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Advanced Power Systems
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Induction Machines_Introduction

When supplied from a balanced three-phase (polyphase) source, the
three-phase (polyphase) stator winding produces a rotating magnetic
field. The magnetic field rotates at a synchronous speed given by:
120 f
ns 
P
This phenomenon applies to synchronous motors as well.

Due to the stator magnetic field, an emf is induced in the rotor
winding. The emf sets up a current in the rotor.

Current-carrying rotor winding (conductors) under the stator
magnetic field produces a torque (force) in the direction of the
rotating stator magnetic field.

An induction machine may be regarded as a transformer. The rotor
(current) frequency is, however, different from the stator (current)
frequency.
Slip
Slip: The difference between the speed of the rotating flux produced
by the stator and the speed of the rotor is called slip speed, and the
ratio of slip speed to synchronous speed is called slip.
n  nr
S s
ns
where:
ns = synchronous speed
nr = rotor speed (r/min)
s = slip
The slip depends on the mechanical load connected to the rotor shaft.
Increasing the shaft load decreases the rotor speed, thus increasing
the slip.
Modeling Induction Machines_1
Kundur, p.279

Motor Mode of Operation: At no-load, the machine operates at
negligible slip. If a mechanical load is applied, the slip increases
such that the induced voltage and current produce the torque
required by the load. The machine thus operates as a motor.

Generator Mode of Operation: If the rotor is driven by a primemover at a speed greater than that of the stator field, the slip is
negative. The polarities of the induced voltages are reversed so
that the resulting torque is opposite in direction to that of rotation.
The machine thus operates as a generator.
Modeling Induction Machines_2
In developing the model of induction machines, following aspects
will differ from those of synchronous machines:

The d- and q-axis equivalent circuits are identical as the rotor has
symmetrical structure.

The rotor speed is not fixed but varies with load. This has an
impact on the selection of the d-q reference frame.

There is no excitation source to the rotor windings. Consequently,
the dynamics of the rotor circuits are determined by slip.

The current induced in the shorted rotor windings produce a field
with the same number of poles as that produced by the stator
windings. Rotor windings may therefore be modeled by an
equivalent three-phase winding.
Stator and Rotor Circuits
  r t
 1  s  s t
v a 
v  
 b
 v c 
 a   R s
p  b   
  c  
v A 
v  
 B
vC 
 A   R r
p  B   
 C  
Rs
 i a 
 i 
 b 
R s  ic 
Rr
 i A 
 i 
 B 
Rr  iC 
Basic Machine Equations in d-q Reference Frame
Stator voltages:
vds  Rs ids   s  qs  p ds
vqs  Rs iqs   s  ds  p qs
Transient term
Speed voltage term
Stator winding voltage drop term
Rotor voltages:
vdr  Rr idr  s s  qr  p dr
vqr  Rr iqr  s s  dr  p qr
Power input to the stator:
Ps  vds ids  vqsiqs
Power input to the rotor:
Pr  vdr idr  vqr iqr
Rotor speed voltages created in the
rotor windings moving at slip speed w.r.to.
the synchronously rotating flux wave.
Power and Electromagnetic Torque
The electromagnetic torque developed is obtained as the power associated
with the speed voltages divided by the shaft speed in mechanical radian per
second. The speed voltage terms associated with the rotor:
vdr _ speed _ voltage   s s  qr
vqr _ speed _ voltage  s s  dr
Power associated with the speed voltage:
Pspeed _ voltage  vdr _ speed _ voltage idr  vqr _ speed _ voltage iqr


  dr iqr   qr idr  s s 
The electromagnetic torque:
Te 
Pspeed _ voltage
 r   s s 
  qr idr   dr iqr
Acceleration Equation
The electromagnetic torque developed by the motor drives the
mechanical load. If there is a mismatch between the electromagnetic
torque (Te) and the mechanical load torque (Tm), the differential
torque accelerates the rotor mass. Consequently,
dr
1
Te  Tm 

dt
2H
where, H is the combined moment of inertia of the rotor and the
connected load and r is the angular velocity of the rotor.
Representation in Stability Studies
For representation in power system stability studies, the transient voltage
terms are neglected in the stator voltage equations. With the stator
transients neglected and rotor windings shorted, the per unit induction
motor electrical equations can be summarized as follows:
vds  Rs ids   s  qs
Stator voltages
vqs  Rs iqs   s  ds
vdr  0  Rr idr  s s  qr  p dr
Rotor voltages
vqr  0  Rr iqr  s s  dr  p qr
 ds   Ls  Lm ids  Lm idr
Flux linkages
 qs   Ls  Lm iqs  Lm iqr
 dr  Lm ids   Lr  Lm idr
 qr  Lm iqs   Lr  Lm iqr
Representation in Stability Studies (cont’d)
To reduce the previous equations to a form suitable for implementation in
a stability program, the rotor currents can be eliminated and the
relationship between the stator current and voltage in terms of a voltage
behind the transient reactance can be expressed as follows:
idr 
iqr 
 ds
 qs
 dr  Lm ids
 Lr  Lm 
 qr  Lm iqs
 Lr  Lm 

  dr  Lm ids   Lm 
L2m 
  
 dr   Ls  Lm  
  Ls  Lm ids  Lm 
 ids
L

L
L

L
L

L


r
m   r
m
r
m 

  qr  Lm iqs   Lm 
L2m 
  
 qr   Ls  Lm  
  Ls  Lm iqs  Lm 
 iqs
L

L
L

L
L

L

r
m   r
m
r
m 

Representation in Stability Studies (cont’d)
vds
 Lm 

L2m  

 qr  Ls  Lm 
 Rs ids   s 
iqs 


L

L
L

L
m
r
m  
 r

2


L
m
 iqs   s  Lm  qr
 Rs ids   s  Ls  Lm 
L L 
Lr  Lm 
 r
m

 Rs ids  X 's iqs  v'd
vqs
2
 Lm 

 
L
m
 ids 
 dr   Ls  Lm 
 Rs iqs   s 


L

L
L

L
m
r
m  

 r
2


L
m

 ids   s  Lm  dr
 Rs ids   s  Ls  Lm 
L L 
Lr  Lm 
 r
m

 Rs ids  X 's ids  v'q
vds  Rs ids  X 's iqs  v'd
vqs  Rs ids  X 's ids  v'q
Where,
 Lm 
 qr
v'd    s 
L

L
 r
m
 Lm 
 dr
v'q   s 
L

L
 r
m
X 's
2


L
m


  s  Ls  Lm 
Lr  Lm 

Transient reactance of the
induction machine
Representation in Stability Studies (cont’d)
The stator voltages:
vds  Rs ids  X 's iqs  v'd
vqs  Rs ids  X 's ids  v'q
The stator voltage equation can be combined and can be
expressed in the following form:



vds  jvqs  Rs  jX 's ids  jids   v'd  jv'q

The above relationship can be expressed in the following form:


Vs  Rs  jX 's I s  V '
Rs
+
Vs
X’s
+
Is
V’
Vs: stator terminal voltage
V’: voltage behind
aaatransient impedance
Induction machine transient equivalent circuit.
Representation in Stability Studies (cont’d)
By eliminating the rotor currents and expressing the rotor flux linkages in
terms of vd’ and vq’ , following can be obtained:
idr 
 qr  Lm iqs
 dr  Lm ids
; iqr 
 Lr  Lm 
 Lr  Lm 
   Lm ids 
  s s  qr  p dr
vdr  0  Rr  dr
L

L

r
m 
  L

  L 
1  L 
or , p  s m  dr    '  s m  dr  Lm ids   s s  s m  qr
T0  Lr  Lm 
 Lr  Lm

 Lr  Lm 
 
or , p v'q  
1
T0'
v  X
'
q
s
 
 X 's ids  s s v'd
  qr  Lm iqs 
  s s  dr  p qr
vqr  0  Rr 
L

L
r
m 

  L

  L 
1  L 
or , p  s m  qr    '  s m   qr  Lm iqs  s s  s m  dr
T0  Lr  Lm 
 Lr  Lm

 Lr  Lm 

 
or , p v'd  
1
T0'
v  X
'
d
s
 
 X 's iqs  s s v'q

Representation in Stability Studies (cont’d)
Rotor circuit dynamics:
 
p v'd  
 
p v'q  

v  X
T
1
'
0
'
d

v  X
T
1
'
0
where ,
L  Lm
T0'  r
Rr
'
q
X s   s  Ls  Lm 
 
s
 X 's iqs  s s v'q
s
 X 's ids  s s v'd
 
Transient open-circuit time
constant which characterizes the
decay of the rotor transients when
stator is open circuited.
Rotor Electromagnetic Torque
The electromagnetic torque:
Te   qr idr   dr iqr
  qr  Lm iqs 
  dr  Lm ids 

   dr 
  qr 
 Lr  Lm 
 Lr  Lm 
 Lm 
 Lm 


 iqs
   qr 
ids   dr 

 Lr  Lm 
 Lr  Lm 
1 '

vd ids  v'q iqs
s


 v'd ids  v'q iqs with  s  1.0 pu
Alternative Rotor Constructions



High efficiency at normal operating conditions requires a low rotor
resistance.
On the other hand, a high rotor resistance is required to produce a high
starting torque and to keep the magnitude of the starting current low and
the power factor high.
The wound rotor is one way of meeting the above mentioned need for
varying the rotor resistance at different operating conditions. Woundrotor motors are, however, more
expensive than squirrel-cage motors.
Effect of the rotor resistance
the torque-slip curves.
Double Squirrel-Cage Rotor Construction





Following double squirrel-cage arrangements can also be used to
obtained a high value of effective resistance at starting and a low value
of the resistance at full-load operation.
It consists of two layers of bars, both short-circuited by end rings.
The upper bars are small in cross-section and have a high resistance.
They are placed near the rotor surface so that the leakage flux sees a path
of high reluctance; consequently, they have a low leakage inductance.
The lower bars have a large cross-section, a lower resistance and a high
leakage inductance.
Double squirrel-cage rotor bars
Double Squirrel-Cage Rotor Construction (cont’d)


At starting, rotor frequency is high and very little current flows through
the lower bars; the effective resistance of the rotor is then the high
resistance upper bars.
At normal low slip operation, leakage reactances are negligible, and the
rotor current flows largely through the low resistance lower bars; the
effective rotor resistance is equal to that of the two sets of bars in
parallel.
Double squirrel-cage rotor bars
Deep-Bar Rotor Construction




The use of deep, narrow rotor bars produces torque-slip characteristics
similar to those of a double-cage rotor.
Leakage inductance of the top cross-section of the rotor bar is relatively
low; the lower sections have progressively higher leakage inductance.
At starting, due to the high rotor frequency, the current is concentrated
towards the top layers of the rotor bar.
At full-load operation, the current distribution becomes uniform and the
effective resistance is low.
Deep-bar rotor construction
Equivalent Circuit with a Double Cage or Deep Bar Rotor
Equivalent circuit of a singlecage induction motor (with one
rotor winding).
Equivalent circuit of a doublecage induction motor (two rotor
windings).
Equivalent Circuit_Single Rotor Circuit Representation
For system studies, the rotor should be
represented by a single rotor circuit whose
parameters vary as a function of slip, s.

m 2  ms 2  R1
Rr 0 

Rr s   Rr 0
m2  s2
Rr 0  mR1 
R2 

X r s   X 1 
m2  s2
where,
RR
Rr 0  1 2
R1  R2
m
R1  R2
X2