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The voltage/current model in field-oriented AC drives at very

low flux frequencies

Citation for published version (APA):

Burgt, van der, J. J. A. (1996). The voltage/current model in field-oriented AC drives at very low flux frequencies. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR458508

DOI:

10.6100/IR458508

Document status and date: Published: 01/01/1996 Document Version:

Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication:

• A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website.

• The final author version and the galley proof are versions of the publication after peer review.

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in Field-Oriented AC Drives

at Very Low Flux Frequencies

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The Voltage

I

Current Model

in Field-Oriented AC Drives

at Very Low Flux Frequencies

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in Field-Oriented AC Drives

at Very Low Flux Frequencies

PROEFSCHRIFf

ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de Rector Magnificus, prof.dr. J.H. van Lint, voor

een commissie aangewezen door het College van Dekanen in het openbaar te verdedigen

op dinsdag 16 april1996 om 16.00 uur

door

JOSEPHUS JOHANNES ANTONIUS VANDER BURGT

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Dit proefschrift is goedgekeurd door de promotoren: prof.dr.ir. A.J.A. Vandenput

en

prof.dr.ir. P.P.J. van den Bosch

copromotor: Dr.-Ing. F. Blaschke

CIP-DATA KONINKLUKE BIBLIOTHEEK, DEN HAAG Burgt, Josephus Johannes Antonius van der

The voltage/current model in field-oriented AC drives at very low flux frequencies I Josephus Johannes Antonius van der Burgt.

-Eindhoven: Eindhoven University of Technology. -Fig., Tab. Thesis Technische Universiteit Eindhoven.- With ref. -With summary in Dutch.

ISBN 90-386-0120-4 NUGI 832

Subject headings: variable speed drives I AC drives ; field-orientation.

Druk: Universitaire Drukkerij TU Eindhoven Copyright © by Jos van der Burgt

All rights reserved. No part of this thesis may be reproduced, stored in a retrieval system or transmitted in any form, by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the author.

The author makes no warranty, that the methods, calculations and data in this book are free from error. The application of the methods and results is at the user's risk and the author disclaims all liability for damages, whether direct, incidental or consequential, arising from such application or from other use of this book.

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Preface

First of all I would like to thank my parents for taking care of me for the past 28 years and for giving me the opportunity to study. Further, I would like to thank all those who contributed in one way or another to my work.

I especially wish to thank my promotor Prof. Vandenput and my co-promotor Dr. Blaschke for our fruitful discussions. Dr. Blaschke showed me that "smart control" does not necessarily mean digital control, but rather advanced control on any (analog) control system.

A special word of thanks is directed to Sjoerd Bosga, my fellow "promovendus". He supported me in my theoretic surveys and together we built the experimental setup including the digital control system. He brought my attention to many small and large problems and most of the time also provided the proper solutions. I also thank him for his help with the writing and printing of this thesis: long live the Mac!

I wish to thank all members of the Electromechanics and Power Electronics group of the university for their support, be it scientific, technical or other. And finally I thank all students who have directly or indirectly helped me during the research.

Greetings to my family and my friends.

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Abstract

For field-oriented control of an induction machine it is necessary to determine the angle of the magnetic flux vector inside the induction machine with respect to the stator frame. When this is to be done without a flux sensor and without a rotor speed/position sensor, the voltage/current model, or uti-model, is the means of determining this flux angle.

The uti-model experiences drift problems, which are problematic especially at low flux frequencies. By analyzing the uti-model in field-oriented coordinates instead of fixed stator coordinates, the drift problem is transformed into a stability problem. The stability improvement is done by two feedback methods inside the uti-model: flux magnitude feedback and flux derivative feedback. The uti-model is first analyzed apart from the induction machine (basic analysis) and finally in a closed loop with the induction machine.

With flux magnitude feedback a steady-state angle and magnitude error will occur. This error increases with decreasing flux frequency. It can be eliminated by utilizing the flux command value in the feedback loop. The steady-state error does not occur with flux derivative feedback. For high flux frequencies, flux derivative feedback with constant gain is the best solution. At low frequencies flux magnitude feedback with constant gain should be used. At very low frequencies however, the flux magnitude feedback must be extended by an additional feedback gain, which results in vector instead of scalar feedback. This extension considerably improves the steady state behaviour and the stability of the uti-model. Finally, at very low frequencies the two flux magnitude feedback gains must be taken in function of the speed and the load of the induction machine.

The two internal feedback methods still leave an operation area to be solved: the area around zero flux frequency. In this small frequency range the high-frequency magnetizing current injection method can provide a way of determining the flux angle, even at zero flux frequency. This method is based upon saturation effects inside the induction machine and its basic functioning is introduced in this thesis. Further research should contain the complete theoretical background and prove and optimize its practical utilization.

An important issue that is not fully studied in this thesis is the effect of parameter detuning, especially stator resistance detuning. The negative effects (in steady state and dynamically) of stator resistance detuning seem to be minimized by flux magnitude feedback with proper feedback gains. This phenomenon is still to be studied in more detail.

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Table of contents

List of symbols

1 Introduction

1.1 Variable speed control

1.2 Field orientation of induction machines 1.3 Motivation of the thesis

1.4 Outline

2 Model of the induction machine

xiii

1

1 2 3 5 2.1 Basic equations of the induction machine; notation and assumptions 5

2.2 Choice of the reference frame 7

2.3 Equation system of the IA 8

2.4 Equation system of the U A 14

2.5 Steady state of the IA 17

2.6 Steady state of the UA 19

2. 7 Voltage I frequency control of the U A 21

3 Field orientation

3.1 Introduction

3.2 Variable frequency control; open loop constant current control 3.3 Flux control

3.4 Vector control; field-oriented control 3.5 Determination of the flux angle <ps

4 The uli-model

25

25 25 28 31 32

37

4.1 Fundamental structure of the u/i-model 37

4.2 Equation system of the u/i-model in the stator flux reference frame 39 4.3 Linearized model of the stator flux calculator 42 4.4 Change of state variables in the linearized model 45

4.5 Simulation of the stator flux calculator 48

4.6 Further analysis of the linearized model 52

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4.8 Linear analysis of the SFC containing flux magnitude feedback 61

4.8.1 Ideal case:

[iiis]o

=

'IJ:

61

4.8.2 General case:

[iiis1

pi

'IJ;

63

4.9 Simulation of the SFC containing flux magnitude feedback 66 4.9.1 Ideal case:

[iji

8 ]

0

='I';

66

4.9.2 General case:

[iiis]o

pi

'IJ;

71

4.10 Stabilizing feedback with the derivative of the flux magnitude in the

SFC 74

4.11 Linear analysis of the SFC containing flux derivative feedback 76 4.12 Simulation of the SFC containing flux derivative feedback 80

4.13 Comparison of the feedback loops with d and S2 84

4.14 Behaviour of the feedback loops at zero flux frequency 89

4.15 Conclusion 91

a) Linearized uli-model in stator flux coordinates 91

b) Flux magnitude feedback 91

c) Flux derivative feedback 92

d) Combination of flux magnitude feedback and flux derivative

feedback 92

5 The current-fed induction machine controlled

by

the u/i-model

95

5.1 Simplified model of the lA controlled by the uli-model 95 5.2 Linearized model of the lA controlled by the u/i-model 102

5.3 Simulation of the lA controlled by the uli-model 104

5.4 The u/i-model in "closed loop" containing flux magnitude feedback 108

5.5 Linear analysis of the u/i-model in "closed loop" containing flux

magnitude feedback Ill

5.6 Flux magnitude feedback according to Ohtani 112

5. 7 Extension of the flux magnitude feedback loop I 1 4

5.8 Linear analysis of the extended flux magnitude feedback loop 115 5.9 Simulation of the lA control with flux magnitude feedback 119 5.9.1 Idealcase:

l=l

=::>

[iiis]o='i';=['i's]o

120

5.9.2 General case:

i

,..1 =::>

[iiis]o ,..

'IJ; ,.. [ 'IJ

8 ]0 125

5.10 The u/i-model in "closed loop" containing flux derivative feedback;

linear analysis 129

5.11 Relation between the flux derivative feedback vector (S 1, S2) and the

operation point of the machine 131

5.12 Simulation of the lA control with flux derivative feedback 136 5.13 Combined feedback of the flux magnitude and the flux derivative 140

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Table of contents

5.14 High-frequency magnetizing current injection: sensorless flux determination based upon saturation effects

5.14.1 Basic method based upon saturation effects 5.14.2 Fundamental measuring method

5 .14.3 Measuring method in practice using the stator voltage 5.15 Conclusion

6 Real-time simulations and experiments

6.1 Introduction 6.2 DSP system

6.3 Experimental drive setup 6.4 Software

6.5 Real-time simulations and experiments 6.6 Experimental results in the "open loop" case

6.6.1 No internal feedback in the uli-model 6.6.2 Flux magnitude feedback with gain d

a) Ideal case:

[ilis]o

==

w:

b) General case:

[ilis]o ..

w:

6.6.3 Flux derivative feedback with gains (S1, S2)

141 142 145 148 150

153

153

154 156

158

159 160 162

165

165 169 172 6.6.4 Combination of flux magnitude and flux derivative feedback 176 6.7 Experimental results in the "closed loop" case 176 6. 7.1 No internal feedback in the u/i-model 176 6.7.2 Flux magnitude feedback with gains (S3, S4) 178

a) b)

General

Flux magnitude feedback at very low frequency

178

181

c) Main inductance error:

i ..

l 187

6.7.3 Flux derivative feedback with gains (Sl, S2) 190 6. 7.4 Combination of flux magnitude and flux derivative feedback 192 6.8 High-frequency magnetizing current injection 193

6.9 Conclusion 196

6.9.1 "Open loop" system 196

6.9.2 "Closed loop" system 197

7 Conclusions and recommendations

7.1 Conclusions

7 .l Recommendations and prospects

201

201 203

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A Per unit system (PUS)

207

B

Transformation of voltages and currents from

2

to

3

phases and

reverse

209

C Machine parameters

211

References

213

SUID~Dary

217

1 The u/i-model in flux coordinates 217

1 a) Drift problem in the uti-model 217

1 b) The uti-model in flux coordinates: stability problem 217 1 c) Flux derivative feedback versus flux magnitude feedback 218 2 Results of "open loop" analysis

3 "Closed loop" analysis 3 a) Simplified model 3 b) High frequency analysis 3 c) Low frequency analysis 3 d) Zero flux frequency 3 e) Resulting vector field plot 3 f) Steady-state error and saturation

4 High-frequency magnetizing current injection

5 Parameter detuning

Sam en vatting

Curriculum vitae

218 219 219 220 220 221 221 222 222 223

225

227

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List of symbols

1. Abbreviations AC C/P CSI DC lA p.u. PUS PIC RFC SFC UA uti-model VSI

vvc

2. Notation Alternating Current

Cartesian to Polar transformation Current Source Inverter

Direct Current

Current source inverter fed Asynchronous machine per unit

Per Unit System

Polar to Cartesian transformation Rotor Flux Calculator

Stator Flux Calculator

Voltage source inverter fed Asynchronous machine voltage/current model

Voltage Source Inverter Voltage Vector Calculator

General description of the notation used in this thesis: at;i a == quantity;

b =system in which the quantity originates (stator, rotor, air gap); c

=

reference coordinate system from which the quantity is viewed; i

=

l: parallel component; i = 2: perpendicular component.

x scalar quantity

x vector quantity

x

*

command value of variable x

x

estimated value of variable or parameter x (in VVC) x model value of variable or parameter x (in u/i-model)

[ x

J

value of variable x in the operating point where linearization is done [ x ]0 stationary value of variable x

[ x lnit initial value of variable x x time derivative of variable x

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3. Symbols

u voltage

i currrent

e time derivative of a flux linkage

'lj.l flux linkage (abb.: flux)

r resistance

1 inductance

m torque

t time

a phase angle of a voltage vector

E phase angle of a current vector

<p phase angle of a flux vector

p rotor position angle

e

moment of inertia

Sl, S2, S3, S4 feedback gains in the u/i-model d = S3 (used in chapter 4)

j imaginary unit

A System matrix of linearized model

R Rotation matrix

4. Superscripts (coordinate system)

s sl,s2

r

rl,r2

stator coordinate system

stator sl (parallel), s2 (perpendicular) component of an equivalent two-phase system

rotor coordinate system

rotor rl (parallel), r2 (perpendicular) component of an equivalent two-phase system

stator flux coordinate system

parallel and perpendicular component with respect to the stator flux vector rotor flux coordinate system

magnetizing component, i.e.:

parallel component with respect to the rotor flux vector torque-producing component, i.e:

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5. Subscripts

s stator winding system r rotor winding system (cage) 1 linked with the air gap

el electromagnetic

load load

!.A. magnetizing

cr leakage

6. Special symbols

cr8 = relative stator leakage inductance

1

crr = 1or relative rotor leakage inductance

1

10 = 108 + (lor ) total leakage inductance of the induction machine l+o,

7.Examples

l

08 estimated value of the stator leakage inductance

List of symbols

u~sl u/i-model value of the stator voltage component parallel with the stator flux vector

i~ rotor current vector in the stator reference frame

i~rZ* command value of the stator current torque-producing component

~ ~

stator flux angular velocity in the stator reference frame

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CHAPTER

1

Introduction

1.1

Variable speed control

For more than 20 years there is a competition between DC machines and AC machines in variable speed drive applications. Previously the DC drives, especially the separately excited DC machine, were dominating the field of variable speed drive systems. The DC machine is nowadays still an appropriate solution for special, highly dynamic, four-quadrant operation. The aim of the development of variable speed AC drives has always been to match the DC drive performance in a certain operation area, starting from rated frequency and advancing to lower frequencies and finally zero frequency.

The major reason to change from a DC to an AC machine is the difference in mechanical construction: the DC machine has a mechanical commutator that needs periodic maintenance and, in general, is vulnerable to external influences. This commutator also limits the amount of overspeed and overload that is permitted. AC machines in general and induction machines in particular have a much more robust construction, because they do not have a commutator. The induction machine is the most robust electrical machine.

The disadvantage of the induction machine is the fact that it needs a more complicated controller to reach the performance of variable speed DC drives. A lot of research has been done in the last decades to develop variable speed controllers for induction machine drives. Very important contributions to the progress of these controllers were the increasing developments in power electronics and digital processors, enabling arbitrary voltage and current shapes and frequencies and complex (real-time) control algorithms. With these tools an endless variation of variable speed controllers has been manufactured, from simple voltage/frequency control to complex model based control systems with parameter adaptation.

1.2

Field orientation of induction machines

One of the basic principles that is found in most induction machine controllers is the concept of field orientation. With field orientation the control signals (voltage and/or current) are

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orientation the flux vector in the induction machine and especially its angle has to be known. This flux angle can be found in three different ways:

1) By m~suring the magnetic field inside the machine with Hall sensors or sensing coils. These sensors are mechanically vulnerable and costly to install.

2) By a machine model, the so called current model, fed by the stator current and the rotor speed or angle. This requires a rotor shaft encoder that is often undesired, because it is an extra mechanical component with separate wiring.

3) By a machine model that only needs the stator voltage and current as its inputs. This is the voltage/current model or uli·model and enables a sensorless field·oriented control, because no additional flux or speed/position sensors are necessary.

The sensorless field-oriented control by means of the voltage/current model is the ideal way of highly dynamic induction machine control. The field orientation enables a simple decoupling control like the DC machine control and serves three purposes: to maintain a constant flux level; to limit the current; to stabilize the AC machine.

1.3

Motivation of the thesis

It is generally known that the voltage/current model, or uti-model, in its original form has a poor performance at low frequencies, because of drift problems, parameter sensitivity and measurement error sensitivity. Therefore the sensorless field-oriented control of induction machines is not used in the low frequency range. A simple alternative in this frequency range is the current model with a shaft encoder. The sensorless aspect is lost, but the performance is superior to the u/i-model at low frequencies and zero frequency control is possible. A sensorless alternative could be the extension of the simple uli·model with complex additional models. This model extension is combined with complex control algorithms, like observers, adaptive control, etc.

The aim of this thesis is to find a relatively simple solution to the problems of the u/i-model at low frequencies. The solution is sought in the internal structure of the u/i-model. This has led to two kinds of feedback inside the u/i-model instead of complex control structures around the u/i-model.

The analysis tools that are used are: modelling of the uli-model and the field-oriented control system; linearization of these models and eigenvalue analysis of the linearized models; simulation of the non-linear systems; experiments on a field-oriented induction machine drive system.

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Introduction

1.4

Outline

Chapter 2 introduces the equation system and the block diagrams of the current-fed and voltage-fed induction machine. First the simplifications and assumptions that are used in the uti-model analysis are described.

Chapter 3 gives a short summary about induction machine control. It shows the disadvantages of scalar control and the benefits of field-oriented control.

fu chapter 4 the uti-model is introduced. It is analyzed using eigenvalue analysis and digital simulation. The uti-model is only considered as a flux estimator without feedback to the induction machine control. This simple structure enables a comprehensive analysis of the basic functioning of the uti-model. fu this chapter the two feedback loops, that should improve the performance of the uti-model, are presented.

The u/i-model is further analyzed in chapter 5. fu this chapter the uti-model is part of the complete field-oriented control system of the induction machine, this means that it is functioning in a closed loop with the induction machine. The results of the "open loop" analysis of the former chapter are used as a starting point for the "closed loop" analysis in this chapter. The analysis is focused on the low frequency behaviour.

Chapter 6 presents the experimental verification of the theoretic analysis. A comparison is made between real-time simulations and experiments. This chapter is divided into two parts. The first part consists of a comparison with chapter 4, the second part verifies chapter 5.

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CHAPTER

2

Model of the induction machine

2.1

Basic equations of the induction machine;

notation and assumptions

In general, the induction machine is used to transform electrical energy to mechanical energy or vice versa. The induction machine consists of electric circuitry, electromagnetic circuitry and electromechanic circuitry. The main objective of the induction machine modelling is to build a simple but sufficient model that describes these circuits and their interconnections. The main path in the model is the relation between the voltage and/or the current of the stator phases on the input side, the magnetic flux inside the machine and the electromagnetic torque on the output side. Normally, the induction machine is a three-phase machine with a squirrel-cage rotor. The model that is chosen in this thesis is the symmetric orthogonal two-phase equivalent model. This model can be translated back to the three-phase squirrel-cage machine, but also to a symmetric machine with an arbitrary number of stator and rotor phases. The transformation from the symmetric three-phase system to the symmetric two-phase system and vice versa is described in appendix B.

The assumptions and simplifications that are made in modelling the induction machine are listed below:

• The model is a symmetric orthogonal two-phase model without homopolar components; • The air gap has a constant width: effects of saliency and stator and rotor slots are

neglected;

• There is no magnetic saturation nor hysteresis; the magnetic conductivity of the iron is infinite;

• The equivalent number of turns in the stator and the rotor windings is equal; • Space harmonics of the winding systems are neglected;

• The leakage inductance of the windings is split into an equal stator and rotor part; • The number of pole pairs is one.

The induction machine can either be supplied by a voltage source or a current source. These sources are assumed to be ideal, this means that they can supply any desired voltage or current without losses. The current-fed induction machine (or current-fed asynchronous machine) is abbreviated lA and will be discussed in section 2.3, the voltage-fed induction

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machine (or voltage-fed asynchronous machine) is abbreviated UA and is presented in section 2.4.

The equations are written in the per unit system (PUS). This has the advantage that all variables and equations are without units. Furthermore, the parameters of the machine do not change too much with the power range. The PUS that is used here is described in appendix A.

The notation of the machine variables is somewhat different from the common notation in the literature. This notation is chosen to build a consistent set of variables. The details are found in [Blaschke'96]. The basic form of the notation is described in the list of symbols found at the beginning of this thesis. At the end of this list some examples are given.

In the modelling the modular model structure of [Blaschke'96] is copied. The basic equations of the induction machine are the stator voltage equation (2.1 ), the rotor voltage equation (2.2), the equations of the flux linkages of the stator and the rotor winding systems, (2.3) and (2.4), and the torque equation (2.5}.

S •S 0 S

Us= rs1s

+'II'

s (2.1)

r O •r • r

Dr "' = rrlr

+ 'II'

r (2.2)

'II': ='II'!

+ lcrsi: (2.3)

'II'~==

'11'1

+

lcrri~ (2.4)

[ (n) a

r

·a

mel= R 2 '11'1 Is (2.5)

The equations are given in vector format. Every vector variable (written in bold) has two components: the first component is parallel with the reference frame axis and the second component is perpendicular to the reference frame axis. The symbols are: u voltage, i -current,

'II'

flux linkage, r- resistance and 10 - leakage inductance; subscript s indicates the stator winding system, r the rotor winding system; superscript s indicates the stator reference frame (stator coordinates), r the rotor reference frame and a any arbitrary reference frame;

'11' 1 is the magnetic flux in the air gap of the machine: the air gap flux; mel is the

electromagnetic torque;

R(-T)

is a rotation matrix over n/2 radians; the dot

t>

indicates a time derivative. All symbols are described in the list of symbols.

The stator voltage equation is restricted to the stator reference frame because of the differentiation of the stator flux vector, the rotor voltage equation is however restricted to the

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Model of the induction machine

rotor reference frame because of the differentiation. The flux equations and the torque equation can be defined in any reference frame, denoted by superscript a.

Before these equations are completed to obtain the induction machine model, the reference frame of the model has to be chosen. Some considerations about this are given in the next section.

2.2

Choice of the reference frame

The reference frame of the simulation model must be a coordinate system in which all quantities in a steady state are at rest, i.e. in which all quantities, appearing as DC quantities, are constants. The direction of the stator or rotor current vector and the direction of any flux vector may be considered as reference axis, because these vectors are rotating at the same speed, i.e. synchronously, with respect to each other in a steady state. One should not choose the rotor axis or the stator axis as the reference axis, because all vectors are moving with respect to the rotor and the stator in a general and also in a stationary operation condition.

The second consideration in choosing the reference frame is the desired decoupling of the cartesian coordinates of the stator current vector. This decoupling of the stator current components means that the two components can be controlled separately and independently of each other. This way a control strategy similar to the DC machine control can be achieved. Figure 2.1 shows some of the different reference axes that can be chosen.

Figure 2.1. Different coordinate axes in AC machine modelling.

The flux linkage of the rotor windings, or abbreviated rotor flux, can be considered as a central quantity because this flux is responsible for producing the current in the rotor windings (the rotor current) and for producing the electromagnetic torque, which will be

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shown in the .. next section. Therefore it is reasonable to choose the rotor flux axis as the reference coordinate axis, see also [Blaschke'74] and [Blaschke'96].

Zhu shows that only in the rotor flux reference frame the two stator current components are completely decoupled [Zhu'93]. This is not the case when choosing the stator flux axis or the air gap flux axis as the reference axis (however, simple measures can overcome this problem, see also [Doncker'94]). This simple decoupling strategy is another reason to choose the rotor flux reference frame.

The rotor flux vector appears in its own coordinate system only with its value as its first coordinate; the second coordinate is zero. Therefore, the superscript ljJrl can be omitted for the sake of simplicity. The rotor flux can be written as:

'i''ljlr

=

(ljJ~rll

=

(lPr)

r ljJ~r2 0 (2.6)

The stator flux is always

1J1~r

= [

ljJ~rl ljJ~r

2

r

To emphasize the coordinate transformation to the rotor flux reference frame, the cartesian field-oriented coordinate parallel with the rotor flux vector is called the magnetizing

coordinate and the one perpendicular to the rotor flux vector the torque-producing

coordinate. It must be stressed that these coordinates (with respect to the induced voltage of the machine) do not correspond with the generally known reactive and active components of the stator current (with respect to the terminal voltage of the machine) which are used to determine the power factor at the machine terminals.

2.3

Equation system of the lA

The inputs of the current-fed induction machine (lA) model are the stator currents. The three stator currents of a three-phase induction machine are transformed into an orthogonal phase system connected with the fixed stator (see appendix B). Inside the lA model the two-phase stator currents are transformed from the stator reference frame to the rotor flux reference frame. This way the magnetizing component and the torque-producing component of the stator current are obtained. These field-oriented stator currents are the input variables of the IA model. Output variables are the rotor flux inside the machine, the slip frequency of the rotor flux, the angle of the rotor flui with respect to the stator reference frame (the field orientation angle), the electromagnetic torque, the rotor frequency and angle and the stator terminal voltages.

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Model of the induction machine

The equation system of the lA in vector notation starts with the voltage vector equation (2.1) of the stator windings (in stator coordinates) and the voltage vector equation (2.2) of the rotor windings (in rotor coordinates). The stator flux linkage (abb. stator flux) and the rotor flux linkage (abb. rotor flux) are defmed in (2.3) and (2.4).

The total magnetizing current i~1 is defined as the air gap flux divided by the main inductance 1:

~1 = ~ +i~

With (2.4) an alternative magnetizing current i~r can be defined:

11'~ = l·i~

·a

:a

(l

)"a

•t-tr = "s + +Or lr where lor O r = -1

is the rotor leakage factor.

The mechanical equations of motion are:

Ps= jpsdt

with the electromagnetic torque from (2.5).

(2.7) (2.8) (2.9) (2.10) (2.11) (2.12) (2.13)

The superscript a means that the equation is valid in any reference coordinate system. Such an equation (without a derivative) can be copied to the rotor flux reference frame simply by substituting a by 'lj>r_

The rotor flux vector becomes a scalar in its own reference frame:

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Transformation of the derivative of the rotor flux vector from rotor coordinates (equation (2.2)) to rotor flux coordinates is done by rotating this vector over the angle -<p~ (minus the angle of the rotor flux vector with respect to the rotor reference frame). This leads to:

,j,

~= gt (

R(

<p~

)'1'

~r)

=

R(<pn,p~r+~~·R(<p~)R(~)'~'~r

=

R(<p~)( ~r) +~~-

R(<pn(

:J

=

R(

<p~

)( •

~

r ]

<pr"'ll'r

(2.15)

With the help of the last equation, vector equation (2.2) yields in field oriented coordinates:

(2.16)

(2.17)

This means that the magnetizing rotor current (i.e. the induced magnetizing current component in the rotor windings) originates from the variation of the flux value and the torque-producing rotor current originates due to the rotation of the flux vector with respect to

the rotor. The frequency ~ ~ of the rotor flux with respect to the rotor is called the rotor flux

slip frequency.

The rotor flux magnitude is calculated by integrating the first equation of (2.17):

(2.18)

Equations (2.9) and (2.1 0) yield the relation between the rotor current and the stator current:

1

-i'lj.lrl

=

1 ( i'ljlrl _

~)

r ~s 1 -'ljlt2 I -'ljlr2 -lr

=

l+a, ls (2.19)

(29)

Model of the induction machine

With (2.4) the electromagnetic torque of (2.5) can be written as a function of the rotor flux: m _1·\jJrl," ljJr2 + 1·1jJr2,1, ljJrl

el = s 't'l s 't'l

_ -iljJrl(o _1 iljJr2) + iljJr2(•" _1 iljJrl)

- s or r s 't'r or r

(2.20)

This can be simplified with (2.19):

= iljJr2.11 _ lor iljJr2 'tlJr

s 't'r t+o, s 1 (2.21)

_ _ l_·ljJr2

- l+a 1s 'tlJr r

When the rotor flux slip frequency is added to the rotor frequency the rotor flux frequency with respect to the stator is obtained:

(2.22) This frequency is integrated to obtain the field orientation angle <p;:

(2.23) This angle is used to transform the input current vector from the stator frame to the rotor flux reference frame:

(2.24)

with the rotation matrix

(2.25)

The equations mentioned above belong to the equation system of the current fed induction machine or IA. A block diagram of the IA has been drawn in figure 2.2.

(30)

r---;:;.,- ---,

:

0.

:

I

~I

I I I I I I I I ·0.~---+---,

]

le

I I I I I I I I I I I I

i=

L--- ~---I I I I I I I I I I I I I I I I I I I I I I I I I I I I

'

I

L---...

"''"'

&

(31)

Model of the induction machine

The stator voltage of the induction machine is calculated in a supplementary block added to the lA. This block is the voltage vector calculator or VVC. In the VVC the stator voltage is calculated from the rotor flux and the stator current in rotor field oriented coordinates. The calculation starts from the stator voltage equation (2.1 ):

(2.26) With (2.4) and (2.3) this yields:

S •S d ( S 1 •S 1 •S)

Us = rsls + dt

lJt

r - orlr + osls (2.27)

This equation is transformed to rotor flux coordinates by the rotation over -qJ~ (minus the field orientation angle (2.23)). First the equation is rewritten:

After the rotation over -qJ~ this yields:

ljJr •\j)r • ljJr !ljJr l !'ljlr • sR(:n:lf ljJr ·\j)r 1 •\j)r]

Us =rsls +"i'r -lorlr + osls +qJr 2JL"i'r -lor1r + osls

Split into cartesian coordinates, this equation yields:

I

Ug'ljlrl

=

rsir1

+~

r-lor

i;.vr

1+ las Itrl_

~;(

-lcrit2 +lasir2)

l

Ug\jl'2

=

rsitr2 -lor

i

r2 +las

i

r2 +

~;('\j!

r

-loci~rl

+

las~rl)

And with (2.19) to eliminate the rotor current:

After rearranging, this equation finally becomes:

(2.28)

(2.29)

(2.30)

(2.31)

(32)

u ljJrl r.iljJrl +

1P

r

s "' s s l+cr,

(2.33)

By defining the total leakage inductance transformed to the stator side (abbreviated: the leakage inductance)

(2.34) the stator voltage equation in rotor flux coordinates finally becomes:

(2.35)

The stator voltage vector in stator coordinates is obtained by the rotation over the field orientation angle <p~:

(2.36)

The VVC is shown in figure 2.3.

2.4

Equation system of the UA

The induction machine supplied by a voltage source is the voltage fed induction machine (or voltage fed asynchronous machine, abb.: UA). The UA model looks very much like the IA model with the VVC. The IA model is again used as part of the UA to calculate the rotor flux and its slip frequency from the field oriented stator current. But in the UA the stator current is not an independent variable but depends on the stator voltage that is now the independent input variable of the UA model.

In the UA the equations of the VVC have to be rewritten to produce the stator current as a function of the stator voltage. Therefore, (2.35) changes into:

(33)

Model of the induction machine

This can be rearranged into:

j'l'rl=_l_[u'lflrl_ri'~'rl_ ~rl

+.;:,si'\jlr2

s l s s s l+a "'r s

cr r

j'l'r2= -•-[u'\jlr2 -tgi'\jlr2 _

'i'r~~]-~si'\jlrl

s la s s l+a, r s

The block diagram of the U A has been drawn in the figure 2.4.

~r

'i'r

rs t - - - , I 1

vvc:

L---J

• s tJ.lr

Figure 2.3. Block diagram of the Voltage Vector Calculator (VVC).

(34)

r---·-·---1

=

---,

I I I I I I I I I I I I I I I

---

---

---~ r -1 I I I I I I I I I I

!S:

~~

~---~---

....

"t"'

---,

I I I I I I I r I I r I I I ---~

(35)

Model of the induction machine

2.5

Steady state of the lA

The steady state of the machine is denoted by square brackets around the variables and subscript 0 (zero): [ ]o. In the steady state of the IA the rotor flux magnitude and the rotor flux slip frequency are constant:

J [

'l.jJ r ]0 = constant

l[

<P

~

1

=constant

(2.39)

Therefore the rotor flux derivative is zero:

(2.40) This means that the rotor magnetizing current is zero (equation (2.17)) and the stator magnetizing current is equal to the main magnetizing current (equation (2.19)):

(2.41)

The rotor magnetizing current being zero means that the current vector of the rotor system [ ir

1

is perpendicular to the main magnetizing current vector [

i~tr

]

0 and the rotor flux vector [

1J1

r )0 .

The stator and rotor torque-producing currents follow from (2.19), (2.17) and (2.39):

(2.42)

Instead of the rotor flux slip frequency also the torque-producing stator current can be taken as an independent steady-state constant. In that case, (2.42) becomes:

[j'IJ!r2] r 0 = -=L[i'ljlr2] l+o, s 0 (2.43)

(2.44)

With a constant magnetizing and torque-producing component of the stator current also its magnitude and its angle with respect to the rotor flux are constant:

(36)

called the loading angle. Because this angle is constant in steady state, its derivative is zero: (2.46)

(2.47) this leads to:

[ •r]

"so= IPro+

[•r] [•'ljlr] Es

o"" IPro

[•r] (2.48)

This means that the slip frequency of the stator current (with respect to the rotor) is equal to the slip frequency of the rotor flux.

A common, simple control strategy is to maintain the stator current magnitude at a constant value. The steady-state values of the magnetizing and torque-producing current components, the flux magnitude, the electromagnetic torque and the slip frequency as a function of the loading angle are in that case:

{

[i~rll

=[is]o

·cos(e~r]o

[ir

2 ]

0 =

[i

8 ]0 ·sin[

e~r

]0

[m 1] "'

1

l·[i ]2 ·cos[e'ljlr] ·sin[e'ljlr] = ! l ·[i ]2 ·sin[2e*]

e 0 1 +Or s 0 s 0 s 0 1 +Or s 0 s 0

[ • r] [ • r]

r [ 'ljlr] Es 0 •

IPr

0

=

(1+~r}·l

tan Es 0 with (2.49) (2.50) (2.51) (2.52) (2.53)

This case is discussed in more detail in section 3.2 of chapter 3. This case results in a varying flux magnitude in function of the loading angle as can be seen in (2.50). A much more preferable situation is the case in which the flux magnitude is kept constant. In that case the steady-state values of the field oriented stator current components are:

(37)

Model of the induction machine

(2.54)

(2.55)

And the steady-state torque is:

[m

I]

= - 1

-·[1J1 ]

·[i'ljlrZ] =

[1!Jr]o

2

·tan[e'ljlr]

e 0 l+Or r 0 s 0 l(l+or) s 0 (2.56)

When the steady-state current and flux values have been obtained the steady-state voltage values of the VVC can be calculated. This is done in the next section. This yields the terminal voltages of the IA in steady state. These are the same as the steady-state voltages of the UA, if a constant flux value is taken as the basic condition.

2.6

Steady state of the UA

In the steady state of the UA the rotor flux magnitude and the rotor flux slip frequ_ency are assumed to be constant, like in the case of the lA. According to the same calculations as in the case of the lA, also the field-oriented stator current components and consequently the electromagnetic torque and the current slip frequency are constant. The steady-state values in the IA and the UA are therefore the same, because in both cases a constant flux is assumed.

In the UA the steady-state stator voltages have to be calculated from the current and flux values of section 2.5. This is done by using the stator voltage equations of the VVC.

Equation (2.35) of the VVC results with (2.40) and

{

[ itri]o

= 0

(2.57)

[ itr2]0

= 0

in the steady-state stator voltage equations:

(2.58)

The factor-( 1 ) can be eliminated from [utr2] of (2.58) with (2.41):

(38)

(2.59)

This means that the steady-state perpendicular component [ ut2 ]

0 of the stator voltage vector

is independent of the rotor leakage inductance lor .

Because all quantities in these equations to the right of the equal signs are constants, (

u~rl

]

0

and [

u~rz ]

0, i.e. the field oriented cartesian coordinates of the stator voltage vector, are

constant in the steady state. Therefore the polar coordinates of the stator voltage vector with respect to the rotor flux vector are also constant:

{

[ u8 ]0 = constant

[

a.~r

]

0 = constant

(2.60)

And because the angle is constant, the frequency is zero:

r~~rl-o

(2.61)

The relation between the stator voltage slip frequency :X~ and the rotor flux slip frequency ~ ~ is (equivalent with (2.47)):

•1jlr

(•s •s) •r •r •r

a.s • a.s-P -!pr=a.s-IPr

With (2.61) this leads to an equality in the steady state:

And therefore (with (2.52)):

[ • r] a.s 0

=

[ • r] Es 0

=

[ • r] IPr 0

=

(l+~r)·l tan

r [ 1jlr] Es 0

From (2.62) and (2.61) it is also clear that:

(2.62)

(2.63)

(2.64)

(2.65) This means that not only the current vectors but also the stator voltage vector and the rotor flux vector rotate in synchronism.

Now the stator voltage components [u~r1

]

0

and [u~r2

]

0

can be calculated in the steady state from (2.59):

(39)

Model of the induction machine

[u~rl]o·['Vr]o{~

-

[~:]o·tan[e~r1{os+(l:~r))}

[

u~r

2

]

0

= ['Vr ]0 { ~ ·tan[

E~r

]

0 + [

~:]

0

·(1 + os)}

(2.66)

(2.67) From these cartesian coordinates the polar coordinates can be calculated, using two new variables Ot and Oz:

o

1 ·{rs _

[~s]

·tan[E"'r] ·(o

+~)}

1 s 0 s 0 s

(1

+Or) (2.68)

02·{~·tan[Et1

+

[~~]

0

·(1+os)}

(2.69)

[ us]o •

['Vr]o~(Ot)

2 + (02)2 (2.70)

(2.71)

2.7

Voltage

I frequency control of the UA

The most simple way of supplying the UA is to feed it with a voltage vector with a constant frequency :X~ and a constant magnitude us. The voltage frequency determines the speed of the motor (if the slip is neglected) and the voltage value determines the flux value in the machine according to (2.70), that can be converted into:

['V ] _ [us]o

r 0-

~(Ot)2

+ (02)2

(2.72)

The factor

t/

~(0

1

)

2 +(02)2 in this equation depends on the voltage frequency and on the

loading angle [

e~r

]

0 (and therefore on the load), according to (2.68) and (2.69). For a

machine with per unit parameters (see Appendix A for the per unit system):

r8 = 0.04 1 =3

10s = 0.15 ==:> 08 = 0.05

10r

=

0.15 ==:>Or

=

0.05

and a maximum tangent of the loading angle (with (2.53)):

(40)

the relation between flux and voltage of (2.72) can be simplified into:

['i'r]o...

[us]o

t.o5·l[

<X~11

This is valid as long as the frequency

~ ~ is large compared to { ·tan[

e~r

]

0 ... 0.04.

Equation (2.75) can be rewritten into the next equation:

where'$; is the command value of the rotor flux.

(2.74)

(2.75)

(2.76)

This means that the voltage has to change linearly with the frequency to maintain a desired constant flux in the machine. This is the so called voltage/frequency control. The performance of this control scheme follows from the combination of (2.76) and (2.72) which yields the actual machine flux magnitude:

{2.77)

A plot of this equation as a function of the loading angle with the stator voltage frequency as a parameter and '$;=I is shown in figure 2.5a. From figure 2.5 it is clear that the voltage/frequency control functions well at high frequencies, that is high rotor frequencies (in case of normal slip values). It does not function very well at low frequencies because of the relatively large influence of the stator resistance, that is neglected. At a positive loading angle the flux and the torque-producing current decrease and therefore the torque decreases and the machine will slow down (if the load torque is constant). At a negative loading angle the flux increases and the torque-producing current becomes more negative and consequently the torque becomes more negative. This again causes the machine to slow down.

(41)

a)

r----c=s~-~;:::::=-T~[~~]o

0 20 40 60 0.1 0.05

(

e~r]o

('')

Model of the induction machine

-1.5-60 -40 -20 0 20 40 60 [e~r]oe) b) -1·5 -60 -40 -20 0 20 40 60

[ef]

0

C)

c)

Figure 2.5. Steady-state characteristics of the V/f controlled UA: rotor flux and torque-producing current as a function of the loading angle with the stator voltage frequency as a parameter (to the right of the figures): a) rotor flux, equation (2.77); b) torque-producing stator current, equation (2.55); c) electromagnetic torque [me1

Jo

=

-1

-·['i'r]o

·[i~r

2

J

.

(42)
(43)

CHAPTER

3

Field orientation

3.1

Introduction

The easiest way to drive an induction machine is to connect it directly to the mains supply. This way it is fed with a constant voltage and a constant frequency. This is only acceptable for induction machines that must run at a constant frequency and small load changes (and varying rotor speed because of the variable slip) without fast dynamic responses asked for. There is a long history of analysis of induction machines supplied this way, showing vast transients at switch on, load changes, rotor blocking etc. [Leonhard'85, Kovacs'84]. In the next sections these transients in currents, flux and torque are tried to be suppressed or excluded by a more sophisticated way of induction machine control and supply.

Induction machines have specific characteristics, e.g. badly damped dynamics, which can be very disadvantageous concerning the driving system. On the other hand, the induction machine has no commutator, so it is mechanically robust and the maintenance is very low. Nevertheless, the flux vector and the current vector are not perpendicular to each other, as they are in a compensated DC machine. In such a DC machine, the torque is controlled by the armature current and the flux is controlled by the excitation current and there is complete decoupling, i.e. the variation of one adjustable quantity has no influence on the other.

3.2

Variable frequency control;

open loop constant current control

To have more control over the machine, the induction machine will generally be connected to a power electronic converter. In most of the applications a pulse width modulated voltage

source inverter (PWM-VSI) is used, i.e. an inverter with a variable voltage and frequency output. With this inverter it is possible to control the current by adding a current feedback loop [Vas'90, Leonhard'9la]. This way the current regulated PWM-VSI or CR-PWM is created. However, in this chapter a current source inverter (CSI) will be considered, i.e. an inverter with which the current value i8 and the current angle £~ (the angle of the stator

current vector i~ with respect to the stator coordinate system) can be chosen. This CSI can be considered to be an ideal CR-PWM-VSI. The CSI is used here to make the analysis of the machine control simpler and more comprehensive, see also [Blaschke'96].

(44)

in this chapter, so 'lj!r = 'lj!1 = 'lj!. The current angle e~ can follow a command value e~* or it can be driven by a frequency command value

~

r

and an integrator:

e~

==

J

~

t

dt. The easiest way of operating a CSI fed induction machine (lA) is the situation where the current i; and the frequency

~

;*

are fixed to constant values. In the rotor flux coordinate system the

constant current can be split into a magnetizing component i t and a torque-producing component i~2 as a function of the current-to-flux or loading angle

e;.

This is shown in figure 3.1.

flux axis

-~---[cs

1o

=

0

Figure 3. I. Circle diagram of the IA with constant stator current in the flux coordinate system.

The steady-state values of the rotor flux, the stator current components and the torque are shown in figure 3.2, where i; = 1 (the rated value) is assumed.

Figure 3.2 shows that this control mechanism is very unsatisfactory: the flux decreases when the torque-producing current increases (figure 3.2a) and therefore the torque does not increase with the torque-producing current, but reaches a maximum at

e;

45° (figure 3.2b). Furthermore the calculations of figure 3.2 are unrealistic, because flux values much larger than 1 p.u. are unrealistic in a real machine. And flux values smaller than I p.u. are not desired (at least not below rated rotor speed). This leaves only the small area where the flux is about 1 as an operating range. But around ( 'lj! ]

0

=

I the stationary operating point is not stable

because of the negative slope of the torque curve. From figure 3.2 it is clear that the induction machine control with

i;

= 1 is not useful.

(45)

Field orientation

-45

a) b)

Figure 3.2. Steady-state load characteristics of the IA with i; =I and 1 = 3:

a) field-oriented stator current components and flux; b) electromagnetic torque.

The same calculations for [ 1jJ

]max

= [ '\jl

]N

= I (i.e. a rated flux value at the no-load condition) lead to i;

= [

1jJ ]N

=

.!.

and (with 1 = 3) a maximum torque of only 0.167 is reached, see

1 1

figure 3.3a and b.

a) b)

Figure 3.3. Steady-state load characteristics of the lA with i:

=.!.

1

a) field-oriented stator current components and flux; b) electromagnetic torque.

This steady-state analysis shows that induction machine control with a constant current value is very unsatisfactory. Also transient analysis shows this unsatisfactory behaviour, because of the weakly damped electromagnetic and mechanic oscillations that occur at frequency (speed) or load changes. An example is shown in figure 3.4 where a torque step of 15% is simulated.

(46)

~~~~--~~--~--~ps ===="'===~==I tV , :nii.ItnnfNC\.ci;=='==""'=~ E ljJ (rad) s 500 1000 1500 2000 2500 3000 500 1000 1500 2000 2500 3000 t (pu) t(pu) a) b) 500 1 000 1500 2000 2500 3000 1 (pu) c)

Figure 3.4. Dynamic response of the IA with constant current control at a 15% torque step: a) field-oriented stator current components; b) flux, loading angle and rotor speed; c) electromagnetic torque and load torque.

3.3.

Flux control

The disadvantages of a constant current controlled induction machine occur because in steady state with constant current the flux decreases strongly when the load increases. Therefore, it is important to keep the flux constant when the load increases. This can be done by increasing the current magnitude, according to equation (2.50):

(3.1)

The graphs of the flux, the stator current field oriented components and the torque are drawn in figure 3.5.

(47)

Field orientation

-45 0 45

E~n- 90

a) b)

Figure 3.5. Steady-state load characteristics of the flux controlled IA:

a) field-oriented stator current components and flux; b) electromagnetic torque.

A linear

mel-;~

graph results, that guarantees stable steady-state operating points for every slip frequency ; ; . When this flux control must also operate in transient load changing conditions, equation (3 .1) is still valid and E; has to be known to calculate

i;:

*

·*

'ljJ Is= l·cosElJ! s {3.2)

To know

E;,

information about the rotor flux vector 1Jis in stator coordinates is needed. This can be done in several ways and will be discussed in section 3.5.

If the current is controlled according to equation (3.2), the current will become very large at heavy loads, i.e. when

E;

becomes large. But the current must be limited to prevent damage to the controlled system, especially the power converter. The easiest way to do this is to limit the current magnitude i;. What happens then is drawn in figure 3.6 where the current is limited to its rated value. Because the current is again constant at its limit, the flux will decrease exactly in the same way as in figure 3.2. Also the torque will decrease, so at loads larger than the rated load value, the machine will lose its torque capability, demagnetize and stop, see figure 3.7. It is clear why this control method is called deadbeat controL This is of course a good limiting mechanism when the overload will only occur in emergency situations and the best or the only solution is to stop and shut off the machine. But when the machine has to be slowed down smoothly while keeping the flux at rated level (e.g. elevator or crane applications), other ways must be found to limit the current.

(48)

3~==~~==~~~~~~

t.2

"ljl,

•s

1~~~---~ a)

90

b)

Figure 3.6. Steady-state load characteristics of the flux controlled lA with current magnitude limiting: a) stator current components and flux; b) electromagnetic torque.

a) 1 00 200 300 400 500 600 t (pu) 0 100 200 300 400 500 600 t (pu) b) 100 200 300 400 500 600 t (pu) c)

Figure 3.7. Dynamic simulation of the deadbeat control of the lA:

a) field-oriented stator current components; b) electromagnetic torque, load torque and rotor speed; c) flux.

(49)

Field orientation

With the control mechanisms mentioned up till now, where only the magnitude of the stator current was controlled, the transient behaviour of the induction machine is still determined by the dynamics of the machine itself. This causes slowly damped oscillations with eigenvalues that are dependent on the steady-state conditions of the machine.

To improve the behaviour of the induction machine, not only the stator current magnitude, but also the loading angle e;' must be controlled, creating a so called vector control. This is

the topic of the next section.

3.4

Vector control; field-oriented control

To obtain a satisfactory control structure for the induction machine, the two components of

the stator current vector must be controlled. This can be either the current magnitude and the loading angle or the magnetizing and the torque-producing components of the current. From (2.41) it can be seen that the magnetizing stator current must be controlled in a way to keep the flux in the machine constant. And the torque-producing stator current must be controlled to generate a desired torque (equation (2.21)). When the magnetizing and torque-producing stator current command values (the cartesian coordinates) are present, they can be transformed into the polar coordinates: the stator current magnitude and the loading angle.

Because the current vector is oriented with respect to the flux reference frame (and for example not the stator reference frame) by adjusting the loading angle e;', this control strategy is called the flux oriented control or field oriented control. It is a vector control

because both the magnitude and the angle of the current vector are controlled.

To illustrate the advantage of the vector control the simulation of figure 3.7 is repeated. But in this case the current magnitude is not limited, but the torque-producing current component

is limited. The magnetizing current component is kept constant independently on the torque-producing component. When the load torque is higher than the maximum electromagnetic torque of the machine, the machine will slow down, but maintain its rated flux level and a constant torque level. This is illustrated in figure 3.8.

(50)

200 400 600 800 1000 200 400 800 800 1000

t (pu) t (pu)

a) b)

Figure 3.8 Dynamic simulation of the vector-controlled lAunder too heavy load:

a) field-oriented stator current components and flux; b) electromagnetic torque, load torque and rotor speed.

3.5.

Determination of the flux angle

q:J8

To feed the induction machine with the stator current vector command values the field oriented stator current coordinates must be transformed to the coordinates in the stator reference frame. This can be done by adding the flux angle qJ8 to the loading angle e;':

(3.3) This way the current vector command values are the current magnitude i; and the current angle e~* with respect to the stator reference frame. These two quantities will be used to control the inverter that feeds the induction machine. The determination of the flux angle qJs

is the base of every field oriented control. But this is not easy.

The best way to determine the flux angle is to measure the magnetic field inside the machine, using Hall sensors or extra induction coils [Blaschke'74], see figure 3.9. This direct way of detennining the flux angle is called the direct field oriented control. But the sensors are

expensive and difficult to build into a standard induction machine. And the extra sensors make the induction machine, which is by itself a very robust mechanical structure, mechanically vulnerable.

The second way to determine the flux angle is to measure the mechanical rotor angle ps (using an optical encoder or a resolver) and add it to the flux-to-rotor angle qJr [Hasse'72]:

(51)

Field orientation

This is the indirect field orientation (orientation with respect to the rotor angle). The flux-to-rotor angle cpr can be calculated from a model

i

of the induction machine as shown in figure 3.10. This model needs the stator current and the measured rotor position as its inputs and is therefore called the current/rotor-angle model or i I p8

-model.

Figure 3.9. Direct field orientation by measurement of the flux inside the AC machine.

- - - - -+-+--...-t I

I

I

~ I

I

I L _____________ J

Figure 3.10. Indirect field orientation with the i I p5

-model.

(52)

This control scheme needs a rotor position sensor or speed sensor, that is again a mechanically vulnerable part to be added to the induction machine. A second disadvantage is the need of the parameter rr (the rotor resistance) in the model. The real rotor resistance rr depends on the temperature, that can vary largely when the output power of the machine is changing a lot. The varying rr causes estimation errors in the flux angle, causing wrong field orientation which in turn results in a bad static and dynamic control performance, especially at high loads.

The third way to determine the flux angle is to calculate the rotor flux vector from the stator voltage and the stator current with the use of a model of the induction machine, see figure 3.11. The advantage is that there are no sensors needed inside the machine or on the rotor shaft, but only voltage and current sensors. This type of field orientation is also a direct field

orientation. The model that is used is called the voltage/current model or model. The

u/i-model is the subject of this thesis. The structure of the u/i-u/i-model is explained in chapter 4.

lA

sl s2 Us Us -s <p

u/i-model

(53)
(54)
(55)

CHAPTER

4

The u/i-model

4.1

Fundamental structure of the u/i-model

The determination of the flux angle <p~ is the base of every field-oriented control. In this chapter the u/i-model will be presented as an instrument to determine this flux angle. In case of field orientation, the command values of the coordinates of the stator voltage (or current) vector are coordinates with respect to the rotor flux. To feed the induction machine with the command values, these field-oriented coordinates must be transformed to the coordinates in the stator reference frame. This is done by adding the rotor flux angle <p~ to the voltage angle agwr (or the current loading angle

ef).

A plot of these vectors and angles is shown in figure 4.1.

u

stator axis

Figure 4.1. The angles of the stator voltage and current vectors with respect to the rotor flux and the fixed stator reference frame.

In the u/i-model the rotor flux vector (value and angle) of the induction machine is calculated from the measured stator currents and stator voltages. The mathematical equation system of the uJi-model starts from the vector equations (2.1), (2.3), (2.4), (2.7) and (2.8) of the machine of chapter 2. The stator terminal voltage u8 minus the stator resistance voltage drop r5

·is,

that equals the stator flux derivative, is defined as e5 :

(4.1) (4.2) (4.3)

(56)

(4.4)

To determine the rotor flux from the measured stator voltage and the measured stator current the next steps are followed in the u!i-model (the tilde,~, indicates u!i-model values):

7's_ s -·s_--s 'l's-us-rs1s = es

(4.5)

i:=

f

•:dt

(4.6)

is =is

1 S

-l

m

·s

1s

(4.7)

-s -;-s-

:!1..

•S

(4.8)

1r- - -1s 1 -s -s --;-s 'l'r

=

'1'1

+

lorir

(4.9)

The block diagram of the u!i-model in vector format, constructed from these equations, is drawn in figure 4.2.

s

Us

~s

~1 •

Figure 4.2 Block diagram of the u!i-model in stator coordinates.

This block diagram can be simplified to the one drawn in figure 4.3. This is done by eliminating the air-gap flux and the rotor current from (4.7) - (4.9), resulting in the next equation:

(4.10) where, according to (2.34),

l

0 is the total leakage inductance transformed to the stator side:

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