Lecture Notes
M.Kaliamoorthy AP/EEE, PSNACET
3-Phase Induction Machines - Dynamic Modeling Using Reference Frame Theory
Winding arrangement for a 2-pole, 3-phase, wye-connected symmetrical induction
machine is shown in Fig.1. Stator windings are identical, sinusoidally distributed
windings, displaced by120°, with Ns equivalent turns and resistance rs. Consider the case
when rotor windings are also three identical sinusoidally distributed windings, displaced
by120°, with Nr equivalent turns and resistance rr.
Fig1. 2 pole Three Phase Induction Machine
In abc reference frame, voltage equations can be written as
Vabcs = rs iabcs + pλabcs
Vabcr = rr iabcr + pλabcr
( f abcs )T = [ f as
f bs
f cs ], ( f abcr )T = [ f ar
f br
f cr ]
s: denotes variables and parameters associated with the stator circuits and r: denotes
variables and parameters associated with the rotor circuits.
Lecture Notes
M.Kaliamoorthy AP/EEE, PSNACET
λabcs   Ls
λ  = ( L ) T
 abcr   sr
Lsr  iabcs 
Lr  iabcr 
Where,

 Lls + Lms
 1
L s =  − Lms
 2
−1L
 2 ms
1
− Lms
2
Lls + Lms
1
− Lms
2
1

− Lms 
2

1
− Lms ,
2

Lls + Lms 


 Llr + Lmr
 1
L r =  − Lmr
 2
−1L
 2 mr
1
− Lmr
2
Llr + Lmr
1
− Lmr
2





Llr + Lmr 

1
− Lmr
2
1
− Lmr
2
Lls and Lms are, respectively, the leakage and magnetizing inductance of the stator
windings. Llr and Lmr are, respectively, the leakage and magnetizing inductance of the
rotor windings.
2π
2π 

cos(θ r +
) cos(θ r − )
 cos θ r
3
3

2π
2π 
cos θ r
cos(θ r +
) ,
L sr = L rs = Lms cos(θ r − )
3
3 

cos(θ + 2π ) cos(θ − 2π )

cos θ r
r
r


3
3
“Lsr” is the amplitude of the mutual inductances between stator and rotor windings. A
majority of induction machines are not equipped with coil-wound rotor windings;
instead, the current flows in copper or aluminum bars which are uniformly distributed
in a common ring at each end of the rotor. This type of rotor is referred to as a
squirrel-cage rotor. Rotor variables can be referred to the stator windings by
appropriate turn’s ratio.
2
′
iabcr
Also,
N 
N
N
N
′ = s Vabcr , λabcr
′ = s λabcr , Lms =  s  Lsr
= r iabcr , Vabcr
Ns
Nr
Nr
 Nr 
2π
2π 

cos(θ r +
) cos(θ r −
)
 cos θ r
3
3 

2π 
cos θ r
cos(θ r +
) ,
[L′sr ] = N s [L sr ] = Lms cos(θ r − 2π )
3
3 
Nr

cos(θ + 2π ) cos(θ − 2π )

cos θ r
r
r


3
3
2
N 
Lmr =  r  Lms ,
 Ns 
2


[L′r ] =  N r  [L r ]
 Ns 
Lecture Notes
M.Kaliamoorthy AP/EEE, PSNACET

 Llr′ + Lmr

[L′r ] =  − 1 Lms
 2
−1L
 2 ms
−
1
Lms
2
Llr + Lms
−
1
Lms
2





Llr + Lms 

1
− Lms
2
1
− Lms
2
Where,
N
Llr′ =  s
 Nr
2

 Llr

Flux linkage may be expressed as
λabcs   L s
 λ ′  =  ( L′ ) T
 abcr   sr
L′sr  iabcs 
′ 
L′r  iabcr
Voltage equations expressed in terms of machine variables referred to the stator windings
may be written as
Vabcs  rs + pL s
V ′  =  p (L′ )T
sr
 abcr  
pL′sr  iabcs 
′ 
rr′ + pL′r  iabcr
Where,
N
rr′ =  s
 Nr
2

 rr

Energy stored in the coupling field may be written as
1
(iabcs )T (L s − L ls I )iabcs +
2
1
′ + (iabcr
′ )T (L′r − L′lr I )iabcr
′
(iabcs )T (L sr )iabcr
2
Wc = W f =
Where, I: identity matrix
Voltage equations expressed in terms of machine variables referred to the stator windings
may be written as
P ∂Wc (i j , θ r )
Te (i j , θ r ) =
2
∂θ r
Since Ls and Lr are functions of θr, the above equation for the electromagnetic torque
yields.
Lecture Notes
M.Kaliamoorthy AP/EEE, PSNACET
p
∂
′
Te = ( )(iabcs )T
[L′sr ]iabcr
2
∂θ r


1
1
1
1
1
1

′ − ibr
′ − icr
′ ) + ibs (ibr
′ − iar
′ − icr
′ ) + ics (icr
′ − ibr
′ − iar
′ ) sin θ r 
ias (iar
2
2
2
2
2
2
P



= − Lms 

2
 3

′
′
′
′
′
′
+ 2 [ias (ibr − icr ) + ibs (icr − iar ) + ics (iar − ibr )]cos θ r

The torque and rotor speed are related by
2
Te = J   pωr + TL
P
Equations of Transformation for Rotor Circuit
In the analysis of induction machines it is desirable to transform the variables associated
with the symmetrical rotor windings to the arbitrary reference frame.

cos β
2
( f qd′ 0 r )T = f qr′ f dr′ f 0′r
K r =  sin β
3
′ )T = [ f ar′ f br′ f cr′ ]
( f abcr
 1
 2
where, β=θ-θr from figure below
′
f qd′ 0 r = K r f abcr
[
]
2π
2π 
) cos( β +
)
3
3 
2π
2π 
sin( β − ) sin( β +
),
3
3 
1
1


2
2
cos( β −
Fig. 2. Axis of 2-pole, 3-phase Symmetrical machine.
t
θ r = ∫ ωr (t )dt +θ r (0)
0
Lecture Notes
M.Kaliamoorthy AP/EEE, PSNACET


 cos β
sin β
1


(K r )−1 = cos(β − 2π ) sin( β − 2π ) 1
3
3


2
2
π
π
cos( β +
) sin( β +
) 1

3
3

“r” subscript indicates the variable, parameters and transformation associated with
rotating circuits.
Voltage Equations in Arbitrary Reference Frame Variables
For two-pole, 3-phase symmetrical induction,
Vabcs = rs iabcs + pλabcs
′ = rr′iabcr
′ + pλabcr
′
Vabcr
′
λabcs = (Ls )iabcs + (Lsr′ )iabcr
Vabcs = K sVqd 0 s , iabcs = K s iqd 0 s
′ = (Lsr′ ) iabcs + (Lr′ )iabcr
′
λabcr
′ = K rVqd′ 0 r , iabcr
′ = K r iqd′ 0 s
Vabcr
T
Using the above transformation equations, we can transform the voltage equations to an
arbitrary reference frame rotating at speed of ω.
Vqd 0 s = rs iqd 0 s + ωλqds + pλqd 0 s
′ + pλqd
′ 0r
Vqd′ 0 r = rr′iqd 0 r + (ω − ω r )λqdr
[
where, (λqds )T = λds
− λqs
]
[
′ )T = λdr′
0 , (λqdr
− λqr′
0
]
Flux linkage equations in abc reference frame can be transformed to qd axes using Ks and
Kr transformation matrices.
λqd 0 s   K s L s (K s )−1 K s L′sr (K r )−1  iqd 0 s 

=


′ 0 r 
′ 0 r  K r L′sr (K s )−1 K r L′r (K r )−1  iqd
λqd
Where
K s L s (K s )
−1
K r L′r (K r )
−1
 Lls + M
=  0
 0
 Llr′ + M
=  0
 0
0
Lls + M
0
0
Llr′ + M
0

3
0 , M = Lms
2
Lls + M 
0

3
0 , M = Lms
2
Llr′ + M 
0
Lecture Notes
M.Kaliamoorthy AP/EEE, PSNACET
M 0 0 
K s L′sr (K r ) = K r (L′sr ) (K s ) =  0 M 0 
 0 0 M 
Voltage equations written in expanded form can be expressed as
−1
T
−1
Vqs = rs iqs + ωλds + pλqs
Vqr′ = rr′iqr′ + (ω − ωr )λdr′ + pλqr′
Vds = rs ids − ωλqs + pλds ,
Vdr′ = rr′idr′ − (ω − ωr )λqr′ + pλdr′
V0 s = rs i0 s + pλ0 s
V0′r = rr′i0′ r + pλ0′r
Flux linkage equations are
λqs = Lls iqs + M (iqs + iqr′ )
λds = Lls ids + M (ids + idr′ )
λ0 s = Lls i0 s
λqr′ = Llr′ iqr′ + M (iqs + iqr′ )
λdr′ = Llr′ idr′ + M (ids + idr′ )
λ0′r = Llr′ i0′ r
Since machine and power system parameters are nearly always given in ohms or percent
or per unit of a base impedance, it is convenient to express the voltage and flux linkage
equations in terms of reactances rather than inductances.
Let
ϕ = λωb
Then
ω
ϕ ds + pϕ qs
ωb
ω
p
Vds = rs ids − ϕ qs + ϕ ds ,
ωb
ωb
Vqs = rs iqs +
V0 s = rs i0 s +
p
ωb
ϕ0s
(ω − ωr ) ϕ ′
p
ϕ qr′
ωb
ωb
(ω − ωr ) ϕ ′ + p ϕ ′
Vdr′ = rr′idr′ −
qr
dr
ωb
ωb
Vqr′ = rr′iqr′ +
V0′r = rr′i0′ r +
dr
p
ωb
+
ϕ0′ r
And flux linkages become flux linkages per second with the units of volts.
ϕ qs = X ls iqs + X m (iqs + iqr′ )
ϕ qr′ = X lr′ iqr′ + X m (iqs + iqr′ )
ϕ ds = X ls ids + X m (ids + idr′ ),
ϕ 0 s = X ls i0 s
ϕ dr′ = X lr′ idr′ + X m (ids + idr′ )
ϕ 0′ r = X lr′ i0′ r
Lecture Notes
M.Kaliamoorthy AP/EEE, PSNACET
Equivalent circuits of a 3-phase, symmetrical induction machine with rotating q-d axis at
speed of ω.
Equivalent circuits of a 3-phase, symmetrical induction machine with rotating q-d axis at
speed of ω.
Equivalent circuits of a 3-phase, symmetrical induction machine with rotating q-d axis at
speed of ω.
Lecture Notes
M.Kaliamoorthy AP/EEE, PSNACET
Electromagnetic torque in terms of arbitrary reference frame variables may be obtained
by substituting the equations of transformation in
Te =
P
′
(iabcs )T ∂ (Lsr′ )iabcr
2
∂θ r
[
]
T ∂
P
(Lsr′ )(K r ) −1iqd′ 0r
(K s ) −1 iqd 0 s
2
∂θ r
After some work, we will have the following:
=
 3  P 
′ − ids iqr
′ )
Te =    M (iqs idr
 2  2 
Where, Te is positive for motor action. Other expressions for the electromagnetic torque
of an induction machine are
 3  P 
Te =   (λqr′ idr′ − λdr′ iqr′ )
 2  2 
 3  P 
Tem =   (λds iqs − λqs ids )
 2  2 
 3  P  1 
Te =    (ϕ qr′ idr′ − ϕ dr′ iqr′ )
 2  2  ωb 