

rrr

ms

s

rr,abcs

i

i

i

cos3

2cos3

2cos

32coscos32cos

32cos

32coscos

LNN


17/41

cr

br

ar

rrr

rrr

rrr

ms

s

rr,abcs

i

i

i

cos3

2cos3

2cos

32coscos

32cos 3

2cos

3

2coscos

LNN

stator, b

stator, c

stator, a

rotor, b

rotor, a

rotor, c

r



18/41

DYNAMIC MODEL, 2-PHASE MODEL

It is easierto look on dynamic of IM using two-phase

model. This can be constructed from the 3-phasemodel using Parks transformation

Three-phaseTwo-phase equivalent

There is magnetic coupling

between phases

There is NO magnetic

coupling between phases


19/41



stator, b

rotor, b

rotor, a

rotor, c

stator, c

stator, a

r

rrotating

Three-phase

r

stator, q

rotor,

rotor,

stator, d

rotating

r

Two-phase equivalent



20/41



r

stator, q

rotor,

rotor,

stator, d

rotating

r

Two-phase equivalent

However coupling still exists between

stator and rotor windings



21/41

All the 3-phase quantities have to be transformed to 2-

phase quantities

In general ifxa,xb, andxc are the three phase quantities, the

space phasor of the 3 phase systems is defined as:

c2ba xaaxx3

2x , where a = ej2/3

qd jxxx

cbac

2bad x

2

1x

2

1x

3

2xaaxx

3

2RexRex

cbc2baq xx3

1xaaxx

3

2ImxImx



22/41

bai

c2ia

ai

All the 3-phase quantities have to be transformed into

2-phase quantities

c2bas iaaii3

2i

ai3

2bai

3

2

c2ia

3

2

si

qsdss jiii

dsi

qsi si

d

q


cbac2

basds i2

1i

2

1i

3

2iaaii

3

2ReiRei

cbc2basqs ii3

1iaaii

3

2ImiImi


23/41

The transformation is given by:

c

b

a

31

31

31

3

1

3

1

31

31

32

o

q

d

i

i

i

0

i

i

iFor isolated neutral,

ia + ib + ic = 0,

i.e. io =0

idqo = Tabc iabc

The inverse transform is given by:

iabc = Tabc-1

idqo



24/41


vabcs = Rsiabcs + d(abcs)/dtvabcr = Rrriabcr + d(abr)/dt

IM equations :

r

stator, q

rotor,

rotor,

stator, d

rotatingr

3-phasevdq = Rsidq + d(dq)/dtv = Rrri + d()/dt

2-phase


25/41

r,dqss,dqs

dqwhere,

qs

ds

qqqd

dqdd

s,dqs

i

i

LL

LL

r

r

qq

dd

r,dqs

i

i

LL

LL

r

r

r,r i

i

LL

LL

qs

ds

qd

qd

s,r i

i

LL

LL

s,rr,r

Express in stationary frame

Express in rotating frame

vdq

= Rs

idq

+ d(dq

)/dt

v = Rri + d()/dt



26/41

Note that:

Ldq = Lqd = 0 L = L = 0

Ldd = Lqq L = L

The mutual inductance between stator and rotor depends on rotor

position:

Ld = Ld = Lsr cos r Lq = Lq = Lsr cos r

Ld = Ld = - Lsr sin r Lq = Lq = Lsr sin r



27/41

Ld = Ld = Lsr cos r Lq = Lq = Lsr cos r

Ld = Ld = - Lsr sin r Lq = Lq = Lsr sin r

r

stator, q

rotor,

rotor,

stator, d

rotatingr



28/41

r

r

sq

sd

rrsrrsr

rrsrrsr

rsrrsrdds

rsrrsrdds

q

d

i

ii

i

sLR0cossLsinL

0sLRsinsLcossLcossLsinsLsLR0

sinsLcossL0sLR

v

vv

v

In matrix form this an be written as:

The mutual inductance between rotor and stator depends on

rotor position



29/41

Magnetic path from stator linking the

rotor winding independent of rotor

position mutual inductance

independent of rotor position

rotor, q

rotor, d stator, d

stator, q

r

Both stator and rotor

rotating or stationary

The mutual inductance can be made

independent from rotor position byexpressing both rotor and stator in

the same reference frame, e.g. in the

stationary reference frame



30/41

If the rotor quantities are referred to stator, the following can bewritten:

rq

rd

sq

sd

rrrrmmr

rrrrmrm

mss

mss

rq

rd

sq

sd

i

i

i

i

sL'RLsLL

LsL'RLsL

sL0sLR0

0sL0sLR

v

v

v

v

Lm, Lrare the mutual and rotor self inductances referred to stator, and Rr isthe rotor resistance referred to stator

Ls = Ldd is the stator self inductance

Vrd, vrq, ird, irq are the rotor voltage and current referred to stator



31/41

How do we express rotor current in stator (stationary) frame?

rc2rbrar iaaii3

2i

In rotating frame this can be written as: rjrr eii

r

r

ir

In stationary frame it be written as:

rjrsr eii

jrei

rqrd jii

qs

dsird

irq

is known as the space vectorof

the rotor currentri



32/41

It can be shown that in a reference frame rotating at g, theequation can be written as:

rq

rd

sq

sd

rrrrgmmrg

rrgrrmrgm

mmgsssg

mgmsgss

rq

rd

sq

sd

i

i

i

i

sL'RL)(sLL)(

L)(sL'RL)(sL

sLLsLRL

LsLLsLR

v

v

v

v



33/41

DYNAMIC MODEL Space vectors

IM can be compactly written using space vectors:

gsg

gsg

ssgs j

dt

diRv

grrg

grg

rr )(jdt

diR0

grm

gss

gs iLiL

gsm

grr

gr iLiL

All quantities are written in generalreference frame


34/41

Product of voltage and current conjugate space vectors:

DYNAMIC MODEL Torque equation

csbs2ascs2bsas*ss aiiai3

2vaavv

3

2iv

It can be shown that for ias + ibs + ics = 0,

cscsbsbsasas*ss iviviv3

2ivRe

Pin

vas

ias

vbs

ibs

vcs

ics

3

2Re v

si

s

*


35/41


Pin

vas

ias

vbs

ibs

vcs

ics

3

2Re v

si

s

*

Pin

3

2Re v

sis

* 32

Re (vd

jvq)(i

d ji

q) 3

2v

di

d v

qi

q

vi2

3

Pt

in

If

q

d

v

vv and

q

d

i

ii


36/41

The IM equation can be written as:

v R i L si G ri F g i

pi

3

2it V

3

2it R i it L si it G ri it F g i

The input power is given by:


Power

Losses in winding

resistance

Rate of change

of stored

magnetic energy

Mech powerPowerassociated with

g upon

expansion gives

zero


37/41

pmech

mT

e

3

2it G ri


rq

rd

sq

sd

rrmr

rrmr

i

i

ii

0L0L

L0L0

00000000

rq

rd

sq

sd

rrrrgmmrg

rrgrrmrgm

mmgsssg

mgmsgss

rq

rd

sq

sd

i

i

i

i

sL'RL)(sLL)(

L)(sL'RL)(sL

sLLsLRL

LsLLsLR

v

v

v

v


38/41

r

rdrsdm

rqrsqm

t

rq

rd

sq

sd

em

iLiL

iLiL

0

0

i

i

i

i

2

3

T

pmech mTe 32 it G ri


We know that m = r/ (p/2),

Te

3

2

p

2L

mi

sqi

rd i

sdi

rq


39/41

Te

3

2

p

2L

mi

sqi

rd i

sdi

rq

Te

3

2

p

2L

mIm i

si

r

'*

rmsss iLiL but sssrm iLiL

Te

3

2

p

2Im i

s

s

* Lsi

s

*

Te

3

2

p

2Im i

s

s

* Te 32

p

2

s i

s



40/41

DYNAMIC MODEL

Simulation

Re-arranging with stator and rotor currents as state space

variables:

sq

sd

m

m

r

r

sr

2

m

rq

rd

sq

sd

srsrrmssmr

srrsrsmrms

mrrmrrs

2

mr

rmrmrsq

2

mrrs

sr

2

m

rq

rd

sq

sd

v

v

L0

0LL0

0L

LLL1

i

ii

i

LRLLLRLL

LLLRLLLRLRLLLRL

LLLRiLLR

LLL1

i

ii

i

The torque can be expressed in terms of stator and rotor currents:

Te

3

2

p

2L

mi

sqi

rd i

sdi

rq

Te

TL

Jdmdt


41/41

Which finally can be modeled using SIMULINK:

Documents

Inductionmotor Dyn 2