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SPEED CONTROL OF
SWITCHED RELUCTANCE MOTORS
by
ARDESHIR MOTAMEDI-SEDEH
A thesis submitted to the Department of Electx-ical and Cornputer Engineering
in conformity with the requirements for
the degree of Master of Science ( Engineering )
Queen's University
Kingston, Ontario, Canada
March 1998
copyright O Ardeshir Mo tamedi-Sedeh, 1998
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ABSTRACT
A speed control systern for a switched reluctance motor ( SRM ) drive using
proportional-integral control strategy is designed and presenred in this t hesis. Performance
is based on the ability to provide each phase with pulses of current during the torque
productive pends ( the zone of increasuig inductance ) by selecting precietermined switching
angles. The control system maintains the desired speed in the face of variations of the load
or the rnotor parameters. Variable speed can be achieved using this control system.
After reviewing the operating principles of the SRM and its different control
strategics, modeling has k e n done under the assurnption of linear magnetic characteristics.
The torque characteristics and the dymmic differen tial equa tions. nevertheless. are nonhear.
The control structure consists of a fdforward controuer and a propoflional-integral speed
controller. A hysteresis controller is also employed t~ maintain the current within a pre-set
band based on a defmed algorithm.
Stability of the nonlinear control system has been investigated using the second
method of Lyapunov and the method of Krasovski
In this thesis, two control schemes; narnely current-source switched reluctance motor
( CSSRM ) and voltage-source switched reluctance motor ( VSSRM ) are designed, analyzed,
and simulated under different working conditions using the Matiab software package to
investigate the steady-state operation and dynamic response of the system. The rnotor
performance and simulation results are discussed.
. . Il
To my family
for their support and encouragement
Fim and foremost, 1 wish to achowiedge my sincere gratitude and deep indebtedness
to my research supervisor Dr. M. M. Bayoumi whose kindness, friendly guidance, and
financial support made it possible for me to continue with my research endeavor.
1 would also like to express my appreciation to others who helped:
To rny wife, Laleh, and my daughter, Sepideh, for their understanding, patience, and
moral support while 1 was struggling with th& thesis.
To my parents, and my brother. Javad, for their continual encouragement and support.
To my feiiow coileagues in the Control and Robotics Laboratory. especialiy Gino
Labinaz, for their advice, cooperation, and valuable discussions.
TABLE OF CONTENTS
. . ABSTRACT ......................................................... u
ACKNOWLEDGEMENTS ............................................. iv
TABLEOFCONTENTS ............................................... v
LISTOFRGURES ................................................... ix
... LISTOFSYMBOLS ................................................ xrti
C m R 1 ......................................................... 1
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .*.. . . . . . . . . . . . . . . . . . 1
............................................ 1 .1 INTRODUCTION 1
1.2 SRM CONSTRUCTION AND PRINCIPLE OF OPERATTON .......... 2
1.3 APPLICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1-4ENERGIZATION ............................................ 5
........................ 1.5 DESIGN AND MODELING OF THE SRM 7
...................................... 1.6 POUrER CONVERTERS 1 1
................................. 1.7 ROTOR POSITION SENSING 15
1.8 CONTROL OF THE SWKCHED RELUCïANCE MOTOR . . . . . . . . . . 16
. . . . . . . . . . . . . . . . . . . . . . . . 1 -9 OB JECTlVES OF THE PRESENT WORK SI
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10 THESIS OUTLINE 22
CHAITER2 ........................................................ 25
MODELINGOFTHESRM ............................................ 25
2-1INTRODUaON ........................................... 25
2.2 MODEL OF SWITCHED RELUCTANCE MOTORS . . . . . . . . . . . . . . . . 27
.................................... 2.2.1 Inductance Profile 28
2.2.2 Instantaneous Torque Expression . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.2.3 Mechanical Subsystern Dynamics ......................... 38
2.3 SRM DRIVE OPERATION AND CONTROL VARIABLES . . . . . . . . . . 39
CHAPTER3 ........................................................ 40
DESIGN OF A SPEED CONTROLLER FOR THE SRM . . . . . . . . . . . . . . . . . . . . . . 40
3.1INTRODUCTION ........................................... 40
3.2 PERFORMANCE AND SPECIFICATIONS ....................... 41
3.3 SPEED CONTROLLER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
CHAPTER4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
STABLITY ANALYSIS OF THE SPEED CONTROL SYSTEM . . . . . . . . . . . . . . . 50
........................................... 4.1 INTRODUCTION 50
4.2 THE SECOND METHOD OFLYAPUNOV . . . . . . . . . . . . . . . . . . . . . . . 51
4.3 STABILITY OF THE CLOSED-LOOP ERROR SY STEM . . . . . . . . . . . . 52
4.4 KRASOVSKI'S METHOD .................................... 69
CHAPTERS ........................................................ 75
SIMULATION AND PERFORMANCE OF THE SPEED CONTROL SYSTEM .... 75
5.1 INTRODUCTION ........................................... 75
5.2 MODE OFCONTROL ....................................... 75
.............................................. 5.3SRMSUPPLY 76
..................................... 5.4 SRM DRIVE ANALYSIS 77
. . . . . . . . . . . . . . . . . . . 5.5 SWITCHING S'i"RATEGY FOR SIMULATION 80
.......................... 5.6 EQUATIONS USED IN SIMULATION 81
................................. 5.7 HYSTERESIS CONTROLLER 83
................................ 5.8 SPEED CONTROL SCHEMES 84
............ 5.9 PERFORMANCE OF THE SPEED CONTROL SY STEM 90
.................................. 5.9.1 S tartïng Performance 91
................... 5.9.2 Effect of Step Change in Speed Demand 92
........................... 5.9.3 Effect of Parameter Variations 92
. . . . . . . . . . . . . . . . . . . . . . 5.9.4 Effect of Luad Torque Disturbances 93
...................... 5.9.5 Effsct of Switching Angle Variations 93
................................... 5.9.6 VSSRM Simulation 95
CHAPTER6 ....................................................... 120
4 ..................................... SUMMARY AND CONCLUSIONS 120
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 INTRODUCTION 120
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 SUMMARY OFTHE THESIS 121
vii
6.3 ADVANTAGES AND DISADVANTAGES OF P-1 CONTROLLER ... 122
. . . . . . . . . . . . . . . . . . . . . . . 6.4 SUGGESTIONS FOR FURTHER WORK 124
REFERENCES ..................................................... 125
.................................. MOTOR DATA AND PARAMETERS 132
DEEUVATION OF THE CONDITIONS OBTAINED ........................ FROM KRASOVSKI'S METHOD IN CHAPTER 4 134
VITA ........................................................... 140
LIST OF FIGURES
Figure 1.1
Figure 1.2
Figure 2.3
Figure 1.4
Figure 1.5
Figure 1.6
Figure 1.7
Figure 1.8
Figure 2.1
Figure 2.2
Figure 2.3
Figure 2.4
Figure 3.1
Figure 3.2
Figure 4.1
Figure 4.2
Cross sectional view of a four-phase SRM showing switching circuit for only one phase.
Variation of inductance for one phase.
Typical diagram of the phase current for a CSSRM drive.
Typical diagram of the phase current for a VSSRM drive.
SRM fesding schemes. (a) Voltage-source feeding. (b) Current-source feeding.
Basic power converter circuit configurations.
Basic functional block diagram of an SRM drive.
Block diagram of the proposed SRM speed control system.
Schematic diagram of a four-phase SRM.
Inductance profde for one phase.
Stmcture of a four-phase SRM.
Inductance profiles of the SRM phases.
Block diagram of the speed control system.
Feedforward controller.
System configuration for stability study ( linear analysis ).
Block diagram of the closed-loop system with the nonlinear ekment.
Figure 4.3 System configuration ( simplifed block diagram ) for stability study ( nonlinear anaiysis ).
Figure 4.5 The region satisfying the sufficient conditions.
Figure 5.1 Control scheme of the CSSRM drive.
Figure 5.2 SRM electrical subs ystem for the control scheme of the CSSRM drive.
Figure 5.3 Control scheme of the VSSRM drive.
Figure 5.4 SRM electrical subsystem for the control scheme of the VSSRM drive.
Figure 5.5 Speed response of the CSSRM dnve for a step change in the reference speed.
Figure 5.6 CSSRM torque produced when the motor accelerates from rest to the reference speed.
Figure 5.7 Speed error of the CSSRM drive for a step change in the reference speed.
Figure 5.8 Phase current waveform of the CSSRM drive when the motor accelerates from rest to the reference speed.
Figure 5.9 Phase voltage waveform of the CSSRM dnve when the motor accelerates from rest to the reference speed.
Figure 5.10 Speed response of the CSSRM drive when the reference speed is doubled at t=0.1 sec.
Figure 5.1 1 Speed response of the CSSRM drive when the moment of inertia J is doubled.
Figure 5.12 Speed response of the CSSRM dnve when the phase resistance r is increased by 20%.
Figure 5.13 Speed response of the CSSRM drive under varying load torque disturbances.
Figure 5.14 Motor Torque response of the CSSRM drive under varying load torque disturbances.
Figure 5.15 Speeû response of the CSSRM drive when the conduction angle is decreased by 2 degrees.
Figure 5.16 Torque response of the CSSRM drive when the conduction angle is decreased by 2 degrees.
Figure 5.17 Speed response of the VSSRM drive for a step change in the reference speed.
Figure 5.18 VSSRM torque produced when the motor accelerates from rest to a reference speed.
Figure 5.19 Speed error of the VSSRM drive for a step change in the reference speed.
Figure 5.20 Phase current waveform of the VSSRM dnve when the motor accelerates from rest to a reference speed.
Figure 5.2 1 Phase voltage waveform of the VSSRM drive when the motor accelerates from rest to a reference speed.
Figure 5.22 Speed response of the VSSRM drive when the reference speed is doubled at t=0.1 sec.
Figure 5.23 Speed response of the VSSRM dnve when the moment of inertia J is doubled.
Figure5.24 SpeedresponseoftheVSSRMdrivewhen the phase resistance r is increased by 20%.
Figure 5.25 Speed response of the VSSRM drive under varying load torque disturbances.
Figure 5.26 Motor Torque response of the VSSRM drive under varying load torque disturbances.
Figure 5.27 Speed response of the VSSRM drive when the conduction angle is decreased by 2 degrees.
Figure 5.28 Torque response of the VSSRM drive when the conduction angle is decreased by 2 degrees.
LIST OF SYMBOLS
B coefficient of friction
e speed error
1 current
id, i' demand current
h hysteresis band
Idc dc iink current
Im saturation current
rz current amplitude
J moment of inertia
k suffix denoting phases
K dope of the inductance profüe
K~ integral gain of the speed controiler
K~ proportional gain of the speed controuer
L , L ( 0 ) phaseinductance
=a aligned phase inductance
Lu unaligned phase induc tance
number of rotor poles
number of stator poles
number of phases
phase ra i s t ance
complex variable of the Laplace transform
tirne
electromagnetic torque
demand torque
load torque
controller output
voltage
dc link voltage
Lyapunov function
mechanical power
rotor pole pitch
rotor pole arc
stator pole arc
darnping ratio
rotor position
CSSRM
P-1
SRM
VSSRM
extinction angle
""on", "off' angle
flux linkage
actual speed of the motor
demand speed of the motor
undamped natural frequency
current source switched reluctance rnotor
proportionai-integral
switched reluctance mo tor
voltage source switched reluctance motor
CHAPTER 1
INTRODUCTION
1.1 INTRODUCTION
Electricai drives play an important role in modern indusuies. In the last two decades
a new type of electrical dnve named the switched reluctance motor ( SRM ) drive has been
introduced and investigated. The switched reluctance rnotor drive has now reached a level of
matunty that allows it to be used in industry as an efficient brushless drive with cost
advantages, a wide speed range, and inherent simplicity and ruggedness. Actually. the SRM
is not a new concept. EarIy inventors of electromagnetic engines understood the switched
reluctance principle but were unsuccessful in their attempts to build a motor due to poor
electrornagneiic and mechanical designs and to the unavailability of suitable switching devices.
An interest in switched field machines was revived in the 1960's with the advent of the
thyristors. In the early 1980ts, the first SRM dnve system using the new technology became
commercially available as a result of work done by a group at Leeds and Nottingham
Universities in the U.K. [1,2].
In recent years, this machine has seen a revival of interest for applications in 10 w and
medium power drives. Such a machine is emerging as an attractive solution for variable-speed
applications due to several advantages. Most prominent among these are the simple structure
of the motor with coils on the stator and no windings or bnishes on the rotor. and the reduced
component count of the converter due to the unidirectional current requirernent. The machine
has saliencies on both the stator and rotor. A shaft position sensor is usually necessary to
control switching of the power semiconductor devices.
1.2 SRM CONSTRUCTION AND PRINCIPLE OF OPERATION
The SRM is characterized by its singly excited doubly salient geometry. The rotor is
made of a stack of steel laminations cut to yield the desired number of rotor poles. The rotor
carries no rotor windings, squirrel cage bars or permanent rnagnets. The stator is constructed
of steel laminations cut to give the required number of stator poles. The rotor and stator need
not have the sarne number of poles. Windings are phced over the stator teeth and coils are
connected to forrn phases. The structure of a four-phase switched reluctance rnotor is
depicted in Figure 1.1, where for simplicity, only the coi1 of one phase has b e n drawn.
The construction of the SRM is very simple, rugged, and reliable compared to that
of conventional machines such as dc or ac machines [3]. The advantageous features of an
S R M are as foilows:
i ) The SRM has simple laminations, wide stator slots for the phases and no rotor windings.
permitting inexpensive manufacturing .
ü ) With no rotor winding, the SRM exhibits only minoirotor heating. Most of the heating
occurs in the stator where the machine is most easily cooled, thus reducing its thermal
limitations,
-7 1'
l One Stator Phase
Figure 1.1 Cross sectional view of a four-phase SRM showing switching circuit for ody one phase.
iii ) The SRM requires only unipolar currents. permitting the design of an inexpensive
converter with one controllable power switch per phase.
iv ) Extreme high speeds are possible.
The SRM aiso has sorne c h disadvantages. The most important is the pulsed. or at
least non-uniform. nature of the torque produced which leads to torque ripples and may
contribute to acoustic noise.
The excitation is switched sequentiaily fiorn phase to phase as the rotor moves. When
a phase is energized, a reluctance torque, which tends to align the stator and rotor poles, is
produced. The torque is a nonlinear hinction of the phase current and the rotor position. In
fact. the torque is developed through the tendency of the magnetic circuit to adopt a
configuration of minimum reluctance, ie., for the rotor poles to move into line with the stator
poles and to maximize the inductance of the excited coils [4]. Figure 1.2 shows a cyclic
variation of inductance as rotation occm. A motoring torque is produced if a phase is excited
during the interval when the inductance is increasing.
For an operating speed and torque, the stator current must be initiated and
cornmutated at p r e s c n i rotor angek. Correct timing of the excitation of the phase winding
depends on the position of the rotor. Therefore, a direct or indirect rotor position sensor is
an essential elernent of the conuol system in order to position the current pulse appropriately.
1.3 APPLICATION
The switched reluctance motor is a simple and robust electrical machine which is
4
fuding applications over a wide speed and power range. The application potential Cor SRM
drives is considerable. S witched reluctance mo tor drives exhibit many attractive performance
features for positioning applications 15.61. They possess high reliability and efficiency ( in
excess of 85% for the motor and converter together ), which are desirable for automotive
applications. The high speeds achievable using SRMs are suitable for fan and pump drives.
The robust bmshiess construction and good thermal features ( heat loss is largely
confined to the stator ) make the drive attractive for mining and flameproof applications in
addition to traction and aûcraft engine applications 17). The simple motor construction and
the low cost, fault tolerant power ektronics have made the switched reluc tance mo tor drive
a strong contender for many applications in industrial, aerospace, automotive, robotics and
domestic applications.
1.4 ENERGIZATION
The excitation of a phase magnetizes both the stator and the rotor. This produces a
toque, causing the rotor to align its poles with those excited on the stator. Thus, sequential
phase excitation causes rotor motion, which synchronously aligns the rotor poles with those
excited on the stator.
The switched reluctance motor cm be fed either by a current source or by a voltage
source supply. With a current source supply, a current generator of variable amplitude forces
pulses of c m n t into the phases at certain rotor positions. Figure 1.3 shows a typical diagram
of the current for the motoring operation of a current source SRM drive. This figure shows
Lu C
Rotor Angle 0 1 2 4
Rotor and stator poIes aligneci
Figure 1.2 Variation of inductance for one phase.
Figure 1.3 Typical diagram of the phase axrrent for a CSSRM drive.
that the current is forced into the phase at 8, and removed at eOfl , with 0, and Bd lying
in the zone of increasing inductance.
With a voltage source supply. a voltage generator of constant amplitude is applied
across the pharas at certain rotor positions. The actual cumnt is forced to follo w the demand
c m n t between an upper and a lower limit in the hysteresis band by a hysteresis controller.
In a voltage source SRM drive. various waveforms c m be obtained for the current. A typical
supply stroke for one phase in the motoring operation is Uustrated in Figure 1.4. Pulsed
shapes of the remaining phases are shifted by ar/q ( in th& thesis 15') from each other.
where a, and q are rotor pole pitch and the number of phases. respectively.
The energizing interval goes 6rom 0, to ûO6. At 8,, the voltage generator is fùlly
applied across the phase to establish a current into it. From zero to e6, two operating
modes. A and B, can exist. Mode A occurs when the back emf is las than the source voltage.
In chis mode, the phase current is regulated at the desired amplitude by chopping the voltage
source. Mode B occurs when the back emf is greater than the generator voltage. The de-
energizing interval goes from e O , to 8,. At 8, the current extinguishes and no voltage is
applied across the phase.
1.5 DESIGN AND MODELING OF THE SRM
The nonlinear nature of the machine and the switching circuits, the wide range of
possible configurations of motor and power circuits, and the difficulties in modeling the
machine, make the SRM drive a particularly dficuIt one to design weli (81. Fundamental
7
(a) Chopping mode (Mode A)
@) Single puise mode (Mode B)
F i ~ e 1.4 Typical diagram of the phase m e n t for a VSSRM drive.
design considerations and design aspects, such as the number of phases to be emplo yed, the
ratio of the number of rotor poles to the number of stator poles, and the values of rotor pole
arc and stator pole arc are dixussed in [9].
A helpfd approach is to divide the work into design methods based on :
( a ) linear methods, ie., based on the analysis of circuits with current-independent
parameters.
( b ) nonlinear methods, Le., based on circuits with cument-dependent parameters.
( c ) hite-element solutions of the magnetic fields. leading either to a direct solution of the
performance or to the calculation of parameters for circuits [IO]. As computing power has
become greater and more accessible, attention has k e n tumed to the application of Mte-
elernent ( FE ) analysis to switched reluctance geometries. One of the principal contributors
to this area is G. E. Dawson [Il].
In contrast to the SRM's simple and straightforward construction, mathematical
models for this machine tend to be complex and difficult to develop. The phases of the
switched reluctance motor are usuaily modeled by a resistance in series with an inductance
which depends on the rotor angle. The behavior of the machine may be determinecl by solving
the dinerential equations associated widi this quivalent circuit. Models generally fa11 into two
classes: those which neglect the magnetic saturation and those which take this into
consideration.
Simplifed models based on iinear magnetization characteristics have been reported
by [12.13,14]. Models baçed on noniinear magnetization characteristics have b e n published
by severai authors e.g. [ 151. Research on a hear version of the SRM for low speed systems
was initiated at Queen's University in the mid 1980's [ 161. This involved investigation of the
magnetic configuration and control aspects of the linear SRM using finite element analysis and
the construction of test models in conjunction with extensive analysis of a rotary SRM [ 171.
Pattison 1171 presents two models for the SRM: a simplifled mode1 which negiects magnetic
saturation and a more complete mode1 which includes a noniînear magnetintion
characteristic. He has concluded that the performance of an SRM is suongly dependent on
the control algo nthm and the energization scheme emplo yed.
The SRM can be represented by rneans of its magnetic characteris tics, which give the
flux Jr linked with a motor phase as a îünction of the current i into the phase as weU as the
rotor positions 8. The Jr-i plane is divided into two main regions, linear and saturated.
according to the level of current. The linear region is located from zero up to a current equal
to its saturation value 1,; the saturated region is located beyond Im. If the machine is
operated below the magnetic saturation, then the relationship between the flux-linkage and
the current at the instantaneous rotor position 0 is a straight line whose slope is the
instantaneous inductance L. If there is magnetic saturation. then the torque should be derived
as the derivative of coenergy or field stored energy.
Saturation has the two main effects of increasing the motor size required for a given
torque, and at the same t h e decreasing the KVA per horsepower requirement of the power
electronic device for a given torque and speed. While saturation is desirable from this point
of view, at the same time it reduces the energy conversion capability of a motor of given
dimensions. evaluated at a given peak current. Saturation therefore generally leads to a
srnaïier converter but a large motor.
An accurate and cornprehensive noniinear mode1 of SRM. however. is extremely
complicated and is computationally intensive to be implemented in real time for control
purposes. The more complete model would be a good model for the design process of
doubly-salient motors. .
The assumption of a iinear magnetization characteristic leads to a model which has
reasonable accuracy and which requires o d y a small amount of development effort and can
be simulated quickly. This model would be usehil in investigations of motor and control
scheme performance.
It is apparent that a iinear model, which does not describe saturation, cannot be
reliable for motor design. although it has been used for exploring control strategies and
converter ratings.
1.6 P O M R CONVERTERS
In switched reluctance motors, the motor torque is proportional to the square of the
current so that the converter has to supply only unidirectional currents. Therefore, converters
feeding the SRM are of the unipolar type and they generally use one switch per phase. The
function of the converter is to switch the current from one phase to another in synchronism
with the rotor position so as to provide positive current in the phases with increasing
inductance.
The commutation in the SRM is a current commutation. This means that at every
commutation, the current in the off-going phase must be reduced to zero and the current in
the on-going phase must build up b m zero. Various converter configurations for S M have
ken proposed and studied [2,18-231. These converters dBer kom each O ther by the feeding
scheme and by the energy recovery technique used during the commutation from one phase
to the next,
For switched reluctance motors, two feeding schemes are possible: voltage-source
feeding and current-source feeding. Figure 1.5 shows these two dEerent feeding schemes.
Current-source feeding is partkularly suitable for low-speed operation. On the other hand,
voltage-source f&g is suitable for high-speed operation when the counter emf is high and
it may thereby be dficult to maintain constant currents [21]. The voltage control can be
achieved by chopping the bus voltage using the switching devices.
AU converter configurations comprise two essential elements: frstly, a controlled
switch or switches to connect the direct voltage source to the winding to build up the current;
secondly, an alternative path for the current to take when the switch is turned off The most
appropriate choice of circuit configuration depends on the drive power level, the supply
voltage and the application. Some of the more cornmonly used converter circuits are shown
in Figure 1.6.
In Figure 1.6 ( a ), the upper and lower switches are turned on and off together and
the phases are completely independent. Figure 1.6 ( b ) shows the simple bifîlar winding
circuit which achieves unipolar operation with only one switch per phase. However. the
leakage inductance of the phase windings causes high voltage spikes during current turn off
[Ml. In Fgure 1.6 ( c ), a split level dc source is used. For a four phase motor, there are four
switches and four diodes, giving one switch per phase. However, in this circuit, the phases
- Voltage source
1s Current source
Figure 1.5 SRM feeding schemes. (a) Voltage-source feeding. (b) Current-source feeding.
Figure 1.6 Basic power converter circuit configurations.
14
are not completely independent. The circuit in Figure 1.6 ( d ) is a new converter design
described in [18]. This circuit is called the C-durnp circuit because the trapped energy is
dumped into a capacitor before king returned to the dc source. In addition, this circuit has
oniy one switch per phase, and it uses a single rail dc supply.
1.7 ROTOR POSITION SENSING
Switched reluctance motors are controlled by switching the phase currents in
synchronism with the regions of rising stator inductance. As the rotor poles move from the
unaligned position io the aligned position, with a stator pole, the inductance of that stator coi1
varies fkom a minimum value to a maximum value. Since the coi1 inductance varies with the
rotor position, SRM commutation ha always been associated with instantaneous rotor
position As a result, it is apparent that the performance of an SRM drive system ( effective
control of speed, torque and torque pulsations ) depends strongly on the accurate placement
of the phase c m n t pulses relative to the machine rotor angle. Therefore, the rotor position
information is essential for the control section in order to generate the commutation signais.
A variety of position sensors are commercially available for detecting the rotor
position. Rotor position information is obtained from one of the many types of available
physical position sensors, such as an optical encoder, which consists of an optical sensor and
an interrupting type disc [24]. The switched reluctance motor commutation strategy may use
the rotor position feedback ( ie., signal ftom opto-interrupters ) to derive the commutating
signais for the inverter switches.
E n o r in rotor position detection could have a direct impact on the commutation
positions, and may resdt in shifts of the phase conduction periods. The accuracy of position
information determines the efkiency and torque capability as well as the torque pulsation of
ihe drive. The high cost and low reliability of discrete position sensors have led researchers
to seek ways of eliminating these sensors [25]. In many indusuial environments the position-
senshg devices rnay no t be highly reliable due to effects such as elec tromagnetic interference.
high temperature and dust. In addition, for fractional horse power drives the cost of the
position sensor becornes a substantial part of the o v e r d dnve system cost. In such
applications indirect rotor-position sensing is an attractive alternative.
Some of the reported sensorles schemes [26] consist of estùnating the rotor position
using phase irnpedance measurements. Some researchers have used a state observer to
estimate the rotor position fiom measurements of the stator voltage and the coil currents [27].
1.8 CONTROL OF THE SWITCHED RELUCTANCE MOTOR
S witched reluctance motor drives are emerging as an attractive solution for variable-
speed applications on account of their numerous advantages. Most prominent among these
are the simple structure of the motor and the reduced component count of the power
converter due to the unidirectional current requirement. which contribute to lowering the cost
of the SRM dnve system and incrcase its reliability. However, the developed torque is a
nonlinair k t i o n of the rotor position and the stator current. Control of the motor is made
more complicated by the nonlineantks found not only in the motor, but &O in the switching
converters which arise due to the power switches.
The prirnary disadvantage of a switched reluctmce motor is the high torque ripple
content compared with conventionai machines. This usually contributes to acoustic noise and
vibration. The shafi torque in an SRM is the sum of the torques generated by each of the
stator phases, which are controki independently. Torque pulsations are most significant
during the commutation periods w*kn the torque production mec hanism is transferred from
one active phase to another. There are essentially two primary approaches for reducing the
toque pulsations [28]: One rnethod is to improve the magnetic design of the motor, whereas
the other is to use sophisticated electronic controL
Most of the literature on mvitched reluctance motor drives focuses on the design and
modeling of the motor and on the configuration of power converters. There are several papers
that address the control aspects of the SRM.
Open-loo p strategies have been suggested and investigations using angle and current
amplitude regdation were reported. The simplest and most commonly used control strategy
for the SRM involves the use of constant input voltage [29]. The phases are tumed off and
on based on the switching imposeù by the electronic comrnutator. Since this is an open-loop
control strategy. improvement of the dynamic behavior can only be achieved by a better
design of the motor itself and its electronic commutator. Such a scheme suffers from the fact
that the response is not robust when the motor is exposed to interna1 or external disturbances.
Dynamic perf'ormances can be improved by introducing feedback control The basic
hinctional block diagram of a switched reluctance mo tor drive system is given in Figure 1.7.
Closed-loop control strategies for switched reluctance motors are proposed in [29-331. [29]
inuoduced for the ksst tirne the application of feedback iinearization to electnc drives. They
reported the work on control of SRM for trajectory tracking in robotics applications. In this
work they developed a nonlinear mathematical mode1 based on experirnentally rneasured data.
In fact. this approach introduces a transformation to convert a nonlinear system into an
equivalent h e a r one. It is assumed that ali plant parameten are known and that the motor
acceleration is measurable. Unfortunately, their controkr is computationdy quite complex
and. moreover, requires the accurate mode1 of extemal loads in addition to that of the SRM.
In [30], the equations of the SRM are expresseci in a rotating reference frarne.
Constant speed as well as piecewise Iinearized inductance curves and known parameters are
assumed ho wever. Instantaneous torque control is taken into account, while magnetic
saturation is ignored. A Floquet transformation is constructeci to express the equations of the
switched reluctance motor. The paper shows that. under the assumption of constant velocity,
the switched reluctance motor may be modeled as a linear tirne-varying system. A change of
variables was suggested which enabled it to be descnbed by a linear the-invariant system for
which the control design is much easier. This design gives good results only for switched
reluctance motors operating at low s p d . Moreover. a tradeoff is considered between peak
torque and torque ripple.
In [31] a fuzzy logic controiier is used to control the speed of a switched reluctance
motor drive. Fuzzy logic controkrs are based on the theory of fuzzy sets and fuzzy logic. The
design of a fuay logic controiler does not require an accurate mode1 of the plant. This is
appreciated in motor drive applications where the mo tor andfor the mechanical load are
usually dexnbed by a set of nonlinear. differential equations or are partially unknown. A
conventional controuer adjusts the system control parameters on the basis of a set of
differential equations which represents a model of the process dynamics. In a h i u y logic
controller, these adjustments are handled by a f ü q rule-based expert system. While the
overd hizy behavior avoids any speed overshoot. the ripples usually related to the switching
are still unavoidable.
in 1321 a model-reference adaptive control of a variable reluc tance motor for low-
s p d , high-toque mode of operation suitable for robotrs applications has been considered.
The spline h t i o n s are used to model the variabIe reluctance motor characteristics relating
the electric toque to the rotor angular position and the winding current. This mode1 leads to
an adaptive controiler. This work concem itseif only with the low-speed operation. In this
study, instead of using the fun-order system. a reduced-order mode1 is considered. The paper
also assumes that the rnagnetic saturation is due ro the winding current only and is not a
function of the rotor position.
In [33] the design of a feedback linearization controuer for switched reluctance motor
is presented. The mathematical model of the motor takes the magnetic saturation into
account. To reduce the torque ripples in the drive system the design of a robust controller
based on the second method of Lyapunov for speed tracking application has been presented.
The main idea behind the feedback linearization control scheme is to transform the nonlinea.
system dynamics into a linear one.
The feasibiiity of using artifcial neural networks to develop discrete t h e dynamic
models for a switched reluctance motor drive system that includes bo th fault y and fault-Free
behaviors is examined in [34]. SRMs are capable of operating despite the presence of faults
such as partial phase shorts or extemal faults in the power converter. making SRMs irnpo rtant
in high-reliability applications. Faults impart transient changes to machine inductances that
are nifficult to model &+y. Afkr this transient period. SRMs are capable of functioning
at a reduaxi level of performance. nie purpose of this work is to present a methodology that
can be used to construct an artincial neural network capable of modeling the dynarnics of
SRM drive systems, or any dynamical systerns. The approach employed in this paper.
however, appears to consume considerable computational resources.
While proportional-integral convol scheme represents a simple strategy, its
implementation for driving an SRM is rather novel, and its good performance is apparent. In
the cornpetition with more advanced controllers, the P-I controuer has generally been
regarded as one of the most likely candidates to succeed in industrial applications. The main
reasons for thk include: simpücity. lower cost, ability to achieve zero steady-state error. ease
of irnplernentation. robustness, good speed of response, good stability, and other desirable
feature~. P-1 controllers are extensively used in many drives where speed control is desired.
In this thesis, a speed controuer based on the proportional-integrai control strategy is used.
1.9 OBJECTIVES OF THE PRESENT WORK
The objectives of this thesis are:
1. To develop a simple, but reliable model of the switched reluctance mo tor that can be used
to simulate. and evaluate the closed-Ioop performance of the speed control system.
2. To design and apply a controkr based on the proportional-integrai control strategy in the
speed control of the switched reluctance motor.
3. TO study and investigate the stability of the nonlinear control systern ushg the second
method of Lyapunov. and also the Krasovski's method.
4. TO sirnulate the propose. speed control system scherne sho wn in Figure 1.8 under different
w o h g conditions. and for two dinerent methods. using the Matlab software package and
also to investigate the performance and report on the results.
1.10 THESIS OUTLINE
In chapter 1, the history, structure. operation, applications, energization. design and
modehg, power converters, rotor position sensing. control aspects of the switched
reluctance motor. and the objectives of the present studies are reviewed.
In chapter 2, a sirnplified mode1 of the switched reluctance motor is developed.
Expressions for inductance, voltage, and the instantaneous torque are derived. The control
variables are identifiai.
In chapter 3, a speed controller for the switched reluctance motor using the
proportional-integral control technique is developed. The function of t h controller is to force
the motor speed to foiiow pre-specined tracks without excessive overshoots or oscillations.
In chapter 4, stability of the nonIinear control system is investigated using the second
method of Lyapunov. In addition, we apply another approach named Krasovski's method.
which is also based on the second method of Lyapunov, in order to examine the stability of
the noniineu control system. Krasovski's method takes both the equations of the electrical
and mechanical subs ystems into account. Suficient conditions are O btained for global
asymptotic stabiüty of the nonlinear system.
In chapter 5. the switching strategy and the control system schemes for the SRM
s p d control ( two different methods ) are designeci and presented and the relevant equations
used in the simulation are identified. Simulations are carried out under different working
conditions. and the performance and results are presented as weiL
In chapter 6, a summary of this thesis, the conclusions drawn from the present work
as well as suggestions for fiirther work are presented.
CHAPTER 2
MODELING OF THE SRM
2.1 INTRODUCTION
One of the important points in any control system design is to develop a good
mathematicai mode1 which represents the plant d e r various operating CO nditio ns. Therefo re.
to properly evaluate the switched reluctance mo tor performance and the effectiveness of
dserent control schemes, a reliable model is required. The model should represent the static
and dynamic characteristics of tke machine with adequate accuracy.
The purpose of this chapter is to obtain a mathematical model for the switched
reluctance motor used for the design of the speed controuer. The mode1 is represented by a
set of differential equations which are obtained using standard electric circuit and machine
theory.
In order to include ail relevant dynamics, the plant is modeled ( like any
electrornechanical system ) as a combination of two subsystems: an electric subsystem derived
fiom Kirchhoff s laws, which describe the dynamic behavior of the stator. and a mechanical
subsystem derived fkom Newton's law, which accounts for the mechanical load and the rotor
dynamics. The two subsystems are coupled together by energy conversion consideratiom.
Figure 2.1 depicts the schematic diagram of a four-phase SRM.
The inductance p r o f i and the instantanmus toque expression are developed assurning hear
magnetic characteristics. A h , the modes of operation are described and the control variables
are i d e n W .
2.2 MODEL OF SWITCFIED RELUCTANCE MOTORS
Voltage relations for a switched reluctance motor are the same as for any other
electrical machine. ie., the voltage appiied to the stator terminais equals the sum of a voltage
drop due to resistive losses and the induced voltage due to flux Linkage variations. Therefore.
the voltage equation for the k phase of an SRM [17] will be:
where
v, = phase voltage
i, = phase current
r = phase winding raistance
qt = flux M a g e
O, = rotor position
t = time
Since Jr, is a function of 8, and ik, equation ( 2.1 ) c m be expanded using the partial
denvatives of Jr, to yield:
2.2.1 Inductance Profile
A crucial ciifference between the switched ductance motor (a doubly salient machine)
and nomsalient machines lies in the fact that the dependence of flux linkage on the current
does not Vary saiusoidaIly with the rotor position In generai, it could be any periodic function
of 8, [30]. Due to the symmetric location of the poles, the mutual inductance between phases
can be neglected [33]. As a result, Jr, is only a function of i, ( in addition to the rotor
position 8, ) and is not a function of any other winding currents. The flux M a g e for a phase
winding, q,. varies cyclicaIiy with the rotor position 8, and is a noniïnear function of the
phase current i,, when parts of the magnetic circuit become saturated. The relationship
between the flux linkage and the winding inductance is % ( Bk,îk ) = i, Lk ( 8,,ik) .
As discussed in Chapter 1, it is essential for designing and building the SRM to
recognkre the dependence of the phase inductance on the instantaneous phase current as well
as the rotor position. An accurate and cornprehensive nonlinear mathematical modeling of
SRM, however, is extremely compkated and is computationally intensive to be implemented
in real tirne for control purposes.
TO alleviate these dfliculties, the simplified mode1 is presented. The linear mode1 does
not take into account the dependence of the phase inductance on the phase cun-ent caused by
the iron saturation In other words, for control purposes, the model does not need to contain
the fine details that would be required for designing and building the machine.
A simplined model for the SRM is obtained by modifying equation ( 2.2 ) with the
following assumptions:
i ) There is negligible mutual flux linkage between phase windings:
ii ) The ferromagnetic material in the machine has a hear B-H characteristic.
Although the rnodel precision is reduced as a result of these assumptions, the model
development effort, the rnodel cornplexity, and the simulation run tirne are greatly reduced
by ignoring the saturation [17].As a side benefit, the sirnplined model is easy to understand
and to deal with because it is expressed in the farniliar elements of induc tance and rais tance.
A linear magnetization characteristic i m p b that the flux Mage , for a given angle, varies
linearly with the current Under these conaitons, it is convenknt to use the inductance ins tead
of the flux linkage directly.
In general, the simplified model is a good model when:
1. Dif'ferent control strategies and motor performance can be investigated.
2. Design and building of the motor itself is not required.
3. Reaso na ble precisio n is ac hieved.
4. Modest computing resources are needed.
5. Fast simulation run tirne is required.
In addition, the use of a Iinear model is justifïed on the ground of providing sufficiently
accurate c m n t waveforms for control purposes. In cases where higher precision is needed,
the model should include the magnetic saturation. The higher precision may be achieved by
a substantial increase in the model complexity and in the simulation run tirne.
Under the commonly made assumption of magnetic linearity, the flux M a g e of phase
k would be:
where Lk is the phase inductance.
This dennition may be substituted into equation ( 2.2 ) to give equation ( 2.4 ).
4 Since the term - is the angular speed o. thus dt
It is sometimes usefùl to reformulate equation ( 2.4 ) as:
30
When the voltage is applied to the winding, the rate at which the current builds up.
and the maximum value it attains, depends upon the relative positions of the poles at which
the voltage is applied and the speed of the motor when it is applied. Assuming constant speed,
dL, then based on equation ( 2.7 ). it is desirable to switch the voltage on when - is essentidy 4
zero, and when Lk (0,) is a minimum. These conditions allow a very high rate of change of
dik becomes signifïcant, - decreases
dt
( i p dk( 0,)
) overcomes ( vk - rik 4
4 current; the current reaches the desired level quickly. When Lk(8,) increases and - and may become negative if the motional emf
). When the voltage is set to zero, the motional e d
forces the current to zero.
The actuai shape of the current pulse is detennined by the switching points and the
magnitude of the voltage. This control scheme has k e n suggested by [17] and [9]. If the
motional emf is insuEcient to lirnit the current. then the voltage must be manipulated or
chopped to keep the current w i t h the machine and control constraints.
It is seen that the performance of the machine is desctibed in t e m of the parameter
Lk ( 8,) . Assurning that L, ( 8,) follows the trapezoidal current-independent variation. the
idealized inductance profile over one rotor pole pitch is shown in Figure 2.2. The structure
of a four-phase switched reluctance motor is depicted in Figure 2.3, where for simplicity, the
coù of only one phase has been drawn. In Figure 2.2. four zones can be recognUed dong the
pitch. namely the zones fiom -0, to O, from O to P, . frorn P, to P r , and from P r to
Pr + P, = EL, - 0 , . In zone 1 , ( d g n e d zone ). the rotor pole does not face the stator pole
and ihe inductance is minimum ( L, ); in zone 2, there is a rise in the overlapping of the pole
surfaces and the inductance increases; in zone 3 ( aligned zone ). the rotor pole faces the
stator pole and the inductance is maximum (La ); in zone 4, there is a shonage in the
overlapping of the pole surfaces and the inductance decreases.
The inductance is a periodic hinction of 0 with period ar . The equation of the
inductance profde for phase 1 can be approxirnated over a period by
and that of phase k by
L k ( 0 ) = L , [ O - - ( k - l ) ] 4
Rotor Angle
Rotor and stator ples alignai
Figure 2.2 Inductance profile for one phase.
Figuré 2.3 Structure of a four-phase SRM.
where q is the number of phases, and K is the dope of the profde in the zone of increasing
inductance, that is,
The inductance profiles of phases for the switched reluctance motor of Figure 2.3 are shown
in Figure 2.4.
2.2.2 Instantaneous Torque Expression
The operation of the switched reluctance motor is described by the voltage and torque
equations of each phase. For convenience. let us write Lk(8 , ) as L,. The voltage drop on
the phase resistance r can be neglected as it is much less than the induced voltage. Assuming
magnetic linearity, the phase voltage will be:
The above equation may be rewritten as
di, i, dLk ik dLk Vk = 4- + -- + --
dt 2 dt 2 dt
The expression for the instantaneous power will be:
d l de The texm in brackets rnay be rewritten as - ( - L, i i ) and the term - is the mo tor speed. dt 2 dt
nus
1 The term (- L, i f ) is recognized as the increase in stored magnetic energy. and the term 2
i,' d ~ , -- o as the power converted to mechanical work But. the mechanical power developed 2 de
per phase is:
where Tk is the toque devdoped per phase. Equating equation ( 2.15 ) with the second term
in equation ( 2.14 ) we get:
Equation ( 2.16 ) shows that the instantaneous torque per phase, even with the
simplifying assumptions, is a quaciratic function of the phase current and is determined by the
current mjected into the phase winding and is positive in the rotor position interval at which
the phase inductance is gicreasmg as the rotor position. 8, increases. This torque is negative
when the phase inductance is decreasing as 8 increases and it is zero where the phase
inductance is constant. Substituting ( 2.8 ) into ( 2.16 ) yields the instantaneous torque
developed by phase 1 over the four sectors in Figure 2.2 respectively as:
where K is given by equation ( 2.10 ). The instantaneous torque developed by the motor is
the sum of the instantaneous toques developed by the individual phases, Le.,
For the switched reluctance motor rotating in the positive direction, a motoring torque
is therefore developed when the rotor travels dong the zone of increasing inductance. No
torque is developed when the rotor travels along the unaligned or the aligned zones. A
braking torque is developed when the rotor travels along the zone of decreasing inductance.
A more accurate analysis, including flux fringing and saturable iron, would predict a
torque pulse dinerent from the one approximated by equation ( 2.16 ). To include flux
fringing and iron saturation in the torque analysis. the magnetic fields must be nurnerically
computed using a f~te-element method [35].
2.2.3 Mechanical Subsystern Dynamics
Application of Newton's law yields the dserential equation of the mechanicd
subsystem This describes the interaction of the load torque, Tl .the electromagnetic torque,
T, and the rotor speed o. Le..
where J is the moment of inertia of the system, B is the viscous friction coefficient, and T, is
the load torque. T is the electncd torque. The mechanicd variables, the rotor position and
the rotor velocity influence the d ynamics of the elec trical su bs ys tem t hroug h the derivative
of the flux Mage.
2.3 SRM DRIVE OPERATION AND CONTROL VARIABLES
Having developed a mode1 for the switched reluctance mo tor. we need to fmd out the
control characteristics of the SRM before designhg a speed control system for the motor. The
SRM can be energized either by a current source or a voltage source supply.
In a current source SRM drive, a current generator of variable arnplitude forces puises
of m n t into the SRM phases at certain positions of the rotor. The control variables of the
current source SRM drive are the amplitude of the current generator and the CO nduc tion angle
( - Brn ) . Without loss of generaluy. the "on" angle can be set at zero and the CO nduction
angle would thus coincide with the "off' angle [15].
A convenient strategy for controllhg a current source SRM drive relies on using the
current amplitude to adjust the torque and setting the "off' angle at its maximum value. The
maximum "off' angle is q u a i to the angular shift between two adjacent phases so as to
prevent their contemporaneous conduction.
In a voltage source switched reluctance mo tor drive, a voltage generator of constant
amplitude is applied across the SE2M phases at certain positions of the rotor. A current loop
is built up around the phases and. when required. it regulates the current by chopping the
voltage generator. The control variables of a voltage source SRM drive are the on / O ff angles
and the current amplitude.
A convenient strategy for controhg a voltage source SRM drive relies on using the
current amplitude to adjust the torque and selecting the "off' angle to rnaximize the torque
for every level of current and speed.
CHAPTER 3
DESIGN OF A SPEED CONTROLLER FOR THE SRM
3.1 INTRODUCTION
In this chapter. the design of a speed controlier for the switched reluctance motor
using Roportional-Integral control strategy is presented. While Proportional-Integral control
represents one of the sirnplest strate&, its implementation for driving an SRM has not been
extensively investigated before, and its good performance is apparent. In the cornpetition with
more advanced controllers. the P-1 controller has generally been regarded as the one most
likely to succeed in indusuial applications. The main reasons for this have k n simplicity.
lower cost, zero steady-state error, ease of irnplementation, robustness, good speed of
response, good stability, and other desirable features. P-1 controilers are extensively used in
many drives where speed control is desired. A P-1 control law is chosen only because of the
above-mentioned reasons and is not intended to infer that this is the best control law.
Our control objective is to track a reference speed trajectory. Each phase is provided
with current during the torque productive pend of rotation, and the amplitude of the current
is adjusted such that the desired level of torque is obtained. This desired level is dictated by
an extenial feedback loop for controhg the speed. The controuer parameters are checked
in such a way to guarantee stability of the plant. that is, to make a Lyapunov candidate V a
non-increasing function of tirne.
Switched reluctance motors have a pronounced nonlineiuity. At current levels below
the flux saturating values, the torque developed is proportional to the square of the coi1
current. As we are trying to track the mo tor speed with a linear control law, we cm expect
this nonlinearity to afféct the performance. A compensation for the nonlùiearity is not to
drive the motor with U ( controller output ) but rather with dm. This compensation is
correct when current is used as the controL
A feedback control system is employed for speed control of the switched reluctance
motor drive io provide the desirable performance. This consists of a P-I controller which
compares the actual speed of the drive and the required speed signal set by the user. The
controller output is the torque reference.
P-I controkr has a proportional as well as an integral term in the fonvard path. The
integrai component of this controller wiU ensure that the steady state error is zero for a step
change. This is a very useful feature Cor many applications. The P-I control is applied to the
SRM drive with the motor working in the linear region of its magnetic characteristics. This
is a good approach for investigating the effectiveness of the control system as the torque
npple in the linear region is significant and the torque characteristics are nonhear.
3.2 PERFORMANCE AND SPECIFICATIONS
The expected performance of a drive depends on the type of application. The
performance requirements are stability, accuracy, speed of response, and ro bustness. All the
above terms can be specifred for step changes in the input cornmand to the control system.
The satisfactory relative stability means the absence of large values of overshoo t and
thus adequately damped response. Accuracy is measured in terms of the steady state error
between the step command input and the output variable. Speed of response is measured in
t e m of the rise and settling time of the controUed variable. The response time of a drive
system is de6ned as the t h e taken by the system output to reach about 90% of the input
s@ reference cornrnand signai In other words. it shows ho w fast a system can respond to
changes in the input conditions. Robustness requires that the overail control system be
relatively insensitive to extemal andor intemal disturbances.
nie speed reguiation ( speed tracking capabilities ) of a drive system is the ability of
the drive to maintah the preset s p e d under varying load conditions. In many industrial
applications it is irnperative that the system respond quickly to a load disturbance and
maintain a steady constant speed.
For certain processes and certain given inputs. the proportional-integral controller cm
Iead to zero steady state error, good speed of response and stability if the controller
parameters are weIl tuned. However. in some situations. it may be sensitive to changes in the
system parameters andior external disturbances as the y are strongly based on the d ynamics
of the systern king controlled. Dynarnic system parameters can change due to operating
conditions and also due to component ageing over the lifetime of the system. The load cm
also Vary, and sornetimes one controkr must be able to handle a variety of different loads.
The tmnsient performance cm be adjusted to satisfy the systems specifications by adjusting
the proportional and integral gain constants of the controller. Le.. K p and K,.
3.3 SPEED CONTROLLER
The parameters of the proportional-integral controller need to be adjusted to
efféctively control the process whose dynamics are descriaed by a set of differential equations.
The drive incorporates four hysteresis control loops for the phase currents. By properly
designing the P-1 controkr, it is possibb to make the transient response to a step input
exhibit reiatively small or no overshoot. In the design of the speed controller. the time
response of the current loop wiu be assumed to be so fast that the current reference WU be
considered to coincide with the actual currents. However. the dynamics of the current loop
is taken into account during the simulation of the drive system.
The response of a system cm be easily analyzed if the system could be approximated
by a second order systern Therefore, for simplicity in the analysis. the elec trical time constant
of the motor is neglected, as the mechanical Ume constant of that motor is usually much larger
than its electrical time constant. A schematic of the sped loop is given in Figure 3.1.
A feedforward controller as shown in Figure 3.2 translates the demand t orque Td into
appropriate instantaneous phase currents whic h WU cause the switched reluctance mo tor to
meet the dernand. The feedforward controller consists of a torque-to-current transformation
block and a commutation systern The fùnction of the commutation system is to designate the
appropriate phases for torque development and commutation.
In designing the speed controkr we aim to have no more than 5% overshoo t and that
the rise time should be iess than 5 milliseconds. A darnping ratio C = 0.7 of the linear mode1
wiil meet the overshoot requirement and for this damping ratio. a rise time of about 5
Load 9 Toque
Figure 3.1 Block diagram of the speed control system.
- Wd - 7
t
S w k M Rekiclance Motor
Cl SC ' FC SRM
I
bt 4 - - - Speed Controllei
9 b U=Td
. b Feediomard Contmller T- b t C2
*
- 1 )
miUiseconds suggesu an undamped naturd frequency of about on = 400 radsec. Most
systems king anaiyzed for control system design are much more complicated than the basic
second-order system Approximating the system by a second order mode1 would be used oniy
for guidance.
Since the mechanical time constant of the mo tor is much higher than the elec trical tirne
constant, the goveming electromechanicd equation of the switched reluctance motor ( the
dynamic equation of the mechanical subsystem ) can be simpEed as [8]:
For the closed-loop system, we get:
w here
o = the actual speed of the motor
o, = the demand speed of the motor
I = the moment of inertia
B = the coefficient of friction
Kp = the proportional gain of the speed controiler
K, = the integral gain of the speed controkr
Tl = the Load torque
T, = the cornmand torque
It is assumed that the load toque T, and the demand speed ad are constant and thus
their derivatives are zero. After differentiating equation ( 3.2 ) with respect to tirne, we
O btain:
A state-space representation of the whole cbsed-loop drive system using the phase
variables is adopteci. niis can be written as:
where
The output variable is the mechanical angular velocity.
The transfer function between the input and the output of the closed-loop system is
given b y
where
1 and s are the identity matrix and the complex variable of the Laplace transform, respec tively.
Therefore. for the whole system ( closed-loop drive system ) we have:
48
Using B and J as in Appendix 1, then the characteristic equation is given by:
Knowing C and 0,. one can easily calculate K p and KI. For C = 0.7 and on = 400 radsec,
we get Kp= 0.892 and KI= 256. If a different design criterion is selected, the above
parameters would correspondingly be changed. When a smaller overshoot in s p d is
required, the damping ratio of the system is increased. As it was stated earlier, the transient
performance of the system can be adjusted to satisQ the system specifcations by tuning the
controuer constants K p and K,.
CHAPTER 4
STABILITY ANALYSIS OF THE SPEED CONTROL SYSTEM
4.1 INTRODUCTION
For a given control system, stability is a basic requirement that needs to be
guaranteed. If the system is linear and tirne invariant. more than one method is available to
check whether the system is stable or not. Among these methods are the Nyquist stability
criterion and the Routh-Hurwitz stability criterion If the system is nonlinear, or if it is linear
but tirne varying, however, then these methods of checking stability cannot be applied.
Lyapunov technique, which plays an important role in control and system theory, is
a technique for anaimg the stability of nonlinear as weil as linear dynamic sys terns based on
the state space representation of systems described by ordinary differential equations. This
technique is in principle very general and powemil Lyapunov developed a fundamental
method of determining the stabaty of a dynamic system based on a generalization of energy
consklerations. Lyapunov presented two methods, called the fnst and the second methods of
Lyapunov, for detemiining the stability of d ynamic systems described by ordinary differential
equations.
The ksst method assumes that we can obtain an explicit fom of the solutions of the
diEerentid equations and t hese are used for the analysis.
The second rnethod of Lyapunov, which is aiso called the direct method of Lyapunov,
will be used in this study. Stability of the nonlinear control system is investigated using this
method. In addition, we apply another approach; narnely Krasovski' s method, which is based
on the second method of Lyapunov. in order to examine stability of the nonlinear control
system Kmovski's method takes both the ektrical and mechanical subsystem equations into
account. Sufficient conditions are obtained for global asyrnptotic stability of the nonlinear
systern
4.2 THE SECOND METHOD OF LYAPUNOV
The second method of Lyapunov is the most general approach for the determination
of stabihty of nonlinear andor linear tirne-varying systems. This method, also applies to iinear,
the- invariant systerns. The second method does not require the explicit solution of the
differential equations. ~ e . , by using the second method of Lyapunov. we can determine the
stability of a system without solvhg the state equations. The system dynarnics must be
describai by a state-space mode1 The state-space mode1 description of a given system is not
unique but depends on which variables are chosen as state variables.
If the total enecgy of a mechanical or electrical system is continuously dissipated. then
the system whether Linear or nonlinear. must eventudiy settle down to an equilibnum point.
Thus, we may conclude that a certain equilibrium point of a s ystem is stable by exarnining the
variation of a single scalar fùnction caiied the Lyapunov function ( V-fùnction ). From the
properties of the V- hinction. one should be able to answer the question of stability or
5 1
instability of chat equilibrium point. The V-function is no t unique. and dilferent choices may
in general lead to different domains of stability.
The major problem in applying this method to nonlinear system is the construction
of Lyapunov huictions [36]. It should be noted that Lyapunov theorerns on stability of
equilibrium points in nonhear systems fi~mish only sunicient conditions. Therefore, failure
to construct a Lyapunov function for a given system does not imply that the such equilibrium
points are unstable. The nonlinear control system is simulated as a final check on stability.
4.3 STABILITY OF THE CLOSED-LOOP ERROR SYSTEM
One of the most powemil methods for studying the stability properties of a system is
the Lyapunov's second rnethod. It consists in the use of an auxiiiary function, which
generalizes the role of the energy concept in mechanical systems. The auxiliary or Lyapunov
function needs to satisfy certain conditions [36]. In fact, this approach deterrnines stability
without actually having to solve the dfierential equations. In this method a function of the
state variables having special properties is formed. and the derivative of the function with
respect to tirne is taken If this derivative is negative dong the trajectories of the system, it
can be s h o w that the equilibrium point under consideration is asymptoticaily stable.
For control purposes, we model a switched reluctance mo tor driving a constant load.
Since the mechanical t h e constant of the motor is rnuch higher than the electrical tirne
constant. the goveming electromechanical equaiion ( the dynamic model of the mechanical
subsystem ) of the switched reluctance motor can be simplified as:
52
where
o(t) = a scalar representing the angular speed of motor
J = the rotor inertia
B = the coefficient of fiction
T, = the load torque, and
T, = the command torque.
In fact, equation ( 4.1 ) represents the h e a r approximation to the system nonlinearity. Yet.
there is no general way of hding Lyapunov functions for nonlinear systems. This is a
fundamental drawback of the direct method. Therefore, faced with specific systerns. one has
to use experience. intuition. and physical Lisights to search for an appropriate Lyapunov
fùnc tion.
Stability anaiysis for linear time-invariant sys terns is well known. Lyapunov func tions
for combinations of subsystems may be derived by adding the Lyapunov functions of the
subsystems. Since nonlinear control systems may inchde Linear components ( whether in p h t
or in controuer ). we should be able to describe iinear systems in the Lyapunov formalkm
[36]. Figure 4.1 shows the speed control system configuration for hear stability analysis.
Given the dynamics of the motor, Our objective is to synthesize a controiler which
would ensure a global asyrnptotic stability result for the motor speed traclcing error. In other
words, the control scheme should be designed so as to guarantee global asymptotic
s tability of the equilibrium point of the error differential equatio n. With this in mind, wr: de fuie
the motor speed tracking emor to be the Merence between the reference speed and the plant
actual speed. Thus the error is given by:
where o, is a scalar representing the desired motor speed trajectory. And o is the actuai
motor speed. The primary concem of the controller is to adjust the input signal such that the
plant actual speed follows the reference speed. In other words, e is required to be kept as
close as possible to the point e = O. This c m be accomplished by d e f h g a positive definite
function V ( e ) and getting conditions such that the tirne denvative of V ( e ) is negative
de finite.
Equation ( 4.1 ) can be put in the following form:
Given the above dynamic equation, we can design a controller which would achieve the stated
control objectives if the term on the right-hand side of the above equation ( 4.3 ) is assumed
to be the control input which is given by:
The command toque Td for any speed error signal e is calculated by the speed controiler
using equation ( 4.4 ).
To formulate the dinerential equation describing the speed tracking error e . we
rewrite equation (4.3) in t e m of the speed tracking error. Assuming that a, is a constant.
we take the denvative of equation ( 4.2 ) with respect to t h e which yields:
We rewrite the above equation using the expression for the control hw. Le.. equation (4.4)
in order to obtain the following equation:
After differentiating with respect to the. we get:
The above equation is the closed-loop characteristic equation of the error signal e . This
equation rnay be rewritten as foIlows:
w here
and
The tracking error e will asyrnptoticaiiy go to zero provided that the parameters K,
and K2 are such that equation ( 4.10 ) is stable. The necessary and su fficient condition for the
stability of this linearized system is that KI >O and K2>0.
As stated earkr, we wili continue to investigate the stability of the assumed linear
system using Lyapunov method in order to obtain a candidate Lyapunov hinction V which.
in tuni, wdi be used for checking the stability of the nonlinear system by the Lyapunov
ap proach.
Now. the stability of the linearized system is investigated by formulating a positive
dennite Lyapunov function V for the speed tracking error equation, and ensuring that the time
derivative of the Lyapunov function is negative definite.
The speed tracking error equation is rewritten in the state space as:
where
X, = e
and
X* = é
For this mathematical mode1 ( linearized system ). if a Lyapunov function can be found
and shown to decrease with tirne dong the trajectory, then the system is stable. A class of
scalar hurtions that piay an important role in Lyapunov analysis is the quadratic form. As a
possible Lyapunov fünction we choose
and its derivative with respect to time is given by
2 Y = 4K~~12~1 + 2(pll - Klp12 - K2pu)xix2 + 2(p1, - ~ , p , ) x Z ~ ( 4.16 )
w here
and
We select P ( a real constant symrnetric matrix ) appropriately as a positive definite
ma& satisfjing the foIlowing Lyapunov equation ( Sylvester's theorem ):
Thus equation ( 4. f 5 ) may be rewritten as:
Q may be chosen to be the identity matrix. Since Q is assumed to be positive defmite, is
a negative dehite quadratic hnction and the system is asymptotically stable.
In order for V to qualify as a Lyapunov hinction. V must be positive d e f ~ t e and Y
must be negative defite. Sylvester criterion provides conditions on P for the positive
defjniteness of V = X PX [37]. According to this criterion, V is positive d e f ~ t e if and only
if al principle minon of P are positive. Such a matrix P is then a positive d e f ~ t e matrix.
Using the Matlab package software. we $et the r n a h P as:
Therefo re,
and
Since accordhg to Sylvester's theorem, the conditions on P are satisfied, hence the iinearized
system ( rnotor speed tracking error ) is globally asymptotically stable.
As we stated earlier. this positive defuite V-hinction is now used as a candidate
61
Lyapunov function for investigating the stability of the nonlinear system. Figure 4.2 shows
the block diagram of the closed-loop system with the nonlinear elemenr. The nonlinear
element includes the nonlinear dinerential equations governing the dynamics of the system and
also the noniinear rektionship between torque and current in the switched reluctance motor.
It will be assumed that the input-output characteristics of the noniinear element can be
described by a function r = g ( rd) ; where s, is the input and s is the output of the no nlinear
element Figure 4.3 ülustrates the speed control system configuration for the stability analysis
of the nonlinear system ( s i m p W i block diagram of the closed-loop system with the
nonlinear eiement ). The system to which the procedure of stability analysis is applicable can
be represented by this simplified block diagram. In Figure 4.3, h ( e ) represents the nonlinear
element of the system combined with the transfer hinction of the P-1 speed controller.
The system equation for the motor speed tracking error will be
Choosing the state variables x, = e . x2 = è. the state equations wiU be
Load Torque 7
Figure 4.2 Block diagram of the closed-loop system with the nonlinear element.
Load Toque 7 SRM
Figure 4.3 System configuration ( simplified block diagram ) for stabiiity study ( nonlinear analysis ).
Knowing V. it wili be found that
Equation ( 4.26 ) cm be rewritten in the following form:
Therefo re,
Y = - 0.006dx: - (2.5d - 6 4 0 . 0 0 6 ) ~ ~ ~ ~ - 0.01~:
The new matrix for P wiU be
Sylvester's theurem yields the foUowing conditions for Y to be negative defhte. A necessary
and su8inent condition for y to be negative dennite is that the determinant of P be positive
if the system order is even and negative if system order is odd. and that the successive
principal minors of even order be positive and the successive principal minors of odd order
be negative; that is, we m u t have
From the roots of the above quadratic equation. the sufficient conditions for global
asymptotic stability for the nonlinear system will be:
It should be emphasized that by emplo ying a dinerent Lyapunov function candidate. we rnight
be able to obtain dinerent stability conditions. V-functions that provide wider bounds usually
exist. But to obtain this Mprovement other than by trial and error requires more advanced
methods [37].
In inequality ( 4.31 ). f ( x , ,x,) represents the variation with respect to t he of the
motor toque which is a function of the speed error e and its tirne denvative è . Figures 4.4
and 4.5 graphically illustrate the region satisfjing the suficient conditions and the bounds
determined by inequality 4.3 1 and for the reference speed of 80 rad/sec. This is the reg ion for
which V is positive dennite and Y is negative definite. If the trajectory is restncted to lie
wiîhin the region illustrateci in Figure 4.4 wwhe the ssufncint conditions hold. then the system
is globally asymptoticaiiy stable.
4.4 KRASOVSKI'S METHOD
We may examine asymptotic stability of equilibrium States of nonlinear systems
without linearization Several methods based on the second method of Lyapunov are available
for this purpose. One of these approaches is Krasovski's method for testing suficient
conditions for asymptotic stability of nonlinear systems [38]. In this thesis. Krasovski's
method takes both electncal and mechanical subsystems equations of the nonlinear sys tem
under consideration into account. Sufficient conditions are obtained for global asymptotic
stability of the nonlinear system.
According to KmsovMs theorem, if the system is descnbed by x = f ( X ) . then we
define
where F ( X ) is the Jacobian matrix of the system and F T ( x ) is the transpose of F ( X ) .
Krasovski's method suggests a simple form of Lyapunov function candidate for noniinear
systems. namely, V ( X ) = f T ( x ) f ( x ) . The negative definiteness of F(x) irnpks the
negative denniteness of y [36,38]. Therefore, the equilibnum state is asymptoticdy stable.
If in addition. f T ( x ) f ( x ) - .. as 1x0 - a. then the equilibrium state is globally
asymptoticdy stable.
W e choose phase currents and motor speed as state variables. Thus,
The Jacobian matrix for the system is
As we obtamed in chapter 2, the system equations ( electrical and mechanical ) are given by
and
- - do - l ( T - T, - Ba) dt J
Equation ( 4.36 ) may be rewritten in the following form:
Mer taking partial denvatives. we get
r + Kx, - -- Kx4 4 Lr,
Then the transpose o f F ( X ) , denoted by F =(XI. is as foliows:
Then F ( x ) which is the sum of F(X) and F T ( x ) is equal to:
According to Krasovski's theorem, if @y) is negative definite, then the equiiibnum state is
asyrnptoticaIly stable. As the switching strategy adopted in this study assumes that only one
phase is energized at a time. toque must be developed by a single phase when it is energized.
Sylvester's theorem yields the following conditions for to be negative defmite:
and
The above conditions may be rewntten in the foilowing form:
r + K o > O
and
The condition deterrnined by ( 4.44 ) may be rewntten as
where T, is the electromagnetic torque developed by the k" phase of the motor. The
detailed derivation of the above conditions is s h o w in Appendix II. The system will be
asyrnptoticany stable in the large if V ( X ) = f T (x)f( X) - as 1 X 1 - m . If electrical and
mechanical parameters of the nonlinear system change under different operating conditions
such that the above inequalities remain satisfied. then the noniineu system is globally
asyrnptoticaiiy stable.
It should be noted that Lyapunov and Krasovski theorerns on stability of nonlinear
systems fu-h only sutficient conditions. Therefore, for points outside the regions s a t i s w g
the sufficient conditions. for a given system we cannot imply that the control system is
unstable. The nonlinear control system is simulated as a h a l check on system stability.
CHAPTER 5
SIMULATION AND PERFORMANCE OF THE SPEED CONTROL
SYSTEM
5.1 INTRODUCTION
Control of a switched reluctance motor drive with a feedback loop is quite involved.
and this complexity increases as higher performance is demanded. The main reason for this
compiexity is that SRM dynamics are represented by nonlinear equations.
The scherne which will be used for controllhg the switched reluctance mo tor drive
is based on curent regulation by means of a hysteresis band algorithm. This leads to a control
system in which the speed is controlled by adjusting the reference current.
The SRM drive system is simulated using two d8erent methods, namely; the current
source as weii as the voltage source supply. The cornplete simulation results are ako
presented in this chap ter.
5.2 MODE OF CONTROL
The control objective is to provide each phase with current over an appropnate angle
of rotation ( the torque productive period of rotation ) and to adjust the amplitude of the
current such that the desired level of torque is obtained. The desired level of torque will
usuaLly be dictated by a feedback loop for con t rohg the speed of the rnotor.
5.3 SRM SUPPLY
The switched reluctance motor c m be fed either by a current source or by a voltage
source ~upply. With a cunent source supply, a current generator of variable amplitude forces
pulses of current into the phases at the certain positions of the rotor, Le., the current is forced
into the phase at ern and then dowed to go to rero and hence be removed ai eO6, whereby 8,"
and e 0 , must lie in the zone of increasing inductance. The control variables of a current
source SRM drive are the amplitude of the c m n t generator 1, and the "on" and "ofE" angles
Bo,, and e 0 , . The current amplitude is controlled from zero to I , and the "on" and "off'
angles from zero to P,, with 0, s OOr. Current-source feeding is particularly suitable for
low-speed operation when the motor torque must be closely controlled with minimum ripple.
With a voltage source supply, a voltage generator of constant amplitude is applied
across the phases at the certain positions of the rotor. In practice, a rotor position sensor
provides a signal corresponding to the torque productive region for each phase of the mo tor.
This signal is usai to apply voltage to the motor winding. The control variables of a voltage
source SRM are the onloff angles 0, and €log and the c m n t amplitude 1,. Voltage-source
feed8ig is suitable for high-speed operation when the counter emf is high and it is difficult to
maintain constant currents.
Chopping the voltage generator c m be executed to apply voltage levels less than the
generator voltage. In practice. a current transducer provides a signal which is proportional
to the winding current. This is compared to a reference signai in a feedback control system
causing the winding current to be chopped so as to maintain a constant current over the
torque productive region
5.4 SRM DRIVE ANALYSIS
Analysis of the control characteristics of the switched reluctance motor drive is carx-ied
out at the steady state and under the assumption of constant motor speed. In addition. the
operation of the SRM drive is done under the assurnption that the motor works in the hear
region of its magnetic characteristics. The hear region is distinguished by a current level
which is less than the saturation value 1, .
The switched reluctance motor drive develops a positive torque when the rotor travels
along the zone of increasing inductance. If phase 1 were supplied by a current pulse of
amplitude i when the rotor moves from zero to a,/q and similarly with respect to the other
phases. the instantaneous torque developed by the rnotor would be:
As the switching strategy adopted in this study assumes that only one phase is energized at
a the. torque must be developed by a single phase when it is energized.
77
The characteristics of dl the phases are the same excep t that they are phase s hifted by
ar/q ( in thû study 15 mechanical degrees for an 8/6 four-phase SRM ). The maximum
conduction interval, however, must be equal tu:
to prevent the contemporaneous conduction of two adjacent phases. During the conduction
interval the ins tantaneous to rque is
and zero elsewhere.
The torque characteristics are determineci with reference to the average torque
developed by the motor. which can be expressed as the number of phases tirnes the average
torque developed by phase 1, that is,
where T, is k hemtanmus torque of phase 1 and given by equation ( 2.17 ). For equations
78
( 5.3 ) and ( 5.4 ). the average torque will be
Equation ( 5.5 ) indicates that there are two effective control variables, namely, the current
amph& and the conduction intervaL With no loss of generality, the "on" angle may be set
at zero. and equations ( 5.2 ) and ( 5.5 ) would then be reduced to
and
Therefore, the average torque of the SRM dive is a quadratic hnction of the current
amplitude. It is, however, linear with respect to the "off' angle. #en the "off' angle is equal
to a,/q , then the average torque wiU be given by
The average torque would coincide with the instantaneous torque and would be a quadratic
hinction of the current amplitude.
A convenient strategy for controlling the drive is obtained by using the current
amplitude to adjust the torque and setting the "off' angle to its maximum value. The
maximum "'off' angle is equal to the angular s h . ktween two adjacent phases so as to
prevent their contemporaneous conduction, and is given by equation ( 5.6 ).
In a voltage source switched reluctance motor a current loop is built up around the
phases and, when required, the current is regulated by chopping the voltage generator. The
motor is supplied by a two-level voltage. At €Ion, the voltage generator is fuUy applied across
the phase to establish a current through it. The chopping interval extends to the "off' angle
eo,. During this intervai, the current is regulated at the demanded level by chopping the
supply voltage. Again, a convenient strategy for con t rohg the voltage source switched
reluctance motor drive is by using the current amplitude to adjust the torque and then
selecting the 'off' angle to maximize the torque.
5.5 SWITCHING STRATEGY FOR SIMULATION
The speed control of a switched reluctance mo tor drive requises the identification of
the switching strategy. The "on" and the "off' angles are switched every ar/q degree. The
current can be regulated at the demanded level by chopping the voltage source. From the
Linear analysis, we know that the toque produced is proportional to the square of the current
through the winding. Hence. the torque and, in tum. the speed c m be controlled by
controhg the current.
The motor under investigation has four phases which must be energized and
cornmutateci at the correct rotor angles if it is to operate effectively. In the energization
SC h e m emplo yed here, a single phase is energ ized at a t h e . The 'on'' and "off' angles are
nxed at On and Bof. The actual phase to be energized is detennined by the rotor position.
A fdforward controller, which consists of a torque-to-current transformation block and a
commutation system. translates the demand torque T, into the appropriate phase currents
which wiU cause the SRM to meet the demand ( Figure 3.2 ).
5.6 EQUATIONS USED IN SIMULATION
The speed control system introduced in chapter 3 ( Figure 3.1 ) is simulated using the
equations described in the previous chapters. The feedback control system employed for
speed control of the switched reluctance rnotor drive. consists of a Proportional-Integrai
controller which compares the actuai speed of the drive and the required speed signal
provided by the user. Upon speed demand, the speed controller cornputes the value of the
control signal U using equation ( 4.4 ), which yields the torque demand T, ( Figure 3.1).
In this closed-Ioop conirol system, the output of the speed controller generates a
manipulated signai which is interpreted as the torque demand. As the switching strategy
adopted in this study assumes that only one phase is energized at a tirne, the torque T, must
be developed by a single phase when it is energized. The current required to produce this
torque is obtained fiom equation ( 2.17 ) and is given below:
where K is given by equation ( 2.10 ).
The system equations that are used in the simulation are as follows:
- - du - 1( T - T, - Bo) dt J
where
k = the suffix denoting phases
v, = the phase voltage
r = the phase winding resistance
T = the motor torque
o= the actual speed
J = the moment of inertia
i, = the actual phase current
0, = the rotor position
T, = the load torque
B = the coefficient of fiction
L, = the phase inductance
i , = the demand phase current
The actual current that produces the torque is obtained fiom equation ( 5.12 ).
5.7 HYSTERESIS CONTROLLER
The hysteresis controiier maintains the current waveform between an upper and a
lower E t in the hysteresis band. The actual current is fed back to be compared with the
reference current. Whenever the actual current exceeds the upper band. the output voltage
changes fkom V to O and the c m n t starts to decay. As soon as the current crosses the lower
band E t , the voltage changes back again and the actual current starts to increase.
By simulating a hysteresis controler, the actual current is forced to follow the
demand current in equation ( 5.9 ). In hysteresis control, the foliowing control Iaw is applied:
w here
i, = the dowed hysteresis band
V,, = the power converter dc bus voltage
5.8 SPEED CONTROL SCHEMES
The drive system has b e n designed to operate in the speed control mode. A speed
e m r obained h m the reference speed and position informaion generates a reference torque
signal using a P-1 algorithm. This cornmand is used to generate the desired currents. The
huiction of the commutator is to designate the appropriate phases for torque development and
commutation. The algorithm used by the commutator maximizes the conduction region in
order to produce maximum motor torque. Performance is based on the ability to provide each
phase with pulses of current during the torque productive periods of rotation and to adjust
their amplitude either through chopping or by selecting predetermined switching angles. The
torque magnitude is controlled by adjusting the current amplitude.
The speed control system is sirnulated using two different control schemes of Figure
5.1 and Figure 5.3. In some wes, it is the steady-state operation that is most important. For
example, this rnay be the case in air-conditioning and in pump drives. In O ther cases, such as
in robots and actuators, dynarnic performance rnay be more important.
Figure 5.1 shows the control configuration of a current source switched reluctance
motor drive and is suitable for investigating the steady-state operation of the drive. In one
revolution each p h conducts as many pulses of unidirectional current as there are rotor
poles, so that there are qN, pulses or working strokes per revolution. The shape of the
current pulse varies with torque and speed. An important special case is. however. shown in
di Figure 5.1, in which the current waveform is flat-topped. Since - = O during the d t
conduction angle, the sum of the motor back-emf in this interval and the voltage drop across
the phase resistance must be equal to the appiied voltage.
Rectangular current pulses can be realized at low speeds if the phase is switched on
at the unaligned position and off at the aligned position. and the current is maintained
constant. In this configuration, the manipuiated signal ( output of the speed controller ) which
is a torque demand is entered into a hnctional block based on equation ( 5.8 ) to deliver the
current reference for the supply. The phases to be energized depend on the value of the rotor
position obtained fiom equation ( 5.10 ). In simulation, four look-up tables set up the current
references for phases using the related 8, and €lof angles. Figure 5.2 shows the electrical
subsystem of the switched reluctance motor simulated based on the scheme of Figure 5.1.
Figure 5.3 shows the control corQuration of a voltage source switched reluctance
motor drive and takes into account the dynamic performance of the drive. In this
configuration, a current loop is built up around each phase and. when required, the current
is regukted using the hysteresis controller. which manipulates the phase voltage. The SRM
system has been divided into two electrical and mechanical subsystems. Figure 5.4 shows the
electrkal subsystem of the SRM ba.sed on equation ( 5.12 ). In Figure 5.4. as we can see, the
electrical dynamic rnodeling of the motor is constructed and simulated using different
components and blocks. The mechanical subsystem of the SRM is also set up based on
equation ( 5.11 ). The variation of phase inductance with the rotor position is obtained from
Td *
Id + f(u1
Fcn
Producl
Figure 5.1 Control scheme of the CSSRM drive.
Figure 5.4 SRM electrical subsystem for the control scheme of the VSSRM drive.
equation ( 2.8 ). In the simulation, four look-up tables ( Figure 5.4 ) build the variation of
phase inductances with the rotor position
5.9 PERFORMANCE OF THE SPEED CONTROL SYSTEM
In order to test the proposai control suategies ( schemes of Figures 5.1 and 5.3 ) and
to mvestigate the steady-state and dynamic responses of the system, the switched reluctance
motor drive has been simuhted under dEerent working conditions on a digital cornputer
using the Matlab software package. The results of the simulation are also presented in this
chapter. Data and parameters of the SRM are found in Appendix 1.
Performance is based on the ability to provide each phase with pulses of current
during the torque productive periods of rotation. For torque production. it is assumed that
a constant current could be established instantaneously, could be held constant over the
required zone of rising inductance and could then be removed as quickly as commutation
allows. In practice, this can be approximated very well at low speeds. Considering low-speed
operation of the system, current is established in a phase winding by closing the switches. The
current rises rapidly to the desired value because the appIied voltage is large in relation to the
hpedance, and at low speed the back-emf is low as weU At low speeds. the phase penod is
sufficiently long to enable shaping of the current pulse by repeated switching within this
perbd, whkh has been called chopping controL For low speeds, the switching angles are kept
at constant values that maximVe the motor torque with minimum ripple.
The primary disadvantage of a switched reluctance motor is the high torque npple
content compared with conventional machines. This would contribute to acoustic noise and
vibration The origin of toque pulsations in a switched reluctance motor is due to the highly
nonlinear and discrete nature of the torque production mechanism. In addition, the torque
rippk is due to dre sharp edges in the profiles of the inductance. The sharpness in the edges
onginates from the approximation of the inductance profiles given by equaiion ( 2.8 ). The
ripple should be smaller with practical motors since their profiles are smoother.
5.9.1 Starting Performance
To examine the steady-state performance of the switched reluctance mo tor the
following simulation tests are camed out based on the control scheme of Figure 5.1 ( current
source switched reluctance motor ). Figure 5.5 illustrates the simulated speed response when
the motor is allowed to accelerate fiom rest to a reference speed of 40 radsec. with a load
torque of 10 Nm. It can be seen that the speed response is quite good, that is, with a very
smaii oveahoot ( about 1.07% ) and oscillations. The correspondhg torque, speed error.
phase current, and phase voltage for the above - mentioned transition are shown in Figures
5.6 - 5.9, respectively. M e r an initial transient, the speed and the torque converge to their
tinal values. The time required for the speed to rise from zero to one hundred percent of the
demand speeù ( rise tirne t, ) is about 5 milliseconds. The way in which the speed error
changes with tirne for a step change in the speed reference. from a s tandstill condition, is
shown in Figure 5.7. The speed error is high initially after which it approaches zero as the
output speed tracks the rekrence speed. The prirnary fùnction of a speed controiler is to make
this speed error zero as quickly as possible, with a minimal overshoot in the actual speed.
Figures 5.8 and 5.9 illustrate the rec tangular cunent and voltage pulses. respec tively.
5.9.2 Effect of Step Change in Speed Demand
Figure 5.10 inutrates the operation of the drive due to the effect of a step change in
speed dernand h m 40 rad/sec to 80 radsec at t = 0.1 second while the Io ad torque is held
constant at 10 Nm The control system is shown tu be capable of controlling the motor under
the variabk speeds of the drive. The amount of overshoot is about 2.86%. when the reference
speed is changed from 40 rad/sec t O 80 radlsec.
5.9.3 Effect of Parameter Variations
The sensitivity of the system when the motor is subjected to uncertainties in the
mechanicd and the electrical parameters ( rotor inertia and phase resistance ) is investigated.
Here the rotor inertia J is doubled and the phase resistance is increased by 20%. Figures 5.1 1
and 5.12, respective1y. show the effect of these variations on the speed response of the drive.
Inspection of t k e figures indicates that there is no significant change in the speed response
from that of Figure 5.5. The amount of overshoot in Figure 5.11 and Figure 5.12 is about
1.96% and 11.6196, respectively.
5.9.4 Effect of Load Torque Disturbances
One of the desirable characteristics of a high-performance dnve is that the drive m u t
show robustnes during load torque disturbances. In many v a r i a b l e - s e drive applications,
it is important that the systern respond quickly to a load disturbance and maintain a steady
speed. A simulation test is carried out to observe the response of the dnve under load torque
disturbance. The load torque is changed fiom 10 Nm to 8 Nrn at t = 0.07 second and
subsequently it is increaçed to 10 Nm at t = 0.14 second for a reference speed of 40 radsec.
The correspondmg speed response is as shown in Figure 5.13. It can be seen that at t = 0.07
second when the load torque is decreased to 8 Nm there is a very slight increase in the motor
speed but the controller is able to hold the motor speed close to the reference value.
However, when the load torque is increased to 10 Nm at t = 0.!4 second there is a very
slight dip in the motor speed, but the controller is able to track the reference speed quite
closely. The correspondhg rnotor torque response for the above-mentioned load torque
disturbances is as shown in Figure 5.14. As we can see. the moto r torque fo llo ws the required
demand,
5.9.5 Effect of Switching Angle Variations
The dependency of motor performance on the switch-on angle with respect to the unaligned
position and switch-off angle with respect to the aligned position is also investigated. The
selection of the switch-on and switch-off angles dictate the motor performance at different
bads and speeds. S k the switched reluctance motor inherently produces a pulse of torque
due to excitation of any single phase, the overall torque ripple depends on the me thod used
to cornmutate the mo tor. that is. to switch the torque-producing excitation fro m O ne phase
to the next.
For motoring operation the puises of phase current must coincide with a period of
increasing inductance. ia. when a pair of rotor poles is approaching alignment with the stator
poles of the excited phase. The timing and dweii of the current pulse are both important in
determining the torque, the eaiciency. and the smoothness of the operation. SRM
commutation has always k e n associateci with instantaneous rotor position. As a result. it is
apparent that the performance of an SRM drive system ( effective control of speed, torque
and torque pulsations ) depends strongly on accurate placement of the phase current pulses
relative to the machine rotor angle. Therefore. the rotor position information is essentid as
a feedback for the control section in order to generate the commutation signals.
Error in rotor position detection has a direct impact on the coinmutation positions,
and shifts the phase conduction periods. The accuracy of position information determines the
efkkncy and torque capabiiity as weii as torque pulsation of the drive. The high cost and low
reliabiky of discrete position sensors have led researchers to seek ways of eliminating these
sensors. In many industrial environments the position-sensing devices prove to be unreliable
due to eEects such as eiectromagnetic interference, high temperature, number of connections.
and d u t Therefore, the sensoriess techniques for switched reluctance motors must satisQ an
accuracy requirement in position estimation.
In thû study, the maximum conduction angle for each phase is 15 mechanical degrees
in order to get maximum torque with minimum npple. To examine the effect of switching
an& variation, a simulation test is carried out when the conduction angle is decreased by 2
degrees. As we can see Erom Figures 5.15 and 5.16 the reduction in conduction angle
increases npple in the motor speed and torque.
5-9-6 VSSRM Simulation
The same simulation tests as for the CSSRM case are carried out for the VS SRM case
based on the control scheme iIiustrated in Figure 5.3 ( the voltage source switched reluctance
motor ) to examine the dynamic performances of the switched reluctance motor. Figure 5.17
shows the sirnulated transient speed response, when the motor is ailowed to accelerate from
rest to a reference speed of 40 radlsec. with a load torque of 10 Nm. It c m be found that the
speed response is still quite good, that is. with a reasonable overshoot ( about 25% ) and
oscillations. The current hysteresis band is set at 0.5 A As before, see the CSSRM drive
performances. the sarne simulation tests under dEerent working conditions are carried out
and Figures 5.18 - 5.28 illustrate the corresponding system responses. In Figure 5.22 the
amount of overshoot is about 1 1.9 1 %, when the reference speed is changed from 40 radsec
to 80 rad/sec. Figure 5-17 shows an increase in the ripple and higher overshoot when
compared with Figure 5.5. As we can see from Figure 5.23 the larger values of moment of
inertia J reduces the npple. Figures 5.27 and 5.28 shows the effect of the switching angle
variation The reduction in conduction angle ùicreases ripple in the motor speed and torque.
Time (sec)
Figure 5.5 Speed response of the CSSRM drive for a step change in the reference speed.
Time (sec)
Figure 5.6 CSSRM torque produced when the motor accelerates from rest to the reference speed.
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Time (sec)
Figure 5.7 Speed error of the CSSRM drive for a step change in the reference speed.
oL 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
Time (sec)
Figure 5.10 S peed response of the CSSRM drive when the reference speed is doubled at t= O. 1 sec.
I 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Time (sec)
Figure 5.11 Speed response of the CSSRM drive when the moment of inertia J is doubled.
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Time (sec)
Figure 5.12 Speed response of the CSSRM drive when the phase resistance r is increased by 20%.
load torque in Nm
Time (sec)
Figure 5.13 Speed response of the CSSRM drive under varying load torque disturbances.
bad torque in Nm 10 ' v A h . r A -
v
O I 1 I I I 1 1 I 1 1
O 0.01 0.03 0.05 0.07 0.09 0.105 0.12 0.14 0.16 0.18 0.2 Time (sec)
Figure 5.14 Motor Torque response of the CSSRM drive under varying load torque dis turbances.
0.1 Time (sec)
Figure 5.15 S m response of the CSSRM drive when the conduction angle is decreased by 2 degrees.
Time (sec)
Figure 5.16 Torque response of the CSSRM drive when the conduction angle is decreased by 2 degrees.
0.08 0.1 0.12 Time (sec)
Figure 5.17 Speed response of the VSSRM drive for a step change in the reference s p d .
0.08 0.1 0.12 Time (sec)
Figure 5.18 VSSRM torque produced when the motor accelerates from rest to a reference speed-
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Time (sec)
Figure 5.19 Speed error of the VSSRM drive for a step change in the reference speed.
Time (sec)
Figure 5.20 Phase current waveform of the VSSRM drive when the motor accelerates fiom rest to a reference speed.
0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Time (sec)
Figure 5.22 Speed response of the VSSRM drive when the reference speed is doubled at t= O. 1 sec.
- loL 1 I 1 1 1 1 1 1 1 I O 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
Time (sec)
Figure 5.23 Speed response of the VSSRM drive when the moment of inenkt J is double..
- I O L O 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
Time (sec)
Figure 5.24 Speed response of the VSSRM drive when the phase resistance r is increased by 20%.
-
ioad torque in Nm
I
-
Time (sec)
Figure 5.25 Speed response of the VSSRM drive under varying load torque disturbances.
M load torque in Nm
Time (sec)
Figure 5.26 Motor Torque response of the VSSRM drive under varying load torque dis tur bances.
Figure 5.27 Speed response of the VSSRM drive when the conduction angle is decreased by 2 degrees.
- loL O 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
Time (sec)
Time (sec)
Figure 5.28 Torque response of the VSSRM drive when the conduction angle is decreased by 2 degrees.
CHAPTER 6
SUMMARY AND CONCLUSIONS
6.1 INTRODUCTION
Switched reluctance motors have gained increasing popularity in variable-speed drives
and have k n found competitive with traditional ac and dc drives becausr of their simple
structure, mggedness, 10 w cost, and the simpbty of the associated unipolar po wer converter.
There is a growing dernand for SRM drive systerns for such diverse applications as aerospace,
defense, robotics, traction and automotive, mining, oil and gas industries, and products such
as pumps. fans, and appliances. Moreover, the switched reluctance motor cm be designed for
low speed and high torque capability. These characteristics make the switched reluctance
mo tor attractive for direct-drive applications. Ho wever, its nonlinear elec tromechanical
characteristics and its periodic variation of inductance in each phase result in torque ripple and
hence speed ripple.
The control of the switched reluctance motor is not a perfected art and design of
controuer for the SRM is quite a chdenging problem. P-1 controuers are widely used for
motor control because they are robust and they give good performance at low cost. However,
control theory has advanced beyond conventional controllers and new techniques have ken
developed which can be applied to motor control In this concluding chapter, the research
work is Summarized, Avantages and disadvantages of P-1 controllers are explained, and some
suggestions for further research are presented.
6.2 SUMMARY OF THE THESIS
An introduction and a literature review of the switched reluctance rnotor were
presented in chapter 1. After reviewing the operation of the SRM and its difEerent control
strategies, a simple but reliable mode1 of the SRM, with the assumption of linear magnetic
characteristics, is used to simulate and evaluate the closed-loop performance of the speed
control system. This mathematical mode1 of the switched reluctance motor, which gives a
better appceciation of the operating characteristics and the control purposes for the machine,
was developed in chapter 2.
A speed controiler system based on a proportional-integral control strategy was
designed in chapter 3. Our control objective is to track a reference speed trajectory. A P-1
control law is chosen only because of its very good and desirable features and is not intended
to infer that this is the best control law. A feedback control system was employed for speed
control of the switched reluctance motor drive to provide the desirable performance. This
consists of a P-1 controiler which compares the actual speed of the drive and the required
speed signal provided by the user. The transient performance can be adjusted to satisfy the
system spe~~ca t ions by adjusting the controller constants K, and K,.
Performance is based on the ability of the control system to provide each phase with
pulses of current during the torque productive periods by selecting predetennined switching
angles. Stability of the control system using the second method of Lyapunov and Krasovski's
method was studied and investigated in chapter 4. Suficient conditions for global asymptotic
stability of the nonluiear system were derived.
The proposeci speed control system schemes ( CSSRM drive and VSSRM drive
control schemes ) were designed and simulated for steady-state operation and dynamic
performance under different working conditions. This was reported in chapter 5. The
performance and results were presented as well. Moreover. the relative robustness of the
control system was investigated by changing the system parameters and by applying extemal
dis turbances.
It was s h o w that even with sinplQing assumptions, the torque is a quadratic
huiction of the phase cucrent. The f d f o w a r d controkr, whkh translates the demand torque
T, into appropriate phase currents, consists of the toque-to-current transformation block and
the commutation system The hnction of the commutation system is to designate the
appro priate phases for torque development and commutation. In the feedfo rward contro 1
scheme employed in this study, a single phase is energized at a tirne. The "on" and "off'
angles are fixeci at €3, and ûd. The actual phase to be energized is determined by the rotor
position-
6.3 ADVANTAGES AND DISADVANTAGES OF P-1 CONTROLLER
Proportional-Integral controllers have been widely used in industry. In cornpetition
with more advanced controliers, P-1 controllers have generally been regarded as more
122
successhl in industrial applications .The main reasons for this have been simplicity. lower
cost. zero steady-state error. ease of implementation. robustness, accuracy, good speed of
response, and abiiity to achieve stability.
A P-1 controkr has a proportional as weii as an integral term in the forward path. The
integrai component of this controbr wi.ü ensure that the steady-state error is zero for a step
change. This is a very usehi feature for many applications. Although a P-1 controiIer makes
the steady-state error zero, the transient response rnay not be acceptable. For example, It
could increase the s e t h g tirne. This might not be tolerated in some high-performance
applications such as robotics. Nevertheless, a P-1 controller can be designed to respond very
quickly, but in this case an undesirable speed overshoo t may occur. Hence, one has to choose
a design suitable to a particular application.
P-i controkrs can fdter out high-frequency noise, because they are essentidy low-
pass füters. P-1 controliers are examples of good classical control design techniques. One
drawback is that they are strongly based on the dynamics of the system king controlled and
in some situations they may be sensitive to changes in the piant parameters andlor extemd
disturbances.
At the present tirne, many applications use P-1 control strategies for speed control.
despite some of its disadvantages. However, due to the high nonlinearity of the switched
reluctance motor. more advanced control iaws could be employed where supenor dynmic
performance is desired under aiI motor operating conditions.
6.4 SUGGESTIONS FOR FURTHER WORK
The potential for m e r research work in this area is vat. In this thesis. a simplified
SRM dynamic mode1 which neglects magnetic saturation was developed and simulated. The
model does not need to contain the £ine detaüs that would be required for design and
construction of the machine. For futther work, an SRM model which inchdes a nonlineu
magnetization characteristics can be developed. This model c m improve the accuracy,
particularly for high performance applications. The cost of higher precision is a substantial
increase in model complexity and in the simulation tirne.
Some high-performance industrial applications such as robotics and machine tools
require variable speed drive systerns which need to be relatively insensitive to parameter
variations andor extemal disturbances. They also require drive system that are highly
accurate. robust, and fast To provide such a high degree of performance, more advanced and
complex control strategies may be needed. These could be adaptive. hizzy logic-. or neural
networks-based control strategies which could be employed to make the system highly robust
and accurate.
In this study, the research involved modeking, system design, and simulation. The next
step would be to experimentally test these theoretical results. This experirnental approach
could also be an area for tùture research.
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APPENDM 1
MOTOR DATA AND PARAMETERS
7.5 k W
460v
32 A
8 A
1900 rpm
10 mH
110 m H
1 .O5 rad
0.35 rad
0.42 rad
0.95 R
132
0.0016 kg. rn2
0.004 Nmlradlsec
APPENDM II
DERIVATION OF THE CONDITIONS OBTAINED FROM
KRASOVSKI'S METHOD IN CHAPTER 4
(II. 1 )
According to Krasovski's theorem. if F ( x ) is negative d e f ~ t e . then the equilibrium state is
asymptoticaily stable. Sylvester's theorem yields the following conditions for F(x) to be
negative definite. A necessary and sufEicient condition for F(x) to be negative d e f ~ t e is that
the determinant of F(x) be positive if the system order is even and negative if system order
is odd. and that the successive principal minors of even order be positive and the successive
principal minors of odd order be negative; that is,
Since the phase inductances of the SRM are positive, the four above conditions are equivalent
to one condition, Le,,
As x, = o , therefore,
r + Kx, O - 2
4
Then inequality ( II.8 ) becornes as foiiows:
[ $ % - - Kr, =3 K KI4 =4 J B
yX4 - - -2-
- K x , r + Kx5
L2 1
LI
Hence,
Afier simplification. we get
This inequality may be rewritten as foiIows:
Since x, = ik. the above condition may be rewritten in the foiiowing form:
1 As Tk = - ~ i : , the condition determined by ( II. 10 ) may be rewtitten as:
2
where T, is the electromagnetic torque developed by the k th phase of the motor.
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