Rotating Reference Frame Control of Switched Reluctance

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ROTATING REFERENCE FRAME CONTROL OF SWITCHED RELUCTANCE

MACHINES

A Thesis

Presented to

The Graduate Faculty of The University of Akron

In Partial Fulfillment

of the Requirements for the Degree

Master of Science

Tausif Husain

August, 2013 ROTATING REFERENCE FRAME CONTROL OF SWITCHED RELUCTANCE

MACHINES

Tausif Husain

Thesis

Approved: Accepted:

______Advisor Department Chair Dr. Yilmaz Sozer Dr. J. Alexis De Abreu Garcia

______Committee Member Dean of the College Dr. Malik E. Elbuluk Dr. George K. Haritos

______Committee Member Dean of the Graduate School Dr. Tom T. Hartley Dr. George R. Newkome

______Date

ABSTRACT

A method to control switched reluctance motors (SRM) in the dq rotating frame is proposed in this thesis. The torque per phase is represented as the product of a sinusoidal inductance related term and a sinusoidal current term in the SRM controller. The SRM controller works with variables similar to those of a synchronous machine (SM) controller in dq reference frame, which allows the torque to be shared smoothly among different phases. The proposed controller provides efficient operation over the entire speed range and eliminates the need for computationally intensive sequencing algorithms.

The controller achieves low torque ripple at low speeds and can apply phase advancing using a mechanism similar to the flux weakening of SM to operate at high speeds. A method of adaptive flux weakening for ensured operation over a wide speed range is also proposed. This method is developed for use with dq control of SRM but can also work in other controllers where phase advancing is required. The proposed adaptive control method uses the command and actual currents to adaptively determine the required amount of flux weakening.

Through the unique features of dq controls, the proposed method provides an analogous control to synchronous machines for SRM while achieving a lower ripple and efficient and wide speed range of operation depending on the motors application. This has been verified through simulations and experiments.

iii

DEDICATION

I dedicate my work to my family, friends and advisors who supported me throughout the course of my degree.

ACKNOWLEDGMENTS

I wish to express my sincere gratitude to my advisor, Dr. Yilmaz Sozer, for his guidance, encouragement and support during my graduate studies. His technical knowledge, managerial skills and human qualities have been a source of inspiration.

I’m so grateful to my professors, Dr. Malik Elbuluk for his help and guidance during my studies at the University of Akron. I wish also to thank my committee member, Dr. Tom Hartley for his help in different moments through my research and for serving his advice whenever needed.

I would like also to acknowledge my colleague students for their continuous help and for providing the best educational atmosphere and enjoyable moments that we spent together during our work.

Finally and most importantly, I would like to thank my family for their unconditional help and encouragement all the time.

TABLE OF CONTENTS

Page

LIST OF TABLES ...... x

LIST OF FIGURES ...... xi

CHAPTER

I. INTRODUCTION ...... 1

1.1. Switched Reluctance Motors ...... 1

1.2. Switched Reluctance Motor Configurations...... 1

1.3. Advantages and Disadvantages of SRM...... 3

1.4. Motivation for Research ...... 5

1.5. Thesis Outline ...... 6

II. LITERATURE REVIEW...... 7

2.1. Introduction...... 7

2.2. Principle of Operation...... 8

2.3. Converter Topologies ...... 15 vi

2.4. SRM Modeling ...... 18

2.5. SRM Control and their Objectives ...... 20

2.5.1.Torque ripple minimization ...... 23

2.5.2.Excitation parameter control ...... 27

2.6. Conclusions...... 29

III. DQ CONTROL OF SWITCHED RELUCTANCE MACHINES ...... 31

3.1. Introduction...... 31

3.2. Motor Control in the dq Reference Frame...... 32

3.3. Proposed dq Control Method ...... 37

3.3.1.Negativity removal...... 38

3.3.2.Non-linearity block ...... 43

3.3.3.Phase advancing using dq ...... 46

3.3.4.Torque estimation using dq...... 48

3.4. Negativity Removal Block Outputs from Different Commands ...... 49

3.5. Conclusions...... 51

IV. ADAPTIVE FLUX WEAKENING OF SRM USING DQ CONTROL ...... 52

4.1. Introduction...... 52 vii

4.2. Theory of Demagnetization and Demagnetization Curves...... 53

4.3. Adaptive Flux Weakening of SRMs ...... 56

4.3.1.Selection of optimum position for the controller ...... 58

4.3.2.Threshold current selection ...... 61

4.3.3.Adaptive method ...... 65

4.4. Conclusions...... 69

V. MODELING AND SIMULATION RESULTS ...... 71

5.1. Introduction...... 71

5.2. Modeling ...... 72

5.3. Simulation Results ...... 74

5.3.1.dq control ...... 76

5.3.2.Flux weakening controller ...... 85

5.3.3.Efficiency of dq controller ...... 93

5.4. Conclusions...... 98

VI. EXPERIMENTAL SETUP AND RESULTS ...... 99

6.1. Introduction...... 99

6.2. Experimental Hardware ...... 99 viii

6.2.1.Experimental SRM modeling ...... 100

6.2.2.Inverter used for experimental validation ...... 103

6.2.3.Interfacing circuitry...... 104

6.2.4.dSPACE controller...... 107

6.3. Dynamometer and System Setup ...... 107

6.4. Experimental Results ...... 108

6.4.1.dq control of SRM ...... 111

6.4.2.Adaptive flux weakening using dq control ...... 124

6.5. Conclusions...... 128

VII. CONCLUSION AND FUTURE WORK ...... 129

7.1. Conclusions...... 129

7.2. Future Work ...... 131

REFERENCES ...... 133

LIST OF TABLES

Table Page

5.1 SRM parameters of 110 kW 12/8 SR machine ...... 73

5.2 Data from conventional excitation angle control at 2000 rpm ...... 94

5.3 Data from dq control at 2000 rpm...... 94

5.4 Data from conventional excitation angle control at 5000 rpm ...... 95

5.5 Data from dq control at 5000 rpm...... 96

5.6 Data from conventional excitation control at 7000 rpm ...... 97

5.7 Data from dq control at 7000 rpm...... 97

6.1 SRM parameters of 300 W 12/8 experimental SR machine ...... 101

LIST OF FIGURES

Figure Page

1.1. Internal structure of a 12-8 SRM ...... 2

2.1. A typical SRM drive system with feedback ...... 8

2.2. Flux linkage for aligned and unaligned position ...... 9

2.3. Air gap inductance for two electrical cycles ...... 10

2.4. Energy partitioning in standstill ...... 10

2.5. Energy partitioning when rotor moves from unaligned to aligned position...... 11

2.6. λ-i-θ Characteristics of a 12-8 110 kW machine ...... 14

2.7. T-i-θ Characteristics of a 12-8 110 kW machine...... 15

2.8. Classic bridge converter ...... 16

2.9. Different modes of operation for the classic bridge converter...... 17

2.10. Different method of SRM modeling ...... 19

2.11. Voltage controlled drive...... 21

2.12. Current controlled drive ...... 22

2.13. T-i-θ characteristics of two adjacent phases ...... 24

3.1. SM torque production (values in pu) ...... 34

3.2. abc and dq reference frame vectors ...... 35

3.3. Torque versus 휃 profile of an SRM for different current levels...... 36

3.4. Block diagram of the proposed controller ...... 37

3.5. Conversion of sinusoidal waves to non-sinusoidal wave ...... 39

3.6. Graphical representation of f’ix, fx(θ) and fix with respect to time...... 40

3.7. Torque versus 휃 profile of an SRM and a sinusoidal component fx(θ) ...... 43

3.8. Block diagram regarding the consideration of machine non linearity ...... 44

3.9. Flowchart for offline calculation of Gx(θ) ...... 45

3.10. Current wave shapes with and without phase advancing...... 47

3.11. Block diagram of torque estimator...... 49

3.12. fix a for varying command parameters...... 50

4.1. Demagnetization curve ...... 54

4.2. Flowchart to obtain the DM curves ...... 55

4.3. T-i-θ characteristics ...... 57

4.4. ΔT -i-θ characteristics ...... 58

4.5. Rotor and stator poles’ position at 휃푡ℎ...... 59

4.6. Machine structure illustrating rotor and stator pole widths ...... 60

4.7. Torque, current and rotor position for a phase ...... 61

4.8. Torque, current and rotor position of one phase for the three cases...... 62

4.9. Phase vector of torque command ...... 65

4.10. Block diagram of flux weakening controller ...... 66

4.11. Torque and current of a phase illustrating the main objective of the algorithm .....66

4.12. Flowchart of adaptive flux weakening controller using hill climbing method ...... 68

4.13. The heuristic convex nature of maximum torque per amp w.r.t PA ...... 69

xii

5.1. T-i-θ characteristics of the 110 kW SR using analytic modeling ...... 73

5.2. T-i-θ characteristics of the 110 kW SR using finite element modeling ...... 74

5.3. Torque with traditional control at 1000 rpm ...... 75

5.4. Current with traditional control at 1000 rpm...... 75

5.5. Graphical representation of 퐺푥 (휃) ...... 77

s 5.6. Graphical representation of f ix ...... 77

5.7. Torque using the dq controller with proper tuning at 1000 rpm ...... 78

5.8. Current using the dq controller with proper tuning at 1000 rpm...... 78

5.9. Torque with different machine model at 1000 rpm ...... 80

5.10. Current using the dq controller with different machine model at 1000 rpm ...... 80

5.11. Torque with dq using the FEM based method...... 81

5.12. Torque with dq using the FEM based method...... 81

5.13. Torque with dq using coupled simulation ...... 82

5.14. Current with dq using coupled simulation...... 83

5.15. Torque with dq with a rotor pole width of 21 degrees ...... 84

5.16. Torque with dq with a rotor pole width of 18 degrees ...... 84

5.17. Torque with dq with a rotor pole width of 22 degrees ...... 85

5.18. Torque with dq with a rotor pole width of 22 degrees ...... 85

5.19. Torque with no phase advancing ...... 86

5.20. Phase torque with no phase advancing ...... 86

5.21. Phase current with no phase advancing ...... 87

5.22. Torque with phase advancing ...... 88

xiii

5.23. Phase torque with phase advancing ...... 88

5.24. Phase current with phase advancing ...... 89

5.25. Variation of fq and fd with step response in torque at a speed of 5000 rpm...... 90

5.26. Variation of fq and fd with step response in reference speed at a torque of 65 Nm….91

5.27. Total torque during single pulse mode operation ...... 92

5.28. Phase current during single pulse mode operation ...... 92

5.29. Phase torque during single pulse mode operation ...... 93

5.30. Torque-speed envelope with and without phase advancing ...... 93

6.1. a) Stator of the experimental SRM ...... 101 b) Rotor of the experimental SRM ...... 101 c) Stator and rotor separately of the experimental SRM ...... 101 d) Experimental SRM assembled ...... 101

6.2. T-i-θ characteristics using Arthur Raduns’ model...... 102

6.3. T-i-θ characteristics using finite element analysis...... 102

6.4. Inverter used for experimental implementation ...... 103

6.5. Gate driver circuit block diagram ...... 104

6.6. Encoder interface circuit ...... 105

6.7. Current conditioning circuit ...... 106

6.8. Hardware Circuits ...... 106

6.9. Complete experimental setup...... 108

6.10. Conventional control method ...... 109

6.11. Currents from acquisition method using oscilloscope...... 110

6.12. Currents from acquisition method using control desk...... 110

6.13. Total estimated torque from conventional control ...... 112

xiv

6.14. Estimated phase torques from conventional control ...... 112

6.15. Currents from dq using oscilloscope ...... 113

6.16. Currents from dq using control desk ...... 113

6.17. Total estimated torque from dq controller ...... 114

6.18. Estimated phase torque from dq controller ...... 115

6.19. Reference current commands from dq controller ...... 115

6.20. Actual currents with dq controller ...... 116

6.21. Reference command currents with zero component commanded ...... 117

6.22. Actual currents with zero component commanded ...... 117

6.23. Phase torques with zero component commanded...... 118

6.24. Total torque with zero component commanded ...... 118

6.25. Phase reference current commands with no advancing at 1700 rpm ...... 119

6.26. Phase currents with no advancing at 1700 rpm ...... 120

6.27. Phase torque with no advancing at 1700 rpm...... 120

6.28. Total torque with no advancing at 1700 rpm...... 121

6.29. Reference current commands with phase advancing at 1700 rpm ...... 122

6.30. Actual currents with phase advancing at 1700 rpm...... 122

6.31. Phase torque with phase advancing at 1700 rpm...... 123

6.32. Total estimated torque with phase advancing at 1700 rpm ...... 123

6.33. Response of fq, fd and resulting θPa with load step responses at 5 s., 17 s., 22 s., 31 s., 38 s., 42 s. and 54 s...... 124

6.34. Response of fq, fd and resulting θPa with speed step responses at 8 s., 17 s., 38 s. and 60s...... 125

6.35. Phase currents at a load of 0.3 Nm at 2500 rpm with adaptive control...... 126

6.36. Phase currents at a load of 0.5 Nm at 2500 rpm with adaptive control...... 127

6.37. Phase currents (ch1-3) and total torque (ch4) while running the machine at 2000 rpm without flux weakening ...... 127

6.38. Phase currents (ch1-3) and total torque (ch4) while running the machine at 2000 rpm with flux weakening ...... 127

xvi

CHAPTER I

INTRODUCTION

1.1. Switched Reluctance Motors

Switched Reluctance Motors (SRM) were first invented in the mid-1800s but failed to become a viable solution in industrial needs at that time as it needed a drive to operate it and the mechanical switches at that period were not very efficient. However since the invention of modern power semiconductor switches in the 1960s it started to realize its potential. With further improvements in power electronics, microcontrollers and computer aided design of electric machinery switched reluctance motor performance levels have risen to levels comparable with DC and AC machines. The machine continues to be an area of intrigue for researchers because SRMs with their inherent construction simplicity and ruggedness provide a low cost solution. [1]

1.2. Switched Reluctance Motor Configurations

In terms of construction, SRMs are doubly salient machines with independent phase windings on the stator which are usually made of magnetic steel lamination. The salient nature of the machine means that entire torque is produced on the principle of reluctance.

The rotor is also simple in nature as it is simply a stack of steel laminations without any windings or permanent magnets. Thus, there is smaller losses on the rotor in comparison

to AC induction machines. This also means that there is no need to cool the rotor. It is only required to cool the stator which is easier to achieve. SRMs exist in various configurations in terms of the number of rotor and stator poles and phases depending on its application and design objectives. The basic three phase machine has six stator poles and four rotor poles more commonly known as a 6-4 SRM. The three phase, 12-8 machine as shown in

Fig. 1.1 is a two repetition version of the basic 6-4 version. Thus the fundamental switching frequency of one phase is given by:

휔 푓 = 푚 ∗ 푁 퐻푧 (1.1) 1 60 푟

Figure 1.1: Internal structure of a 12-8 SRM.

Windings exist in the stator of the SRM and are either connected in series or in parallel or a combination of them. In a series connection all the coils are the same and the supply voltage is divided equally among the coils where as in a parallel connection the voltages are the same but the current is divided equally. The choice of a particular configuration is based on application, for example in low voltage applications it is better

to use a parallel structure so that we can apply full voltage across each of the coils.

Energizing a stator phase results in the most adjacent rotor pole-pair being attracted toward the energized stator to minimize the reluctance of the magnetic path. Hence, if we energize the phases in succession we will be generating reluctance torque in either direction of rotation.

SRMs are generally multiphase machines where the phases are electrically and magnetically independent. This is a major advantage as it makes SRMs a more fault tolerant reliable machine. The machines unique nature of independent phases also requires the use of a different type of converter in comparison to the converters for AC induction machines. The most widely used converter topology provides a greater fault tolerance as it is not possible to short the DC bus.

SRMs inherent construction also has some disadvantages as it produces a large torque ripple, high acoustic noise and lower power density in comparison to its competitors the Permanent Magnet Synchronous Machines and induction machines [59].

Rising costs of permanent magnets led to SRMs receiving a lot of attention in the research arena. This led to the development in SRMs which place them at a competitive advantage.

Thus they are being used in a large number of applications such as pumps, vacuum blowers, starter/generators, electric and hybrid vehicles, electric power steering systems and various others devices. [1]-[3]

1.3. Advantages and Disadvantages of SRM

The major advantages [2, 4] of using SRMs are summarized below:

 Simple mechanical structure provides a lower cost of production

 Absence of rotor windings results in lowering copper losses and helps make the

cooling loop simpler

 The independent nature of the phases and converter means that this machine

has higher fault tolerance and robustness in comparison to AC machines. This

is because the loss of one stator phase does not prevent operation of the drive

and it is not possible to have shoot through faults in the converter.

 The currents in SRMs are unidirectional so a lower number of power switches

are required.

 Has a high starting torque

 Low rotor inertia inducing a high torque/inertia ratio

 Can operate over a wide speed range with large constant power regions.

 High efficiency at high speeds

The disadvantages of SRM are:

 The independent nature of the phases also results in need for complex torque

sharing methods to ensure ripple free operation. Thus SRMs inherently suffer

from high torque ripple.

 The ripple in SRMs and other radial forces cause a lot of vibration resulting in

high acoustic noise. 4

1.4. Motivation for Research

Over the years the research trend has moved towards developing more environmentally friendly systems. Electric motors have been an essential component in this regard as they are source of most of the energy conversion and are a key enabling technology for a greener world. SRMs with their inherent construction simplicity, higher reliability and good power density provide a viable solution particularly with the rising price of their closest competitor the permanent magnet motors. However, they are limited in their servo type applications due high torque ripple and the need for complex and computationally intensive control strategies. The complexities in the control comes due to the inherent torque ripple which needs to be reduced and an excitation strategy of the motor for maximizing efficiency.

One reason for complex control strategies is that it is not possible to directly connect an AC supply to the motor leads and run it like other AC machines, i.e. a drive is a must for SRMs and another problem is that we cannot use the popular control strategies of AC machines as we cannot transform the SRM to the rotating reference frame due to its unique features of unipolar currents and independent phases. This research is aimed at developing the suitable control strategy in the rotating reference frame for the SRMs by utilizing the quasi-sinusoidal nature of the torque. This would simplify the control considerably while also providing a systematic procedure for sharing the torque, which would help in ripple reduction by means of a simple control strategy. The next step in this research was to develop a suitable strategy for operation in high speeds by taking advantage of control in the rotating reference frame. 5

1.5. Thesis Outline

This thesis is structured in to seven chapters. Chapter I provides a brief overview of SRMs, their advantages and disadvantages and the motivation and objectives of this thesis.

Chapter II takes an in-depth look at SRMs providing the structure of the machines, its principles of operation and the topology of converters used. This chapter also contains a review of SRM control strategies and the objectives they fulfill.

Chapter III introduces the dq control of SRMs. It first describes control of machines in the rotating reference frame and then introduces the proposed dq control strategy.

Chapter IV introduces how flux weakening will take place in SRMs using the dq control strategy. It describes the theory of demagnetization and how the adaptive flux weakening scheme operates for maximum efficiency.

Chapter V discusses how the SRM has been modeled, both analytically and through finite element analysis for simulation verification. It then describes the digital controller design and its implementation.

Chapter VI presents the experimental setup of the SRM. This setup is used to design, develop, test and validate the proposed scheme.

In Chapter VII the thesis is summarized and its features are highlighted again. This chapter also gives ideas for future work in terms of hardware and software to improve the performance of the controllers.

CHAPTER II

LITERATURE REVIEW

2.1. Introduction

The concept of SRMs were first proposed about a 150 years ago but they were limited in their commercial development. The advent of better switching technology and work done by Professor Lawerence[5] in the 1970s established the design fundamentals and operating principles for commercial development of SRMs. In his work other than describing the fundamentals of SRM design and its drive circuits he also discussed the flux variations, nonlinearities and control of the machine.

SRM drives unlike other AC machines require a power converter and associated control system. This was the main reason why SRMs were not suitable for commercial application until the 1960s. A typical SRM drive usually consists of a power converter and control system. The power converter is connected to a DC supply that applies positive dc voltage to magnetize the machine or apply negative dc voltage to demagnetize the machine.

The control system basically consists of two controllers. One being the outer loop controller. This can be a speed or a torque controller which aims for zero error between the reference command and a feedback. The second one is an inner loop controller which aims to regulate the current and voltage by commanding the necessary gate signals. Another

block which is very important and unique to SRMs is the commutation block which is responsible for phase sequencing and firing. A typical SR drive system is shown in Fig.

2.1.

Control System

Command Outer Loop Reference Inner Loop Gate Phase POWER CONVERTER SRM Controller Current Controller Signals Voltages Feedback

Current or Voltage Feedback Turn on Angle Commutation Position Feedback Turn off angle

Figure 2.1: A typical SRM drive system with feedback.

This chapter describes the principle of operation for SRM and the converter topologies that are generally used. Following that is a literature review on SRM modeling and the method used in this thesis. After that a literature review is presented on general

SRM control and the many primary objective functions. Particular attention is paid to torque ripple minimization schemes and excitation parameter control for SRMs.

2.2. Principle of Operation

Switched Reluctance Motor as its name suggests operates on the principle of reluctance as it is the tendency of an electromagnetic system to attain a stable equilibrium position of minimum reluctance. When a phase is excited the flux induced in the stator pole flows through the rotor structure. This results in the rotor being attracted towards the stator to achieve minimum reluctance. The movement of the rotor poles with respect to the stator 8

poles results in gradual increase and decrease of the reluctance and flux linkage. The minimum reluctance position, also known as the aligned position, is where the inductance and flux linkage is maximum. The rotor has minimum inductance and flux linkage when the rotor and stator is completely unaligned i.e. the rotor is exactly in between two stator poles. The unequal number of stator and rotor poles are important since this ensures that not all poles are aligned or unaligned at the same instant. The flux linkage at completely aligned and unaligned position is shown in Fig. 2.2. The phase inductance variation for two electrical cycles is shown in Fig. 2.3. The energy conversion process that takes place in

SRMs for the production of torque can be explained with the concept of stored magnetic energy W and co-energy W’ as shown in Fig. 2.4.

)

 Completely Unaligned

Completely Aligned FluxLinkage (

0 Current (A)

Figure 2.2: Flux linkage for aligned and unaligned position.

Aligned Inductace Aligned Position Inductance (mH)

Unalgined Unaligned Position Inductace

0 0 Electrical Degrees

Figure 2.3: Air gap inductance for two electrical cycles.

W' FluxLinkage

0 Current

Figure 2.4: Energy partitioning in standstill.

At a constant current, neglecting losses, the electrical energy input into the SRM is equal to the sum of stored magnetic energy W, and the energy converted to mechanical 10

work represented by the co-energy W’. The stored magnetic energy is not lost in the energy conversion process and can be retrieved by the electrical system by using an appropriate converter. In SRMs it is desirable to run the machine at currents high enough to push the machine into magnetic saturation as this results in greater energy conversion ratios. The energy conversion ratio (ER) is given by:

푊 퐸푅 = (2.1) 푊+푊′

W ) 

W' FluxLinkage (

Completely Unaligned Completely Aligned

0 Current (A)

Figure 2.5: Energy partitioning when rotor moves from unaligned to aligned position.

In an SRM with the position changing from the completely unaligned position to the completely aligned position as shown in Fig. 2.5 it can be seen that the energy W’ is the energy being converted to mechanical work and W being the energy that is fed back to

the converter. For constant excitation the incremental mechanical work done can represented as:

푊′ = ∫ 휆(휃, 푖)푑푖 = ∫ 퐿(휃, 푖)푖푑푖 (2.2)

Where the inductance L and flux linkage λ are functions of the rotor position and current. This change in the co-energy occurs for every change in rotor position. Thus the air gap torque in terms of co-energy can be represented as:

푑푊′((휃,𝑖) 𝑖 푇푒 = | (2.3) 푑휃 𝑖=푐표푛푠푡푎푛푡

If the inductance is considered to vary linearly then be using Eqn. 2.2 and 2.3 a simplified equation for the torque production in an SRM can be derived as:

푑퐿((휃,𝑖) 𝑖2 푇 = ∗ (2.4) 푒 푑휃 2

The instantaneous torque of SRM is not constant and the total torque of the machine is given by the sum of individual phase torques. In Eqn. 2.4 the torque is proportional to the square of the current as a result the current can be unipolar to produce unidirectional torque. This unipolar current requirement has several advantages as it means that only one switch is required for controlling the current in a phase making the drive more economical.

This also results in the motor resembling a dc series motor resulting in good starting torque.

An elementary equivalent circuit for SRM for phase voltages in the stator windings can be written as the sum of the resistive voltage drop in the coil and the rate of change of flux linkage with respect to time as shown in Eqn. 2.5

푑휆 푉 = 푖푅 + (2.5) 푝ℎ 푑푡

The flux linkage given in Eqn. 2.5 is a nonlinear function of position and current and can also be written in terms of the inductance as shown in Eqn. 2.6

휆 = 휆(휃, 푖) = 퐿(휃,푖)푖 (2.6)

Thus the phase voltage Eqn. in 2.5 can be written in terms of the phase inductance and speed of the machine as

푑휆 푑퐿(휃,𝑖)𝑖 푑𝑖 푑휃 푑 퐿(휃,𝑖) 푉 = 푖푅 + = 푖푅 + = 푖푅 + 퐿(휃,푖) ∗ + 푖 (2.7) 푝ℎ 푑푡 푑푡 푑휃 푑푡 푑휃

Here the three terms on the right hand side represent the resistive voltage drop, inductive voltage drop and the induced back-emf voltage and the result is similar to the series excited dc motor voltage equation. However the back emf is produced in different ways. In dc machines the back-emf voltage is produced by the rotating magnetic field where as in SRMs it is dependent on the instantaneous rate of change of phase flux linkage.

The back-emf voltage equation is given by

푑휃 푑퐿(휃,𝑖) 푒 = 푖 (2.8) 푑푡 푑휃

The equation given in Eqn. 2.7 can also be used to derive the torque by using the power balance relationship. Multiplying Eqn 2.7 with current results in the power being

1 푑( 퐿 𝑖2 ) 1 푑 퐿(휃,𝑖) 푃 = 푖푉 = 푖 2푅 + + 푖 2 휔 (2.8) 푝ℎ 2 푑휃 2 푑휃

With the first term representing the stator winding loss, the second term representing the rate of change of magnetic stored energy and the third term represents the mechanical power, which is a product of the torque and speed. We can observe from Eqn.

2.8 and 2.4 that the term for the torque production is the same.

The nonlinearity associated with the electromagnetic profile of SRM means that the machines characteristics is heavily dependent on the static λ-i-θ and T-i-θ characteristics of the machine as shown in Fig. 2.6 and 2.7. These characteristics are used in the development of most SRM algorithms.

.35 Aligned Position .3

.25 )  .2

.15 Flux Linkage ( 1

Unalgined 0.05 Position

0 0 40 80 120 160 200 240 280 Current (A)

Figure 2.6: λ-i-θ characteristics of a 12-8 110 kW machine.

200

150

100

Torque (Nm) Torque -50

-100

-150

-200 0 5 10 15 20 25 30 35 40 45 Rotor position mechanical ()

Figure 2.7: T-i-θ characteristics of a 12-8 110 kW machine.

2.3. Converter Topologies

SRMs unique structure makes it necessary for the use of a converter unlike AC machines. The unipolar nature of the currents coupled with the stator phases being electrically isolated means that the power converters used in SRMs is quite different from those used in AC machines. This lead to the development of a wide variety of converter topologies. The type of converter used is closely dependent on the number of phases and the application of the machine.

In literature [1, 2, 5] several converter topologies have been proposed. The converter with the most versatility and greater industrial acceptance is the classic bridge converter [5] as shown in Fig. 2.8. The main advantage of using a classic converter is that it is easier to control while providing greater flexibility. It allows for independent control 15

of phases which is important when phase overlap is desired. Phase overlap being an essential component for high performance drives. It is suitable for high power, high performance drives. The only disadvantage of this drive is that it requires two switches per phase unlike the other converter topologies.

A B C

Figure 2.8: Classic bridge converter.

The classic converter has four basic conditions under which it can work. The first one is the magnetization mode. In this case both the switches are turned on. This causes the current to pass through switches keeping the diodes off. Thus the phase receives the entire DC bus voltage magnetizing the phase. This increases the current flowing through the phase. In the second mode both the switches are turned off. This results in the current that has already been built in the phase windings to discharge back into the source through the diodes. The diodes being turned on results in a negative DC bus voltage being applied across the phase windings. This causes the current to decrease sharply.

The third and fourth modes of operation are the freewheeling modes. During freewheeling either the bottom or the top switch is turned on. One diode is also turned on for freewheeling. Freewheeling is a method of decreasing the current slowly. The freewheeling paths are alternated so that the switching frequency is reduced. This is beneficial for high-current rated devices. The connections of one phase for the four different modes are shown in Fig. 2.9.

A A

(b) Demagnetization (a) Magnetization

A A

d) Freewheeling 2 c) Freewheeling 1

Figure 2.9: Different modes of operation for the classic bridge converter.

Other converter topologies such as the split-capacitor converter, Miller converter energy- efficient converter I and energy-efficient converter II also exist as discussed in [2].

The prime objectives of these converters are to reduce the number of switches being used.

The disadvantage of the split capacitor is that it can only be used for machines with even number of phases and also apply only half the DC bus voltage. Miller converters are limited in their applications as they cannot sustain demagnetization in one phase simultaneously.

The energy efficient converters allows demagnetization through a separate demagnetization circuitry. There are other topologies that also exist but have not been discussed in the scope of this thesis as the classic bridge converter was used. Details on other converters can be found in [1, 2, 5, 6].

2.4. SRM Modeling

SRMs are always operated under magnetic saturation to maximize the energy conversion ratio. However this has a drawback and makes the machine highly nonlinear for modeling purposes. The nonlinearity of the machine makes the use of linear models unsuitable for high performance applications. Thus a nonlinear model of the machine which can calculate or predict the magnetic characteristics for any rotor position and phase current is necessary. This helps in performance predictions, simulations, design and real time control applications. Researchers have addressed the problem in many ways and can generally be classified as follows:

SRM Model

FEA analysis Geometry Based Nonlinear Models based look up Analytical Models table models

Figure 2.10: Different methods of SRM modeling.

There are various techniques to adapt the machines magnetic characteristics using nonlinear models [7 – 15]. In [7] one of the very early nonlinear SRM models which takes in to account the magnetic saturation was proposed. In this model the flux linkage is selected directly as a model variable rather than considering it as the product of the nonlinear inductance and current [7-9]. This though fail to represent the physical machine model and requires the computation of constants from data already derived from finite element or experimental machine models. A model based on the decomposition of flux linkage into vector functions of position and current was proposed in [10]. Other modeling methods involve a nonlinear representation of the phase inductance machine profile

[11,12]. Fuzzy logic and artificial neural networks (ANN) which are suitable for nonlinear system modeling have also been developed [13-15]. However this is processor intensive and requires a large number of data for training the system.

Another branch of modeling techniques provides an analytical model of the SRM based on machine geometry [16- 20]. In [16-18] Radun developed an analytical model based on the machine geometry and electromagnetic laws. This model is very useful in formulating the machine characteristics and can also be utilized for simulating the physical

machine during the design process. However this model is very complex and cannot be used in real time applications and also is not very accurate at high speed. In [19,20] geometry based analytical models based on Fourier series was proposed. These models were simpler in nature and could be converted for real time controller implementation.

However, they were not accurate enough for physical machine dynamic simulations.

The third method is based on designing machine on a finite element software and then generating the static characteristics of the machine from there. These static characteristics are typically very close to experimental static characteristics. The characteristics are then stored in lookup tables and can be used for dynamic simulation as well as using them in real time controllers. However the large memory allocation requirements limit their use for real time controller implementation. This however is not a major problem and has been used along with the method proposed in [16] for simulation verification throughout this thesis.

2.5. SRM Control and their objectives

Switched reluctance motors are similar to series-excited dc and synchronous reluctance machines in terms of machine characteristics and certain model equations however in control it is very remotely connected to them or any other machine. Thus SRM control strategy is very unique and is not similar to most machines. SRM control can generally be classified in two categories, one for low performance and another one for high performance drives [1,2]. The basic control parameters for SRMs both for high performance and low performance drives are the same. They are the turn-on, turn-off

angles and the phase current. The complexity of choosing these parameters determine whether the drive is high performance or low performance. Turn-on angle is generally described as the position where phase should start being excited. Turn-off angle corresponds to the point at which the phase should start the demagnetization process so that the phase can demagnetize without producing any negative torque. The values of turn- on and turn-off are optimized based on the speed and commanded current. At higher speed it is necessary to start conducting and demagnetizing sooner as the back emf reduces the rate of rise of current initially and increases the rate of current fall during demagnetization.

Vdc Duty V* Cycle Gate Signals ω* Outer Loop PWM Electronic Vph Converter SRM + Controller Controller Commutator - ω θ Angle θon Calculator θoff Speed Calculator

Figure 2.11: Voltage controlled drive.

In low performance drives one method of SRMs is by having a voltage controlled drive. Here a fixed frequency PWM with variable duty cycle is used. The block diagram of this method is shown in Fig. 2.11. Here the angle calculator generates the values of the turn-on and turn-off angles depending on the speed. The PWM controller adjusts the duty cycle based on the voltage command which is determined by the outer loop controller. The electronic commutator generates the gate signals to the converter. The SRM is also

equipped with a position sensor for feedback to the system and is an essential part of all

SRM control as its control highly dependent on rotor position. Sometimes a current sensor is incorporated into the converter for over current protection and feedback.

High performance drives for SRMs are generally torque controlled drives where a torque command is executed by regulating the current in the inner loop. The block diagram for a current controlled drive is shown in Fig 2.12. The reference current is generated based on the load characteristics, speed and control strategy being used. A current is fed back into the current regulator with the help of current sensors in each phase. This allows for greater transient response of the of control system. The turn-on and turn-off angles generated by the angle commutator block varies according to the control strategy as well. High performance drives mainly focuses their control strategies in the torque controller and angle controller blocks.

Vdc

Gate Signals ω* Outer Loop Tref Torque Iref Current Vph Converter SRM Controller Controller Regulator - ω Iph Angle θon Electronic Calculator Commutator Speed θoff Calculator θ

Figure 2.12 Current controlled drive. 22

A higher performance in terms of torque ripple minimization, torque per ampere maximization, efficiency maximization is required in certain applications. Typically all these high performance controllers are current controlled devices. For torque ripple minimization torque controllers are used with advanced torque sharing methods as described the literature [20 – 35]. There are other controllers which are aimed at torque per ampere maximization and efficiency maximization by adjusting the excitation parameters as discussed in literature [36 – 44]. Some of these controllers achieve a complete high grade operation by satisfying both the required objectives [21, 22]. The subsections that follows gives a greater literature review on torque ripple minimization and excitation parameter control.

2.5.1. Torque ripple minimization

Torque ripple is inherent in SRMs due to their doubly salient structure. The magnetization characteristics of individual phases along with the T-i-θ characteristics of the motor dictate the amount of torque ripple in the machine. Hence it is important to first discuss the source of the torque ripple and how to define it. The prime source of the ripple if we look at Fig. 2.13 is the torque dip that exists between two phases that are commutating at the same time. The torque dip means that for ripple free operation some overlap must exist and a form of torque sharing between the two commutating phases need to be established for reducing ripple.

3.5 Phase A Phase B

3 Torque dip

2.5

1.5

1 Phase Torque in N.m Phase Torque

0.5

-0.5 -30 -25 -20 -15 -10 -5 0 5 10 15 Rotor Position in degrees

Figure 2.13: T-i-θ characteristics of two adjacent phases [23]

Several techniques of ripple reduction have been proposed in the literature. In [23,24] the authors provided a brief review of all the techniques that are currently available and how they compare against each other. The reduction techniques can broadly be divided into two basic methods, one involving instantaneous torque control [25 – 27], another involving artificial neural networks and fuzzy controllers [28 -31] and lastly the most widely used method based on current profiling and torque sharing functions [32 – 35].

The instantaneous torque control methods described in [25 -27] attempts to minimize the torque ripple by controlling the torque production of the phases through some form of torque sharing. In [26] a direct instantaneous torque control (DITC) system was proposed. DITC has a simple structure but its applications like other instantaneous torque control methods is limited in their applications due to complex switching rules during the commutation region. The work done in [27] provides an iterative learning method which

negates some of the controllers disadvantages in terms of its model dependencies but still suffers from complex switching rules.

To negate the problem of model dependencies researchers have proposed methods based on fuzzy logic and neural networks [28-31]. These methods are more robust in terms of parametric variations in the machine in comparison to DITC based methods. Fuzzy logic based controllers proposed in [28,29] produces a smooth torque output. These controllers adapt themselves online and also is capable of adjusting excitation angles at higher speeds.

However, the online adaptation methods suffers from the fact that high initial currents will arise.

Neural-network based current minimization techniques have also been proposed in literature in [30,31]. These methods generate the appropriate phase currents using neural networks to model the machine. Although this methods solves the problem of high initial currents it still has a disadvantage as the neural networks require extensive offline training.

The third method that will be reviewed is based on torque sharing functions and current profiling. In [24] a detailed study of torque sharing functions (TSFs) have been presented. TSFs are specified to ideally provide torque sharing between individual phases to meet the primary objective of torque ripple. TSFs are generally classified into linear, sinusoidal, cubic or exponential sharing functions [25,32-34]. The author in [24] also discussed about the secondary objectives the TSFs provide and then proposed a method which directly translated the reference torque to a reference current waveform based on

analytical expressions. An optimization criteria is applied to the TSF using both the primary and secondary objectives.

The technique presented by [32] minimized the peak current requirements of each phase and adopted a linear torque sharing during commutation. This was among the first current profiling based techniques which proved low ripple. In [33] the authors presented a technique using sinusoidal TSFs and a fixed frequency PWM current regulator with the duty cycle being varied. This method however was limited to low current applications as it had fixed excitation regions. In [34] provided another TSF optimization technique based on optimizing the turn-on angles and overlap regions as well as the core torque sharing function. Online tuning of the current profiles have been presented in [35]. This method offers very low ripple and is not highly dependent on machine models. However it requires machines to be designed with higher rotor pole widths and is computationally intensive which requires high bandwidth controllers as well as a large memory to operate efficiently.

Certain researchers have adopted completely different routes in torque ripple minimization [36,37]. In [36] a bio inspired method for torque ripple reduction is proposed.

This method developed a simple model of the SRM using intelligent control system based on the computational model of a mammal’s limbic system and emotional processes.

Another approach was that of adapting the rotating reference frame control of AC machines to SRM as done in [37]. Here the SRM inductance was modeled in the dq reference frame which was used with a complex switching strategy to establish smooth torque control. The method was based primarily on producing a smooth inductance profile to generate smooth

torque which is its main drawback. This thesis presents another method of torque ripple minimization in the rotating reference frame.

2.5.2. Excitation parameter control

SRMs require control of its excitation parameters namely the turn-on and turn-off angle position for proper operation. This unique nature of SRMs requiring control for excitation of its phases has led to research in optimization of these excitation parameters

[38-47]. These can be classified as analytical methods, self-tuning methods artificial intelligence methods and lookup table based methods.

The first concept for the need of changing the excitation angles was proposed by

[38]. In [38] the authors observed changing speeds and DC voltage levels had an effect in the back-emf voltage of the SRM. This back-emf voltage affected the rise and fall times of the currents such that it had to be adjusted in different speeds. The turn-on angles are ideally selected so that the actual currents reach the reference current at the onset of pole overlap.

The turn-off angle is generally selected so that so that phase demagnetizes to zero before producing any negative torque. As the speed of the machine increases the back-emf voltage prevents the current from rising up to the reference point at the onset of overlap. It also prevents the current from being demagnetized to zero before the onset of negative torque.

Thus a form of phase advancing is required. In [38] analytical expression for advancing the angle is given. But this requires calculation of the motors unaligned inductance.

The authors in [39] provided another analytical method for determining the turn-on and turn-off angles. The turn-on angle was determined from the nonlinear model and

electrical equations of the SRM rather than the linear model which greatly improves the accuracy. The turn-off angle was determined from the derived turn-on angle and the flux linkage. This method though analytical sounds suffers from the disadvantage of not having a feedback and the system modeling inaccuracies that exist.

A closed loop form of excitation angle was proposed in the research [40-42, 57,

58]. Here the authors presented a method to adjust the turn-on angle to place the first peak of the reference current command at the pole overlap position. This is a closed loop method as the peak of the actual currents are detected and its position is placed at the pole overlap position and the turn-on position advanced accordingly. The method is robust and operates under a wide speed range. The turn-off angle proposed by this research was based on calculating optimum turn-off angles for maximum torque per ampere through experimental sweeps. The data was then used to form a simple equation dependent on the speed and dc bus voltage. The main disadvantage with this method is that its turn-off angle requires huge offline calculations and is highly machine dependent.

Self-tuning methods based have also been proposed in the literature [43-45]. In [43] a self-tuning method was proposed which takes the basic turn-on and turn-off angles proposed in [38] and adapts them by including the effects of the machine geometry, namely the rotor and stator pole arcs and then optimizes them to reduce the copper losses thereby increasing the system efficiency. Artificial intelligence methods have been proposed in [44,

45]. In [44] the authors use a neural network based optimization criteria where the inputs are the speed and the reference current and the turn-on and turn-off angles are the outputs.

An adaptive neuro-fuzzy method was proposed in [45]. This took advantage of both the fuzzy systems expert knowledge and neuro systems learning capabilities.

The importance of optimizing the excitation parameters have been highlighted in

[45] which showed that it can improve the power factor of system. This is a very important factor in the acceptance of SRMs in the industry. However there is a trade-off between optimizing the turn-on and turn-off angles for maximum torque per ampere and efficiency and the torque ripple. Researchers have also focused in finding the optimum trade-off between the two. Optimum criteria fulfilling both these requirements have been proposed in [21, 28, 46, 47]. The authors in these papers proposed optimizing the parameters for low torque ripple at low speed and maximizing efficiency at higher speeds. In [47] the author presented a promising combination of DITC and an analytical switching scheme which adjusts the turn-on and turn-off dependent on the operating conditions. However the analytical equations used does not take the machines nonlinearity into account.

This thesis is aimed at providing at rotating reference frame based control of the

SRMs and like its AC counter parts the need for advancing the turn-off angle for flux weakening to enable high speed operation is necessary. A closed loop adaptive method using minimal machine characteristics is proposed to enable the rotating reference frame control of SRMs over a wide speed range.

2.6. Conclusions

In this chapter the principles of operation of SRMs and their performance characteristics are provided. It is important to understand these as it forms the fundamentals

of work carried out in this thesis. The chapter also discusses some of the converter topologies that are used for SRMs. The particular converter used for this research is discussed in greater detail. It is essential to have an accurate model of the SRM for verification of the control strategies thus a short literature review on the SRM modeling techniques have been included in this chapter. It discusses the advantages and disadvantages of some common methods for real time controller implementation and dynamic simulations. Finally this chapter presents an overview and literature review of

SRM control. Particular attention is paid in the review of methods used for reducing the torque ripple and optimizing the excitation parameters of SRMs. From the review it is concluded that there is no suitable control methods available for controlling the SRM with the objective of torque ripple minimization. This thesis in the next chapters presents a novel method of dq control of SRMs and an adaptive method of performing flux weakening on

SRM with the main objective of providing an analogous control to synchronous machines.

CHAPTER III

dq CONTROL OF SWITCHED RELUCTANCE MACHINES

3.1. Introduction

In this thesis, the torque-sharing among the phases is implemented in the dq rotating reference frame in the controller. This controller is analogous to that used in synchronous machines (SM) [48], and thus, it distributes the torque production responsibility smoothly among the different phases. The method simplifies the required control scheme and makes it very efficient and effective. The proposed SRM dq control is developed for applications with the requirements of reducing torque ripple at low speeds and supporting high speed operation [49]. In previous research, the SRM inductance was modeled in the dq-reference frame [37, 50], although the main objective in the SRM control is to produce smooth ripple free torque and not to obtain a smooth inductance profile. Therefore, it would be more practical to concentrate on utilizing the torque production scheme obtained by the dq modeling rather than the inductance modeling in that reference frame. In this research, the torque is represented in the controller as the product of a sinusoidal inductance related term and current dependent terms. This sinusoidal inductance related terms are achieved by correction terms such that they become like a sine wave. Then, by commanding the current related terms as sinusoidal components, the produced torque will be smooth like that produced by the SM 31

machines. The dq control scheme for SRM has several advantages. First, the angle decoder needed by SRM controller is removed since the dq control automatically enable the appropriate phases. Second, the method inherently reduces the torque ripple. Third, phase excitation duration can be manipulated which helps in producing higher torques at higher speeds. Finally, phase advancing could be implemented easily without the need of switching between different controllers. The proposed controller is much simpler than the conventional SRM controller despite its ability to perform all the major control functions that are typically required.

The chapter is organized with Section 3.2 describing the principle of motor control in the dq reference frame for synchronous machines and how it can be related to the SRM. Following that Section 3.3 goes into further details of the proposed controller and the theory behind its operating principles.

3.2. Motor Control in the dq Reference Frame

The advancement of machines has continued to increase in the past century with the induction motor being the motor of choice for most of the century. This led to greater investigation and analysis of these machines, which in turn lead to the development of the reference frame theory. With this mathematical tool it was possible to transform the phase variables into an arbitrary reference of choice which would make the control of the motors easier. Fortunately this was not limited to just induction machines, but it could also be applied to permanent magnet synchronous machines (PMSM) and synchronous machines (SM).

The common thing among these machines was that their phases were coupled and this lack of coupling in SRM limited its application there. A novel method has therefore been developed in this thesis to control the SRM using dq transformation. This method involves a transformation of the SRMs variables into the rotating frame. The controller treats the commands from the outer loop controller as signals for a synchronous machine.

A virtual SM to SRM converter block converts the signals into the SRMs reference frame for the current regulator. This method reduces the amount of control complexity of the

SRM controller, eliminates the need for phase encoder and removes the torque ripple through a collaborative torque production among the phases.

Consider a three phase SM with the phases denoted as A, B and C. The torque production in SM is quite smooth as shown in Fig. 3.1. From the figure it can be observed that torque is produced based on an interaction between two fluxes. The first flux is the rotor flux which has fixed value and is attached with the rotor. The second flux is the stator flux which is derived by the SM control system. To maintain fixed smooth output torque, the stator flux and the angle between the rotor and stator fluxes has to be regulated. This task of maintaining a fixed angle between the two fluxes of SM machine can be very difficult to achieve if the control functions are implemented directly in the abc reference frame voltages as the three voltages have to be controlled collectively while observing the rotor flux position. Control in the transformed dq reference frame is a better alternative. In this method, two rotating perpendicular axes known as the q and the d -axes are used. All the variables – voltages, currents and fluxes- are projected in these

axes and the control functions are implemented on those projected q and d components.

Later, the required control action is transformed back to the abc reference frame.

Current 1 Ia Ib 0 Ic

Current P.U Values Current -1 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Time (s) Flux 1 a b 0 c P.U Values  -1 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Time (s) Torque 2 Ta Tb 1 Tc Total Torque

Torque P.U Values Torque 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Time (s)

Figure 3.1: SM torque production (values in pu).

In three-phase non-salient SM, the produced torque is given by [51]

푇 = 퐾(휆푎푠휆푎푟 + 휆푏푠휆푏푟 + 휆푐푠휆푐푟)

2휋 2휋 = 퐾 (sin2 휃 + sin2 (휃 − ) + sin2 (휃 + )) (3.1) 3 3

where T is the torque, λxy is normalized flux of stator (y is s) or rotor (y is r) phase is x, K is constant and 휃 is rotor position. As λxy is sinusoidal, the sum of the torques from the three phases becomes constant. As mentioned earlier an easy approach to control SM is to convert all the variables to the dq reference frame where these are treated as DC variables as [51]. This helps develop an analogous control between AC and DC machines.

′ 푇 = 퐾 (휆푞푠휆푑푟 − 휆푑푠휆푞푟) (3.2)

′ 퐾 is a constant, and 휆푞푗 , 휆푑푗 is flux component in the q and d axes, respectively 푗 = 푠, 푟.

Generally, the stator flux is adjusted by controlling the stator current. The transformation from the abc domain to the dq frame is given by Eqn. 3.3 and represented graphically in

Fig. 3.2.

2 푓 = 푓 − 푗푓 = 푒−푗휃[푓 + 푎. 푓 + 푎2 푓 ] (3.3) 푞푑푠 푞 푑 3 푎 푏 푐

푗2휋 where f stands for voltage, current or flux, 푎 = 푒 3 and 휃 is the rotor position.

Fb Fq

θ Fa

Figure 3.2: abc and dq reference frame vectors.

In SRM, the torque is produced by the interaction between the current (or the flux) and the rate of change of inductance with the rotor position (the angle(휃)) [7]. For three- phase SRM, the torque (T) can be represented by

푇 = 푓 푓 (휃) + 푓 푓 (휃) + 푓 푓 (휃) 푖푎 휃푎 푖푏 휃푏 푖푐 휃푐 (3.4)

푓 푓 where 푖푥 is a function related to the current and 휃푥 is a nonlinear function related the rotor position where x=a, b or c. The typical torque versus θ profile of an SRM for different current levels is shown in Fig. 3.3. The torque curve looks like a distorted sine

푓 푓 wave depending on 휃푥 while the amplitude of the curve depends on the current. If 휃푥 is

푓 푓 taken as a pure sine signal, then making 푖푥 in Eqn. (3.4) a sine wave in phase with 휃푥 would produce constant torque as in the case of SM in Eqn. (3.1). However, there are

푓 non-idealities in the case of SRM that defy this similarity. First 휃푥 is not a pure sine wave and second 푓푖푥 should be always be non-negative.

300 I=40A I=80A I=120A 200 I=160A I=200A I=240A 100 I=280A I=320A

Torque (Nm) Torque Region Region II Region III I -100

-200

-300 0 5 10 15 20 25 30 35 40 45 Rotor Postion Mechanical 

Figure 3.3: T-i-θ profile of a 110 kW SRM. 36

푓 푓 푑푞 By representing the terms 푖푥 and 휃푥 as sinusoidal terms, the control of SM machine can be applied for controlling SRM. If this controller could be implemented, it would be possible to control the torque of individual phases collectively and eliminate the

푓 significant torque dips during the commutation. Moreover, most of the distortion in 휃푥 occurs around θ=0 and θ=π; the dq control command currents are close to zero around these regions having reduced distortion effect. Finally, in Region III, the commanded current decays smoothly which suits the demagnetization region characteristics.

3.3. Proposed dq Control Method

SM to SRM conversion Block

f’ia Adaptive Flux fq dq fia Non Iiaref Sa Va Negativity Iibr Current Weakening to f’ib fib Linear Sb Inverter Vb SRM Removal ef Regulator Controller fd abc f’ic fic Model S V fq c c Speed ω ref Controller θ Iphases Mechanical ω Load

Figure 3.4 Block Diagram of the proposed controller.

The block diagram of the proposed controller is shown in Fig. 3.4 The torque controller commands fq which is distributed among the three phases through the

푓 푓 components of 푖푥 which are in phase with 휃푥. The phase advancing can be achieved by the term fd as needed above base speed. The components fq and fd are converted to the abc domain by the dq to abc conversion block to produce f’ia, f’ib and f’ic. The conversion can be defined by the following matrix:

푐표푠휃 푠푖푛휃 1 푓′푖푎 2휋 2휋 푓푞 2 푐표푠 (휃 − ) 푠푖푛 (휃 − ) 1 [푓′ ] = [ 3 3 ][푓 ] (3.5) 푖푏 3 푑 2휋 2휋 푓′푖푐 푐표푠 (휃 + ) 푠푖푛 (휃 + ) 1 푓0 3 3

Those variables have negative values and thus cannot be implemented using Eqn. 3.4.

Thus, the negativity removal block manipulates them by distributing the torque portion of the negative one(s) between the other phases. The negativity removal block thus produces the non-negative variables fia, fib and fic to produce the same required torque. As stated

푓 fix previously, the term 휃푥 in Eqn. 3.4 is a distorted sine wave, but the commanded term

푓 is produced assuming a pure sine 휃푥 . The nonlinear model block then makes the

fix 푓 adjustments in to compensate for the distortion in 휃푥 and to generate the currents commands ia,ref, ib,ref and ic,ref. The remaining blocks in Fig. 3.4 are the same as any SRM controller. The descriptions of the negativity removal, nonlinear model and phase advancing blocks are explained in detail in the following subsections.

3.3.1. Negativity removal

The variable f’ix is generated as a sinusoidal function. However, the actual value fix should be positive, and then negative values of f’ix should be manipulated. This is achieved by distributing any negative commanded current between the other phases that has positive current command as shown in Fig 3.5. At a particular instant either one or two phases produce negative values, since the sinusoidal signals are 120 degrees phase shifted from each other in a three phase machine. The SRM controller aims to maintain

푑퐿(휃,푖) the sinusoidal current terms aligned with , i.e. to be in phase with inductance 푑휃

푑퐿(휃,푖) variation. Therefore, positive torque is produced when the signal is positive. 푑휃

60 120

40 100

20 Negativity 80

0 60 fix f'ix Removal -20 Block 40 -40 20

-60 0 Time Time

Figure 3.5: Conversion of sinusoidal waves to non-sinusoidal waves.

For simplicity, the torque production can be written as:

2휋 2휋 푇 = 푓′ cos 휃 + 푓′ cos (휃 − ) + 푓′ cos (휃 + ) (3.6) 푖푎 푖푏 3 푖푐 3

2휋 with phase shifts between the three phases and where 푓′ depends on the required 3 푖푥 torque. The portions of 푓′푖푥 that is negative must be distributed to the other phases which have positive currents since the SRM current has to be always positive.

As the controller reference frame is aligned with the positive torque production region, given that the torque is cosine function producing positive torque when the cosine function is positive, the sharing implemented can be explained by the following example where suppose 푓′푖푎 푓′푖푏 are negative at certain instant then the portion of torque taken by 푓푖푐 would be given by

2휋 ′ 푓′ 푐표푠 (휃− ) 푓 푖푎 푐표푠 (휃) 푖푏 3 푓 = 푓′ + 2휋 + 2휋 (3.7) 푖푐 푖푐 푐표푠(휃+ ) 푐표푠(휃+ ) 3 3 39

If only 푓′푖푎 is negative then

2휋 푓′푚 푐표푠푚 (휃+ )∗푓′ 푐표푠 (휃) ′ 푖푐 3 푖푎 푓푖 = 푓 푖 + 2휋 2휋 2휋 (3.8) 푐 푐 (푓′푚 푐표푠푚 (휃+ )+ 푓′푚 푐표푠푚 (휃− ))∗푐표푠(휃+ ) 푖푐 3 푖푏 3 3

Where m the value of m determines whether a sinusoidal TSF or linear TSF is to be implemented. When the value of m is 1 a linear TSF is implemented and a value of 2 indicates a sinusoidal TSF. Increasing the order of m results in higher order sinusoidal

TSFs but the tradeoff for the control complexity involved in getting a slightly better sharing is not profitable at higher orders. These equations ensure that the required torque will be implemented using only positive values at 푓푞 with m = 2 is presented in Fig. 3.6.

The equations presented here are the core basic equations and can be simplified to remove all divisions making it easy to implement in hardware. After the negativity cancellation the new values of the signals are 푓푖푥. These values maintain the torque sharing smoothness discussed previously.

100

0 f'ix

-100 1 a

 0 f

-1

100

fix 50

0 Time

Figure 3.6: Graphical representation of f’ix, fx(θ) and fix with respect to time. 40

One such simplification with m=2 is possible by considering that the d axis current is always zero. i.e.

푓′푖푎 = 퐼 cos 휃 (3.9)

Now for example, assume 푓′푖푐 is negative, and its role is needed to be distributed to the positive current phases. Consider first that the negative role of phase C is entirely taken by phase A. This can be obtained by adding to 푓′푖푎 another term 푓푥푖푎 such that

2휋 푓푥 cos 휃 = 푓′푖 cos (휃 + ) (3.10) 푖푎 푐 3

It follows that

2휋 2휋 푓′ cos(휃+ ) 푓′ cos(휃+ ) ′2 푖푐 3 퐼 푖푐 3 푓푖푐 푓푥푖푎 = = = ′ (3.11) cos휃 퐼 cos 휃 푓 푖푎

Similarly, if the role of phase C is to be taken by phase B only the required correction should be:

′2 푓푖푐 푓푥푖푎 = ′ (3.12) 푓 푖푏

The role of phase C can be shared by both phases A and B. The fraction that has to be shared by every phase should be selected to improve the efficiency. One way to do that is to give the phase a fraction of the phase C torque based on the magnitude of its corresponding sine value. In this way, the phase that can produce more torque will have larger portion which means torque/ampere maximization. Then the new values of the

̃ ̃ correction terms 푓푖푎 and 푓푖푏 are

푓′2 푓′2 푓′2 ̃ 푖푐 푖푎 푖푐 푓푖푎 = ′ ′2 ′2 = ′2 ′2 푓′푖푎 (3.13) 푓 푖푎 푓푖푎 +푓푖푏 푓푖푎 +푓푖푏

푓′2 푓′2 푓′2 ̃ 푖푐 푖푏 푖푐 푓푖푏 = ′ ′2 ′2 = ′2 ′2 푓′푖푏 (3.14) 푓 푖푏 푓푖푎 +푓푖푏 푓푖푎 +푓푖푏

The new total signals applied by the phases will be

′2 ′2 ′2 ′2 푓푖푐 푓푖푎 +푓푖푏 +푓푖푐 푓푖푎 = 푓′푖푎 + ′2 ′2 푓′푖푎 = 푓′푖푎 ′2 ′2 (3.15) 푓푖푎 +푓푖푏 푓푖푎 +푓푖푏

′2 ′2 ′2 ′2 푓푖푐 푓푖푎 +푓푖푏 +푓푖푐 푓푖푏 = 푓′푖푏 + ′2 ′2 푓′푖푏 = 푓′푖푏 ′2 ′2 (3.16) 푓푖푎 +푓푖푏 푓푖푎 +푓푖푏

푑퐿 Suppose now phases B and C have negative value at position; hence, their roles have 푑휃 to be carried out by phase A only. By following the same derivation, the new total signal of phase A will be:

′2 ′2 ′2 ′2 ′2 푓푖푏 푓푖푐 푓푖푎 +푓푖푏 +푓푖푐 푓푖푎 = 푓′푖푎 + ′ + ′ = 푓′푖푎 ′2 (3.17) 푓 푖푎 푓 푖푎 푓푖푎

A compact way to represent these relations can be obtained by defining the following notation

1 푥 > 0 푢(푥) = { (3.18) 0 푥 ≤ 0

Then the negativity cancellation can be obtained using:

푓′2+푓′2+푓′2 푓푖 = 푢(푓′푖 ) 푓′푖 푖푎 푖푏 푖푐 푎 푎 푎 ( ′ ) ′2 ( ′ ) ′2 ( ′ ) ′2 (3.19) 푢 푓 푖푎 푓푖푎 +푢 푓 푖푏 푓푖푏 +푢 푓 푖푐 푓푖푐

푓′2+푓′2+푓′2 푓푖 = 푢(푓′푖 ) 푓′푖 푖푎 푖푏 푖푐 푏 푏 푏 ( ′ ) ′2 ( ′ ) ′2 ( ′ ) ′2 (3.20) 푢 푓 푖푎 푓푖푎 +푢 푓 푖푏 푓푖푏 +푢 푓 푖푐 푓푖푐 42

푓′2+푓′2+푓′2 푓푖 = 푢(푓′푖 ) 푓′푖 푖푎 푖푏 푖푐 푐 푐 푐 ( ′ ) ′2 ( ′ ) ′2 ( ′ ) ′2 (3.21) 푢 푓 푖푎 푓푖푎 +푢 푓 푖푏 푓푖푏 +푢 푓 푖푐 푓푖푐

These equations ensure that the required torque will be implemented using only positive

푓 values at 휃푥 and also with high efficiency. After the negativity cancellation the new values of the signals are 푓푖푥. These values maintain the torque sharing smoothness discussed previously. However this simplification is limited to the d-axis command being zero all the time.

3.3.2. Nonlinearity block

This part of the controller brings most of the complications in conventional SRM control methods. However, the proposed SRM dq control uses very simple operations for this block. The components of this block are shown in Fig. 5. For phase A, the term

푓 (휃) 휃푎 in Eqn. 3.6 as shown in Fig. 3.6 is a distorted sine wave and not perfectly sinusoidal as considered in the previous equations in the previous section.

1 fx cos 0.5

0 Normalized Value Normalized -0.5

-1 0 50 100 150 200 250 300 350 Rotor Position  Electrical

Figure 3.7: Torque versus 휃 profile of an SRM and a sinusoidal component fx(θ)

푓 (휃) Then 휃푎 can be represented by:

푓 ( ) 푓 = 퐺 (휃) cos 휃 => 퐺 (휃) = 휃푎 휃 (3.22) 휃푎(휃) 푎 푎 cos휃

퐺 (휃) 푓 fia where 푎 is the distortion term in 휃푎(휃). The commanded term is produced by the dq 푓 controller assuming a pure sinusoidal 휃푎(휃) and since this is not the case, a correction is needed to be made in the term fia using 퐺푎(휃). The value of 퐺푎(휃) is calculated from the T-i-θ characteristics of the machine. For determination of the 퐺푎(휃) values, the T-i-θ graphs for a particular current was normalized and then divided by a sinusoidal function.

This operation results in a correction term. The 퐺푎(휃) tables were found in a similar way for different current levels and then compared with each other. The resulting correction factors were found to be similar, and hence, a single-dimension (1-D) lookup table (LUT) could be used for the correction factor dependent on the rotor position. Fig. 6 shows the flow chart of the process for obtaining 퐺푥 (휃).

fix

I fix xref X LUT

θ 1/Gx(θ ) LUT

Figure 3.8: Block diagram illustrating regarding consideration of machine nonlinearity.

The second LUT is also a 1-D which accounts for the relationship between the commanded torque and the current required to generate that torque. This current serves

for the normalized T-i-θ curve used in the LUT. This table is independent of the rotor position as it was found that the all the normalized T-i-θ characteristic curves were similar and the effect of the position was taken care of by the 퐺푥 (휃).

푠 푠 푓푖푎 Assume now the current related function fia is updated to be 푓푖푎 given by 푓푖푎 = . By 퐺푎 (휃)

s commanding f ia for torque production in Eqn. 3.6, the torque for phase A (Ta) becomes:

푠 푓푖푎 푇푎 = 푓푖푎 푓휃 (휃) = 퐺푎(휃) cos 휃 = 푓푖푎 cos 휃 (3.23) 푎 퐺푎(휃)

This is the required expression to make smooth torque production. A similar discussion can be made for phases B and C. The function 퐺푎(휃) can be represented in a LUT to make the controller simpler.

T-i-θ Characteristics

Normalize T-i-θ for a particular current

Divide by Sinusoidal function

Save Gx(θ)

Figure 3.9: Flowchart for offline calculation of Gx(θ). 45

s Since f ix does not depend on θ, one point per curve (the peak value) can be used to make

s another look-up table to relate the function f ix with the corresponding current value. This approach simplifies the current controller significantly.

3.3.3. Phase advancing using dq

In SM, flux weakening for high speed operation is achieved by commanding a current in the d axis [22]. The suggested SRM dq control can achieve a similar effect

(phase advancing [16]) by commanding fd in the d axis. Consider a case where fq and fd in

fix 푓 Fig. 3.4 are nonzero. Then, the waveform of will be advanced from the 휃푥 as shown in

Fig. 3.10. The advanced fix faces lower back-emf voltage and the current would be able to increase faster. The following equation gives the phase current rate of change for phase

푑휆 푑푙 푑푖 푉푑푐−푅푎푖푎− 휔 푎 = 푑푙 푑휃 (3.24) 푑푡 푑휆 푑푖

푑휆 Here 푖 is phase A current, 푉 is the DC bus voltage, is the flux rate of change with 푎 푑푐 푑푙

푑푙 푑휆 the inductance, is the inductance rate of change with position, 휔 is the speed and is 푑휃 푑푖 the flux rate of change with current.

1 f'ix fx

0.5

Normalized P.U Normalized Values -0.5

-1 0 50 100 150 200 250 300 350 Rotor position Electrical 

Figure 3.10: Current wave shapes with and without phase advancing.

At high speeds, 휔 will have large value and this reduces the current rate of change

푑푙 since 푉 is fixed. However, at small values for 휃, the term has small value. 푑푐 푑휃

Therefore, by performing the magnetization (applying 푉푑푐 across the phase) at early

푑휆 푑푙 angle, the term 휔 will have small value and the current can grow higher and faster. 푑푙 푑휃

Then by commanding a certain value for fd more current will build up at early angles which are useful for torque production. Moreover, fix goes to zero earlier such that the demagnetization will be performed before approaching the generation or negative torque region which is required during high speed operation. The sinusoidal components in (5) and (6) are the components which help in aligning the phases properly with phase advancing while avoiding negative torque production. In this case however it was necessary to put a limit as it is necessary to stop the sharing functions whenever the sinusoidal functions aligned themselves to produce large negative torques at the

beginning of each phase conduction. The value of this limit is discussed in the next chapter.

Therefore, it is now necessary to modify equations from its simplified form as explained above to account for the phase advancing after negativity removal. In simplification the equations for real time implementation the value of m was chosen to be

1. In this case through simple trigonometric simplifications the equations presented in

Eqn. 3.7 and 3.8 reduced to:

3 2휋 3 2휋 푓푖 = 푓′푖 + 푓′푖 ∗ (−.5 + √ ∗ tan (휃 + ) + 푓′푖 ∗ (−.5 − √ ∗ tan (휃 + ) (3.25) 푐 푐 푎 2 3 푏 2 3

With 푓′푖푎 and 푓′푖푏 negative and if only 푓′푖푎 is negative then

′ ′ 푓푖푐 = 푓 푖푐 − 푓 푖푎 (3.26)

These equations can now be easily implemented in a real time controller with the tan tables being stored in a lookup table. This section described how flux weakening with the dq control method can be achieved. However the method for determining the optimum fd has not been discussed yet and would be dealt with in greater detail in the next chapter where an adaptive method closed loop control method is proposed for phase advancing.

3.3.4. Torque estimation using dq

With the presented method it is possible to go in a reverse direction to achieve torque estimation. It can be done by using the tables in the non-linearity block. For the estimation actual phase currents are multiplied with the cosine functions representing the

equivalent phases and then the non-linearity of the phases handled. The steps are shown in the following Fig. 3.11.

퐼푥 푓푖푥 LAT

퐶표푠휃 푇푒푠푡 푥 X

퐺푥 (휃) 휃푥 LAT

Figure 3.11: Block diagram of torque estimator.

3.4. Negativity Removal Block Outputs from Different Commands

It is worth taking a careful look at how the outputs after the negativity removal block is by varying the input commands of fd, fq and f0 to study how they affect the commands into the current regulator.

In Fig. 3.12 the current related command shapes fix after negativity removal block for varying commands of fd and f0 with fq at a constant command of 80 to produce a smooth 80Nm torque at the output.

푓 (휃). 푓 (휃) Figure 3.12(a) shows the shape of 휃푥 When 휃푥 is positive any positive fix will produce positive torque. Figure 3(b) shows the fix shape after the negativity 49

removal block if no phase advancing is applied and complete conduction region is used, i.e. fd and f0 are both 0. The smoothness of the command results in low torque ripple.

 0 f

-1

0 36 72 108 144 180 216 252 288 324 360 (a)

100 fq=80 A, fd = 0 A,f0 =0 A

50 fix (f(A)) 0 0 36 72 108 144 180 216 252 288 324 360 (b) 200 fq=80 A, fd = -40 A,f0 =0 A fq=80 A, fd = -10 A,f0 =0 A 100 fq=80 A, fd = -80 A,f0 =0 A fix (f(A))

0 0 36 72 108 144 180 216 252 288 324 360 (c)

100 fq=80 A, fd = 0 A,f0 =20 A

50 fix (f(A))

0 0 36 72 108 144 180 216 252 288 324 360 (d) Rotor Position 

Figure 3.12:f ix a for varying command parameters

The conduction region can be manipulated by the commands fd and f0. In Fig. 3.12(c), it is observed that the phase is shifted towards the left with the value of fd controlling the amount of phase shift, larger fd results in more phase advancing, here fq was 80 and f0 was

0. The negativity removal algorithm also reshapes the command fix to have higher commands at the beginning of the conduction region. This minimizes the effect of back-

EMF which is similar to SM where fd is applied for flux weakening at higher speeds. The

control designer can thus command fd using this dq controller to achieve phase advancing for increased torque production at higher speed, thereby performing type of flux weakening for the SRM.

It can be seen that when only the f0 is command along with fq the conduction region is reduced as shown in Fig. 3.12(d) Incrementing f0 would further reduce the conduction region. In all the cases the fq, fd and f0 are applied simultaneously.

3.5. Conclusions

In this chapter, a novel SRM controller structure was proposed suitable for traction applications. The proposed controller is similar to the SM controller in the dq rotating reference frame. In the SRM dq controller, the torque component is transformed into the dq frame instead of the inductance or flux like other previous attempts at dq control for SRM. It has a simple structure and removes the need for angle decoding blocks from the SRM control structure. Using only approximate look-up tables for the torque control, the proposed controller can achieve lower torque ripple without any additional torque ripple minimization algorithm. Moreover, the same dq control structure was capable of performing flux weakening like in SM. The controller had one extra feature where the zero component could be varied to change the conduction periods thereby acting as a source of control of system efficiency by trading off some torque ripple minimization capabilities. The next chapter presents a novel method of controlling the d-axis commands for controlling the flux weakening of an SRM.

CHAPTER IV

ADAPTIVE FLUX WEAKENING OF SRM USING DQ CONTROL

4.1. Introduction

SRMs are multiphase machines where the phases are independent of each other and are independently excited to generate torque. The sequence and timing of the phase excitations are utilized to optimize a control objective when commutating the torque production from one phase to another. During the commutation period, the torque production is shared by two phases. Some controllers share torque in a way that maximizes efficiency [22] while others minimize torque ripple [23]. In this chapter, the torque will be shared between the phases through control in the rotating reference frame which is analogous to that used in SM as it was described in the previous chapter and in [53]. The method simplifies the required control scheme and makes it very efficient and effective.

Like most machines, SRMs also face the problem of higher back-emf voltage at higher speeds affecting its control over a wide speed range. The problem was first addressed in [38] where the authors used an analytical method for calculating the amount of phase advance required at higher speed. In [54], the authors presented a method of operation over a wide speed range using the machine geometry information, while in [42] the excitation regions were determined through a mixture of offline calculations for turn- off angle and an automatic method of determining the turn-on angle. All these methods 52

perform the phase advance through open loop control. By making the phase advancing as a part of closed loop control, it can become more robust against parametric variations and external disturbance. The second chapter of this thesis contains a more detailed review of the methods for determining the optimal angle control.

The main objective of this paper is to enhance the high speed operation of the rotating reference frame (dq) based SRM control developed in the previous chapter. The proposed method tries to regulate the phase current at the position where the phase inductance starts to have a sharp decrease (θth) through adjusting fd command for the dq controller. Incorporating the unique features of dq control, the proposed method increases the average torque production of SRMs over a wide speed range.

This chapter first discusses about the demagnetization principle of SRMs and how it should be determined theoretical. It goes on to propose a look table based solution to the problem of determining the optimal turn-off angles. The section following that presents the adaptive method that is proposed in this thesis. This section is sub divided in sections explaining the selection of the optimum positions and threshold current values. Following that is a section on the algorithms adaptive. Following which is the concluding section of this chapter.

4.2. Theory of Demagnetization and Demagnetization Curves

The longer time an SRM phase is excited the higher will be the torque production.

However, if the phase is excited for too long, it may not be possible to make the current reach zero by demagnetization before the phase enters the negative torque region. In this research, a demagnetization curve (DM) is defined as a function that takes the speed and 53

the current as inputs and gives as output the latest angular position to start the demagnetization such that negative torque production can be avoided. For every speed, the

DM relations are presented as curves of current vs. rotor position [55].

Region (b) Region (a)

Figure 4.1: Demagnetization curve.

Consider the two regions in Fig. 4.1 which contains a DM curve for a specific speed. If the current exceeds the DM curve with the operating point in region (b) of Fig. 4.1, the current cannot reach zero before negative torque region. This region starts after angle θe and if the current does not become zero by the time the rotor reach this angle negative torque will be produced. The demagnetization operation is achieved by applying the negative DC bus voltage (–Vdc) across the phase windings. There is a need for a method to generate the DM curve and a control algorithm for using it. This curve can be generated using the current- flux-position (i-λ-θ) curves. The relation between the flux and the applied DC bus voltage during demagnetization is given by:

푑휆 = −푉 − 𝑖 푅 (4.1) 푑푡 푑푐 where 𝑖 is the phase current and 푅 is its winding resistance.

For the generation of the demagnetization the flux-current-position (λ-i-θ) curves of the

SRM is first generated using Finite Element Simulation software. This source of information is utilized in this research to obtain the DM curves that could maximize the torque production capability of the SRM without generating negative torque. The flow chart to obtain the DM curve is shown in Fig. 4.2.

ω=ω*, k=0

λk=ε, θk= θe

ik=fλ-i-θ(λk, θk)

F=Vdc+ ik R

λk+1= λk+∆t F

θk+1= θik> -∆t i ω k M No Yes END

Figure 4.2: Flowchart to obtain the DM curves.

As stated previously, there will be a DM curve for every speed. To generate the DM curve, a case is assumed that the demagnetization started at some angle θ* when the current was in its maximum value. The angle θ* should be selected such that the current value reaches zero only at the angle θe defined earlier. The DM curve is constructed by moving backward starting at the angle θe and assuming the flux is zero (λ0=0) then. The DM curve can be constructed by performing discrete time approximation of the flux differentiation with a

time step of ∆t at a particular speed. Starting from λ0 and θe, the current can be obtained using the i-λ-θ at λ0 and θe (where the current will be zero). By backward differentiation, the rotor position and the flux value can be updated using the following relations:

휆푘+1 = 휆푘 + Δ푡(푉푑푐 + 𝑖푅) (4.2)

휃푘+1 = 휃푘 − Δ푡휔, (4.3)

where 휆푘 and 휃푘 are the flux and the rotor position at sample number k. Then using the i-

λ-θ as the function, the DM current for that speed at θk+1 is given by:

𝑖푘+1 = 푓𝑖−휆−휃(휆푘+1, 휃푘+1) (4.4)

The process continues until the current reaches its maximum value iM. The values for θk+1 and ik+1 are saved as the DM curve for that speed. This algorithm is used offline to generate the DM curves for different speeds which can be stored in a 2-D lookup table to be used in the controller. However, lookup tables require large memory banks and also needs a suitable interpolation techniques, hence to negate this problem an adaptive method which utilizes the principles of demagnetization is explained in the next section.

4.3. Adaptive Flux weakening of SRMs

The adaptive flux weakening method has been developed considering the T-i-θ characteristics of the machine as shown in Fig. 4.3. The concept of phase advancing/flux weakening is used to counter the effect of speed on the back-emf voltage. The rate of change of current per phase can be described as

푑퐿푥 (휃) 푑𝑖 푉푥 −푅푥 𝑖푥−𝑖푥 휔 푥 = 푑푡 (4.5) 푑푡 퐿푥 (휃)

푑퐿 (휃) where 𝑖 is the phase current, 푉 is the phase voltage, 푥 is describing the rate of change 푥 푥 푑푡 of inductance, and 퐿푥 (휃) is the non-linear inductance profile of the machine. Here the back- emf voltage of the machine can be represented by

푑퐿 (휃) 푒 = 𝑖 푥 휔 (4.6) 푥 푑푡

As the speed increases, the back-emf of the machine also increases, which results in the current not rising fast enough to desired values in Region I of Fig. 4.3. The current also decreases slowly resulting in an extension of the Region III further into the generating region. It can be observed that in Region III, a large current will result in a higher average torque but due to the higher back-emf voltage the demagnetization will be slow which results in negative torque production. Thus a tradeoff is required between higher average torque and torque per ampere to determine the optimum amount of phase advancing.

300 I=40A I=80A I=120A 200 I=160A I=200A I=240A 100 I=280A I=320A

Torque (Nm) Torque Region Region II Region III I -100

-200

-300 0 5 10 15 20 25 30 35 40 45 Rotor Postion Mechanical 

Figure 4.3: T-i- θ characteristics.

4.3.1. Selection of optimum position for the controller

The proposed method relies on measuring the phase current at a specified rotor position (θth) and comparing it to a targeted phase current (Ith) to determine the optimum amount of advancing. Therefore the choice of this threshold angle is of great importance.

The main requirements on this position is that it should remain constant in terms of load and speed variations. The controller stability could also be greatly increased if the angle remains the same in terms of parametric changes that the machine faces with prolonged use.

80 I=40A I=80A 60 I=120A I=160A I=200A 40 I=240A I=280A 20 I=320A

0 Delta Torque (Nm) Torque Delta -20

-40

-60 0 5 10 15 20 25 30 35 40 45 Rotor Position 

Figure 4.4: ΔT-i- θ characteristics.

This angle is picked to be the point where the maximum rate of change of torque production takes place. Reviewing the static T-i-θ characteristics in Fig. 4.3, it can be argued that torque changes are more prominent around the aligned and unaligned positions.

This method is based on adapting the turn-off angles such that the rate of change of torque production is particularly higher for motoring operation in Region III when commutation occurs. For further analysis the rate of change of torque with respect to rotor position has been calculated using the static characteristics yielding Fig 4.4. The figure shows that the rate of change of torque production is significant near the aligned and unaligned positions.

Here we can observe that the maximum change happens at the same point for all different current values thereby confirming a multiplicity and independence of this position with respect to the load torque. The angle from which the rate of change of torque starts decreasing also corresponds to the rotor and stator poles’ position where pole overlap starts decreasing as shown in Fig. 4.5. The threshold position can generally be given by

훽 −훽 휃 = 휃 + 푟 푠 (4.7) 푡ℎ 푎푙𝑖푔푛푒푑 2 where 휃푎푙𝑖푔푛푒푑 is the completely aligned rotor position and 훽푟 is the rotor pole width and

훽푠 is the stator pole width as shown in Fig 4.6.

Figure 4.5: Rotor and stator poles’ position at 휃푡ℎ.

BETAS BETAR

Rsh R3 R0

R2 R1

Figure 4.6: Machine structure illustrating rotor and stator pole widths

This position also corresponds to the negative torque peak when the machine is running as shown in Fig 4.7. This position is valid for all currents since the T-i-θ curves shows that torques from the complete aligned position to this angle is linear and has the highest slope which contributes to negative torque production at a rapid rate with small changes in current. Therefore, at the peak of ΔT the negative torque peak would occur while the current is decreasing and the phase is being demagnetized.

In the control algorithm, if the turn-on angle is to be coupled with the turn-off angle for torque ripple minimization then the turn-on angle should be limited. The new parameter

θonlimit is introduced which would set the limit up to which the turn-on angles can be advanced. The value of θonlimit can be found by the same method as that of finding θth by investigating the T-i-θ characteristics of the motor. In this case, the rate of change of torque in Region I is considered. The plots in Fig. 10 show that the maximum change happens at the same position for different current values since SRMs are designed in a symmetric

manner. The focus here is to limit the turn-on angle to avoid any negative peaks in Region

I. The turn-on angle limit is given by:

훽 −훽 휃 = 휃 − 푟 푠 (4.8) 표푛푙𝑖푚𝑖푡 푢푛푎푙𝑖푔푛푒푑 2

Advancing the turn-on limit further would result in large negative torque production at the start of the conduction. The value of this θonlimit was incorporated into the negativity removal block as we would be using the dq control for this proposed adaptive turn-off angle control.

150 80

100 40

Negative Torque Peak at th Current (A) Current

20 (Nm) Torque Actual Current at th 50

Desired Current(Ith) at th

0 -20 3.5 4 4.5 5 5.5 Time (S) -3 x 10

Figure 4.7: Torque, current and rotor position for a phase.

4.3.2. Threshold current selection

The threshold value for the current is the second important feature of this adaptive system. The threshold current is compared with the actual currents at the threshold position.

The result of this comparison is inputs to the adaptive control system. It is necessary for the threshold current to be suitable for all possible loads and torque. Hence, to determine the optimum value of the threshold a data analysis based method was adopted. A prototype nonlinear simulation of the system was performed at a fixed high speed and torque command for three cases. The cases are:

I. No PA

II. Excess PA

III. Optimum PA found through sweeping

The cases plotted in Fig 4.8 is then analyzed in greater detail for the determination of the threshold current.

Legend 200 A Iopt Topt. 150 TnoPA

B  InPA IexPA 100 TexPA data7 Rotor PositionRotor Phase Current (A)/ Phase Current Phase Torque(Nm)/ 50 IcmdEx IcmdOpt IcmdnoPA 0 Position

Simulation Time

Figure 4.8: Torque, current and rotor position of one phase for the three cases.

The green lines in Fig 4.8 represent case I which is with no phase advancing being applied by control system. It can be seen that at higher speeds when the current is stopped

at the start of the negative torque production region the current cannot decrease fast enough so a large amount of negative torque is produced as the current cannot demagnetize fast enough due to the back emf. The high current in this region also doesn’t produce a high positive torque at the angle shown by line A as the current cannot rise fast enough to the reference due to the back emf again. Thus for increasing efficiency in terms of higher Avg.

Torque it is necessary to reduce the conduction angle by advancing the turn off region.

This is not the ideal case as the turn off angle is never at this point but this example helps illustrate the problem related with not advancing the turn off at higher speeds.

The blue lines in Fig 4.8 represent case II which is with excessive phase advancing being applied by control system. Now to reduce the negative torque production which reaches its peak at the threshold angle as discussed in the earlier sections we advance the turn off angle. The results is that the negative torque production of that phase reaches to almost zero and we can achieve a higher average torque. This situation would require us to choose an Ith for which the minimum torque is approximately less than -0.5 Nm or equal to 0. We would chose it to be -0.5 Nm instead of 0 as even a large current would produce very little torque. So by decreasing the current too much to zero we would end up decreasing the average torque by sacrificing torque in the area around line A. This situation is more preferable than with no phase advancing present.

Another advantage of choosing Ith as zero is that at this point would satisfy the theory presented by the demagnetization curves in section 4.5. Thus it is possible to negate the need of look up tables with interpolation or analytical equations making the system simpler. This also results in making the system robust against parametric variations due the

difference between the actual flux-i-theta characteristics and those calculated by finite element analysis.

This however is not the optimum point for the Ith for phase advancing. Study was carried for a wide range of turn of angles and the optimum position with maximum torque per ampere was found. This position is represented by the red lines showing the optimum phase advancing for maximum torque per ampere. At this operating situation the amount of negative torque removed from torque production is less than the amount of positive torque production reduced due to advancing. It was found that the Ith at this optimum situation to be approximately the same for a wide speed range when the maximum torque is commanded for that speed. Approximately meaning that it was found to be from 32 to

36. So by fixing the Ith to a value of 34 A the system can work over a wide speed range. It is however preferable to have an Ith that varies according to load and speed conditions for higher system performance. Further analysis of the collected data yields Eqn. 4.6.

푓(퐼푐푚푑) 퐼푡ℎ =푓푎푐푡표푟∗휔∗ (4.9) 푉퐷퐶

Here an increase in the speed increases the Ith if everything else is constant. Whereas an increase in Command Increases the Ith. This also has a nonlinear relations ship as the Ith saturates near the higher command regions as demonstrated by 푓(퐼푐푚푑). It is also depends on the DC bus voltage where an increase in VDC results in a reduction of Ith. The factor is dependent on the machine that is working and needs to be found by just running the machine for one optimum operating point. An improved solution would be to remove the nonlinearities and use an adaptive method for varying Ith as well. This however is not covered in the scope of this thesis and remains something to be explored in the future. The 64

method with a constant Ith was found to provide satisfactory results keeping with the main objective of having a simple controller and has thus been used in the controller implementations proposed in the thesis.

4.3.3. Adaptive method

The adaptive phase advancing method proposed in this thesis advances the turn off angle while reducing the conduction region as shown in Fig 3.11. The phase vectors of the control components are shown in Fig. 4.9. As the speed increases, the fd is commanded to produce the advancing angle θPA. This method of phase advancing is analogous to flux weakening in SM where the total torque being applied is the f. In this system, the amount of advancing angle is limited to prevent the system to operating below its optimum efficiency at above critical high speeds when single pulse mode operation occurs. The value of θPA is defined by Eqn. 4.10

푓푑 휃푃퐴 = arctan ( ) (4.10) 푓푞

Phase Advancing Limit

θpa Speed increases fd

Figure 4.9: Phase vector of torque command.

The block diagram for the system is shown in Fig 4.10. The command for fq can be determined from either speed or torque controller and PA block determines the amount of fd iteratively. The inputs to the adaptive algorithm is the torque command, and the actual currents and position. It is also fed back a previous value of the fd which it commanded.

The algorithm aims to match the actual current with the threshold current at the threshold position as shown in Fig 4.11 [60].

Adaptive Flux Weakening Controller

fq ωref Speed Torque ωactual Controller Limiter fd Adaptive Phase Iactual Advancing Algorithm θ dq SRM fd-1 Controller

Figure 4.10: Block diagram of flux weakening controller

150 80

100 40

Negative Torque Peak at th Current (A) Current

20 (Nm) Torque Actual Current at th 50

Desired Current(Ith) at th

0 -20 3.5 4 4.5 5 5.5 Time (S) -3 x 10 Figure 4.11: Torque and current of a phase illustrating the main objective of the algorithm. 66

This adaptive algorithm used in this method is a simple hill climbing algorithm

[56]. Hill climbing method is a mathematical optimization technique. It is a very old and simple system which is used to find the local optimum positions of a problem. The relative simplicity of this algorithm makes it attractive for this application. The algorithm is an iterative one where the principle of the method is to compare the current outputs which is defined by fd in this problem with the optimum objective function which is:

퐼푎푐푡 = 퐼푡ℎ (4.11)

At,

휃 = 휃푡ℎ (4.12)

Here,

퐼푎푐푡 = 푓(푓푑푞 ) (4.13)

The result of the comparison is either the control output get closer to our object or move further away from the object. This helps determine whether to increment or decrement the out of fd. The algorithm increments the output if the actual current is greater than the threshold and decrements if the actual is less than the threshold. This shown in greater detail in the flow chart presented in Fig. 4.12. The adaptive system also has a limiters incorporated into it. This is done to ensure that the system does not go out of bounds. The k in the flowchart represents any number whose value determines the resolution of the advancing angle, with a higher k giving a lower resolution and vice versa.

fd=fd-1

θ =θ th

N Iactual>Ith Y

fd=fd-1 + k fd=fd-1 - k

Fd<-(fq + fd>0 0.25* f Y q) N N Y

fd=-(fq + f =0 f =f f =f d d d d d 0.25* fq)

Figure 4.12: Flowchart of adaptive flux weakening controller using hill climbing method.

This hill climbing method is a local search method where the objectives to find the local optimum of a system. It has disadvantage as it only finds the local maximum. So this will never reach the global maximum unless the system is heuristic convex. This is however the case as shown in Fig 4.13. The phase advance angle is swept across a wide range and we see that the curve has a convex shape thereby negating the disadvantage of hill climbing methods. This however was found for one particular machine that was investigated in the development of the algorithm. If the system is not heuristic convex in nature then it can be adjusted by having random restarts or using another system for adaptation such as stochastic hill climbing and simulated annealing[56].

0.8

0.75

0.7 Corresponds to advance angle 0.65 where, Iact=Ith at th

0.6 Iact>Ith Iact<=Ith 0.55 Maximum Torque Per Ampere (Nm/A) MaximumPer Ampere Torque 0.5 0 10 20 30 40 50 60 70 Phase Advance angle in Electrical 

Figure 4.13 : The heuristic convex nature of maximum torque per amp with respect to PA.

The unique feature of this control method is that it can adapt itself to adjust the amount of phase advancing irrespective of the operating conditions. The benefits of this system when used in conjunction with the dq control method is to allow high speed operation through adaptive phase advancing while preserving the dq controllers inherent torque ripple minimization feature.

4.4. Conclusions

In this chapter, an adaptive flux weakening control for SRMs based on the dq method is proposed. The proposed controller has a simple structure and removes the need for extensive offline calculations or lookup tables for flux weakening and wide speed range operation of the SRM. It compares the actual currents with a threshold current at a threshold voltage and uses the hill climbing adaptive method for determining the amount

of phase advancing. This makes a closed loop adaptive system which is not heavily dependent on the machine parameters beforehand.

The next chapter is about the finite element modeling and simulation on the proposed methods. An analysis was performed on the efficiency of the dq and flux weakening controller by varying the third component of f0 and how it can be a source for the future work. It also compares how the dq controller matches up with traditional angle controller.

CHAPTER V

MODELING AND SIMULATION RESULTS

5.1. Introduction

In chapter III a control strategy based on the rotating dq reference frame for SRMs was presented. Then in chapter IV an adaptive flux weakening controller which advances the turn off angles and compliments the dq controller was presented. These chapters developed the theory and algorithms for simplifying the SRM control using the dq controller and automating the turn-off angle selection with the help of the adaptive flux weakening section. This chapter uses those algorithms in computer simulations for validation. This chapter also discusses about parametric sensitivity of the proposed controllers. Machine design limitations have also been briefly discussed.

Several simulations were conducted for the verification process. The simulations were conducted on a 110 kW test SRM for validating the control. The machine was modeled using Arthur Raduns’ model presented in [16 - 18] and using the finite element analysis based model. Both the models were used in the verification process. The next sections present the simulation results verifying the dq controller and the adaptive flux weakening controller. The preceding section contains a brief analysis of the efficiency using the dq controller compared to traditional excitation angle based controllers.

5.2. Modeling

For proper verification of the proposed control methods it was necessary to have a good nonlinear model of the SRM. The first model that was used for initial verification of the dq controllers torque sharing ability, i.e. negativity removal block, was the Spong model presented in [7]. The motivation for using this model was that it could address the

SR machines nonlinearity without introducing a high degree of complexity. This method however suffered as it was difficult to model a real machine as it had no geometric parameters associated with it. Hence Arthur Radun model presented in [16-18] was used for further verification. This modeling method is based on the machines geometry and magnetic properties. This model dynamically generates the solutions for the static torque and flux linkage. This geometry based model allows greater flexibility in the design and study of the control algorithms as it is easier to vary machine parameters. Thus this model is used as the primary simulation source for development. The problem this model faced was at high speeds. So for developing the flux weakening for a wide speed range operation a look up table based method was used.

The main motor used during the verification process was a three phase 110 kW

SRM with a 12/8 pole configuration. The general machine parameters are given in Table

5.1. It was modeled using the machines geometric parameters using the Arthur Raduns’ model. The model also generated the static T-i-θ characteristics shown in Fig. 5.1 that are used in the controller design. Small changes were incorporated into the machine model by changing certain parameters to observe the system performance. This model also provided the static curves used in the development of the theory for flux weakening. 72

Table 5.1 SRM parameters of 110 kW 12/8 SR machine.

No. of Phases 3

Power 110 KW

Max. Speed 8000 rpm

Max. Current 150A

DC Link Voltage 600V

Phase resistance 0.3 Ohm

Max. Torque 110 Nm

Stator Poles 12

Rotor Poles 8

200

150

100

0 Torque (Nm) Torque -50

-100

-150

-200 0 5 10 15 20 25 30 35 40 45 Rotor Position Mechanical 

Figure 5.1: T-i-θ characteristics of the 110 KW SR using analytic modeling.

This method was suitable for simulations at low speed and matched the T-i-θ characteristics from finite element simulation which is shown in Fig. 5.2. However at higher speeds, as the flux model of the analytical method is not very accurate, look up tables were used in the modeling process.

300

200

100

0 Torque (Nm) Torque

-100

-200

-300 0 5 10 15 20 25 30 35 40 45 Rotor Postion Mechanical 

Figure 5.2: T-i-θ characteristics of the 110 kW SR using finite element modeling.

5.3. Simulation Results

The control algorithms were implemented in Matlab/Simulink environment. A 110 kW,

600 V SRM is modeled based on the modeling approach provided in the previous section.

The motor parameters are shown in the Table 5.1. The SRM was first tested using the conventional control then the proposed methods were tested. The control algorithm used for the conventional control is described by Fig 2.12.

Figure 5.3 and 5.4 shows the torque and current curves using the conventional SRM controller operating at 1000 rpm. In this controller, the torque production responsibility is 74 devoted to one phase at a time before it gets transferred to the next phase. Clearly, there is a high torque ripple during the commutation periods, where the upcoming phase at the beginning of its cycle cannot carry the load of producing all the torque immediately.

Therefore, the lack of the adequate toque sharing leads to high torque ripples.

200

150

100 Torque (Nm) Torque

0 0.01 0.015 0.02 0.025 0.03 0.035 0.04 Time (S)

Figure 5.3: Torque with traditional control at 1000 rpm.

200

150

100 Current (A) Current

0 0.01 0.015 0.02 0.025 0.03 0.035 0.04 Time (S)

Figure 5.4: Current with traditional control at 1000 rpm. 75

Furthermore, this controller suffers for need to adjust its turn-on and turn-off angles as these needs to be adapted with varying loads and speeds. Therefore the angle commutator blocked described in Fig 2.12 would require the commands for the turn-on and turn-off angle to be controlled dynamically. The proposed method of using the dq controller removes the need for using the angle commutator block making the control simpler and more analogous to synchronous machines and also brings with it some added advantages of minimizing ripple. However the amount of ripple being reduced is subject to the accuracy of the rotor position and lookup tables used. Results from the two main contributions of this thesis is presented in the next subsections. Following which is an efficiency analysis for the dq controller.

5.3.1. dq control

The look up tables for the dq controller was generated using the data from the 110 kW SRM being modelled using the methods mention in the preceding section. The lookup table of 퐺푥 (휃) is represented graphically in Fig. 5.5. Here the position θ is aligned with cos(θ) being aligned with the positive torque production regions of the T-i-θ characteristics.

s Fig. 5.6 contains the graphical representation of f ix .

2 Gain

-1 0 20 40 60 80 100 120 140 160 180 Rotor Position 

Figure 5.5: Graphical representation of 푮풙(휽).

4 Current Output Current

0 0 5 10 15 20 25 30 35 Input fix command

s Figure 5.6: Graphical representation of f ix .

The torque and currents curves for the proposed SRM dq controller are shown in Fig.

5.7 and 5.8. The proposed controller produces very little torque ripple if the lookup tables

77 and machine design is optimum as the figure shows. This small ripple could be acceptable for traction applications, and through further adjustments in the look up tables, the torque ripple can be improved further.

200

150

100 Torque (Nm) Torque

0 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 0.022 Time (S)

Figure 5.7: Torque using the dq controller with proper tuning at 1000 rpm.

200

150

100 Current (A) Current

0 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 0.022 Time (S)

Figure 5.8: Current using the dq controller with proper tuning at 1000 rpm.

The controller proposed here is dependent on the machine characteristics and is sensitive to changes in the machine parameters. To investigate the parameter vacation effects on the controller performance, a sensitivity analysis has been carried out for this system. The system is mostly dependent on the 푮풙(휽) look-up table which is calculated from the T-i-θ characteristics of the motor. These values for a particular machine can be found through finite element analysis or through experiments. For our analysis we varied some parameters of the machine and the resulting T-i-θ characteristics of the motor are obtained and used to generate the look-up table of 푮풙(휽). In one case, the motor model considered to design the controller had a motor with a rotor pole width of 19 degrees while the actual controlled motor had a rotor pole width of 21 degrees. After implementing the designed controller for that machine, it was found that the mismatch in the machine design has a more prominent ripple effect as shown in Fig.5.9. It can be seen from Fig. 5.8 and

5.10 that the current shapes were similar to a comparative ideal case but the torque had more ripple. The second look-up table had its effect only on the response time of the system as the commanded torque was never reached by the system due to the linear nature of this table. This problem could be solved by tuning the outer loop controller. Therefore, it can be concluded that only one part (one look-up table) was parameter sensitive to the operation of the entire system.

200

150

100 Torque (Nm) Torque

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 Time (S)

Figure 5.9: Torque with different machine model at 1000 rpm.

200

150

100 Current (A) Current

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 Time (S)

Figure 5.10: Current using the dq controller with different machine model at 1000 rpm.

This case led to test the machine further using the finite element data in look up tables. The results at the same speed is shown in Fig 5.11 and Fig 5.12.

200

180

160

140

120

100

Torque (Nm) Torque 80

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 Time (S)

Figure 5.11: Torque with dq using the FEM based method

200

150

100 Current (A) Current

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 Time (S)

Figure 5.12: Torque with dq using the FEM based method

It can be seen from the Fig 5.11 that the ripple is higher when comparing graphically the previous model case shown in Fig. 5.7. The reasons for this is that the data in look up tables from the finite element analysis based model did not have a high resolution and a larger band gap for the current regulation was used. Still we can see that the ripple during

81 commutation is low. By performing Fast Fourier Transform (FFT) analysis on this result the torque ripple was found to be around 5% at the pole passing frequencies.

Coupled simulation showed that the concept of dq and its ability to reduce torque ripple and machine operation in a simple manner by negating the need for the angle commutator. However the differences in results from the two simulation models merited further investigation so a coupled simulation (Finite Element Analysis (FEA) software working harmoniously with the control and circuit simulator) was performed. The SRM dq controller is implemented with the motor running at 1000 rpm. The obtained curves for the torque and the currents are shown in figs. 5.13 and 5.14, respectively where the output torque has very little ripple on it. The results show that the control strategy is effective in this simulation which takes into account the delays in the inverter as well as a more realistic

SRM motor model.

200

180

160

140

120

100

Torque (Nm) Torque 80

0 2 4 6 8 10 12 14 16 18 Time (mS)

Figure 5.13: Torque with dq using coupled simulation.

200

180

160

140

120

100

Current (A) Current 80

0 2 4 6 8 10 12 14 16 18 Time (mS)

Figure 5.14: Current with dq using coupled simulation.

The dq controller was then tested to check its dependency on the motor design parameters. The controller was found to be dependent on the rotor pole width quite highly.

The torque outputs for the different pole widths and proper look up tables are shown in Fig

5.15 to 5.18. It can be observed that the system has an optimum range in the rotor pole widths for operation with minimum ripple. Outside that range the currents are not able to follow the commands so there is a higher peak. This could be mitigated to a certain degree by increasing the DC bus voltage or tuning the 퐺푥 (휃) look up table.

200

150

100 Torque (Nm) Torque 50

0 0.005 0.01 0.015 0.02 0.025 Time (S)

Figure 5.15: Torque with dq with a rotor pole width of 21 degrees.

200

150

100 Torque (Nm) Torque 50

0 0.005 0.01 0.015 0.02 0.025 Time (S)

Figure 5.16: Torque with dq with a rotor pole width of 18 degrees.

250

200

150

100 Torque (Nm) Torque

0 0.005 0.01 0.015 0.02 0.025 Time (S)

Figure 5.17: Torque with dq with a rotor pole width of 22 degrees.

200

150

100 Torque (Nm) Torque 50

0 0.005 0.01 0.015 0.02 0.025 Time (S)

Figure 5.18: Torque with dq with a rotor pole width of 19 degrees.

5.3.2. Flux weakening controller

The adative flux weakening controller was developed to advance the turn off angles of the controller at higher speeds as it was not possible demagnetize the curves fast enough due to a high back-emf voltage. This results in large negative torque and the average torque

85 being less and also results in a reduction of system efficiency. The torque is shown in Fig.

5.19 has an average of 38.96 Nm and we can see that there is a large amount of negative torque being produced as demonstrated by Fig. 5.20. The currents responsible for large negative torque is shown in 5.21.

100

40 Torque (Nm) Torque

0 1 2 3 4 5 6 7 8 9 10 Time (mS)

Figure 5.19: Torque with no phase advancing.

100

Torque (Nm) Torque 20

-20 1 2 3 4 5 6 7 8 9 10 Time (mS)

Figure 5.20: Phase torque with no phase advancing.

150

100

Current (A) Current 50

0 1 2 3 4 5 6 7 8 9 10 Time (mS)

Figure 5.21: Phase current with no phase advancing.

The optimum phase advancing for this speed was found as by a parametric sweep as described in chapter 4 and then used program the adaptive controler. We can observe the Torque shown in Fig. 5.22 has a higher average value of 46.59 Nm. This is primarily because the negative torque production has beeen reduced as demonstrated by Fig. 5.23 in comparision to Fig 5.20. The negative peak also occurs at θth as shown in Fig 5.23 in accordance with controllers base as described in chapter IV. This occurred as the turn-off angle was advanced whose effects we can observe by look at the currents in Fig 5.21 and

Fig. 5.24. The currents in Fig. 5.24 reaches zero faster due to the advanced turn-off.

100

40 Torque (Nm) Torque

0 1 2 3 4 5 6 7 8 9 10 Time (mS)

Figure 5.22: Torque with phase advancing.

100

40 Torque (Nm) Torque 20

1 2 3 4 5 6 7 8 9 10 Time (mS)

Figure 5.23: Phase torque with phase advancing.

150

100 Current (A) Current 50

0 1 2 3 4 5 6 7 8 9 10 Time (mS)

Figure 5.24: Phase current with phase advancing.

With the concept of phase advancing and the capability of the controller to reach a a desired fd at a desired load and speed established the algorithims robustness and overall effectivenes was tested by dynamically varying the load torque and speed. The result of step changes in load torque is shown in Fig 5.25. The results of changes in reference speed is in Fig 5.26. From the figures it can be observed that fd adapted well with respect to the transient conditions. With the fd reaching optium efficiency values to match the Ith with the

Iactual at θth. When this was not possible at very high load torque demands at a certain speed it reached its limit for phase advancing.

125 100 75 50 25 0 Load Torque (Nm) Torque Load 0.5 1 1.5 2 2.5 3 Time (S) 125 100 75 50 fq (A) 25 0 0.5 1 1.5 2 2.5 3 Time (S) 0 -25 -50 -75 fd (A) -100 -125 0.5 1 1.5 2 2.5 3 Time (S) 75 60 45 30 15 PA (Degrees)

 0 0.5 1 1.5 2 2.5 3 Time (S)

Figure 5.25: Variation of fq and fd with step response in torque at a speed of 5000 rpm. Nm. This control thus can also adapt it self to single pulse mode operation. In Fig 5.27 the total torque of single pulse mode operation is region. Fig. 5.28 and 5.29 demonstrates the currents and phase torques. It can be observed that some negative phase torque being produced at the start of the conduction region but this compensates for current to rise up to its peak before the back emf increases too much. Thus producing an overall increase in the average torque.

8000 6000 4000

Reference 2000 Speed (rpm) Speed 0 0.5 1 1.5 2 2.5 3 Time (S) 125 100 75

fq (A) 50 25 0 0.5 1 1.5 2 2.5 3 Time (S) 0 -25 -50

fd (A) -75 -100 -125 0.5 1 1.5 2 2.5 3 Time (S) 75 60 45 30

PA (degrees) 15  0 0.5 1 1.5 2 2.5 3 Time (S) Figure 5.26: Variation of fq and fd with step response in reference speed at a

torque of 65

100

50 Torque (Nm) Torque

0 1 1.5 2 2.5 3 3.5 4 Time (mS)

Fig 5.27 Total torque during single pulse mode operation.

150

100 Current (A) Current 50

0 1 1.5 2 2.5 3 3.5 4 Time (mS)

Fig 5.28 Phase current during single pulse mode operation.

100

25 Torque (Nm) Torque

-25 1 1.5 2 2.5 3 3.5 4 Time (mS)

Fig 5.29 Phase torque during single pulse mode operation.

The inclusion of fd command thus resulted in an increase in the average torque production over the entire speed range. This is illustrated in Fig 5.30.

150 dq with Phase Advancing Traditional SRM Control

100

50 Torque (N-m) Torque

0 0 2000 4000 6000 8000 Speed (rpm)

Fig 5.30: Torque-speed envelope with and without phase advancing.

5.3.3. Efficiency of dq controller

In the scope of this thesis a new controller which serves as an analogues control method to AC machines for SRMs have been proposed. This method is simple enough to serve as an alternative to the conventional excitation angle control. Thus it is important to compare the performance of both the control methods on the same platform. The two 93 methods were tasted in three speed ranges. The low speed range where it is paramount to have low torque ripple. The medium speed range where it is still possible to reach the command torque but some advancing of the conduction regions are required due to back emf. The third being the high speed region where it is necessary to operate in single pulse mode operation as there is a very high back-emf voltage. The command current of 140 A for a commanded torque of 100 Nm was used in all three cases and then the turn-on and turn-off angles adjusted.

The first case is at a speed of 1000 rpm. The resulting output parameters from the two control methods which are of importance to us are tabulated in Table 5.2 and 5.3.

Table 5.2: Data from conventional current control at 2000 rpm

Iref Turn-on Turn- off Average Torque Phase Torque per phase (A) (electrical (electrical Torque Ripple current RMS current degrees) degrees) (Nm) from FFT RMS (Nm/A) (%) (A) 140 18 145 96.92 6.56 83.4 1.16

140 20 156 102.15 11.06 86.28 1.18

Table 5.3: Data from dq control at 2000 rpm

fq fd f0 Average Torque Phase Torque per phase Torque Ripple current RMS current (Nm) from FFT RMS (Nm/A) (%) (A) 100 0 0 99.47 4.79 92.47 1.08

100 0 37.5 100.67 6.74 85.76 1.17

From the two tables the first case is with both the controllers having inputs for having low torque ripple and then for maximum torque per ampere. In terms of torque ripple we can observe that the dq control method has a lower torque ripple with a 26.9% improvement on the conventional case. It does this by achieving a higher average torque as well. Here however the traditional case as a better torque per ampere when look at the phase currents and gives 6.8% better torque per ampere. In the second case the controller is optimized for achieving maximum torque per ampere. Here in the dq we can increase the efficiency by sacrificing some torque ripple. The same average torque is maintained but at higher efficiency and higher torque ripple. In the conventional method it produces a maximum torque per ampere in terms of phase currents of 1.18 Nm/A which is .8% higher than the dq but at a higher torque ripple of 11% compared to 6.74% by the dq controller.

Thus it can be shown that depending on the control designer’s requirements the control parameters of the dq controller can also be adapted to achieve the objectives but at a higher efficiency or lower torque ripple.

The second case is at a speed of 5000 rpm which is a medium speed range with a high back-emf voltage. The resulting output parameters from the two control methods which are of importance are tabulated in table 5.4 and 5.5.

Table 5.4: Data from conventional current control at 5000 rpm

Iref Turn-on Turn- off Average Torque Phase current Torque per (A) (electrical (electrical Torque Ripple RMS phase RMS degrees) degrees) (Nm) from FFT (A) current (%) (Nm/A) 140 -12 128 93.57 10.20 86.21 1.08

Table 5.5: Data from dq control at 5000 rpm

fq fd f0 Average Torque Phase current Torque per Torque Ripple RMS phase RMS (Nm) from FFT (A) current (%) (Nm/A) 100 0 0 92 6.89 91.41 1.01 100 40 30 93.92 8.70 86.26 1.09

Here it can be observed from the conventional control that some advancing is required as there is back emf built up. Thus it was optimized for giving maximum torque per ampere. We can observe that it had a torque ripple of 10% and torque per ampere of

1.08 Nm/A. In the dq control without any fd or f0 command we can see that the controller produces a lower torque ripple of 6.89% but has lower torque per ampere in terms of phase

RMS currents. The dq controller then had a fd and f0 commanded so that it is possible to increase the efficiency to levels comparable with the conventional control in terms of torque per ampere. Here we observed that similar efficiencies with a difference of 0.5% with the dq controller being higher whilst having a lower torque ripple. Hence it can concluded that at medium speeds also the dq controller is more efficient than conventional controller while maintaining the same simplicity.

The third case is at a speed of 7000 rpm which is a high speed range with a very high back-emf voltage. The resulting output parameters from the two control methods which are of importance are tabulated in Table 5.6 and 5.7. Here the objectives of the control system is to achieve as high a torque as possible due to single pulse mode operation.

Table 5.6: Data from conventional current control at 7000 rpm

Iref Turn-on Turn- off Average Torque Phase Torque per (A) (electrical (electrical Torque Ripple current phase RMS degrees) degrees) (Nm) from FFT RMS current (%) (A) (Nm/A) 140 -22 128 62.39 19.51 74.44 0.84

Table 5.7: Data from dq control at 7000 rpm

fq fd f0 Average Torque Phase Torque per Torque Ripple current phase RMS (Nm) from FFT RMS current (%) (A) (Nm/A) 100 80 25 63.33 20.35 74.5 0.85

At this high speed it is no longer possible to reduce ripple due to single pulse mode operation it is important to produce as high a torque as possible with higher efficiencies.

Here also we can see that by changing the control parameters associated with the dq control we can achieve a higher torque per ampere.

Thus validating the purpose of the dq control as an alternative simple control strategy which is analogues to AC machines and is capable of operation over a wide speed range. The dq control does this be offering the flexibility of chasing either low torque ripple operation or high torque per ampere operation without extensive calculations of offline angles in terms of torque per ampere and highly complex control strategies in terms of lower torque ripple.

5.4. Conclusions

In this chapter the modeling of the SR machine for simulation has been discussed.

The chapter then presents simulation studies which validates the proposed control strategies and their operation. This chapter then provides an efficiency comparison between the proposed method and the conventional current control method with optimized excitation parameters.

The next chapter of this thesis is about the experimental setup and results validating these control methods.

CHAPTER VI

EXPERIMENTAL SETUP AND RESULTS

6.1. Introduction

In chapter V the proposed control strategies were verified through computer simulations. In this chapter the methods are verified experimentally. For experimental verification a low power setup using a small 300 W machine and a commercially available converter with some modifications has been used. An interface circuit has also been built to communicate between the controller and the converter. The experimental results presented here have been carried out on a dSPACE controller. The following sections in this chapter is about the experimental setup and experimental results.

6.2. Experimental Hardware

The development of the hardware and test setup for experimental validation presented the main challenge in this thesis. The motor and converter for a low power experimental validation of the system was procured. The machine and converter was used in an old commercial washing machine from Maytag. Hence the machines characteristics could not be obtained from the manufacturers and had to be obtained through unconventional methods. The following subsections describes how the experimental machine has been modeled for torque estimators and controller design, structure of the

inverter that has been used and the circuits that has been implemented for interfacing with the controller and the dSPACE controller and the structure of the program.

6.2.1. Experimental SRM modeling

The machine used for the verification in simulation was not ready in time for experimental verification by the time this thesis had to be submitted so an alternative 300

W SRM which was used as the front load motor of Maytag Neptune washing machines was used for experimental verification. This provided a low power solution for testing out the algorithm initially but presented its own set of problems in terms of modeling the machine as no concrete machine parameters were available for the motor. Hence, after calculating the initial parameters the motor was opened and its internal structure analyzed and measured to determine the parameters for modeling. The disassembled motor is shown in Fig. 6.1. The next problem was that of determining the number of turns which is harder to measure without unwinding the entire machine. For this we ran the motor at a fixed load current of 3A and measured the average torque using a torque transducer. Then by using those geometric parameters in the Arthur Raduns’ model a static T-i-θ characteristic curve was generated. The number of turns was then manipulated using trial and error method with the first guess being an educated one based on the diameter of one winding and the width of a bunch of windings together. These geometric parameters shown in Table 6.1 were then used in the finite element software for a more accurate modeling of this machine.

The T-i-θ characteristics of the machine are shown in Fig. 6.2 and 6.3.

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Figure 6.1: a) Stator of the experimental Figure 6.1: b) Rotor of the experimental SRM SRM

Figure 6.1: c) Stator and rotor separately Figure 6.1: d) Experimental SRM of the experimental SRM assembled

Table 6.1 SRM parameters of 300 W 12/8 experimental SR machine

No. of Phases 3 Stack Length 1.889 inch

No. of Stator Poles 12 Air Gap 0.015 inch No. of Rotor Poles 8 Radius to air gap at rotor 3.2724 inch

Power 300 W Radius to outside rotor yoke 2.4 inch

Phase Resistance 2.2 Ohms Radius to inside stator yoke 3.8665 inch

Minimum Inductance 6.553 mH No. of Turns 150

Maximum Inductance 28.25 mH Rotor Pole width 15 degrees

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0 Torque Nm Torque -2

-4

-6

-8

-10 0 20 40 60 80 100 120 140 160 180 Rotor Position 

Figure 6.2: T-i-θ characteristics from Arthur Raduns’ model.

0 Torque Nm Torque -2

-4

-6

-8

-10 0 20 40 60 80 100 120 140 160 180 Rotor Position 

Figure 6.3: T-i-θ characteristics from finite element analysis. 102

6.2.2. Inverter used for experimental validation

The inverter that was procured shown in Fig. 6.4 is a classic bridge inverter as described in Fig. 2.8 with gate drivers incorporated into them. The inverter is rated at 300

W and had a rectifier to convert 120V-AC to 170 V DC. This section of the circuit was however bypassed so that we could apply our on DC bus voltage giving a better control of the entire system. The converter also came equipped with its own controller for running the SRM and this also had to be removed so that we could give our control command signals separately. The MOSFETs used here is IRF 644 which could handle a peak Vds of

250 V and a continuous current of 14 A at room temperature. The gate driver circuitry in this section used an IR2101 high-low gate driver IC with its standard circuit configuration.

Figure 6.4: Inverter used for experimental implementation. 103

6.2.3. Interfacing circuitry

For proper communication between the inverter and the controller an interface was built. The interface circuits mainly consisted of three parts. They are

 Gate Drive interface circuitry

 Encoder interface circuitry

 Current conditioning circuits

The gate driver interface circuit takes the gate inputs from the dSPACE controller and then passes them through a buffer and then opto-coupler to inverters gate driver inputs.

The block diagram of this circuit is given in Fig 6.5. The opto-couplers are used to isolate the ground of the power supply from the controller.

Gate Signals Buffer Buffered OptoCoupler Isolated From dspace SN74LVC541A gate signals ACPL4800 gate signals

Figure 6.5: Gate driver circuit block diagram.

The second interface circuit that had to be built is an encoder interface circuit. The diagram for this circuits is shown in Fig. 6.6. The motor was fitted with an HEDS5505A06 which was an optical encoder with two channels and index. This encoder needed 5 V supply and had to have a pull up resistor connect to the output channels to drive the one TTL load.

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Hence a buffer was used here as well to increase the loading capacities per output. The IC used as a buffer here is actually a level shifting buffer so as to make this system capable of being run from a DSP as well as dSPACE but here the same supply voltage was applied on both the ends essentially making it a buffer. The circuit diagram for this is shown in Fig

6.6.

+5V Supply

R R R CH. A CH. A Encoder CH. B CH. B MC14504BCP HEDS5505 A06 CH. I CH. I

Figure 6.6: Encoder interface circuit.

The third part was that of a current conditioning circuit. The currents sensor used is an

LEM 55. The current sensor has a much higher rating than the current that are passing through hence three turns were given to the turn on sensors. The value outputs from the current sensor had a high noise and also needed to have their levels shifted. Hence a conditioning circuit was used. The conditioning circuit shown in Fig. 6.7 consists of a voltage follower, a passive low pass filter and a level shifter. The low pass filter has the following cut off frequency

1 푓 = (6.1) 푐 2휋푅퐶 105

with 푅 = 33 Ω 푎푛푑 퐶 = 0.1 µ퐹 푓푐 = 48.23 푘퐻푧

10KΩ 10KΩ Vref

- 10KΩ - Input to Current Sensor Controller 33Ω Ouput + + 220Ω 0.1µF

Figure 6.7: Current conditioning circuit.

Figure 6.8: Hardware circuits.

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The interface circuitry of the gate driver and encoder discussed here has been implemented on a proto board and the current conditioning circuit on a PCB. Fig 6.8 is a picture showing the completed interface circuit coupled with the inverter.

6.2.4. dSPACE controller

dSPACE controller was used for all the control algorithm implementations as it allows for algorithms to be implemented on Simulink making it easier for hardware implementation and algorithm development from the initial stages. The system used here was a dSPACE micro Autobox II system which communicated with the host computer through the Ethernet port. This system has an IBM PPC running at 900 MHz and is equipped with ADC with 16 bit resolution and PWM generation with a range of 0.0003 Hz to 150 KHz. The system is also equipped with digital capture units for use with an encoder.

This made the entire dSPACE system ideal for initial algorithm development.

6.3. Dynamometer and System Setup

After successful testing of the hardware, controller and motor at no load conditions it was connected to a Magtrol Eddy current hysteresis dynamometer. The dynamometer was controlled using the Magtrol DSP6000 controller unit for the application of load torque. Several power supplies have been used to power the different components of the setup. Data acquisition and real time control of the system was performed using the

DSPACE Control Desk 3.7.4. Oscilloscopes and current probes where also used for observing the current wave shapes. The experiments were carried out at DC bus voltage of

120 V and the switching was limited to 30 Khz. The entire control system had a control

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loop of 30 Khz speed and the Euler method was used as the primary integration method for the solver. A PI based speed controller was implemented for speed control with the gains being selected through a trial and error method until a satisfactory performance was achieved. The focus of this thesis is not speed control hence this portion didn’t get much attention. The overall system performance would also improve if a better PI controller or any other speed controller is implemented. Fig. 6.9 shows a picture of the entire experimental platform used for testing.

Figure 6.9: Complete experimental setup.

6.4. Experimental Results

The first step for experimental validation was to implement the traditional control system based on the control strategy shown in Fig 6.10.

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Vdc

Gate Signals ω* Iref Current Vph PI Speed Controller Converter SRM Regulator - ω Iph Control θon Electronic Angle Inputs Commutator Speed θoff Calculator θ

Figure 6.10: Conventional control method.

The speed calculator here was based on tracking the position difference every

300µ seconds and then dividing it by time and adjustment factors as shown by Eqn. 6.2.

With the factor 500 being used as the encoder has 500 counts per complete mechanical revolution.

휃 푆푝푒푒푑 = 60 ∗ ( 푑푖푓푓 ) ) (6.2) 300µ푆∗500

A PI speed controller was implemented to attain the commands required for running the machine at the reference speed. The gains of the controller was designed by trial and error method with a proportional gain of 0.0005 and an integral gain of 0.015. These gains values provided a satisfactory speed response with minimal oscillations suitable for testing the algorithms. However a better speed control method with more accurate gains would greatly improve system performance but that remains a part of future plans.

The system was run at speed of 500 rpm and the corresponding current shapes are shown in Fig 6.11 and Fig 6.12. With Fig 6.11 being generated from the scopes and Fig

6.12 from the data acquired with control desk acquisition software. Some spikes from the

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data acquired from the scope can be observed but this problem is negated when as the current shapes with the data from the acquisition software as there is a low pass filter in the current conditioning circuit.

Figure 6.11: Currents from control method using oscilloscope.

Current (A) Current 4

0 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 Time(S)

Figure 6.12: Currents from Control Method using control desk.

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6.4.1. dq control of SRM

The dq control method is basically a torque controlled method whose inputs are in terms of torque which it later transforms to reference currents. Furthermore one of the benefits of dq control methods is its inherent ability to reduce the amount of torque ripple and serve as an alternative analogous control method to synchronous machines with the help of dq to synchronous conversion block. Hence it was important to have a high bandwidth torque estimator. The torque estimator used here is a LUT based method where the static T-i-θ characteristics of the experimental SRM is used in 2D LUT with the actual currents and rotor position as the input. The Fig 6.13 and 6.14 shows phase torques and total torque from the traditional control system at speed of 500 rpm and load of 1 Nm applied by DSP6000 controller. The turn-on and turn-off angles were optimized for providing a low torque ripple but we can still observe a torque dip during commutation as there is no sharing method being implemented here. The time scale is the same in both of these two figures.

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4.5

3.5

Torque (Nm) Torque 2.5

1.5

1 0.03 0.035 0.04 0.045 Time (S)

Figure 6.13: Total estimated torque from conventional control.

4.5

3.5

2.5

2 Torque (Nm) Torque

1.5

0.5

0 0.03 0.035 0.04 0.045 Time (S)

Figure 6.14: Estimated phase torques from conventional control.

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Then the dq control method was implemented at the same speed of 500 rpm and a certain load torque from the DSP6000 Controller. The currents from scope again has some noise as the current probes where to close to each other and this could not be avoided to the close proximity of the phase conductors in the setup.

Figure 6.15: Currents from dq using oscilloscope.

Current (A) Current 4

0 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 Time (S)

Figure 6.16: Currents from dq using control desk. 113

We can observe that the current is higher in the initial conduction regions for each of the phase to ensure proper torque sharing with during commutation. From Fig 6.17 and

6.18 cit an be observed that there is no longer any torque dips presented here. Furthermore with the torque dips eliminated we can see that a lower torque is demanded from the machine at the same load torque. Another reason for this that the output of the PI controller now is the reference torque term fq and no longer the Iref it was in the previous case. The torque sharing has been incorporated as a different reference current is now being generated with some shaping involved. The reference current with shaping is shown in Fig. 6.19 and we can see that the actual currents can follow the reference as demonstrated in Fig 6.20.

The time scales in Fig 6.17 to Fig 6.20 is the same and in the same range.

2 Torque (Nm) Torque 1

-1 0.4 0.405 0.41 0.415 0.42 0.425 0.43 Time (S)

Figure 6.17: Total estimated torque from dq controller.

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2 Torque (Nm) Torque 1

-1 0.4 0.405 0.41 0.415 0.42 0.425 0.43 Time (S)

Figure 6.18: Estimated phase torque from dq controller.

Reference Current (A) Current Reference 3

0 0.4 0.405 0.41 0.415 0.42 0.425 0.43 Time (S)

Figure 6.19: Reference current commands from dq controller.

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Current (A) Current 4

0 0.4 0.405 0.41 0.415 0.42 0.425 0.43 Time (S)

Figure 6.20: Actual currents with dq controller.

After the successful implementation of the basic dq control strategy an experiment was run at the same speed of 500 rpm and load with a zero component being commanded.

This according to simulation studies and theory is supposed to decrease the conduction region there by increasing the system torque per ampere but at the cost of increasing the torque ripple. By comparing the Fig. 6.19 and Fig. 6.21 it can be observed that the commutation time has decreased when a zero component is injected. As a result the actual current shapes are also different. With a decrease in the overlap region between the two commutating phases we now have less freedom in terms of our controller for sharing the torque to reduce the torque ripple. As a result from Fig. 6.22 and Fig. 6.23 it can be see that there is an increase in the torque ripple. This also results in requirement of a higher torque for operating at that same speed and load conditions. This thereby results in a higher torque

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per ampere according to the simulation and theoretical studies presented in the previous chapters.

Reference Current (A) Current Reference 3

0 0.03 0.032 0.034 0.036 0.038 0.04 0.042 0.044 0.046 0.048 0.05 Time (S)

Figure 6.21: Reference command currents with zero component commanded.

Current (A) Current 4

0 0.03 0.032 0.034 0.036 0.038 0.04 0.042 0.044 0.046 0.048 0.05 Time (S)

Figure 6.22: Actual currents with zero component commanded.

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4.5

3.5

2.5

2 Torque (Nm) Torque

1.5

0.5

0 0.03 0.032 0.034 0.036 0.038 0.04 0.042 0.044 0.046 0.048 0.05 Time (S)

Figure 6.23: Phase torques with zero component commanded.

2 Torque (Nm) Torque 1

-1 0.03 0.032 0.034 0.036 0.038 0.04 0.042 0.044 0.046 0.048 0.05 Time (S)

Figure 6.24: Total torque with zero component commanded.

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The next validation for the dq controller was done at high speed and how it performs there. The tests were performed at a speed of 1700 rpm and high load. First the experiment was run without any fd commanded i.e. no phase advancing. In this case it can be seen that the actual currents shown in Fig 6.26 can no longer support the reference current commands shown in Fig 6.25 due to a high back-emf voltage. As a result there is some negative torque production as shown in Fig 6.27. This results in the total torque having more ripple and a decrease in the overall system efficiency.

Reference Current (A) Current Reference 3

0 0.03 0.035 0.04 0.045 Time (S)

Figure 6.25: Phase reference current commands with no advancing at 1700 rpm.

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Current (A) Current 4

0 0.03 0.035 0.04 0.045 Time (S)

Figure 6.26: Phase currents with no advancing at 1700 rpm.

2 Torque (Nm) Torque 1

-1 0.03 0.035 0.04 0.045 Time (S)

Figure 6.27: Phase torque with no advancing at 1700 rpm.

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2 Torque (Nm) Torque 1

-1 0.03 0.035 0.04 0.045 Time (S)

Figure 6.28: Total torque with no advancing at 1700 rpm.

The experiment was then run with some fd i.e. phase advancing being applied. The actual currents shown if Fig. 6.30 now can follow the reference current commands shown in Fig.

6.29 more closely. This results in a higher torque build up in the initial region and a removal in the production of negative torque as shown in Fig. 6.31. As a result the total torque shown in Fig. 6.32 has lower ripple and can achieve the load torque with a lower average torque and current requirements thereby increasing the overall system efficiency. Thus proving that the dq control can work over a wide speed range with the fd being commanded similar to flux weakening in synchronous machines. The control designer also has the option of designing is system for either low torque ripple or with increased efficiency by commanding the zeroth component.

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Reference Current (A) Current Reference 3

0 0.03 0.032 0.034 0.036 0.038 0.04 0.042 0.044 Time (S)

Figure 6.29: Reference current commands with phase advancing at 1700 rpm.

Current (A) Current 4

0 0.03 0.035 0.04 0.045 Time (S)

Figure 6.30: Actual currents with phase advancing at 1700 rpm.

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2 Torque (Nm) Torque 1

-1 0.03 0.032 0.034 0.036 0.038 0.04 0.042 0.044 Time (S)

Figure 6.31: Phase torque with phase advancing at 1700 rpm.

2 Torque (Nm) Torque 1

-1 0.03 0.032 0.034 0.036 0.038 0.04 0.042 0.044 Time (S)

Figure 6.32: Total estimated torque with phase advancing at 1700 rpm.

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6.4.2. Adaptive flux weakening using dq control

The second proposed technique in this thesis is that of an adaptive flux weakening where the controller adaptively varies the fd component in an adaptive manner for achieving maximum torque per ampere at medium and high speeds by changing the level of advancing required. This adaptive fd component generation also allows the machine to run over a wide speed range.

1.2 1 0.8 0.6 0.4 0.5 0 0 10 20 30 40 50 60 Load Torque (Nm) Torque Load 4 Time (S) 3 2

fq (A) 1 0 0 10 20 30 40 50 60 0 Time(S) -2 -4

fd (A) -6 -8 0 10 20 30 40 50 60 Time (S) 100 75 50 25

PA degrees 0  0 10 20 30 40 50 60 Time (S)

Figure 6.33: Response of fq, fd and resulting θPa with load step responses at 5 s., 17 s., 22 s., 31 s., 38 s., 42 s. and 54 s. 124

2500 2000 1500 1000 500 Speed (rpm) Speed 0 20 40 60 80 Time (S) 6 4 2 fq (A) 0 0 20 40 60 80 Time (S) 0 -2 -4 -6 fd (A) -8 -10 0 20 40 60 80 Time (S) 100 75 50 25

PA degrees 0  0 20 40 60 80 Time (S)

Figure 6.34: Response of fq, fd and resulting θPa with speed step responses at 8 s., 17 s., 38 s. and 60s.

The adaptation of fd with step changes in load at a fixed speed of 1700 rpm is shown in

Fig. 6.33. Figure 6.34 shows the response of fd with step changes in speed with the load torque being constant at 0.5 Nm. The test results suggest that the proposed controller works well during transient and steady state conditions.

Two experiments at a high speed was also conducted to demonstrate that at a high speed of 2500 rpm with the adaptive phase advancing. The experiments were first conducted at a load torque of 0.3 Nm where the current can regulated. The second

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experiment was conducted at the same speed of 2500 rpm but with a higher load torque of

0.5 Nm making the system go into single pulse mode operation. The currents from the first experiment is shown in Fig. 6.35. Here we can observe that the adaptive phase advancing method allows for current regulation by advancing the turn on and turn off angles. With the torque values increased as shown in the second experiment the load currents can no longer match the reference hence the phase is advanced to enable single pulse mode operation as shown by the phase currents in Fig. 6.36. Another experiment was run at a speed of 2000 rpm at a load of 0.26 Nm. Figure 6.37 shows the current shapes and the torque output without fd component. It can be seen that the commanded speed is maintained

푑푖 at the commanded load torque for an RMS current of 1.85 A and the 푥 is also low. The 푑푡

푑푖 adaptive controller demonstrated a higher 푥 as shown in Fig. 6.38. The commanded load 푑푡 and speed is maintained with an RMS phase current of 1.68 A with adaptive flux weakening control. This demonstrates that flux weakening with the fd command increases the torque

푑푖 per ampere and 푥 which would enable a wide speed range of operation. 푑푡

Figure 6.35: Phase currents at a load of 0.3Nm at 2500 rpm with adaptive control.

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Figure 6.36: Phase currents at a load of 0.5Nm at 2500 rpm with adaptive control.

Figure 6.37: Phase currents (ch1-3) and total torque (ch4) while running the machine at 2000 rpm without flux weakening

Figure 6.38: Phase currents (ch1-3) and total torque (ch4) while running the machine at 2000 rpm with flux weakening

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6.5. Conclusions

In this chapter a low power experimental setup has been described. The hardware circuitry has been built and tested step by step with the conventional control method. Then the proposed dq controller was implemented carefully. The experimental results then verified the effectiveness of the proposed control strategy. Experimental work was also carried out on the adaptive controller for wide speed range operation with the dq controller and it was also working as predicted. Thus this thesis presented a novel control strategy and its operation in simulation and hardware demonstrated. The method was successful in ripple minimization at low speed and capable of wide speed range operation. The method could also increase efficiency with the zero component commands.

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CHAPTER VII

CONCLUSION AND FUTURE WORK

7.1. Conclusions

This thesis presented a novel control strategy based on the rotating reference for switched reluctance machines. Previous work was done for rotating reference frame control of SRM based on the inductance. This thesis presented a reference frame control with respect to the phase torque. Control in the dq reference frame removes the need for a computationally intensive electronic commutator. Through the proposed SM to SRM converter block an analogous control strategy to SM for SRMs has been established. The proposed controller’s dependency on the actual machine model has been investigated briefly and future work would include a greater study in this regard.

Chapter II presented a brief discussion on the basics of the switched reluctance machine. It also presented a review on the modeling techniques for SRMs. A detailed literature review was conducted for high performance SR drives in terms of torque ripple reduction and excitation parameter controls.

Chapter III presented the basic dq control strategy which incorporated some transformation blocks based on finite element data from a machines design. These transformation blocks presented the SRM as a synchronous machine from the control engineers’ perspective. Thereby providing an analogous control for SRMs to SMs.

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Chapter IV presented an adaptive flux weakening control method for advancing the turn-off angles for SRMs. This method was independent on the machines parameters and facilitated operation over a wide speed and load range dynamically. Though the method is primarily designed for use with the dq controller it can also operate on traditional control methods.

Chapter V and VI presented simulation and experimental validation of the proposed control strategies. An experimental platform, based on a 300 W SRM using a dSPACE controller, was setup for this purpose.

The main contributions of this thesis can be summarized as:

 A literature review was conducted on existing SRM modeling and control

methods.

 A novel control strategy for SRMs has been developed making the SR

control analogous to that of AC machines.

 A rotating reference frame based control in the dq reference frame for

SRM has been presented. This made the SRM torque control similar to

DC machine control.

 A negativity removal block has been developed for converting bi-polar

signals from the dq to abc block to unipolar commands.

 A nonlinearity correction block has been developed to help minimize

torque ripple.

 A brief sensitivity analysis has been carried out on the dq controller

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 The controller removed the need for an excitation angle sequencer and

additional ripple minimization algorithms.

 An adaptive phase advancing algorithm for flux weakening was

established.

 An experimental platform was established for rapid controller

implementation and testing.

7.2. Future Work

This research opened the doors for a lot of further research in SRMs. A list of future research avenues is listed below.

 Experimentally verify the control algorithms on the high power

experimental platform based on a 110 kW SRM.

 Investigate the parameter sensitivity of the dq controller in greater detail.

 Incorporating an adaptive or artificial intelligence based model in the non-

linearity block to remove the need for the machine model and lookup

tables for ripple minimization.

 Develop a speed controller for optimal performance on the dq control

method.

 Investigate the avenues for incorporating torque ripple minimization and

sensorless control algorithms used in synchronous machines as the

machine is now like synchronous machine from the dq0 axis command

perspective.

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 Develop adaptive turn on angle controller for the machine instead of the

hard limit that is currently being used.

 Develop a controller for generating the optimal zero sequence component

for maximum torque per ampere.

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REFERENCES

[1] R. Krishnan, “Switched Reluctance Motor Drive: Modeling, simulation, Analysis, Design and application,” Magna Physics Publishing, 2001. [2] T.J.E. Miller, “Switched Reluctance Motors and their Control,” Magna Physics Publishing and Clarendon Press, Oxford, 1993. [3] M. Krishnamurthy, C.S Edrington, A. Emadi, P. Asadi, M. Ehsani, B. Fahimi, "Making the case for applications of switched reluctance motor technology in automotive products," IEEE Trans. Power Electron., vol.21, no.3, pp.659-675, May 2006. [4] K.M. Rahman, B. Fahimi, G. Suresh, A.V. Rajarathnam, M. Ehsani, "Advantages of switched reluctance motor applications to EV and HEV: design and control issues," IEEE Trans. Ind. Appl., vol.36, no.1, pp.111-121, February 2000. [5] P. J. Lawrenson, J. M. Stephenson, P. T. Blenkinsop, J. Corda and N. N. Fulton, “Variable-Speed Reluctance Motors,” IEE Proc., Part B, vol. 127, no. 4, pp. 253-265, 1980. [6] S. Mir, I. Husain, and M.E. Elbuluk, “Energy-Efficient C-Dump Converters for Switched Reluctance Motors,” IEEE Trans. Power Electron., vol. 12, no. 5, pp. 912- 921, September 1997. [7] M. Iilic’-Spong, R. Marino, S. Peresada, and D. Taylor, “Feedback linearizing control of a switched reluctance motors,” IEEE Trans. Autom. Control, vol. AC-32, no. 5, pp. 371–379, May 1987. [8] D. A. Torrey, X. M. Niu, and E. J. Unkauf, “Analytical modeling of variable reluctance machine magnetization characteristics,” IEEE proc. Electr. Power Appl., vol. 142, no. 1, pp. 14–22, January 1995. [9] H. J. Chen , D. Q. Jiang , J. Yang and L. X. Shi "A new analytical model for switched reluctance motors", IEEE Trans. Magn., vol. 45, no. 8, pp.3107 -3113, August 2009. [10] F. R. Salmasi and B. Fahimi, “Modeling switched-reluctance machines by decomposition of double magnetic saliencies,” IEEE Trans. Magn., vol. 40, no. 3, pp. 1556–1561, May 2004. [11] B. Fahimi, S. Gopalakrishnan, J. Mahdavi, and M. Ehsani, “A new approach to model switched reluctance motor drive application to dynamic performance, prediction, control and design,” IEEE proc. PESC, vol. 2, pp. 2097–2102, 1998. 133

[12] G. Suresh, B. Fahimi, K. M. Rahman, and M. Ehsani, “Inductance based position encoding for sensorless SRM drives,” IEEE proc. PESC, vol. 2, pp. 832–837, 1999. [13] A. D. Cheok and Z. Wang, “Fuzzy logic rotor position estimation based switched reluctance motor DSP drive with accuracy enhancement,” IEEE Trans. Power Electron., vol. 20, no. 4, pp. 908–921, July 2005. [14] W. Lu, A. Keyhani, and A. Fardoun, “Neural network-based modeling and parameter identification of switched reluctance motors,” IEEE Trans. Energy Convers., vol. 18, no. 2, pp. 284–290, June 2003. [15] Z. Y. Lin, D. S. Reay, B. W. Williams, and X. N. He, “Online modeling for switched reluctance motors using B-spline neural networks,” IEEE Trans. Ind. Electron., vol. 54, no. 6, pp. 3317–3322, December 2007. [16] A.V. Radun, "Design considerations for the switched reluctance motor," IEEE Trans. Ind. Appl., vol.31, no.5, pp.1079-1087, September/October 1995. [17] A.V. Radun, “Analytically computing the flux linked by a switched reluctance motor phase when the stator and rotor poles overlap,” IEEE Trans. Magn., vol. 36, no. 4, pp. 1996–2003, July 2000. [18] [A. Radun, “Analytical calculation of the switched reluctance motor’s unaligned inductance,” IEEE Trans. Magn., vol. 35, no. 6, pp. 4473–4481, November 1999. [19] S. A. Hossain and I. Husain, “A geometry based simplified analytical model of switched reluctance machines for real-time controller implementation,” IEEE Trans. Power Electron., vol. 18, no. 6, pp. 1384–1389, November 2003. [20] A. Khalil and I. Husain "A Fourier series generalized geometry-based analytical model of switched reluctance machines", IEEE Trans. Ind. Appl., vol. 43, no. 3, pp.673 -684 May/June 2007. [21] K. M. Rahman and S. E. Schulz, “High-performance fully digital switched reluctance motor controller for vehicle propulsion,” IEEE Trans. Ind. Appl., vol. 38, no. 4, pp. 1062–1071, July/August 2002.

[22] .C. Kjaer, J.J. Gribble and T.J.E. Miller: "High-Grade Control of Switched Reluctance Machines," IEEE Trans. Ind. Appl., vol. 33, no. 6, pp. 1585-1593, November/December 1997. [23] I. Husain, "Minimization of Torque Ripple in SRM Drives", IEEE Trans. Ind. Electr., vol. 49,no. 1, pp. 28-39, February 2002. [24] V. P. Vujicic "Minimization of torque ripple and copper losses in switched reluctance drive", IEEE Trans. Power Electron., vol. 27, no. 1, pp.388 -399, January 2012.

134

[25] M. Ilic-Spong , T. J. E. Miller , S. R. MacMinn and J. S. Thorp "Instantaneous torque control of electric motor drives", IEEE Trans. Power Electron., vol. 2, no. 1, pp.55 - 61 January 1987. [26] R. B. Inderka and R. W. A. A. De Doncker, “DITC-direct instantaneous torque control of switched reluctance drives,” IEEE Trans. Ind. Electron., vol. 39, no. 4, pp. 1046– 1051, July/August 2003. [27] K. Russa, I. Husain, M.E. Elbuluk, "A self-tuning controller for switched reluctance motors”, IEEE Trans. Power Electron., vol.15, no.3, pp.545-552, May 2000. [28] S. Mir, M.E. Elbuluk and I. Husain, "Torque Ripple Minimization in Switched Reluctance Motors using Adaptive Fuzzy Control,", IEEE Trans. on Ind. Appl., vol.35, No.2, pp.461-468, March/April 1999. [29] S. K. Sahoo , S. K. Panda and J. X. Xu "Iterative learning control based direct instantaneous torque control of switched reluctance motors", IEEE proc. PESC, pp.4832 -4837 June 2004 [30] D. S. Reay, T. C. Green, and B. W. Williams, “Neural networks used for torque ripple minimization from a switched reluctance motor,” IEEE proc. EPE, vol. 6, pp. 1–6. September 1993. [31] J. G. O’Donovan J, P. J. Roche, R. C. Kavanagh, M. G. Egan, and J. M. D. Murphy, “Neural network based torque ripple minimization in a switched reluctance motor,” Ind. Electron., Control and Inst., 1994. IECON '94., 20th Int. Conf. on , vol.2, no., pp.1226, September 1994. [32] D. S. Schramm , B. W. Williams and T. C. Green T "Torque ripple reduction of switched reluctance motors by PWM phase current optimal profiling,” IEEE proc. PESC, vol. 2, no. 29, pp.857 -860, July 1992. [33] I. Husain and M. Ehsani, "Torque Ripple Minimization in Switched Reluctance Motor Drives by PWM Current Control," IEEE Trans. Power Electron., vol.11, no. l, pp.83- 88, January 1996. [34] X. D. Xue , K. W. E. Cheng and S. L. Ho "Optimization and evaluation of torque- sharing functions for torque ripple minimization in switched reluctance motor drives", IEEE Trans. Power Electron., vol. 24, no. 9, pp.2076 -2090, September 2009. [35] R. Mikail, I. Husain, Y. Sozer, M.S. Islam, T. Sebastian, "Torque-Ripple Minimization of Switched Reluctance Machines Through Current Profiling," IEEE Trans. Ind. Appl., vol.49, no.3, pp.1258-1267, May-June 2013. [36] E. Daryabeigi, G. Arab Markadeh, C. Lucas, A. Askari, "Switched reluctance motor (SRM) control, with the developed brain emotional learning based intelligent controller (BELBIC), considering torque ripple reduction", Proc. IEEE Conf. IEMDC, pp. 979-986, 3-6 May 2009. 135

[37] J.N. Nagel and R.D. Lorenz: "Rotating Vector Methods for Smooth Torque Control of a Switched Reluctance Motor Drive," IEEE Trans. Ind. Appl., vol. 30 no. 2, pp. 540- 548, March/April 2000. [38] S. R. MacMinn and J. W. Sember, “Control of a switched-reluctance aircraft starter- generator over a very wide speed range”, Proc. Inter. Energy Conv. Eng. Conf., pp. 631-638, August 1989.

[39] Y. Z. Xu, R. Zhong, L. Chen, S.L. Lu, "Analytical method to optimise turn-on angle and turn-off angle for switched reluctance motor drives," Electric Power Appl., IET , vol.6, no.9, pp.593,603, November 2012. [40] E. Mese, Y. Sozer, J.M. Kokernak, D.A. Torrey, “Optimal excitation of a high speed switched reluctance generator”. IEEE Appl. Power Electron. Conf. and Expo.,APEC 2000, vol.1, no., pp.362-368, 2000. [41] Y. Sozer, D. A. Torrey, E. Mese, “Automatic control of excitation parameters for switched-reluctance motor drives,” IEEE Trans. Power Electron., vol.18, no.2, pp.594-603, Mar 2003. [42] Y. Sozer, D.A. Torrey, “Optimal turn-off angle control in the face of automatic turn- on angle control for switched-reluctance motors,” IET Electr. Power Appl., vol.1, no.3, pp.395-401, May 2007. [43] S. A. Fatemi, H. M. Cheshmehbeigi, E. Afjei, “Self-tuning approach to optimization of excitation angles for switched-reluctance motor drives,” European Conf. on Circuit Theory and Design, pp.851,856, August 2009. [44] B. Fahimi, G. Suresh, J. P. Johnson, M. Ehsani, M. Arefeen, I. Panahi, “Self-tuning control of switched reluctance motors for optimized torque per ampere at all operating points,” IEEE Appl. Power Elec. Conf. and Expo. APEC 1998, pp.778-783 vol.2, February 1998. [45] M.M Beno, N.S. Marimuthu, N.A. Singh, “Improving power factor in switched reluctance motor drive system by optimizing the switching angles,” IEEE TENCON Region 10 Conf., pp.1-5, November 2008. [46] C. Mademlis, I. Kioskeridis “Performance optimization in switched reluctance motor drives with online commutation angle control”, IEEE Trans. Energy Convers., 2003, vol.18, no.3, pp.448,457, September 2003. [47] N. Nakao, K. Akatsu, "Torque control of a switched reluctance motor by using torque estimation and excitation angle control," IEEE Energy Convers. Cong. and Expo.(ECCE), 2012 IEEE , pp.4314,4321, September 2012. [48] T.M. Jahns, "Motion control with permanent-magnet AC machines," Proc. IEEE, vol. 82, no. 8, pp. 1241–1252, August 1994.

136

[49] M. Ehsani, Y. Gao; J.M. Miller, "Hybrid Electric Vehicles: Architecture and Motor Drives," Proc. IEEE, vol. 95, no. 4, pp. 719–728, Apr. 2007 [50] K.R. Geldhof, T.J Vyncke, F.M.L.L De Belie, J.A.A Melkebeek, L. Vandevelde, "A Space Vector Strategy for Smooth Torque Control of Switched Reluctance Machines", in Proc. IEMDC '07. 2007, vol.2, pp.1269-1275, 3-5 May 2007. [51] D. W. Novotny and T. A. Lipo, “Vector Control and Dynamics of AC Drives”, Clarendon Press, Oxford, 1996. [52] Y.Sozer, D.A. Torrey, “Adaptive flux weakening control of permanent magnet synchronous motors. “ Ind. Appl. Conf. 1998, vol. 1, pp. 475-482, October 1998. [53] T. Husain, A. Elrayyah, Y. Sozer and I. Husain, “Dq Control of Switched Reluctance Machines,” IEEE Appl. Power Elec. Conf. and Expo. APEC 2013, pp.1537-1544, 17- 21 March 2013. [54] M. S. Islam, M. N. Anwar ad I. Husain, “Design and control of switched reluctance motors for wide-speed-range operation,” Electric Power Appl., IEE Proc. - , vol. 150, no. 4, pp. 425- 430, July 2003. [55] T. Husain, A. Elrayyah, Y. Sozer and I. Husain, “An Efficient Universal Controller for Switched Reluctance Machines,” IEEE Appl. Power Elec. Conf. and Expo. APEC 2013, pp.1530-1536, 17-21 March 2013. [56] J.R. Stuart, P. Norvig,“Artificial Intelligence: A Modern Approach” Prentice Hall, 2010. [57] M. Kaplan, J.M. Kokernak, E. Meese, Y. Sozer, D.A. Torrey “ Method for operating a switched reluctance electrical generator using data mapping”, U.S. patent, November 2004. [58] D.A. Torrey, Y. Sozer “Closed loop control of excitation parameters for high speed switched-reluctance generators”, U.S. patent, July 2006. [59] Y. Zou, M. Elbuluk, Y. Sozer, “A complete Modeling and Simulation of Induction Generator Wind Power Systems”, Industry Appl. Soc. Ann. Meet.(IAS), 2010 IEEE, pp.1,8,3-7, October 2010. [60] T. Husain, A. Elrayyah, Y. Sozer and I. Husain, “Adaptive Flux Weakening Control of Switched Reluctance Machines in Rotating Reference Frame,” IEEE Energy Conv.. Cong. and Expo. ECCE 2013, in press.

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