A NOVEL PARAMETER COMPENSATION SCHEME FOR INDIRECT …

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.


if 0 I> ::-<Pf D. C. MOTOR

ia 0 I> t>T e

Figure 1. Schematic of a Separately Excited DC Motor.


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.


Rs JX1s JX1 r

+ [>- + t> + I~ Is Ir

vs JXm Em Er Rr s

Figure 2. Steady State Equivalent Circuit of an Induction Motor.


• 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


~~---'-----_.;..i>'-----------C> \1/r

Stator Reference Frame

Figure 3. Phasor Diagram of an Induction Motor.


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.


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.


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


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-


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)


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 )


(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)


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.


+ --t>

Is

Figure 7. An Equivalent Circuit of Induction Motor.


+

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.


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.


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-


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 :


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.


* -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)


(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)


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


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.


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,


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


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


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


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-


1 de

+ Tl T3 T5

t------t:,------ '---------· ._ ___ a -----b

T4 C T2

D4 D6 D2

Figure 12. The Inverter


START

RETURN

Figure 13. Flow Chart for Link Current Computation


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.


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.


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


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


(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


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.


- 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)


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.


-. ::> 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)


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.


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


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


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)


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.


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.


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


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.


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


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.


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


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


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


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


-...... 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


..... ::::::,

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 ..,_ ______ __, ____________ ..


Figure 25. Effects of Step Change in Flux Command


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


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



Figure 26. Effects of Step Change in Load Torque ( Speed Loop )


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.


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


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


The vita has been removed from the scanned document

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A NOVEL PARAMETER COMPENSATION SCHEME FOR INDIRECT …