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.
v
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
Fd<-(fq +
0.25* fq)
N
N
Y
fd=fd
fd=fd
fd=-(fq +
0.25* fq)
Figure 4.12: Flowchart of adaptive flux weakening controller using hill climbing method.
This hill climbing method is a local search method where the objectives to find the
local optimum of a system. It has disadvantage as it only finds the local maximum. So this
will never reach the global maximum unless the system is heuristic convex. This is however
the case as shown in Fig 4.13. The phase advance angle is swept across a wide range and
we see that the curve has a convex shape thereby negating the disadvantage of hill climbing
methods. This however was found for one particular machine that was investigated in the
development of the algorithm. If the system is not heuristic convex in nature then it can be
adjusted by having random restarts or using another system for adaptation such as
stochastic hill climbing and simulated annealing[56].
68
Maximum Torque Per Ampere (Nm/A)
0.8
0.75
0.7
Corresponds to advance angle
where, Iact=Ith at th
0.65
0.6
Iact>Ith
Iact<=Ith
0.55
0.5
0
10
20
30
40
50
60
Phase Advance angle in Electrical 
70
Figure 4.13: The heuristic convex nature of maximum torque per amp with respect to PA.
The unique feature of this control method is that it can adapt itself to adjust the amount of
phase advancing irrespective of the operating conditions. The benefits of this system when
used in conjunction with the dq control method is to allow high speed operation through
adaptive phase advancing while preserving the dq controllers inherent torque ripple
minimization feature.
4.4.
Conclusions
In this chapter, an adaptive flux weakening control for SRMs based on the dq
method is proposed. The proposed controller has a simple structure and removes the need
for extensive offline calculations or lookup tables for flux weakening and wide speed range
operation of the SRM.
It compares the actual currents with a threshold current at a
threshold voltage and uses the hill climbing adaptive method for determining the amount
69
of phase advancing. This makes a closed loop adaptive system which is not heavily
dependent on the machine parameters beforehand.
The next chapter is about the finite element modeling and simulation on the
proposed methods. An analysis was performed on the efficiency of the dq and flux
weakening controller by varying the third component of f 0 and how it can be a source for
the future work. It also compares how the dq controller matches up with traditional angle
controller.
70
CHAPTER V
MODELING AND SIMULATION RESULTS
5.1.
Introduction
In chapter III a control strategy based on the rotating dq reference frame for SRMs
was presented. Then in chapter IV an adaptive flux weakening controller which advances
the turn off angles and compliments the dq controller was presented. These chapters
developed the theory and algorithms for simplifying the SRM control using the dq
controller and automating the turn-off angle selection with the help of the adaptive flux
weakening section. This chapter uses those algorithms in computer simulations for
validation. This chapter also discusses about parametric sensitivity of the proposed
controllers. Machine design limitations have also been briefly discussed.
Several simulations were conducted for the verification process. The simulations
were conducted on a 110 kW test SRM for validating the control. The machine was
modeled using Arthur Raduns’ model presented in [16 - 18] and using the finite element
analysis based model. Both the models were used in the verification process. The next
sections present the simulation results verifying the dq controller and the adaptive flux
weakening controller.
The preceding section contains a brief analysis of the efficiency
using the dq controller compared to traditional excitation angle based controllers.
71
5.2.
Modeling
For proper verification of the proposed control methods it was necessary to have a
good nonlinear model of the SRM. The first model that was used for initial verification of
the dq controllers torque sharing ability, i.e. negativity removal block, was the Spong
model presented in [7]. The motivation for using this model was that it could address the
SR machines nonlinearity without introducing a high degree of complexity. This method
however suffered as it was difficult to model a real machine as it had no geometric
parameters associated with it. Hence Arthur Radun model presented in [16-18] was used
for further verification. This modeling method is based on the machines geometry and
magnetic properties. This model dynamically generates the solutions for the static torque
and flux linkage. This geometry based model allows greater flexibility in the design and
study of the control algorithms as it is easier to vary machine parameters. Thus this model
is used as the primary simulation source for development. The problem this model faced
was at high speeds. So for developing the flux weakening for a wide speed range operation
a look up table based method was used.
The main motor used during the verification process was a three phase 110 kW
SRM with a 12/8 pole configuration. The general machine parameters are given in Table
5.1. It was modeled using the machines geometric parameters using the Arthur Raduns’
model. The model also generated the static T-i-θ characteristics shown in Fig. 5.1 that are
used in the controller design. Small changes were incorporated into the machine model by
changing certain parameters to observe the system performance. This model also provided
the static curves used in the development of the theory for flux weakening.
72
Table 5.1 SRM parameters of 110 kW 12/8 SR machine.
No. of Phases
3
Power
110 KW
Max. Speed
8000 rpm
Max. Current
150A
DC Link Voltage
600V
Phase resistance
0.3 Ohm
Max. Torque
110 Nm
Stator Poles
12
Rotor Poles
8
200
150
100
Torque (Nm)
50
0
-50
-100
-150
-200
0
5
10
15
20
25
Rotor Position Mechanical 
30
35
40
45
Figure 5.1: T-i-θ characteristics of the 110 KW SR using analytic modeling.
73
This method was suitable for simulations at low speed and matched the T-i-θ
characteristics from finite element simulation which is shown in Fig. 5.2. However at
higher speeds, as the flux model of the analytical method is not very accurate, look up
tables were used in the modeling process.
300
200
Torque (Nm)
100
0
-100
-200
-300
0
5
10
15
20
25
Rotor Postion Mechanical 
30
35
40
45
Figure 5.2: T-i-θ characteristics of the 110 kW SR using finite element modeling.
5.3.
Simulation Results
The control algorithms were implemented in Matlab/Simulink environment. A 110 kW,
600 V SRM is modeled based on the modeling approach provided in the previous section.
The motor parameters are shown in the Table 5.1. The SRM was first tested using the
conventional control then the proposed methods were tested. The control algorithm used
for the conventional control is described by Fig 2.12.
Figure 5.3 and 5.4 shows the torque and current curves using the conventional SRM
controller operating at 1000 rpm. In this controller, the torque production responsibility is
74
devoted to one phase at a time before it gets transferred to the next phase. Clearly, there is
a high torque ripple during the commutation periods, where the upcoming phase at the
beginning of its cycle cannot carry the load of producing all the torque immediately.
Therefore, the lack of the adequate toque sharing leads to high torque ripples.
200
Torque (Nm)
150
100
50
0
0.01
0.015
0.02
0.025
Time (S)
0.03
0.035
0.04
Figure 5.3: Torque with traditional control at 1000 rpm.
200
Current (A)
150
100
50
0
0.01
0.015
0.02
0.025
Time (S)
0.03
0.035
Figure 5.4: Current with traditional control at 1000 rpm.
75
0.04
Furthermore, this controller suffers for need to adjust its turn-on and turn-off angles
as these needs to be adapted with varying loads and speeds. Therefore the angle
commutator blocked described in Fig 2.12 would require the commands for the turn-on
and turn-off angle to be controlled dynamically. The proposed method of using the dq
controller removes the need for using the angle commutator block making the control
simpler and more analogous to synchronous machines and also brings with it some added
advantages of minimizing ripple. However the amount of ripple being reduced is subject
to the accuracy of the rotor position and lookup tables used. Results from the two main
contributions of this thesis is presented in the next subsections. Following which is an
efficiency analysis for the dq controller.
5.3.1. dq control
The look up tables for the dq controller was generated using the data from the 110
kW SRM being modelled using the methods mention in the preceding section. The lookup
table of 𝐺𝑥 (𝜃) is represented graphically in Fig. 5.5. Here the position θ is aligned with
cos(θ) being aligned with the positive torque production regions of the T-i-θ characteristics.
Fig. 5.6 contains the graphical representation of f six .
76
5
4
Gain
3
2
1
0
-1
0
20
40
60
80
100
Rotor Position 
120
140
160
180
Figure 5.5: Graphical representation of 𝑮𝒙 (𝜽).
9
8
7
Current Output
6
5
4
3
2
1
0
0
5
10
15
20
Input fix command
25
30
35
Figure 5.6: Graphical representation of f six .
The torque and currents curves for the proposed SRM dq controller are shown in Fig.
5.7 and 5.8. The proposed controller produces very little torque ripple if the lookup tables
77
and machine design is optimum as the figure shows. This small ripple could be acceptable
for traction applications, and through further adjustments in the look up tables, the torque
ripple can be improved further.
200
Torque (Nm)
150
100
50
0
0.006
0.008
0.01
0.012
0.014
Time (S)
0.016
0.018
0.02
0.022
Figure 5.7: Torque using the dq controller with proper tuning at 1000 rpm.
200
Current (A)
150
100
50
0
0.004
0.006
0.008
0.01
0.012
0.014
Time (S)
0.016
0.018
0.02
0.022
Figure 5.8: Current using the dq controller with proper tuning at 1000 rpm.
78
The controller proposed here is dependent on the machine characteristics and is
sensitive to changes in the machine parameters. To investigate the parameter vacation
effects on the controller performance, a sensitivity analysis has been carried out for this
system. The system is mostly dependent on the 𝑮𝒙 (𝜽) look-up table which is calculated
from the T-i-θ characteristics of the motor. These values for a particular machine can be
found through finite element analysis or through experiments. For our analysis we varied
some parameters of the machine and the resulting T-i-θ characteristics of the motor are
obtained and used to generate the look-up table of 𝑮𝒙 (𝜽). In one case, the motor model
considered to design the controller had a motor with a rotor pole width of 19 degrees while
the actual controlled motor had a rotor pole width of 21 degrees. After implementing the
designed controller for that machine, it was found that the mismatch in the machine design
has a more prominent ripple effect as shown in Fig.5.9. It can be seen from Fig. 5.8 and
5.10 that the current shapes were similar to a comparative ideal case but the torque had
more ripple. The second look-up table had its effect only on the response time of the system
as the commanded torque was never reached by the system due to the linear nature of this
table. This problem could be solved by tuning the outer loop controller. Therefore, it can
be concluded that only one part (one look-up table) was parameter sensitive to the operation
of the entire system.
79
200
Torque (Nm)
150
100
50
0
0.002
0.004
0.006
0.008
0.01
Time (S)
0.012
0.014
0.016
0.018
0.02
Figure 5.9: Torque with different machine model at 1000 rpm.
200
Current (A)
150
100
50
0
0.002
0.004
0.006
0.008
0.01
Time (S)
0.012
0.014
0.016
0.018
0.02
Figure 5.10: Current using the dq controller with different machine model at 1000 rpm.
This case led to test the machine further using the finite element data in look up
tables. The results at the same speed is shown in Fig 5.11 and Fig 5.12.
80
200
180
160
Torque (Nm)
140
120
100
80
60
40
20
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
Time (S)
Figure 5.11: Torque with dq using the FEM based method
200
Current (A)
150
100
50
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
Time (S)
Figure 5.12: Torque with dq using the FEM based method
It can be seen from the Fig 5.11 that the ripple is higher when comparing graphically
the previous model case shown in Fig. 5.7. The reasons for this is that the data in look up
tables from the finite element analysis based model did not have a high resolution and a
larger band gap for the current regulation was used. Still we can see that the ripple during
81
commutation is low. By performing Fast Fourier Transform (FFT) analysis on this result
the torque ripple was found to be around 5% at the pole passing frequencies.
Coupled simulation showed that the concept of dq and its ability to reduce torque
ripple and machine operation in a simple manner by negating the need for the angle
commutator. However the differences in results from the two simulation models merited
further investigation so a coupled simulation (Finite Element Analysis (FEA) software
working harmoniously with the control and circuit simulator) was performed. The SRM dq
controller is implemented with the motor running at 1000 rpm. The obtained curves for the
torque and the currents are shown in figs. 5.13 and 5.14, respectively where the output
torque has very little ripple on it. The results show that the control strategy is effective in
this simulation which takes into account the delays in the inverter as well as a more realistic
SRM motor model.
200
180
160
Torque (Nm)
140
120
100
80
60
40
20
0
2
4
6
8
10
Time (mS)
12
14
Figure 5.13: Torque with dq using coupled simulation.
82
16
18
200
180
160
Current (A)
140
120
100
80
60
40
20
0
2
4
6
8
10
Time (mS)
12
14
16
18
Figure 5.14: Current with dq using coupled simulation.
The dq controller was then tested to check its dependency on the motor design
parameters. The controller was found to be dependent on the rotor pole width quite highly.
The torque outputs for the different pole widths and proper look up tables are shown in Fig
5.15 to 5.18. It can be observed that the system has an optimum range in the rotor pole
widths for operation with minimum ripple. Outside that range the currents are not able to
follow the commands so there is a higher peak. This could be mitigated to a certain degree
by increasing the DC bus voltage or tuning the 𝐺𝑥 (𝜃) look up table.
83
200
Torque (Nm)
150
100
50
0
0.005
0.01
0.015
Time (S)
0.02
0.025
Figure 5.15: Torque with dq with a rotor pole width of 21 degrees.
200
Torque (Nm)
150
100
50
0
0.005
0.01
0.015
Time (S)
0.02
Figure 5.16: Torque with dq with a rotor pole width of 18 degrees.
84
0.025
250
Torque (Nm)
200
150
100
50
0
0.005
0.01
0.015
Time (S)
0.02
0.025
Figure 5.17: Torque with dq with a rotor pole width of 22 degrees.
200
Torque (Nm)
150
100
50
0
0.005
0.01
0.015
Time (S)
0.02
0.025
Figure 5.18: Torque with dq with a rotor pole width of 19 degrees.
5.3.2. Flux weakening controller
The adative flux weakening controller was developed to advance the turn off angles
of the controller at higher speeds as it was not possible demagnetize the curves fast enough
due to a high back-emf voltage. This results in large negative torque and the average torque
85
being less and also results in a reduction of system efficiency. The torque is shown in Fig.
5.19 has an average of 38.96 Nm and we can see that there is a large amount of negative
torque being produced as demonstrated by Fig. 5.20. The currents responsible for large
negative torque is shown in 5.21.
100
Torque (Nm)
80
60
40
20
0
1
2
3
4
5
6
Time (mS)
7
8
9
10
9
10
Figure 5.19: Torque with no phase advancing.
100
Torque (Nm)
80
60
40
20
0
-20
1
2
3
4
5
6
Time (mS)
7
8
Figure 5.20: Phase torque with no phase advancing.
86
Current (A)
150
100
50
0
1
2
3
4
5
6
Time (mS)
7
8
9
10
Figure 5.21: Phase current with no phase advancing.
The optimum phase advancing for this speed was found as by a parametric sweep
as described in chapter 4 and then used program the adaptive controler. We can observe
the Torque shown in Fig. 5.22 has a higher average value of 46.59 Nm. This is primarily
because the negative torque production has beeen reduced as demonstrated by Fig. 5.23 in
comparision to Fig 5.20. The negative peak also occurs at θth as shown in Fig 5.23 in
accordance with controllers base as described in chapter IV. This occurred as the turn-off
angle was advanced whose effects we can observe by look at the currents in Fig 5.21 and
Fig. 5.24. The currents in Fig. 5.24 reaches zero faster due to the advanced turn-off.
87
100
Torque (Nm)
80
60
40
20
0
1
2
3
4
5
6
Time (mS)
7
8
9
10
9
10
Figure 5.22: Torque with phase advancing.
100
80
Torque (Nm)
60
40
20
0
1
2
3
4
5
6
Time (mS)
7
8
Figure 5.23: Phase torque with phase advancing.
88
150
Current (A)
100
50
0
1
2
3
4
5
6
Time (mS)
7
8
9
10
Figure 5.24: Phase current with phase advancing.
With the concept of phase advancing and the capability of the controller to reach a a desired
f d at a desired load and speed established the algorithims robustness and overall
effectivenes was tested by dynamically varying the load torque and speed. The result of
step changes in load torque is shown in Fig 5.25. The results of changes in reference speed
is in Fig 5.26. From the figures it can be observed that f d adapted well with respect to the
transient conditions. With the f d reaching optium efficiency values to match the Ith with the
Iactual at θth . When this was not possible at very high load torque demands at a certain speed
it reached its limit for phase advancing.
89
Load Torque (Nm)
fq (A)
125
100
75
50
25
0
0
-25
-50
-75
-100
-125
PA (Degrees)
fd (A)
125
100
75
50
25
0
75
60
45
30
15
0
0.5
1
1.5
Time (S)
2
2.5
3
0.5
1
1.5
Time (S)
2
2.5
3
0.5
1
1.5
Time (S)
2
2.5
3
0.5
1
1.5
Time (S)
2
2.5
3
Figure 5.25: Variation of f q and f d with step response in torque at a speed of 5000 rpm.
Nm.
This control thus can also adapt it self to single pulse mode operation. In Fig 5.27
the total torque of single pulse mode operation is region. Fig. 5.28 and 5.29 demonstrates
the currents and phase torques. It can be observed that some negative phase torque being
produced at the start of the conduction region but this compensates for current to rise up to
its peak before the back emf increases too much. Thus producing an overall increase in the
average torque.
90
Reference
Speed (rpm)
8000
6000
4000
2000
0
0.5
1
1.5
Time (S)
2
2.5
3
0.5
1
1.5
Time (S)
2
2.5
3
0.5
1
1.5
Time (S)
2
2.5
3
0.5
1
1.5
Time (S)
2
2.5
3
125
fq (A)
100
75
50
25
0
0
fd (A)
-25
-50
-75
-100
-125
PA (degrees)
75
60
45
30
15
0
Figure 5.26: Variation of f q and f d with step response in reference speed at a
torque of 65
91
100
Torque (Nm)
75
50
25
0
1
1.5
2
2.5
Time (mS)
3
3.5
4
Fig 5.27 Total torque during single pulse mode operation.
150
Current (A)
100
50
0
1
1.5
2
2.5
Time (mS)
3
3.5
Fig 5.28 Phase current during single pulse mode operation.
92
4
100
Torque (Nm)
75
50
25
0
-25
1
1.5
2
2.5
Time (mS)
3
3.5
4
Fig 5.29 Phase torque during single pulse mode operation.
The inclusion of f d command thus resulted in an increase in the average torque production
over the entire speed range. This is illustrated in Fig 5.30.
Torque (N-m)
150
dq with Phase Advancing
Traditional SRM Control
100
50
0
0
2000
4000
6000
Speed (rpm)
8000
Fig 5.30: Torque-speed envelope with and without phase advancing.
5.3.3. Efficiency of dq controller
In the scope of this thesis a new controller which serves as an analogues control
method to AC machines for SRMs have been proposed. This method is simple enough to
serve as an alternative to the conventional excitation angle control. Thus it is important to
compare the performance of both the control methods on the same platform. The two
93
methods were tasted in three speed ranges. The low speed range where it is paramount to
have low torque ripple. The medium speed range where it is still possible to reach the
command torque but some advancing of the conduction regions are required due to back
emf. The third being the high speed region where it is necessary to operate in single pulse
mode operation as there is a very high back-emf voltage. The command current of 140 A
for a commanded torque of 100 Nm was used in all three cases and then the turn-on and
turn-off angles adjusted.
The first case is at a speed of 1000 rpm. The resulting output parameters from the
two control methods which are of importance to us are tabulated in Table 5.2 and 5.3.
Table 5.2: Data from conventional current control at 2000 rpm
Iref
(A)
Turn-on
(electrical
degrees)
Turn- off
(electrical
degrees)
Average
Torque
(Nm)
140
18
145
96.92
Torque
Ripple
from FFT
(%)
6.56
140
20
156
102.15
11.06
Phase
current
RMS
(A)
83.4
Torque per phase
RMS current
(Nm/A)
1.16
86.28
1.18
Table 5.3: Data from dq control at 2000 rpm
100
0
0
99.47
Torque
Ripple
from FFT
(%)
4.79
100
0
37.5
100.67
6.74
fq
fd
f0
Average
Torque
(Nm)
94
Phase
current
RMS
(A)
92.47
Torque per phase
RMS current
(Nm/A)
1.08
85.76
1.17
From the two tables the first case is with both the controllers having inputs for
having low torque ripple and then for maximum torque per ampere. In terms of torque
ripple we can observe that the dq control method has a lower torque ripple with a 26.9%
improvement on the conventional case. It does this by achieving a higher average torque
as well. Here however the traditional case as a better torque per ampere when look at the
phase currents and gives 6.8% better torque per ampere. In the second case the controller
is optimized for achieving maximum torque per ampere. Here in the dq we can increase
the efficiency by sacrificing some torque ripple. The same average torque is maintained
but at higher efficiency and higher torque ripple. In the conventional method it produces a
maximum torque per ampere in terms of phase currents of 1.18 Nm/A which is .8% higher
than the dq but at a higher torque ripple of 11% compared to 6.74% by the dq controller.
Thus it can be shown that depending on the control designer’s requirements the control
parameters of the dq controller can also be adapted to achieve the objectives but at a higher
efficiency or lower torque ripple.
The second case is at a speed of 5000 rpm which is a medium speed range with a
high back-emf voltage. The resulting output parameters from the two control methods
which are of importance are tabulated in table 5.4 and 5.5.
Table 5.4: Data from conventional current control at 5000 rpm
Iref
(A)
140
Turn-on
(electrical
degrees)
Turn- off
(electrical
degrees)
-12
128
Average
Torque
(Nm)
93.57
Torque
Ripple
from FFT
(%)
10.20
95
Phase current
RMS
(A)
86.21
Torque per
phase RMS
current
(Nm/A)
1.08
Table 5.5: Data from dq control at 5000 rpm
100
0
0
92
Torque
Ripple
from FFT
(%)
6.89
100
40
30
93.92
8.70
fq
fd
f0
Average
Torque
(Nm)
Phase current
RMS
(A)
91.41
Torque per
phase RMS
current
(Nm/A)
1.01
86.26
1.09
Here it can be observed from the conventional control that some advancing is
required as there is back emf built up. Thus it was optimized for giving maximum torque
per ampere. We can observe that it had a torque ripple of 10% and torque per ampere of
1.08 Nm/A. In the dq control without any f d or f 0 command we can see that the controller
produces a lower torque ripple of 6.89% but has lower torque per ampere in terms of phase
RMS currents. The dq controller then had a f d and f 0 commanded so that it is possible to
increase the efficiency to levels comparable with the conventional control in terms of
torque per ampere. Here we observed that similar efficiencies with a difference of 0.5%
with the dq controller being higher whilst having a lower torque ripple. Hence it can
concluded that at medium speeds also the dq controller is more efficient than conventional
controller while maintaining the same simplicity.
The third case is at a speed of 7000 rpm which is a high speed range with a very
high back-emf voltage. The resulting output parameters from the two control methods
which are of importance are tabulated in Table 5.6 and 5.7. Here the objectives of the
control system is to achieve as high a torque as possible due to single pulse mode operation.
96
Table 5.6: Data from conventional current control at 7000 rpm
Iref
(A)
140
Turn-on
(electrical
degrees)
Turn- off
(electrical
degrees)
-22
128
Average
Torque
(Nm)
62.39
Torque
Ripple
from FFT
(%)
19.51
Phase
current
RMS
(A)
74.44
Torque per
phase RMS
current
(Nm/A)
0.84
Table 5.7: Data from dq control at 7000 rpm
fq
100
fd
80
f0
25
Average
Torque
(Nm)
63.33
Torque
Phase
Ripple
current
from FFT
RMS
(%)
(A)
20.35
74.5
Torque per
phase RMS
current
(Nm/A)
0.85
At this high speed it is no longer possible to reduce ripple due to single pulse mode
operation it is important to produce as high a torque as possible with higher efficiencies.
Here also we can see that by changing the control parameters associated with the dq control
we can achieve a higher torque per ampere.
Thus validating the purpose of the dq control as an alternative simple control
strategy which is analogues to AC machines and is capable of operation over a wide speed
range. The dq control does this be offering the flexibility of chasing either low torque ripple
operation or high torque per ampere operation without extensive calculations of offline
angles in terms of torque per ampere and highly complex control strategies in terms of
lower torque ripple.
97
5.4.
Conclusions
In this chapter the modeling of the SR machine for simulation has been discussed.
The chapter then presents simulation studies which validates the proposed control
strategies and their operation. This chapter then provides an efficiency comparison between
the proposed method and the conventional current control method with optimized
excitation parameters.
The next chapter of this thesis is about the experimental setup and results validating
these control methods.
98
CHAPTER VI
EXPERIMENTAL SETUP AND RESULTS
6.1.
Introduction
In chapter V the proposed control strategies were verified through computer
simulations. In this chapter the methods are verified experimentally. For experimental
verification a low power setup using a small 300 W machine and a commercially available
converter with some modifications has been used. An interface circuit has also been built
to communicate between the controller and the converter. The experimental results
presented here have been carried out on a dSPACE controller. The following sections in
this chapter is about the experimental setup and experimental results.
6.2.
Experimental Hardware
The development of the hardware and test setup for experimental validation
presented the main challenge in this thesis. The motor and converter for a low power
experimental validation of the system was procured. The machine and converter was used
in an old commercial washing machine from Maytag. Hence the machines characteristics
could not be obtained from the manufacturers and had to be obtained through
unconventional methods. The following subsections describes how the experimental
machine has been modeled for torque estimators and controller design, structure of the
99
inverter that has been used and the circuits that has been implemented for interfacing with
the controller and the dSPACE controller and the structure of the program.
6.2.1. Experimental SRM modeling
The machine used for the verification in simulation was not ready in time for
experimental verification by the time this thesis had to be submitted so an alternative 300
W
SRM which was used as the front load motor of Maytag Neptune washing machines
was used for experimental verification. This provided a low power solution for testing out
the algorithm initially but presented its own set of problems in terms of modeling the
machine as no concrete machine parameters were available for the motor. Hence, after
calculating the initial parameters the motor was opened and its internal structure analyzed
and measured to determine the parameters for modeling. The disassembled motor is shown
in Fig. 6.1. The next problem was that of determining the number of turns which is harder
to measure without unwinding the entire machine. For this we ran the motor at a fixed load
current of 3A and measured the average torque using a torque transducer. Then by using
those geometric parameters in the Arthur Raduns’ model a static T-i-θ characteristic curve
was generated. The number of turns was then manipulated using trial and error method
with the first guess being an educated one based on the diameter of one winding and the
width of a bunch of windings together. These geometric parameters shown in Table 6.1
were then used in the finite element software for a more accurate modeling of this machine.
The T-i-θ characteristics of the machine are shown in Fig. 6.2 and 6.3.
100
Figure 6.1: a) Stator of the experimental
SRM
Figure 6.1: b) Rotor of the experimental
SRM
Figure 6.1: c) Stator and rotor separately
of the experimental SRM
Figure 6.1: d) Experimental SRM
assembled
Table 6.1 SRM parameters of 300 W 12/8 experimental SR machine
No. of Phases
3
Stack Length
1.889 inch
No. of Stator Poles
12
Air Gap
0.015 inch
No. of Rotor Poles
8
Radius to air gap at rotor
3.2724 inch
Power
300 W
Radius to outside rotor yoke
2.4 inch
Phase Resistance
2.2 Ohms
Radius to inside stator yoke
3.8665 inch
Minimum Inductance
6.553 mH
No. of Turns
150
Maximum Inductance
28.25 mH
Rotor Pole width
15 degrees
101
10
8
6
4
Torque Nm
2
0
-2
-4
-6
-8
-10
0
20
40
60
80
100
Rotor Position 
120
140
160
180
160
180
Figure 6.2: T-i-θ characteristics from Arthur Raduns’ model.
10
8
6
4
Torque Nm
2
0
-2
-4
-6
-8
-10
0
20
40
60
80
100
Rotor Position 
120
140
Figure 6.3: T-i-θ characteristics from finite element analysis.
102
6.2.2. Inverter used for experimental validation
The inverter that was procured shown in Fig. 6.4 is a classic bridge inverter as
described in Fig. 2.8 with gate drivers incorporated into them. The inverter is rated at 300
W and had a rectifier to convert 120V-AC to 170 V DC. This section of the circuit was
however bypassed so that we could apply our on DC bus voltage giving a better control of
the entire system. The converter also came equipped with its own controller for running
the SRM and this also had to be removed so that we could give our control command
signals separately. The MOSFETs used here is IRF 644 which could handle a peak Vds of
250 V and a continuous current of 14 A at room temperature. The gate driver circuitry in
this section used an IR2101 high-low gate driver IC with its standard circuit configuration.
Figure 6.4: Inverter used for experimental implementation.
103
6.2.3. Interfacing circuitry
For proper communication between the inverter and the controller an interface was
built. The interface circuits mainly consisted of three parts. They are

Gate Drive interface circuitry

Encoder interface circuitry

Current conditioning circuits
The gate driver interface circuit takes the gate inputs from the dSPACE controller
and then passes them through a buffer and then opto-coupler to inverters gate driver inputs.
The block diagram of this circuit is given in Fig 6.5. The opto-couplers are used to isolate
the ground of the power supply from the controller.
Gate Signals
From dspace
Buffer
SN74LVC541A
Buffered
gate signals
OptoCoupler
ACPL4800
Isolated
gate signals
Figure 6.5: Gate driver circuit block diagram.
The second interface circuit that had to be built is an encoder interface circuit. The diagram
for this circuits is shown in Fig. 6.6. The motor was fitted with an HEDS5505A06 which
was an optical encoder with two channels and index. This encoder needed 5 V supply and
had to have a pull up resistor connect to the output channels to drive the one TTL load.
104
Hence a buffer was used here as well to increase the loading capacities per output. The IC
used as a buffer here is actually a level shifting buffer so as to make this system capable of
being run from a DSP as well as dSPACE but here the same supply voltage was applied on
both the ends essentially making it a buffer. The circuit diagram for this is shown in Fig
6.6.
+5V Supply
R
R
R
CH. A
CH. A
CH. B
CH. I
Encoder
HEDS5505 A06
MC14504BCP
CH. B
CH. I
Figure 6.6: Encoder interface circuit.
The third part was that of a current conditioning circuit. The currents sensor used is an
LEM 55. The current sensor has a much higher rating than the current that are passing
through hence three turns were given to the turn on sensors. The value outputs from the
current sensor had a high noise and also needed to have their levels shifted. Hence a
conditioning circuit was used. The conditioning circuit shown in Fig. 6.7 consists of a
voltage follower, a passive low pass filter and a level shifter. The low pass filter has the
following cut off frequency
𝑓𝑐 =
1
2𝜋𝑅𝐶
105
(6.1)
with 𝑅 = 33 Ω 𝑎𝑛𝑑 𝐶 = 0.1 µ𝐹 𝑓𝑐 = 48.23 𝑘𝐻𝑧
10KΩ
10KΩ
Vref
Current Sensor
Ouput
220Ω
+
10KΩ
+
33Ω
0.1µF
Figure 6.7: Current conditioning circuit.
Figure 6.8: Hardware circuits.
106
-
Input to
Controller
The interface circuitry of the gate driver and encoder discussed here has been
implemented on a proto board and the current conditioning circuit on a PCB. Fig 6.8 is a
picture showing the completed interface circuit coupled with the inverter.
6.2.4. dSPACE controller
dSPACE controller was used for all the control algorithm implementations as it
allows for algorithms to be implemented on Simulink making it easier for hardware
implementation and algorithm development from the initial stages. The system used here
was a dSPACE micro Autobox II system which communicated with the host computer
through the Ethernet port. This system has an IBM PPC running at 900 MHz and is
equipped with ADC with 16 bit resolution and PWM generation with a range of 0.0003 Hz
to 150 KHz. The system is also equipped with digital capture units for use with an encoder.
This made the entire dSPACE system ideal for initial algorithm development.
6.3.
Dynamometer and System Setup
After successful testing of the hardware, controller and motor at no load conditions
it was connected to a Magtrol Eddy current hysteresis dynamometer. The dynamometer
was controlled using the Magtrol DSP6000 controller unit for the application of load
torque. Several power supplies have been used to power the different components of the
setup. Data acquisition and real time control of the system was performed using the
DSPACE Control Desk 3.7.4. Oscilloscopes and current probes where also used for
observing the current wave shapes. The experiments were carried out at DC bus voltage of
120 V and the switching was limited to 30 Khz. The entire control system had a control
107
loop of 30 Khz speed and the Euler method was used as the primary integration method for
the solver. A PI based speed controller was implemented for speed control with the gains
being selected through a trial and error method until a satisfactory performance was
achieved. The focus of this thesis is not speed control hence this portion didn’t get much
attention. The overall system performance would also improve if a better PI controller or
any other speed controller is implemented. Fig. 6.9 shows a picture of the entire
experimental platform used for testing.
Figure 6.9: Complete experimental setup.
6.4.
Experimental Results
The first step for experimental validation was to implement the traditional control system
based on the control strategy shown in Fig 6.10.
108
Vdc
Gate Signals
ω*
Iref
PI Speed Controller
-
Current
Regulator
Vph
Converter
SRM
ω
Iph
Speed
Calculator
Control
Angle Inputs
θon
θoff
Electronic
Commutator
θ
Figure 6.10: Conventional control method.
The speed calculator here was based on tracking the position difference every
300µ seconds and then dividing it by time and adjustment factors as shown by Eqn. 6.2.
With the factor 500 being used as the encoder has 500 counts per complete mechanical
revolution.
𝑆𝑝𝑒𝑒𝑑 = 60 ∗ (
𝜃𝑑𝑖𝑓𝑓 )
300µ𝑆∗500
)
(6.2)
A PI speed controller was implemented to attain the commands required for running
the machine at the reference speed. The gains of the controller was designed by trial and
error method with a proportional gain of 0.0005 and an integral gain of 0.015. These gains
values provided a satisfactory speed response with minimal oscillations suitable for testing
the algorithms. However a better speed control method with more accurate gains would
greatly improve system performance but that remains a part of future plans.
The system was run at speed of 500 rpm and the corresponding current shapes are
shown in Fig 6.11 and Fig 6.12. With Fig 6.11 being generated from the scopes and Fig
6.12 from the data acquired with control desk acquisition software. Some spikes from the
109
data acquired from the scope can be observed but this problem is negated when as the
current shapes with the data from the acquisition software as there is a low pass filter in
the current conditioning circuit.
Figure 6.11: Currents from control method using oscilloscope.
10
9
8
Current (A)
7
6
5
4
3
2
1
0
0
0.005
0.01
0.015
0.02
Time(S)
0.025
0.03
Figure 6.12: Currents from Control Method using control desk.
110
0.035
6.4.1. dq control of SRM
The dq control method is basically a torque controlled method whose inputs are in
terms of torque which it later transforms to reference currents. Furthermore one of the
benefits of dq control methods is its inherent ability to reduce the amount of torque ripple
and serve as an alternative analogous control method to synchronous machines with the
help of dq to synchronous conversion block. Hence it was important to have a high
bandwidth torque estimator. The torque estimator used here is a LUT based method where
the static T-i-θ characteristics of the experimental SRM is used in 2D LUT with the actual
currents and rotor position as the input. The Fig 6.13 and 6.14 shows phase torques and
total torque from the traditional control system at speed of 500 rpm and load of 1 Nm
applied by DSP6000 controller. The turn-on and turn-off angles were optimized for
providing a low torque ripple but we can still observe a torque dip during commutation as
there is no sharing method being implemented here. The time scale is the same in both of
these two figures.
111
5
4.5
Torque (Nm)
4
3.5
3
2.5
2
1.5
1
0.03
0.035
0.04
0.045
Time (S)
Figure 6.13: Total estimated torque from conventional control.
5
4.5
4
Torque (Nm)
3.5
3
2.5
2
1.5
1
0.5
0
0.03
0.035
0.04
Time (S)
Figure 6.14: Estimated phase torques from conventional control.
112
0.045
Then the dq control method was implemented at the same speed of 500 rpm and a certain
load torque from the DSP6000 Controller. The currents from scope again has some noise
as the current probes where to close to each other and this could not be avoided to the close
proximity of the phase conductors in the setup.
Figure 6.15: Currents from dq using oscilloscope.
10
9
8
Current (A)
7
6
5
4
3
2
1
0
0
0.005
0.01
0.015
0.02
0.025
0.03
Time (S)
0.035
0.04
Figure 6.16: Currents from dq using control desk.
113
0.045
0.05
We can observe that the current is higher in the initial conduction regions for each
of the phase to ensure proper torque sharing with during commutation. From Fig 6.17 and
6.18 it can be observed that there is no longer any torque dips presented here. Furthermore
with the torque dips eliminated we can see that a lower torque is demanded from the
machine at the same load torque. Another reason for this that the output of the PI controller
now is the reference torque term f q and no longer the Iref it was in the previous case. The
torque sharing has been incorporated as a different reference current is now being generated
with some shaping involved. The reference current with shaping is shown in Fig. 6.19 and
we can see that the actual currents can follow the reference as demonstrated in Fig 6.20.
The time scales in Fig 6.17 to Fig 6.20 is the same and in the same range.
5
4
Torque (Nm)
3
2
1
0
-1
0.4
0.405
0.41
0.415
Time (S)
0.42
0.425
Figure 6.17: Total estimated torque from dq controller.
114
0.43
5
4
Torque (Nm)
3
2
1
0
-1
0.4
0.405
0.41
0.415
Time (S)
0.42
0.425
0.43
Figure 6.18: Estimated phase torque from dq controller.
10
9
Reference Current (A)
8
7
6
5
4
3
2
1
0
0.4
0.405
0.41
0.415
Time (S)
0.42
0.425
0.43
Figure 6.19: Reference current commands from dq controller.
115
10
9
8
Current (A)
7
6
5
4
3
2
1
0
0.4
0.405
0.41
0.415
Time (S)
0.42
0.425
0.43
Figure 6.20: Actual currents with dq controller.
After the successful implementation of the basic dq control strategy an experiment
was run at the same speed of 500 rpm and load with a zero component being commanded.
This according to simulation studies and theory is supposed to decrease the conduction
region there by increasing the system torque per ampere but at the cost of increasing the
torque ripple. By comparing the Fig. 6.19 and Fig. 6.21 it can be observed that the
commutation time has decreased when a zero component is injected. As a result the actual
current shapes are also different. With a decrease in the overlap region between the two
commutating phases we now have less freedom in terms of our controller for sharing the
torque to reduce the torque ripple. As a result from Fig. 6.22 and Fig. 6.23 it can be see that
there is an increase in the torque ripple. This also results in requirement of a higher torque
for operating at that same speed and load conditions. This thereby results in a higher torque
116
per ampere according to the simulation and theoretical studies presented in the previous
chapters.
10
9
Reference Current (A)
8
7
6
5
4
3
2
1
0
0.03
0.032
0.034
0.036
0.038
0.04 0.042
Time (S)
0.044
0.046
0.048
0.05
Figure 6.21: Reference command currents with zero component commanded.
10
9
8
Current (A)
7
6
5
4
3
2
1
0
0.03
0.032
0.034
0.036
0.038
0.04 0.042
Time (S)
0.044
0.046
0.048
0.05
Figure 6.22: Actual currents with zero component commanded.
117
5
4.5
4
Torque (Nm)
3.5
3
2.5
2
1.5
1
0.5
0
0.03
0.032
0.034
0.036
0.038
0.04 0.042
Time (S)
0.044
0.046
0.048
0.05
Figure 6.23: Phase torques with zero component commanded.
5
4
Torque (Nm)
3
2
1
0
-1
0.03
0.032
0.034
0.036
0.038
0.04 0.042
Time (S)
0.044
0.046
0.048
Figure 6.24: Total torque with zero component commanded.
118
0.05
The next validation for the dq controller was done at high speed and how it performs
there. The tests were performed at a speed of 1700 rpm and high load. First the experiment
was run without any fd commanded i.e. no phase advancing. In this case it can be seen that
the actual currents shown in Fig 6.26 can no longer support the reference current commands
shown in Fig 6.25 due to a high back-emf voltage. As a result there is some negative torque
production as shown in Fig 6.27. This results in the total torque having more ripple and a
decrease in the overall system efficiency.
10
9
Reference Current (A)
8
7
6
5
4
3
2
1
0
0.03
0.035
0.04
0.045
Time (S)
Figure 6.25: Phase reference current commands with no advancing at 1700 rpm.
119
10
9
8
Current (A)
7
6
5
4
3
2
1
0
0.03
0.035
0.04
0.045
Time (S)
Figure 6.26: Phase currents with no advancing at 1700 rpm.
5
4
Torque (Nm)
3
2
1
0
-1
0.03
0.035
0.04
Time (S)
Figure 6.27: Phase torque with no advancing at 1700 rpm.
120
0.045
5
4
Torque (Nm)
3
2
1
0
-1
0.03
0.035
0.04
0.045
Time (S)
Figure 6.28: Total torque with no advancing at 1700 rpm.
The experiment was then run with some f d i.e. phase advancing being applied. The actual
currents shown if Fig. 6.30 now can follow the reference current commands shown in Fig.
6.29 more closely. This results in a higher torque build up in the initial region and a removal
in the production of negative torque as shown in Fig. 6.31. As a result the total torque
shown in Fig. 6.32 has lower ripple and can achieve the load torque with a lower average
torque and current requirements thereby increasing the overall system efficiency.
Thus
proving that the dq control can work over a wide speed range with the f d being commanded
similar to flux weakening in synchronous machines. The control designer also has the
option of designing is system for either low torque ripple or with increased efficiency by
commanding the zeroth component.
121
10
9
Reference Current (A)
8
7
6
5
4
3
2
1
0
0.03
0.032
0.034
0.036
0.038
Time (S)
0.04
0.042
0.044
Figure 6.29: Reference current commands with phase advancing at 1700 rpm.
10
9
8
Current (A)
7
6
5
4
3
2
1
0
0.03
0.035
0.04
Time (S)
Figure 6.30: Actual currents with phase advancing at 1700 rpm.
122
0.045
5
4
Torque (Nm)
3
2
1
0
-1
0.03
0.032
0.034
0.036
0.038
Time (S)
0.04
0.042
0.044
Figure 6.31: Phase torque with phase advancing at 1700 rpm.
5
4
Torque (Nm)
3
2
1
0
-1
0.03
0.032
0.034
0.036
0.038
Time (S)
0.04
0.042
0.044
Figure 6.32: Total estimated torque with phase advancing at 1700 rpm.
123
6.4.2. Adaptive flux weakening using dq control
The second proposed technique in this thesis is that of an adaptive flux weakening
where the controller adaptively varies the f d component in an adaptive manner for achieving
maximum torque per ampere at medium and high speeds by changing the level of
advancing required. This adaptive f d component generation also allows the machine to run
Load Torque (Nm)
4
3
2
1
0
PA degrees
fd (A)
1.2
1
0.8
0.6
0.4
0.5
0
fq (A)
over a wide speed range.
0
-2
-4
-6
-8
100
75
50
25
0
0
10
20
30
Time (S)
40
50
60
0
10
20
30
Time(S)
40
50
60
0
10
20
30
Time (S)
40
50
60
0
10
20
30
Time (S)
40
50
60
Figure 6.33: Response of f q , f d and resulting θ Pa with load step responses at 5 s., 17 s., 22
s., 31 s., 38 s., 42 s. and 54 s.
124
Speed (rpm)
fq (A)
6
4
2
0
PA degrees
fd (A)
2500
2000
1500
1000
500
0
-2
-4
-6
-8
-10
100
75
50
25
0
0
20
0
20
0
20
0
20
40
Time (S)
60
80
60
80
40
Time (S)
60
80
40
Time (S)
60
80
40
Time (S)
Figure 6.34: Response of f q , f d and resulting θPa with speed step responses at 8 s., 17 s.,
38 s. and 60s.
The adaptation of f d with step changes in load at a fixed speed of 1700 rpm is shown in
Fig. 6.33. Figure 6.34 shows the response of f d with step changes in speed with the load
torque being constant at 0.5 Nm. The test results suggest that the proposed controller works
well during transient and steady state conditions.
Two experiments at a high speed was also conducted to demonstrate that at a high
speed of 2500 rpm with the adaptive phase advancing. The experiments were first
conducted at a load torque of 0.3 Nm where the current can regulated. The second
125
experiment was conducted at the same speed of 2500 rpm but with a higher load torque of
0.5 Nm making the system go into single pulse mode operation. The currents from the first
experiment is shown in Fig. 6.35. Here we can observe that the adaptive phase advancing
method allows for current regulation by advancing the turn on and turn off angles. With
the torque values increased as shown in the second experiment the load currents can no
longer match the reference hence the phase is advanced to enable single pulse mode
operation as shown by the phase currents in Fig. 6.36. Another experiment was run at a
speed of 2000 rpm at a load of 0.26 Nm. Figure 6.37 shows the current shapes and the
torque output without f d component. It can be seen that the commanded speed is maintained
at the commanded load torque for an RMS current of 1.85 A and the
adaptive controller demonstrated a higher
𝑑𝑖𝑥
𝑑𝑡
𝑑𝑖𝑥
𝑑𝑡
is also low. The
as shown in Fig. 6.38. The commanded load
and speed is maintained with an RMS phase current of 1.68 A with adaptive flux weakening
control. This demonstrates that flux weakening with the f d command increases the torque
per ampere and
𝑑𝑖𝑥
𝑑𝑡
which would enable a wide speed range of operation.
Figure 6.35: Phase currents at a load of 0.3Nm at 2500 rpm with adaptive control.
126
Figure 6.36: Phase currents at a load of 0.5Nm at 2500 rpm with adaptive control.
Figure 6.37: Phase currents (ch1-3) and total torque (ch4) while running the machine at
2000 rpm without flux weakening
Figure 6.38: Phase currents (ch1-3) and total torque (ch4) while running the machine at
2000 rpm with flux weakening
127
6.5.
Conclusions
In this chapter a low power experimental setup has been described. The hardware
circuitry has been built and tested step by step with the conventional control method. Then
the proposed dq controller was implemented carefully. The experimental results then
verified the effectiveness of the proposed control strategy. Experimental work was also
carried out on the adaptive controller for wide speed range operation with the dq controller
and it was also working as predicted. Thus this thesis presented a novel control strategy
and its operation in simulation and hardware demonstrated. The method was successful in
ripple minimization at low speed and capable of wide speed range operation. The method
could also increase efficiency with the zero component commands.
128
CHAPTER VII
CONCLUSION AND FUTURE WORK
7.1. Conclusions
This thesis presented a novel control strategy based on the rotating reference for
switched reluctance machines. Previous work was done for rotating reference frame
control of SRM based on the inductance. This thesis presented a reference frame control
with respect to the phase torque. Control in the dq reference frame removes the need for a
computationally intensive electronic commutator. Through the proposed SM to SRM
converter block an analogous control strategy to SM for SRMs has been established. The
proposed controller’s dependency on the actual machine model has been investigated
briefly and future work would include a greater study in this regard.
Chapter II presented a brief discussion on the basics of the switched reluctance
machine. It also presented a review on the modeling techniques for SRMs. A detailed
literature review was conducted for high performance SR drives in terms of torque ripple
reduction and excitation parameter controls.
Chapter III presented the basic dq control strategy which incorporated some
transformation blocks based on finite element data from a machines design. These
transformation blocks presented the SRM as a synchronous machine from the control
engineers’ perspective. Thereby providing an analogous control for SRMs to SMs.
129
Chapter IV presented an adaptive flux weakening control method for advancing
the turn-off angles for SRMs. This method was independent on the machines parameters
and facilitated operation over a wide speed and load range dynamically. Though the
method is primarily designed for use with the dq controller it can also operate on
traditional control methods.
Chapter V and VI presented simulation and experimental validation of the
proposed control strategies. An experimental platform, based on a 300 W SRM using a
dSPACE controller, was setup for this purpose.
The main contributions of this thesis can be summarized as:

A literature review was conducted on existing SRM modeling and control
methods.

A novel control strategy for SRMs has been developed making the SR
control analogous to that of AC machines.

A rotating reference frame based control in the dq reference frame for
SRM has been presented. This made the SRM torque control similar to
DC machine control.

A negativity removal block has been developed for converting bi-polar
signals from the dq to abc block to unipolar commands.

A nonlinearity correction block has been developed to help minimize
torque ripple.

A brief sensitivity analysis has been carried out on the dq controller
130

The controller removed the need for an excitation angle sequencer and
additional ripple minimization algorithms.

An
adaptive
phase
advancing
algorithm for
flux
weakening was
for
rapid
established.

An
experimental
platform
was
established
controller
implementation and testing.
7.2.
Future Work
This research opened the doors for a lot of further research in SRMs. A list of
future research avenues is listed below.

Experimentally
verify
the
control
algorithms
on
the
high
power
experimental platform based on a 110 kW SRM.

Investigate the parameter sensitivity of the dq controller in greater detail.

Incorporating an adaptive or artificial intelligence based model in the nonlinearity block to remove the need for the machine model and lookup
tables for ripple minimization.

Develop a speed controller for optimal performance on the dq control
method.

Investigate the avenues for incorporating torque ripple minimization and
sensorless control algorithms used in synchronous machines as the
machine is now like synchronous machine from the dq0 axis command
perspective.
131

Develop adaptive turn on angle controller for the machine instead of the
hard limit that is currently being used.

Develop a controller for generating the optimal zero sequence component
for maximum torque per ampere.
132
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