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CHAPTER 1
INTRODUCTION
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Introduction
Till recently, in applications variable speed operation was required, only DC motors were
used due to the ease, which with one could control them. Separately excited DC motors
were particularly popular in applications where fast torque response was required.
However DC motors have some generic disadvantages like
requirement of periodic maintenance, unstable in explosive or corrosive environments due to sparking problem commutation is difficult high currents and voltages, and hence its use is use is
limited to low power, low speed motors
These problems can be overcome by using Induction Motors that have a simple and
rugged structure. Further, they have a lower weight to output power ratio compared to
their DC counterparts.
1.1. Vector Control of Induction Motor
The idea behind the vector control or field oriented control is to control the
Induction Motors in the similar for DC motor control .The flux and torque, in the case of
DC machines, can be controlled independently controlling the field and armature currents
respectively. It is because of this inherent decoupling between the flux and the armature
currents; one is able to achieve very good torque dynamics from DC machines. Unlike
DC machines, there is no inherent decoupling between the flux and the torque producing
components of the stator current in AC machines. Therefore, achieving good torque
dynamics in AC machines is not easy. However, nowadays field orientation control or
vector control techniques have been employed, which result in good torque dynamics of
AC motors.
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1.2. Sensorless Vector Control of Induction Motor
Today, vector controlled induction motor has been established as the core servo-
drive system for industry applications, and has been widely applied almost in all
industrial fields. However, in some applications, the necessity of the speed sensor for
vector control may become the defect of the ,or make the users hesitate to apply this
excellent drive to their systems. The effort of engineers has solved this difficulty, and the
vector control of induction motor can be now implemented without speed sensor.
Hereafter, this implementation is briefly named as sensorless control.
Induction Motor drives without shaft sensor, sensorless drives, are increasingly
applied in many industrial processes involving lower cost and higher performance
specifications. To achieve sensorless control requires either flux measurement using flux
sensors, flux estimation, or speed identification. it is worthy of note that both voltage and
current sensors are required for the implementation of flux estimation and speed
identification.
1.3. Sensorless Vector Control of Induction Motor at Zero Frequency
The sensorless drive at low speed and in the regenerating operation still remains
an unsolved problem . For the stable sensorless control at low speed including zero
frequency, a new control scheme using secondary speed-emf estimation was presented in
this dissertation work, instead of the flux or excitation current. Especially at zero stator
frequency, the secondary speed emf is estimated under fluctuated reference of the
secondary flux to assure the stability of the estimation, and the stable sensorless drive is
realized.
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CHAPTER 2
MOTOR CONTROL STRATERGIES
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Motor control stratergies
2.2.1. Direct field oriented control
In this mode of control the flux measurement can be made using either the hall
sensors or the stator search (sense) coils. If the stator coils are used, then the voltage
sensed from the coils will have to be integrated to obtain the air gap flux linkages. The
measured air flux linkage components are used to calculate the required (rotor, stator or
air gap) flux linkage space phasor magnitude and position V. The value ofV thus
computed is used to align the arbitrary axis along the flux linkage space phasor to achieve
decoupled control of the torque and flux producing components of the stator current and
space phasor.
The flux sensing devices are placed in the air gap of the machine, which will
determine the air gap flux space phasor. Any other flux space phasor can be calculated as
it has an algebraic relationship with the air gap flux space phasor. The air gap flux sensed
by either hall-effect devices or stator search coils suffer from the disadvantage that a
specially constructed induction motor is required. Further, hall sensors are very sensitive
to temperature and mechanical vibrations and the flux signal is distorted by large slot
harmonics that can not be filtered effectively because their frequency varies with motor
speed. In the case of stator search (sense) coils, they are placed in the wedges close to the
stator slots to sense the rate of change of air flux. The induced voltage in the search coil
is proportional to the rate of change of flux. This induced voltage has to be integrated to
obtain the air gap flux. At low speeds below about 1HZ, the induced voltage will be
significantly low which would give rise to in accurate flux sensing due to presence of
comparable amplitudes of noise and disturbances in a practical system. As an alternative,
indirect flux estimation techniques are preferred as explained in the next sub-section.
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2.2.2. Indirect field oriented control:
In an Indirect Field Oriented Control (IFOC) a flux estimator is used to estimate the
required flux linkage space phasor magnitude and angular positiona
U . The shaft position
is usually needed for estimating flux linkage space phasor position. If the shaft transducer
is a position encoder, then the position informationr
U can be directly used. But if the
shaft transducer is a speed transducer like a tacho, then speed has to be integrated to
obtain the shaft position. In the case of shaft transducer being a position encoder, the
speed feedback is obtained by differentiating the shaft position information.
Indirect sensing of flux space phasors give a more versatile drive system that can
be used with standard commercial motors, but this approach would generally result in a
more complex control system. Since it is generally desirable to have a scheme which is
applicable for all induction motors, the indirect field oriented has emerged as the more
popular method. In the indirect method of field orientation the flux linkage space phasor
is estimated from the motor model as will be discussed in next section. As a consequence
all indirect methods are sensitive to variations in some machine parameter like the stator
or rotor time constants. For example, in the rotor flux oriented control, the indirect rotor
flux estimator is sensitive to the rotor time constant Xr, of the motor. In the case of stator
flux oriented control, the indirect stator flux estimator is sensitive to the stator time
constant of the motor. In the air gap flux oriented control, the indirect air gap flux
estimator is sensitive to both the stator and the rotor time constants. Therefore, if the
value of the motor parameter varies, the desired decoupled of the flux and the torque
components of the stator current space phasor is not achieved and this leads to
deterioration in the dynamic behavior of the drive system.
2.3 Sensorless ControlSensorless control is another extension to the FOC algorithm that allows
induction motors to operate without the need for mechanical speed Sensorless control
is another extension to the FOC algorithm that allows induction motors to operate
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without the need for mechanical speed sensors. These sensors are notoriously prone to
breakage so removing them not only reduces the cost and size of the motor but improves
the drives long term accuracy and reliability. This is particularly important if the motor
is being used in a harsh, inaccessible environment such as an oil well.
Instead of physically measuring certain values control engineers can calculate
them from a systems state variables. This is known as the state space modeling approach
and is a powerful method for analyzing and controlling complex non-linear systems with
multiple inputs and outputs. In high performance sensorless motor drives the two main
control techniques used are open loop estimators and closed loop observers. In early
literature the terms observer and estimator are often used interchangeably however most
recent papers define estimators as devices that use a model to predict the speed using the
phase currents and voltages as state variables. Observers also use a model to estimate
values, however these estimates are improved by an error feedback compensator that
measures the difference between the estimated and actual values. The predicted value of
speed is then used by the FOC to adjust the PWM waveform in exactly the same way as
an actual measured value.
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CHAPTER 3
DYANAMIC MODEL OF INDUCTION MOTOR
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Dyanamic model of IM
3.1. Introduction
In developing the dynamic model of the induction motor, the following assumptions
will be made without affecting the validity of the model.
y The motor has symmetrical three phase windings.y The mmf wave is sinusoidally distributed in space.y The stator and rotor iron have infinite permeability.y Skin effect and core losses are neglected.y The motor is operating in the linear region of B-H characteristic.
In order to understand and analyze vector control, the dynamic model of the
induction motor is necessary. It has been found that the dynamic model equations
developed on a rotating reference frame is easier to describe the characteristics of
induction motors. It is the objective of this chapter is to derive and explain induction
motor model in relatively simple terms by using the concept of space vectors and d-q
variables. It will be shown that when we choose a synchronous reference frame in which
rotor flux lies on the d-axis, dynamic equations of the induction motor is simplified and
analogous to a DC motor. Traditionally in analysis and design of induction motors, the
per-phase equivalent circuit of induction motors shown in Fig. 3.1 has been widely
used. In the circuit, Rs (Rr) is the stator (rotor) resistance and Lm is called the
magnetizing inductance of the motor. Note that stator (rotor) inductance Ls (Lr) is defined
by
Ls = Lls + Lm, Lr= Llr+ Lm (3.1)
where Lls(Lrs) is the stator (rotor) leakage inductance. Also note that in this equivalent
circuit, all rotor parameters and variables are not actual quantities but are quantities
referred to the stator . Parameters of the circuit are determined from no-load test and
locked rotor test. It is also known that induction motors do not rotate synchronously to
the excitation frequency. At rated load, the speed of induction motors is slightly (about 2
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-7% slip in many cases) less than the synchronous speed. If the excitation frequency
injected into the stator ise
[ and the actual speed converted into electrical frequency unit
isr
[ , slip s is defined by
s = (e
[ r
[ )/e
[ =sl
[ /e
[ (3.2)
andsl
[ is called the slip frequency which is the frequency of the actual rotor current. In
the steady-state AC circuit, current and voltage phasors are used and they are denoted by
the underline. In Fig. 3.1, power consumption in the stator is interpreted as Is2Rs, while
Ir2Rr/s represents both power consumption in the rotor and the mechanical output
(torque). By subtracting rotor loss Ir2Rr from Ir
2Rr/s, produced torque (mechanical power
divided by the shaft speed) is given by
Te = ir2Rr(P/2) (1-s) / (swr) = ir
2Rr[ P / (2we )], (3.3)
where P is the number of poles. Although the per-phase equivalent circuit is useful in
analyzing and predicting steady-state performance, it is not applicable to explain dynamic
performance of the induction motor.
Fig. 3.1 Conventional Per-phase Equivalent Circuit
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3.2. Dynamic Model in Space Vector Form
In an induction motor, the 3-phase stator windings are designed to produce
sinusoidally distributed mmf in space along the airgap periphery. Assuming uniform
airgap and neglecting the effects of slot harmonics, distribution of magnetic flux will also
be sinusoidal. It is also assumed that the neutral connection of the machine is open so that
phase voltages, currents and flux linkages are always balanced and there are no zero
phase sequence component in the system. For such machines, the notation in terms of the
space vector is very useful. For 3-phase induction motors, the space vector Yss
of the
stator voltage, current and flux linkage is defined from its phase quantities by
Yss
= (2/3) (Ya + kYb + k2Yc ), (3.4)
where k = exp(j 2/3). The above transform is reversible and each phase quantities can
be calculated from the space vector by,
Ia = Re (Y s ), Ib = Re (k2Ys ), Ic = Re (kY s ). (3.5)
For a sinusoidal 3-phase quantity of constant rms value, the corresponding space
vector is a constant-magnitude vector rotating at the frequency of the sinusoid with
respect to the fixed (stationary) reference frame. Note that the space vector is at vector
angle 0 when
A-phase signal (Ya) is at its sinusoidal peak value in steady-state. With space vectornotation, voltage equations on the stator and rotor circuits of induction motors are,
vs
= Rs is
+ ps (3.6)
vr = Rrir + pr = 0 (3.7)
It is very convenient to transform actual rotor variables (Vr, ir, r) from Eq.
3.7 on a rotor reference frame into a new variables ( Vr, ir
, r
) on a stator reference
frame as in the derivation of conventional steady-state equivalent circuit. The Space
Vector diagram for induction motor is shown in fig 3.2 Let the stator to rotor winding
turn ratio be n and the angular position of the rotor be r, and define
ir
= (1/n) exp(jr) ir, r
= n exp(j r) r (3.8)
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VS
is iqs
ids
ir
1
a r
fig.3.2 Space Vector diagram for induction motor
Also, by defining referred rotor impedances as Rr = n2Rr, etc., we have
vs = Rs is + ps (3.9)
0 = Rr irs
+ (p jr) r (3.10)
Where r= pr, is the speed of the motor in electrical frequency unit and
s
= Lsis
+ Lmir (3.11)
r
= Lmis
+ Lrir (3.12)
The above 4 equations (Eq. 3.9 - 3.12) constitute a dynamic model of the induction motor
on a stationary (stator) reference frame in space vector form. These model equations may
be simplified by eliminating flux linkages as
vs
= (Rs + Lsp) is+ Lm pir
(3.13)
0 = (Rr+ Lr(p jr)) ir
+ Lm (p jr) is
(3.14)
Stator axis
Rotor axis
Arbitrary axis
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From the equations. 3.13-3.14, the dynamic equivalent circuit model on a stationary
reference frame can be drawn as in Fig. 3.3.
Fig. 3.3.Dynamic Equivalent Circuit on a Stationary Reference Frame
For steady-state operation with excitation frequency e, p in Eq. 3.13-3.14 may be
replaced by je and after some algebraic manipulation, we get
vs
= (Rs + jeLs ) is
+ Lm pir (3.15)
0 = (Rr/ s + jeLr) ir
+ je Lm is
. (3.16)
which exactly describes the conventional steady-state equivalent circuit of Fig. 3.1.
Now, the previous procedure can be generalized so that the dynamic model is
described on an arbitrary reference frame rotating at a speed a, where Eq. 3.15 -3.16 is a
special case with a,= 0 . To do that, define the new space vector on the arbitrary frame
as
Ya
= exp(- j a)Ys (3.17)
and reconstruct all the model equations in terms of the new space vectors. In the arbitrary
reference frame, Eqs. 3.6-3.8 are modified to
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vsa
= (Rs + Ls p) isa
+ Lm pira
+ j a sa
(3.18)
0 = (Rr+ Lrp) ira
+ Lm p Isa
+ j (a -sl) ra (3.19)
With new flux linkage equations defined by,
sa
= Ls isa
+ Lm ira (3.20)
ra
= Lm isa
+ Lrira (3.21)
By substituting Eqs. 3.20-3.21 into Eqs. 3.14-3.15, we have
vsa
= ((Rs + Ls (p + a)) isa
+ Lm (p + ja )ira
(3.22)
0 = ((Rr+ Lr(p + ja jr) ira
+ Lm (p + ja - jr)isa
(3.23)
where eliminated flux linkage variables are eliminated.
Normalized equivalent circuit on a arbitrarily rotating frame based on Eq. 3.18-
3.23 is shown in Fig. 3.4. Now, depending on a specific choice of a, many forms of
dynamic equivalent circuit can be established. Among them, the synchronous frame form
can be obtained by choosing a = e. This form is very useful in describing the concept
of vector control of induction motors as well as of PM synchronous motors because at
this rotating frame, space vector is not rotating, but fixed and have a constant magnitude
in steady-state. Since space vectors in the synchronous frame will frequently be used,
they are denoted without any superscript indicating the type of frame. Another possible
reference frame used in vector control is the rotor reference frame by choosing a = o
which is , in fact, the reverse step of Eq. 3.8 with n =1.
Dynamic Equivalent Circuit on an Arbitrary Reference Frame Rotating at a.
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3. 3.D-QEquivalent Circuit
In many cases, analysis of induction motors with space vector model is
complicated due to the the fact that we have to deal with variables of complex numbers.
For any space vectorY, define two real quantities Sq and Sdas,
S= Sq+ j Sd (3.24)
In other words, Sq = Re (S) and Sd = Im (S). Fig. 3.5 illustrates the relationship
between d-q axis and complex plane on a rotating frame with respect to stationary a-b-c
frame. Note that d- and q-axes are defined on a rotating reference frame at the speed of a= pa with respect to fixed a-b-c frame.
Definition of d-axis and q-axis on an arbitrary reference frame
With the above Eq. 3.22-3.23 can be written the following 4 equations of real variables
( ) ds qsa a a a a
ds s s s a m dr a m qrv R p i i p i i[ [! (3.25)
( )a a a a a
qs s s qs s a ds m qr a m drv R pL i L i pL i L i[ [! (3.26)
0 ( )qs ds
a a a a
r s dr sl m m sl r qrR pL i L i pL i L i[ [! (2.27)
0 ( )ds qs
a a a a
r s qr sl m m sl r drR pL i L i pL i L i[ [! (2.28)
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The above 4 equations are expressed in a matrix form as follows:
ds
qsqs
aa
ds s s s a a
aa
s a s s a
a sl r s sl r
dr
asl sl r r qr
iv p p
ip pvp p io
p p io
[ [
[ [[ [
[ [
- - -
(3.29)
where sl a r[ [ [! 3.29a
For future reference, the above matrix equation simplified for popular reference
frames in analysis and design of vector control will be introduced. For stationary
reference frame, by substituting a = 0, the above equation is reduced to
0 0
0 0=
0
0
ds s s
ds
s s qsqs
r r s r r dr
r
r r r qr
v p p i
p p iv
p p i
p p i
E E
EE
E
E
[ [
[ [
- - -
(3.30)
Some implementation of vector control drive includes calculation in rotor reference
frame (frame is attached to the rotor rotating at r ). In this case, we can substitute all a
in Eq. (3.29) by r, which makes simplified rotor voltage equations. Moreover, for
synchronous frame, we have
e eds s s s e m e m ds
ees e s s e m m qsqs
e
m sl m r s sl r dr
e
sl m m sl r r qr
v p p i
p p iv
p p io
p p io
[ [
[ [
[ [
[ [
- - -
(3.31)
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As mentioned before, each variable (voltage, current or flux linkage) in the synchronous
frame is stationary and fixed to a constant magnitude in steady-state. Based on Eq. 3.4,
dynamic d-q equivalent circuit is shown in Fig. 3.2.
Fig. 3.4 D-axis equivalent circuit on a arbitrary frame
Fig. 3.5Q-axis equivalent circuit on a arbitrary frame
Expression for the Electromagnetic Torque
The electro magnetic torque Te can be expressed in terms of the stator, rotor or air gap
flux linkages as follows:
ird
vqs
isd
Lr ([a-[r) qrP RrRs [a qsP Ls
Lm
drP
vds
irq
vqr
i Lr ([a-[m) drP RrRs -[a dsP Ls
mL
qrP qsP vqs
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32 2
mPe dr qs qr ds
r
Li i
LX P P ! - (3.32)
32 2
P
e ds qs qs dsi iX P P ! - (3.33)
32 2
P
e md qs mq dsi iX P P ! - (3.34)
3.4. Sensorless vector controller model based on Secondary Speed
Emf on Secondary Speed Emf
3.4.1. Induction Motor Model Based on Secondary Speed Emf
The voltage equation of Induction Motor is rewritten as follows:
stator voltage equation:
+ + ps s s s sv i [P P (3.35)
rotor voltage equation:
0 + + pr r sl r i r[ P P (3.36)
Stator flux equation:
=Ls s s m r i L iP (3.37)
rotor flux equation:
+r m s r r i iP (3.38)
Substituting the equations (3. 37) & (3.38) into voltage equations, equations (3.35) &
(3.36) can be written as follows:
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( ( )) + ( +p)s s s s m rv p i i[ [ (3.39)
0 ( + ( ) + (p+ )r sl r m sl s
p i i[ [ (3.40)
From the equations (3.37) & (3.38) d-axis and d-axis flux linkage equations can be
written as follows:
+ds s ds m dr
i iP (3.41)
=Lqs s qs m qri L iP (3.42)
=Ldr m ds r dr
i L iP (3.43)
+qr m qs r qri iP (3.44)
Separating the d-axis and q-axis voltages, the voltage equations becomes as follows
= R
R
ds dss s s e m e m
qsqs s e s s e m m
drm sl m r s sl r
qrsl m sl r r
v iR pL L pL L
iv L R pL L pL
ipL L pL Lo
iL pLm L pLo
[ [
[ [
[ [[ [
- - -
(3.45)
The secondary fluxes ,d qJ J and the corresponding excitation currents ,d qi iJ J are in (3.47)
and (3.48)
Here
,dr d
q r q
P J
P J
!
! (3.45)
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Therefore, the equations (3.41-3.44) becomes
d ds dr d
m r m
q qs qr q
i i i
L L Li i i
J
J
J
J
! ! - - - - (3.46)
d ds drr
q qs qrm
i i i
i i i
J
J
!
- - - (3.47)
The vectors of the stator voltage sv , stator current si , rotor current ri ,the secondary flux J
and the secondary excitation current iJ, in (3.44),(3.46) and (3.47) are as follows:
Letds
s
qs
vv
v
!
- ,
ds
s
qs
ii
i
!
- ,
dr
r
qr
ii
i
!
- (3.48)
,d dq q
ii
iJ
J
J
JJ
J
! !
- -
from equations (3.45)-(3.48), the vector representation of the voltage equation using the
secondary excitation current iJ
is obtained in the following equation (3.49)
2 2
2 22 2
( )
0( / ) { ( / ) }
m ms e e
sr rs
m mr m r r m r sl
r r
p I J p I J iv
iI p I J
W W
J
[ [
[
! - - -
(3.49)
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where , ,L I JW
areasfollows
2
1 0 0 1,
0 1 1 0
ms
r
LL L
L
I J
W !
! !
- -
(3.50)
Fig3.6 equivalent circuit of the IM
Fig3.6 shows the equivalent circuit of the induction motor based on (3.49). Since the
secondary flux J and the excitation current iJ
are indefinite at the angular frequency
0e[ !
,the sensorless algorithm based on JoriJ can not assure the stable operation in the
low speed region. To solve this problem, the authors propose a new algorithm based on
the
Secondary speed emf er ,is defined as follows
2
mr r
r
Le J i
LJ[! (3.51)
2
2
mr q
dr r
qr mr d
r
Li
e L
e Li
L
J
J
[
[
! -
-
(3.52)
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The Secondary speed emf er, leads excitation current iJ by the angle of / 2T , and
the magnitude is proportional to the rotor speed r[ .therefore, the exact estimation of the
secondary peed emf er is leads to the estimation of the secondary flux position and the
rotor speed. From equations (3.49) and (3.50), the voltage equation using the secondary
peed emf er , is obtained in the following equation (3.53)
2={(R ) } ( / ) ( )s s s r m r s r v p I J i R L L i i eJW [W
(3.53)
For the equation (3.53) the space vector diagram is shown the following fig 3.9
fig 3.7 space vector diagram of the IM
3.4.2. Estimation of Secondary Speed Emf
The Secondary Speed Emf re is estimated by assuming the error between the actual
excitation current iJ and its reference*
iJ is small enough, that is
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*
*
*0
d
q
i Ii i
i
J J
J J
J
! !
- - ;
(3.54)
The fig3.8 shows theSecondary Speed Emf re estimation system. Since the actual
Fig 3.8Speed Emf estimation
position of d-q axis is unknown in the controller, the sensorless algorithm is based on the
estimated position of dc-qc axis. Equation (3.53) for the actual motor is effective even on
the dc-qc axis frame. Since the only difference between the actual motor model is
secondary speed emf rMe on the dc-qc axis frame in the controller is defined as follows.
rMd
rM
rMq
ee
e
!
- (3.55)
The motor model is given in (3.56) by replacing the actual secondary speed emf erM in
(3.53)
2= -(R ) ( / ) ( )s s s e s r m r s r p i v I J i R L L i i eJW [ W (3.56)
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The model voltage sMp iW is given in (3.54) can be calculated from the know
values*, , ( )s sv i i iJ J! and re .On the otherhand, the actual voltage sp iW across the
leakage inductance can be obtained by calculating the current difference between the
detected stator currents si at the two adjusting sampling points.
The estimation error re between the actual emf re in the equation and the model
emf re in eq (3.53) is represented by using the voltage difference sp iW across the
leakage inductance between the actual motor and model as follows
r r rM e e e( !
s sp i p iW W! (3.57)
From the relation between re( and sp iW( in the above equation, the model emf rMe can
be estimated in by the following equation by using the estimation gain KJ
r
se K p i dtJ W! ( (3.58)
0
0
dK
K K
J
JJ
! - (3.59)
From the (3.56) and (3.58), the transfer function from re to re( can be
obtained as follows;
1( )r re sI K seJ
( ! (3.60)
0
0
ddr dr
qr qr
q
s
s Ke e
e es
s K
J
J
( ! (- -
-
(3.61)
The time constants for the convergence of the secondary speed emf errorsdre( and qre(
In( 3.61) are given by1
qKJ
and1
dKJ
, respectively
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3.5. Sensorless Control System Configuration
isd
- d-controller
ids*
+
iqs* +
_ q-controller
iqs
iaisq -
isd ib - ic
wr*+
+ -
-wre +
pLisMd -
pLisMd pLisMq+
wsl*+
pLisMq wre +
+M
erMd+
M
erMq
wre
Fig.3.9 Schematic Block Diagram of Sensorless Vector Control System
Decoupling
network
vd*e
iM
vq*
2-Ph3-Ph
Sinusoidal
PWM
Voltagesource
Inverter
AC toDC
3-PhAC
IM
3-Ph2-Ph
sL
sL
M
odel
Speed
emf
Estimato
r
Flux position
com
Speed
E timat
speed controller
LrLm
2id
*
Rr
Lr id*
1/s
vq*
*
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3.5.1 Estimation of the rotor speed re[ :
From the relation in (3.52) speed qre , the estimated rotor speed re[ is obtained in the
following eq (3.62) using the estimated q component r
qe in the r
e and the exciting
current reference *diJ .
2 *
rre rMq
m d
Le
L iJ
[ !(3.62)
From the rotor speed error between the rotor speed reference*
r[ and the estimated rotor
speed re[ ,the torque ref*
X is determined through the PI controller. From the relation in
eq (5.75) between the motor torqueX and the stator current qsi under the condition that
the q-axis component of the excitation current diJ equals to zero, the reference of the q-
axis current *qsi is the determined in (3.64)
T qsK iX ! (3.63)
* *1qs
T
i
K
X! ,2
*mT d
r
LK i
LJ
! (3.64)
3.5.2Estimation of slip angular speed:
The voltage equation of the IM is
2 2
2 22 2
(R )
0( / ) { ( / ) }
m ms e e
sr rs
m mr m r r m r sl
r r
L Lp I J p I J
iL Lv
iL LR L L I R L L p I J
L L
J
W W[ [
[
! - - -
(3.65)
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From the second row of the equation (3.65), one can be written as follows:
2 22 2
1 0 1 0 0 1( / ) { ( / ) } 0
0 1 0 1 1 0 0
ds m mr m r r m r sl
qs r r
i IL LR L L R L L p
i L L
J[
! - - - - -
(3.66)
Simplifying the above equation, yields:
2 22 2
0( / ) { ( / ) } 0
0
ds m mr m r r m r sl
qs r r
i IL LR L L R L L p
i IL L
J
J
[
! - - -
(3.67)
The second row of the equation (3.67), yields
2 22 * 2 *
22 * *
( / ) { ( / ) }0 0
( / ) 0
m mr m r qs r m r sl d
r r
mr m r qs s l d
r
L L R L L i R L L p i
L L
L R L L i i
L
J
J
[
[
!
!(3.68)
Replacing the qsi , diJ with*
di
J, *qsi in the q-axis component in the second row of the
equation (3.68) becomes
2 22 * 2 *( / ) { ( / ) }0 0m mr m r qs r m r sl d
r r
L LR L L i R L L p i
L LJ[ ! (3.69)
ie.
22 * *( / ) 0m
r m r qs sl d
r
L R L L i i
LJ[ ! (3.70)
From the (3.70) slip can be calculated in the equation (3.71)
* *
*
rsl qs
r d
R iL iJ
[ ! (3.71)
By adding the estimated speed re[ to the slip angular speed reference*
sl[ , the angular
speed e[ is determined as follows:
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*
e sl r e[ [ [! (3.72)
On the otherhand, the d-axis component of the estimated speed emf r de represents the
position estimation error U( between the actual position U and estimated positionM
U as
shown in fig 3.10. By using equations(3.52) and (3.54), the estimation error U( can be
obtained in (3.73) under approximation of tan U( ; = U(
2*m
rMd rMq d
r
Le e tan i
LJU [ U! ( (; (3.73)
Fig 3.10 Estimated axis (dc-qc) and speed emf
From the relation in the equation in (3.73), the position compensation term MU( can be
calculated in (3.74) by using the compensation gain KU
2 *
rM rMd
m d
L K e dt
L iU
J
U( ! (3.74)
The estimated axis positionM
U is given in equation (3.75)
M e MdtU [ U! ( (3.75)
According to (3.73) & (3.74), the compensation system of the axis position error is
shown in fig 3.11.the transfer function of the position estimation error U( is obtainedfrom fig 3.11 as follows;
( / )es
ss K
U
U U [( !
(3.76)
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Fig 3.11compensation system of axis position error
The time constant for the convergence of the position estimation error U( is1
KU
.
Under the constant secondary excitation current iJ (= IJ ), the d-axis stator current
reference *sdi is the constant value IJ in (3.54) and q-axis stator current reference*
qsi is
given in (3.64).Using the current control errors between the *qsi ,*
dsi and the detected
currents ,qs dsi i , the compensation voltages*
dv( and*
qv( for stator current are calculated
through PI controllers as shown in fig 3.11. These compensation voltages *dv( and*
qv(
are the compensation terms of voltage drop ( )s sp iW across stator resistance and the
leakage inductance.
3.5.3Calculation of decoupling terms:
Replacing * *,d qv v( ( , dIJ , qsI with ( )s sp iW ,*
dIJ ,*
qsI in the first row of (3.65), the
voltage references are*
dsv and*
qsv obtained as follows;
ie.
2 2
{( ) } { }m ms s e s er r
L Lv p I J i p I J iL L
JW W[ [! (3.77)
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2 2
* * * *
** * * 2 2
0( )
( ) 0
m me
r rs eds ds qs d
ds eqs qs ds m mse
r r
L Lp
L Lpv I I I
Ipv I I L Lp
L L
J
J
[W W[
W W[[
! - - - - - - - - -
(3.78)
2* * * *( ) mds s ds e qs d
r
Lv p I I p I
LJW W[! (3.79)
2* * * *
( ) mqs s qs e ds e dr
Lv p I I I
LJW W[ [! (3.80)
2* * * *mds ds e qs d
r
Lv v i p i
LJ[ W! ( (3.81)
* *
ds ds dov v v! ( (3.82)
2* *
0m
d e qs d
r
Lv i p i
LJ[ W! (3.83)
2* * * *mqs ds e ds d
r
Lv v i i
LJ[ W! ( (3.84)
* *
qs qs qov v v! ( (3.85)
2* *m
qo e ds d
r
Lv i i
LJ[ W! (3.86)
where 0dv , qov are the decoupling terms
3.6. Sensorless Control at Zero Frequency
Fig3.12. Vector Diagram at Zero Angular Frequency
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Fig 3.63 shows the vector diagram based on (3.54) at zero angular frequency of e[ , at
zero angular frequency of e[ , the secondary speed emf re in (3.51) can be modified by
using slip
equation and the second row of (3.49)
22( ) ( )m mr r r s
r r
L Le J i R i i
L LJ J[! ! (3.87)
From equation (3.87), the secondary speed emf re and the term2( ) ( )m
r s
r
LR i i
LJ
are
canceled out each other. In this case, the voltage equation (3.52) results in only the
voltage drop across the stator resistance as follows;
r s se R i! (3.88)
Since the term of the secondary speed emf re is not included in equation (3.88),
the estimation of secondary speed emf re is impossible. For the estimation of the
secondary speed emf re at zero angular frequency, the sinusoidal component with the
amplitude IJ
( and the angular frequency d[ is super imposed to secondary excitation
current reference *diJ as follows ;
*
*
*sin
0
d d
q
i I I tii
J J J
J
J
[ ( ! ! - -
(3.89)
Fig.3.13 Vector Diagram under Fluctuating Excitation
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in (3.89).since re and2( ) ( )m r s
r
LR i i
LJ
terms are not canceled out, the stable
estimation of re is possible. Since the motor control is realized at the stator side, the
stator current reference to obtain the fluctuating excitation current iJ
in (3.89) is needed.
By substituting (3.89) into the second row in (3.49),the d-axis stator current reference *ds
i
is obtained as follows;
* *(1 )rsd d
r
Li i
RJ
!
21 ( ) sin( )d rd d
r
Li t
R
J J
[[ U! ( (3.90)
Where,1tan ( )d r
d
r
L
R
[U !
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CHAPTER 4
BLOCK SCHEMATIC OF SENSORLESS VECTOR
CONTROL
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4.1. Introduction
The purpose this section is to discuss the basic steps involved in the development
of simulation blocks for the Sensorless Vector Control of the induction motor. All the
simulation blocks are developed in MATLAB6.1/SIMULINK.This Schematic of
Sensorless Vector Control of IM Drive System consist of the following basic parts:
a. Induction Motor Drivesb. Three phase to two-phase transformation (a, b, c to , )c. Stator to synchronous reference frame transformation (E,F d,q)d. Sensorless vector control algorithme. Decoupling Networkf. d-q to a, b, c transformation(Two Phase to three phase transformation)g. Sine-Triangle PWM of Three Phase Inverters
In the practical implementation of the Sensorless Vector Controlofthe induction
motor is fed from a voltage source inverter with fast current control loops. This approach
is used in high performance induction motor servo drives for Machine tool, Rotary press,
Storrer, Pressor and Winder applications. Sensorless vector controlled induction motor
has been used widely used from the standpoints of cost, size and reliability.
In field oriented control system, the induction motor behaves like a dc machine
under both steady state and transient conditions. Consequently, similar drive control
strategies can be employed. Below base speed, the magnetizing current of the induction
motor representing the rotor flux magnitude is maintained constant at its maximum
possible value to achieve constant torque operation. Above base speed, the flux is
reduced thereby giving the field-weakening region or the constant horse power region of
operation. In Sensorless vector controlled induction motor speed is estimated from the q
axis component of the secondary speed emf in the synchronous reference frame
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A typical Sensorless vector controlled drive system consists of an induction motor,
which is driven by a voltage source inverter, speed controller ,current controller , speed
e.m.f estimator, flux position estimator, speed estimator and Park & Clark
transformations
4.2. Induction Motor Drives
Power electronic devices known as motor drives are used to operate AC motors at
frequencies other than that of the supply. These consist of two main sections, a controller
to set the operating frequency and a three-phase inverter to generate the required
sinusoidal three-phase system from a DC bus voltage. The model of the Induction Motor
is developed as per the equations which is shown in fig 4.1
4.3 Three phase to two-phase transformation (a, b, c to , )
Mode
of the Induction Motor
6
vq s
5
vd s4
we r
3
T e2
iq s
1
id s
0 w
1
sth
1
J.s+B
speed
v a
v b
v c
th
v qs
v ds
abc--dqs
2 /pW e r
ids
iqs
lam qr
lam dr
Te
Torque
v qs
iqs
w
v ds
ids
lamda ds
lamda qs
Stator Fluxes
iqr
idr
w
wr
lam qr
lam dr
Rotor fluxes
lam ds
lam qs
lam dr
lam qr
ids
iqs
idr
iqr
Current s
4
T l
3vcn
2
vb n
1
va n
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The three stator currents isa, isb andiscthat are measured, are first transformed to
an equivalent two-phase system (isand is) because the induction motor is represented as
equivalent two-phase machine. The three- phase to two-phase transformation (3-2) is
carried out in the stator reference frame.
This transformation is a general transformation that can be applied to any variable of
the induction motor like the stator voltages, stator currents, flux linkages etc.
1 1
3 3
1 0 0
0
a
b
c
ii
ii
i
E
F
!
(4.1)
The block for the three phase to two-phase transformation (a, b, c to ,) is developed
as per the equation(4.1) which is shown in fig 4.2.
>
2
Ibet
1
ia l
1 /sqrt(3)
1 /sqrt(3)/1
1/ sqrt(3)
1 /sqrt(3)
3
ic n
2
ia n
1
ib n
Fig 4.2 Block diagram for the Three Phase To Two-Phase Transformation (a, b, c to E,F)
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4.4. Stator to synchronous reference frame transformation (E,F d,q)
The two-phase stator currents that are in the stator reference frame are
transformed to a synchronous reference frame. The choice of the synchronous
reference frame is dependent on the flux along which the orientation is to be
performed. If the arbitratory is oriented along the rotor flux linkage space phasor,
then the synchronous reference frame would be the rotor flux reference frame and if
the arbitrary axis is to be oriented along the stator flux linkage space phasor, then the
synchronous reference would be the stator flux reference frame etc. If the angle V,
represents the instantaneous position of the synchronous reference frame along which
the arbitrary axis is aligned, then the transformation from the stator to synchronous
reference frame. The inputs to this block are is , is and the rotor flux positionV. The
outputs of this block are isdand isq.
cos sin
sin cos
ds
qs
i i
i i
E
F
V V
V V
! (4.2)
The block diagram for the (E,F d,q) transformation is developed as per the
equation(4.2) which is shown in fig 4.3
fig 4.3. PARK TRASFORMATION(2Ph-->2Ph)
2
iq s
1
id s
sin
sin
cos
cos
Ibet*sin(th)
Ibet*cos(th)
Ial*sin(th)
Ial*cos(th)
3
theta
2
ib t
1
ia l
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4.5. Sensorless vector control algorithm
y Estimation of Secondary Speed Emf:y The magnitude of theSecondary Speed Emf can be estimated for Sensorless
Vector control is in equation in (4.3)
rM se K p i dt
JW! ( (4.3)
The block diagram is developedfor the Secondary Speed Emf Estimation as per the
equation(4.3) which is shown in the following fig.
!
2
e rM q
1
e rM d
K* u
i * p d e l i d q s
uK p h i
K p h i
1
s
I n t e g ra t o r 1
m
2
P s i g d l i q s
1
P s i g d l i d s
Fig4.4. Secondary Speed Emf Estimation Model
y Estimation of Rotor Speed re[ Rotor speed can be calculated using the following eq (4.4) which simulation block is
shown in the following fig4.5
2 *
rre rMq
m d
Le
L iJ[ ! (4.4)
SPEED ESTIMATOR
1
Wre
Lr/(M ^2)
ErM q*Lr/M ^2
u(1)/u(2)
(Lr/(M 2*Ip hidref))*erM q
2
erM q
1
Iphidref
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Fig4.5. Estimation of Rotor Speed re[ Model
y Estimation of slip *sl[ :Slip angular speed *sl[ can be calculated using the following eq (4.5) which simulation
block is shown in the following fig4.6* *
*
rs l qs
r d
Ri
L iJ
[ ! (4.5)
"
# $ %
& '
#
&
( #
'
) $
0
1
1
Ws
u(1)/u(2)
Rr/LR*idsref/ Iphdref
Rr/(Lr)
Ga i n2
Iphiref
1
iqsref
Fig4.6.Block diagram forEstimation of slip *sl[
Synchronous speed can be calculated using the following equation (4.6)
*e sl r e[ [ [! (4.6)
y Estimation of flux position compensation:
The flux position compensation term is estimated using the following equation
2 *
rM rMd
m d
LK e dt
L iU
J
U( ! , which simulation block is in the following fig
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Position Estimator
1
delt at hM
-K -
Lr/(M ^2*iphdref)1
1
s
Int egrat or
u( 1) /u( 2)
Fc n
2
Iphidref
1
erM d
Fig4.6.1. Model for Estimation of flux position compensation
The estimated axis position MU is given in eq (4.7)
M e MdtU [ U! ( (4.7)
where2 *
rM rMd
m d
LK e dt
L iU
J
U( !
4.6. Speed and Current Controllers
The reference speed refis compared with the estimated speed re which is
estimated from equation (3.62). The speed error is passed through a zero steady state
error controller like a PI controller to obtain the command value for the quadrature
component of the stator current *qsi (i.e. Torque reference*
X ), in the synchronous
reference frame.
The reference for the direct current *dsi , of the stator current space phasor in the
case of the vector control can be a constant value up to base speed operation of the motor.
For the operation of the motor above the base speed, the *dsi is decreases in such a manner
to maintain the power constant i.e. by weakening the field. The command values *dsi and
*
qsi are compared with the feedback values of the stator currents ids and iqs in the
synchronous reference frame. The current errors thus obtained are passed through PI
controllers which form the current controllers of the drive system.
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In feed back control systems a controller may be introduced to modify the error
signal and to achieve better control action. The introduction of controllers will modify the
transient response and steady state error of the system.
The simulation blocks for the speed and current controllers are shown in following figs
4.7,4.8&4.9
speed-controller
1
out_1
1
s
sat=70
T re flS um
Kps
Kps
Kis
1/ T i1
spe e d error
Fig4.7. Model for speed controller
d-controller
1
De lVqsrefvsqre fl
1
s
sat=12 0
S um
Kp
P
Ki
1 /T i1
dc- e rror
Fig4.8. Model for d- controller
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q_controller
1
de l Vqsrefvsdre fl
1
s
sat
Ki
ki
S um
Kp
KP
1
qerrr
Fig4.9.Model for q- controller
y Calculation of the d-axis stator current reference *dsi The d-axis stator current reference *dsi is calculated as for the equation (3.90)
which simulation block is shown in fig 4.10
CALCULATION OF Iqsref
1
iqsref
Lr / (M ^ 2 )
Lr / (M ^ 2 * i phdr e f)
u ( 1 ) / u ( 2 )
(Lr * T re f)/ (Lm ^ 2 * I p hi dr e f)2
I p h i d r e f
1
T ref
Fig4.10.Model for *dsi
y Calculation of the *d
iJ
:
The reference magnetizing current *diJ is calculated as for the equation (3.89)
which simulation block is shown in fig 4.11
CALCULATION O F Iphidref at ZERO FREQUENCY
1
Iphidref
0
delIphi
Product2Product1
5
Iphi
sin(u )
Fcn
wd
Constant
Clock
Fig4.10.Mdel for *diJ
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4.7. Decoupling Network:
It can be noted that under proper vector control, the stator current components ids
and iqs decoupled, and hence the outputs of the current controllers can be used as
command values for the current source inverter. However, in the case of the voltage
source inverter, the stator voltage command values Vds and Vqs are not decoupled.
Hence, decoupling networks are necessary to generate Vdsref and Vqsref in the
synchronous reference frame, if a voltage source inverter is used. In the present work,
voltage source inverter is used to drive the induction motor. Therefore, suitable
decoupling terms will have to be incorporated to the outputs of the current controllers.
As discussed in the earlier, the d-axis stator circuit loop has a coupling term
(2
* *me qs d
r
Li p i
L
J[ W )from the quadrature axis and the q-axis stator circuit loop has a
coupling term (2
* *m
e ds d
r
Li i
LJ
[ W )from the direct axis. If the coupling terms are not
compensated, then the torque and the flux components of the stator current will not be
decoupled. Therefore, feed forward terms, Vdo for d-axis voltage compensation and vqo
for q-axis voltage compensation, must be added to the output of the current controllers.
Vdo and Vqo are given by :
2
* *
0m
d e qs d
r
Lv i p i
LJ
[ W! and2
* *mqo e ds d
r
Lv i i
LJ
[ W! respectively
(4.8)
The feed forward terms, Vdo for d-axis voltage compensation and vqo for q-axis
voltage compensation are estimated based on equaion (4.8) in the sensorless vector
control model simulation block diagram which block diagram is shown fig4.11
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Fig4.11.Modelforse
nsorlessvectorcontrolmodel
vsref
vsref
vsref
psigmadlis
SENSORLESS VECTOR CONTRO
2
pLi
pLi
0
zero
K* u
vs
K* u
is
K* u
e rM
sigma
We
Rs
Rs1K* u
Rs
Product2
Product1
u
u
u
u
u
K* u
M atrix
Ga in3
m
(M /Lr) 2*R r
Constant1
0
Cons
0
0
8
iphidref
7
erM q
6
erM d
5we
4
iq s
3
id s
2
vdsref
1
vqsref
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4.8. d-q to a, b, c transformation(Two Phase to three phase
transformation)
The vdsref and vqsref thus obtained in the synchronous reference frame are first
converted into two phase stator reference frame and then to three phase stator
reference using the following transformations .Using the general variable x ,the
transformations are given by
d,q to ,E F transformation:
cos sin
sin cos
d
q
x x
x x
E
F
V V
V V
!
- - - (4.10)
1 0
1 3
2 2
1 3
2 2
a
b
c
xx
xx
x
E
F
! - -
(4.11)
The three reference voltages thus obtained after the transformations are used as
reference in pulse width modulator to obtain the switching pattern for the inverter
switches.
The block diagram for d,q to ,E F transformation and ,E F a,b,c are developed as
per the equation s(4.10) &(4.11) whose simulation block diagram is shown in the fig4.11
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Invpark_TF(2-Ph-->2-Ph)
2
valsref
1
vbtaref
sin
sin
cos
cos Vqsref*costheta
Vdsref*sintheta
Product4
Product1
3
theta
2
vdsref
1
vqsref
InvclarkTF(2Ph-->3Ph)
3
vcref
2
vbref
1
varef
. 8 66
sqrt(3)/2
-.5
-.5
2
valsref
1
vbtsref
Fig4.11.Model for d,q ,E F transformation and ,E F a,b,c
4.9. Sine-Triangle PWM of Three Phase Inverters
Although the basic MOSFET circuitry for an inverter may seem simple,
accurately switching these devices provides a number of challenges for the power
electronics engineer. The most common switching technique is called Pulse Width
Modulation (PWM) which involves applying voltages to the gates of the six MOSFETS
at different times for varying durations to produce the desired output waveform. In Figure
4.12, Q1 to Q6 represents the six MOSFETS and a,a,b,b,c,c represent the respective
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control signals. In practice each switching leg may consist of more than two MOSFETs
in order to reduce switching losses by paralleling the on resistance.
Figure 4.12 - Basic Three-Phase Voltage Source Inverter
In the following equations logic values that are equal to 1 when the MOSFET is
on and 0 represent the control signals when it is off. In AC induction motor control when
the upper MOSFET is switched on i.e. a,b,c is 1 the corresponding lower MOSFET is
switched off i.e. a,b,c = 0. Using complementary signals to drive the upper and lowerMOSFETS prevents vertical conduction providing that the control signals dont overlap.
From the states of a,b,c the phase voltages connected to the motor winding can be
calculated using the following matrix representation:
Knowing the phase voltage for a given switching state is important for the
technique known as sine triangle Pulse Width Modulation which will be discussed in
detail in section 4.9.
(4.12)
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Sinusoidal Pulse Width Modulator
One commonly used PWM scheme is called carrier based modulation. This uses a
carrier frequency usually between 10 to 20 kHz to produce positive and negative pulses
of varying frequency and varying width. The pulse width and spacing is arranged so that
their weighted average produces a sine wave. The sine-triangle PWM model is shown in
fig4.12
Sintriangle Pulse Width Modulated nverter
3
V cn
2
V bn
1
V a n
Tga
Tgb
Tgc
Va n
Vb n
Vc n
Three P hase V oltage Source Inverter
Varef
Vbref
Vcref
Tga
Tgb
Tgc
Sintriangle Pulse Width M odulator3 Vcref
2 Vbref
1 Varef
Sintriangle Pulse Width Modulator
3
T gc
2
T gb
1
T ga
Rel ay 3
Rel ay 2
Re l ay 1
6 0 * 2 1
Fsw
Ac
F s wCwav e
CARRI ER WAVE
3 0 0
Ac
3 Vcref
2 Vbref
1 Var ef
Fig4.13. model for sine-triangle PWM
Fig4.14.Shows sine-triangle PWM Inverter model
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Three Phase Voltage S ource Inverter
3
V cn
2
V b n
1
V a n
1 /3
V zs
1 7 5
Vdc /2
V co
V b o
V a o
3
T g c
2
T g b
1
T g a
Fig4.14.1. Model for sine-triangle PWM Inverter
In sine-triangle PWM a triangular carrier waveform of frequency fs establishes the
inverter switching frequency. This is compared with three sinusoidal control voltages that
comprise the three phase system. The output of the comparators produces the switching
scheme used to turn particular inverter MOSFETS on or off. These three control voltages
have the same frequency as the desired output sine wave which, is commonly referred to
as the modulating frequency, fm. The modulation ratio is equal to mf= fm/fs. The value of
mf should be an odd integer and preferably a multiple of three in order to cancel out the
most dominant harmonics as these are responsible for converter losses. One limitation of
the sine triangle method is that it only allows for a limited modulation index, so it doesnt
fully use the DC bus. The modulation index can be increased by using distorted
waveforms that contain only triplen (multiples of three) harmonics. These form zero
sequence systems where the harmonics cancel out resulting in no iron losses .It is
discussed in detail in the following section.
To obtain balanced 3-phase output voltages from the 3-phase PWM inverter, the
same triangular voltage waveform is compared with three sinusoidal control voltages that
are 1200
out of phase, as shown in the fig 4.15. The comparison of V control with
triangular wave form results in the following logic signals to control the switches in legs
A,B,C.
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If Va ref > Vtri , Vao =2
d cV
else Vao = -2
d cV
If Vbref > Vtri , Vbo = 2d cV
else Vbo = -2
d cV
If Vcref > Vtri , Vco =2
d cV
Else Vao = -2
d cV
0 0.002 0.004 0 .006 0 .008 0 .01 0 .012 0 .014 0 .016 0.018 -300
-200
-100
0
100
200
300
Time t in sec
3-Ph
RefVoltages
Fig 4.15. Reference voltages and carrier wave forms
The common mode voltage is given in (4.19)
i.e. 1
3no ao bo coV V V V ! (4.19)
The output phase voltages can be calculated by subtracting the common mode voltage
from the pole voltages.
a n a o n oV V V! (4.20)
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b n b o n oV V V! (4.21)
c n c o n oV V V! (4.22)
The pole voltages are shown in fig 4.16 ,phase voltage Van is shown in fig 4.17 and line-
line voltage Vab is shown in fig 4.18
0 0 .0 02 0 .0 04 0 .0 06 0 .0 08 0 .0 1 0 .0 12 0 .0 14 0 .0 16 0 .0 18 -200
0
20 0
Vao
0 0 .0 02 0 .0 04 0 .0 06 0 .0 08 0 .0 1 0 .0 12 0 .0 14 0 .0 16 0 .0 18 -200
0
20 0
Vbo
0 0 .0 02 0 .0 04 0 .0 06 0 .0 08 0 .0 1 0 .0 12 0 .0 14 0 .0 16 0 .0 18 -200
0
20 0
time t in sec
Vco
-Vdc/2
+Vdc/2
Fig4.16. Pole voltages Vao,Vbo and Vco Waveforms
0 0 . 0 0 2 0 . 0 0 4 0 . 0 0 6 0 . 0 0 8 0 . 0 1 0 . 0 1 2 0 . 0 1 4 0 . 0 1 6 0 . 0 1 8 -2 50
-2 0 0
-1 50
-1 0 0
-5 0
0
50
1 0 0
1 50
2 0 0
2 50
tim e t in s e c
phase
voltage
V
an
V a n
Fig 4.17 Phase
voltage Van Waveform
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0 0 .0 02 0 .00 4 0 .0 06 0 .0 08 0 .0 1 0 .01 2 0 .0 14 0 .0 16 0 .0 18 -400
-300
-200
-100
0
100
200
300
400
time in sec
L-L
Volt
ge
Vab
V ab
V dc
-Vdc
Fig 4.18 line-line voltage Vab Waveform
These 3-phase voltages will now be fed to the induction motor. In the 3- phase
inverters, only the harmonics in the line-to-line voltages are concerned. The harmonics in
the output (Van) of any one of the legs are identical to the harmonics in Vao, where only
the odd harmonics exist as side bands , centered around m f and its multiples, provided mf
is odd.. only considering the harmonics at mf ( the same applies to its odd multiples), the
phase difference between the mf harmonic in Van and Vbn is (120mf)0
. This phase
difference will be equivalent to zero (a multiple of 3600 ) if mf is odd and a multiple of 3.
As a consequence, the harmonic at mf is suppressed, in the line-to-line voltage Vab . The
same argument applies in the suppression of harmonics at the odd multiples of mf , if mf
is chosen to be an odd multiple of 3 ( where the reason for choosing mf to be odd
multiple of 3 is to keep mfodd and hence, eliminate even harmonics ). Thus some of the
dominating harmonics in the one-leg inverter can be eliminated from the line-to line
voltage of a 3 phase inverters.
In the linear modulation (ma e 1.0) the fundamental frequency component in
the output voltage varies linearly with the amplitude modulation ratio ma. The peakvalue of fundamental frequency component in one of the inverter legs is
(van)1 = ma v vd/2
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For low values of mf (mf e 21) to eliminate the even harmonics, a
synchronized PWM (mf be an integer) should be used and mf should be an odd integer.
Moreover, mfshould be a multiple of 3 to cancel out the most dominant harmonics in the
line to line voltage. The reason for using the synchronous PWM inverter is that the
asynchronous PWM (where mf is not an integer) results in sub harmonics (of
fundamental frequency) that are very undesirable in most applications.
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CHAPTER 5
SIMULATED RESULTS AND CONCLUSIONS
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5.1. Description of proposed Scheme
Fig 5.1 shows the simulated schematic simulation block diagram for Sensorless
Vector Control of induction motor drive system, whose specifications and the parameters
of sensorless control scheme are shown in Appendix A.
The system consists of an induction motor which is driven by a voltage source
inverter. The dc-link to the inverter is obtained from the output of ac-dc converter which
is fed from the three phase mains. The inverter switching is controlled by the speed and
current controllers as shown in fig 5.1
The ac-dc converter consists of a three phase bridge rectifier followed by a
capacitor which output is fed to the three phase voltage source inverter. The control
signals for the inverter switches are obtained form the sine triangle modulator block. The
six power switches output are three phase pulse modulated voltages which are fed to the
induction motor. The three phase stator currents ias,ibs and ics are measured(sensed by
using hall effect sensors),are transformed to,
i iE F
in the stationary reference frame ,,ds e qs e
i i
are calculated from,
i iE F
in the synchronous reference(stator flux reference frame) by using
estimated positionm
U .
The voltage across the leakage inductances
pL iW can be obtained by calculating
the current difference between the detected stator currentssi at the two adjacent sampling
points. the model voltage across the leakage inductancesm
pL iW
can be obtained(eq
(3.54)) from the know values *, , ( )s sv i i iJ J! and rMe .The model secondary speed
emf rMe can be estimated(eq (3.58)) by using speed emf estimation gain KJ .
From the d-axis component of the secondary speed emf, flux position compensation term
MU( in equation (3.74), and q-axis component of the secondary speed emf, rotor speed
re[ in equation (3.62), can be estimated.
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Fig5.1Simulatedschematicsim
ulationblockdiagramforSenso
rlessVector
Controlofinductionmotordrivesystem
The
referencespeed
ref
[
iscomparedwiththeestimatedrotorspe
ed
re
[
andthe
speederrorthuspassedthroughaspeedco
ntroller,whichisaPIcontrollerandservesthe
threepurposes-stabilizesthedriveandadjuststhedampingratioatdesire
dvalue,makes
Schematic block diagram of the Sensorless Vector Ccontrol scheme
speed-con troller
softsta rt
q_ c ontroller
d-con troller
Wre f
1
s
W e
delv dsref
Idsref
delv qsref
Iqsref
we
Iphidref
Vqsref
Vdsref
V C D e c oup l
Psigdlids
Psigdliqs
erMd
erMq
Spee d em f Estim a tor
Iphidref
erMqW re
S p e e d E stim a tor
iqsref
Iphidef
W s
S L I P
erMd
Iphidrefde lta thM
Positi on E stim a tor
pL
pL
v qsref
v dsref
idse
iqse
we
erMd
erMq
iphidref
pLiMd
pLiMq
M O DEL
. 1 8 2
.00351s+1
LP Fq
. 1 8 2
.00351s+1
LP Fd
Tref
Iphidrefiqsref
Iqsref
IphidrefIphire f
v qsref
v dsref
the ta
v bta ref
va lsref
Invp a rk_T F
v b
va
I
Idsref
Idsref
2 /p
2 /p
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the steady state error close to zero by integral action, and filters out noise gain .The
output of the PI controller is applied to the limiter which sets a torque producing
componentqsref
i .From the torque producing componentqsref
i , slip speedsl
[ can be
estimated using equation (3.71),which is added to the estimated speedre
[ to get the
synchronous speede
[ ,which sets the inverter frequency. The inverter frequency is
adjusted to make the actual speed equal to the reference speed. The reference for the
direct componentdsref
i of stator current space phasor is estimated by using the equation
(3.90).
The command valuesdsref
i andqsref
i are compared with the feedback values of the
stator currents dsei and qsei , which are in the synchronous frame. The current errors thus
obtained passed through a current controllers, which are the PI controllers, which serves
the same three purposes just described. The decoupling terms0, 0d q
v v are calculated from
the equations (3.83)&(3.86) and added to the output of the current
controllers *dsv( and*
qsv( to get stator voltage command values*
dsv ,*
qsv .
The stator voltage command values *dsv ,*
qsv are first converted to two phase stator
reference frame then three phase synchronous reference frame .These three reference
voltages are used as reference signals to a sine triangle pulse width modulator to obtain
the switching pattern for the inverter switches. Finally, the output of the sine triangle
pulse width modulated voltage source inverter is fed to the three phase induction motor.
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5.2. Simulation Results
The fig 5.2 shows the speed response at reference speed Wref = 188.57 rad/sec which
shows that the estimated speedre
[ is coincident with the speedr
[ actual.
0 0 .5 1 1 .5 0
10 0
20 0
Wref
0 0 .5 1 1 .5 -100
0
10 0
20 0
Wre
0 0 .5 1 1 .5 -200
0
20 0
40 0
time(sec)
Wr
Fig.5.2.Speed response vs time
The voltage response Vds, Vqs are shown in fig 5.3 and locus of the voltages Vds and Vqsare shown in fig 5.4.The reference voltages to the PWM modulator are shown in fig 5.5.
Stator voltages van, vbn & vcn are shown in fig 5.6.
1 . 4 1 . 4 1 1 . 4 2 1 . 4 3 1 . 4 4 1 . 4 5 1 . 4 6 1 . 4 7 1 . 4 8 1 . 4 9 1 . 5 - 4 0 0
- 2 0 0
0
2 0 0
4 0 0
t im e ( s e c )
Vd
s
1 . 4 1 . 4 1 1 . 4 2 1 . 4 3 1 . 4 4 1 . 4 5 1 . 4 6 1 . 4 7 1 . 4 8 1 . 4 9 1 . 5 - 4 0 0
- 2 0 0
0
2 0 0
4 0 0
Vqs
V qs ,V ds
fig 5.3 Voltage waveform at steady state
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-400 -300 -200 -100 0 100 200 300 400 -400
-300
-200
-100
0
10 0
20 0
30 0
40 0
Vd s
Vqs
Vds Vs Vqs
fig 5.4 Locus of the voltages Vds and Vqs at steady state
2 2.05 2.1 2.15 2.2 2.25 2.3 2.35 2.4 2.45 2.5 -400
-200
0
20 0
40 0
tim 2 (sec)
Vbref
2 2.05 2.1 2.15 2.2 2.25 2.3 2.35 2.4 2.45 2.5 -400
-200
0
20 0
40 0
Varef
volatage response
Fig 5.5 Reference voltages waveforms to the PWM Modulator at steady state
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Stator voltages of the motor are as shown in fig 5.6.
1. 4 1. 41 1. 42 1. 43 1. 44 1. 45 1. 46 1. 47 1. 48 1. 49 1. 5
-200
0
200
Vbn
1. 4 1. 41 1. 42 1. 43 1. 44 1. 45 1. 46 1. 47 1. 48 1. 49 1. 5
-200
0
200
t3
4
5
(6 5 7
)
Vcn
1. 4 1. 41 1. 42 1. 43 1. 44 1. 45 1. 46 1. 47 1. 48 1. 49 1. 5
-200
0
200
Van
Fig 5.6 Stator voltages of the induction motor at steady state
Simulated results at zero frequency are shown in the following figs
0 0. 5 1 1 . 5 2 2. 5 3-60
-40
-20
0
W
8
0 0. 5 1 1 . 5 2 2. 5 30
20
40
60
t@
A B
(C B
c)
W
D
E
0 0. 5 1 1 . 5 2 2. 5 3-50
0
50
100
WF
,WF G
,WH
I
W
D
0 0 . 5 1 1 . 5 2 2 . 5 3-6 0
-4 0
-2 0
0
W
P
Q
0 0 . 5 1 1 . 5 2 2 . 5 30
2 0
4 0
6 0
tR
S
T (U
T c)
W
V
W
0 0 . 5 1 1 . 5 2 2 . 5 3-5 0
0
5 0
10 0
WX
,WX Y
,W` a
W
V
Fig 5.7.Actual motor speedr
[ , estimated rotor speedre
[ and slip speed
sl[ characteristics at zero Frequency
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0 0.5 1 1.5 2 2.5 3-5 0
-4 0
-3 0
-2 0
-1 0
0
Wsl
W sl,W e a t zero frequenb
y
0 0.5 1 1.5 2 2.5 3-1 0
-5
0
5
10
15
We
time(sec)
Fig5.8.The stator angular frequencye
[ andsl
[ characteristics at zero Frequency
0 0.5 1 1.5 2 2.5 3-10
-5
0
5
10q-axis current response at zero frequency
Iqse
0 0.5 1 1.5 2 2.5 3-15
-10
-5
0
time(sec)
Iqsre
f
0 0.5 1 1.5 2 2.5 3-10
-5
0
5
10
Idse
Idse,idseref at Zero frequency
0 0.5 1 1.5 2 2.5 34.5
5
5.5
6
6.5
7
time(sec)
Idseref
Fig.5.9 dq axis currents characteristics at zero Frequency
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0 0 5 1 1 5 2 2 5d
f
0
40
20
0
20
erM
g
h i p
eq
se cr
0 0 5 1 1 5 2 2 5d
f
4
2
0
erM
s
e rMt
u e rMv
Fig 5.10.Estimated speed emf characteristics at zero frequency
The fig 5.7 shows that the estimated speedre
[ is coincident with the actual motor
speedr
[ . The fig 5.8 shows the stator angular frequencye
[ is fluctuating around the
zero with the amplitude of 10r/min and the angular frequency of 2 2d
[ T! v rad /sec.
The fig 5.10 shows the estimated speed emfrq
e is also fluctuating with the amplitude of
10 % of the average emf and angular frequency ofd
[ .From these results, the stable
sensorless control at zero frequency is realized.
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5.3. Conclusions
In this dissertation work for the stable low speed drive, a new sensorles control
scheme, which is based on the secondary speed emf estimation under fluctuating
excitation current is presented. The sensorles vector control scheme of the induction
motor at low speed region including zero stator frequency can be successfully controlled
regardless of the load and even zero frequency is approached without losing stability.
Constant operation at zero frequency is not possible, but stable crossing is very well
possible, even at a reasonably slow rate.
The proposed drive can compete with a speed-sensor equipped drive if
continuous operation at ac excitation and high load is not required. The simulated
characteristics of the sensorles control scheme were verified using a 4-ploe 2.2kW
induction motor. Even at the zero stator angular frequency, the stable sensorless drive is
realized in the speed range of more than 40 r/min.
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5.4. Further work
All further work is summarized schematically in the following ideas:
Development of fuzzy controllers to achieve better performance. Practical implementation of this sensorless vector control using DSP
controllers (TMS 320 F240, TMS 320 F243).
Try to find suitable parameter adoption schemes for Vector Control undervarious operating conditions.
Application of modern control techniques for design of optimum speedand current controllers for reducing EMI and for increasing energy savings
from the mains.
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APPENDIX: B
INTRODUCTION TO SIMULINK
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INTRODUCTION
In this project MATLAB6.1/SIMULINK software is used for sensorless vector control of
induction motor drive. In past, high-level programming languages such as FORTRAN or
C have been used for carrying out simulations. The writing of source code requires much
greater skill and knowledge on the part of the user. For example, proper integrations
routines must be selected and written, even simple mathematical manipulations have to
be programmed. These programs typically produce results, which must be post-processed
to derive visual impressions. This is a two-step process, and typically results in large files
of data, which must be stored before processing.
Computer simulation plays an important role in the design, analysis, and
evaluation of power electronic converters and their controllers. Designing and developing power electronic circuits without suitable computer simulation is extremely laborious,
error-prone, time-consuming, and expensive. Therefore, it is essential to teach, at the
undergraduate level, power converter modeling and simulation, together with the
dynamic behavior of the converter, using a theoretical framework suited for controller
design and development.
Nowadays, a variety of software tools, such as SPICE, EMTP, SABER,
CASPOC, SIMPLORER, SPECTRE, etc., is available to simulate electrical and
electronic circuits. The most used simulators are SPICE or PSPICE, user-friendly
programs designed to perform analysis of low power analog electronic circuits. Several
power electronics professors have used SPICE to simulate the behavior of power
electronics converters.
SIMULINK is a window-oriented dynamics modeling software package built on
top of the MATLAB numerical workspace. An advantage is that models are entered as
block diagrams with an intuitive graphical interface when the corresponding
mathematical descriptions are available for the target systems. This application is not
difficult to do for basic topologies of dcdc switching converters. Furthermore, a set of
blocks with signal interconnections could be masked as a subsystem for convenience in
the SIMULINK environment. The parameters of masked subsystems are then entered in
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dialog windows and can be changed interactively during a simulation. Simulation results
can be viewed during the simulation via a virtual oscilloscope and then exported to the
MATLAB workspace for subsequent off-line analysis. The SIMULINK modeling
environment provides make construction of simple dynamical systems quite easy. This
construction is also true for the design and verification of feedback controllers for
dynamical systems. If the mathematical way of using Kirchhoffs laws to construct the
corresponding dynamical systems is not favored, the MATLAB environment can also be
used to develop mathematical models from inputoutput data.
MATLAB/SIMULINK software is widely used for the simulation of almost all
types of dynamic systems. This software package is also valuable for teaching and
learning since it provides a series of standard routines and software toolboxes, such as a
control toolbox, system identification blocks, nonlinear control design block set, and
neural networks block set, which enable students to perform system simulation,
identification, and control.
The latest versions of MATLAB/SIMULINK include a Power System Blockset
This toolbox features electrical models of power semiconductors and the most commonly
used power devices (machines, transformers, power lines, voltage sources), and allows
simulation of power systems and power electronics. This package is valuable for
imulating well-known topologies several of which are included as demonstrations, but it
tends to generate too many algebraic loops on more complex or novel power topologies.
These algebraic loops are difficult to handle (because they are inherent to the modeling
method) and are time consuming, often preventing simulation convergence.
Furthermore, this toolbox does not easily allow open-loop or closed-loop
simulation of series associations of power rectifiers, nor does it study the steady and the
transient-states in cases of unbalanced or distorted and/or polluted power supply.
Considering the approach of with PSPICE and SIMPLORER, the authors think that a
system-level simulation, considering only the ideal switching and functional behavior of
power semiconductors, would be desirable for MATLAB/SIMULINK. The system-level
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simulation is fast enough and free of algebraic loops and convergence problems
(SIMULINK has built-in integration methods suited to deal with stiff systems).
Therefore, it could avoid the problems of the Power System Block set mentioned
above. Additionally, the system-level derived models to implement in SIMULINK can be
used for closed-loop controller design, since they are switched state-space models. This
advantage is lost when using the Power System Blockset or SIMPLORER.
Considering the increasing capabilities of MATLAB/SIMULINK for the
simulation of dynamic systems, it is advantageous to adapt the ideal models of
semiconductors and simulation methods presented here for this software since only one
software package is needed. The simulation time is short (a few seconds); an excellent
graphical interface is available with parametric identification of the system and the ability
to choose the numerical integration method and toolboxes for closed-loop control. In
addition, the SIMULINK package offers the benefits of a hierarchical structure and uses
MATLAB as its mathematical engine. If required, the modeling method here proposed
could be adapted to other programs. Since the goal is to teach nonlinear mathematical
modeling and control and the simulation of power converters, this paper shows, in
Section II, how to write system-level models of power electronics circuits. In Section III,
examples of pulse width modulation (PWM) ac/dc and dc/ac power electronic converters
are given.
The simulation models described are quite suitable to study power electronics
converters in drives or other applications whose simulation times are not too long, since
only the ideal behavior of the power switches is considered. This work was initially
developed for research in the area of new topologies for power electronics. However,
further developments allowed its use as a valuable teaching aid. Therefore, this work
presents a new way to teach undergraduate students the dynamic behavior of powerelectronics circuits without cutting down the analytic skills needed to learn and
synthesize power converter controllers. The new method can also be used as verification
of analytical methods, allowing students to check their mathematical work quickly and
use it for power converter behavior and controller development.
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BIBLOGRAPHY
[1] Takaharu Takeshita, Yoshiki Nagatoshi, and Nobuyuki Masti, Sensorles Vector
Control of Induction Motor at Zero Frequency. Power Conversion Conference,
2002. PCC Osaka 2002.Proceedings of the, Volume:2, 2-5Apri 2002 Pages:510 -
515 vol.2
[2] K.Nagasaka, Y.Nagatoshi, T.Takesita, N.Matsui, Sensorles Vector Controlled
Induction Motor Drive at Zero Frequency 2001 National Convention Record
IEEJ,4-115,p.1376.
[3] Y.Nagatoshi, K.Nagasaka, T.Takesita, N.Matsui, Sensorles Vector Control Scheme
of Induction Motor Drive at Zero Frequency 2001 Conference Record of IEEJ,
SPC-01-65.[4] Y.S.Lai, C.N.Lui, K.Y.Luo, C.I.Lee, and C.H.Liu, Sensorless Vector Controllers for
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[5] L. Umanand, Modeling & Simulation studies & Digital controller Synthesis for
Vector Controlled A.C. Drives., I.I.Sc, Bangalore, 1994.
[6] Scott Wade, Barry W. Williams, Modeling and Simulation of Induction Motor
Vector Control with Rotor Resistance Identification, IEEE Trans. Power
Electronics, vol.12, no.3, May 1997,pp495-506.
[7] P.Vas, Vector Control ofACMachines. London, U.K: Oxford Univ. Press, 1990.
[8] I. Boldea&Naser, Vector control of AC Drives.
[9] B.K Bose, power Electronics and AC Drives. Englewood cliffs, NJ Prentice-hall,
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[10] G.K.Dubey, Power semiconductor Controlled Drives.
[11] Ned Mohan, Tore M. Undeland, William P. Robbins. Power Electronics
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[12] Finch, J.W. Atkinson, D.J. General Principles of Vector Control for Induction