Nano and Microelectromechanical Systems P5

3.4. INDUCTION MOTORS
In this section, the following variables and symbols are used:
u u
as bs
,
and
u
cs
are the phase voltages in the stator windings as, bs and cs;
u u
qs ds
,
and
u
os
are the quadrature-, direct-, and zero-axis components of
stator voltages;
i i
as bs
,
and
i
cs
are the phase currents in the stator windings as, bs and cs;
i i
qs ds
,
and
i
os
are the quadrature-, direct-, and zero-axis components of

stator currents;
ψ ψ
as bs
,
and
ψ
cs
are the stator flux linkages;
ψ ψ
qs ds
,
and
ψ
os
are the quadrature-, direct-, and zero-axis components of
stator flux linkages;
u u
ar br
,
and
u
cr
are the voltages in the rotor windings ar, br and cr;
u u
qr dr
,
and
u
or
are the quadrature-, direct-, and zero-axis components of

rotor voltages;
i i
ar br
,
and
i
cr
are the currents in the rotor windings ar, br and cr;
i i
qr dr
,
and
i
or
are the quadrature-, direct-, and zero-axis components of
rotor currents;
ψ ψ
ar br
,
and
ψ
cr
are the rotor flux linkages;
ψ ψ
qr dr
,
and
ψ
or
are the quadrature-, direct-, and zero-axis components of

rotor flux linkages;
r
ω
and
rm
ω
are the electrical and mechanical angular velocities;
r
θ
and
rm
θ
are the electrical and mechanical angular displacements;
T
e
is the electromagnetic torque developed by the motor;
T
L
is the load torque applied;
s
r
and
r
r
are the resistances of the stator and rotor windings;
L
ss
and
rr
L

are the self-inductances of the stator and rotor windings;
ms
L
is the stator magnetizing inductance;
ls
L
and
lr
L
are the stator and rotor leakage inductances;
N
s
and
N
r
are the number of turns of the stator and rotor windings;
P is the number of poles;
m
B
is the viscous friction coefficient;
J
is the equivalent moment of inertia;
ω
and
θ
are the angular velocity and displacement of the reference frame.
© 2001 by CRC Press LLC
3.4.1. Two-Phase Induction Motors
Two-phase induction motors, shown in Figure 3.4.1, have two stator and
rotor windings.

r
s
+
−
+−
L
ss
r
s
L
ss
i
bs
u
bs
u
as
i
as
N
s
ω
r e
T,
T
L
θ ω θ
r r r
t= +
0

B
m
as'
bs
as
bs'
br
br'
ar
ar'
as Magnetic Axis
ar Magnetic Axis
br Magnetic Axis
bs Magnetic Axis
r
r
+
−
+
−
L
rr
r
r
L
rr
i
br
u
br

u
ar
i
ar
N
r
Stator
Rotor
Load
Magnetic Coupling
Figure 3.4.1. Two-phase symmetrical induction motor
To develop a mathematical model of two-phase induction motors, we
model the stator and rotor circuitry dynamics. As the control and state
variables we use the voltages applied to the stator (as and bs) and rotor (ar
and br) windings, as well as the stator and rotor currents and flux linkages.
Using Kirchhoff’s voltage law, four differential equations are
u r i
d
dt
as s as
as
= +
ψ
,
u r i
d
dt
bs s bs
bs
= +

ψ
,
u r i
d
dt
ar r ar
ar
= +
ψ
,
u r i
d
dt
br r br
br
= +
ψ
.
Hence, in matrix form we have
u r i
abs s abs
abs
d
dt
= +
ψψ
,
u r i
abr r abr
abr

d
dt
= +
ψψ
, (3.4.1)
© 2001 by CRC Press LLC
where
u
abs
as
bs
u
u
=






,
u
abr
ar
br
u
u
=







,
i
abs
as
bs
i
i
=






,
i
abr
ar
br
i
i
=







,
ψψ
abs
as
bs
=






ψ
ψ
, and
ψψ
abr
ar
br
=






ψ
ψ

are the phase voltages, currents, and
flux linkages;
r
s
s
s
r
r
=






0
0
and
r
r
r
r
r
r
=







0
0
are the matrices of the
stator and rotor resistances.
Studying the magnetically coupled motor circuits, the following matrix
equation for the flux linkages is found
ψψ
ψψ
abs
abr
s sr
sr
T
r
abs
abr






=













L L
L L
i
i
,
where
L
s
is the matrix of the stator inductances,
L
s
ss
ss
L
L
=






0
0

,
L L L
ss ls ms
= +
,
L
N
ms
s
m
=
ℜ
2
;
L
r
is the matrix of the rotor inductances,
L
r
rr
rr
L
L
=







0
0
,
L L L
rr lr mr
= +
,
L
N
mr
r
m
=
ℜ
2
;
L
sr
is the matrix of the
stator-rotor mutual inductances,
L
sr
sr r sr r
sr r sr r
L L
L L
=
−







cos sin
sin cos
θ θ
θ θ
,
L
N N
sr
s r
m
=
ℜ
.
Using the number of turns in the stator and rotor windings, we have
i i
abr
r
s
abr
N
N
'
=
,
u u
abr

s
r
abr
N
N
'
=
, and
ψψ ψψ
abr
s
r
abr
N
N
'
=
.
Then, taking note of the turn ratio, the flux linkages are written in matrix
form as
ψψ
ψψ
abs
abr
s sr
sr
T
r
abs
abr

'
'
' '
'






=












L L
L L
i
i
, (3.4.2)
where
L L

r
s
r
r
rr
rr
N
N
L
L
'
'
'
=






=






2
0
0

,
L L L
rr lr mr
' ' '
= +
;
© 2001 by CRC Press LLC
L L
sr
s
r
sr ms
r r
r r
N
N
L
'
cos sin
sin cos
=






=
−







θ θ
θ θ
,
L
N
N
L
ms
s
r
sr
=
,
L
N
N
L
mr
s
r
mr
'
=







2
,
L L
N
N
L
mr ms
s
r
sr
'
= =
,
L L L
rr lr ms
' '
= +
.
Substituting the matrices for self- and mutual inductances
L
s
,
L
r
'
and

L
sr
'
in (3.4.2), one obtains
ψ
ψ
ψ
ψ
θ θ
θ θ
θ θ
θ θ
as
bs
ar
br
ss ms r ms r
ss ms r ms r
ms r ms r rr
ms r ms r rr
as
bs
ar
br
L L L
L L L
L L L
L L L
i
i

i
i
'
'
'
'
'
'
cos sin
sin cos
cos sin
sin cos












=
−
−

























0
0
0
0
.
Therefore, the circuitry differential equations (3.4.1) are rewritten as
u r i
abs s abs
abs

d
dt
= +
ψψ
,
u r i
abr r abr
abr
d
dt
' ' '
'
= +
ψψ
.
where
r r
r
s
r
r
s
r
r
r
N
N
N
N
r

r
'
'
'
= =






2
2
2
2
0
0
.
Assuming that the self- and mutual inductances
L L L
ss rr ms
, ,
'
are time-
invariant and using the expressions for the flux linkages, one obtains a set of
nonlinear differential equations to model the circuitry dynamics
( ) ( )
L
di
dt

L
d i
dt
L
d i
dt
r i u
ss
as
ms
ar r
ms
br r
s as as
+ − = − +
' '
cos sinθ θ
,
( ) ( )
L
di
dt
L
d i
dt
L
d i
dt
r i u
ss

bs
ms
ar r
ms
br r
s bs bs
+ + = − +
' '
sin cosθ θ
,
( ) ( )
L
d i
dt
L
d i
dt
L
di
dt
r i u
ms
as r
ms
bs r
rr
ar
r ar ar
cos sin
'

'
' ' '
θ θ
+ + = − +
,
( ) ( )
− + + = − +L
d i
dt
L
d i
dt
L
di
dt
r i u
ms
as r
ms
bs r
rr
br
r br br
sin cos
'
'
' ' '
θ θ
.
Cauchy’s form of these differential equations is found. In particular, we

have the following nonlinear differential equations to model the stator-rotor
circuitry dynamics for two-phase induction motors
© 2001 by CRC Press LLC
di
dt
L r
L L L
i
L
L L L
i
L L
L L L
i
r
L
L L
L L L
i
r
L
L
L L L
u
L
L L L
u
as rr s
ss rr ms
as

ms
ss rr ms
bs r
ms rr
ss rr ms
ar r r
r
rr
r
ms rr
ss rr ms
br r r
r
rr
r
rr
ss rr ms
as
ms
ss rr ms
r ar
= −
−
+
−
+
−
+







+
−
−






+
−
−
−
+
'
' '
'
'
'
'
'
'
'
'
'
'

'
' '
'
sin cos
cos sin cos
2
2
2 2
2 2 2
ω ω θ θ
ω θ θ θ
L
L L L
u
ms
ss rr ms
r br
'
'
sin ,
−
2
θ
di
dt
L r
L L L
i
L
L L L

i
L L
L L L
i
r
L
L L
L L L
i
r
L
L
L L L
u
L
L L L
u
bs rr s
ss rr ms
bs
ms
ss rr ms
as r
ms rr
ss rr ms
ar r r
r
rr
r
ms rr

ss rr ms
br r r
r
rr
r
rr
ss rr ms
bs
ms
ss rr ms
r ar
= −
−
−
−
−
−
−






+
−
+







+
−
−
−
−
'
' '
'
'
'
'
'
'
'
'
'
'
'
' '
'
cos sin
sin cos sin
2
2
2 2
2 2 2
ω ω θ θ

ω θ θ θ
L
L L L
u
ms
ss rr ms
r br
'
'
cos ,
−
2
θ
di
dt
L r
L L L
i
L L
L L L
i
r
L
L L
L L L
i
r
L
L
L L L

i
L
L L L
u
L
L L L
u
L
L
ar ss r
ss rr ms
ar
ms ss
ss rr ms
as r r
s
ss
r
ms ss
ss rr ms
bs r r
s
ss
r
ms
ss rr ms
br r
ms
ss rr ms
r as

ms
ss rr ms
r bs
ss
' '
'
'
' '
'
'
' '
sin cos cos sin
cos sin
= −
−
+
−
+






−
−
−







−
−
−
−
−
−
+
2 2 2
2
2 2 2
ω θ θ ω θ θ
ω θ θ
ss rr ms
ar
L L
u
'
'
,
−
2
di
dt
L r
L L L
i
L L

L L L
i
r
L
L L
L L L
i
r
L
L
L L L
i
L
L L L
u
L
L L L
u
L
L
br ss r
ss rr ms
br
ms ss
ss rr ms
as r r
s
ss
r
ms ss

ss rr ms
bs r r
s
ss
r
ms
ss rr ms
ar r
ms
ss rr ms
r as
ms
ss rr ms
r bs
ss
' '
'
'
' '
'
'
' '
cos sin sin cos
sin cos
= −
−
+
−
−







+
−
+






+
−
+
−
−
−
+
2 2 2
2
2 2 2
ω θ θ ω θ θ
ω θ θ
ss rr ms
br
L L
u

'
'
.
−
2
(3.4.3)
The electrical angular velocity
ω
r
and displacement
θ
r
are used in
(3.4.3) as the state variables. Therefore, the torsional-mechanical equation of
motion must be incorporated to describe the evolution of
ω
r
and
θ
r
. From
Newton’s second law, we have
T T B T J
d
dt
e m rm L
rm
∑
= − − =ω
ω

,
d
dt
rm
rm
θ
ω=
.
The mechanical angular velocity
ω
rm
is expressed by using the
electrical angular velocity
ω
r
and the number of poles P. In particular,
ω ω
rm r
P
=
2
.
The mechanical and electrical angular displacements
θ
rm
and
θ
r
are
related as

θ θ
rm r
P
=
2
.
Taking note of Newton’s second law of motion, one obtains
© 2001 by CRC Press LLC
d
dt
P
J
T
B
J
P
J
T
r
e
m
r L
ω
ω= − −
2 2
,
d
dt
r
r

θ
ω=
.
To find the expression for the electromagnetic torque developed by two-
phase induction motors, the coenergy
( )
rabrabsc
W θ,,
'
ii
is used, and
( )
r
rabrabsc
e
WP
T
∂θ
θ∂ ,,
2
'
ii
=
.
Assuming that the magnetic system is linear, one has
(
)
( )
W W L L
c f abs

T
s ls abs abs
T
sr abr abr
T
r lr abr
= = − + + −
1
2
1
2
i L I i i L i i L I i
' ' ' ' ' '
.
The self-inductances
L
ss
and
L
rr
'
, as well as the leakage inductances
L
ls
and
L
lr
'
, are not functions of the angular displacement
θ

r
, while the
following expression for the matrix of stator-rotor mutual inductances
L
sr
'
was derived
L
sr ms
r r
r r
L
'
cos sin
sin cos
=
−






θ θ
θ θ
.
Then, for P-pole two-phase induction motors, the electromagnetic torque
is given by
(
)

[ ]
(
)
(
)
[ ]
T
P
W
P P
L i i
i
i
P
L i i i i i i i i
e
c abs abr r
r
abs
T
sr r
r
abr ms as bs
r r
r r
ar
br
ms as ar bs br r as br bs ar r
= = =
− −

−












= − + + −
2 2 2
2
∂ θ
∂θ
∂ θ
∂θ
θ θ
θ θ
θ θ
i i
i
L
i
, ,
( )
sin cos

cos sin
sin cos .
'
'
'
'
'
' ' ' '
(3.4.4)
Using (3.4.4) for the electromagnetic torque
T
e
in the torsional-
mechanical equations of motion, one obtains
(
)
(
)
[ ]
d
dt
P
J
L i i i i i i i i
B
J
P
J
T
r

ms as ar bs br r as br bs ar r
m
r L
ω
θ θ ω= − + + − − −
2
4 2
' ' ' '
sin cos
d
dt
r
r
θ
ω=
. (3.4.5)
Augmenting differential equations (3.4.3) and (3.4.5), the following set
of highly nonlinear differential equations results
© 2001 by CRC Press LLC
,sincossincos
cossin
''
'
'
'
'
'
'
'
'

'2'
brr
ms
arr
ms
as
rr
r
rr
r
rrbr
rrms
r
rr
r
rrar
rrms
rbs
ms
as
srras
u
L
L
u
L
L
u
L
L

L
r
i
L
LL
L
r
i
L
LL
i
L
L
i
L
rL
dt
di
θθθθω
θθωω
ΣΣΣΣ
ΣΣΣ
+−+









−+








+++−=
,cossincossin
sincos
''
'
'
'
'
'
'
'
'
'2'
brr
ms
arr
ms
bs
rr
r

rr
r
rrbr
rrms
r
rr
r
rrar
rrms
ras
ms
bs
srrbs
u
L
L
u
L
L
u
L
L
L
r
i
L
LL
L
r
i

L
LL
i
L
L
i
L
rL
dt
di
θθθθω
θθωω
ΣΣΣΣ
ΣΣΣ
−−+








++









−−−−=
,sincos
sincoscossin
''
2
'
''
ar
ss
bsr
ms
asr
ms
rbr
ms
r
ss
s
rrbs
ssms
r
ss
s
rras
ssms
ar
rssar
u

L
L
u
L
L
u
L
L
i
L
L
L
r
i
L
LL
L
r
i
L
LL
i
L
rL
dt
di
ΣΣΣΣ
ΣΣΣ
+−−−









−−








++−=
θθω
θθωθθω
,cossin
cossinsincos
''
2
'
''
br
ss
bsr
ms
asr

ms
rar
ms
r
ss
s
rrbs
ssms
r
ss
s
rras
ssms
br
rssbr
u
L
L
u
L
L
u
L
L
i
L
L
L
r
i

L
LL
L
r
i
L
LL
i
L
rL
dt
di
ΣΣΣΣ
ΣΣΣ
+−++








++









−+−=
θθω
θθωθθω
( ) ( )
[ ]
d
dt
P
J
L i i i i i i i i
B
J
P
J
T
r
ms as ar bs br r as br bs ar r
m
r L
ω
θ θ ω= − + + − − −
2
4 2
' ' ' '
sin cos
,
d
dt

r
r
θ
ω= ,
(3.4.6)
where
L L L L
ss rr msΣ
= −
' 2
.
In matrix form, a set of six highly coupled nonlinear differential
equations (3.4.6) is
© 2001 by CRC Press LLC
di
dt
di
dt
di
dt
di
dt
d
dt
d
dt
L r
L
L r
L

L r
L
L r
L
B
J
as
bs
ar
br
r
r
rr s
rr s
ss r
ss r
m
'
'
'
'
'
'
ω
θ

































=
−
−

−
−
−































Σ
Σ
Σ
Σ
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 1 0



























i
i
i
i
as
bs
ar
br
r
r
'
'
ω
θ
+
+ +







+ −






− − −






+ +






L
L
i
L L
L
i
r
L

L L
L
i
r
L
L
L
i
L L
L
i
r
L
L L
L
i
r
L
L L
ms
bs r
ms rr
ar r r
r
rr
r
ms rr
br r r
r
rr

r
ms
as r
ms rr
ar r r
r
rr
r
ms rr
br r r
r
rr
r
ms
2
2
Σ Σ Σ
Σ Σ Σ
ω ω θ θ ω θ θ
ω ω θ θ ω θ θ
'
'
'
'
'
'
'
'
'
'

'
'
'
'
'
'
sin cos cos sin
cos sin sin cos
(
)
ss
as r r
s
ss
r
ms ss
bs r r
s
ss
r
ms
br r
ms ss
as r r
s
ss
r
ms ss
bs r r
s

ss
r
ms
ar r
ms as ar bs br r
L
i
r
L
L L
L
i
r
L
L
L
i
L L
L
i
r
L
L L
L
i
r
L
L
L
i

P
J
L i i i i
Σ Σ Σ
Σ Σ Σ
ω θ θ ω θ θ ω
ω θ θ ω θ θ ω
θ
sin cos cos sin
cos sin sin cos
sin
'
'
' '
+






− −






−
−







+ +






+
− +
2
2
2
4
(
)
[ ]
+ −








































i i i i
as br bs ar r
' '
cosθ
0
+



















































+
− +
− −

−
L
L
L
L
L
L
L
L
u
u
u
u
L
L
u
L
L
u
L
L
u
L
L
u
L
L
rr
rr
ss

ss
as
bs
ar
br
ms
r ar
ms
r br
ms
r ar
ms
r br
ms
'
'
'
'
' '
' '
cos sin
sin cos
cos
Σ
Σ
Σ
Σ
Σ Σ
Σ Σ
Σ

0 0 0
0 0 0
0 0 0
0 0 0
0 0 0 0
0 0 0 0
θ θ
θ θ
θ
r as
ms
r bs
ms
r as
ms
r bs
L
u
L
L
u
L
L
u
L
L
u
P
J
T

−
−



































−































Σ
Σ Σ
sin
sin cos
θ
θ θ
0
0
0
0
0
0
2
0
(3.4.7)
© 2001 by CRC Press LLC
Modeling Two-Phase Induction Motors Using the Lagrange Equations
The mathematical model can be derived using Lagrange’s equations.
The generalized independent coordinates and the generalized forces are
q
i
s
as

1
=
,
q
i
s
bs
2
=
,
q
i
s
ar
3
=
'
,
q
i
s
br
4
=
'
,
q
r5
= θ
,

and
Q u
as1
=
,
Q u
bs2
=
,
Q u
ar3
=
'
,
Q u
br4
=
'
,
Q T
L5
= −
Five Lagrange equations are written as
d
dt q q
D
q q
Q
∂
∂

∂
∂
∂
∂
∂
∂
Γ Γ Π
& &
1 1 1 1
1






− + + =
,
d
dt q q
D
q q
Q
∂
∂
∂
∂
∂
∂
∂

∂
Γ Γ Π
& &
2 2 2 2
2






− + + =
,
d
dt q q
D
q q
Q
∂
∂
∂
∂
∂
∂
∂
∂
Γ Γ Π
& &
3 3 3 3
3







− + + =
,
d
dt q q
D
q q
Q
∂
∂
∂
∂
∂
∂
∂
∂
Γ Γ Π
& &
4 4 4 4
4







− + + =
,
d
dt q q
D
q q
Q
∂
∂
∂
∂
∂
∂
∂
∂
Γ Γ Π
& &
5 5 5 5
5






− + + =
.
The total kinetic, potential, and dissipated energies are
,cossin

sincos
2
5
2
1
2
4
'
2
1
2
3
'
2
1
542532
2
2
2
1
541531
2
1
2
1
qJqLqLqqqLqqqL
qLqqqLqqqLqL
rrrrmsms
ssmsmsss
&&&&&&&

&&&&&&
+++++
+−+=Γ
Π = 0
,
( )
D r q r q r q r q B q
s s r r m
= + + + +
1
2
1
2
2
2
3
2
4
2
5
2
& & & & &
' '
.
Thus,
∂
∂
Γ
q
1

0
=
,
∂
∂
Γ
&
& &
cos
&
sin
q
L q L q q L q q
ss ms ms
1
1 3 5 4 5
= + −
,
∂
∂
Γ
q
2
0
=
,
∂
∂
Γ
&

& &
sin
&
cos
q
L q L q q L q q
ss ms ms
2
2 3 5 4 5
= + +
,
∂
∂
Γ
q
3
0
=
,
∂
∂
Γ
&
& &
cos
&
sin
'
q
L q L q q L q q

rr ms ms
3
3 1 5 2 5
= + +
,
∂
∂
Γ
q
4
0
=
,
∂
∂
Γ
&
& &
sin
&
cos
'
q
L q L q q L q q
rr ms ms
4
4 1 5 2 5
= − +
,
© 2001 by CRC Press LLC

( ) ( )
[ ]
,cossin
sincos
cossin
5324154231
542532
541531
5
qqqqqqqqqqL
qqqLqqqL
qqqLqqqL
q
ms
msms
msms
&&&&&&&&
&&&&
&&&&
−++−=
−+
−−=
Γ
∂
∂
∂
∂
Γ
&
&

q
Jq
5
5
=
,
∂
∂
Π
q
1
0=
,
∂
∂
Π
q
2
0=
,
∂
∂
Π
q
3
0=
,
∂
∂
Π

q
4
0=
,
∂
∂
Π
q
5
0=
,
∂
∂
D
q
r q
s
&
&
1
1
=
,
∂
∂
D
q
r q
s
&

&
2
2
=
,
∂
∂
D
q
r q
r
&
&
'
3
3
=
,
∂
∂
D
q
r q
r
&
&
'
4
4
=

,
∂
∂
D
q
B q
m
&
&
5
5
=
.
Taking note of
&
q i
as1
=
,
&
q i
bs2
=
,
&
'
q i
ar3
=
,

&
'
q i
br4
=
and
&
q
r5
= ω
, one
obtains
(
)
(
)
L
di
dt
L
d i
dt
L
d i
dt
r i u
ss
as
ms
ar r

ms
br r
s as as
+ − + =
' '
cos sinθ θ
,
(
)
(
)
L
di
dt
L
d i
dt
L
d i
dt
r i u
ss
bs
ms
ar r
ms
br r
s bs bs
+ + + =
' '

sin cosθ θ
,
( ) ( )
L
d i
dt
L
d i
dt
L
di
dt
r i u
ms
as r
ms
bs r
rr
ar
r ar ar
cos sin
'
'
' ' '
θ θ
+ + + =
,
( ) ( )
− + + + =L
d i

dt
L
d i
dt
L
di
dt
r i u
ms
as r
ms
bs r
rr
br
r br br
sin cos
'
'
' ' '
θ θ
,
(
)
(
)
[ ]
J
d
dt
L i i i i i i i i B

d
dt
T
r
ms as ar bs br r as br bs ar r m
r
L
2
2
θ
θ θ
θ
+ + + − + = −
' ' ' '
sin cos
For P-pole induction motors, by making use of
r
r
dt
d
ω
θ
=
, six
differential equations, as found in (3.4.6), result.
Control of Induction Motors
The angular velocity of induction motors must be controlled, and the
torque-speed characteristic curves should be thoroughly examined. The
electromagnetic torque developed by two-phase induction motors is given by
equation (3.4.4). To guarantee the balanced operating condition for two-

phase induction motors, one supplies the following phase voltages to the
stator windings
(
)
u t u t
as M f
( ) cos= 2 ω
,
(
)
u t u t
bs M f
( ) sin= 2 ω
,
© 2001 by CRC Press LLC
and the sinusoidal steady-state phase currents are
(
)
i t i t
as M f i
( ) cos= −2 ω ϕ
and
(
)
i t i t
bs M f i
( ) sin= −2 ω ϕ
.
Here the following notations are used:
u

M
is the magnitude of the
voltages applied to the as and bs stator windings;
i
M
is the magnitude of the
as and bs stator currents;
ω
f
is the angular frequency of the applied phase
voltages,
ω π
f
f
=
2
;
f
is the frequency of the supplied voltage;
ϕ
i
is the
phase difference.
The applied voltage to the motor windings cannot exceed the admissible
voltage
u
M max
. That is,
u u u
M M Mmin max

≤ ≤
.
The motor synchronous angular velocity
ω
e
is found using the number
of poles as
ω
π
e
f
P
=
4
. It is evident that the synchronous velocity
ω
e
can be
regulated by changing the frequency f. To regulate the angular velocity, one
varies the magnitude of the applied voltages as well as the frequency. The
torque-speed characteristic curves of induction motors must be thoroughly
studied. Performing the transient analysis by solving the derived differential
equations (3.4.6), one can find the steady-state curves
( )
ω
r T e
T= Ω
by
plotting the angular velocity versus the electromagnetic torque developed.
The following principles are used to control the angular velocity of

induction motors.
Voltage control. By changing the magnitude
u
M
of the applied phase
voltages to the stator windings, the angular velocity is regulated in the stable
operating region, see Figure 3.4.2.a. It was emphasized that
u u u
M M Mmin max
< <
, where
u
M max
is the maximum allowed (rated)
voltage.
Frequency control. The magnitude of the applied phase voltages is
constant
u
M
constant
, and the angular velocity is regulated above and below the
synchronous angular velocity by changing the frequency of the supplied
voltages f. This concept can be clearly demonstrated using the formula
P
f
e
π
ω
4
=

. The torque-speed characteristics for different values of the
frequency are shown in Figure 3.4.2.b.
Voltage-frequency control. The angular frequency
ω
f
is proportional to
the frequency of the supplied voltages,
ω π
f
f= 2
. To minimize losses, the
applied voltages applied to the stator windings should be regulated if the
frequency is changed. In particular, the magnitude of phase voltages can be
decreased linearly with decreasing the frequency. That is, to guarantee the
© 2001 by CRC Press LLC
constant volts per hertz control one maintains the following relationship
const
f
u
i
Mi
=
or
const
u
fi
Mi
=
ω
. The corresponding torque-speed

characteristics are documented in Figure 3.4.2.c. Regulating the voltage-
frequency patterns, one shapes the torque-speed curves. For example, the
following relation
const
f
u
i
Mi
=
can be applied to adjust the magnitude
u
M
and frequency f of the supplied voltages. To attain the acceleration and settling
time specified, overshoot and rise time needed, the general purpose (standard),
soft- and high-starting torque patterns are implemented based upon the
requirements and criteria imposed (see the standard, soft- and high-torque
patterns as illustrated in Figure 3.4.2.d). That is, assigning
(
)
ω ϕ
f M
u=
with
domain
u u u
M M Mmin max
< <
and range
ω ω ω
f f f

min max
< <
, one
maintains
u
f
Mi
i
=
var
or
u
Mi
fi
ω
= var
. For example, the desired torque-speed
characteristics, as documented in Figure 3.4.2.e, can be guaranteed.
ω
r
slip,
ω
e
1
0
1
2
ω
e
1

2
T
e
T
e start
u
M max
ω
r
slip,
ω
e
1
0
T
e
f
fmax max
,ω
u
M1
u
M 2
Voltage Control
u u u
M M Mmax
> >
1 2
f
f1 1

,ω
f
f2 2
,ω
f
f3 3
,ω
Frequency Control
f f f f
f f f f
max
max
> > >
> > >
1 2 3
1 2 3
ω ω ω ω
ω
r
slip,
ω
e
1
0
T
e
T
e start
Voltage Frequency Control
u

f
const
u
const
Mi
i
Mi
fi
−
= =,
ω
u f
M max max
u f
M1 1
u f
M 2 2
u f
M 3 3
ω
r
slip,
ω
e
1
0
T
e
T
e start

u
M
u
M max
f
min
Voltage Frequency Patterns−
u
M min
General Purpose Pattern
u
f
const
Mi
i
−
=
f
max
f
High Torque Pattern
Soft Torque Pattern
Stable Operating
Region
Variable Voltage Frequency Control
u
f
u
Mi
i

Mi
fi
−
= =var, var
ω
a
b
c
d
e
ω
r max
ω
r max
ω
r max
0 0
0
0
ω
f max
ω
f min
ω
f
0
u f
M 3 3
u f
M 2 2

u f
M1 1
u f
M max max
Figure 3.4.2. Torque-speed characteristic curves
( )
ω
r T e
T= Ω
:
a) voltage control; b) frequency control;
c) voltage-frequency control: constant volts per hertz control;
d) voltage-frequency patterns;
e) variable voltage-frequency control
© 2001 by CRC Press LLC
S-Domain Block Diagram of Two-Phase Induction Motors
To perform the analysis of dynamics, to control induction machines, as
well as to visualize the results, it is important to develop the s-domain block
diagrams. For squirrel-cage induction motors, the rotor windings are short-
circuited, and hence
0
''
==
brar
uu
. The block diagram is built using
differential equations (3.4.6). The resulting s-domain block diagram is shown
in Figure 3.4.3.
L
ms

sL
Σ
+ L'
rr
r
s
i
as
L
ms
sL
Σ
+ L'
rr
r
s
i
bs
L
ms
L
ms
sL
Σ
+ L
ss
r'
r
i'
ar

L
ms
sL
Σ
+ L
ss
r'
r
i'
br
L'
rr
L
ms
u
as
L'
rr
L
ms
u
bs
L
ms
+
-
sin
cos
1
s

ω
r
θ
r
r'
r
L
rr
L'
rr
X
r'
r
L
rr
L'
rr
X
X
X
r
s
L
ss
L
ss
r
s
L
ss

L
ss
X
X
X
X
L
ms
L
ms
XX
XX
X
X
X
X
1
sJ + B
m
P
2
X
X
-PL
ms
2
T
L
+
+

+ +
-
+
+
+
-
+
-
+
-
+
+
+
+
-
+
+
+
-
+
+
+
+
+
+
+
+
-
+
-

i'
ar
i
bs
i
as
i'
br
i'
ar
i
as
i
bs
i'
br
-
X
X
X
X
X
X
X
X
Figure 3.4.3.S-domain block diagram of squirrel-cage induction motors
© 2001 by CRC Press LLC
3.4.2. Three-Phase Induction Motors
Dynamics of Induction Motors in the Machine Variables
Our goal is to develop the mathematical model of three-phase induction

motors, as shown in Figure 3.4.4, using Kirchhoff’s and Newton’s second
laws.
ω
r e
T,
T
L
θ ω τ τ θ
r r
t
t
r
d= +
∫
( )
0
0
B
m
as'
bs
as
bs'
br
br'
ar
ar'
as Magnetic Axis
ar Magnetic Axis
cs Magnetic Axis

bs Magnetic Axis
cs'
cs
Stator
cr
Rotor
+
L
ss
r
s
u
as
i
as
N
s
+
L
ss
r
s
u
bs
i
bs
+
L
ss
r

s
i
cs
u
cs
+
L
rr
r
r
u
ar
i
ar
N
r
+
L
rr
r
r
u
br
i
br
+
L
rr
r
r

i
cr
u
cr
cr'
Load
Magnetic Coupling
Figure 3.4.4. Three-phase symmetrical induction motor
Studying the magnetically coupled stator and rotor circuitry, Kirchhoff’s
voltage law relates the abc stator and rotor phase voltages, currents, and flux
linkages through the set of differential equations.
For magnetically coupled stator and rotor windings, we have
u r i
d
dt
as s as
as
= +
ψ
,
u r i
d
dt
bs s bs
bs
= +
ψ
,
u r i
d

dt
cs s cs
cs
= +
ψ
,
u r i
d
dt
ar r ar
ar
= +
ψ
,
u r i
d
dt
br r br
br
= +
ψ
,
u r i
d
dt
cr r cr
cr
= +
ψ
. (3.4.8)

It is clear that the abc stator and rotor voltages, currents, and flux
linkages are used as the variables, and in matrix form equations (3.4.8) are
rewritten as
© 2001 by CRC Press LLC
u r i
abcs s abcs
abcs
d
dt
= +
ψψ
,
u r i
abcr r abcr
abcr
d
dt
= +
ψψ
, (3.4.9)
where the abc stator and rotor voltages, currents, and flux linkages are
u
abcs
as
bs
cs
u
u
u
=











,
u
abcr
ar
br
cr
u
u
u
=











,
i
abcs
as
bs
cs
i
i
i
=










,
i
abcr
ar
br
cr
i
i
i
=











,
ψψ
abcs
as
bs
cs
=










ψ
ψ
ψ

,
and
ψψ
abcr
ar
br
cr
=










ψ
ψ
ψ
.
In (3.4.9), the diagonal matrices of the stator and rotor resistances are
r
s
s
s
s
r
r

r
=










0 0
0 0
0 0
and
r
r
r
r
r
r
r
r
=











0 0
0 0
0 0
.
The flux linkages equations must be thoroughly examined, and one has
ψψ
ψψ
abcs
abcr
s sr
sr
T
r
abcs
abcr






=













L L
L L
i
i
, (3.4.10)
where the matrices of self- and mutual inductances
L
s
,
L
r
and
sr
L
are
L
s
ls ms ms ms
ms ls ms ms
ms ms ls ms
L L L L

L L L L
L L L L
=
+ − −
− + −
− − +










1
2
1
2
1
2
1
2
1
2
1
2
,
L

r
lr mr mr mr
mr lr mr mr
mr mr lr mr
L L L L
L L L L
L L L L
=
+ − −
− + −
− − +










1
2
1
2
1
2
1
2
1

2
1
2
,
(
)
(
)
(
)
(
)
(
)
(
)
L
sr sr
r r r
r r r
r r r
L=
+ −
− +
+ −











cos cos cos
cos cos cos
cos cos cos
θ θ π θ π
θ π θ θ π
θ π θ π θ
2
3
2
3
2
3
2
3
2
3
2
3
.
Using the number of turns
N
s
and
N
r

, one finds
u u
abcr
s
r
abcr
N
N
'
=
,
i i
abcr
r
s
abcr
N
N
'
=
and
ψψ ψψ
abcr
s
r
abcr
N
N
'
=

.
The inductances are expressed as
© 2001 by CRC Press LLC
L
N
N
L
ms
s
r
sr
=
,
L
N N
sr
s r
m
=
ℜ
, and
L
N
ms
s
m
=
ℜ
2
.

Then, we have
(
)
(
)
(
)
(
)
(
)
(
)
L L
sr
s
r
sr ms
r r r
r r r
r r r
N
N
L
'
cos cos cos
cos cos cos
cos cos cos
,= =
+ −

− +
+ −










θ θ π θ π
θ π θ θ π
θ π θ π θ
2
3
2
3
2
3
2
3
2
3
2
3
and
L L
r

s
r
r
lr ms ms ms
ms lr ms ms
ms ms lr ms
N
N
L L L L
L L L L
L L L L
'
'
'
'
= =
+ − −
− + −
− − +










2

2
1
2
1
2
1
2
1
2
1
2
1
2
,
L
N
N
L
lr
s
r
lr
'
=
2
2
.
From (3.4.10), one finds
ψψ
ψψ

abcs
abcr
s sr
sr
T
r
abcs
abcr
'
'
' '
'






=













L L
L L
i
i
. (3.4.11)
Substituting the matrices
L
s
,
L
sr
'
and
L
r
'
, we have
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( )
.
coscoscos
coscoscos
coscoscos
coscoscos
coscoscos

coscoscos
'
'
'
'
2
1
2
1
3
2
3
2
2
1
'
2
1
3
2
3
2
2
1
2
1
'
3
2
3

2
3
2
3
2
2
1
2
1
3
2
3
2
2
1
2
1
3
2
3
2
2
1
2
1
'
'
'










































+−−+−
−+−−+
−−++−
−++−−
+−−+−
−+−−+
=





















cr
br
ar
cs
bs
as
mslrmsmsrmsrmsrms
msmslrmsrmsrmsrms
msmsmslrrmsrmsrms
rmsrmsrmsmslsmsms
rmsrmsrmsmsmslsms
rmsrmsrmsmsmsmsls
cr
br
ar
cs
bs
as
i
i
i
i
i
i
LLLLLLL

LLLLLLL
LLLLLLL
LLLLLLL
LLLLLLL
LLLLLLL
θπθπθ
πθθπθ
πθπθθ
θπθπθ
πθθπθ
πθπθθ
ψ
ψ
ψ
ψ
ψ
ψ
Using (3.4.9) and (3.4.11), one obtains
u r i r i L
i L i
abcs s abcs
abcs
s abcs s
abcs sr abcr
d
dt
d
dt
d
dt

= + = + +
ψψ ( )
' '
,
u r i r i L
i L i
abcr r abcr
abcr
r abcr r
abcr sr
T
abcs
d
dt
d
dt
d
dt
' ' '
'
' ' '
' '
( )
= + = + +
ψψ
, (3.4.12)
where
r r
r
s

r
r
N
N
'
=
2
2
.
Matrix equations (3.4.12) in expanded form using (3.4.11) are rewritten
as
© 2001 by CRC Press LLC
( )
( )
( )( ) ( )( )
,
coscos
cos
3
2
'
3
2
'
'
2
1
2
1
dt

id
L
dt
id
L
dt
id
L
dt
di
L
dt
di
L
dt
di
LLiru
rcr
ms
rbr
ms
rar
ms
cs
ms
bs
ms
as
mslsassas
π

θ
π
θ
θ
−
+
+
++
−−++=
( )
( )( )
( )
( )( )
,
cos
cos
cos
3
2
'
'
3
2
'
2
1
2
1
dt
id

L
dt
id
L
dt
id
L
dt
di
L
dt
di
LL
dt
di
Liru
rcr
ms
rbr
ms
rar
ms
cs
ms
bs
msls
as
msbssbs
π
θ

θ
π
θ +
++
−
+
−++−=
( )
( )( ) ( )( )
( )
,
cos
coscos
'
3
2
'
3
2
'
2
1
2
1
dt
id
L
dt
id
L

dt
id
L
dt
di
LL
dt
di
L
dt
di
Liru
rcr
ms
rbr
ms
rar
ms
cs
msls
bs
ms
as
mscsscs
θ
π
θ
π
θ
+

−
+
+
+
++−−=
( )
( )( ) ( )( )
( )
,
coscos
cos
'
2
1
'
2
1
'
'
3
2
3
2
'''
dt
di
L
dt
di
L

dt
di
LL
dt
id
L
dt
id
L
dt
id
Liru
cr
ms
br
ms
ar
mslr
rcs
ms
rbs
ms
ras
msarrar
−−++
+
+
−
++=
π

θ
π
θ
θ
( )( )
( )
( )( )
( )
,
cos
cos
cos
'
2
1
'
'
'
2
1
3
2
3
2
'''
dt
di
L
dt
di

LL
dt
di
L
dt
id
L
dt
id
L
dt
id
Liru
cr
ms
br
mslr
ar
ms
rcs
ms
rbs
ms
ras
msbrrbr
−++−
−
++
+
+=

π
θ
θ
π
θ
( )( ) ( )( )
( )
( )
.
cos
coscos
'
'
'
2
1
'
2
1
3
2
3
2
'''
dt
di
LL
dt
di
L

dt
di
L
dt
id
L
dt
id
L
dt
id
Liru
cr
mslr
br
ms
ar
ms
rcs
ms
rbs
ms
ras
mscrrcr
++−−
+
+
+
−
+=

θ
π
θ
π
θ
Cauchy’s form differential equations, given in matrix form, are found to
be
© 2001 by CRC Press LLC
di
dt
di
dt
di
dt
di
dt
di
dt
di
dt
L
r L r L r L
r L r L r L
r L r L r L
r L r L r L
as
bs
cs
ar
br

cr
L
s m s ms s ms
s ms s m s ms
s ms s ms s m
r m r ms r ms
'
'
'

































=
− − −
− − −
− − −
− − −
1
0 0 0
0 0 0
0 0 0
0 0 0
0
1
2
1
2
1
2
1
2

1
2
1
2
1
2
1
2
Σ
Σ
Σ
Σ
Σ
0 0
0 0 0
1
2
1
2
1
2
1
2
− − −
− − −










































r L r L r L
r L r L r L
i
i
i
i
i
i
r ms r m r ms
r ms r ms r m
as
bs
cs
ar
br
cr
Σ
Σ
'
'
'
(
)
(
)

( ) ( )
( ) ( )
(
)
(
)
( )
+
+ −
− +
+ −
− +
+
1
0 0 0
0 0 0
0 0 0
0 0 0
2
3
2
3
2
3
2
3
2
3
2
3

2
3
2
3
2
3
L
r L r L r L
r L r L r L
r L r L r L
r L r L r L
r L r L
L
r ms r r ms r r ms r
r ms r r ms r r ms r
r ms r r ms r r ms r
s ms r s ms r s ms r
s ms r s ms
Σ
cos cos cos
cos cos cos
cos cos cos
cos cos cos
cos cos
θ θ π θ π
θ π θ θ π
θ π θ π θ
θ θ π θ π
θ π θ
( )

( ) ( )
r s ms r
s ms r s ms r s ms r
as
bs
cs
ar
br
cr
r L
r L r L r L
i
i
i
i
i
i
cos
cos cos cos
'
'
'
θ π
θ π θ π θ
−
− +














































2
3
2
3
2
3
0 0 0
0 0 0
(
)
(
)
(
)
(
)
(
)
(
)

+
− + −
− − +
− + −
1
0 13 13
13 0 13
13 13 0
2 2 2
3
2
3
2 2 2
3
2
3
2 2 2
3
2
3
L
L L L L L
L L L L L
L L L L L
L
ms r ms r ms r r ms r r ms r r
ms r ms r ms r r ms r r ms r r
ms r ms r ms r r ms r r ms
Σ
Σ Σ Σ

Σ Σ Σ
Σ Σ Σ
. . sin sin sin
. . sin sin sin
. . sin sin
ω ω ω θ ω θ π ω θ π
ω ω ω θ π ω θ ω θ π
ω ω ω θ π ω θ π ω
(
)
(
)
(
)
(
)
(
)
(
)
r r
ms r r ms r r ms r r ms r ms r
ms r r ms r r ms r r ms r ms r
ms r r ms r r ms r r ms r ms
L L L L L
L L L L L
L L L L L
sin
sin sin sin . .
sin sin sin . .

sin sin sin . .
θ
ω θ ω θ π ω θ π ω ω
ω θ π ω θ ω θ π ω ω
ω θ π ω θ π ω θ ω
Σ Σ Σ
Σ Σ Σ
Σ Σ Σ
− + −
+ − −
− + −
2
3
2
3
2 2
2
3
2
3
2 2
2
3
2
3
2 2
0 13 13
13 0 13
13 13 ω
r

as
bs
cs
ar
br
cr
i
i
i
i
i
i
0













































'
'
'

( ) ( )
( ) ( )
( ) ( )
( )
+
+ − − + − −
+ − − − − +
+ − + − − −
− − − −
1
2
2
2
1
2
1
2
2
3
2
3
1
2
1
2
2
3
2
3
1

2
1
2
2
3
2
3
2
3
L
L L L L L L L
L L L L L L L
L L L L L L L
L L L
L
ms lr ms ms ms r ms r ms r
ms ms lr ms ms r ms r ms r
ms ms ms lr ms r ms r ms r
ms r ms r ms r
Σ
'
'
'
cos cos cos
cos cos cos
cos cos cos
cos cos cos
θ θ π θ π
θ π θ θ π
θ π θ π θ

θ θ π θ
( )
( ) ( )
( ) ( )
+ +
− + − − − +
− − − + − +




































2
3
1
2
1
2
2
3
2
3
1
2
1
2
2
3
2
3

1
2
1
2
2
2
2
π
θ π θ θ π
θ π θ π θ
L L L L
L L L L L L L
L L L L L L L
u
u
u
u
u
u
ms lr ms ms
ms r ms r ms r ms ms lr ms
ms r ms r ms r ms ms ms lr
as
bs
cs
ar
br
cr
'
'

'
'
'
'
cos cos cos
cos cos cos









.
(3.4.13)
Here, the following notations are used
( )
L L L L
L ms lr lrΣ
= +
3
' '
,
L L L
m ms lrΣ
= +
2
'

,
L L L L
ms ms ms lrΣ
= +
3
2
2 '
.
Newton’s second law is applied to derive the torsional-mechanical
equations, and the expression for the electromagnetic torque must be
obtained.
For P-pole three-phase induction machines, as one finds the expression
for coenergy
( )
W
c abcs abcr r
i i, ,
'
θ
, the electromagnetic torque can be
straightforwardly derived as
( )
T
P
W
e
c abcs abcr r
r
=
2

∂ θ
∂θ
i i, ,
'
.
© 2001 by CRC Press LLC
For three-phase induction motors we have
(
)
(
)
(
)
W W L
c f abcs
T
s ls abcs abcs
T
sr r abcr abcr
T
r lr abcr
= = − + + −
1
2
1
2
i L I i i L i i L L I i
' ' ' ' ' '
θ
Matrices

L
s
and
L
r
'
, as well as leakage inductances
L
ls
and
L
lr
'
, are
not functions of the electrical displacement θ
r
. Therefore, we have
( )
[ ]
( ) ( )
( ) ( )
( ) ( )
(
)
T
P
P
L i i i
i
i

i
P
L i i i i i i i i
e abcs
T
sr r
r
abcr
ms as bs cs
r r r
r r r
r r r
ar
br
cr
ms as ar br cr bs br ar
=
= −
+ −
− +
+ −





















= − − − + − −
2
2
2
2
3
2
3
2
3
2
3
2
3
2
3
1
2
1

2
1
2
1
2
i
L
i
∂ θ
∂θ
θ θ π θ π
θ π θ θ π
θ π θ π θ
'
'
'
'
'
' ' ' ' '
sin sin sin
sin sin sin
sin sin sin
(
)
(
)
[ ]
{
}
cr cs cr br ar r

as br cr bs cr ar cs ar br r
i i i i
i i i i i i i i i
' ' ' '
' ' ' ' ' '
sin
cos .
+ − −
+ − + − + −

























1
2
1
2
3
2
θ
θ
(3.4.14)
Using Newton’s second law and (3.4.14), the torsional-mechanical
equations are found to be
( ) ( ) ( )
[ ]
{
}
d
dt
P
J
T
B
J
P
J
T
P
J

L i i i i i i i i i i i i
i i i i i i i i i
B
J
r
e
m
r L
ms as ar br cr bs br ar cr cs cr br ar r
as br cr bs cr ar cs ar br r
m
r
ω
ω
θ
θ ω
= − −
= − − − + − − + − −
+ − + − + − −

























2 2
4
2
1
2
1
2
1
2
1
2
1
2
1
2
3
2

' ' ' ' ' ' ' ' '
' ' ' ' ' '
sin
cos
−
P
J
T
L
2
,
d
dt
r
r
θ
ω=
. (3.4.15)
Augmenting differential equations (3.4.13) and (3.4.15), the resulting
model for three-phase induction motors in the machine variables, is found.
Mathematical Model of Three-Phase Induction Motors in the
Arbitrary Reference Frame
The abc stator and rotor variables must be transformed to the
quadrature, direct, and zero quantities. To transform the machine (abc)
stator voltages, currents, and flux linkages to the quadrature-, direct-, and
zero-axis components of stator voltages, currents and flux linkages, the direct
Park transformation is used. In particular,
u K u
qdos s abcs
=

,
i K i
qdos s abcs
=
,
ψψ ψψ
qdos s abcs
= K
, (3.4.16)
where the stator transformation matrix
K
s
is given by
© 2001 by CRC Press LLC
(
)
(
)
(
)
(
)
K
s
=
− +
− +











2
3
2
3
2
3
2
3
2
3
1
2
1
2
1
2
cos cos cos
sin sin sin
θ θ π θ π
θ θ π θ π
. (3.4.17)
Here, the angular displacement of the reference frame is
(

)
θ ω τ τ θ= +
∫
t
t
d
0
0
.
Using the rotor transformations matrix
K
r
, the quadrature-, direct-,
and zero-axis components of rotor voltages, currents, and flux linkages are
found by using the abc rotor voltages, currents, and flux linkages.
In particular,
u K u
qdor r abcr
' '
=
,
i K i
qdor r abcr
' '
=
,
ψψ ψψ
qdor r abcr
' '
=

K
, (3.4.18)
where the rotor transformation matrix is
( )
(
)
(
)
(
)
(
)
(
)
K
r
r r r
r r r
=
− − − − +
− − − − +











2
3
2
3
2
3
2
3
2
3
1
2
1
2
1
2
cos cos cos
sin sin sin
θ θ θ θ π θ θ π
θ θ θ θ π θ θ π
.(3.4.19)
From differential equations (3.4.12)
u r i
abcs s abcs
abcs
d
dt
= +
ψψ

,
u r i
abcr r abcr
abcr
d
dt
' ' '
'
= +
ψψ
,
by taking note of the inverse Park transformation matrices
K
s
−1
and
K
r
−1
,
we have
( )
K u r K i
K
s qdos s s qdos
s qdos
d
dt
− −
−

= +
1 1
1
ψψ
,
( )
K u r K i
K
r qdor r r qdor
r qdor
d
dt
− −
−
= +
1 1
1
' ' '
'
ψψ
. (3.4.20)
Making use of (3.4.17) and (3.4.19) one finds inverse matrices
K
s
−1
and
K
r
−1
. In particular,

(
)
(
)
(
)
(
)
K
s
−
= − −
+ +










1
2
3
2
3
2
3

2
3
1
1
1
cos sin
cos sin
cos sin
θ θ
θ π θ π
θ π θ π
,
and
( ) ( )
(
)
(
)
(
)
(
)
K
r
r r
r r
r r
−
=
− −

− − − −
− + − +










1
2
3
2
3
2
3
2
3
1
1
1
cos sin
cos sin
cos sin
θ θ θ θ
θ θ π θ θ π
θ θ π θ θ π

.
© 2001 by CRC Press LLC
Multiplying left and right sides of equations (3.4.20) by
K
s
and
K
r
,
one has
u K r K i K
K
K K
qdos s s s qdos s
s
qdos s s
qdos
d
dt
d
dt
= + +
−
−
−1
1
1
ψψ
ψψ
,

u K r K i K
K
K K
qdor r r r qdor r
r
qdor r r
qdor
d
dt
d
dt
' ' ' '
'
= + +
−
−
−1
1
1
ψψ
ψψ
. (3.4.21)
The matrices of the stator and rotor resistances
r
s
and
r
r
'
are diagonal,

and hence,
K r K r
s s s s
−
=
1
and
K r K r
r r r r
' '−
=
1
.
Performing differentiation, one finds
(
)
(
)
(
)
(
)
d
dt
s
K
−
=
−
− − −

− + +










1
2
3
2
3
2
3
2
3
0
0
0
ω
θ θ
θ π θ π
θ π θ π
sin cos
sin cos
sin cos

,
( )
( ) ( )
(
)
(
)
(
)
(
)
d
dt
r
r
r r
r r
r r
K
−
= −
− − −
− − − − −
− − + − +











1
2
3
2
3
2
3
2
3
0
0
0
ω ω
θ θ θ θ
θ θ π θ θ π
θ θ π θ θ π
sin cos
sin cos
sin cos
.
Therefore,
K
K
s
s
d

dt
−
= −










1
0 1 0
1 0 0
0 0 0
ω
and
( )
K
K
r
r
r
d
dt
−
= − −











1
0 1 0
1 0 0
0 0 0
ω ω
.
One obtains the voltage equations for stator and rotor circuits in the
arbitrary reference frame when the angular velocity of the reference frame
ω
is not specified. From (3.4.21) the following matrix differential equations
result
u r i
qdos s qdos qdos
qdos
d
dt
= + −











+
0 0
0 0
0 0 0
ω
ω ψψ
ψψ
,
u r i
qdor r qdor
r
r qdor
qdor
'
d
dt
' ' ' '
= +
−
− +











+
0 0
0 0
0 0 0
ω ω
ω ω ψψ
ψψ
. (3.4.22)
© 2001 by CRC Press LLC
From (3.4.22), six differential equations in expanded form are found to
model the stator and rotor circuitry dynamics. In particular,
u r i
d
dt
qs s qs ds
qs
= + +ωψ
ψ
,
u r i
d
dt
ds s ds qs
ds

= − +ωψ
ψ
,
u r i
d
dt
os s os
os
= +
ψ
,
(
)
u r i
d
dt
qr r qr r dr
qr
' ' ' '
'
= + − +ω ω ψ
ψ
,
(
)
u r i
d
dt
dr r dr r qr
dr

' ' ' '
'
= − − +ω ω ψ
ψ
,
u r i
d
dt
or r or
or
' ' '
'
= +
ψ
. (3.4.23)
Using the matrix equation for flux linkages
ψψ
ψψ
abcs
abcr
s sr
sr
T
r
abcs
abcr
'
'
' '
'







=












L L
L L
i
i
we have
ψψ
abcs s abcs sr abcr
= +L i L i
' '
and
ψψ

abcr
'
sr
T
abcs r abcr
= +L i L i
' ' '
.
These equations can be represented using the quadrature, direct, and
zero quantities. Employing the Park transformation matrices one has
K L K i L K i
s qdos s s qdos sr r qdor
− − −
= +
1 1 1
ψψ
' '
and
K L K i L K i
r qdor sr
T
s qdos r r abcr
− − −
= +
1 1 1
ψψ
' ' ' '
.
Thus
ψψ

qdos s s s qdos s sr r qdor
= +
− −
K L K i K L K i
1 1' '
,
ψψ
qdor r sr
T
s qdos r r r abcr
' ' ' '
= +
− −
K L K i K L K i
1 1
. (3.4.24)
Taking note of the Park transformation matrices and applying the
derived expressions for
s
L
,
'
sr
L
and
'
r
L
, by multiplying the matrices we
have

K L K
s s s
ls
ls
ls
L M
L M
L
−
=
+
+










1
0 0
0 0
0 0
,
K L K K L K
s sr r r sr
T

s
M
M
' '− −
= =










1 1
0 0
0 0
0 0 0
,
© 2001 by CRC Press LLC
and
K L K
r r r
lr
lr
lr
L M
L M
L

'
'
'
'
−
=
+
+










1
0 0
0 0
0 0
,
M L
ms
=
3
2
.
In expanded form, the flux linkage equations (3.4.24) are

ψ
qs ls qs qs qr
L i Mi Mi
= + +
'
,
ψ
ds ls ds ds dr
L i Mi Mi= + +
'
,
ψ
os ls os
L i=
,
ψ
qr lr qr qs qr
L i Mi Mi
' ' ' '
= + +
,
ψ
dr lr dr ds dr
L i Mi Mi
' ' ' '
= + +
,
ψ
or lr or
L i

' ' '
=
. (3.4.25)
Using the expressions (3.4.25) in (3.4.23), the differential equations
result
(
)
(
)
u r i L i Mi Mi
d L i Mi Mi
dt
qs s qs ls ds ds dr
ls qs qs qr
= + + + +
+ +
ω
'
'
,
(
)
(
)
u r i L i Mi Mi
d L i Mi Mi
dt
ds s ds ls qs qs qr
ls ds ds dr
= − + + +

+ +
ω
'
'
,
(
)
u r i
d L i
dt
os s os
ls os
= +
,
(
)
( )
( )
u r i L i Mi Mi
d L i Mi Mi
dt
qr r qr r lr dr ds dr
lr qr qs qr
' ' ' ' ' '
' ' '
= + − + + +
+ +
ω ω
,
(

)
( )
( )
u r i L i Mi Mi
d L i Mi Mi
dt
dr r dr r lr qr qs qr
lr dr ds dr
' ' ' ' ' '
' ' '
= − − + + +
+ +
ω ω
,
(
)
u r i
d L i
dt
or r or
lr or
' ' '
' '
= +
.
Cauchy’s form of differential equations is
© 2001 by CRC Press LLC
( )
[
( )

]
,
1
''
''2
2
qrqsRMrdrRMds
qrrdsRMSMqssRM
RMSM
qs
MuuLiLMiM
iMriMLLirL
MLL
dt
di
−++−
+−−−
−
=
ω
ω
( )
[
( )
]
,
1
''
''2
2

drdsRMrqrRMqs
drrdssRMqsRMSM
RMSM
ds
MuuLiLMiM
iMrirLiMLL
MLL
dt
di
−+++
+−−
−
=
ω
ω
(
)
di
dt L
r i u
os
ls
s os os
= − +
1
,
( )
[
( )
]

,
1
''
'2''
2
'
qrSMqsrdrRMdsSM
drRMSMqrrSMqss
RMSM
qr
uLMuiLMiL
iMLLirLiMr
MLL
dt
di
+−++
−−−
−
=
ω
ω
( )
[
( )
]
,
1
''
'''2
2

'
drSMdsrqrRMqsSM
drrSMqrRMSMdss
RMSM
dr
uLMuiLMiL
irLiMLLiMr
MLL
dt
di
+−+−
−−+
−
=
ω
ω
( )
di
dt
L
r i u
or
lr
r or or
'
'
' ' '
= − +
1
, (3.4.26)

where
L L M L L
SM ls ls ms
= + = +
3
2
and
L L M L L
RM lr lr ms
= + = +
' '
3
2
.
One concludes that the nonlinear differential equations are found to
describe the stator-rotor circuitry transient behavior. To complete the model
developments, the torsional-mechanical equations
T B T J
d
dt
e m rm L
rm
− − =ω
ω
,
d
dt
rm
rm
θ

ω=
, (3.4.27)
must be used.
The equation for the electromagnetic torque must be obtained in terms of
the quadrature- and direct-axis components of stator and rotor currents.
Using the formula for coenergy

(
)
(
)
( )
W L L
c abcs
T
s ls abcs abcs
T
sr r abcr abcr
T
r lr abcr
= − + + −
1
2
1
2
i L I i i L i i L I i
' ' ' ' '
θ
,
one finds

( )
T
P
W
P
e
c abcs abcr r
r
abcs
T
sr r
r
abcr
= =
2 2
∂ θ
∂θ
∂ θ
∂θ
i i
i
L
i
, ,
( )
'
'
'
.
Hence, we have

© 2001 by CRC Press LLC
( )
(
)
(
)
T
P P
e s qdos
T
sr r
r
r qdor qdos
T
s
T
sr r
r
r qdor
= =
− − − −
2 2
1 1 1 1
K i
L
K i i K
L
K i
∂ θ
∂θ

∂ θ
∂θ
'
'
'
'
.
By performing multiplication of matrices, the following formula results
(
)
T
P
M i i i i
e qs dr ds qr
= −
3
4
' '
. (3.4.28)
Thus, from (3.4.27) and (3.4.28), one has
(
)
d
dt
P
J
M i i i i
B
J
P

J
T
r
qs dr ds qr
m
r L
ω
ω= − − −
3
8 2
2
' '
,
d
dt
r
r
θ
ω=
. (3.4.29)
Augmenting the circuitry and torsional-mechanical dynamics, as given
by differential equations (3.4.26) and (3.4.29), the model for three-phase
induction motors in the arbitrary reference frame results.
We have a set of eight highly coupled nonlinear differential equations
( )
[
( )
]
,
1

''
''2
2
qrqsRMrdrRMds
qrrdsRMSMqssRM
RMSM
qs
MuuLiLMiM
iMriMLLirL
MLL
dt
di
−++−
+−−−
−
=
ω
ω
( )
[
( )
]
,
1
''
''2
2
drdsRMrqrRMqs
drrdssRMqsRMSM
RMSM

ds
MuuLiLMiM
iMrirLiMLL
MLL
dt
di
−+++
+−−
−
=
ω
ω
(
)
di
dt L
r i u
os
ls
s os os
= − +
1
,
( )
[
( )
]
,
1
''

'2''
2
'
qrSMqsrdrRMdsSM
drRMSMqrrSMqss
RMSM
qr
uLMuiLMiL
iMLLirLiMr
MLL
dt
di
+−++
−−−
−
=
ω
ω
( )
[
( )
]
,
1
''
'''2
2
'
drSMdsrqrRMqsSM
drrSMqrRMSMdss

RMSM
dr
uLMuiLMiL
irLiMLLiMr
MLL
dt
di
+−+−
−−+
−
=
ω
ω
( )
di
dt
L
r i u
or
lr
r or or
'
'
' ' '
= − +
1
,
(
)
d

dt
P
J
M i i i i
B
J
P
J
T
r
qs dr ds qr
m
r L
ω
ω= − − −
3
8 2
2
' '
,
d
dt
r
r
θ
ω=
. (3.4.30)
© 2001 by CRC Press LLC