Power Electronic Converters E-Book

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 572

Veröffentlichungsjahr: 2013

Beschreibung

A voltage converter changes the voltage of an electrical power source and is usually combined with other components to create a power supply. This title is devoted to the control of static converters, which deals with pulse-width modulation (PWM) techniques, and also discusses methods for current control. Various application cases are treated. The book is ideal for professionals in power engineering, power electronics, and electric drives industries, as well as practicing engineers, university professors, postdoctoral fellows, and graduate students.

Leseprobe

Table of Contents

Introduction

Chapter 1. Carrier-Based Pulse Width Modulation for Two-level Three-phase Voltage Inverters

1.1. Introduction

1.2. Reference voltages νaref, νbref, νcref

1.3. Reference voltages Paref, Pbref, Pcref

1.4. Link between the quantities νa, νb, νc and Pa, Pb, Pc

1.5. Generation of PWM signals

1.6. Determination of the reference waves νarefk, νbrefk, and νcrefk from the reference waves νarefk, νbrefk, νcrefk

1.7. Conclusion

1.8. Bibliography

Chapter 2. Space Vector Modulation Strategies

2.1. Inverters and space vector PWM

2.2. Geometric approach to the problem

2.3. Space vector PWM and implementation

2.4. Conclusion

2.5. Bibliography

Chapter 3. Overmodulation of Three-phase Voltage Inverters

3.1. Background

3.2. Comparison of modulation strategies

3.3. Saturation of modulators

3.4. Improved overmodulation

3.5. Bibliography

Chapter 4. Computed and Optimized Pulse Width Modulation Strategies

4.1. Introduction to programmed PWM

4.2. Range of valid frequencies for PWM

4.3. Programmed harmonic elimination PWM

4.4. Optimized PWM

4.5. Calculated multilevel PWM

4.6. Conclusion

4.7. Bibliography

Chapter 5. Delta-Sigma Modulation

5.1. Introduction

5.2. Principle of single-phase Delta-Sigma modulation

5.3. Three-phase case: vector DSM

5.4. Conclusion

5.5. Bibliography

Chapter 6. Stochastic Modulation Strategies

6.1. Introduction

6.2. Spread-spectrum techniques and their applications

6.3. Description of stochastic modulation techniques

6.4. Spectral analysis of stochastic modulation

6.5. Conclusion

6.6. Bibliography

Chapter 7. Electromagnetic Compatibility of Variable Speed Drives: Impact of PWM Control Strategies

7.1. Introduction

7.2. Objectives of an EMC study

7.3. EMC mechanisms in static converters

7.4. Time-domain simulation

7.5. Frequency-domain modeling: a tool for the engineer

7.6. PWM control

7.7. Comparison of sources for different carrier-based PWM strategies

7.8. Space vector PWM

7.9. Structure for minimizing the common mode voltage

7.10. Conclusion

7.11. Bibliography

Chapter 8. Multiphase Voltage Source Inverters

8.1. Introduction

8.2. Vector modeling of voltage source inverters

8.3. Inverter as seen by the multiphase load

8.4. Conclusion

8.5. Bibliography

Chapter 9. PWM Strategies for Multilevel Converters

9.1. Introduction to multilevel and interleaved converters

9.2. Modulators

9.3. Examples of control signal generators for various multilevel structures

9.4. Conclusion

9.5. Bibliography

Chapter 10. PI Current Control of a Synchronous Motor

10.1. Introduction

10.2. Model of a synchronous motor

10.3. Typical power delivery system for a synchronous motor

10.4. PI current control of a synchronous motor in the fixed three-phase coordinate system of the stator

10.5. PI current control for a synchronous motor in a rotating coordinate system (d, q)

10.6. Conclusion

10.7. Bibliography

Chapter 11. Predictive Current Control for a Synchronous Motor

11.1. Introduction

11.2. Minimum-switching-frequency predictive control strategies

11.3. Limited-switching-frequency predictive control strategies

11.4. Limited-switching-frequency predictive current control strategies for a synchronous motor

11.5. Conclusion

11.6. Bibliography

Chapter 12. Sliding Mode Current Control for a Synchronous Motor

12.1. Introduction

12.2. Sliding mode current control for a DC motor

12.3. Sliding mode current control of a synchronous motor

12.4. Conclusion

12.5. Bibliography

Chapter 13. Hybrid Current Controller with Large Bandwidth and Fixed Switching Frequency

13.1. Introduction

13.2. Main types of discrete-output current regulators

13.3. Tools for limit cycle analysis

13.4. Conclusion

13.5. Bibliography

Chapter 14. Current Control Using Self-oscillating Current Controllers

14.1. Introduction

14.2. Operating principle of the self-oscillating current controller

14.3. Improvements to the SOCC

14.4. Characteristics of the SOCC

14.5. Extensions to the SOCC concept

14.6. Conclusion

14.7. Bibliography

Chapter 15. Current and Voltage Control Strategies Using Resonant Correctors: Examples of Fixed-frequency Applications

15.1. Introduction

15.2. Current control with resonant correctors

15.3. Voltage control strategy

15.4. Conclusion

15.5. Appendix: transformer parameters

15.6. Bibliography

Chapter 16. Current Control Strategies for Multicell Converters

16.1. Introduction

16.2. Multilevel conversion topology

16.3. Modeling and analysis of degrees of freedom for control

16.4. Analysis of degrees of freedom available to the control algorithm

16.5. Classification of control strategies

16.6. Indirect control strategy for a single-phase leg

16.7. Direct control strategy for a single-phase leg

16.8. Command strategy, three-phase approach

16.9. Features of multicell converters: need for an observer

16.10. Conclusions and outlook

16.11. Bibliography

List of Authors

Index

First published 2011 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc. Adapted and updated from Commande rapprochée de convertisseur statique 1&2 published 2009 in France by Hermes Science/Lavoisier © LAVOISIER 2009

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

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© ISTE Ltd 2011

The rights of Eric Monmasson to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Cataloging-in-Publication Data

Commande rapprochee de convertisseur statique. English

Power electronic converters: PWM strategies and current control techniques / edited by Eric Monmasson.

p. cm.

Includes bibliographical references and index.

ISBN 978-1-84821-195-7

1. Electric current converters. 2. Electric motors--Electronic control. I. Monmasson, Eric. II. Title.

TK7872.C8C66 2011

621.3815′322--dc22

2010051719

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

ISBN 978-1-84821-195-7

Introduction

The present-day soul searching over modern society’s dependency on oil has resulted in considerable attention being given to alternative, renewable energy sources; this has emphasized how important electrical energy will be in the future, and its potential for reducing environmental impact.

This is particularly true for electrical energy conversion, a field that has seen continual progress over the last 30 years. The primary reasons for this have been the development of power switches operating at increasingly higher speeds and ever-increasing power levels. At the same time digital systems, able to act as controllers for power electronics, have opened up new possibilities both in terms of ease of use and increased performance.

Thus static converters and their accompanying controllers have become critical to modern power conversion devices. This fact has rapidly led controller designers to pay close attention to the electrical output from static converters (voltages and currents) since these have a direct effect on the quality of the higher-level control variables such as torque and velocity (in the case of an actuator) or active and reactive power flows (in the case of a generator connected to the grid).

This observation has naturally led designers to organize their controllers in a hierarchical manner. The lowest level is known as the current and/or voltage controller. This inner-loop controller ensures that electrical quantities output from the converter are correctly regulated, thus ensuring a high quality of energy transfer between the source upstream of the converter and its downstream load. The next highest level accompanies the inner loop controller. This second level is often referred to as “algorithmic control” because it is generally implemented in a microprocessor (DSP, RISC, etc.). This more sophisticated element of the control focuses on controlling the variables directly relevant to the final application (speed of a motor, etc.). The resultant servo loops are known as outer-loop controllers and their control quantities act as references to the inner-loop controllers.

The current-voltage control of static converters is an important technical consideration since it is crucial to the correct functioning of the entire energy conversion system.

The dynamic characteristics that are required span over a considerable range and the specifications are continually being tightened in response to technological advances. Thus, a great deal of research effort has been devoted to this topic, both in universities and in industry, and we will attempt to summarize that work in this book, discussing not only the reference algorithms but also the current trends in innovation within this field.

In this book on current and voltage control of static converters, we will focus on the following two central themes:

– Chapters 1 to 9 focus on pulse width modulation (PWM) techniques that enable a static converter to continuously generate variable output voltages in response to binary orders sent to the static converter, with both the output amplitude and frequency being controllable in terms of instantaneous mean values;

– Chapters 10 to 16 focus on electric current control techniques.

In order to better introduce PWM techniques, we should recall a few important characteristics of electronic power devices. These are high-efficiency devices since they only operate in fully blocking states (zero current) or fully conducting states (zero voltage). The transition from a blocking state to a conducting state, and vice versa, occurs in response to a switching action.

However, this switching method of energy conversion can only produce a limited number of different voltage levels, in other words, it is a quantized process. Consequently, in order to achieve acceptable accuracy for the amplitudes and frequencies of the resultant voltage waveforms we must modulate the duration of the voltage pulses applied to the gates of the power switches.

The modulation will be more effective when its associated frequency is higher. However, this modulation frequency can only be increased up to a certain limit beyond which it leads to unacceptably high switching losses in the power switches. Another factor limiting increases in switching frequency is associated with an increase in conducted and radiated interference, which can cause damage to equipment near the static converter: this is a problem of electromagnetic compatibility.

Thus, this inevitable compromise between an increase in the modulation frequency and the drawbacks associated with this increase has led researchers to develop a wide range of modulation techniques.

The quality criteria commonly specified for a PWM system include minimization of the total harmonic distortion of the electric current, maximization of the linear range of the fundamental harmonic voltage, minimization of torque harmonics (in the case of motor control), reduction of losses within the static converter, and minimization of the common mode voltage that is produced.

The aim of the first nine chapters of this book is to highlight the wide variety of PWM techniques that are available. Focus will be on the case of the voltage source inverter because of its importance in industrial applications.

Chapters 1 and 2 can be treated as reference chapters. They discuss in depth the two main families of PWM strategies: carrier-based PWM strategies and space vector PWM strategies. In both these cases we will study a two-level voltage inverter intended to feed a three-phase inductive load such as an electric motor. We will emphasize the conceptual similarities between these two approaches despite their different implementations. The degree of freedom introduced by the addition of a zero-sequence component to the modulated voltage enables us to meet a range of challenges (e.g. maximization of the linear range and limitation of losses).

Chapter 3 considers overmodulation of three-phase voltage inverters, a very important mode of operation in case of variable speed drive applications. We will discuss modulation strategies for when the required voltage is close to or greater than the maximum possible value, with the main objective to maximize the total power while restricting the effects of low frequency harmonic components.

Chapter 4 discusses high-power systems for which the modulation frequency is necessarily restricted. The idea here is to work with modulation frequencies that are synchronized with the fundamental harmonic, and to optimize the harmonic content of the modulated voltage waveforms by careful choice of the exact switching times. We will consider the case of a three-level inverter as well. We will also present an original configuration for a multi-level power supply using active filtering based on two, two-level inverters (one supplying the requisite power and the other operating as an active filter). This construction enables the harmonic content of the power supply to be optimized.

Chapter 5 describes the Delta-Sigma modulation strategy. The main advantages of this modulation are its robustness, the possibility of reducing the ratio between the switching frequency and the modulation frequency, and the possibility of operating with either variable or fixed switching frequency.

Chapter 6 considers stochastic modulation methods. The main advantage is that they can broaden the spectrum of the modulated signals, thus reducing electromagnetic and acoustic interference. The latter will be the subject of a detailed study at the end of the chapter.

Chapter 7 continues from the previous chapter. It focuses on analyzing conducted electromagnetic interference produced by the modulated voltages output from a voltage source inverter driving an electric motor.

Chapters 8 and 9 offer an introduction to the study of modulation strategies for energy conversion devices where the power delivery is distributed. There is a strong interest, presently, in designing energy conversion structures with multiple windings or multiple levels. These structures are inevitably more complex than a traditional three-phase motor driven by a two-level inverter. Indeed, such structures enable fault-tolerant architectures to be developed, exploiting the redundancies present in such a system; they can also be used to distribute between multiple components the power that the device must deliver, thus reducing the stress on the power switches and increasing the lifetime of the equipment.

Chapter 8 introduces an extension of the space vector PWM technique to multiphase systems through a formalism based on linear algebra.

Chapter 9 provides a generic discussion of PWM applied to common multilevel converter topologies. In particular, we show how redundancies in the voltage levels can be exploited to optimize additional objectives such as voltage balancing across flying capacitors.

Chapters 10 to 16 are devoted to current regulation techniques. It seems worthwhile recalling the main reasons that have led designers to integrate this type of regulation in a fairly systematic manner into the design of inner-loop control systems for static converters. The main objectives are to ensure accurate control of the instantaneous current waveforms to protect the static converter from any potential current surges, to reject disturbances caused by the load, to be robust with regard to parametric variations and to nonlinearities within the converter, and also to offer excellent control dynamics. From a quantitative point of view these criteria result in a minimal static error, maximize the bandwidth, and offer an optimized modulation depth and a minimum level of distortion.

These fundamental requirements for current regulation are often accompanied, where possible, with additional regulation requirements such as control over the switching frequency in the case of hysteresis-based control, balancing of intermediate voltages in the case of a multicell converter, etc.

As we will see in this part of the book, current control structures are also tightly integrated with the applications with which they are associated. It is for this reason that we have included a range of studied examples although we make no claims of being exhaustive. These examples will of course include the combination of a voltage inverter and motor, a very popular case and one that has seen the greatest range of experimental work in terms of current control. Nevertheless, the current control techniques that are brought together in this volume are also relevant to other types of applications such as high performance power-in-the-loop emulation, the generation of electrical energy both for the electrical grid and for isolated networks, DC/DC power delivery, and high-power applications based on multilevel converters.

In terms of their operating principles, current regulation methods for static converters can be divided into two main families:

– direct control, also known as amplitude control, for which the outputs of the current regulators directly control the associated static converter. These control strategies are exclusively nonlinear. Their main advantage is that they ensure excellent system dynamics along with a high robustness in the face of parametric variations and model uncertainties. Their main drawbacks are variation in switching frequency and the emergence of limit cycles at steady state;

– indirect control, also known as PWM control, for which the outputs of the current regulators act as inputs to a PWM modulator (Chapters 1 to 9). These control strategies may be either linear or nonlinear and offer the possibility of controlling the converter in a very accurate manner and at a fixed frequency, thus avoiding any risk of limit cycles appearing. However, the dynamics that can be achieved using such methods are generally not up to the standard of those obtained using direct command.

Chapters 10 to 12 discuss a single application: current control for a synchronous motor supplied by a three-phase voltage inverter. For this application, control of the current is equivalent to control of the torque. Chapter 10 discusses indirect control using a PI controller in a rotating reference frame. The quantities being controlled are constant at steady state, which makes servo control of those quantities easier. This first control method is linear and will be treated as a reference method since it is so widely used in industry.

Chapter 11 discusses direct and indirect predictive control, the principle of which involves calculating the most suitable voltage vector to apply in each sampling period. This control strategy is relatively demanding in terms of computation time but can be implemented in a highly parallel manner. It is therefore very well suited to implementation in an FPGA (Field Programmable Gate Array).

Chapter 12 describes direct and indirect sliding mode control. The design principle behind these two sliding mode control methods is described in detail and their contrasting strengths in terms of dynamics and precision are clearly demonstrated.

Chapter 13 discusses hysteresis-based control. The aim is to explain the fundamentals of this type of control using theoretical tools developed for the study of nonlinear systems. Without focusing on any specific application, this chapter offers a different perspective on the qualities of this direct control method focusing on the concept of modulated hysteresis control, which combines the excellent dynamic performance of hysteresis control with guaranteed fixed frequency operation.

Chapter 14 discusses current and voltage control using a self-oscillating regulator, known as SOCC (self-oscillating current control) and SOVC (self-oscillating voltage control). This innovative direct control technique is protected by patents. It relies on self-oscillation within the control loop, a property that guarantees fixed frequency operation at the same time as promising excellent performance in terms of dynamics and robustness. Here, the application area is for power-in-the-loop emulation of electrical loads.

Chapter 15 introduces the principle of resonant control at fixed frequency. This type of control makes it possible to introduce an infinite gain at a precisely known frequency, which results both in elimination of any tracking error and rejection of any disturbance at this specific frequency. This control strategy, which is a sensitive form of regulation, is very promising since it is ideally suited to distributed energy generation networks. Here the authors demonstrate the qualities of these resonant regulators through an example of control of a wind turbine able to operate both on a power grid and on an isolated network. This is an indirect, linear type of control.

Finally, Chapter 16 presents a state of the art for current control of multicell converters. The number of degrees of freedom available in such conversion structures is useful for studies into multi-objective current control (both tracking of commanded current references and balancing of internal voltages). The application area here is for high-power equipment. We also show how the principle control paradigms presented earlier can be adapted to the case of multilevel converters.

Eric MONMASSON

February 2011

Chapter 1

Carrier-Based Pulse Width Modulation for Two-level Three-phase Voltage Inverters1

1.1. Introduction

Two-level three-phase voltage inverters are very widely used for feeding alternating current electrical machines serving as actuators with variable input voltages (controllable for amplitude and frequency). However, they are also increasingly being used as sinusoidal current absorption rectifiers. A chapter from an earlier book [LAB 04] has already introduced these topics from a modeling perspective. Figure 1.1 recalls the basic principles of a two-level three-phase voltage inverter feeding a balanced three-phase load connected in a star configuration with isolated neutral; the diagram introduces the notations we will use; the input reference voltage is taken to be the mid-point between the direct current bus rails.

We can present the problem of control via PWM in the following manner:

– starting with the reference voltages varef, vbref, vcref to be imposed on terminals of the different phases of load, the first step is to determine the voltages Pa, Pb, Pc produced by the legs of the inverter, suitable reference

Figure 1.1.Schematic diagram showing notations used

values Paref, Pbref, Pcref, such that the actual output voltages Pa, Pb, Pc, lead to the desired values of the voltages va, vb, vc;

In case of carrier-based modulation, which is the subject of this chapter, the transformation of the reference signals Pj ref into binary signals xj is achieved by comparing these signals to a carrier wave vp (triangular or saw-toothed) whose frequency determines the intervals over which we want < Pj> to match Pj ref (Figure 1.2). We have xj equal to 1 and therefore:

Figure 1.2.Sawtooth carrier modulation

if:

Alternatively, we have xj equal to 0 and therefore:

if:

Carrier-based modulation also refers to any sort of modulation where the intervals where xj is equal to 1 and those where xj is equal to 0 are produced by a microcontroller or FPGA [MON 08] using a computation that emulates the intersection process between the reference and carrier as described earlier.

We will show how the reference voltages varef, vbref, vcref to be applied to the load can be represented, and then describe the conversion of these values to the reference voltages Paref, Pbref, Pcref for each leg, and finally show how these values are transformed into binary control signals (PWM signals) for the switches. We will see that this enables us to:

– derive the various intersective modulation strategies described in the literature (sine-triangle modulation, sub-optimal modulation, centered modulation, and flat-top and flat-bottom modulation);

– establish the similarities between certain types of intersective modulation and other modulation strategies such as space vector modulation.

1.2. Reference voltages varef, vbref, vcref

Traditionally, when discussing intersective PWM [KAS 91, LAB 95, MOH 89, and SEG 04] it is assumed that the reference values have the form:

[1.1]

where Vref is the desired amplitude for the voltages and θref is an angular coordinate obtained by integrating the desired reference pulsation for the voltages:

[1.2]

We will introduce the rotation matrices P(θ) and the Clarke submatrix C32 [SEM 04]:

Equation [1.1] can then be written as:

[1.3]

or be represented by the diagram shown in Figure 1.3.

Figure 1.3.Generation of reference wavesva ref, vb ref, vc reffrom the desired amplitudeVrefand pulsationωref

We can consider Vref and θref to represent a vector rotating with speed ωref and whose projection onto three axes mutually separated by 2π/3 gives the reference voltages va ref, vb ref, vc ref (Figure 1.4).

Figure 1.4.“Vector” representation of the generation of reference wavesva ref, vb ref, vc ref

In the steady state case Vref and ωref have fixed values. In the transient case they may vary as a function of time.

This classical approach takes the two degrees of freedom required to fix the reference voltages va, vb, vc to be their amplitude Vref and their pulsation ωref (equal to 2π times their reference frequency fref).

[1.4]

This is equivalent to taking (Figure 1.5a):

[1.5]

It is then more helpful to consider Vref and φref as the control parameters (Figure 1.5b) since, in contrast to ∆ref, φref is not required to be zero in the steady state regime but only to have a constant value.

Figure 1.5.Generation of reference wavesva ref, vb ref, vc refwhen their steady state pulsation is determined by the load; a) expression [1.4]; b) expression [1.5]

On the basis of Figure 1.5b we can write:

[1.6]

Equation [1.6] can then be written as:

[1.7]

so that finally:

[1.8]

Figure 1.6.Generation of reference wavesva ref, vb ref, vc reffrom their components dq

Equation [1.7] can be observed to be equivalent to the diagram in Figure 1.6, where the reference quantities are taken to be:

and:

instead of:

and:

since:

where I is the 2 × 2 identity matrix, the voltages vd ref and vq ref can be determined from the desired voltages va ref, vb ref, vc ref if we left-multiply both sides of [1.7] by:

then:

[1.9]

[1.10]

which shows that the matrix:

1.3. Reference voltages Pa ref, Pb ref, Pc ref

In contrast to the voltages va ref, vb ref, vc ref, the voltages Pa ref, Pb ref, Pc ref are not required to have a sum of zero. We therefore have three degrees of freedom in defining these voltages. If we introduce the homopolar component of Pa ref, Pb ref, Pc ref:

then the quantities:

sum up to zero, just like the voltages va ref, vb ref, vc ref do. Making use of the definition of P0ref we can write:

[1.11]

The matrix , which acts as an identity matrix with respect to three quantities whose sum is zero (Equation [1.10]), acts to eliminate the homopolar component when it is applied to three quantities that do not sum up to zero. Using a process similar to that used for the quantities va ref, vb ref, vc ref, we can represent the quantities Pa−h ref, Pb−h ref, Pc−h ref in terms of two reference quantities Pd ref, Pq ref (Figure 1.7):

[1.12]

Figure 1.7.Generation of reference wavesPa−h ref, Pb−h ref, Pc−h ref, from their dq components

By defining the matrix:

[1.13]

we can express the reference quantities Pa ref, Pb ref, Pc ref as a function of the reference quantities Pd ref, Pq ref, P0ref (Figure 1.8):

Figure 1.8.Generation of reference wavesPa ref, Pb ref, Pc reffrom their d-q-0 components

[1.14]

1.4. Link between the quantities va, vb, vc and Pa, Pb, Pc

Referring to the notations in Figure 1.1 we can write:

[1.15]

[1.16]

Therefore, the difference between Pi and vi stems from the homopolar component: the Pi values include a homopolar component while the vi values do not.

1.5. Generation of PWM signals

In order to determine the states of the switches of each leg from the reference waves Pj ref, we will consider sequentially:

– the case where these waves are compared to a reverse sawtooth carrier;

– the case where these waves are compared to a conventional sawtooth carrier;

– the case where these waves are compared to a triangular carrier.

We will assume that the carrier is normalized and varies between −1 and +1, and that the reference waves also vary in this manner since they are divided by U/2:

[1.17]

1.5.1. Reverse sawtooth wave

Each leg then undergoes a transition from S′j closed to Sj closed at time tjk when the reference wave Pj ref,n intersects the carrier and takes a value greater than that of the carrier (Figure 1.9).

The order in which the switches commutate depends on the order in which the carrier intersects the reference waves.

There are six possible sequences: a, then b, then c; a, then c, then b; b, then c, then a; b, then a, then c; c, then a, then b; and c, then b, then a.

Figure 1.9.Modulation with a reverse sawtooth carrier

Each commutation causes one of the components of the vector (xa, xb, xc) to transition from 0 to 1; the vector starts with the value (0, 0, 0) at the start of the period, with all closed; it ends the period with the value (1, 1, 1).

The time tjk, where Pjref ,n intersects with the carrier is the root of the following equation:

[1.18]

[1.19]

(NOTE Remember that Pj ref,n is normalized using equation [1.17]). Equation [1.19] shows that the PWM process, over which Pj takes the value −U/2 and then +U/2, results in the mean value of Pj over the modulation period being equal to the value of Pj ref at one particular point within an interval of this period: the point where the carrier intersects Pj ref,n. If the reference waves Pj ref vary only slightly over a modulation period, the sequence of samples Pj ref,n (tjk) will provide a good representation of the reference waves. The same goes for the mean values < Pj >k of the voltages Pj.

Figure 1.10.Modulation using a reverse sawtooth wave with synchronous sampling of the reference waves

For the rest of this section we will assume that the reference waves are sampled at the start of each modulation period, so that we have:

[1.20]

Using equation [1.14] we can write:

[1.21]

[1.22]

Equation [1.15], which connects the voltages va, vb, vc to the instantaneous values of the voltages Pa, Pb, Pc can also be applied to the mean values (over each modulation period) of these quantities. This gives us:

[1.23]

Substituting [1.22] into [1.23] we obtain:

[1.24]

given that:

, where I is (as mentioned earlier) the 2 × 2 identity matrix:

1.5.2. Conventional sawtooth carrier

Over each modulation period the carrier now varies linearly from −1 to +1 and returns from +1 to −1 at the boundary between one period and the next. Over the (k + 1)th modulation period from:

to:

every switch Sj will close at tk since at this moment the carrier takes the value −1 and therefore has a value smaller than that of each of the reference waves, implying that xa, xb and xc are equal to 1.

Each leg then undergoes a transition from Sj closed to S′j closed at the time tjk when the reference wave Pj ref,n intersects the carrier vp (Figure 1.11). Each transition causes one component of the vector (xa, xb, xc) to move from 1 to 0, starting from a value [1,1,1] at tk and finishing with a value [0,0,0] at the end of the period.

The voltage is U/2 from tk to tjk over the interval where xj is 1 and Sj is closed. It is −U/2 from tjk to tk+1 over the interval where xj is zero and S′j is closed. The voltages uj are linked to the voltages Pj by equation [1.15].

The time tjk when Pj intersects the carrier is the solution of the following equation:

[1.25]

The mean value of Pj over the (k+1) th modulation period is therefore:

[1.26]

If we adopt a synchronous sampling scheme for the reference waves, we obtain:

Figure 1.11.Modulation by a conventional sawtooth carrier

[1.27]

When the reference waves are sampled at the beginning of the modulation period, equations [1.20] to [1.24] apply equally well to the case of a conventional sawtooth carrier as to the reverse sawtooth carrier.

1.5.3. Triangular carrier

Modulation by a triangular carrier can be considered as equivalent to repeated modulation, first by a reverse sawtooth wave and then by a conventional sawtooth wave.

The period of the carrier is twice the duration Tp / 2 of each of the ramps (first decreasing and then increasing) that constitute the carrier (Figure 1.12).

If the period starts with modulation by a decreasing ramp, at the start of the period all the switches S′j are closed since at this point in time the carrier has a value greater than that of every reference wave.

Each leg then undergoes a transition from S′j closed to Sj closed at the time when the corresponding reference wave crosses the carrier wave; by the end of the decreasing ramp all the switches Sj are closed.

During the increasing ramp each arm undergoes a transition from Sj closed to S′j closed, such that at the end of the period the situation is once again when all the S′j switches are closed.

There is no longer, as was the case with sawtooth waves, a moment where all the legs commutate simultaneously at the point between one modulation period and the next.

If the reference waves are sampled, this may occur:

– at the start of each modulation period, as was the case with sawtooth carriers (Figure 1.13);

Figure 1.12.Modulation by a triangular carrier

Figure 1.13.Modulation by a triangular carrier with synchronous reference sampling at the start of each carrier period

– at the start of each sawtooth component of the carrier, which means that the waves Pj match the mean values of their reference waves Pj ref on the scale of every half-period of the modulation (Figure 1.14).

Figure 1.14.Modulation by a triangular carrier with synchronous reference sampling at the start of each half-period

[1.28]

[1.29]

Over each half-period of the carrier we have, as with sawtooth carriers, six possible switching sequences, depending on the values of Pa ref k, Pb ref k, and Pc ref k over this half-period.

For a half-period consisting of an upward ramp, the vector (xa, xb, xc) moves gradually from (1,1,1) to (0,0,0) with the transitions from Sj closed to S′j closed, acting first on the leg whose reference voltage is smallest, then on the one with the middle reference voltage, and finally on the leg with the largest reference voltage.

It can be seen that the twelve switching sequences we have just defined are identical to those that are obtained using space vector modulation (Chapter 2 and reference [LAB 98]).

Modulation by a triangular carrier has the property that it is indiscernible in terms of the switching sequences from space vector modulation.

1.5.4. Note

A modulation based on a random carrier is sometimes used, selecting in a non-deterministic manner for each period, either a conventional or a reverse sawtooth.

1.6. Determination of the reference waves Pa ref k, Pb ref k, and Pc ref k from the reference waves va ref k, vb ref k, vc ref k

As we saw in section 1.5, with PWM, Pj will only match Pj ref when averaged over a given period of modulation. The same clearly applies to the voltages va, vb, vc with respect to the reference waves va ref, vb ref, vc ref.

Here again, the problem is to determine over each modulation period the values of the waves Pa ref k, Pb ref k, Pc ref k (or the Pd ref k, Pq ref k, P0 ref k components of these waves) such that we obtain:

[1.30]

Substituting [1.30] into [1.24] we obtain the equation that must be used to link the reference values for the voltages of each phase and the reference values for the dq components of the voltages in each leg:

[1.31]

If we multiply both sides of [1.31] by:

we obtain:

[1.32]

We then substitute [1.32] into [1.21] to obtain:

[1.33]

Since the matrix is equivalent to an identity matrix for the quantities va ref k, vb ref k, vc ref k that sum up to zero, equation [1.33] can be reduced to:

[1.34]

Equation [1.34] shows that the reference values va ref, vb ref, vc ref fix the values of Pa ref k, Pb ref k, Pc ref k except for their homopolar component P0ref k, which is a remaining degree of freedom, which can be manipulated to optimize the modulation to match some desired quality criterion. This result is consistent with the statement given at the end of section 1.4.

1.6.1. “Sine” modulation

“Sine” modulation is obtained if in equation [1.34] we take the homopolar component P0ref of the reference waves Pj ref to have a value zero, which makes these waves equal to the reference waves vj ref:

[1.35]

In the steady state case the Pj ref waves then form a balanced three-phase system of sinusoidal voltages, just like the waves vj ref, and therefore the term “sine” modulation is given (Figure 1.15).

Figure 1.15.Sinusoidal modulation with triangular carrier

The amplitude of the Pj ref waves, and hence the amplitude of the sinusoidal waves that can be produced in the steady state case at the three-phase terminals of the load, cannot be greater than U/2 if we wish to avoid the emergence of an effect known as overmodulation1.

Compared to full-wave control, which gives voltages at the three-phase terminals of the load whose fundamental component has amplitude, “sine” modulation incurs a reduction in amplitude of:

[1.36]

or a reduction of 21%. This reduction in amplitude is known as “voltage drop due to pulse width modulation” [LAB 95].

1.6.2. “Centered” modulation

“Centered” modulation is when a value for the homopolar component in equation [1.34] is taken to be equal to + minus half the sum of the largest and smallest of the reference waves va ref k, vb ref k, vc ref k.

If max(vj ref k) denotes the operation of selecting the largest of the reference waves va ref k, vb ref k, vc ref k and min(vj ref k) the operation for selecting the smallest of the waves, we obtain:

[1.37]

It can be seen that the value of the homopolar component has the effect of causing the largest and smallest of the reference waves Pj ref k to lie symmetrically on each side of the horizontal axis, and hence the term “centered” is given. In the case where the reference waves vj ref form a balanced three-phase system with sinusoidal values of amplitude V and pulsation ω:

equation [1.37] gives the following voltages Pj ref (Figure 1.16):

and so on.

Figure 1.16.Centered modulation with a triangular carrier

The amplitude of the reference waves Pa ref, Pb ref, Pc ref is never greater than U/2 and there are no saturation effects as long as:

or when:

The voltage drop is not more than 9% [LAB 95]. We observe that the increase in amplitude for Vref relative to U/2 when using this technique is the same as with space vector modulation [LAB 98].

In addition, with synchronous sampling of the reference waves, centering gives (over each period of the carrier in the case of sawtooth carriers or over each half-period of the carrier in the case of triangular carriers) the same duration for the time over which the vector (xa, xb, xc) is (0,0,0) and the time over which it is (1,1,1), in other words, the time interval over which all the switches S′j are closed and the time interval over which all the switches Sj are closed.

As a result, centered modulation using a triangular carrier and synchronous sampling of the reference waves over each half-period is indiscernible from space vector modulation in the case of a two-level three-phase voltage inverter [LAB 98].

1.6.3. “Sub-optimal” modulation

This method can produce a result close to that of centered modulation in terms of maximum amplitude that can be achieved for the reference waves when they form a balanced three-phase system of sinusoidal voltages. It takes the homopolar component of the voltages Pj ref to be a sinusoidal wave of amplitude 0.09 U whose pulsation is three times that of the reference waves [LAB 95] (Figure 1.17):

[1.38]

Figure 1.17.Sub-optimal modulation with a triangular carrier

Vref can then achieve amplitudes of up to 1.15 U/2 without introducing any overmodulation effects [LAB 95].

1.6.4. “Flat top” and “flat bottom” modulation

Flat top modulation involves setting the largest of the reference waves Pj ref to be equal to 1, by requiring the homopolar component to have a value equal to:

[1.39]

This strategy (Figure 1.18) is intended to reduce switching losses by avoiding any switching from taking place in a given leg over the time period where its reference voltage Pj ref n is largest.

Setting the voltage Pj ref n equal to 1 over this interval is equivalent to keeping Sj constantly closed, since we must have:

Figure 1.18.Flat-top modulation with a triangular carrier

Similarly, flat-bottom modulation sets the most negative of the reference waves Pj ref equal to −1 by setting the homopolar component equal to:

[1.40]

which means that S′j is kept constantly closed for each leg over the intervals where Pj ref is most negative (Figure 1.19).

Figure 1.19.Flat-bottom modulation with a triangular carrier

Flat-top modulation (or flat-bottom modulation) implies an unequal distribution of current between the two switches of each leg, since current flows in switches Sj (or S′j) over an interval equivalent to one third of the period of reference waves uj ref in the case of a balanced sinusoidal three-phase system.

This drawback can be addressed by combining these two types of modulation: the most positive (largest) of the reference waves Pj ref is set to 1 when this wave is greater than the absolute value of the smallest of those waves, and the most negative (smallest) of the reference waves is set to −1 when its absolute value is greater than that of the largest of the reference waves (Figure 1.20).

Figure 1.20.Combined flat-top-flat-bottom modulation

1.7. Conclusion

In this chapter we have derived the equations connecting the desired reference values for the phase voltages with the reference values for the leg voltages in case of a two-level three-phase voltage inverter feeding a balanced three-phase load connected in a star configuration when the legs are controlled using carrier-based PWM.

In particular, we have shown that centered PWM with a triangular carrier is indistinguishable from space vector PWM and that the flat-top and flat-bottom strategies can be used to reduce switching losses at a given PWM frequency by avoiding the need to switch for each leg during certain intervals.

We have not considered issues such as harmonic content of the voltages produced using these techniques and the influence on this content of the type of modulation chosen (sine, centered, sub-optimal, or flat-top-flat-bottom) or of the type of carrier wave used2 (conventional or reverse sawtooth, triangular, or random). Discussion of these issues would require a dedicated chapter on the topic.

1.8. Bibliography

[BOO 88] BOOST M.A., ZIOGAS P.D., “State-of-the-art carrier PWM techniques: a critical evaluation”, IEEE Trans. Ind. Appl., 24(2), 271–280, 1988.

[HAU 99] HAUTIER J.P., CARON J.P., Convertisseurs statiques: méthodologie causale de modélisation et de commande, Edition Technip, Paris, 1999.

[HOL 93] HOLZ J., “On the Performance of optimal pulse width modulation technique”, EPE Journal, 3, (1), 17–6, 1993.

[HOU 84] HOULDSWORTH J.A., GRANT D.A., “The use of harmonic distorsion to increase the output of a three-phase PWM inverter”, IEEE Trans. Ind. Appl., 20(5), 1224-1228, 1984.

[KAS 91] KASSAKIAN J.G., SLECHT M.F., VERGHESE G.C., Principles of Power Electronics, Addison Wesley, Reading, MA, 1991.

[KAZ 94] KAZMIERKOWSKI M.P., DZIENAKOWSKI M.A., “Review of Current Regulation technique for three-phase PWM Inverter”, IEEE-IECON, Bologne, vol. 1, p. 567–575, 1994.

[LAB 95] LABRIQUE F., BAUSIÈRE R., SÉGUIER G., Les convertisseurs de l’électronique de puissance 4: la conversion continu-continu, Lavoisier, Paris, 1995.

[LAB 98] LABRIQUE F., SÉGUIER G., BUYSE H., BAUSIÈRE R., Les convertisseurs de l’électronique de puissance 5, Lavoisier, Paris, 1998.

[LAB 04] LABRIQUE F., LOUIS J.P., Modélisation des onduleurs de tension en vue de leur commande en MLI, Chapter 4. In: LOUIS J.P. (ed.), Modèles pour la commande des actionneurs électriques, p. 185–213, Hermès, Paris, 2004.

[LOU 04a] LOUIS J.P. (ed.), Modélisation des machines électriques en vue de leur commande: Concepts généraux, Hermes, Paris, 2004.

[LOU 04b] LOUIS J.P. (ed.), Modèles pour la commande des actionneurs électriques, Hermes, Paris, 2004.

[LOU 95] LOUIS J.P., BERGMANN C., “Commande numérique des ensembles convertisseurs-machines, (1) Convertisseur-moteur à courant continu”, Techniques de l’ingénieur, D 3641 and D 3644, 1995, “(2) Systèmes triphasés: régime permanent”, Techniques de l’ingénieur, D 3642, 1996, “(3) Régimes intermédiaires et transitoires”, Techniques de l’ingénieur, D 3643 and D 3648, 1997.

[MOH 89] MOHAN N., UNDELAND T., ROBBINS W., Power Electronics, John Wiley & Sons, Chichester, 1989.

[MON 93] MONMASSON E., HAPIOT J.C., GRANDPIERRE M., “A digitalc Control system based on field programmable gate array for AC drives”, EPE Journal, vol. 3, n° 4, p. 227–234, 1993.

[MON 08] MONMASSON E., CIRSTEA M.N., “FPGA Design Methodology for Industrial Control Systems-A Review”, IEEE Transactions on Industrial Electronics, vol. 54, n° 4, p. 1824–1842, 2007.

[SEG 04] SÉGUIER G., BAUSIÈRE R., LABRIQUE F., Electronique de puissance, 8th edition, Dunod, Paris, 2004.

[SEM 04] SEMAIL E., LOUIS J.P., Propriétés vectorielles des systèmes électriques triphasés, chapitre 4. In: LOUIS J.P. (ed.), Modélisation des machines électriques en vue de leur commande: Concepts généraux, p. 181–246, Hermes, Paris, 2004.

1 Chapter written by Francis LABRIQUE and Jean-Paul LOUIS.

2. In all cases we have taken the example of a triangular carrier wave.

Chapter 2

Space Vector Modulation Strategies1

2.1. Inverters and space vector PWM

2.1.1. Problem description

In variable speed control, the purpose of an inverter is to control the power delivered to the (synchronous or induction) motor by means of values averaged over a switching period Td. The structure of such a converter is shown in Figure 2.1.

It can clearly be seen that this structure is very similar to the structure of a full-bridge chopper used to feed a DC current motor, which is not surprising considering the nature of the sources connected to the converter: DC bus (voltage source) as input and machine (current sink) as output.

We will make a number of assumptions in our study of this structure:

– we will treat the DC voltage bus as ideal (zero impedance, voltage E=Const);

– we will assume perfect switches;

2.1.2. Inverter model

2.1.2.1. Initial equations

The traditional way of studying such an inverter involves introducing the connectivity functions associated with each leg of the inverter [LAB 04].

Figure 2.1.Schematic of a three-phase voltage inverter

The connection function ci is associated with leg :

such that:

[2.1]

[2.2]

We can express the phase voltages as a function of ca, cb, and cc:

[2.3]

[2.4]

We can therefore replace the third equation in our initial system [2.3] with equation [2.4]. This gives us an equation with the form:

[2.5]

where:

[2.6]

2.1.2.2. Transformation 3/2

2.1.2.2.1. Property

Here we will make use of a property of the matrix T32:

[2.7]

2.1.2.2.2. Application

Expression [2.6] for (v3s) can therefore be written in the following form:

[2.8]

We now transform this into the basis (α, β) using the Concordia transform. This is written as:

[2.9]

where:

The assumption that the zero sequence component is zero enables us to simplify equation [2.9]:

[2.10]

We can then replace (v3s) with this expression in equation [2.8]:

[2.11]

such that:

[2.12]

Table 2.1.Normalized voltage vectors αβ as a function of inverter status

We will use this equation to calculate the values of υα and υβ that can be produced by the inverter at a given instant. These values (in “normalized” form) are listed in Table 2.1 as a function of the possible combinations of switch states (and hence connection functions ci).

The eight available combinations for the three connection functions result in seven accessible points in the plane (υα, υβ), as can be seen in Figure 2.2.

Figure 2.2.