Lecture Notes
M.Kaliamoorthy AP/EEE, PSNACET
1. Voltage and torque equations of a three phase symmetrical induction machine.
The winding arrangement for a 2 pole 3 phase wye connected symmetrical induction
machine is shown in the figure below.
The stator windings are identical, sinusoidally distributed windings, displaced by 120 degrees
with Ns equivalent turns and resistance rs. The rotor windings are also identical sinusoidally
distributed windings, displaced 120 degrees with Nr equivalent turns and resistance rr. The
voltage equations in machine variables may be expressed as,
Lecture Notes
M.Kaliamoorthy AP/EEE, PSNACET
Vabcs  rs iabcs  pabcs ...............................(1)
Vabcr  rr iabcr  pabcr ..............................(2)
In the above expression s and r subscripts denote variables associated with the stator and
rotor windings, respectively. For magnetically linear system, the flux linkages may be
expressed as
abcs   Ls
   L T
 abcr   sr
Lsr  iabcs 
         (3)
Lr  iabcr 
The winding inductances are derived in Lecture notes (Refer Lecture notes on Inductance
of symmetrical Induction Machines)
In particular Stator inductance matrix, rotor inductance matrix and mutual inductance
between stator and rotor is given by,
L
L
L
s
r
sr

 Lls  Lms
 1
   Lms
 2
1L
 2 ms

 Llr  Lmr
 1
   Lmr
 2
1L
 2 mr
1
 Lms
2
Lls  Lms
1
 Lms
2
1
 Lmr
2
Llr  Lmr
1
 Lmr
2

cos  r


2

 Lsr cos r 
 
3

2
cos r 
3
 






1

 Lms 
2

1
 Lms ..................( 4)
2

Lls  Lms 

1

 Lmr 
2

1
 Lmr ..................(5)
2

Llr  Lmr 

2 
2


cos r 
 cos r 
3 
3


2

cos  r
cos r 
3

2 

cos r 
cos  r

3 





........(6)




In the above inductance equations, Lls and Lms are respectively the leakage and magnetizing
inductances of stator windings respectively and Llr and Lmr are respectively the leakage and
magnetizing inductances of rotor windings respectively. The inductance Lsr is the amplitude
of the mutual inductance between stator and rotor windings. When expressing the voltage
equations in machine variable form, it is convenient to refer all rotor variables to the stator
windings by appropriate turn’s ratio.
Lecture Notes
M.Kaliamoorthy AP/EEE, PSNACET
Nr
iabcr ............................................(7)
Ns
i ' abcr 
V ' abcr 
Ns
Vabcr .........................................(8)
Nr
' abcr 
Ns
abcr ..........................................(9)
Nr
Lms 
Ns
Lsr ..............................................(10)
Nr
Thus we define
L' sr 
Ns
Lsr
Nr

cos  r


2

 L ms cos r 
 
3

2

cos r 
3
 






2 
2


cos r 
 cos r 
3 
3


2


cos  r
cos r 
3

2



cos r 
cos  r

3 





......(11)




Also
2
N
Lmr   r
 Ns

 Lms

N
L r   s
 Nr
2
'
L' r

 Lr

 '
 L lr  Lms
 1
   Lms
 2
1L
 2 ms
N
Where L lr   s
 Nr
'
1
 Lms
2
L'lr  Lms
1
 Lms
2



......(12)

'
L lr  Lms 

1
 Lms
2
1
 Lms
2
2

 Llr ..................................(13)

The flux linkages may now be expressed as
 abcs   Ls
'    '
 abcr   L sr
 
T
L' sr   iabcs 
  ' ....................................(14)
L' r  i abcr 
Lecture Notes
M.Kaliamoorthy AP/EEE, PSNACET
The voltage equations expressed in terms of machine variables
 Vabcs  rs  pLs
T
V '   
'
 abcr   pL sr 
pL' sr   iabcs 
  ' .........(15)
r ' r  pL' r  i abcr 
2
Where
N 
r r   s  rr ..............................(16)
 Nr 
'
Torque Equations in Machine Variables
The energy stored in the coupled field may be written as
Wf 

1
iabcs T Ls  Lls I iabcs  iabcs T L' sri ' qdr  1 i ' abcr
2
2
 L
T
'
r

 L'lr I i ' abcr ........(17)
Since Llr is only function of rotor position the torque equation can be written by omitting the
first two terms of the above equation and is as given below
Te 
 
P
iabcs T  L' sr i ' abcr ........................................(18)
2
 r
The above equation may be now simplified as
   ' 1 ' 1 ' 

 ' 1 ' 1 '
 ' 1 ' 1 ' 
ias  iar  ibr  icr   ibs  ibr  icr  iar   ics  icr  ibr  iar  sin  r 
2
2 
2
2 
2
2 
P





Te   Lms 

2


3
'
'
'
'
'
'

ias ibr  icr  ibs icr  iar  ics iar  ibr cos  r


2


 
 
 
