A NOVEL PARAMETER COMPENSATION SCHEME FOR INDIRECT VECTOR CONTROLLED
INDUCTION MOTOR DRIVES
by
Dhaval B. Dalal
Thesis submitted to the Faculty of the
Virginia Polytechnic Institute and State University
in partial fulfillment of the requirements for the degree of
Master of Science
Charles. E. Nunnally
in
Electrical Engineering
APPROVED:
Krishnan Ramu, Chairman
July, 1987
Blacksburg, Virginia
Mansell H. Hopkinl!f
A NOVEL PARAMETER COMPENSATION SCHEME FOR INDIRECT VECTOR CONTROLLED
INDUCTION MOTOR DRIVES
by
Dhaval B. Dalal
Krishnan Ramu, Chairman
Electrical Engineering
(ABSTRACT)
Indirect vector controlled induction motor drives are gaining acceptance because
they allow the induction motor to be controlled like a separately excited de motor, i.e.
they achieve decoupling of torque and flux producing currents. But, effectiveness of
these drives is lost as they are highly parameter sensitive. Studies have indicated
that the decoupling of the torque and the flux channels is lost when machine param-
eters change with temperature, saturation etc. Many schemes have been proposed
to overcome these parameter sensitivity effects. But most of these schemes them-
selves are parameter dependent and hence inapplicable to high precision control
applications. A new parameter compensation scheme which uses air gap power
equivale.nce for sensing parameter changes is developed in this thesis. It is shown
that this scheme is independent of key motor parameters and requires no additional
transducers for implementation.
Acknowledgements
I wish to express my sincere gratitude and appreciation to Dr. Krishnan Ramu, whose
constant encouragement, support and guidance from the conception of this project till
the end were tremendous. I have benefitted a great deal from working with him and
I hope to get an opportunity to work with him again.
I also greatly appreciate the time and attention of the other members of my graduate
committee, Dr. Nunnally and Dr. Hopkins.
This thesis would not have been complete but for the support provided by some
friends. I would like to thank Ramesh Kanekal, Dilip ldate, Aravind and Joe for their
lending a helping hand.
Acknowledgements iii
Table of Contents
1.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Vector Control Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Parameter Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Parameter Compensation ............................................. 3
1.4 Literature Survey .................................................... 3
1.5 Proposed Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.0 Vector Control Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Separately Excited DC Motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Steady State Analysis of Induction Motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Vector Control Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.5 Classification of Vector Control Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.6 Indirect Vector Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.7 Tuning of Vector Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.0 Parameter Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Table of Contents iv
3.1 Parameter Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3 Torque Controlled Drive (Speed Loop Open) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3.1 Torque Expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3.2 Flux Linkage Expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.3.3 Ranges of a. and . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.3.4 Analysis and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.4 Speed Controlled Drive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4,0 Parameter Compensation Scheme .. , .. , ....................... , . . . . . . . . 34
4.1 Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.2 Modified Reactive Power (MRP) Compensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.3 Proposed Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.3.1 Advantages .................................................... ·-. 41
4.3.2 Disadvantages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.4 An Alternate Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.0 Steady State Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.1 Steady State Equivalent Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.2 Changes in Rotor Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.3 Mutual Inductance Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.4 Compensation Algorithm ............................................. 52
6.0 Dynamic Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
6.1 The Drive System Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
6.1.1 The Induction Motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
6.1.2 The Vector Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
6.1.3 Transformation Circuit ............................................ 56
Table of Contents y
6.1.4 Inverter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
6.1.5 Parameter Compensation Block. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
6.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
6.2.1 Changes in Rotor Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
6.2.2 Mutual Inductance Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
7.0 Conclusions .... • .................................. , . . . . . . . . . . . . . . . . 72
7.1 Scope For Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Blbllography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
Appendix A. Induction Motor Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Appendix B. List of Symbols .............................................. ·· 77
VITA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
Table of Contents vi
List of Tables
Table 1. Inverter Logic ........................................... 58
List of Tables vii
List of Illustrations
Figure 1. Schematic of a Separately Excited DC Motor. . ................... 8
Figure 2. Steady State Equivalent Circuit of an Induction Motor. . ........... 10
Figure 3. Phaser Diagram of an Induction Motor ........................ 12
Figure 4. Control Block Diagram of an Induction Motor Drive. . ............ 14
Figure 5. Vector Controlled Induction Motor Drive. . .................... 15
Figure 6. Schematic of Indirect Vector Controller ........................ 21
Figure 7. An Equivalent Circuit of Induction Motor. . ..................... 22
Figure 8. Torque Controlled Induction Motor Drive ....................... 28
Figure 9. Block Diagram of MRP Compensation Scheme ................. 36
Figure 10. Block Diagram of Proposed Scheme ......................... 39
Figure 11. Block Diagram of Complete Drive System Including Compensation .. 40
Figure 12. The Inverter ............................................ 42
Figure 13. Flow Chart for Link Current Computation ...................... 43
Figure 14. Block Diagram for an Alternate Implementation .......... ; ...... 46
Figure 15. Steady State Equivalent Circuit for Current Regulated Inverter ...... 48
Figure 16. Effects of Rotor Resistance Variations (Steady State) ............. 51
Figure 17. Effects of Changes in the Mutual Inductance (Steady State) ........ 53
Figure 18. Stator Phase Currents for Hysteresis Controller. . ............... 59
Figure 19. Voltage Switching Waveform. . .............................. 60
List of Illustrations viii
Figure 20. Plots for Step Change in Rotor Resistance ..................... 63
Figure 21. Plots for Linear Change in Rotor Resistance ................... 64
Figure 22. Effects of Increase in Mutual Inductance ...................... 66
Figure 23. Effects of Decrease in Mutual Inductance ...................... 67
Figure 24. Effects of Step Change in Torque Command ................... 68
Figure 25. Effects of Step Change in Flux Command ...................... 69
Figure 26. Effects of Step Change in Load Torque ( Speed Loop ) ............ 70
List of Illustrations ix
1.0 Introduction
Induction motor drives are very popular in industries because of their low cost and
mechanical robustness over other types of motor drives. The induction motors are
attractive from a mechanical point of view as they require almost no maintenance,
have greater overload capacity and are capable of operating at higher speeds than
other motors. Despite such superior mechanical characteristics, the induction motor
drives were not used in high performance applications for a long time due to a lack
of control strategy.
1.1 Vector Control Scheme
Introduction of vector control scheme resulted in transforming the induction motor
(more generally, any ac motor) into an equivalent separately excited de motor for
control purposes. Although it was introduced in the early 1970's[1], the implementa-
tion of the vector controller remained a complex task till recently. It was difficult to
generate the comman<:I signals for accurate control and amplify those command
signals accurately at voltage/current levels, in real time. Rapid developments in
microprocessor technology on the one hand and in switching power devices on the
other have facilitated real time implementation of accurate vector controllers for in-
duction motors with reasonable cost. The increase in processor speeds allows
reconfigurable software implementation of the controller and helps generate accurate
values of the command currents/voltages to the converter with minimum delay [2,3].
Introduction 1
The software implementation in the microprocessor helps in making the controller
more interactive and easy to tune. Higher switching frequencies in the inverters en-
able the instantaneous control of the input currents/voltages to the motor.
The implementation of the vector controller requires a knowledge of instantaneous
position of the rotor flux. This can be either measured or estimated. Direct vector
controllers measure the rotor flux using Hall probes or search coils. The direct vector
control scheme is not very popular as it involves modifications to the existing motor .
or addition of sensors and precision problems in measurements at low speeds. The
indirect vector control scheme uses a real time model of the motor to predict the in-
stantaneous flux position. This scheme is widely used for vector controlled induction
motor drives.
1.2 Parameter Sensitivity
In order to achieve the decoupling of the torque and flux producing components, the
motor model incorporated in the vector controller has to be properly tuned. The pa-
rameters of the motor change with changes in temperature, magnetic saturation and
frequency. These variations in motor parameters cause deterioration of both the
steady state and dynamic operation of the induction motor drive. The steady state
performance degradation is in the form of input-output torque nonlinearity and satu-
ration of the machine. The effects on the dynamic performance include low frequency
torque and flux oscillations with a large settling time. These effects are highly unde-
sirable and t_he essence of the vector control scheme is lost if the parameter vari-
ations in the motor are not tracked by the motor model in the vector controller.
Introduction 2
1.3 Parameter Compensation
Ideally, there should be a one-to-one correspondence between the actual motor pa-
rameters and the parameters incorporated in the vector controller. Changes in indi-
vidual parameter values should be sensed and compensated accordingly in the
controller. But, it is expensive and complex to monitor each parameter individually.
Some parameters are more sensitive to operating conditions than others. On the
other hand, the drive performance (decoupling control) is dependent only on some
of the motor parameters, viz., the rotor parameters. This illustrates the need to sense
and compensate only a few select motor parameters in order to maintain decoupling
control achieved by vector control scheme. It will be shown later that it is sufficient
to monitor and correct the rotor resistance or the rotor time constant of the motor.
The sensing of the parameter variations using thermal models of the machine or
thermal sensors is ruled out because of the complexity involved.
1.4 Literature Survey
Several methods have been proposed to overcome the effects of parameter sensi-
tivity in indirect vector control schemes. Some schemes involve the measurement
of rotor resistance or rotor time constant. These schemes can be classified as direct
parameter compensation schemes. One such scheme [4] determines the rotor time
constant using injection of negative sequence voltages into the motor. Another
method [5] was proposed for rotor time constant identification through correlation.
Most of the parameter compensation schemes are, however, indirect in the sense
Introduction 3
that no parameters are monitored directly. Following are more prominent of the in-
direct parameter compensation schemes.
1. Modified _Reactive Power compensation scheme (MRP) [6]
2. Estimation of flux [7]
3. Parameter Adaptation Controller (PAC) using real and reactive values of power.
[8]
4. Model Reference Adaptive System (MRAS) [9]
These schemes determine a particular machine variable from terminal measure-
ments and compare it with its reference value. Any discrepancy between the two is
interpreted as a mismatch between the motor and the controller. A corresponding
change is incorporated in the controller to track the motor parameters. In a closed
loop system, the motor model in the controller is updated till the measured value of
the variable equals its reference value. When both are equal, the controller and the
motor are properly tuned and no correction signal is generated. Choice of such a
physical variable depends on the following factors:
1. The variable should be easily measurable. (Minimum number of transducers in-
volved / quantities measured.)
2. It should give a good indication of parameter variations, i.e., it should be highly
sensitive to parameter variations.
3. The reference value of the variable should not be sensitive to parameter vari-
ations. Otherwise, the error between the reference and the measured values may
not give a true indication of parameter variations in the motor.
Introduction 4
Applying these criteria to the above mentioned schemes, it has been found that most
of them are not ideal for parameter compensation. It is shown in a recent analysis
[1 O] that most of these schemes are themselves parameter dependent and their
sensitivities to parameter variations have been quantified.
1.5 Proposed Scheme
In this context, clearly there is a need for a parameter compensation scheme which
is insensitive to the parameter variations and which minimizes the number of addi-
tional transducers for compensation. The development and verification of such a
scheme is the focus of this thesis. The proposed scheme uses the principle of air gap
power equivalence. Its effectiveness is shown by steady state and dynamic simu-
lations of the entire drive systems. An attractive feature of the proposed scheme is
that it does not use any additional transducers for compensation.
The thesis is organized as follows. Chapter 2 contains a detailed description of the
vector control principle and the derivation of the equations for the indirect vector
control scheme. In chapter 3, the effects of parameter variations on the indirect
vector controlled induction motor drive are presented and the need for a compen-
sation scheme is highlighted. Chapter 4 outlines the requirements for an ideal com-
pensation scheme and contains the proposal of a new compensation scheme based
on the principle of air gap power equivalence. The results of the steady state and
dynamic simulations of the proposed scheme are presented in chapters 5 and 6, re-
spectively. The conclusions are presented in chapter 7.
Introduction 5
2.0 Vector Control Scheme
This chapter contains a detailed description of the principle behind the vector control
scheme and the derivation of equations for the indirect vector controller. From these
equations, the inherent sensitivity of the indirect vector control scheme becomes
obvious.
2.1 Background
For high performance servo drive applications, traditionally de machines are pre-
ferred over ac machines. This is largely due to the difference in control aspects of the
machines. AC motors, particularly the induction motor, are inherently multivariable,
non-linear control plants. The de motors are very easily controllable as the field and
armature can be separately controlled. This enables precise torque and field control
in high performance applications.
DC motors, however, have several mechanical limitations which restrict their appli-
cations. DC motors use brushes and commutators which require continuous mainte-
nance and limit the maximum speed. In addition, the de motors can not be operated
in corrosive or explosive environments thus limiting their use in certain industrial
applications.
Vector Control Scheme 6
The vector control scheme was introduced in the early 1970's as a control solution for
the ac motor drives. The vector control scheme effectively transforms the ac machine
into an equivalent separately excited de motor.
2.2 Separately Excited DC Motor
The ease of control of a separately excited de motor can be illustrated by considering
its schematic diagram shown in Figure 1 on page 8. The inputs to the motor are the
field current, i, and the armature current, i •. The outputs of the motor are the torque,
T. and the flux, cp,. The input-output relationship is given by following equations :
where,
Kt is the torque constant of the motor.
K, is the flux proportional constant.
(2.1)
(2.2)
The field and the armature currents are independently controlled. The flux is pro-
portional to only the field current. If the field current i, is held constant, the torque
becomes directly proportional to the armature current, i •. Due to commutator action,
there is always a quadrature relationship in space between the armature MMF and
the field MMF.
Vector Control Scheme 7
if 0 I> ::-<Pf D. C. MOTOR
ia 0 I> t>T e
Figure 1. Schematic of a Separately Excited DC Motor.
Vector Control Scheme 8
2.3 Steady State Analysis of Induction Motor
The complexity of control of the induction motor compared to the separately excited
de motor becomes apparent when the steady state equivalent circuit of a squirrel
cage induction motor is considered as shown in Figure 2 on page 10. The output
torque equation is :
2 P lrRr T = 3- --e 2 S(l) s
(2.3)
where P is the number of poles, s is the slip, I, is the rotor current, R, is the rotor re-
sistance and w. is the synchronous frequency.
The flux linkages are given by :
Mutual flux linkage = 'I'm = Lm Im (2.4)
Stator flux linkage = 'l's = Lm Im + L1s ls (2.5)
Rotor flux linkage = 'Vr = Lm Im + L1r I, (2.6)
where, Lm is the mutual inductance, L,. is the stator leakage inductance, L,, is the rotor
leakage inductance, Im is the magnetizing branch current and I, is the stator current.
From Figure 2 and equations (2.3-2.6), the following points can be noted :
• Both the rotor field flux and torque are controlled through stator phase currents
only.
Vector Control Scheme 9
Rs JX1s JX1 r
+ [>- + t> + I~ Is Ir
vs JXm Em Er Rr s
Figure 2. Steady State Equivalent Circuit of an Induction Motor.
Vector Control Scheme 10
• Unlike separately excited de motor, there are no separate field and armature
windings to control field and torque independently and hence the control be-
comes complex.
Another significant feature is that in the induction motor drive, accurate knowledge
of the rotor flux position becomes vital for proper control. This can be explained with
reference to Figure 3 on page 12. The angle between the rotor flux and stator current,
0r is known as torque angle and is determined by the load conditions. For an inverter
fed induction motor, the phase angle of the current can be controlled with respect to
the stator reference frames. This phase angle is the sum of the torque angle and the
flux position angle, 0,. With load conditions, the torque angle changes and hence to
adapt to this condition, the phase angle of the current has to be changed accordingly.
For this, the knowledge of flux position becomes important. If the flux position with
respect to the stator currents is known, accurate control can be achieved by adjusting
the phase currents till the required torque angle is obtained.
2.4 Vector Control Principle
Assuming that the rotor flux position 0, is known, then it is possible to resolve the
stator current phaser along the rotor flux and in quadrature to it, as shown in
Figure 3 on page 12. The in-phase component is the flux producing current, i, and the
quadrature component is the torque producing current, ir- If these two components
can be controlled independently, then the induction motor control becomes very
much similar to the control of a separately excited de motor. A block diagram of such
a control scheme is given in Figure 4 on page 14. This process of transforming the
Vector Control Scheme 11
~~---'-----_.;..i>'-----------C> \1/r
Stator Reference Frame
Figure 3. Phasor Diagram of an Induction Motor.
Vector Control Scheme 12
control of the induction motor to that of an equivalent separately excited de motor is
known as vector control or field oriented control.
The function of the vector controller is to generate the flux and torque producing
current commands from the torque and flux commands. The block diagram of a
vector controlled induction motor drive is shown in Figure 5 on page 15. If the
transfer function in the dotted box is given by G(s), then the vector controller has a
transfer function of G- 1(s), thus making the responses equal to their command values.
2.5 Classification of Vector Control Schemes
For proper implementation of the vector controller, the knowledge of the rotor field
angle, 0,, is crucial. The vector control schemes are classified according to the man-
ner in which this field angle is obtained. The field angle is measured in the direct
vector control scheme. The indirect vector controllers use estimation of slip angle or
some terminal measurements and the motor parameters to compute the field angle.
The direct vector control scheme uses either Hall sensors or a set of sensing coils
placed near the air gap and embedded in the stator slots. These are used to measure
instantaneous flux values from which the field angle is derived. The introduction of
Hall sensors or search coils involves modification to existing machine thereby in-
creasing the cost. The Hall sensors are found to be sensitive to temperature and thus
are not very reliable. In addition, the sensing coils produce very low voltage at
standstill and at low speeds. This impairs the accuracy of measurement. Due to
these reasons, the direct vector controller is not very popular. Most of the vector
controllers use the indirect vector control scheme.
Vector Control Scheme 13
i., - ias I -f's - - -Three Phase 1bs "ts IND.JCTION
transformation I . 1cs
- MOT~ - - "ts .... - c1rcu1t I . -,j\
..)
e.,
Figure 4. Control Block Diagram of an Induction Motor Drive.
Vector Control Scheme
_.. -
. -ex: ir ( far canatant t, >
14
if 1 .. a lh,._ ph--ThrN Alue Iftwrwr
- lo'ECT0R Iba . ... 0DfffllCIJ.ER 11 tranafor•atfon .
Indunt-1«:a . - _.,, cfrcuft . Hnor -- -- r •
r '
Figure 5. Vector Controlled Induction Motor Drive.
Vector Control Scheme 15
2.6 Indirect Vector Controller
As indicated earlier, the indirect vector controller uses the machine model in order
to estimate the rotor flux position. In this section, a step-by-step derivation of the
vector controller in dynamic reference frames is given.
The d-q axis equations of an induction motor in synchronously rotating reference
frames are given by [11]:
e Dqs Rs + LsP
e Dds - WsLs
= 0 LmP
0 - (Ws - Wr)Lm
Te - h - Bwr PWr = J
where,
WsLs
Rs + LsP
(Ws - Wr)Lm
LmP
LmP WsLm .e lqs
- WsLm LmP .e 1ds
(2.7) .e .
Rr + LrP {Ws - Wr)Lr lqr
- (ws - Wr)Lr Rr + LrP .e 1dr
(2.8)
(2.9)
u;,, u~. are the stator q and d axes input voltages, ;;,, i~. are the stator q and d axes
currents and ;;,, i~, are the rotor q and d axes_ currents referred to the stator side.
R, and R, are the stator and referred rotor resistances per phase. Lm is the mutual
inductance and L, and L, are the stator and referred rotor self-inductances per phase. "
L,. and L,, are the stator and the referred rotor leakage inductances respectively. w.
and w, are the electrical stator frequency and rotor speed, respectively. Te is the
Vector Control Scheme 16
electromagnetic torque, TL is the load torque, P is the number of poles, B is the
damping factor, J is the moment of inertia and p is the differential operator.
The rotor flux linkages are given by :
(2.10)
(2.11)
For a voltage source inverter fed induction motor, the voltages u:. , u~. can be found
by a simple transformation from the phase voltages which are input to the system.
This transformation is described by Equation (2.12). It should be noted that Tabc is a
standard transformation matrix which can be used for 3-phase to 2-phase transfor-
mation of other variables such as current, flux etc. 0, is the angle between the stator
A phase and the rotor flux position at any time.
where,
_ 2 [cos e, Tabc - 3
sine,
0 27t cos( , - 3 )
sin(0, - ~)
(2.12)
(2.13)
For the present case, a current regulated inverter is assumed. Using this type of
inverter, the stator currents are maintained at their commanded values and they form
the inputs to the system. This has the effect of eliminating stator dynamics from af-
Vector Control Scheme 17
fecting the drive performance. Consequently, the stator equations can be omitted
from the system equations. Thus, the rotor equations become,
(2.14)
(2.15)
Let slip speed w,, be defined as,
(2.16)
and, let the rotor flux linkage be aligned with the d-axis. Thus,
'l'dr = 'I' r (2.17)
'l'qr = 0 = P'l'qr (2.18)
Substituting equations (2.16-2.18) into (2.14-2.15) results in the following rotor
equations:
(2.19)
(2.20)
The relationship between the stator and the rotor d-q axis currents can be derived
by substituting equations (2.17-2.18) into (2.10-2.11),
(2.21)
(2.22)
Vector Control Scheme 18
If we define the rotor time constant T, as,
Lr T =-r R r
the system equations become,
.e Lm lqs ro, =--
s Tr 'Vr
where,
3 P Lm Kt= 2 2 L
r
(2.23)
(2.24)
(2.25)
(2.26)
(2.27)
From these, the torque and flux producing components ir and i, can be identified and
written as :
+ •
2 2 Te Lr =------3 p + + 'V r Lm
The equation for slip speed is,
• L~ i~ (J)s/ = -. -.
Tr 'Vr )
Vector Control Scheme
(2.28)
(2.29)
(2.30)
19
The quantities marked with asterisk indicate the commanded values and the control-
ler instrumented values. Equations (2.28-2.30) describe the function of indirect vector
controller and a schematic of the controller is shown in Figure 6 on page 21. It can
be noted that some proportional terms from equation (2.28) are omitted f~m the
schematic in order to simplify the schematic.
The equations are derived above in dynamic reference frame. In steady state, these
equations are derived using an equivalent circuit of the induction motor shown in
Figure 7 on page 22 (12]. This equivalent circuit is obtained by taking a different re-
ferral ratio'a'from stator to rotor. By choosing the referral ratio to be the stator to rotor N
effective turns ratio ( a = ), the conventional equivalent circuit is obtained. The ' L
equivalent circuit in Figure 7 on page 22 is obtained if a = Lm. This circuit can be r
used to explain the vector control principle in steady state. The stator current is
separated into two distinct components which are at quadrature to each other. The
current flowing through the magnetizing branch reactance is solely responsible for
producing the rotor flux and thus, is equivalent to the current i~ •. Similarly, the current
flowing through the rotor branch produces the torque making it equivalent to the
current i~ •. This enables the induction motor to be controlled like a separately excited
de motor.
The operation of the circuit can be explained using equations as follows :
Since the voltage drops across both the branches are equal,
(2.31)
Vector Control Scheme 20
re- T~ r dt
* +~, * r ..L - lr -- Lm +- --
T * * e "\ . Lr lT ,._N -- - -. - Lm .... / D h
, * . Wsl· Rr - N -- ·.:c . -
D
1
Figure 6. Schematic of Indirect Vector Controller.
Vector Control Scheme 21
+ --t>
Is
Figure 7. An Equivalent Circuit of Induction Motor.
Vector Control Scheme
+
22
Lm LEr Er
Ir= r = L2 Rr Lm Rr m ---- Lr s L2 s r
Defining
Er = j Ws 'I' r
and substituting in equation (2.31), 'I', is obtained as :
Since,
substituting the values equations (2.31-2.33) results in,
2 p Lm T = 3---1,lr e 2 L r
(2.32)
(2.33)
(2.33)
(2.34)
(2.35)
These equations very much resemble the equations for the separately excited de
motor and effectively demonstrate the principle of vector control of induction motor
drives.
Vector Control Scheme 23
2.7 Tuning of Vector Controller
The values of motor parameters incorporated in the vector controller have to be
properly tuned in order to achieve the decoupling of torque and flux. If the parameters
remain constant, then the tuning is simple. However, the rotor parameters change
with temperature, saturation etc. to complicate the tuning of the vector controller. The
effects of parameter variations are to be analyzed to understand the need for tuning
of the vector controlled induction motor drive. Following chapter deals with the pa-
rameter sensitivity of vector controlled induction motor drive.
Vector Control Scheme 24
3.0 Parameter Sensitivity
The indirect vector controller maintains a real time model of the induction machine
in order to compute the rotor flux position. Machine parameters change with tem-
perature and saturation level. This creates a mismatch between the controller and the
motor resulting in degradation of the drive performance. In this chapter, the effects
of parameter variations on the indirect vector controlled induction motor drive are
narrated.
3.1 Parameter Variations
The motor parameters vary with the operating point of the motor. The machine tem-
perature can vary a great deal depending on the operating environment. For example,
in an induction motor, the temperature of the motor can increase from sub-zero tem-
perature to about 200 deg. C when it is started in a very cold environment and al-
lowed to run at full load for a substantial period of time.
The rotor and the stator resistances are affected by the machine temperature. They
can increase by 100 % for an increase of 170/180 deg. C. The increase in rotor re-
sistance has an effect on the mutual flux linkage thereby altering the mutual
inductance of the machine. This can be seen from the Figure 7 on page 22. When a
current source inverter is used, the current I, is maintained at its commanded value
by the inverter. With an increase in R,, the current in rotor branch, I,, will decrease.
Parameter Sensitivity 25
The decrease in I, leads to an increase in Im in order to maintain I, constant. But, the
increase in Im is not accompanied by a linear increase in flux due to saturation. From
equation (2.4) it can be seen that this leads to a decrease in Lm.
The mutual inductance, Lm, and the leakage inductances, L,, and L,,, also change with
changes in the stator current, I,. The stator current magnitude changes with changes
in the torque and flux commands. For each I,, there is a unique value of the flux
linkages. Thus, changes in I, lead to changes in L,,, L,, and Lm. These changes in pa-
rameters lead to a mismatch between the induction motor and vector controller.
3.2 Consequences
From equations (2.28-2.30) it is clear that the vector controller generates commands
for the torque and flux producing currents and slip speed from the input torque and
flux commands using three motor parameters, viz., rotor resistance, R,, rotor self
inductance, L, and the mutual inductance, Lm. In the preceding section it was shown
how these parameters change with changes in operating conditions. The effects of
these variations in steady state are :
• The rotor flux linkage will become different from its commanded value.
• As a result, the electromagnetic torque also deviates from its commanded value.
This produces a non-linear relationship between the actual torque and its com-
manded value.
Dynamically, during load torque changes, an oscillation is caused in both the rotor
flux linkages and torque responses, with a settling time equal to the rotor time con-
Parameter Sensitivity 26
stant. The rotor time constant is in the order of a second and thus the oscillations
have a deleterious effect on the quality of the output.
In a torque controlled drive, these effects are highly undesirable as the output torque
does not match its commanded value. In a speed controlled drive, the commanded
value of the torque is changed till the load torque demand is satisfied maintaining the
speed at its commanded value. But, the parameter variations lead to changes in the
flux and currents, causing considerable heating and derating the motor.
3.3 Torque Controlled Drive (Speed Loop Open)
When the outer speed loop is open in a vector controlled induction motor drive, the
input commands to the system are the flux and the torque commands. The block di-
agram of such a system is shown in Figure 8 on page 28.
3.3.1 Torque Expression
In steady state, the differential term in equation (2.2) becomes zero (p = 0) and the
expression for rotor flux linkage becomes :
(3.1)
The substitution of equation (3.1) into (2.30) gives the slip command as:
* * 1 ir ros, = (3.2)
Tr i,
The torque angle, 0~ and the stator current command, i: are given by :
Parameter Sensitivity 27
T .. ...!.. dt • 'I' I' ..L
L11 INVERTER ..
STRT0R LOGIC • Q.fiRENT
T• . k GEN • . Lm D POSITION, SPEED TRFl&lJCER
R .. D
e ..
Figure 8. Torque Controlled Induction Motor Drive.
Parameter Sensitivity 28
* -1 ir [ * ] Br= tan T (3.3)
(3.4)
Substituting equation (3.3) in (3.2) gives :
(3.5)
(3.6)
(3.7)
The electromagnetic torque command is:
(3.8)
Similarly, the actual electromagnetic torque produced is given as :
T = 1....E.. (Lm)2 (" )2 cos Br sin Br
e 2 2 L 's r (3.9)
In the torque mode, the constraints are :
(3.10)
Parameter Sensitivity 29
(3.11)
Substituting equations (3.5-3.7, 3.10-3.11) into (3.8-3.9) the ratio of torque to its com-
manded value is obtained as :
If we define
T, a=-. T,
and substitute in equation (3.12) with the approximation :
we obtain,
Parameter Sensitivity
(3.12)
(3.13)
(3.14)
(3.15)
(3.16)
30
3.3.2 Flux Linkage Expression
In steauy state, the rotor flux linkages are :
(3.17)
(3.18)
Thus,
'I' r Lm cos 0r -=-
* * cos e; 'I' r Lm * * 2 (3.19)
1 + (w51 Tr) = B * • 2
1 + (a w51 Tr)
3.3.3 Ranges of a and B
As discussed earlier, the rotor resistance can become twice its nominal value as the
rotor temperature increases. A related decrease in Lm can reduce it to around 0.8
times its nominal value. In such an extreme case,
(3.20)
Thus, the least value a can take is 0.4. The upper bound on a is 1.5 which is deter-
mined by errors in the instrumented controller gain in the vector controller and the
increasing value of the self inductance of the rotor at low flux levels. Thus,
0.4 < a< 1.5 (3.21)
Parameter Sensitivity 31
Value of J3 lies between 0.8 in saturation region and 1.2 in the linear region of BH
curve. Thus,
0.8 < J3 < 1.2 (3.22)
3.3.4 Analysis and Results
Detailed simulation results of parameter sensitivity effects for these ranges of a and
.13 are presented in [13) and are summarized here.
When the temperature increases, ( a < 1 ), both the rotor flux linkage and the
electromagnetic torque become greater than their commanded values. When satu-
ration level is increased at ambient temperature ( a > 1 ), the torque and rotor flux
are less than their commanded values. Although the output torque is higher than the
commanded value at higher temperatures, it is not desirable as the relationship be-
tween input-output torques has become non-linear. The drive can not operate as a
torque servo for high performance applications such as robotic drives.
3.4 Speed ControUed Drive
Once the outer speed loop is closed, the electromagnetic torque command, T: will
be modified until the actual output torque equals the load torque in steady state.
Thus, the effect of parameter variations on steady state performance is reduced. The
equations of the actual to commanded values of torque and rotor flux are somewhat
complicated and omitted here as they are peripheral to the objective of the present
Parameter Sensitivity 32
work. Some of the results of parameter sensitivity of the speed controlled drive are
given here to give a proper perspective.
For increased temperature ( a < 1 ), the torque produced is more than the com-
manded value at rated load torque. The rotor flux increases with load torque at a =
0.5. One significant effect of increased temperature in steady state is that the stator
losses increase considerably. Dynamically, there are torque and rotor flux oscil-
lations with a low frequency and large settling time. The oscillations do not appear
on the rotor shaft as speed ripples due to the damping provided by the inertia of the
motor and its load.
The discussion in this chapter illustrates that the performance of the indirect vector
controller suffers due to the parameter variations in the motor. There is deterioration
of both the steady state and transient performance of the drive, especially when the
outer speed loop is open. If the indirect vector control has to be used for high per-
formance applications, the need for some type of parameter compensation scheme
is obvious.
Parameter Sensitivity 33
4.0 Parameter Compensation Scheme
Many schemes have been proposed to overcome the parameter sensitivity aspects
of the indirect vector controlled induction motor. The classification and approaches
of these schemes have already been discussed in chapter 1. In this chapter, the lim-
itations of the existing parameter compensation schemes are highlighted and a novel
parameter compensation scheme which meets the requirements for a high perform-
ance drive system is introduced.
4.1 Requirements
The function of the parameter compensation scheme is to detect any changes in the
motor parameters as a result of changes in the operating environment, and incorpo-
rate these changes in the vector controller. Since this compensation constitutes a
secondary control loop in the drive system, the response time is not of primary im-
portance when compared to the ease of implementation. The bandwidth require..:
ments are anyway not very stringent as the parameter variations are very slow. Thus,
for most applications, the system response is not required to be very fast, which can
be typically of the order of 100 milliseconds. The detection of parameter variations
should involve minimum number of transducers and should be possible using termi-
nal measurements alone. The hardware block for generating the parameter compen-
sation signal to the vector controller should be inexpensive to synthesize. One of the
important requirements is that the effectiveness of the compensation scheme should
Parameter Compensation Scheme 34
be insensitive to parameter variations. Otherwise, the compensation would not be
accurate.
To keep the implementation of the compensation scheme simple, only one parameter
compensation signal is generated to the vector controller. This signal is used to
correct the value of either the rotor time constant or the rotor resistance implemented
in the controller. Since these parameters are the most sensitive and play a crucial
role in the controller functioning, it is sufficient to compensate either of them for the
parameter variations in the motor.
The existing compensation schemes differ in the manner in which detection and
compensation are achieved. A recent study [10] has shown that most of these
schemes are parameter sensitive, and hence, inapplicable to high performance servo
applications. As an example, one of the existing parameter compensation schemes
is outlined in the next section.
4.2 Modified Reactive Power (MRP) Compensation
A block diagram of the parameter adaptation scheme which uses modified reactive
power [6] is shown in Figure 9 on page 36. The modified reactive power is defined
as:
(4.1)
and its commanded value is,
Parameter Compensation Scheme 35
r------., CONTR-OLLER
F" CALCULATOR
Figure 9. Block Diagram of MRP Compensation Scheme
Parameter Compensation Scheme
Signal to correct the rotor time constant
Terminal voltages
Phase currents
36
+
F = (4.2)
The value of F is estimated from the terminal voltages and phase currents as :
(4.3)
where,
(4.4)
The parameter variations in induction motor will change F and it will deviate from F'.
The error between F' and F is amplified through a controller and a correction signal
is obtained to correct either the rotor time constant or the rotor resistance. For a
current source inverter, voltages u •• and u01 change with variations in the motor pa-
rameters such as rotor and stator resistances, leakage inductances and mutual
inductance. However, the change in F is only used to correct the value of R,. The
value of R, is corrected till F equals F'. The parameter dependency of this scheme is
apparent from equation (4.2). F. depends on Lm and L,. If these values do not reflect
actual motor values, the error is not a true indication of parameter variations.
4.3 Proposed Scheme
As a solution to the problems outlined in preceding sections, a new parameter com-
pensation scheme is proposed for the indirect vector controlled induction motor
drive. This scheme uses the principle of air gap power equivalence for the detection
Parameter Compensation Scheme 37
of parameter variations in the motor. If the controller and. the motor are properly
matched, then the power produced in the air gap (synchronous power) should equal
its reference value. Any discrepancy between the two values can be used as an in-
dication of parameter variations in the motor.
A block diagram of the scheme is given in Figure 10 on page 39. Block diagram of a
complete torque controlled drive system incorporating the compensation scheme is
shown in Figure 11 on page 40. The values of reference power and actual power are
computed using the following equations :
(4.6)
(4.7)
P1n = V de Ide (av) (4.8)
(4.9)
where, P. is the actual air gap power, P: is the commanded air gap power, P;" is the
input power, P1c represents the stator copper losses, Pil represents the inverter
losses, Vdc is the DC-Link voltage and Ide is the DC-Link current.
As indicated here, the reference air gap power is computed as a product of the ref-
erence torque and synchronous frequency. The synchronous frequency is the sum
of the rotor electrical speed and slip speed command generated by the vector con-
troller. Actual air gap power is computed by subtracting copper and inverter losses
from the time-averaged value of instantaneous input power. The instantaneous input
Parameter Compensation Scheme 38
w.* • C Sxnch.
Speed)
X
T* e (Torque Command)
~t Inverter- I Losses~
(Stator Res f stance)
LOSS CR.Cl.LATO
A
Figure 10. Block Diagram of Proposed Scheme
Parameter Compensation Scheme
PI ONTROLLER
Parameter Correction Sf gna 1
V X
de CDC Link ~-- Voltage)
~c (Av. DC Link ----'--~ Cur-rent)
LINK _L_J. CURRENT ALCULATOR Inverter Logic
Stator-Phase Currents
39
e,.
ItMRTER LOGIC I. STRTOR 0JIEffS
F1RIETER C0f'FEN5FITIQ>4
SO£t£
flOBJTJON, SPIED TM"EDICER
Figure 11. Block Diagram of Complete Drive System Including Compensation
Parameter Compensation Scheme 40
power is taken as a product of de link current and de link voltage. Due to continuous
inverter switching, the instantaneous current has a highly irregular and discontinuous
waveform. The filter at the input of the inverter averages this current. Thus, there is
a need to average the instantaneous computed link input power. The error between
the reference and actual air gap power is processed through a Pl controller to gen-
erate a rotor resistance correction signal. Figure 12 on page 42 gives details of the
inverter. The link current is synthesized from the inverter logic and the stator phase
currents which are available to the controller logic [14]. The flow chart of the scheme
that is used for the derivation of link current is given in Figure 13 on page 43.
The link voltage is assumed to be constant which is valid if the load does not vary too
much. The inverter losses are computed from the inverter logic and the stator cur-
rent values, neglecting the switching losses.
4.3.1 Advantages
The advantage of the proposed scheme is that it is not dependent on any key pa-
rameters like many other compensation schemes. The parameter dependency of a
compensation scheme can be decided by the parameter dependence of constants
used in the detection block. When the parameters vary, the values of these constants
incorporated in the detection block would not reflect their actual values if they are
parameter dependent. Thus, the detection block wou Id use wrong values of con-
stants to detect these changes in parameters, resulting in deteriorated performance.
For the proposed scheme, the reference value of air gap power is computed as
product of synchronous frequency and torque command. For torque controlled drive,
ros, = ro:, from equation (3.11). This means that the reference value of synchronous
frequency will always equal the actual synchronous frequency of the motor irrespec-
Parameter Compensation Scheme 41
1 de
+ Tl T3 T5
t------t:,------ '---------· ._ ___ a -----b
T4 C T2
D4 D6 D2
Figure 12. The Inverter
Parameter Compensation Scheme 42
START
RETURN
Figure 13. Flow Chart for Link Current Computation
Parameter Compensation Scheme 43
tive of the parameter variations. For computation of actual air gap power, only one
motor parameter, R, is used. Also, it is only used for computation of losses, thus the
effect of that mismatch is minimized.
The other attractive feature of the scheme is that it does not use any additional
transducers for compensation loop. The stator currents are sensed for vector control
and are available directly. The phase voltages are available from the switching logic
of the inverter. The link current is computed from this logic and the stator currents.
The compensation can be performed by the same microprocessor that is used to
implement the vector controller without adding too much computational load on it.
4.3.2 Disadvantages
For all the benefits of the proposed scheme, there are some aspects which limit the
effectiveness of the proposed scheme to some extent. They are :
1. The assumption that link voltage is constant may not apply for all operating con-
ditions. For large load variations, the fluctuations in the link voltage are higher.
A simple remedy to this problem is to have a voltage transducer to measure the
de link voltage.
2. The dependency of the scheme on R, makes it parameter sensitive to some ex-
tent. As pointed out earlier, this effect is minimal and the parameter dependency
of the scheme is much less compared to the existing schemes. One way to rem-
edy it is to have an inner loop for sensing changes in stator resistance and using
the new values for compensation.
Parameter Compensation Scheme 44
3. At higher switching frequencies, inverter switching losses become significant and
have to be taken into account. This can increase the complexity of the air gap
power computation.
4. Due to the fluctuation of the link current, a filter becomes necessary. This means
that the parameter compensation signal can be generated at certain intervals
only (when the average has been taken). Thus, the response time increases. As
indicated earlier, the response time is not crucial. In fact, the averaging is helpful
to the performance as it eliminates response to short transients.
4.4 An Alternate Implementation
If the computation of inverter losses is to be avoided, an alternate implementation is
to compute the input power at the motor end of inverter instead of the de link end.
For this, since the stator voltages and the stator currents are available, there is no
additional transducer necessary. The block diagram of this scheme is given in
Figure 14 on page 46. This scheme adds some computational complexity as the 3
stator phase currents and voltages have to be multiplied and their products added to
find the input power. This power also has to be averaged before it is compared with
the reference value of the air gap power.
Steady state and dynamic performances of the proposed scheme were investigated
through computer simulation and the results are discussed in the next 2 chapters.
Parameter Compensation Scheme 45
w.* I
( S)1"1Ch. Speed)
T* e (Torque Command)
(Stator Res i stance )
LOSS CR.a.LAT
R
Para.meter PI i----:,> Correction
ONTROLLE s i gna 1
INTEG. FILTER
MULTIPLY &.
V de
Stator Phase Currents
CDC Link Volt.age)
Inverter Logic
Figure 14. Block Diagram for an Alternate Implementation
Parameter Compensation Scheme 46
5.0 Steady State Verification
The steady state performance of the proposed scheme was evaluated using simu-
lation techniques. Because of the large time constant of the parameter variations, the
steady state performance provides a very good yardstick to confirm the validity of the
proposed parameter compensation scheme. The results for variations in rotor resist-
ance and mutual inductance are presented in this chapter.
5.1 Steady State Equivalent Circuit
For the steady state simulation of the drive system, there is no need to incorporate
the full system shown in Figure 10 on page 39. Instead, a simplified steady state
equivalent circuit is used as shown in Figure 15 on page 48. It can be noted from
Figure 15 that the stator side components are omitted from the equivalent circuit.
This is justified under the assumption that a current regulated inverter is used to
drive the motor. In steady state, it is assumed that the inverter switching is fast
enough to regulate the stator phase currents to their commanded values, generated
by the vector controller and the transformation circuitry. Thus, the inverter block can
be omitted from simulation.
The vector controller equations (2.28-2.30) are used to generate the commands for the
torque and flux producing currents and the slip frequency. From these equations, the
synchronous frequency, ros, the slip, s, and the stator phase current command, i;, are
generated using :
Steady State Verification 47
1 m
s
Figure 15. Steady State Equivalent Circuit for Current Regulated Inverter
Steady State Verification 48
(5.1)
(5.2)
(5.3)
The air gap power is computed using :
(5.4)
where,
(5.5)
The commanded air gap power is :
(5.6)
These equations are used to. simulate the steady state vector controlled induction
motor drive. For simulation purposes, a 5-hp induction motor was used throughout.
The motor details are given in Appendix A.
When the motor parameters are at their nominal values which are incorporated in the
vector controller, the computed value of air gap power equals commanded air gap
power and the error is zero. Any changes in the motor parameters are reflected as
variations in the air gap power The error between the computed and the commanded
Steady State Verification 49
value of the air gap power is processed to generate a parameter correction signal to
the controller.
5.2 Changes in Rotor Resistance
The rotor resistance is the most sensitive and critical parameter incorporated in the
vector controller. To study the effects of temperature variations, value of R, was
changed and its effect on the air gap power was studied. With the drive running at
1000 RPM the change applied in R, was from 0.8 R; to 2.0 R;, where R; is its nominal
value incorporated in the controller. For an uncompensated drive system, the air gap
power varied from 0.8 times to 1.3 times its commanded value as a result of this
variation. This effect is similar to the result obtained in [13].
When the compensation scheme was used, the error in the air gap power was used
to correct the rotor resistance value in the vector controller till the error became zero.
In this manner, the tuning between the controller and the motor was achieved. Figure
16(a) shows the results of rotor resistance variations on the air gap power for the
compensated and uncompensated systems. As can be seen, for the compensated
system, the error is uniformly zero, thus the compensating scheme provides a pa-
rameter insensitive drive performance. To indicate the tuning achieved by the com-
pensation scheme, the controller rotor resistance is plotted against the actual rotor
resistance in Figure 16(b). The linearity and the symmetry of this curve confirm the
effectiveness of the proposed scheme.
Steady State Verification 50
- 1. 2 . :::> . 0. -z
1.0 < 0.
0.8 0.8 1.0 1.2 1.4 1.6 1. 8 2.0
RR (P.U.)
W/ COMPENSATION ,.. "' W/0 COMPE NSA T I ON
2.0
1.8
- 1. 6 . :::> 0. 1.4 -(.)
1.2 a:: a::
1.0
0.8 0.8 1. 0 1 . 2 1. 4 1. 6 1.8 2.0
RR (P.U.)
Figure 16. Effects of Rotor Resistance Variations (Steady State)
Steady State Verification 51
5.3 Mutual Inductance Variation
Another crucial parameter in the vector controller which is sensitive to operating
conditions is the mutual inductance of the motor. To study the effects of saturation,
the value of Lm was varied from 0.8 L~ to 1.2 L~. The effects of these variations on the
air ~ap power are plotted in Figure 17(a) for the compensated and the uncompen-
sated systems. For uncompensated drive, the air gap power changes from 90 % to
110 % of its commanded value. For the compensated drive, the error is uniformly
zero, once again proving the validity of the proposed scheme. Figure 17(b) shows the
corrections applied to R, by the compensating algorithm for changes in Lm. Since the
compensation algorithm applies changes only to R, even for changes in Lm, the motor
and the controller are not exactly tuned in this case. However, the torque and the flux
levels obtain their commanded values as a result of parameter correction algorithm.
In this way, the drive is insensitive to variations in mutual inductance.
5.4 Compensation Algorithm
As mentioned earlier, the air gap power error is used for correcting the value of rotor
resistance in the controller. The control algorithm for this signal generation is
proportional-and-integral (Pl) type. From the plots for the uncompensated system, it
can be seen that the air gap power error is not a linear function of the parameter
variations, thus a simple linear or proportional controller can not be used. The Pl
controller improves the order of the system by one. As a result, the steady state error
reduces to zero.
Steady State Verification 52
-. ::> a.. -z < a..
1.0
0.8 0.8
-- WI COMPENSATION
-::> a.. 1.0 -u a:: a::
0.8
1. 0 LM (P.U.)
1.2
+-----+ W/0 COMPENSATION
1.0
LM (P.U.) 1.2
Figure 17. Effects of Changes in the Mutual Inductance (Steady State)
Steady State Verification 53
The controller has to be designed for satisfactory transient response. If properly de-
signed, the Pl controller can provide a transient response with little or no overshoot
and oscillations. This is a desirable feature for the system under investigation. On the
other hand, the Pl controller increases the rise time of the system. As discussed
earlier, the rise time is not a very critical factor for many industrial applications and
anyway, the parameter variations are very slow. For the purpose of current work, the
proportional and the integral gains were arrived at by trial and error. Given the com-
plexity of the system, design of the gains was ruled out. The transient response of the
system is analyzed in the next chapter.
Steady State Verification 54
6.0 Dynamic Performance Evaluation
The dynamic response of the proposed compensation scheme has been verified us-
ing a dynamic model of the complete indirect vector controlled induction motor drive
system. This model includes the vector controller, the transformation circuit, a
hysteresis type current regulated inverter, the induction motor and the parameter
adaptation block. In this chapter, the results of the dynamic simulations are presented
to demonstrate the transient response of the proposed compensation scheme.
6.1 The Drive System Simulation
The entire drive system has been simulated in a digital computer in FORTRAN. The
approach used for simulating each block is presented here.
6.1.1 The Induction Motor
For simulating the induction motor, the dynamic d-q axis equations for the induction
motor (2.7-2.9) are used. These equations are non-linear and hence numerical tech-
niques have to be used for solving them. Of the various numerical techniques avail-
able, the Runge-Kutta method [15) was used as it can give accurate results to
simulate the performance of the induction motor. For the present, fourth order
Runge-Kutta system was used to solve the induction motor dynamic equations. The
integral time step h is a critical variable for simulation of these equations. If the value
Dynamic Performance Evaluation 55
of h is too high, the accuracy of the solution is affected. A time step of 5 micro-
seconds was used for the present work and it was found to be satisfactory.
6.1.2 The Vector Controller
The inputs to the vector controller are the torque and the flux commands. The vector
controller generates the commands of the torque and flux producing currents from
these inputs. To simulate the vector controller, the equations (2.28-2.30) are used. In
the absence of any compensation scheme, the values of rotor parameters incorpo-
rated in these equations are their nominal values. In cases of changes in the motor
parameters, it is the function of the compensation scheme to identify these changes
and incorporate them in the vector controller.
6.1.3 Transformation Circuit
The 2-phase d-q axis current commands generated by the vector controller have to
be transformed into 3-phase stator current commands before they can be input to the
inverter logic. The transformation circuit uses the inverse of the transformation matrix
Tabc to accomplish this task. The output of this block is fed to the inverter logic.
6.1.4 Inverter
The inverter is simulated using the Figure 12 on page 42. A hysteresis type current
controller in the inverter is also simulated. The inverter is a current regulated
inverter with a hysteresis band of ± 0.5 Amp. The actual and reference values of
each phase current are compared and error signals are generated. If the current error
Dynamic Performance Evaluation 56
is not within the hysteresis band, then the inverter phases are switched according
to the inverter logic. Inverter logic for phase A is shown in Table 1. Similar logic is
applied to the other two phases. From this logic, the phase voltages are calculated.
This logic ensures that whenever the current level of a particular phase is below the
hysteresis band, then . positive bus voltage is applied to that phase to bring the cur-
rent upto the reference value. Similarly, when the phase current is above the
hysteresis band, then that phase is switched to negative bus voltage to reduce the
current. Figure 18 on page 59 shows the correspondence between the commanded
and the actual values of the stator phase current. The actual value follows the com-
manded value with negligible error. This justifies the assumption that in steady state,
the current input to the stator equals its commanded value generated by the vector
controller. The problem with the hysteresis controllers is that they can increase the
switching frequency of the inverters. The practical inverters may not be able to
switch at such high frequencies as demanded by the hysteresis controller. To illus-
trate this point, the switching waveform for one phase of the inverter is shown in
Figure 19 on page 60. From the figure, it is can be noted that the during some peri-
ods, switching takes place at every time step, i.e., every 5 micro-seconds. This im-
plies a switching frequency of 200 KHz or greater. To restrict the switching frequency,
PWM controllers which have constant switching frequency can be used. The PWM
controllers do not give instantaneous current control as the hysteresis controllers
[16]. For the present work, only the hysteresis controller was used.
6.1.5 Parameter Compensation Block.
The inputs to the compensation block are stator phase currents, inverter logic, the
synchronous frequency and the torque command. From these inputs, logic described
Dynamic Performance Evaluation 57
Table 1. Inverter Logic
,:. i •• T1 T4 v,.
::2:0 i,, :s: U:. - !li) ON OFF + Vd/2
2: 0 i,, ::2: u:. + Iii) OFF OFF - VdJ2 (04 ON)
< 0 i OS 2: u:. + fli) OFF ON - VdJ2
< 0 i,, :s:; u:. - Lii) OFF OFF + VaJ2 (D1 ON)
Dynamic Performance Evaluation 58
15
- 10 . 0.. ::::E < -1-- 5 z L&J a::: a::: :::> 0 u L&J (/)
< :c -5 0..
a::: 0 .... < -10 1--(/)
-15
0 10 20 30 40 50 TIME (MS)
-- ACTUAL VALUE •••COMMANDED VALUE
Figure 18. Stator Phase Currents for Hysteresis Controller.
Dynamic Performance Evaluation 59
300----------------------
-> - '
LaJ 200 -(.!) c( I-...J 0 > LaJ (/) c( ::c 0.
a::: 0 100 I-c( I-(/)
0-... ,---.---,---,---,---.---,---,---.---,--+ 0 100 200 300 400 500 600 100 ·soo 900 1000
TIME (MICRO-SECONDS)
Figure 19. Voltage Switching Waveform.
Dynamic Performance Evaluation 60
in chapter 4 is used to compute the air gap power error. The air gap power error is
processed through the Pl controller described in chapter 5 to generate the rotor re-
sistance correction signal. The correction signal is applied at an interval of 25 ms in
order to eliminate the response due to the hysteresis controller. The selection of the
averaging interval is based on the rotor time constant. It is clear that this interval
should be an order of magnitude small compared to the rotor time constant and at the
same time an order of magnitude large compared to the switching period. It was
found that the filtering is perfect for this value of time constant. This time constant is
related to the operating speed of the machine. The correction signal is applied after
the averaging is performed. This period is small compared to the rotor time constant,
and hence a reasonably good dynamic response is anticipated.
6.2 Results
To verify the effectiveness of the compensation scheme, changes in the rotor resist-
ance were applied, first as a step change from nominal value to 1.5 times the nominal
value ·after 5 seconds and next in a linear fashion from nominal value to twice the
nominal value in 10 seconds. Effects of variations in mutual inductance were also
studied. The torque command was applied after the drive was brought to com-
manded flux level. All the results presented in this section assume this moment as
the starting time (t = 0). The operating speed of the drive was fixed at 1000 rpm.
6.2.1 Changes in Rotor Resistance
To indicate how the controller rotor resistance tracks the rotor resistance, normalized
values of the controller and rotor resistances are plotted in 20 for a step change in
Dynamic Performance Evaluation 61
R,. Plots of normalized values of the rotor flux linkage, torque, flux producing current,
torque producing current, air gap power and air gap power error are also given in
Figure 20 for both compensated and uncompensated systems when the step change
in R, is applied. Similar results for linear change in R, are shown in Figure 21.
The variables plotted are normalized with respect to their reference values in these
plots. The values of the torque and flux commands are input to the drive system.
Actual torque and flux values calculated using the Runge-Kutta algorithm are divided
by these commands to give the normalized values plotted here. ITN and IFN are
normalized values of the torque and flux producing currents. PAN and PEN are the
normalized values of air gap power and air gap power error respectively.
From these plots, it can be seen that the compensation scheme works by reducing
the air gap power error to zero. When the error is zero, the motor performance
matches the commanded values. The step change compensation takes about .75
seconds to reach steady state which is acceptable considering the rotor time con-
stant. The steady state error is not zero due to the fact that in the present imple-
mentation, there is a window for air gap power error. If the error is within this
window, the controller does not generate a correcting signal. This window can be
reduced to reduce the steady state error, but that may lead to more oscillations in the
dynamic response. For linear change, there is an apparent lag in tracking and some
oscillations occur in the beginning as the compensator tries to track the change in
rotor resistance. These oscillations die out as the Pl controller starts tracking the
changes.
Dynamic Performance Evaluation 62
1. 6 =i 1. 4 a. 1. 2 o:: 1. 0 a::
0.8 1.6
:::::,
1.2 >< :::::, ...J IA. 0.8 ,.... 1. 3 :::::,
1. 1 0 a:: 0 1- 0.9
1. 40
=> 1.15
z IA. - 0.90
1. 1
1.0 Q.
; 0.9 1-- 0.8
1.40 :::::, Q. 1. 15 ..... z cc Q. 0.90
0.3 ,... :::::, 0.2 Q. 0. 1 ..... z 0.0 w Q. -0. 1 .
I
0
·- -- ·-.
'I - -··----·--
-/\~
-- • .Ill
r1 l
lj -..
' ' f\J0N\f\f\/ --
. ,(v\NM/V ., -f
-··.
• [\
; \ . ··-----------
t
'
T I I I
2 4 6 8 10
TIME (SECONDS)
Figure 20. Plots for Step Change in Rotor Resistance
Dynamic Performance Evaluation
.........- ACTUAL
-- CONTROLLER
_..,....... W/0 COMPENSATION
-- W/ COMPENSATION
_.._.... W/0 COMPENSATION
--- WI COMPENSATION
-- W/0 COMPENSATION -- WI COMPENSATION
-- W/0 COMPENSATION
-- W/ COMPENSATION
-...... W/0 COMPENSATION -- WI COMPE NSA T I ON
-- WIO COMPENSATION -- WI COMPENSATION
63
2.00 ..... ::::, 1. 75 CL. .... 1.50 a:: 1.25 a::
1.00 ..... 1.8 ::::,
CL. 1. 5 -->c 1. 2 ::> _,
0.9 ....... 1.4
.... 0 1.0 a:: 0 1- 0.8 ,,... 1. 8 .l.-------------r ::::, 1.5 CL. .... z 1.2
0.9 1. 1 -l------------i
=> t. 0 CL. ._, z 0.9 I-
0.8 1.40 .....
::, CL. 1. 15 ..... z < CL. 0.90 ,,... 0.5 ::, 0.3 a. .... z 0. 1 CL. -0. I
0 2 4 6 8 TIME (SECONDS)
Figure 21. Plots for Linear Change in Rotor Resistance
Dynamic Performance Evaluation
10
-- ACTUAL
-- CONTROLLER
-- WIO COMPENSATION
- WI COMPENSATION
-.......... WIO COMPENSATION
- WI COMPENSATION
-- WIO COMPENSATION - W/ COMPENSATICN
-- W/0 COMPENSATION
- WI COMPENSATION
-- WIO COMPENSATION - WI COMPENSATION
-- W/0 COMPENSATION
- WI COMPENSATION
64
6.2.2 Mutual Inductance Variations
Next, the mutual inductance value was varied and its effects studied. The value of
mutual inductance was first changed from nominal value to 120 % of the nominal
value. For another run, it was reduced by 20 % from its nominal value. This covers
whole range of variations possible for the mutual inductance. The results are pre-
sented in figures 22 and 23 for these 2 cases.
It can be noted that the dynamic performance of the compensation scheme is not as
good in this case as in the response to the rotor resistance variations. This can be
attributed to two factors. First, the dynamic response of the uncompensated system
to the mutual inductance variations is more oscillatory than to the rotor resistance
variations. In view of this, it is difficult to compensate this system as the air gap
power error overshoot leads to oscillations in the correction signal. Second, the Pl
controller gains were optimized for response to rotor resistance variations as it is the
more critical parameter. The steady state response to variations in mutual inductance
has been shown to be accurate in the previous chapter.
To illustrate that the compensation scheme does not interfere with the primary torque
control loop, a change in torque command was applied with the compensation
scheme functioning. The response to this variation is plotted in Figure 24. Similarly,
a change in flux command was applied and the drive system performance was found
to be unaffected by the presence of the compensation scheme as shown by the re-
sults given in Figure 25. To investigate the effect of a load torque variation, the speed
loop was closed and a disturbance was applied to the load torque. Again, the com-
pensation scheme did not interfere with the primary speed loop operation. The re-
sults for this run are given in Figure 26.
Dynamic Performance Evaluation 65
1.2 5 ...... ::::) 1.0 Q.
0 ....,
0.75 ...... 1. 2 ::::)
Q. 1. 1 ..., x 1.0 :::> _, 1.. 0.9
_ 1. 30 ::::)
o.. 1. 15 ...., ...., 1.00
0 ._ 0.85
1.0 ,..,
0.9 a. ...., z 0.8 &.. - 0.7
1. 1 ...... ::::)
--·
.....
' _____ .,
-'11.,,.. r
-K
.
•• .
1.0 f· .,. .......... . .,.... .. ,..,,
z ... - 0.9
..,... 1.30
::::> 1. 15 Q. ..... z 1.00 < o.. 0 .85 _ 0.3 :::> 0.2
0.1 z 0.0 ..... o.. -o. 1
-
~-
'
I
0
.. - ... --
i~ -
!"'\_
-I I . I I . 2 4 6 8
TIME (SECONDS) Figure 22. Effects of Increase in Mutual Inductance
Dynamic Performance Evaluation
10
ACTUAL LM
..-- CONTROLLER RR
..._ W/O COMPENSATION
-- W/ COMPENSATION
_..,....... WIO COMPENSATION
-- WI COMPENSATION
_...... W/0 COMPENSATION -- W/ COMPENSATION
-- WIO COMPENSATION
-- WI COMPENSATION
- WIO COMPENSATION -- WI COMPENSATION
- WIO COMPENSATION -- WI COMPENSATION
66
1.25
:, 1. 00 a. ....,
0.75 ,..., 1.20 =>
0.95 >< => -' .... 0.70 ...... 1. 15 => 1.00 a. ..... 0 0.85 0:: 0 .... 0.70
1. 40 => 0.. 1. 15 ...., z LL. - 0.90
1. 1 ...... => 1.0 a. --z 0.9 ....
0.8 .-. 1. 15 => 1. 05
0.95 0.85
a. 0. 75 0. 1
=> -0. 0 !::, -0.1 z -0.2 w a.. -0. 3
'
.
I
! -/
----- - ·-
I
/
- -
-.,. l . -....,., .
.-/
I - ·- ·----·-
_/
I I I I I I
0 2 4 6 8 10
TIME (SECONDS)
Figure 23. Effects of Decrease in Mutual Inductance
Dynamic Performance Evaluation
ACTUAL LLf
....__ CONTROLLER RR
-- W/O COMPENSATION
- WI COMPENSATION
--. W/0 COMPENSATION
- W/ COMPENSATION
..-- W/0 COMPENSATION -- W/ COMPENSATION
-- W/0 COMPENSATION
-- WI COMPENSATION
--... W/0 COMPENSATION - WI COMPENSATION
-- W/0 COMPENSATION - WI COMPENSATION
67
1. t ..... o. 1. 0 .... 0:: 0::
0.9 ..... 1. 1 ::,
e; 1.0 >< ::, ....I 1.&.. 0.9
..... 1. 15 ::> o. 0. 90 .... 0 0.65 Cl:: 0 1- 0.40
1. 1 ..... ::> 0. 1.0 ..., z I.&.
0.9 2. 1 .....
::::> 1. 7 0. ..., z 1. 3 I-
..... . 0.9 2. I
:::, 1. 7 0. ..., z 1.3 < a. 0.9
1. 1
=> 0.7 0.
-; 0.3 I.a.I 0. -0. 1
.
.
'
-
-
I I I I
0 2 4 6 8 TIME (SECONDS)
-...... ACTUAL
-- CONTROLLER
-- WIO COMPENSATION
- WI COMPENSATION
.........._. WIO COMPENSATION
-- WI COMPENSATION
-- W/0 COMPENSATION
10
-- WI COMPENSATION
_,._ WIO COMPENSATION
-- WI COMPENSATION
_.,..... WIO COMPENSATION
-- WI COMPENSATION
-- WIO COMPENSATION -- WI COMPENSATION
Figure 24. Effects of Step Change in Torque Command
Dynamic Performance Evaluation 68
..... ::::::,
a. .._,
a:: a::
..... :::,
0..
X :::, ...J ..... --:::,
0.. .._, 0 0:: 0 I-
,.... ::::::,
0.. .._,
z ..... ---::::::, 0.. ..., z ....
t. 2
1. 1
t.O
0.9 1.0
0.7
0.4
0. 1 1. 8 L4 1.0 0.6 0.2 2.0 1. 6 1. 2 0.8 0.4 0.0 1. 2
0.8
- 0.4 i.o+=============================i --:::, 1. 6
1. 2 ~------z 0.8 < o.. 0. 4
""' 1 . 0 +=====:::'.::============::! => 0.6
0.2 -0. 2 ~------llllj!W ..., a. -0. 6 ..,_ ______ __, ____________ ..
0 2 4 6 8 TIME (SECONDS)
Figure 25. Effects of Step Change in Flux Command
Dynamic Performance Evaluation
10
-- ACTUAL
-- CONTROLLER
- WIO COMPENSATION
-- WI COMPENSATION
___... WIO COMPENSATION
-- WI COMPENSATION
-- W/0 COMPENSATION -- W/ COMPENSATION
-- W/0 COMPENSATION
-- W/ COMPENSATION
_....... WIO COMPENSATION
-- WI COMPENSATION
- WIO COMPENSATION -- WI COMPENSATION
69
1. l ..... ::::> 0. 1.0 ..... 0:::
o::: 0.9 1.4-t===================~ ..... .
::::> 1. 2 .....
1.0 ..J ..._ 0.8 1.2.n::================~ ::::>
t. 0.8 0 0::: 0 .... 0. 4 -1-------------+
1.4 -------------..... ::::> 1. 2 0. ...... z 1.0 .....
0.8 1. 1 --.
::::> 0. 1.0 ..... z .... - 0. 9 -----------
2. s -------------::::> 2.4 o. 2 .0 .._, 1. 6
1.2 o.0.81-------------•
1. 8 +--------------1" .,... ::::> 1. 4 a. 1. 0 .._., 0.6 a 0.2 a. -0. 2 \,,---.---.---r--.--T-.--.-.....--i-
0 2 4 6 8 10
TIME (SECONDS)
--.,._ ACTUAL
- CONTROLLER
- W/0 COMPENSATION
- W/ COMPENSATION
___..,. W/0 COMPENSATION
-- W/ COMPENSATION
-- W/0 COMPENSATION -- W/ COMPENSATION
___... W/0 COMPENSATION
-- W/ COMPENSATION
-- W/0 COMPENSATION -- W/ COMPENSATION
-- W/0 COMPENSATION -- W/ COMPENSATION
Figure 26. Effects of Step Change in Load Torque ( Speed Loop )
Dynamic Performance Evaluation 70
Thus, the proposed scheme is shown to have satisfactory transient response to both
the rotor resistance and mutual inductance variations. It is also shown that the
scheme does not affect the functioning of the primary torque control loop.
Dynamic Performance Evaluation 71
7 .0 Conclusions
The purpose of this thesis was to develop and verify a parameter compensation
scheme for the indirect vector controlled induction motor that would make the drive
system insensitive to any parameter variations and to make the new scheme inde-
pendent of the key motor parameters while keeping the complexity of the implemen-
tation to a low level.
The complexity of control of the induction motors was shown to be their main draw-
back for applications in servo drives. Use of the vector control scheme to transform
the induction motors into separately excited de motors for control purposes was dis-
cussed. The inherent parameter sensitivity of all indirect vector controlled induction
motor drives was highlighted. The effects of the parameter sensitivity were dis-
cussed. The parameter compensation schemes currently being used to overcome the
parameter sensitivity were presented and their limitations were identified.
A novel approach to the problem of parameter sensitivity was developed and inves-
tigated. The new scheme applies the principle of air gap power equivalence in order
to achieve the tuning between the controller and motor. It was shown by extensive
simulation that the proposed scheme has excellent steady state performance. The
dynamic response of the system was also found to be very good. It was also shown
that the system does not use any additional transducer for the purpose of sensing and
adapting to parameter variations.
Conclusions 72
The effectiveness of the compensation scheme was verified for changes in the rotor
resistance and mutual inductance of the motor. In steady state, the compensation
scheme tracked these changes and maintained a steady state error of zero in air gap
power error. Thus, the steady state performance was excellent. For transient re-
sponse, the scheme was optimized for response to rotor parameter variations and
hence responded very well to these changes. The oscillations were minimized and
steady state error was kept within a specified band. The time taken to reach steady
state was about half a second. The dynamic response to changes in mutual
inductance was also good and steady state error was zero. However, the controller
gain was optimized for response to the rotor resistance variations and hence there
were some oscillations and large settling time for mutual inductance variations.
7.1 Scope For Future Work
The next logical step is to implement the scheme on a vector controlled induction
motor drive system and evaluate its performance. This wou Id test the feasibility of the
system to adapt to the parameter variations in real time without using additional
transducers and extra computational time.
Conclusions 73
Bibliography
[1] F. Blaschke, "Oas Verfahren der Feldorientierung zur Regelung der Asynchronmaschine" , Siemens Forsch -u. Entwickl. - Ber.Bd.1,Nr.1, pp.184-193,1972 ( In German)
[2] T. lwakane, H. lnokuchi, T. Kai and J. Hirai, "AC Servo Motor Drive for Precise Positioning Control" , Conf. Record, IPEC, Tokyo, March 1983, pp. 1453-1464.
[3] K. Kubo, M. Watanabe, T. Ohmae and H. Kamiyama, "A software-based speed regulator for motor drives" , Conf. Record, IPEC, Tokyo, March 1983, pp. 1500-1511.
[4] T. Matsus and T. A. Lipo, "A rotor parameter identification scheme for vector controlled induction motor drives" , Conf. Record, IEEE-IAS Annual Meeting, Oct. 1984, pp. 538-545.
[5] R. Gabriel and W. Leonhard, "Microprocessor control of induction motor" , Conf. record, International Semiconductor Power Converter Conference, Orlando, 1982, pp. 385-396.
[6] L. Garces, "Parameter adaptation for the speed controlled static AC drive with squirrel cage induction motor" , Conf. Record, IEEE-IAS Annual Meeting, Oct. 1979, pp. 843-850.
[7] T. Ohtani, "Torque control using flux derived from magnetic energy in induction motors driven by static converter" , Conf. Record, IPEC, Tokyo, March 1983, pp. 696-707.
[8] Y. Yoshida, R. Ueda and T. Sonoda, "A new inverter-fed induction motor drive with a function of correcting rotor circuit time constanf' , Conf. Record, IPEC, Tokyo, March 1983, pp. 672-683.
[9] K. Ohnishi, Y. Ueda and K. Miyachi, "Model reference adaptive system against rotor resistance variation in induction motor drive " , IEEE Trans. on Ind. Elect., Vol. IE-33, Aug. 1986, pp. 217-223.
[10) R. Krishnan and P. Pillay, "Sensitivity analysis and comparison of parameter compensation schemes in vector controlled induction motor drives" , Conf. Re-cord, IEEE-IAS Annual Meeting, Oct. 1986, pp. 155-161.
[11) R. Krishnan "Analysis of electronically controlled motor drives" , Class Notes, VPl&SU, 1986.
Bibliography 74
[12] J. M. Loehrke, "A digital implementation of feedforward field-oriented control" , M. S. Thesis, University of Wisconsin-Madison, 1985.
[13] R. Krishna'n and F. C. Doran, "Study of parameter sensitivity in high performance inverter-fed induction motor drive systems" , Conf. Record, IEEE-IAS Annual Meeting, Oct. 1984, pp. 510-524.
[14] R. Krishnan and F. C. Doran, "A method of sensing line voltages for parameter adaptation of inverter-fed induction motor servo drives" , Cont. Record, IEEE-IAS Annual Meeting, Oct. 1985, pp. 570-577.
[15] James Singer, "Elements of numerical analysis", Academic Press, 1968.
[16] D. M. Brod and D. W. Novotny, "Current control of VSI-PWM inverters" , Conf. Record, IEEE-IAS Annual Meeting, Oct. 1984, pp. 418-425.
Bibliography 75
Appendix A. Induction Motor Parameters
5 hp, Y-connected, 3 phase, 60 Hz, 4 pole, 200 V.
R, = 0.277 Q
R, = 0.183 Q
Lm = 0.05383 H
L, = 0.0553 H
L, = 0.05606 H
J = 0.01667 Kg - m2
Appendix A. Induction Motor Parameters 76
Appendix B. List of Symbols
All symbols marked with an asterisk indicate the commanded/reference value of the
quantity.
a
ac,AC
B
dc,DC
E,
F
f.
Stator to Rotor Aspect Ratio
Alternating Current
Damping Factor
Direct Current
Voltage Drop Across Effective Rotor Resistance
Modified Reactive Power
Synchronous Frequency (Hz.)
i .. ,ibs,ic,,i.,I. Stator Phase Currents
ia Armature Current
i, Field (Flux Producing) Current
ir Torque Producing Current
i:.i:. d-q Axes Stator Currents
i~,i~, d-q Axes Rotor Currents
Im Magnetizing Branch Current
I, Rotor Phase Current
Ide DC-Link Current
IFN Normalized Flux Current
ITN Normalized Torque Current
J Moment of Inertia
Appendix B. List of Symbols 77
Kt Torque Constant
K, Flux Constant
L,, Rotor Leakage Inductance
L,. Stator Leakage Inductance
Lm Mutual Inductance
L, Rotor Self Inductance
L. Stator Self Inductance
N, Rotor Turns per Phase
N. Stator Turns per Phase
p Differential Operator
p Number of Poles
P. Air Gap Power
p err Air Gap Power Error
Pu Inverter Losses
P;n Input Power
psc Stator Copper Losses
PAN Normalized Air Gap Power
PEN Normalized Air Gap Power Error
R, Rotor Resistance per Phase
R. Stator Resistance per Phase
s Slip
Te Electrical Torque
TL Load Torque
T, Rotor Time Constant
Vdc DC-Link Voltage
Vs.Uaa.'>bs.UCsStator Phase Voltages
Appendix B. List of Symbols 78
X,, Rotor Leakage Reactance
X,. Stator Leakage Reactance
Xm Mutual Reactance
a Ratio of Actual to Commanded Values of Rotor Time Constant
Ratio of Actual to Commanded Values of Mutual Inductance
.!ii Hysteresis Current Window
0, Field Angle (Flux Position Angle)
0, Rotor Position Angle
0,, Slip Angle
Sr Torque Angle
u~.u~. d-q Axes Stator Voltages
<p, Flux
'I'd,, 'lj/q, d-q Axes Flux Linkages
'I'm Mutual Flux Linkage
'I', Rotor Flux Linkage
'Vs Stator Flux Linkage
w, Rotor Electrical Speed
w. Synchronous Speed
w,, Slip Speed
Appendix B. List of Symbols 79
The vita has been removed from the scanned document