Lecture Notes
M.Kaliamoorthy AP/EEE, PSNACET
Solution to Serial Test 2
PART A
1. Given that
I as  2 I a cos et
2 

I bs  2 I b cos et 

3 

2 

I cs  2 I c cos et 

3 

Where Ia, Ib and Ic are unequal constants. We know that MMF is given by
MMFS 
Ns
2

2 
2


 I as cos  s  I bs cos  s  3   I cs cos  s  3





Ns 
2

2 I a cos et. cos  s  2 I b cos et 

2 
3


Ns
2



2


. cos  s 
3


2


  2 I c cos et 
3


2


. cos  s 
3


 I  I  I 

3
 2I  I  I 
I b  I c sin et   s 
2  a b c  coset   s    a b c  coset   s  
2
4
4





The first term in the air gap MMF rotating in the Counter clockwise direction at  e . The
last two terms are air gap MMF that rotates in clockwise direction at  e .
2. Yes in a three phase salient pole synchronous machine the mutual inductances between the
stator windings are function of rotor position under dynamic conditions because air gap and
Flux density of stator windings are function of rotor position. It is evident from the below
equations,
g s   r  
1
1   2 cos 2s   r 
Br s ,  r    0
MMF s 
g s   r 
3. Stator Resistance, Field Resistance, Stator Self Inductance, Field Self Inductance, Mutual
Inductances between stator windings, Mutual Inductances between stator and Field
Inductances are the parameters of Synchronous machines.
N 
3
4. MMFs   s  2 I s   coset   ei 0  s 
2
 2 
5. Concentrated Windings:



Lecture Notes
M.Kaliamoorthy AP/EEE, PSNACET
Depending upon the total number of slots and number of poles, certain slots per pole is
available under each pole. This is denoted by m
For example For 18 slots, 2 Pole, 3 phase machine
m  Slots Pole  / Phase
 18 2  3  3
So we have three slots per pole per phase. Now let X be number of conductors per phase are
to be placed under one pole, but we have 3 slots per pole per phase. If all the X conductors
are placed in one slot keeping other two slots empty then the winding is called
Concentrated Winding.
Distributed Winding:
If all the three slots are used to place the conductors then such a winding arrangement is
called as distributed winding.
Distributed winding are preferred over concentrated winding because the coil belonging to a
phase are well distributed over the entire ‘m’ slots per pole per phase which makes the
waveform of induced emf more sinusoidal.
6. Full Pitch Winding
If coil side in one slot is connected to coil side in another slot which is one pole pitch
distance away from first slot, the winding is said to be full pitched winding and coil is full
pitch coil.
Short Pitch Winding:
If coils are used in such a way that coil span is slightly less than a pole pitch i.e. less than 180
degree electrical than the coils are called short pitched coils or fractional pitched coils.
Advantages of Short Pitched Windings
 The length required for the end connections of the coil is less. So less copper required
and so most economical.
 Short pitch eliminates high frequency harmonics; hence waveform of induced emf of
short pitched winding is more sinusoidal.
7. Significance of Rotating Magnetic Field:
3
2
 The resultant flux is of constant value=  m (i.e) 1.5 times the maximum value of the
flux due to any phase.
 The resultant flux rotates around the stator at synchronous speed given by
120 f
Ns 
.
P
Lecture Notes
M.Kaliamoorthy AP/EEE, PSNACET
8. The Transfer Functions of DC motor is given by
m s 
V s 

Tl s 

m s 
Kb
2
S JLa   sB1 La  JRa   B1 Ra  K b
2




 Ra  sLa 
2
S JLa   sB1 La  JRa   B1 Ra  K b
2
9. MMF of field windings is given by
MMF fd  
Nf
2
i fd sin  r
10. Torque expression of a induction machine is given by
   ' 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


 
 
 

PART B
11. MMF SPACE WAVE OF A CONCENTRATED COIL
A cylindrical rotor with small air gap as shown in the figure below is assumed here.
The stator is imagined to wound for two poles with a single N turn full pitch coil carrying
current I in the direction indicated. From the figure each flux line radially crosses the air
gap twice, Normal to the stator and rotor iron surfaces and it’s associated with constant
mmf Ni. On the assumption that the reluctance of the iron path is negligible, half the mmf
 Ni 
  is consumed to create flux from the rotor to stator in the air gap and the other half
 2 
is used to establish flux from the stator to rotor in the air gap. MMF and flux radially
Lecture Notes
M.Kaliamoorthy AP/EEE, PSNACET
outwards from the rotor to the stator will be assumed to be positive and that from the
stator to rotor is negative.
The physical picture is more easily visualized by the development diagram as shown in the
figure above, where the stator winding is laid down flat with the rotor on the top of it. It has been
  Ni 
seen that the MMF is a rectangular space wave where in MMF of 
 is consumed in setting
 2 
  Ni 
flux from the rotor to stator and MMF of 
 is consumed in setting up flux from the stator to
 2 
the rotor. The MMF space wave of a single concentrated coil being rectangular, it can be split up
into its fundamental and harmonics. It is easily follows from the Fourier series analysis that the
fundamental of the MMF wave as shown in figure above is
Fa1 
4  Ni 
  cos   F1 p cos 
 2 
Where  is the electrical angle measured from the magnetic axis of the coil which coincides
with the positive peak of the fundamental wave.
Note: Amplitude of associated harmonic waves are given by F3 P 
4
4
.....
, F5 P 
3
5
MMF SPACE WAVE OF ONE PHASE DISTRIBUTED WINDING
Consider now a basic 2 pole structure with a round rotor, with 5 slots/pole/phase (SPP) and a 2
layer winding as shown in the figure below.
Lecture Notes
M.Kaliamoorthy AP/EEE, PSNACET
The corresponding development diagram shown in the figure below along with the mmf diagram
which is now a stepped wave –obviously closer to the sine wave than the rectangular mmf wave
of a single concentrated coil.
Here the SPP is odd (5), half the ampere conductors of the middle slot of the phase a and a ’
contribute towards establishment of South Pole and half towards North Pole on the stator. At
each slot the mmf wave has a step jump of 2Ncic ampere conductors Where Nc is the coil turns
and ic is the conductor current.
Let
Nph (Series)= Series turns per parallel path of a phase
A = number of parallel paths of a phase.
Lecture Notes
Then
M.Kaliamoorthy AP/EEE, PSNACET
AT
 AN ph series ic   N ph series ia
Phase
Where i a  phase current  Ai c
It now follows that
 N ph series  
ia Provided the winding is concentrated one. The fundamental
AT / Pole / Phase  
P


peak of the concentrated winding is then
F1 p Concentrated Winding  
4  N ph series  

ia
 
P

Since the actual winding is distributed one, the fundamental peak will be reduced by a factor Kb
(Breadth Factor) Thus
 N ph series  
ia
K b 
 
P

4  N ph series  
ia cos 
Fa1 ( Fundamental )  K b 
 
P

F1 p Distributed Winding  
Hence
4
MMF SPACE WAVE OF ONE PHASE SHORT PITCHED WINDING
The effect of the mmf wave of short pitched coils can be visualized as shown in the figure given

'
'

below in which the stator has two short pitched coils a1a1 , a2 aa for phase a of 2 pole structure.
The mmf of each coil establishes one pole. The development diagram is shown in the figure
given below and it is seen that the mmf wave is rectangular but shorter space length than a pole
pitch. The amplitude of the fundamental peak gets reduced by a factor Kp called pitch factor,
compared to the full pitched rectangular wave. It can be shown from the Fourier analysis that
Lecture Notes
M.Kaliamoorthy AP/EEE, PSNACET
F1 p 
 sp 


Fpole  cos

2 

4
 sp 


Henec K p   cos
2 

In general when the winding is both distributed and short pitched, the fundamental space mmf of
phase a is generalizes to
 N ph series  
ia cos 
K w 

P


Where K w  K b K p  Winding Factor
Fa1 
4
(b) (i) Steps involved in the computation of DC motor Response
Lecture Notes
M.Kaliamoorthy AP/EEE, PSNACET
The basic steps involved in the computation of dc motor response and other quantities of interest
are as follows.
 Reading of machine parameters
 Specification of computational needs such as time response, frequency response, eigen
values for stability
 Use of transfer function to evaluate the responses
 Printing/Displaying of the output results.
11. (b) (ii) 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.
Lecture Notes
M.Kaliamoorthy AP/EEE, PSNACET
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)
(- ve sign for rs because genertaor action)
Vqdr  rr iqdr  pλ qdr ...................(2)
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
rs
0
rr  diag rkq1 rkq 2 rfd rkd 
0
0 ........................................(3)
rs 
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
Lecture Notes
M.Kaliamoorthy AP/EEE, PSNACET
 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 
0
0
Lsfd sin  r
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










.......(8)







The magnetizing inductances are defined as
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
Lecture Notes
M.Kaliamoorthy AP/EEE, PSNACET
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
The flux linkages may be now written as
abcs   Ls
 '    2 L' sr
 qdr   3
 
T
L' sr   i 
  abcs .............(24)
'
L r   i ' qdr 

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










.......(25)







Lecture Notes
M.Kaliamoorthy AP/EEE, PSNACET
 L'lkq1  Lmq

Lmq
'
Lr  

0

0

Lmq
0
L'lkq 2  Lmq
0
0
L lfd  Lmd
0
Lmd
'


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 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


 











Lecture Notes
M.Kaliamoorthy AP/EEE, PSNACET
12. (a) 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,
1
1


 Lls  Lms  2 Lms  2 Lms 


Ls    1 Lms Lls  Lms  1 Lms ..................(4)
2
 2

1
1
 L
 Lms Lls  Lms 
 2 ms

2
1
1


 Lmr 
 Llr  Lmr  2 Lmr
2
 1

1
Lr    Lmr Llr  Lmr  Lmr ..................(5)
2
 2

1
1
 L
 Lmr Llr  Lmr 
 2 mr

2

2 
2


cos  r
cos r 
 cos r 

3 
3



2

Lsr  Lsr cos r  2 
cos  r
cos r 
3 
3



2 
2 

cos r 
cos  r
 cos r 

3 
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 
The voltage equations expressed in terms of machine variables
Lecture Notes
M.Kaliamoorthy AP/EEE, PSNACET
 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


 
 
 

13. (b) Winding inductances of salient pole machine
Refer Lecture Notes in the web site for synchronous induction machiens