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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
DYNAMIC MODEL – 3-phase model
• Flux produced by winding b
• Flux produced by winding c
Let’s look at phase a
The relation between the currents in other phases and the flux produced by these currents that linked phase a are related by mutual inductances
DYNAMIC MODEL – 3-phase model
Let’s 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
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
DYNAMIC MODEL – 3-phase model
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
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:
DYNAMIC MODEL – 3-phase model
Combining the stator and rotor voltage equations,
p is the derivative operator , i.e. p = d/dt
DYNAMIC MODEL – 3-phase model
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
DYNAMIC MODEL – 3-phase model
• 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
DYNAMIC MODEL – 3-phase model
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
DYNAMIC MODEL – 3-phase model
• It can be shown that the mutual inductance between stator phases is given by:
o2soacsbcsabs 120cos
4g
rlNLLL
2
L
8g
rlNLLL ms2soacsbcsabs
DYNAMIC MODEL – 3-phase model
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
DYNAMIC MODEL – 3-phase model
• 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
DYNAMIC MODEL – 3-phase model
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
DYNAMIC MODEL – 3-phase model
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
DYNAMIC MODEL – 3-phase model
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 – 3-phase model
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
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 Parks transformation
r
stator, q
rotor,
rotor,
stator, d
rotatingr
Two-phase equivalent
However coupling still exists between stator and rotor windings
DYNAMIC MODEL – 2-phase model
• 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
DYNAMIC MODEL – 2-phase model
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
DYNAMIC MODEL – 2-phase model
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
DYNAMIC MODEL – 2-phase model
DYNAMIC MODEL – 2-phase model
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
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
DYNAMIC MODEL – 2-phase model
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
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
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
DYNAMIC MODEL – 2-phase model
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
DYNAMIC MODEL – 2-phase model
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
DYNAMIC MODEL – 2-phase model
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
DYNAMIC MODEL – 2-phase model
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 – 2-phase model
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
DYNAMIC MODEL Torque equation
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:
DYNAMIC MODEL Torque equation
PowerLosses in winding resistance
Rate of change of stored magnetic energy
Mech power
Power associated with g – upon expansion gives zero
DYNAMIC MODEL Torque equation
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
DYNAMIC MODEL Torque equation
We know that m = r / (p/2),
rmsss iLiL but sssrm iLiL
DYNAMIC MODEL Torque equation
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: