ROTATING REFERENCE FRAME CONTROL OF SWITCHED RELUCTANCE MACHINES. A Thesis. Presented to. The Graduate Faculty of The University of Akron

ROTATING REFERENCE FRAME CONTROL OF SWITCHED RELUCTANCE MACHINES A Thesis Presented to The Graduate Faculty of The University of Akron In Partial Fu...
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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 Dr. Yilmaz Sozer

_____________________________ Department Chair Dr. J. Alexis De Abreu Garcia

_____________________________ Committee Member Dr. Malik E. Elbuluk

_____________________________ Dean of the College Dr. George K. Haritos

_____________________________ Committee Member Dr. Tom T. Hartley

_____________________________ Dean of the Graduate School Dr. George R. Newkome

_____________________________ Date ii

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.

iv

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.

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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

ix

LIST OF TABLES Page

Table

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

x

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 𝜃 profile of an SRM for different current levels.............................36 xi

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, f x(θ) and f ix with respect to time...........................40

3.7.

Torque versus 𝜃 profile of an SRM and a sinusoidal component f x(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. f ix 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 𝜃𝑡ℎ .....................................................................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 𝐺𝑥 (𝜃) ..........................................................................77

5.6.

Graphical representation of f six ..............................................................................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 f q and f d with step response in torque at a speed of 5000 rpm.............90 5.26. Variation of f q and f d 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 f q , f d 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 f q , f d and resulting θPa with speed step responses at 8 s., 17 s., 38 s. and 60s..............................................................125 xv

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 1

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: 𝑓1 =

𝜔𝑚 60

∗ 𝑁𝑟 𝐻𝑧

(1.1)

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 2

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: 3



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. 6

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 7

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 Controller

Reference Current

Inner Loop Controller

Gate Signals

POWER CONVERTER

Phase Voltages

SRM

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

Flux Linkage ()

energy W and co-energy W’ as shown in Fig. 2.4.

Completely Unaligned Completely Aligned

0

Current (A) Figure 2.2: Flux linkage for aligned and unaligned position.

9

Aligned Inductace

Inductance (mH)

Aligned Position

Unalgined Position

Unaligned Inductace 0 0

Electrical Degrees

Figure 2.3: Air gap inductance for two electrical cycles.

Flux Linkage

W

W'

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: 𝐸𝑅 =

𝑊

(2.1)

𝑊+𝑊′

Flux Linkage ()

W

W'

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

11

the converter. For constant excitation the incremental mechanical work done can represented as: 𝑊 ′ = ∫ 𝜆 (𝜃, 𝑖 )𝑑𝑖 = ∫ 𝐿 (𝜃, 𝑖 )𝑖𝑑𝑖

(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: 𝑇𝑒 =

𝑑𝑊′( ( 𝜃,𝑖) 𝑖 𝑑𝜃

|

𝑖=𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

(2.3)

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: 𝑇𝑒 =

𝑑𝐿( ( 𝜃,𝑖) 𝑑𝜃



𝑖2 2

(2.4)

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 12

𝑉𝑝ℎ = 𝑖𝑅 +

𝑑𝜆

(2.5)

𝑑𝑡

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 𝜆 = 𝜆(𝜃, 𝑖 ) = 𝐿 (𝜃, 𝑖 )𝑖

(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 𝑉𝑝ℎ = 𝑖𝑅 +

𝑑𝜆 𝑑𝑡

= 𝑖𝑅 +

𝑑𝐿 ( 𝜃,𝑖) 𝑖 𝑑𝑡

𝑑𝑖

𝑑𝜃 𝑑 𝐿 ( 𝜃,𝑖)

= 𝑖𝑅 + 𝐿 (𝜃, 𝑖 ) ∗ 𝑑𝜃 + 𝑖 𝑑𝑡

𝑑𝜃

(2.7)

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 𝑒=𝑖

𝑑𝜃 𝑑𝐿 ( 𝜃,𝑖) 𝑑𝑡

(2.8)

𝑑𝜃

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 𝑃 = 𝑖𝑉𝑝ℎ = 𝑖 2 𝑅 +

1 𝑑( 𝐿 𝑖 2 ) 2

𝑑𝜃

13

+

1 2 𝑑 𝐿 ( 𝜃,𝑖) 𝑖 2 𝑑𝜃

𝜔

(2.8)

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

Flux Linkage ( )

.25 .2 .15 1 Unalgined Position

0.05 0

0

40

80

120 160 Current (A)

200

240

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

14

280

200 150

Torque (Nm)

100 50 0 -50 -100 -150 -200

0

5

10

15 20 25 30 Rotor position mechanical ()

35

40

45

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.

16

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. 17

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:

18

SRM Model

Geometry Based Analytical Models

Nonlinear Models

FEA analysis based look up 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 19

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 20

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 turnon 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 Cycle

V*

ω*

+

Outer Loop Controller

-

PWM Controller

Gate Signals

Electronic Commutator

ω

Vph

Converter

SRM

θ Angle Calculator

θon θof f 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 21

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 Controller

-

Torque Controller

Iref

Current Regulator

Vph

Converter

ω

Iph Speed Calculator

Angle Calculator

θon θoff

Electronic Commutator

θ

Figure 2.12 Current controlled drive. 22

SRM

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.

23

3.5 Phase A

Phase B

3

Torque dip

Phase Torque in N.m

2.5 2 1.5 1 0.5 0 -0.5 -30

-25

-20

-15 -10 -5 0 Rotor Position in degrees

5

10

15

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

24

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

25

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

26

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 27

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.

28

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 29

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.

30

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). 32

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

33

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 P.U Values

Current 1 Ia Ib Ic

0

-1

0

0.02

0.04

0.06

0.08

0.1 0.12 Time (s) Flux

0.14

0.16

0.18

 P.U Values

1

Torque P.U Values

a b c

0

-1

0.2

0

0.02

0.04

0.06

0.08

0.1 0.12 Time (s) Torque

0.14

0.16

0.18

0.2

2 Ta Tb Tc Total Torque

1

0

0

0.02

0.04

0.06

0.08

0.1 0.12 Time (s)

0.14

0.16

0.18

0.2

Figure 3.1: SM torque production (values in pu). In three-phase non-salient SM, the produced torque is given by [51] 𝑇 = 𝐾(𝜆 𝑎𝑠 𝜆 𝑎𝑟 + 𝜆 𝑏𝑠 𝜆 𝑏𝑟 + 𝜆 𝑐𝑠 𝜆𝑐𝑟 ) = 𝐾 (sin2 𝜃 + sin2 (𝜃 −

34

2𝜋 3

) + sin2 (𝜃 +

2𝜋 3

))

(3.1)

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 𝜃 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. 𝑇 = 𝐾 ′ (𝜆 𝑞𝑠 𝜆𝑑𝑟 − 𝜆 𝑑𝑠 𝜆 𝑞𝑟 )

(3.2)

𝐾 ′ is a constant, and 𝜆 𝑞𝑗 , 𝜆 𝑑𝑗 is flux component in the q and d axes, respectively 𝑗 = 𝑠, 𝑟. 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

𝑓𝑞𝑑𝑠 = 𝑓𝑞 − 𝑗𝑓𝑑 = 3 𝑒 −𝑗𝜃 [𝑓𝑎 + 𝑎. 𝑓𝑏 + 𝑎2 𝑓𝑐 ] where f stands for voltage, current or flux, 𝑎 = 𝑒

𝑗2𝜋 3

and 𝜃 is the rotor position.

Fb Fq

θ Fa

Fd Fc

Figure 3.2: abc and dq reference frame vectors. 35

(3.3)

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(𝜃)) [7]. For threephase SRM, the torque (T) can be represented by 𝑇 = 𝑓𝑖𝑎 𝑓𝜃𝑎 (𝜃) + 𝑓𝑖𝑏 𝑓𝜃𝑏 (𝜃) + 𝑓𝑖𝑐 𝑓𝜃𝑐 (𝜃)

(3.4)

where 𝑓𝑖𝑥 is a function related to the current and 𝑓𝜃𝑥 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 wave depending on 𝑓𝜃𝑥 while the amplitude of the curve depends on the current. If 𝑓𝜃𝑥 is taken as a pure sine signal, then making 𝑓𝑖𝑥 in Eqn. (3.4) a sine wave in phase with 𝑓𝜃𝑥 would produce constant torque as in the case of SM in Eqn. (3.1). However, there are non-idealities in the case of SRM that defy this similarity. First 𝑓𝜃𝑥 is not a pure sine wave and second 𝑓𝑖𝑥 should be always be non-negative. 300

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

200

Torque (Nm)

100

0 Region I

-100

Region II

Region III

-200

-300

0

5

10

15 20 25 30 Rotor Postion Mechanical 

35

40

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

45

By representing the terms 𝑓𝑖𝑥 and 𝑓𝜃𝑥 as sinusoidal terms, the 𝑑𝑞 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 significant torque dips during the commutation. Moreover, most of the distortion in 𝑓𝜃𝑥 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

fq ωref

Adaptive Flux Weakening Controller

fq

fd

dq to abc

f’ia f’ib f’ic

Negativity Removal

fia fib fic

Non Linear Model

Iiaref Iibr ef

Current Regulator

Sa Sb Sc

Inverter

Va Vb Vc

SRM

Speed Controller

θ

Iphases

ω

Mechanical 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 f q which is distributed among the three phases through the components of 𝑓𝑖𝑥 which are in phase with 𝑓𝜃𝑥. The phase advancing can be achieved by the term f d as needed above base speed. The components f q and f d 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: 37

𝑐𝑜𝑠𝜃 𝑠𝑖𝑛𝜃 𝑓′𝑖𝑎 2𝜋 2𝜋 2 [𝑓′𝑖𝑏 ] = 3 [𝑐𝑜𝑠(𝜃 − 3 ) 𝑠𝑖𝑛 (𝜃 − 3 ) 2𝜋 2𝜋 𝑓′𝑖𝑐 𝑐𝑜𝑠(𝜃 + 3 ) 𝑠𝑖𝑛 (𝜃 + 3 )

1 𝑓 𝑞 1] [𝑓 ] 𝑑

1

𝑓0

(3.5)

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 f ia , f ib and f ic to produce the same required torque. As stated previously, the term 𝑓𝜃𝑥 in Eqn. 3.4 is a distorted sine wave, but the commanded term f ix is produced assuming a pure sine 𝑓𝜃𝑥 . The nonlinear model block then makes the adjustments in f ix to compensate for the distortion in 𝑓𝜃𝑥 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 f ix 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

38

the sinusoidal current terms aligned with

𝑑𝐿(𝜃,𝑖) 𝑑𝜃

, i.e. to be in phase with inductance

variation. Therefore, positive torque is produced when the

𝑑𝐿(𝜃,𝑖 ) 𝑑𝜃

signal is positive.

60

120

40

100

Negativity Removal Block

0 -20

80

fix

f'ix

20

60 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: 𝑇 = 𝑓′𝑖𝑎 cos 𝜃 + 𝑓′𝑖𝑏 cos (𝜃 − with

2𝜋 3

2𝜋 3

) + 𝑓′𝑖𝑐 cos (𝜃 +

2𝜋 3

)

(3.6)

phase shifts between the three phases and where 𝑓′𝑖𝑥 depends on the required

torque. The portions of 𝑓′𝑖𝑥 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 𝑓′𝑖𝑎

𝑓′𝑖𝑏 are negative at certain instant then the portion of torque taken

by 𝑓𝑖𝑐 would be given by

𝑓𝑖𝑐 = 𝑓′𝑖𝑐 +

𝑓 ′𝑖𝑎 𝑐𝑜𝑠 (𝜃) 2𝜋 ) 3

𝑐𝑜𝑠(𝜃+

39

+

2𝜋 ) 3 2𝜋 𝑐𝑜𝑠(𝜃+ ) 3

𝑓 ′ 𝑖𝑏 𝑐𝑜𝑠 (𝜃−

(3.7)

If only 𝑓′𝑖 𝑎 is negative then ′

𝑓𝑖 𝑐 = 𝑓 𝑖 𝑐 +

2𝜋 )∗𝑓 ′ 𝑖𝑎 𝑐𝑜𝑠 (𝜃) 3 2𝜋 2𝜋 ′𝑚 𝑓𝑖𝑏 𝑐𝑜𝑠 𝑚 (𝜃− ))∗𝑐𝑜𝑠(𝜃+ ) 3 3

′𝑚 𝑓𝑖𝑐 𝑐𝑜𝑠 𝑚 (𝜃+ 2𝜋 )+ 3

′𝑚 (𝑓𝑖𝑐 𝑐𝑜𝑠 𝑚 (𝜃+

(3.8)

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 𝑓𝑞 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 𝑓𝑖𝑥. These values maintain the torque sharing smoothness discussed previously.

f'ix

100 0

-100

fa

1 0 -1

fix

100 50 0 Time

Figure 3.6: Graphical representation of f’ix, f x(θ) and f ix 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. (3.9)

𝑓′𝑖 𝑎 = 𝐼 cos 𝜃

Now for example, assume 𝑓′𝑖𝑐 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 𝑓′𝑖 𝑎 another term 𝑓𝑥 𝑖𝑎 such that 𝑓𝑥 𝑖𝑎 cos 𝜃 = 𝑓′𝑖 𝑐 cos (𝜃 +

2𝜋 3

(3.10)

)

It follows that

𝑓𝑥 𝑖𝑎 =

2𝜋 ) 3

𝑓 ′𝑖𝑐 cos(𝜃+ cos𝜃

2𝜋

=

′ 𝐼 𝑓 𝑖𝑐 cos(𝜃+ 3 ) 𝐼 cos 𝜃

=

′2 𝑓𝑖𝑐

𝑓 ′𝑖𝑎

(3.11)

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

′2 𝑓𝑖𝑐

𝑓 ′𝑖𝑏

(3.12)

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 𝑓̃𝑖𝑎 and 𝑓̃𝑖𝑏 are 41

′2

′2 𝑓𝑖𝑎 ′2 ′2 𝑖𝑎 𝑓𝑖𝑎 +𝑓𝑖𝑏

𝑓 𝑓̃𝑖𝑎 = 𝑓′𝑖𝑐

𝑓̃𝑖𝑏 =

′2 𝑓𝑖𝑐

′2 𝑓𝑖𝑏

𝑓 ′2

(3.13)

𝑖𝑐 = 𝑓′2 +𝑓 ′2 𝑓′𝑖 𝑎 𝑖𝑎

𝑖𝑏

𝑓 ′2

′2 ′2 𝑓 ′𝑖𝑏 𝑓𝑖𝑎 +𝑓𝑖𝑏

(3.14)

𝑖𝑐 = 𝑓′2 +𝑓 ′2 𝑓′𝑖 𝑏 𝑖𝑎

𝑖𝑏

The new total signals applied by the phases will be 𝑓𝑖 𝑎 = 𝑓′𝑖 𝑎 +

′2 𝑓𝑖𝑐 ′2 ′2 𝑓𝑖𝑎 +𝑓𝑖𝑏

𝑓′𝑖 𝑎 = 𝑓′𝑖 𝑎

𝑓𝑖 𝑏 = 𝑓′𝑖 𝑏 +

′2 𝑓𝑖𝑐 ′2 ′2 𝑓𝑖𝑎 +𝑓𝑖𝑏

𝑓′𝑖 𝑏 = 𝑓′𝑖 𝑏

Suppose now phases B and C have negative value at

′2 ′2 ′2 𝑓𝑖𝑎 +𝑓𝑖𝑏 +𝑓𝑖𝑐 ′2 ′2 𝑓𝑖𝑎 +𝑓𝑖𝑏

′2 ′2 ′2 𝑓𝑖𝑎 +𝑓𝑖𝑏 +𝑓𝑖𝑐

𝑑𝐿 𝑑𝜃

′2 ′2 𝑓𝑖𝑎 +𝑓𝑖𝑏

(3.15)

(3.16)

position; hence, their roles have

to be carried out by phase A only. By following the same derivation, the new total signal of phase A will be: 𝑓 ′2

𝑓 ′2

𝑖𝑎

𝑖𝑎

𝑓𝑖 𝑎 = 𝑓′𝑖 𝑎 + 𝑓𝑖𝑏 + 𝑓𝑖𝑐′ = 𝑓′𝑖 𝑎 ′

′2 ′2 ′2 𝑓𝑖𝑎 +𝑓𝑖𝑏 +𝑓𝑖𝑐 ′2 𝑓𝑖𝑎

(3.17)

A compact way to represent these relations can be obtained by defining the following notation 1 𝑢(𝑥 ) = { 0

𝑥>0 𝑥 ≤0

(3.18)

Then the negativity cancellation can be obtained using:

𝑢(𝑓 ′

′2 ′2 ′2 𝑓𝑖𝑎 +𝑓𝑖𝑏 +𝑓𝑖𝑐 ′2 ′2 ′ ) ( ) ( ′ ) ′2 𝑖𝑎 𝑓𝑖𝑎 +𝑢 𝑓 𝑖𝑏 𝑓𝑖𝑏 +𝑢 𝑓 𝑖𝑐 𝑓𝑖𝑐

(3.19)

𝑓𝑖 𝑏 = 𝑢 (𝑓′𝑖 𝑏) 𝑓′𝑖 𝑏 𝑢 (𝑓′

′2 ′2 ′2 𝑓𝑖𝑎 +𝑓𝑖𝑏 +𝑓𝑖𝑐 ′2 ′2 ) ( ′ ) ( ′ ) ′2 𝑖𝑎 𝑓𝑖𝑎 +𝑢 𝑓 𝑖𝑏 𝑓𝑖𝑏 +𝑢 𝑓 𝑖𝑐 𝑓𝑖𝑐

(3.20)

𝑓𝑖 𝑎 = 𝑢(𝑓′𝑖 𝑎 ) 𝑓′𝑖𝑎

42

𝑓𝑖 𝑐 = 𝑢 (𝑓′𝑖 𝑐) 𝑓′𝑖 𝑐 𝑢 (𝑓′

′2 ′2 ′2 𝑓𝑖𝑎 +𝑓𝑖𝑏 +𝑓𝑖𝑐 ′2 ′2 ′ ) ( ) ( ′ ) ′2 𝑖𝑎 𝑓𝑖𝑎 +𝑢 𝑓 𝑖𝑏 𝑓𝑖𝑏 +𝑢 𝑓 𝑖𝑐 𝑓𝑖𝑐

(3.21)

These equations ensure that the required torque will be implemented using only positive values at 𝑓𝜃𝑥 and also with high efficiency. After the negativity cancellation the new values of the signals are 𝑓𝑖𝑥 . 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 𝑓𝜃𝑎 (𝜃) 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 

Normalized Value

0.5

0

-0.5

-1

0

50

100 150 200 250 Rotor Position  Electrical

300

350

Figure 3.7: Torque versus 𝜃 profile of an SRM and a sinusoidal component f x(θ) 43

Then 𝑓𝜃𝑎 (𝜃) can be represented by: 𝑓𝜃𝑎 (𝜃) = 𝐺𝑎 (𝜃) cos 𝜃 => 𝐺𝑎 (𝜃) =

𝑓 𝜃𝑎 (𝜃)

(3.22)

cos𝜃

where 𝐺𝑎 (𝜃) is the distortion term in 𝑓𝜃𝑎 (𝜃). The commanded term f ia is produced by the dq controller assuming a pure sinusoidal 𝑓𝜃𝑎 (𝜃) and since this is not the case, a correction is needed to be made in the term f ia using 𝐺𝑎 (𝜃). The value of 𝐺𝑎 (𝜃) is calculated from the T-i-θ characteristics of the machine. For determination of the 𝐺𝑎 (𝜃) 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 𝐺𝑎 (𝜃) 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 𝐺𝑥 (𝜃).

fix

θ

X

fix LUT

Ixref

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 44

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 𝐺𝑥 (𝜃). Assume now the current related function f ia is updated to be

𝑓𝑖𝑎 . 𝑎 ( 𝜃)

𝑓𝑖𝑎𝑠 given by 𝑓𝑖𝑎𝑠 = 𝐺

By

commanding f sia for torque production in Eqn. 3.6, the torque for phase A (Ta ) becomes: 𝑇𝑎 = 𝑓𝑖𝑎𝑠 𝑓𝜃𝑎 (𝜃) =

𝑓𝑖𝑎 𝐺𝑎 ( 𝜃)

𝐺𝑎 (𝜃) cos 𝜃 = 𝑓𝑖𝑎 cos 𝜃

(3.23)

This is the required expression to make smooth torque production. A similar discussion can be made for phases B and C. The function 𝐺𝑎 (𝜃) 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 G x(θ). 45

Since f

s ix

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

another look-up table to relate the function f

s 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 f d in the d axis. Consider a case where f q and f d in Fig. 3.4 are nonzero. Then, the waveform of f ix will be advanced from the 𝑓𝜃𝑥 as shown in Fig. 3.10. The advanced f ix 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 A. 𝑑𝑖𝑎 𝑑𝑡

=

𝑉𝑑𝑐 −𝑅𝑎 𝑖𝑎 − 𝑑𝜆 𝑑𝑖

𝑑𝜆 𝑑𝑙 𝜔 𝑑𝑙 𝑑𝜃

Here 𝑖 𝑎 is phase A current, 𝑉𝑑𝑐 is the DC bus voltage, the inductance,

𝑑𝑙 𝑑𝜃

(3.24)

𝑑𝜆 𝑑𝑙

is the flux rate of change with

is the inductance rate of change with position, 𝜔 is the speed and

the flux rate of change with current.

46

𝑑𝜆 𝑑𝑖

is

1

f'ix

Normalized P.U Values

fx 0.5

0

-0.5

-1

0

50

100 150 200 250 Rotor position Electrical 

300

350

Figure 3.10: Current wave shapes with and without phase advancing. At high speeds, 𝜔 will have large value and this reduces the current rate of change since 𝑉𝑑𝑐 is fixed. However, at small values for 𝜃, the term

𝑑𝑙 𝑑𝜃

has small value.

Therefore, by performing the magnetization (applying 𝑉𝑑𝑐 across the phase) at early angle, the term

𝑑𝜆 𝑑𝑙 𝑑𝑙 𝑑𝜃

𝜔 will have small value and the current can grow higher and faster.

Then by commanding a certain value for f d more current will build up at early angles which are useful for torque production. Moreover, f ix 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

47

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: 𝑓𝑖 𝑐 = 𝑓′𝑖 𝑐 + 𝑓′𝑖 𝑎 ∗ (−.5 +

√3 2

∗ tan(𝜃 +

2𝜋 3

) + 𝑓 ′ 𝑖 𝑏 ∗ (−.5 −

√3 2

∗ tan (𝜃 +

2𝜋 3

) (3.25)

With 𝑓′𝑖 𝑎 and 𝑓′𝑖 𝑏 negative and if only 𝑓′𝑖 𝑎 is negative then 𝑓𝑖 𝑐 = 𝑓 ′ 𝑖 𝑐 − 𝑓 ′ 𝑖 𝑎

(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 48

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

𝐼𝑥

𝑓𝑖𝑥 LAT

𝐶𝑜𝑠𝜃𝑥

𝜃𝑥

X

𝑇𝑒𝑠𝑡

𝐺𝑥 (𝜃) 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 f d, f q and f 0 to study how they affect the commands into the current regulator. In Fig. 3.12 the current related command shapes f ix after negativity removal block for varying commands of f d and f 0 with f q at a constant command of 80 to produce a smooth 80Nm torque at the output. Figure 3.12(a) shows the shape of

𝑓𝜃𝑥 (𝜃). When 𝑓𝜃𝑥 (𝜃) is positive any positive

f ix will produce positive torque. Figure 3(b) shows the f ix shape after the negativity 49

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

f

1 0 -1

fix (f(A))

0

36

72

108

144

252

288

324

360

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

50

0

36

72

108

144

200 fix (f(A))

216

100

0

180 (b)

216

252

288

324

360

fq=80 A, fd = -40 A,f0 =0 A fq=80 A, fd = -10 A,f0 =0 A fq=80 A, fd = -80 A,f0 =0 A

100 0

fix (f(A))

180 (a)

0

36

72

108

144

180 (c)

100

216

252

288

324

360

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

50 0

0

36

72

108

144

180 (d)

216

252

288

324

360

Rotor Position  Figure 3.12: f ix a for varying command parameters The conduction region can be manipulated by the commands f d and f 0. In Fig. 3.12(c), it is observed that the phase is shifted towards the left with the value of f d controlling the amount of phase shift, larger f d results in more phase advancing, here f q was 80 and f 0 was 0. The negativity removal algorithm also reshapes the command f ix to have higher commands at the beginning of the conduction region. This minimizes the effect of backEMF which is similar to SM where f d is applied for flux weakening at higher speeds. The 50

control designer can thus command f d 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 f 0 is command along with f q the conduction region is reduced as shown in Fig. 3.12(d) Incrementing f 0 would further reduce the conduction region. In all the cases the f q , f d and f 0 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.

51

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 turnoff 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 f d 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 currentflux-position (i-λ-θ) curves. The relation between the flux and the applied DC bus voltage during demagnetization is given by: 𝑑𝜆 𝑑𝑡

= −𝑉𝑑𝑐 − 𝑖 𝑅

where 𝑖 is the phase current and 𝑅 is its winding resistance.

54

(4.1)

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= θikk>-∆t iM ω

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 55

time step of ∆t at a particular speed. Starting from λ 0 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: 𝜆 𝑘+1 = 𝜆 𝑘 + Δ𝑡(𝑉𝑑𝑐 + 𝑖𝑅) 𝜃𝑘 +1 = 𝜃𝑘 − Δ𝑡𝜔,

(4.2) (4.3)

where 𝜆 𝑘 and 𝜃𝑘 are the flux and the rotor position at sample number k. Then using the iλ-θ as the function, the DM current for that speed at θk+1 is given by: 𝑖 𝑘+1 = 𝑓𝑖−𝜆−𝜃 (𝜆 𝑘+1 , 𝜃𝑘+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 𝑑𝑖𝑥 𝑑𝑡

=

𝑉𝑥 −𝑅𝑥 𝑖𝑥−𝑖𝑥

𝑑𝐿𝑥 (𝜃) 𝜔 𝑑𝑡

𝐿𝑥 (𝜃)

56

(4.5)

where 𝑖 𝑥 is the phase current, 𝑉𝑥 is the phase voltage,

𝑑𝐿𝑥 (𝜃) 𝑑𝑡

is describing the rate of change

of inductance, and 𝐿 𝑥 (𝜃) is the non-linear inductance profile of the machine. Here the backemf voltage of the machine can be represented by 𝑒 = 𝑖𝑥

𝑑𝐿𝑥 (𝜃) 𝑑𝑡

(4.6)

𝜔

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 I=160A I=200A I=240A I=280A I=320A

200

Torque (Nm)

100

0 Region I

-100

Region II

Region III

-200

-300

0

5

10

15 20 25 30 Rotor Postion Mechanical 

Figure 4.3: T-i- θ characteristics. 57

35

40

45

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 I=120A I=160A I=200A I=240A I=280A I=320A

60

Delta Torque (Nm)

40

20

0

-20

-40

-60

0

5

10

15

20 25 Rotor Position 

30

35

40

45

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. 58

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 𝜃𝑡ℎ = 𝜃𝑎𝑙𝑖𝑔𝑛𝑒𝑑 +

𝛽𝑟 −𝛽𝑠 2

(4.7)

where 𝜃𝑎𝑙𝑖𝑔𝑛𝑒𝑑 is the completely aligned rotor position and 𝛽𝑟 is the rotor pole width and 𝛽𝑠 is the stator pole width as shown in Fig 4.6.

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

59

BETAR

BETAS

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 60

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: 𝜃𝑜𝑛𝑙𝑖𝑚𝑖𝑡 = 𝜃𝑢𝑛𝑎𝑙𝑖𝑔𝑛𝑒𝑑 −

𝛽𝑟 −𝛽𝑠

(4.8)

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

60

Current (A)

40 Negative Torque Peak at th 20

Actual Current at th

Torque (Nm)

100

50 0 Desired Current(Ith) at th 0 3.5

4

4.5 Time (S)

5

-20 5.5 -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. 61

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. II.

No PA 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

Phase Torque(Nm)/ Phase Current (A)/ Rotor Position 

Topt. 150

TnoPA B

InPA IexPA

100

TexPA data7 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 62

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 63

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. 𝐼𝑡ℎ = 𝑓𝑎𝑐𝑡𝑜𝑟 ∗ 𝜔 ∗

𝑓(𝐼𝑐𝑚𝑑) 𝑉𝐷𝐶

(4.9)

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 𝑓(𝐼𝑐𝑚𝑑). 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 f d 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 𝑓

𝜃𝑃𝐴 = arctan( 𝑑 ) 𝑓𝑞

fq

Phase Advancing Limit

f Speed increases

θpa

fd

Figure 4.9: Phase vector of torque command.

65

(4.10)

The block diagram for the system is shown in Fig 4.10. The command for f q can be determined from either speed or torque controller and PA block determines the amount of f d 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 f d 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

ωactual

Speed Controller

Torque Limiter Adaptive Phase Advancing Algorithm

Iactual θ

fd dq SRM Controller

fd-1

Figure 4.10: Block diagram of flux weakening controller 150

80

60

Current (A)

40 Negative Torque Peak at th 20

Actual Current at th

Torque (Nm)

100

50 0 Desired Current(Ith) at th 0 3.5

4

4.5 Time (S)

5

-20 5.5 -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 f d in this problem with the optimum objective function which is: 𝐼𝑎𝑐𝑡 = 𝐼𝑡ℎ

(4.11)

𝜃 = 𝜃𝑡ℎ

(4.12)

𝐼𝑎𝑐𝑡 = 𝑓(𝑓𝑑𝑞 )

(4.13)

At,

Here,

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 f d . 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.

67

fd=fd-1 N θ =θ th Y N

Y

fd=0

Iactual>Ith

Y

fd=fd-1 + k

fd=fd-1 - k

fd>0

FdIth

Iact

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