Dynamic Model of Induction Machine MEP 1522 ELECTRIC DRIVES

Dynamic Model of Induction Machine

MEP 1522

ELECTRIC DRIVES

WHY NEED DYNAMIC MODEL?

• In an electric drive system, the machine is part of the control system elements

• To be able to control the dynamics of the drive system, dynamic behavior of the machine need to be considered

• Dynamic behavior of of IM can be described using dynamic model of IM

WHY NEED DYNAMIC MODEL?

• Dynamic model – complex due to magnetic coupling 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

a

b

b’c’

c

a’Simplified equivalent stator winding

ias

Magnetic axis of phase A

Magnetic axis of phase B

Magnetic axis of phase C

ics

ibs

DYNAMIC MODEL, 3-PHASE MODEL

stator, b

stator, c

stator, arotor, b

rotor, a

rotor, c

r

DYNAMIC MODEL – 3-phase model

Let’s 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


• Flux produced by winding b

• Flux produced by winding c


The relation between the currents in other phases and the flux produced by these currents that linked phase a are related by mutual inductances



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 between phase a of stator and phase b of rotor

Mutual inductance between phase a of stator and phase c of rotor


vabcs = Rsiabcs + d(abcs)/dt - stator voltage equation

vabcr = 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


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



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:


Combining the stator and rotor voltage equations,

p is the derivative operator , i.e. p = d/dt


Windings x and y, with Nx and Ny number of turns, separated by α

x

x’

y

y’

Magnetic axis of winding x

Magnetic axis of winding y

It can be shown that the mutual inductance between winding x and y is


• To get self inductance for phase a of the stator let Nx = Ny = Ns and α = 0

• If we consider the leakage flux, we can write the self inductance of phase a of the

stator Las = Lam + Lls

Las Self inductance of phase a

Lam Magnetizing inductance of phase a

Lls Leakage inductance of phase a


Due to the symmetry of the windings, Lam = Lbm = Lcm ,

Hence

Las = Lbs = Lcs = Lms + Lls

• The magnetizing inductance Lms, accounts for the flux produce by the respective phases, crosses the airgap and links other windings

• The leakage inductance Lls, accounts for the flux produce by the respective phases, but does not cross the airgap and links only itself


• It can be shown that the mutual inductance between stator phases is given by:

o2soacsbcsabs 120cos

4g

rlNLLL

2

L

8g

rlNLLL ms2soacsbcsabs


The mutual inductances between stator 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

L

2

LLL


• The mutual inductances between rotor phases can be written in terms of stator magnetising inductances

Self inductance between rotor windings is when Nx = Ny = Nr and α = 0


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

Self inductance between phase a of rotor and phase a of stator is when Nx = Ns , Ny = Nr and α = θr


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

cr

br

ar

rrr

rrr

rrr

mss

rr,abcs

iii

cos32cos3

2cos3

2coscos32cos

32cos3

2coscos

LNN

cs

bs

as

rrr

rrr

rrr

mss

rs,abcr

iii

cos32cos3

2cos3

2coscos32cos

32cos3

2coscos

LNN


cr

br

ar

rrr

rrr

rrr

mss

rr,abcs

iii

cos32cos3

2cos3

2coscos32cos

32cos3

2coscos

LNN

cr

br

ar

rrr

rrr

rrr

mss

rr,abcs

iii

cos32cos3

2cos3

2coscos32cos

32cos3

2coscos

LNN

stator, b

stator, c

stator, arotor, b

rotor, a

rotor, c

r


DYNAMIC MODEL, 2-PHASE MODEL

• It is easier to look on dynamic of IM using two-phase model. This can be constructed from the 3-phase model using Park’s transformation

Three-phase Two-phase equivalent

There is magnetic coupling between phases

There is NO magnetic coupling between phases

• It is easier to look on dynamic of IM using two-phase model. This can be constructed from the 3-phase model using Parks transformation

stator, b

rotor, b

rotor, a

rotor, c

stator, c

stator, a

r

rrotating

Three-phase

r

stator, q

rotor,

rotor,

stator, d

rotatingr

Two-phase equivalent


• It is easier to look on dynamic of IM using two-phase model. This can be constructed from the 3-phase model using Parks transformation

r

stator, q

rotor,

rotor,

stator, d

rotatingr

Two-phase equivalent

However coupling still exists between stator and rotor windings


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

• In general if xa, xb, and xc are the three phase quantities, the space phasor of the 3 phase systems is defined as:

c2

ba xaaxx3

2x , where a = ej2/3

qd jxxx

cbac

2bad x

21

x21

x32

xaaxx32

RexRex

cbc2

baq xx3

1xaaxx

32

ImxImx


bai

c2ia

ai

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

c2

bas iaaii32

i

ai32

bai32

c2ia

32

si

qsdss jiii

dsi

qsisi

d

q


cbac

2basds i

21

i21

i32

iaaii32

ReiRei

cbc2

basqs ii3

1iaaii

32

ImiImi

• The transformation is given by:

c

b

a

31

31

31

31

31

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



vabcs = Rsiabcs + d(abcs)/dt

vabcr = Rrriabcr + d(abr)/dt

IM equations :

r

stator, q

rotor,

rotor,

stator, d

rotatingr

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

v = Rrri + d( )/dt

2-phase

r,dqss,dqs dq

where,

qs

ds

qqqd

dqdds,dqs i

iLLLL

r

r

qq

ddr,dqs i

iLLLL

r

rr,r i

iLLLL

qs

ds

qd

qds,r i

iLLLL

s,rr,r

Express in stationary frame

Express in rotating frame

vdq = Rsidq + d(dq)/dt

v = Rri + d( )/dt


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


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


r

r

sq

sd

rrsrrsr

rrsrrsr

rsrrsrdds

rsrrsrdds

q

d

iiii

sLR0cossLsinL0sLRsinsLcossLcossLsinsLsLR0

sinsLcossL0sLR

vvvv

In matrix form this an be written as:

• The mutual inductance between rotor and stator depends on rotor position


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 rotorrotating or stationary

The mutual inductance can be made independent from rotor position by expressing both rotor and stator in the same reference frame, e.g. in the stationary reference frame


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

rc2

rbrar iaaii32

i

In rotating frame this can be written as: rjrr eii

r

r

ir

In stationary frame it be written as:

rjr

sr eii

jrei

rqrd jii

qs

dsird

irq

is known as the space vector of the rotor current

ri


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

rq

rd

sq

sd

rrrrmmr

rrrrmrm

mss

mss

rq

rd

sq

sd

iiii

sL'RLsLLLsL'RLsL

sL0sLR00sL0sLR

vvvv

Lm, Lr are the mutual and rotor self inductances referred to stator, and Rr’ is

the 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


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

rq

rd

sq

sd

rrrrgmmrg

rrgrrmrgm

mmgsssg

mgmsgss

rq

rd

sq

sd

iiii

sL'RL)(sLL)(L)(sL'RL)(sL

sLLsLRLLsLLsLR

vvvv


DYNAMIC MODEL Space vectors

IM can be compactly written using space vectors:

gsg

gsg

ssgs j

dtd

iRv

grrg

grg

rr )(jdt

diR0

grm

gss

gs iLiL

gsm

grr

gr iLiL

All quantities are written in general reference frame

Product of voltage and current conjugate space vectors:

DYNAMIC MODEL Torque equation

csbs2

ascs2

bsas*ss aiiai

32

vaavv32

iv

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

cscsbsbsasas*ss iviviv

32

ivRe


vi23

P tin

If

q

d

v

vv and

q

d

i

ii

The IM equation can be written as:

The input power is given by:


PowerLosses in winding resistance

Rate of change of stored magnetic energy

Mech power

Power associated with g – upon expansion gives zero


rq

rd

sq

sd

rrmr

rrmr

iiii

0L0LL0L0

00000000

rq

rd

sq

sd

rrrrgmmrg

rrgrrmrgm

mmgsssg

mgmsgss

rq

rd

sq

sd

iiii

sL'RL)(sLL)(L)(sL'RL)(sL

sLLsLRLLsLLsLR

vvvv

r

rdrsdm

rqrsqm

t

rq

rd

sq

sd

em

iLiLiLiL

00

iiii

23

T


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

rmsss iLiL but sssrm iLiL


DYNAMIC MODEL Simulation

Re-arranging with stator and rotor currents as state space variables:

sq

sd

m

m

r

r

sr2m

rq

rd

sq

sd

srsrrmssmr

srrsrsmrms

mrrmrrs2mr

rmrmrsq2mrrs

sr2m

rq

rd

sq

sd

vv

L00LL00L

LLL1

iiii

LRLLLRLLLLLRLLLRLRLLLRL

LLLRiLLR

LLL1

iiii

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

Which finally can be modeled using SIMULINK:

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Dynamic Model of Induction Machine MEP 1522 ELECTRIC DRIVES