Transcript dynamic model

MEP 1522
ELECTRIC DRIVES
Dynamic Model of
Induction Machine
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
DYNAMIC MODEL, 3-PHASE MODEL
Magnetic axis of
phase B
ibs
a
b’
c’
Magnetic axis of
phase A
ias
b
c
ics
Magnetic axis of
phase C
a’
Simplified
equivalent stator
winding
DYNAMIC MODEL –
3-phase model
stator, b
rotor, a
rotor, b
r
stator, a
rotor, c
stator, c
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
Let’s look at phase a
• Flux
produced
by winding b
The relation between the currents
in other
phases
• Flux produced
by winding c
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
as  as,s  as,r
Lasias  Labsibs  Lacs ics
Mutual inductance
between phase a and
phase b of stator
Las,ariar  Las,bribr  Las,cr icr
Mutual inductance
Mutual inductance
Mutual inductance
Mutual
inductance
between phase
a of stator
between phase a and phase
between
phase a of stator
a of stator
and phase a ofbetween
rotor
phase c of stator
phase c of rotor
and phase b ofand
rotor
DYNAMIC MODEL –
vabcs = Rsiabcs + d(abcs)/dt
3-phase model
- stator voltage equation
vabcr = Rrriabcr + d(abcr)/dt
v abcs
 v as 
  v bs 
 v cs 
v abcr
 v ar 
  v br 
 v cr 
•
•
i abcs
i as 
 i bs 
i cs 
i abcr
i ar 
 i br 
i cr 
- rotor voltage equation
abcs
 as 
 bs 
 cs 
abcr
 ar 
 br 
 cr 
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
abcs  abcs,s  abcs,r
abcr  abcr,r  abcr,s
Flux linking stator winding due to stator current
abcs ,s
 L as
 L abs
 L acs
L abs
L bs
L bcs
L acs  i as 
L bcs  i bs 
L cs  i cs 
Flux linking stator winding due to rotor current
abcs , r
 L as ,ar

 L bs,ar
 L cs ,ar

L as , br
L bs, br
L cs , br
L as ,cr  i ar 

L bs,cr  i br 
L cs ,cr  i cr 
DYNAMIC MODEL –
3-phase model
Similarly we can write flux linking rotor windings caused by rotor and stator currents:
Flux linking rotor winding due to rotor current
abcr , r
 L ar
 L abr
 L acr
L abr
L br
L bcr
L acr  i ar 
L bcr  i br 
L cr  i cr 
Flux linking rotor winding due to stator current
abcr ,s
 L ar ,as

 L br,as
 L cr ,as

L ar , bs
L br, bs
L cr , bs
L ar ,cs  i as 

L br,cs  i bs 
L cr ,cs  i cs 
DYNAMIC MODEL –
3-phase model
Combining the stator and rotor voltage equations,
é
ê
ê
ê
ê
ê
ê
ê
ê
ë
é
vas ù ê Rs + pLas
ú
vbs ú ê pLabs
ú ê
vcs ú ê pLacs
=ê
ú
var
pLar,as
ê
ú
vbr ú ê pLbr,as
ê
vbc úû ê pLcr,as
ë
pLabs
pLacs
pLas,ar
pLas,br
pLas,cr
Rs + pLbs
pLbcs
pLbs,ar
pLbs,br
pLbs,cr
pLbcs
Rs + pLcs
pLcs,ar
pLcs,br
pLcs,cr
pLar,bs
pLar,bs
Rr + pLar
pLar,br
pLar,cr
pLbr,bs
pLbr,cs
pLbr,ar
Rr + pLbr
pLbr,cr
pLcr,bs
pLcr,cs
pLcr,ar
pLcr,br
Rr + pLcr
p is the derivative operator , i.e. p = d/dt
ùé
úê
úê
úê
úê
úê
úê
úê
úê
úûë
ias ù
ú
ibs ú
ú
ics ú
iar ú
ú
ibr ú
icr úû
DYNAMIC MODEL –
3-phase model
Magnetic axis of
winding y
y
x
y’
Magnetic axis of
winding x
Windings x and y, with Nx and Ny
number of turns, separated by α
x’
It can be shown that the mutual inductance between winding x and y is
æ rl öæ p ö
Lxy =
= mo N x N y ç ÷ç ÷ cos a
Ix
è g øè 4 ø
y xy
DYNAMIC MODEL –
3-phase model
æ rl öæ p ö
Lxy =
= mo N x N y ç ÷ç ÷ cos a
Ix
è g øè 4 ø
y xy
• To get self inductance for phase a of the stator let Nx = Ny = Ns and α = 0
æ rl öæ p ö
Lam = mo N ç ÷ç ÷
è g øè 4 ø
2
s
• 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:
Labs  Lbcs  L acs 
2  rl 
oNs  
L abs  Lbcs  L acs 

o
 cos120
 g  4 
2  rl 
oNs  
Lms




2
 g  8 
DYNAMIC MODEL –
3-phase model
The mutual inductances between stator phases can be written in
terms of magnetising inductances
abcs ,s

L ms  L ls
 L
   ms
2

 L ms
  2

L ms

2
L ms  L ls
L ms

2


 ias 
 ibs 
 
 ics 
 L ls 

L ms

2
L ms

2
L ms
DYNAMIC MODEL –
3-phase model
æ rl öæ p ö
Lxy =
= mo N x N y ç ÷ç ÷ cos a
Ix
è g øè 4 ø
y xy
Self inductance between rotor windings is when Nx = Ny = Nr and α = 0
2
æ rl öæ p ö æ N 2 ö
æ
ö
æ
ö
æ
ö
Lmr = mo N ç ÷ç ÷ = ç r ÷ mo N s2 ç rl ÷ç p ÷ = ç N r ÷ Lms
è g øè 4 ø è N s2 ø
è g øè 4 ø è N s ø
2
r
•
The mutual inductances between rotor phases can be written in
terms of stator magnetising inductances
Y abcr,r
é
2
2
2
æ
ö
æ
ö
æ
ö
N
N
L
N
L
ê
r
Lms + Lls
- ç r ÷ ms
- ç r ÷ ms
ç
÷
ê èN ø
è Ns ø 2
è Ns ø 2
s
ê
2
2
2
ê
æ N r ö Lms
æ Nr ö
æ N r ö Lms
= ê -ç ÷
-ç ÷
ç ÷ Lms + Lls
N
2
N
è sø
è sø
è Ns ø 2
ê
ê
2
2
2
æ
ö
æ
ö
æ
ö
N
L
N
L
N
ê
r
ms
- ç r ÷ ms
ç r ÷ Lms + Lls
ê -ç N ÷ 2
è sø
è Ns ø 2
è Ns ø
êë
ù
ú
ú
úé i
úê as
úê ibs
úê
úêë ics
ú
ú
úû
ù
ú
ú
ú
úû
DYNAMIC MODEL –
3-phase model
The mutual inductances between the stator and rotor
windings depends on rotor position
æ rl öæ p ö
Lxy =
= mo N x N y ç ÷ç ÷ cos a
Ix
è g øè 4 ø
y xy
Self inductance between phase a of rotor and phase a of stator is when
Nx = Ns , Ny = Nr and α = θr
æ Nr ö
æ Nr ö
æ ö
æ rl öæ p ö
2 rl æ p ö
Las,ar = mo N s N r ç ÷ç ÷ cosq r = ç ÷ mo N s ç ÷ç ÷ cosq r = ç ÷ Lms cosq r
è g øè 4 ø
è Ns ø
è Ns ø
è g øè 4 ø
DYNAMIC MODEL –
3-phase model
The mutual inductances between the stator and rotor
windings depends on rotor position
abcs ,r
abcr ,s
 cos r

Nr
 L ms cos r  2
3
Ns

cos r  2 3




 cos r

Nr
 L ms cos  r  2
3
Ns

cos r  2 3



cos r  2
cos r
cos   2



r

3
3

cos r  2
cos r
cos   2

r


cos r  2
3
cos r  2
3
cosr


3
3



 i
 i


br 
 i cr 
cos r  2
3
cos r  2
3
cos r
ar
 i
 i


bs 
 i cs 
as
DYNAMIC MODEL –
abcs ,r
 cos r

Nr
 L ms cos r  2
3
Ns

cos r  2 3




3-phase model

cos r  2
cos r
cos   2

r

3
3



cos r  2
3
cos r  2
3
cosr
 i
 i


br 
 i cr 
ar
DYNAMIC MODEL –
abcs ,r
 cos r

Nr
 L ms cos r  2
3
Ns

cos r  2 3




3-phase model

cos r  2
cos r
cos   2

r

3
3



cos r  2
3
cos r  2
3
cosr
stator, b
rotor, a
rotor, b
r
stator, a
rotor, c
stator, c
 i
 i


br 
 i cr 
ar
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
There is magnetic coupling
between phases
Two-phase equivalent
There is NO magnetic
coupling between phases
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
stator, b
stator, q
r
rotor, a
rotor, b
r
rotating
rotating
rotor, 
r
rotor, 
stator, a
r
stator, d
rotor, c
stator, c
Three-phase
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
stator, q
r
rotating
rotor, 
However coupling still exists between
r
stator and
rotor, rotor windings
stator, d
Two-phase equivalent
DYNAMIC MODEL –
2-phase model
• All the 3-phase quantities have to be transformed to 2phase quantities
• In general if xa, xb, and xc are the three phase quantities, the
space phasor of the 3 phase systems is defined as:

2
x  x a  ax b  a 2 x c
3

, where a = ej2/3
x  x d  jxq


1
1 
2
 2
x d  Rex   Re  x a  ax b  a 2 x c    x a  x b  x c 
2
2 
3
 3


1
2

x b  x c 
x q  Imx   Im x a  ax b  a 2 x c  
3
3

DYNAMIC MODEL –
2-phase model
• All the 3-phase quantities have to be transformed into
2-phase quantities
is 

2
ia  aib  a 2ic
3
aib
2
ia
3

is  ids  jiqs
2
ai b
3
q
i ds
d
ia
is
2
a ic
i qs
2 2
a ic
3


is

1
1 
2
 2
ids  Re is  Re ia  ai b  a 2ic    ia  i b  ic 
2
2 
3
 3



1
2

ib  ic 
iqs  Im is  Im ia  ai b  a 2ic  
3
3

DYNAMIC MODEL –
2-phase model
• The transformation is given by:
i d   32  13
i   0 1
3
 q 
1
i o   13
3


1
3
1
3
1
3
 i a 
 
 i b 
  i c 
idqo = Tabc iabc
The inverse transform is given by:
iabc = Tabc-1 idqo
For isolated neutral,
ia + ib + ic = 0,
i.e. io =0
DYNAMIC MODEL –
2-phase model
IM equations :
3-phase
vabcs = Rsiabcs + d(abcs)/dt
2-phase
vdq = Rsidq + d(dq)/dt
vabcr = Rrriabcr + d(abr)/dt
v = Rrri  + d( )/dt
stator, q
r
rotating
rotor, 
rotor, 
r
stator, d
DYNAMIC MODEL –
2-phase model
vdq = Rsidq + d(dq)/dt
Express in stationary frame
Express in rotating frame
v = Rri  + d( )/dt
where,
dq  dqs,s  dqs,r

dqs ,s
r ,r
L dd

L qd
L 

L 
  r ,r   r ,s
L dq  i ds 
L qq  i qs 
L   ir 
L   ir 
dqs,r
L d L d  ir 

 i 
L
L
q   r 
 q
r ,s
L d L q  ids 

 i 
L
L
q   qs 
 d
DYNAMIC MODEL –
2-phase model
Note that:
Ldq = Lqd = 0
Ldd = Lqq
L = L = 0
L = L
The mutual inductance between stator and rotor depends on rotor
position:
Ld = Ld = Lsr cos r
Lq = Lq = Lsr cos r
Ld = Ld = - Lsr sin r
Lq = Lq = Lsr sin r
DYNAMIC MODEL –
2-phase model
stator, q
r
rotating
rotor, 
rotor, 
r
stator, d
Ld = Ld = Lsr cos r
Lq = Lq = Lsr cos r
Ld = Ld = - Lsr sin r
Lq = Lq = Lsr sin r
DYNAMIC MODEL –
2-phase model
In matrix form this an be written as:
 v d   R s  sLdd
  
0
vq   
v    sLsr cosr
  
 v    L sr sinr
0
R s  sLdd
sLsr sinr
sLsr cosr
sLsr cosr
sLsr sinr
R r  sL
0
 sLsr sinr  isd 
 
sLsr cosr  isq 

 ir 
0
  
R r  sL  ir 
• The mutual inductance between rotor and stator depends on
rotor position
DYNAMIC MODEL –
2-phase model
stator, q
rotor, q
The mutual inductance can be made
independent from rotor position by
expressing
both
rotor and stator in
stator, d 
rotor, d
the same reference frame, e.g. in the
Both stator and rotor
stationary reference
frame
rotating or stationary
r
Magnetic path from stator linking the
rotor winding independent of rotor
position  mutual inductance
independent of rotor position
DYNAMIC MODEL –
2-phase model
How do we express rotor current in stator (stationary) frame?
ir 

2
ira  airb  a 2irc
3

ir is known as the space vector of
the rotor current
In rotating frame this can be written as: i r  i r e jr

In stationary frame it be written as:
qs
ir
irq
r

r

ird
ds
irs  i r e jr  
 ir e j
 ird  jirq
DYNAMIC MODEL –
2-phase model
If the rotor quantities are referred to stator, the following can be
written:
v sd  R s  sLs
  
v sq    0
 v rd   sLm
  
 v rq    r L m
0
R s  sLs
r L m
sLm
sLm
0
R r ' sLr
 r L r
0  isd 
 

sLm  isq 

r L r   ird 
  
R r ' sLr   irq 
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:
v sd   R s  sLs
  
v sq    gL s
 v rd  
sLm
  
 v rq  (g  r )L m
 gL s
R s  sLs
 (g  r )L m
sLm
sLm
gL m
R r ' sLr
(g  r )L r
 gL m  isd 
  
sLm
  isq 
 (g  r )L r   ird 
  
R r ' sLr   irq 
DYNAMIC MODEL  Space vectors
IM can be compactly written using space vectors:
g
d

s
v sg  R s isg 
 jg sg
dt
sg  L s isg  L m i rg
d  rg
rg  L r irg  L m isg
0  R r i rg 
dt
 j( g   r ) rg
All quantities are written in general
reference frame
DYNAMIC MODEL  Torque equation
Product of voltage and current conjugate space vectors:
vs is* 

 
2
2
v as  av bs  a 2 v cs
i as  a 2i bs  ai cs
3
3

It can be shown that for ias + ibs + ics = 0,
 
2
Re v i  v as ias  v bs i bs  v cs ics 
3
*
s s
Pin = ( v as ias + v bs ibs + v cs ics ) =
3
Re[ vs is* ]
2
DYNAMIC MODEL  Torque equation
3
3
3
*
Pin = Re[ vs is ] = Re[ (v d + jv q )(id - jiq )] = [ v d id + vq iq ]
2
2
2
If
v d 
v 
v q 
and
3 t
Pin  i v
2
i d 
i 
i q 
Pin = ( v as ias + v bs ibs + v cs ics ) =
3
Re[ vs is* ]
2
DYNAMIC MODEL  Torque equation
The IM equation can be written as:
[] [] [ ]
[]
v = R i+ L si+ G wr i+ F w g i
The input power is given by:
pi =
3 t
3
i V = [ it [ R] i + it [ L]si + it [G ] w r i + it [ F] w g i]
2
2
Power
Losses in winding
resistance
Rate of change
of stored
magnetic energy
Power
Mech power associated with
g – upon
expansion gives
zero
DYNAMIC MODEL  Torque equation
p mech = w m Te =
v sd   R s  sLs
  
v sq    gL s
 v rd  
sLm
  
 v rq  (g  r )L m
3 t
i [G ] w r i
2
 gL s
R s  sLs
 (g  r )L m
sLm
 0
 0

 0

 r L m
0
0
r L m
0
sLm
gL m
R r ' sLr
(g  r )L r
0
0
0
 r L r
0  isd 
 
0  isq 

r L r   ird 
  
0   irq 
 gL m  isd 
  
sLm
  isq 
 (g  r )L r   ird 
  
R r ' sLr   irq 
DYNAMIC MODEL  Torque equation
p mech = w m Te =
isd 
 
3 isq 
mTe 
2  ird 
 
 irq 
t
3 t
i [G ] w r i
2
0




0


 L misq  L r irq  r



L
i

L
i
 m sd r rd 
We know that m = r / (p/2),
Te =
3p
L m ( isq ird - isd irq )
22
DYNAMIC MODEL  Torque equation
Te =
but
( )
3p
L m Im is ir'*
22
s  L s is  L m ir  L m ir  s  L s is
Te =
(
3p
Im is ( y*s - L s i*s )
22
( )
)
3p
3p
*
3
p
\ Te =
Im is ys
T
ys ´ is
e i=
T
=
L
i
i
i
e
m ( sq rd
sd rq ) 2 2
22
22
DYNAMIC MODEL
Simulation
Re-arranging with stator and rotor currents as state space
variables:
isd 
 
1
isq  
ird  L2m  L r L s
 
 irq 
 R sL r

2
 r L m
  R sL m

 r L mL s
 r L2misq
R sL r
r L mL s
 R sL m
 R rLm
r L mL r
R rL s
 r L r L s
 r L mL r  isd 
  
 R r L m  isq 
1

 2
r L r L s   ird  L m  L r L s
  
R r L s   irq 
The torque can be expressed in terms of stator and rotor currents:
Te =
3p
L m [ isq ird - isd irq ]
22
Te - TL = J
dw m
dt
 L r
 0

 Lm

 0
0 
 L r  v sd 
 
0  v sq 

Lm 
Which finally can be modeled using SIMULINK: