Lecture Notes
M.Kaliamoorthy AP/EEE, PSNACET
Operational impedances for a synchronous machine with four rotor windings
Construction
A 2 pole 3 phase wye-connected salient pole synchronous machine is shown in the figure given
below. The stator windings are identical and sinusoid ally distributed windings displaced 120
degrees with Ns equivalent turns and resistance rs. The rotor is equipped with a field winding
and three damper windings. The field winding (fd winding) has Nfd equivalent turns with
resistance rfd.One damper winding has the same magnetic axis as the field winding. This
winding is kd winding, has Nkd equivalent turns with resistance rkd.The magnetic axis of second
and third damper windings, the kq1 and kq2 windings, is displaced 90 degrees a head of the
magnetic axis of the fd and kd windings. The kq1 and kq2 windings have Nkq1 and Nkq2
equivalent turns with resistance rkq1 and rkq2.It is assumed that all the rotor windings are
sinusoidally distributed. Also it is assumed that the direction of positive stator currents is out of
the terminals convenient to describe generator action.
The magnetic axes of the stator windings are denoted by as, bs and cs axes. The q axis is the
magnetic axes of the kq1 and kq2 windings while the d axis is the magnetic axis of the fd and
kd windings. Because the synchronous machine is generally operated as a generator, it is
convenient to assume that the direction of positive stator current is out of the terminals as
shown in the figure above. With this convention the voltage equation in machine variables may
be expressed in matrix form as,
Derivation
Vabcs  rsiabcs  pλ abcs ..............(1)
Vqdr  rr iqdr  pλ qdr ...................(2)
(- ve sign for rs because genertaor action)
Lecture Notes
M.Kaliamoorthy AP/EEE, PSNACET
In the above expression s and r subscripts denote variables associated with the stator and rotor
windings, respectively. Both rs and rr are diagonal matrices. In particular
rs  diag rs rs
rs
rs    0
 0
0
0 ........................................(3)
rs 
0
rs
0
rr  diag rkq1 rkq 2 rfd rkd 
rkq1

0

0

 0
0
rkq 2
0
0
0
0
rfd
0
0

0
.............(4)
0

rkd 
The flux linkages equations becomes
abcs   Ls
   
T
 qdr  Lsr 
Lsr   iabcs 

.............(5)
Lr   iqdr 
We can write Stator inductance matrix Ls (Self and Mutual between stator phases) as (Refer
Lecture Notes on Symmetrical synchronous machine and MMF waveforms)

1

1
 


 LA  LB cos 2 r  
 LA  LB cos 2 r   
 Lls  LA  LB cos 2 r
2
3
2
3 



1

2

1




Ls    LA  LB cos 2 r   Lls  LA  LB cos 2 r 
 LA  LB cos 2 r    ...(6)

 2

3
3
2




 1

1
2 

 LA  LB cos 2 r  
 LA  LB cos 2 r   
Lls  LA  LB cos 2 r 

3
2
3
 2




Similarly self and mutual inductance of the damper windings are as given below; the
inductance matrix Lsr and Lr may be expressed as
 Llkq1  Lmkq1
 L
kq1 kq 2
Lr  

0

0




Lskq1 cos  r


2

Lsr   Lskq1 cos r 

3



 Lskq1 cos r  2

3


Lkq1 kq 2
Llkq 2  Lmkq 2
0
0
0
Llfd  Lmfd
0
L fdkd
Lskq 2 cos  r


..........(7)
L fdkd 

Llfd  Lmfd 
Lsfd sin  r
0
0
Lskd sin  r
2 
2 
2




 Lskq 2 cos r 
 Lsfd sin  r 
 Lskd sin  r 
3 
3 
3




2 
2 
2




 Lskq 2 cos r 
 Lsfd sin  r 
 Lskd sin  r 
3 
3 
3




The magnetizing inductances are defined as






.......(8)







Lecture Notes
M.Kaliamoorthy AP/EEE, PSNACET
3
L A  LB ..............................(9)
2
3
 L A  LB ..............................(10)
2
Lmq 
Lmd
The relation ship between mutual inductances and magnetizing inductances are given as below
(proof of which is given in lecture notes)
 N kq1  2
 Lmq ...........................(11)
Lskq1  
N
 s 3
 N kq 2  2
 Lmq ..........................(12)
Lskq 2  
 Ns  3
 N fd  2
 Lmd .............................(13)
Lsfd  
 Ns  3
N 2
Lskd   kd  Lmd ............................(14)
 Ns  3
2
 N kq1  2
 Lmq ........................(15)
Lmkq1  
 Ns  3
2
Lmkq 2
 N kq 2  2
 Lmq ........................(16)
 
N
s

 3
2
Lmfd
 N fd
 
 Ns
 2
 Lmd ............................................(17)
 3
Lmkd
N
  kd
 Ns
 2
 Lmd ............................................(18)
 3
2
 N kq 2 
N 
 Lmkq1   kq1  Lmkq 2 ....................(19)
Lkq1kq 2  
N 
N 
 kq1 
 kq1 
N 
 N fd 
 Lmkd ..........................( 20)
L fdkd   kd  Lmfd  
N 
N
fd
kd




It is convenient to incorporate the following substitute variables that refer the rotor variables to
the stator windings:
 2  N j
'
i j   
 3  N s

i j .......................(21)

N 
'
v j   s v j ...........................(22)
 Nj 
N 
 j '   s  j ...........................(23)
 Nj 
Where j may be kq1, kq2, fd or kd
Lecture Notes
The flux linkages may be now written as
abcs   Ls
 '    2 L' sr
 qdr   3
M.Kaliamoorthy AP/EEE, PSNACET
L' sr   i 
 abcs .............(24)
L' r   i ' qdr 

 
T
Where Ls is defined in the equation (6) and



Lmq cos  r


2

L' sr   Lmq cos r 

3



 Lmq cos r  2

3


Lmq cos  r
Lmd sin  r
Lmd sin  r
2 
2 
2




 Lmq cos r 
 Lmd sin  r 
 Lmd sin  r 
3 
3 
3




2 
2 
2




 Lmq cos r 
 Lmd sin  r 
 Lmd sin  r 
3 
3 
3




 L'lkq1  Lmq

Lmq
'
Lr  

0

0

Lmq
0
L lkq 2  Lmq
0
0
L lfd  Lmd
0
Lmd
'
'






.......(25)









0
..........(26)
Lmd 

L'lfd  Lmd 
0
The voltage equations in terms of machine variables referred to the stator windings are
Vabcs   rs  pLs
V '    2 p L' sr T
 qdr   3
Where
 
pL' sr   i 
  abcs ..............(27)
'
'
r r  pL r   i ' qdr 

2
 3  N
r j    s
 2  N j

 rj ..........................( 28)


 3  N
L lj    s
 2  N j

 Llj ........................(29)


'
'
2
TORQUE EQUATIONS IN MACHINE VARIABLES
The energy stored in the coupling field of a synchronous machine may be expressed as
Wf 
1
iabcs T Ls  Lls I iabcs  iabcs T L' sri ' qdr  1  3 i ' qdr T L' r  L'lr I i ' qdr ........(30)
2
22
Form the above discussions Ls and L’lr are only function of rotor position and Lr is not a
function of rotor position so we can eliminate last term of (30) and since it is a generator i abcs is
Lecture Notes
M.Kaliamoorthy AP/EEE, PSNACET
negative and for a “P” pole machine the above equation can be modified for torque expression
as follows

 P  1
T 
Ls  Lls I iabcs  iabcs T  L' sr i 'qdr ...........(31)
Te    iabcs 
 r
 r
 2  2

 
In expanded form the above equation becomes

 2

1 2
1 2

 L  L   i as  i bs  i cs  iasibs  iasics  2ibsibs  sin 2 r  
2
2

mq
 md



3
 3 2

i bs  i 2 cs  2iasibs  2iasics cos 2 r



 2




 
P
1
1 
3
'
'
ibs  ics  cos  r  ........(32)
Te   Lmq i kq1  i kq 2  ias  ibs  ics  sin  r 
2
2
2 
2

 


 L i ' fd  i ' kd  i  1 i  1 i  cos   3 i  i sin   
 as
bs
cs
r
bs
cs
r
 md
2
2 
2


 










