Download - Inductionmotor Dyn 2
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Dynamic Model of
Induction Machine
MEP 1522ELECTRIC DRIVES
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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
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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
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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
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stator, b
stator, c
stator, a
rotor, b
rotor, a
rotor, c
r
DYNAMIC MODEL 3-phase model
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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
DYNAMIC MODEL 3-phase model
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Flux produced by winding b
Flux produced by winding c
Lets look at phase a
The relation between the currents in other phasesand the flux produced by these currents that linked
phase a are related by mutual inductances
DYNAMIC MODEL 3-phase model
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Lets look at phase a
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
DYNAMIC MODEL 3-phase model
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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
DYNAMIC MODEL 3-phase model
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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
DYNAMIC MODEL 3-phase model
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DYNAMIC MODEL 3-phase model
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:
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DYNAMIC MODEL 3-phase model
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
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DYNAMIC MODEL 3-phase model
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
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DYNAMIC MODEL 3-phase model
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
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DYNAMIC MODEL 3-phase model
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
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DYNAMIC MODEL 3-phase model
cr
br
ar
rrr
rrr
rrr
ms
s
rr,abcs
i
i
i
cos3
2cos3
2cos
32coscos32cos
32cos
32coscos
LNN
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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
DYNAMIC MODEL 3-phase model
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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
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It is easierto look on dynamic of IM using two-phase
model. This can be constructed from the 3-phasemodel 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
rotating
r
Two-phase equivalent
DYNAMIC MODEL 2-phase model
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It is easierto look on dynamic of IM using two-phase
model. This can be constructed from the 3-phasemodel using Parks transformation
r
stator, q
rotor,
rotor,
stator, d
rotating
r
Two-phase equivalent
However coupling still exists between
stator and rotor windings
DYNAMIC MODEL 2-phase model
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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
DYNAMIC MODEL 2-phase model
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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
DYNAMIC MODEL 2-phase model
cbac2
basds i2
1i
2
1i
3
2iaaii
3
2ReiRei
cbc2basqs ii3
1iaaii
3
2ImiImi
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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
DYNAMIC MODEL 2-phase model
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DYNAMIC MODEL 2-phase model
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
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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
DYNAMIC MODEL 2-phase model
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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
DYNAMIC MODEL 2-phase model
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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
DYNAMIC MODEL 2-phase model
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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
DYNAMIC MODEL 2-phase model
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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
DYNAMIC MODEL 2-phase model
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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
DYNAMIC MODEL 2-phase model
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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
DYNAMIC MODEL 2-phase model
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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
DYNAMIC MODEL 2-phase model
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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
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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
*
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DYNAMIC MODEL Torque equation
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
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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:
DYNAMIC MODEL Torque equation
Power
Losses in winding
resistance
Rate of change
of stored
magnetic energy
Mech powerPowerassociated with
g upon
expansion gives
zero
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pmech
mT
e
3
2it G ri
DYNAMIC MODEL Torque equation
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
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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
DYNAMIC MODEL Torque equation
We know that m = r/ (p/2),
Te
3
2
p
2L
mi
sqi
rd i
sdi
rq
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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
DYNAMIC MODEL Torque equation
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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
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Which finally can be modeled using SIMULINK: