Induction Motor – Vector Control or Field
Oriented Control
By
M.Kaliamoorthy
Department of Electrical Engineering
1
Outline
 Introduction
 Analogy to DC Drive
 Principles of Field Orientation Control
 Rotor Flux Orientation Control
Indirect Rotor Flux Orientation (IRFO)
Direct Rotor Flux Orientation (DRFO)
 Stator Flux Orientation Control
Direct Stator Flux Orientation (DSFO)
 References
2
Introduction
 Induction Motor (IM) drives are replacing DC drives
because:
 Induction motor is simpler, smaller in size, less maintenance
 Less cost
 Capability of faster torque response
 Capability of faster speed response (due to lower inertia)
 DC motor is superior to IM with respect to ease of control
 High performance with simple control
 Due to decoupling component of torque and flux
3
Introduction
Induction Motor Drive
Scalar Control
• Control of current/voltage/frequency
magnitude based on steady-state
equivalent circuit model
• ignores transient conditions
•
•
•
•
• for low performance drives
Simple implementation
Inherent coupling of torque and flux
• Both are functions of voltage and
frequency
Leads to sluggish response
Easily prone to instability
Vector Control or Field Orientation
Control
• control of magnitude and phase of
currents and voltages based on
dynamic model
• Capable of observing steady state
& transient motor behaviour
• for high performance drives
• Complex implementation
• Decoupling of torque and flux
• similar to the DC drive
• Suitable for all applications previously
covered by DC drives
4
Analogy to DC Drive
In the DC motor: Te = k f Ia
f controlled by controlling If
If same direction as field f
Ia same direction as field a
Ia and f always perpendicular
and decoupled
• Hence, Te = k f Ia
•
•
•
•
•
= k’ If Ia sin 90
= k’(If x Ia)
f
a
• Keeping f constant, Te
controlled by controlling Ia
• Ia, If , a and f are space vectors
5
Analogy to DC Motor
• In the Induction Motor:
s
c’
b
r
a
b’
c
Te = kr x s
• s produced by stator currents
• r produced by induced rotor
currents
• Both s and r rotates at
synchronous speed s
• Angle between s and r
varies with load, and motor
speed r
• Torque and flux are coupled.
6
Analogy to DC Motor
• Induction Motor torque equation :
3P
Te 
ψ s  is
22
3 P Lm
Te 
ψr  is
2 2 Lr
(1)
(2)
• Compared with DC Motor torque equation:

Te  k I f I a  k ψ f ia sin 90  k ψ f  i a
'

(3)
• Hence, if the angle betweens orr andis is made to be
90, then the IM will behave like a DC motor.
7
Principles of Field Orientation Control
• Hence, if the angle betweens orr andis is made to be
90, then the IM will behave like a DC motor.
Achieved through orientation (alignment) of rotating dq frame
on r or s
Rotor-Flux
Orientation Control
Stator-Flux
Orientation Control
8
Principles of Field Orientation Control
Rotor-Flux
Orientation Control
Stator-Flux
Orientation Control
qs
qs
qr
qs
is
r
r
isq
i
s
dr
Ψs
i sq
r
sd
ds
3 P Lm
Te 
( rd isq  rq isd )
2 2 Lr
is
ds
Ψs
i sd
ds
3P
Te 
( sd isq  sq isd )
22
9
Principles of Field Orientation Control
• Summary of field orientation control on a selected flux vectorf
(i.e. either r , s or  m):
1
2
3
• In revolving (rotating) dfqf - reference frame, obtain
• isqf* from given rotor speed reference r* (via speed controller)
• isdf* from given flux reference f*
• Determine the angular position f of f (i.e. reference frame
orientation angle)
• used in the dfqf  dsqs conversion from vsdqf* (output of
isdqf* current controller) to vsdqs*.
• In the stationary dsqs - frame, obtain the reference stator voltages
vabcs*
• fed to the PWM inverter feeding the IM from vsdqs* using the
dsqs  abc transformation.
10
Rotor Flux Orientation Control
qs
• d- axis of dq- rotating frame
is aligned with r . Hence,
qr
is
r
r
isq
dr
 rdr   r
(4)
 rq  0
(5)
r
• Therefore,
3 P Lm
r
Te 
( rd isq ) (6)
s
d
2 2 Lr
r
i
 sq = torque producing current Similar to
Decoupled
r
i sd
r
sd
 i = field producing current
ia & if in
DC motor
torque and
flux control
11
Rotor Flux Orientation Control
 From the dynamic model of IM, if dq- frame rotates at general
speed g (in terms of vsd, vsq, isd, isq, ird, irq) :
vsd   Rs  SLs
  
 vsq     g Ls
vrd  
SLm
  
 vrq  ( g  r ) Lm
  g Ls
SLm
Rs  SLs
 g Lm
 ( g  r ) Lm
Rr ' SLr
SLm
( g  r ) Lr
  g Lm
 isd 
  
SLm
  isq  (7)
 ( g  r ) Lr  ird 
  
Rr ' SLr  irq 
 r rotates at synchronous speed s
 Hence, drqr- frame rotates at s
Therefore, g = s
 These voltage equations are in terms of isd, isq, ird, irq
 Better to have equations in terms of isd, isq, rd,  rq
(8)
12
Rotor Flux Orientation Control
• Rotor flux linkage is given by: rdq  Lmisdq  L'r irdq
• From (9):
rdq Lm
irdq 
'
r
L

'
r
L
isdq
(9)
(10)
• Substituting (8) and (10) into (7) gives the IM voltage
equations rotating at s in terms of vsd, vsq, isd, isq, rd, rq:
vsdψr   Rs  SLs
  sLs
 ψr  
Rs  SLs
vsq     sLs
vrdψr   Rr ' Lm Lr '
0
 ψr  
0
 Rr ' Lm Lr '
vrq  
S Lm Lr '
 s Lm Lr '
Rr '
Lr '  S
 sl
  s Lm Lr '   isdψr 
 ψr 

SLm Lr '   isq 
 ψr
  rd 
  sl
  ψr 
Rr ' Lr '  S   rq 
(11)
13
Rotor Flux Orientation Control
• Since  rqr  0 , hence the equations in rotor flux
orientation are:
Lm d ψr (12)
d ψr
ψr
v  R i  Ls isd   sLs isq   s
 rd
dt
Lr ' dt
L
d
(13)
vsqψr  Rs isqψr  Ls isqψr   sLs isdψr   s m  rdψr
dt
Lr '
ψr
sd
ψr
s sd
Rr ψr d ψr Lm
v  0   rd   rd 
Rr isdψr
Lr '
dt
Lr '
ψr
rq
v  0   sl
ψr
rq
ψr
rd
Lm

Rr isqψr
Lr '
Important equations for
Rotor Flux Orientation Control!
(14)
(15)
Note:
Total leakage factor =
2
L
  1 m '
Ls Lr
sl = slip speed (elec.)
14
Rotor Flux Orientation Control
ψr
 rdψr  Lmimrd
• Let
• Using (16), equation (14) can be rearranged to give:
i
ψr
sd
i
ψr
mrd
Lr ' d ψr

imrd
Rr dt
(16)
(17)
ψr
• imrd
is called the “equivalent magnetising current” or “field
current”
Lr '
ψr
ψr
(18)
• Hence, from (17): isd  1  S r imrd where  r  R
r
• Under steady-state conditions (i.e. constant flux):
ψr
isdψr  imrd
(19)
15
Rotor Flux Orientation Control
qs
• r rotates at synchronous speed s
• drqr- frame also rotates at s
• Hence,
qr
is
r
r
isq
r
dr
r
i sd
ds
dq- reference frame
orientation angle
   s dt
r
• For precise control, r must be
(20)
obtained at every instant in time
• Leads to two types of control:
– Indirect Rotor Flux
Orientation
– Direct Rotor Flux
Orientation
16
Indirect Rotor Flux Orientation (IRFO)
• Orientation angle:  r   s dt
• Synchronous speed obtained by adding slip speed and
electrical rotor speed
 r   s dt   sl  r  dt
• Slip speed can be obtained from equation (15):
Lmisqψr
(21)
isqψr
Lm Rr ψr
 sl 
i 
 ψr
ψr sq
ψr
Lr '  rd
 r rd  r imrd
ψr
ψr
• Under steady-state conditions (imrd  isd ):
(22)
sl 
isqψr
i
ψr
r sd
(23)
17
Indirect Rotor Flux Orientation (IRFO)
- implementation
 Closed-loop implementation under constant flux condition:
1. Obtain isdr* from r* using (16):
ψr*

ψr*
ψr*
rd
isd  imrd 
(24)
Lm
Obtain isqr* from outer speed control loop since isqr*
 Tm* based on (6):
*
2
Te
3 P Lm
ψr*
(25)
isq  ψr* where kt 
kt isd
2 2 Lr
Obtain vsdqr* from isdqr* via inner current control
loop.
18
Indirect Rotor Flux Orientation (IRFO)
- implementation
 Closed-loop implementation under constant flux condition:
2. Determine the angular position r using (21) and
(23):
ψr*


i
P
sq
*
sl  r dt    ψr*  m  dt
2
  r isd

   s dt   
r

(26)
where m is the measured mechanical speed of the
motor obtained from a tachogenerator or digital
encoder.
r to be used in the drqr  dsqs conversion of
stator voltage (i.e. vsdqr* to vsdqs* concersion).
19
Indirect Rotor Flux Orientation (IRFO)
- implementation
drqr  dsqs transformation
Rotating frame (drqr)
isdr*
r*
Eq. (24) +
isqr* r* +
PI +
Staionary frame (dsqs)
vsdr*
vsq
ejr
PI
-
-
isdr* isqr*
Eq. (23)
NO field
weakening
(constant flux)
slip
+

vsd
s*
 r
r
+
isdr
isqr
vas*
vsqs*
PI
r*
2-phase (dsqs )
to 3-phase (abc)
transformation
2/3
vcs*
isqs
PWM
VSI
IRFO
Scheme
P/2
m
ias
isds
e-jr
vbs*
3/2
ibs
ics
20
Indirect Rotor Flux Orientation (IRFO)
- implementation
 drqr  dsqs transformation
vsqs*
vsdr*
vsqr*
ejr
vsds*
 xsds  cos  r
 s 
 xsq   sin  r
 sin  r   xsdr 
 
cos  r   xsqr 
 dsqs  drqr transformation
isdr
isqr
isds
e-jr
isqs
 xsdr   cos  r
 r   
 xsq   sin  r
sin  r   xsds 
 
cos  r   xsqs 
21
Indirect Rotor Flux Orientation
(IRFO) - implementation
• 2-phase (dsqs ) to 3-phase (abc) transformation:
vas*
vsqs*
vsd
s*
2/3
1 s
xabc  Tabc
xdq
vbs*
vcs*
• 3-phase (abc) to 2-phase (dsqs ) transform is given by:
ias
isds
isqs
3/2
x  Tabcx abc
s
dq
ibs
ics
where:
Tabc
1 0 0 
 1 1 
0 3  3 
and
1
Tabc
1 0 
  12 23 
 12  23 
22
Example – IRFO Control of IM
• An induction motor has the following
parameters:
Parameter
Symbol
Value
Rated power
Prat
30 hp (22.4 kW)
Stator connection
Delta ()
No. of poles
P
6
Rated stator phase
voltage (rms)
Vs,rat
230 V
Rated stator phase
current (rms)
Is,rat
39.5 A
Rated frequency
frat
60 Hz
Rated speed
nrat
1168 rpm
23
Example – IRFO Control of IM ctd.
Parameter
Symbol
Value
Rated torque
Te,rat
183 Nm
Stator resistance
Rs
0.294 
Stator self
inductance
Referred rotor
resistance
Ls
0.0424 H
Rr’
0.156 
Referred rotor self
inductance
Lr’
0.0417 H
Mutual inductance
Lm
0.041 H
24
Example – IRFO Control of IM ctd.
The motor above operates in the indirect rotor field orientation (IRFO)
scheme, with the flux and torque commands equal to the respective
rated values, that is r* = 0.7865 Wb and Te* = 183 Nm. At the
instant t = 1 s since starting the motor, the rotor has made 8
revolutions. Determine at time t = 1s:
1.
2.
3.
4.
the stator reference currents isd* and isq* in the dq-rotating frame
the slip speed sl of the motor
the orientation angle r of the dq-rotating frame
the stator reference currents isds* and isqs* in the stationary dsqs
frame
5. the three-phase stator reference currents ias*, ibs* and ics*
25
Example – IRFO Control of IM ctd.
• Answers:
26
Indirect Rotor Flux Orientation (IRFO)
– field weakening
• Closed-loop implementation under field weakening
condition:
– Employed for operations above base speed
– DC motor: flux weakened by reducing field current if
vf
Lf d
 if 
if
imrd*
Rf
R f dt
– Compared with eq. (17) for IM:
imrd (rated)
L
'
d
ψr
ψr
isdψr  imrd
 r
imrd
Rr dt
– IM: flux weakened by reducing imrd
r (base)
(i.e. “equivalent magnetising current”
or “field current)
r
27
Indirect Rotor Flux Orientation (IRFO)
– field weakening implementation
With field
weakening
r*
Rotating frame (drqr) Staionary frame (dsqs)
imrd r *
+
imrd
r
1
1  S r 
isd
PI
r* +
r*
PI
-
+
isqr* +
imrdr*r
vsdr*
PI
vsqr*
ejr
PI
-
isq *
Eq. (22)
vsqs*
slip
+

 r
r
Same as
in slide 20
+
isdr
isqr
vsds*
isds
e-jr
isqs
28
Indirect Rotor Flux Orientation
(IRFO) – Parameter sensitivity
 Mismatch between IRFO Controller and IM may occur
 due to parameter changes with operating conditions (eg.
increase in temperature, saturation)
 Mismatch causes coupling between T and  producing
components
 Consequences:
 r deviates from reference value (i.e. r*)
 Te deviates in a non-linear relationship from command
value (i.e. Te*)
 Oscillations occurs in r and Te response during torque
transients (settling time of oscillations = r)
29
Direct Rotor Flux Orientation (DRFO)
• Orientation angle:
  tan
r
1
 rq
s
 rd
s
(27)
obtained from:
1. Direct measurements of airgap fluxes mds and mqs
2. Estimated from motor’s stator voltages vsdqs
and stator currents isdqs
Note that: ψ   s 2   s 2
r
rd
rq
(28)
30
Direct Rotor Flux Orientation (DRFO) –
Direct measurements mds & mqs
1. Direct measurements of airgap fluxes mds and mqs
 mds and mqs measured using:
 Hall sensors – fragile
 flux sensing coils on the stator windings – voltages induced in
coils are integrated to obtain mds and mqs
 The rotor flux r is then obtained from:
s
s
L'r
'
(29)
rdq 
mdq  Llr isdq
Lm
 Disadvantages: sensors are inconvenient and spoil the
ruggedness of IM.
s
31
Direct Rotor Flux Orientation (DRFO) –
Direct measurements mds & mqs
Rotating frame
isdr*
r*
Eq. (24) +
isqr* r* +
PI +
-
r
NO field
weakening
(constant flux)
(drqr)
Stationary frame
vsdr*
vsq
r*
ejr
PI
DRFO  r
Scheme tan-1
isdr
isqr
 r
e-jr
vsd
s*
2/3
vbs*
vcs*
mds
 rds
 rqs Eq. (29) mqs
P/2
PWM
VSI
m
ias
isds
isqs
Flux sensing coils
arranged in quadrature
vas*
vsqs*
PI
(dsqs)
3/2
ibs
ics
32
Direct Rotor Flux Orientation (DRFO) –
Estimated from vsdqs & isdqs
2. Estimated from motor’s stator voltages and currents
 sds and  sqs obtained from stator voltage equations:


sdq   vsdq  Rs isdq  sdq 0
 The rotor flux r is then obtained from:
s
s
s

s
(30)

s
s
L'r
rdq 
sdq  Ls isdq
(31)
Lm
 Disadvantages: dc-drift due to noise in electronic
circuits employed, incorrect initial values of flux vector
components sdq(0)
s
33
Direct Rotor Flux Orientation (DRFO) –
Estimated from vsdqs & isdqs
2. Estimated from motor’s stator voltages and currents
 This scheme is part of sensorless drive scheme
 using machine parameters, voltages and currents to
estimate flux and speed
 sdqs calculations (eq. 30) depends on Rs
 Poor field orientation at low speeds ( < 2 Hz), above 2 Hz,
DRFO scheme as good as IRFO
 Solution: add boost voltage to vsdqs at low speeds
 Disadvantages: Parameter sensitive, dc-drift due to
noise in electronic circuits employed, incorrect initial
values of flux vector components sdq(0)
34
Direct Rotor Flux Orientation (DRFO) –
Estimated from vsdqs & isdqs
Rotating frame (drqr) Stationary frame (dsqs)
isdr*
r*
Eq. (24) +
isqr* r* +
PI +
-
r
NO field
weakening
(constant flux)
vsdr*
PI
vsq
r*
ejr
PI
DRFO  r
Scheme tan-1
isdr
isqr
vas*
vsqs*
 r
e-jr
vsd
s*
2/3
vbs*
vcs*
PWM
VSI
sds
 rds
vsdqs
 rqs Eq. (31) sqs Eq. (30) isdqs
m
P/2
ias
isds
isqs
3/2
ibs
ics
35
Direct Rotor Flux Orientation (DRFO) –
field weakening implementation
With field
weakening
r*
Rotating frame (drqr) Stationary frame (dsqs)
imrd r *
+
imrd
r
1
1  S r 
isd
PI
r* +
r*
PI
-
+
isqr* +
vsdr*
vsqs*
PI
vsqr*
ejr
PI
-
 r
tan-1
isdr
isqr
 r
e-jr
vsds*
 rds
 rqs
r
Same as
in
slide
26 or 29
isds
isqs
36
Stator Flux Orientation Control
• d- axis of dq- rotating frame
is aligned with s. Hence,
qs
qs
is
s
Ψs
i sq
i
ψs
ψsd
 ψs
ds
(32)
ψ 0
ψs
sq
(33)
• Therefore,
Ψs
sd
ds
Ψs
i
 sq = torque producing current
 i sdΨs = field producing current
3P
Te 
( sd isq )
22
Similar to
ia & if in
DC motor
(34)
Decoupled
torque and
flux control
37
Stator Flux Orientation Control
 From the dynamic model of IM, if dq- frame rotates at general
speed g (in terms of vsd, vsq, isd, isq, ird, irq):
vsd   Rs  SLs
  
 vsq     g Ls
vrd  
SLm
  
 vrq  ( g  r ) Lm
  g Ls
SLm
Rs  SLs
 g Lm
 ( g  r ) Lm
Rr ' SLr
SLm
( g  r ) Lr
  g Lm
 isd 
  
SLm
  isq  (7)
 ( g  r ) Lr  ird 
  
Rr ' SLr  irq 
 s rotates at synchronous speed s
 Hence, dsqs- frame rotates at s
Therefore, g = s
 These voltage equations are in terms of isd, isq, ird, irq
 Better to have equations in terms of isd, isq, sd,  sq
(8)
38
Stator Flux Orientation Control
• Stator flux linkage is given by: Ψsdq  Lsisdq  Lmirdq
• From (9):
Ψ sdq L
irdq 
Lm

s
Lm
isdq
(35)
(36)
• Substituting (8) and (36) into (7) gives the IM voltage
equations rotating at s in terms of vsd, vsq, isd, isq, sd, sq:
vsdψs  
Rs
0
 ψs  
0
Rs
vsq   
vrdψs   Ls 1  S r 
 sl rLs
 ψs  
 Ls 1  S r 
vrq     sl rLs
S
s
1  S r 
 sl r
  s   isdψs 


S   isqψs 
 ψs
  sl r   sd 
  ψs 
1  S r   sq 
(37)
39
Stator Flux Orientation Control
• Since ψ  0 , hence the equations in stator flux
orientation are:
d ψs
ψs
ψs
vsd  Rs isd   sd
(38)
ψs
sq
dt
ψs
vsq
 Rsisqψs  s sdψs
vrdψs  0   sdψs   r
(39)
d ψs
d


 sd  Ls  isdψs   r isdψs   sl r Ls isqψs (40)
dt
dt 

d ψs 
 ψs
v  0   Ls  isq   r isq   sl r  sdψs  Ls isdψs 
dt 

ψs
rq
Important equations for
Stator Flux Orientation Control!
(41)
40
Stator Flux Orientation Control
• Equation (40) can be rearranged to give:
1  S r  sdψs  1  S r Lsisdψs  sl r Lsisqψs
(42)
ψs
ψs
ψ
i
•
sd should be independent of torque producing current s q
ψs
ψs
ψ
ψ
i
• From (42), sd is proportional to sd and is qs .
ψs
ψs
ψ
• Coupling exists between sd and is q .
ψ
ψs
s
i
Varying s q to control torque causes change in ψ sd
Torque will not react immediately to isψqs
41
Stator Flux Orientation Control
– Dynamic Decoupling
• De-coupler is required to
ψs
ψs
ψs
ψ
– overcome the coupling between sd and isq (so thatisq has
ψs
ψ
no effect on sd )
ψs*
– Provide the reference value foris d
• Rearranging eq. (42) gives:
ψs*
sq

1   sdψs*
*
 S  
 sl isqψs*
 r  Ls

ψs*
isd 

1 
 S 

 r 

(43)
• i
can be obtained from outer speed control loop
• However, eq. (43) requires sl*
42
Stator Flux Orientation Control
– Dynamic Decoupling
•  sl* can be obtained from (41):
 sl *
ψs*
• ψ sd

1
 S 
 r




 i ψs*
(44)

 isdψs*
Ls
*
ψ
in (43) and (44) is the reference stator flux vector s
ψs*
sd
sq
• Hence, equations (43) and (44) provide dynamic decoupling
ψs
of the flux-producing isψs*
and
torque-producing
i
sq currents.
d
43
Stator Flux Orientation Control
– Dynamic Decoupling
• Dynamic decoupling system implementation:
s*
1
 Ls
S
1
r
isqs*
from speed
controller
1
+
+
S
1
 r
isds*
isqs*
S
1
 r
x
x
sl*
1
ψ
ψs*
 isd
Ls
*
s
44
Stator Flux Orientation Control
 dsqs- frame also rotates at s
 For precise control, s must be
obtained at every instant in time
 Leads to two types of control:
qs
qs
is
s
Ψs
i sq
s
Ψs
i sd
dq- reference frame
orientation angle
ds
 Indirect Stator Flux
Orientation
 Direct Stator Flux
Orientation
 s easily estimated from motor’s
stator voltages vsdqs and stator
ds
currents isdqs
 Hence, Indirect Stator Flux
Orientation scheme
unessential.
45
Direct Stator Flux Orientation (DSFO) implementation
 Closed-loop implementation:
1. Obtain isds* from s control loop and dynamic
decoupling system shown in slide 38.
Obtain isqs* from outer speed control loop since
isqr*  Te* based on (34):
*
i
ψs*
sq
Te
3P
 ψs* where kt 
kt isd
22
(45)
Obtain vsdqs* from isdqs* via inner current control
loop.
46
Direct Stator Flux Orientation (DSFO) implementation
 Closed-loop implementation:
2. Determine the angular position s using:
 ψ  tan
s
s

sq
1
(46)
 sd s
sds and sqs obtained from stator voltage equations:


sdq   vsdq  Rs isdq  sdq 0
s
Note that:
s
s
s2
ψ s   sd  sq
s
s2
(47)
(48)
Eq. (48) will be used as feedback for the s control loop
47
Direct Stator Flux Orientation (DSFO) implementation
 Closed-loop implementation:
3. s to be used in the dsqs  dsqs conversion of
stator voltage (i.e. vsdqs* to vsdqs* concersion).
 s estimated from pure integration of motor’s stator voltages
equations eq. (47) which has disadvantages of:
 dc-drift due to noise in electronic circuits employed
 incorrect initial values of flux vector components sdqs(0)
 Solution: A low-pass filter can be used to replace the pure
integrator and avoid the problems above.
48
Direct Stator Flux Orientation (DSFO) implementation
r
r* +
s*
isqs*
+
-
PI
-
Decoupling
system
+ i s*
sd
1
+
PI
- | |
S
vsqs*
vsds*
1
 r
PI
 s
tan-1
isqs
s
Eq. (48)
sds sqs
ejs
isds
 s
e-js
vsds*
sds
2/3
PWM
VSI
vbs*
vcs*
vsdqs
Eq. (47) isdqs
sqs
ias
isqs
isds
m
vas*
vsqs*
PI
+
-
+
P/2
3/2
ibs
ics
Rotating frame (dsqs ) Stationary frame (dsqs )
49
References
• Trzynadlowski, A. M., Control of Induction Motors, Academic
Press, San Diego, 2001.
• Krishnan, R., Electric Motor Drives: Modeling, Analysis and
Control, Prentice-Hall, New Jersey, 2001.
• Bose, B. K., Modern Power Electronics and AC drives, PrenticeHall, New Jersey, 2002.
• Asher, G.M, Vector Control of Induction Motor Course Notes,
University of Nottingham, UK, 2002.
50