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CONTROL OF THE DOUBLY SALIENT PERMANENT MAGNET
SWITCHED RELUCTANCE MOTOR
David Bruce Merrifield
Thesis submitted to the faculty of the Virginia Polytechnic Institute and State
University in partial fulfillment of the requirements for the degree of
Masters of Science
In
Electrical Engineering
Dr. Krishnan Ramu, Chair
Dr. Douglas Lindner
Dr. William Baumann
May 4, 2010
Blacksburg, Virginia
Keywords:
Permanent Magnet Switched Reluctance Motor, PMSRM, SRM, Firing Angle
Selection Efficiency Based Control, Current Control, Speed Control, Average
Torque Control
CONTROL OF THE DOUBLY SALIENT PERMANENT MAGNET SWITCHED
RELUCTANCE MOTOR
David Bruce Merrifield
ABSTRACT
The permanent magnet switched reluctance motor (PMSRM) is hybrid dc motor which
has the potential to be more effect than the switched reluctance (SRM) and permanent magnet
(PM) motors. The PMSRM has a both a salient rotor and stator with permanent magnets placed
directly onto the face of common pole stators. The PMSRM is wound like the SRM and can be
controlled by the same family of converters. The addition of permanent magnets creates
nonlinearities in both the governing electrical and mechanical equations which differentiate the
PMSRM from all other classes of electric motors.
The primary goal of this thesis is to develop a cohesive and comprehensive control
strategy for the PMSRM so as to demonstrate its operation and highlight its efficiency. The
control of the PMSRM starts with understanding its region of operation and the underlying
torque production of the motor. The selection of operating region is followed by a both linear
and nonlinear electrical modeling of the motor and the design of current controllers for the
PMSRM. The electromechanical model of the motor is dynamically simulated with the addition
of a closed loop speed controller. The speed controller is extended to add an efficiency
searching algorithm which finds the operating condition with the highest efficiency online.
iii
Acknowledgements
Firstly, I would like to thank my advisor Dr. Krishnan Ramu for his continued support
and guidance. He has introduced me to the field of electric motors and drives and has allowed
me to work on controls problems which are both challenging and exciting. I would also like to
thank Nimal Lobo who designed the PMSRM which this thesis is based upon, and whose
assistance has been not only appreciated but also essential to my research. Thanks to Ramu Inc.
for their funding and support of my research. This includes Dr. Gray Roberson and Ethan Swint
who have helped me immensely over the past year on both this research plus much more.
Finally, I would like to thank my family for their unconditional support in all aspects of my life.
iv
Table of Contents
List of Figures………………………………………………………………………………...…...v
List of Tables……………………………………………………………………………………..vi
1 Introduction………………………………………………….………………………………….1
1.1 Introduction to the SRM and PMSRM………………………………………………..1
1.2 Thesis Proposal and Contributions…………………………………………………....4
2 SRM and PMSRM Background………………………………………………………………...6
2.1 Operation of the SRM………………………………………………………………...6
2.2 Operation of the PMSRM……………………………………………………………8
2.3 Converter Topologies for the SRM and PMSRM……………………………………11
2.3.1 The Asymmetric Converter………………………………………………...11
2.3.2 One Switch per Phase Converters………………………………………….12
2.3.3 Pulse Width Modulation………………………………………………...…13
2.4 Control of the SRM…………………………………………………………………..15
3 Control Principle for the PMSRM……………………………………………………………..17
3.1 Startup………………………………………………………………………………..17
3.2 Region of Operation………………………………………………………………….18
3.2.1 Effect of the Advance Angle……………………………………………….20
3.2.2 Effect of the Dwell Angle………………………………………………….21
3.2.3 Effect of Speed on the Firing Angles………………………………………23
3.2.4 Selection of Firing Angles…………………………………………….…...26
3.2.5 Sensitivity Analysis………………………………………………………..31
3.3 Control Overview………………………………………………………………........33
4 Current Control Design and Simulation……………………………………………………….37
4.1 Hysteresis Current Control…………………………………………………………..37
4.2 PI Current Control…………………………………………………………………...39
4.2.1 Linearization of the PMSRM Current Model……………………………...40
4.2.2 PI Control Design………………………………………………………….43
4.2.3 Anti-windup PI Control……………………………………………………46
v
4.2.4 PI Controller Simulation…………………………………………………...48
4.3 Adaptive Current Control……………………………………………………………50
4.3.1 PMSRM System Model with Structured Non-Linearites………………….51
4.3.2 MRAC Current Control……………………………………………………52
4.3.3 Adaptive Current Control Simulation……………………………………...54
5 Speed Control Design and Simulation…………………………………………………………60
5.1 Speed Loop Linearization……………………………………………………………60
5.2 Design of a Speed Feedback Filter…………………………………………………..63
5.3 PI Speed Controller……………………………………………………………….….65
5.4 PI Torque Controller…………………………………..……………………………..70
5.5 Efficiency Searching Algorithm……………………..………………………………71
5.6 Comparison of Speed Control Designs...…………………………………………….77
6 Conclusions……………………………………………………………………………….…....79
6.1 Summary…………………………………………………………………...……..….79
6.2 Future Research…………………………………………………………..………….80
References………………………………………………………………….……………………81
Appendix A: 8/10 4ecore PMSRM Specifications..…………………………………………......83
Appendix B: Deadbeat Current Controller Design….….………………………………..……...84
vi
List of Figures
Figure 1.1: Two-phase 8-10 4ecore SRM…………………………………………………………2
Figure 1.2: Two-phase 8-10 PMSRM………………………………………………………….….3
Figure 2.1: Flux Path of a 4ecore SRM………………………………………………………..….6
Figure 2.2: Operation of the SRM………………………………………………………………...7
Figure 2.3: Flux Path of the PMSRM………………………………………………………….….8
Figure 2.4: Operation of the PMSRM………………………………………………………….…9
Figure 2.5: The Asymmetric Bridge Converter………………………………………………….12
Figure 2.6: The Split-dc Converter………………………………………………………………13
Figure 2.7: PWM Chopping of the SRM………………………………………………………...14
Figure 3.1: PMSRM Cogging Torque …………………………………………………………..18
Figure 3.2: Torque Profile of 4ecore PMSRM………..................................................................19
Figure 3.3: Simulated Current and Torque with Selected Advance Angles………………….…20
Figure 3.4: Simulated Average Torque as a Function
of Advance Angle and Reference Current……………………………………………….21
Figure 3.5: Simulated Current and Torque with Selected Dwell Angles………………………..22
Figure 3.6: Simulated Average Torque as a Function of Dwell Angle and Reference Current...23
Figure 3.7: Simulated Current and Torque for Varying Speeds……………………………...…24
Figure 3.8: Simulated Average Torque as a Function of Dwell Angle and Speed………………24
Figure 3.9: Simulated Average Torque as a Function of Advance Angle and Speed…………...25
Figure 3.10: Simulated Average Torque with Set Speed and Current Command………………26
Figure 3.11: Maximum Average Torque………………………………………………………...28
Figure 3.12: Transformation of the Torque Table……………………………………………….29
Figure 3.13: Lookup Tables for the Dwell and Advance Angles……….…….………………....30
Figure 3.14: Current Waveform with Varying Dc-link Voltage…………………………………31
Figure 3.15: Average Torque as a Function Dwell Angle with Variable Dc-link Voltage……...32
Figure 3.16: Average Torque as a Function Advance Angle with Variable Dc-link Voltage…..33
Figure 3.17: Hardware Overview………………………………………………………………..34
Figure 3.18: General Two Phase PMSRM Control Block Diagram……………………………..35
Figure 3.19: Average Torque Control Block Diagram…………………………………………..36
Figure 4.1: Hysteresis Controller with Asymmetric Converter………………………………….38
vii
Figure 4.2: Simulation of the Hysteresis Current Controller……………………………………39
Figure 4.3: Self Inductance of the PMSRM ……………………………………………………..41
Figure 4.4: Block Diagram of the Linear 2-phase PMSRM Current Controller………………………….43
Figure 4.5: Root Locus of the Linear Electrical Model of the PMSRM…………………………44
Figure 4.6: Small Signal Step Response of the PI Controller …………………………………...45
Figure 4.7: Anti-windup PI Current Controller Block Diagram…………………………………47
Figure 4.8: Simulated Current Response with Anti-Windup PI Current Control ……………….47
Figure 4.9: Simulated phase current and voltage with 6A command……………………………49
Figure 4.10: Simulated Phase Current at 3600rpm……………………………………………....50
Figure 4.11: Block Diagram of One Phase of the PMSRM
Model Reference Adaptive Controller……………………………………..……51
Figure 4.12: Comparison of Actual Nonlinearities to Matched Nonlinearities…………………55
Figure 4.13: Adaptive Current Control Simulations at 1600rpm with 𝑖∗ = 7𝐴……………..…..56
Figure 4.14: Simulated Adaptive Gain Convergence at 1600rpm…………………………...…..57
Figure 4.15: Adaptive Current Control Simulations at 3600rpm with 𝑖∗ = 7𝐴……………...….58
Figure 5.1: Open-loop Small Signal PMSRM Mechanical Model………………………....……62
Figure 5.2: IIR Speed Feedback Filter……………………………………………………….…..64
Figure 5.3: Speed Filter Frequency Response………………………………………………...…63
Figure 5.4: Closed-loop Speed Control Block Diagram………………………………….…...…64
Figure 5.5: Step Response and Root Locus of the PI Compensated Mechanical System….…....67
Figure 5.6: Dynamic Closed Loop Speed Control Simulation with Disturbance Inputs………..68
Figure 5.7: Dynamic Closed Loop Speed Control……………………………………….….…..69
Figure 5.8: Closed-loop Torque Control Block Diagram with Firing Angle Lookup………......70
Figure 5.9: Dynamic Closed Loop Torque Control Simulation with Disturbance Inputs….…...71
Figure 5.10: Efficiency Searching Algorithm Flow Chart…………………………………..…..73
Figure 5.11: Frequency Response of the Power Averaging Filter…………………………….…75
Figure 5.12: Simulation of the Efficiency Searching Algorithm……………………………...…76
viii
List of Tables
Table 4.1: Nominal Inductance values for the 2Hp PMSRM……………………………………42
Table 4.2: Nominal Req Values for the 2Hp PMSRM…………………………………………...43
Table 4.3 Adaptive Parameters for simulation at 1600rpm with 𝑖∗ = 7𝐴……………………….56
Table 4.4: Control Parameters for Adaptive Control at 3600rpm with 𝑖∗ = 7𝐴…………..…….58
Table 5.1: Selected Values of the EMF Constant………………………………………………..63
Table 5.2: Average Cogging Flux ………………………………………………………….……63
Table 5.3: Speed Filter Parameters………………………………………………………………64
Table 5.4: Motor Efficiency of the Different Control Schemes…………………………………77
Table A.1: Dimensions of the 4ecore PMSRM……………………………………………….....83
1
1 Introduction
1.1 Introduction to PMSRM
The switched reluctance motor (SRM) has seen growing interest in high volume
commercial and industrial markets for variable speed motors. The basis for the SRM originated
in the 1850s but was not implemented until 1969 when S.A Nasar proposed that a dc SRM is
“practicable to develop” [1]. His proposals have become more viable due to drastic advances in
both power switching devices and electronic controllers. Since then SRM development has
steadily improved; they can now be found commercially in drives as small as printer servos all
the way to 40-kW compressor drives [2].
Commercially, there are many benefits of the SRM compared to other variable speed
motors. The simple shape of the rotor and stator as well as the simple windings applied lends
itself to inexpensive mass production. By design, SRMs require fewer raw materials, due to
shorter stack lengths and more compact windings while still offering comparable power density
to induction and permanent magnet motors. As with other variable speed motors the SRM
requires an electronic power converter but can require as little as one IGBT and one diode per
phase, compared to a variable speed induction or permanent magnet motor, which need at least
twice as many switches.
The focus of past and present research of SRMs deals with some inherent disadvantages
which must be solved to insure the commercial success of the SRM. Firstly, control of an SRM
is a non-trivial task due to the non-linear inductance and torque profiles from the varying air gap
between the rotor and stator poles. Also, the absolute position of the rotor is necessary for phase
excitation and commutation. In most cases this requires additional hardware in the form of a
position sensor, which can either be magnetic or an optical encoder. Additionally, the SRM
inherently produces a large amount of acoustic noise which can be of great concern in many
commercial and industrial applications.
The SRM has salient poles on both the stator and rotor, with dc windings on the stator but
no magnets or windings on the rotor. The numbers of rotor and stator poles, as well as the
number of phases of the machine are central criterion of the motor design process. The motor is
operated by exciting a phase of the stator, which causes the rotor to come to an aligned position.
2
At that point, the other phase(s) are out of alignment. By then commutating the first phase and
exciting a subsequent phase the rotor will move to an aligned position with the new phase.
When this sequence is properly orchestrated the rotor will spin at a continuous rate generating
torque. [2]
A two-phase SRM with a common pole e-core structure is presented in [3] and [4], and
has been shown to have reduced amounts of steel and copper compared to other SRMs while
increasing the power density and overall efficiency. The e-core SRM stator is comprised of
sections with three poles. The two outer poles have the windings for each of the phases, while
the middle pole (common pole) has no windings and is shared between both phases. This
structure can be used to create a segmental stator with two independent e-cores. Another
structure would be to have a single stator comprised of poles with phase windings alternated with
larger common poles. The e-core design allows for shorter flux paths which allows for reduction
in copper wire and core loss which in turn yield higher efficiency. A two phase e-core SRM
with 8 stator poles and 10 rotor poles is shown in figure 1.1.
Figure 1.1: Two-phase 8-10 4-eore SRM (© Krishnan Ramu)
An ac motor called the duel stator doubly salient permanent magnet motor is presented in
[5] and [6], which has a similar structure to an SRM with segmental stator separated by a pair of
magnets. This machine is shown to have higher torque density and efficiency than other ac
3
motors. It is controlled as an ac machine; therefore it only gets positive torque from the magnets
because the net reluctance torque contribution is zero.
The doubly salient permanent magnet switched reluctance motor (PMSRM), has been
proposed [7], [8] to allow for the torque production of a SRM with the addition of PM torque.
This machine has the same mechanical structure as the segmental e-core SRM with magnets
placed along the face of the common stator poles. The PMSRM is a dc motor with torque and
inductance properties similar to that of the SRM which allow the motor to be controlled with any
drive used to power an SRM. Although, there have been no attempts at controlling the PMSRM,
its similar structure and characteristics allow many of the control techniques used for the SRM to
be adapted to its control.
Figure 1.2: Two-phase 8-10 PMSRM (© Krishnan Ramu)
The increased efficiency given by these magnets comes with additional difficulties in
construction, modeling and control of the motor. The rotor must be carefully inserted as to not
touch the brittle on the surface of the stator poles which could easily shatter. As with the SRM,
the torque of the PMSRM is a nonlinear function of rotor position and current but with the
additional problem of having a non-uniform zero crossing. That is, for each current there is a
unique point at which the positive torquing region begins and ends. The flux of the machine is
the sum of the inductive flux and the magnetic flux which adds additional non-linearity to the
4
machine model. Starting up the PMSRM also can also be difficult since the rotor will be fixed in
one of four possible places due to the magnetic attraction while the machine is at rest.
1.2 Thesis Proposal and Contributions
The central goal of this thesis is to design, simulate and implement a control strategy for
the PMSRM. The region of operation, specified by the motors firing angles, will be analyzed for
torque, speed, and efficiency, all of which are critical measures of performance for a variable
speed motor. Linear and non-linear techniques will be used to analyze the current-voltage
relationship and will be used to design three current controllers for the PMSRM. A linear
mechanical model of the PMSRM will be used to design a speed filter and controller to allow for
variable speed operation. The closed loop speed controller will be augmented with a self-tuning
efficiency controller that will optimizing the firing angles of the motor to find the most efficient
operation at any particular load and speed.
The contributions of this thesis are:
Analysis, simulation and verification of the operating region of the PMSRM based on its
relationship to average torque, speed and efficiency of the motor. A control strategy is
developed to maximize average torque.
Design of a current controller for the PMSRM using both linear and structured non-linear
modeling of the motors electromagnetic equations. This includes the design of a gain
scheduled anti-windup PI controller and model reference adaptive controller.
Design of a speed controller for the PMSRM based on an original linear mechanical
model of a doubly salient permanent magnet motor which accounts for reluctance plus
magnetic torque contributions.
A self-tuning efficiency algorithm for any SR motor which finds the most efficient
combination of firing angles online based on real time feedback in a speed controlled
system.
The simulations in this thesis are based on a 2-HP two phase 4e-core PMSRM motor
designed in N. Lobo’s PhD dissertation [11]. The torque and inductance data was extracted from
5
FEA simulations. The rated operating point of this motor is 3600rpm with a 3.8 Nm load
therefore this operating point will be the primary concern in simulation. The relevant parameters
of this motor are attached in appendix A.
6
2 SRM and PMSRM Background
From a mechanical and a control standpoint the SRM and the PMSRM have many
similarities. Although they have different electrical and mechanical models, both have similar
overall structures, torque production and use the same electronic power converters. Two
common SRM converter topologies are presented for the PMSRM, each of which has specific
advantages. In addition, control methods for the SRM which may be applicable to the PMSRM
are investigated.
2.1 Operation of the SRM
The basic magnetic structure of a 4ecore SRM is shown in figure 2.1. When a phase is
excited, the magnetic flux moves through the rotor pole into the stator pole around to the
common pole and back into the rotor. With an e-core structure the common poles of the SRM
are used to shorten the length of the majority of the flux path. The shorter path results in lower
core losses compared to a traditional SRM structure in which the flux path would travel through
the rotor from one excited stator pole to the other.
Figure 2.1: Flux Path of a 4ecore SRM (Adapted from [4])
When the rotor pole and stator pole are unaligned almost all of the flux is through the air
gap resulting in a minimum inductance value. As the poles overlap the flux path is through the
7
rotor pole into the stator and back through the common pole. In this region the inductance
increases as the two poles move closer to alignment. As the inductance reaches its maximum the
torque production becomes zero. The current through the winding is then turned off during this
region so that negative torque will not be created once the rotor continues to move. As the rotor
pole moves past the stator pole, the slope of the inductance becomes negative and if current is
applied then negative torque is produced.
Figure 2.2: Operation of the SRM
From [2], the electrical model of one phase of an SRM is given by:
𝑣 = 𝑅𝑠𝑖 +𝑑𝜆 𝜃, 𝑖
𝑑𝑡
(2.1)
Where v is the voltage applied across the windings, 𝑅𝑠 is the resistance of the
windings, 𝑖 is the current through the windings and 𝜆 is the flux linkage of each phase, which is
equivalent to the product of the current and the inductance.
𝐿𝑎
𝑖𝑎
Λ𝑎 Λ𝑏
Flux Linkage
Phase A
Rotor Position
𝐿𝑏
𝑖𝑏 Phase B
Torque
8
𝜆 = 𝐿 𝜃, 𝑖 𝑖
(2.2)
where L is inductance. The mechanical model of the SRM is defined by its torque production,
which is given by:
𝑇𝑒 =1
2𝑖2
𝑑𝐿 𝜃, 𝑖
𝑑𝜃
(2.3)
The torque is a function of the current squared which allows for positive torque
regardless of the current polarity allowing for simple converter design. Negative torque occurs
when the inductance has a negative slope.
2.2 Operation of the PMSRM
The stator of the PMSRM is the same as that of the SRM with the only difference being a
small amount of steel removed from the face of the common poles which is replaced by a
permanent magnet. All four of the magnets are placed with the same magnetic direction opposite
to that of the windings. Placing the magnets on the stator poles allows for the construction of a
solid back-iron which is much easier for manufacturing than other doubly salient permanent
magnet designs. The windings of the PMSRM produce unipolar current, thus even under a
winding fault the magnet will not be demagnetized.
(a) (b)
Figure 2.3: Flux Path of the PMSRM (a) Phase A aligned (b) Phase B aligned (Adapted from
[11])
9
The when a winding is excited the flux path of the PMSRM goes through the stator pole,
through the aligned rotor pole. The flux then splits to the two adjacent rotor poles and moves
into each of the adjacent common poles, returning through the active pole. As the rotor pole
passes an excited stator pole the winding flux and inductance begins to rise, as does the flux
contribution from the PM. The inductance reaches its maximum at the aligned position, and then
begins to decrease.
Figure 2.4: Operation of the PMSRM
As with an SRM, the PMSRM is designed to have as little mutual inductance between
phases as possible. With this assumption, the instantaneous torque produced by a two phase
PMSRM is:
𝐿𝑎 𝐿𝑏 Self
Inductance
𝜆𝑎
𝜆𝑝𝑚−𝑎
𝑖𝑎
Λ𝑎 Λ𝑏
Flux Linkage
Phase A
Rotor Position
𝜆𝑏
𝑖𝑏 Phase B
𝜆𝑝𝑚−𝑏
Torque
10
𝑇 =1
2𝑖𝑎 ,𝑏
2𝜕𝐿𝑎 ,𝑏
𝜕𝜃+ 𝑖𝑎 ,𝑏 𝑖𝑝𝑚
𝜕𝐿𝑝𝑚 𝑎 ,𝑏
𝜕𝜃+
1
2𝑖𝑝𝑚
2𝜕𝐿𝑝𝑚
𝜕𝜃
(2.4)
The first term is the torque produced by the self inductance of each winding. This is
represented as either phase a or phase b which assumes that only one phase is producing torque
at a time. The second term is the magnet torque, which is a function of the winding current, the
equivalent current through the permanent magnet, 𝑖𝑝𝑚 , and the change in inductance of the
magnet with respect to the active phase. The third term is the cogging toque, and is a function of
only the rotor position.
The equivalent current of the PM times the position derivative of the PM with respect to
each phase is equal to the position derivative of the PM’s flux, shown as:
𝑖𝑝𝑚𝑑𝐿𝑝𝑚 𝑎 ,𝑏
𝑑𝜃=
𝑑𝜆𝑝𝑚 𝑎 ,𝑏
𝑑𝜃
(2.5)
The cogging torque of one electrical cycle must be zero; therefore, when considering the
actual torque of the machine this term can be ignored. Considering the zero effect of torque and
substituting equation 2.5 into 2.4 results in the following equation for torque of the DSPSRM:
𝑇 =1
2𝑖𝑎 ,𝑏
2𝑑𝐿𝑎 ,𝑏
𝑑𝜃+ 𝑖𝑎 ,𝑏
𝑑𝜆𝑝𝑚
𝑑𝜃
(2.6)
The general electrical model for one phase of the PMSRM is the same as the SRM’s
electrical model given in 2.1. However the flux of the PMSRM is the sum of the reluctance flux
and the flux of the PM:
𝜆 𝜃, 𝑖 = 𝑖𝐿 𝜃, 𝑖 + 𝜆𝑝𝑚 𝜃
(2.7)
The result of combining equations 2.1 and 2.7 is the complete electrical model for one
phase of the PMSRM, which is:
𝑣 = 𝑅𝑖 + 𝑑𝑖𝐿 𝜃, 𝑖
𝑑𝑡+
𝑑𝜆𝑝𝑚 𝜃
𝑑𝑡
(2.8)
The electrical and mechanical models of the PMSRM are very similar to their SRM
counterparts; however in both cases they have an additional effect from the PM. In the
11
mechanical equation the additional nonlinearity comes from the EMF of the PM, which is a
function of the change in inductance times the phase current. For the electrical model, the
voltage term is a function of the change in PM flux with respect to time. Overall control
strategies as well as linear techniques used for SRM control design must be modified to account
for these additional terms.
2.3 Converter Topologies for the SRM and PMSRM
As opposed to induction and other ac motors, the SRM and the PMSRM have
unidirectional voltage from a dc voltage source, typically coming from a rectified ac source.
Any converter used for the SRM can be used for the PMSRM. The converters can range in cost
and functionality, and should be selected based on the application. This section presents two
converters, the asymmetric bridge and the split-dc, both of which are well suited for the control
of the PMSRM. The pulse width modulation (PWM) technique is also presented for sustaining
average current with current control and can be used with either converter type.
2.3.1 The Asymmetric Converter
Shown in figure 2.5, the asymmetric bridge converter is a specialized controller designed
for the SRM. The asymmetric converter has independent phase control which can be
implemented with any number of phases. This converter also allows freewheeling operation, or
the ability to apply zero volts, and can recover mechanical energy with regeneration. Since the
asymmetric converter has two switches and two diodes per phase, it is typically limited to high
power and high performance applications. However it is a suitable choice for the initial
development of the PMSRM.
12
Vdc
D2
D1
Phase A
T2
T1
+
-
+
-
Va
D4
D3
Phase B
T4
T3
+
-
Vb
Figure 2.5: The Asymmetric Bridge Converter
When switches T1 and T2 are turned on the full dc bus voltage is applied to phase A
causing the current in the windings to rise. When the current in phase A is to be commutated,
both switches are then turned off. Since the current remains in the same direction both diodes
D1 and D2 become forward biased and the negative bus voltage is seen across the winding,
creating a rapid decrease in current. While the switches are turned off, the current is circulating
back through the voltage source, which is called regeneration. In addition, while the current is
high, switch T2 can be turned off, causing D1 to be forward biased and giving zero volts across
the windings. Thus the asymmetric converter has three degrees of freedom, and can command a
voltage of ±𝑉𝑑𝑐 or 0V. For most applications the PMSRM will need an ac input, necessitating
the addition of a full bridge rectifier to the converter.
2.3.2 One Switch per Phase Converters
From an electronics standpoint, one of the largest advantages of the PMSRM and the
SRM over other variable speed drives is one switch per phase converters. While two switch per-
phase converters, such as the asymmetric converter, are good for high performance applications
as well as laboratory research and testing, high volume commercial markets demand
configurations with less switches. For mass produced, low cost motor drives the number of
switches can make a significant difference in the overall cost of the system. There are a few
13
possible implementations for one switch per phase, one choice being the split dc supply
converter.
Vdc/2
Vdc/2
D2
D1
Phase B
Phase A
T2
T1
Figure 2.6: The Split-dc Converter
When switch T1 is on, diode D2 is reverse biased and current flows through the winds of
phase A with half of the dc bus voltage applied across its windings. When T1 is off current
flows through the phase A windings, through D2 and back into the second capacitor,
regenerating it. This converter configuration can be augmented with a split leg rectifier to allow
for an ac input source (split-ac converter) which has even less devices than a full bridge rectifier;
however, the capacitor size must be larger for this converter to maintain equal voltages across
both phases.
2.3.2 Pulse Width Modulation
One of the most common methods for current regulation is using the PWM method to
apply an average voltage to each phase, which is maintained by the duty cycle of the power
device. PWM control allows the implementation of controllers which command a voltage
between +Vdc with a duty cycle of one and –Vdc with a duty cycle of zero. A typical control
scheme for the SRM would be to apply full voltage when the inductance is rising to reach the
14
desired current level. The current is regulated at the commanded current for the torque
producing region using PWM chopping. When the inductance begins to decrease, full negative
voltage is applied to bring the current to zero as quickly as possible.
Figure 2.7: PWM Chopping of the SRM
For each cycle the PWM scheme is defined as:
𝑣 = −𝑉𝑑𝑐 0 ≤ 𝑡 ≤ 𝑑𝑇𝑠𝑉𝑑𝑐 𝑑𝑇𝑠 ≤ 𝑡 ≤ 𝑇𝑠
(2.9)
Where 𝑑 is the duty cycle and 𝑇𝑠 is the PWM period. The performance and efficiency of
the PWM switching scheme are both directly related to the PWM frequency. As the frequency
increases the output current ripple, Δ𝑖 , decreases as does the efficiency of the converter since the
total number of turn-on and turn-off losses increase. PWM chopping is vital for the
implementation of current regulators, such as the PI controller presented later, in both the SRM
and the PMSRM. In addition, PWM chopping allows for increased efficiency and control
compared to other methods of current control, such as hysteresis (presented in section 4).
I
d
V
1
0
.5
i*
Vdc
-Vdc
0
Ts
2∆i
15
2.4 Control of the SRM
With the success of SRMs over the past 30 years their control has become a well
established area. The control can be broken into three general subcategories which are not
necessarily separate but highlight distinct areas within the general control structure. Current or
voltage control directly manages the power applied to the motor and is based on the electrical
model of the motor. The time in which voltage is applied to the windings is determined by the
firing angles which control the power available to the machine as well as its efficiency. Torque
or speed control is used to create a closed loop control to regulate the speed of the motor.
The combination of when each phase is turned on or off plus the current reference and the
speed of the motor determine how much torque can be produced by the motor. Any given load
or speed may have any number of combinations of firing angles. Therefore, selection of these
angles can be used to additionally control the efficiency of the motor. In [9] exhaustive
simulations are used to map the optimal firing angles to maximize the efficiency and torque of an
8/6 SRM. In [10], Gribble develops specific formulas to calculate firing angles so as to conserve
energy by maximizing the torque output while minimizing input power through current and
voltage control. Using only the position of mechanical overlap, and the aligned and unaligned
inductances, a general equation for optimal firing angle selection is given. This method uses
general inductance relationships as opposed to exhaustive simulation of exact parameters. The
work of Gribble is expanded in [11] where a “firing angle calculator” is presented to select
angles based only on speed, dc-link voltage, reference current and the aligned and unaligned
inductance. The firing angle calculator is augmented by an efficiency optimizing algorithm
which varies the turn-off angle to obtain the optimal efficiency. In [12] the turn-off angle is
optimized to a curve-fit model while the turn-on angle is adjusted to place the peak of the phase
current at the position where the inductance begins to rise. In all cases, efficiency is maximized
by finding critical points in the relationship between when, how long and how much current is
fired into each phase and their respective output torque and speed.
Current control is an integral part of the overall SRM control system, and developing
high performance current regulators is critical to operation as well the implementation of other
more complicated control systems such as efficiency based control. Hysteresis control is a
simple and effective low performance control which is accepted in industry do to its ease of
16
implementation. Linear PI control design is a well established industrial norm for current
control. However linear models of the SRM will vary with any change in firing angle, speed and
current reference itself which indicates decreased performance during variable speed operation.
In [13] a modified anti-windup PI controller was introduced which used linear gain scheduling
based on current and speeds to improve the overall performance. Also, nonlinear control
strategies, such as model reference adaptive control, have been demonstrated as effective current
control strategies [14]. Similarly, adaptive control has been implemented on ac drives including
the SRMs ac brother, variable reluctance motor [15] and the synchronous permanent magnet
motor. [16].
The torque and speed control methods presented in [2] keep set firing angles while using
current reference as the speed controller’s command which is a single-input single-output
system. For most applications of SRMs the speed control can be relatively low performance in
which case a PI controller will provide sufficient transient response without steady state error. In
any cases, the performance of the speed controller is directly linked to the response of the current
controller.
17
3 Control Principle of the PMSRM
The operating performance of the PMSRM is highly dependent on when the phase
currents are turned on and off which means an absolute knowledge of the rotor position is
required. In addition, the amount of current allowed into each of the windings has a significant
effect on the operation of the motor, and it’s control is also required. The choice of these three
control variables affect the speed, torque, efficiency and the acoustic noise produced by the
motor. Generally, a given operating point with a set load and speed is desired to operate the
PMSRM as efficiently as possible with the least amount of acoustic noise; however, the exact
relationship between the control variables and the efficiency is not very straightforward. In
addition, the acoustic noise is greatly affected by the motor design leaving only so much room
for improvement from a controls standpoint.
The relationships between the controls and the performance of the motor can be
quantified through extensive simulation. In this thesis, two types of dynamic simulations are
used to model the PMSRM. In this chapter and the following chapter, the speed of the motor is
set to a constant rate and the torque produced is measured through a torque lookup table
generated from FEA simulations, which is a function of rotor position and current. The current
is calculated from the governing electrical equation 2.8 where the flux and inductance are looked
up from a separate FEA generated table. This type of simulation allows measuring what the
average torque output of the motor will be while easily varying the control parameters as well as
the speed. The second type of simulation, used in chapter 5, is a dynamic speed simulation. In
this case, the speed is a function of the mechanical model of the motor, which includes the torque
computed from the lookup.
3.1 Startup
For an SRM startup is an important aspect of operation. If the position is known, the
current controller with PWM chopping can apply enough torque to initially spin the rotor. If the
starting position is unknown, firing and holding one phase current will move the rotor to a
known aligned position. From there, the mechanical sensor or position observer can then
compute the position of the rotor with respect to each electrical cycle and the startup can
continue as normal.
18
Figure 3.1: PMSRM Cogging Torque
Unlike the SRM, the PMSRM has cogging torque as a result of the PMs on the stator.
Under no load the rotor will be locked in any one of four positions. However with any sort of
external cogging forces the position will not necessarily be one of the four cogging positions.
For the sake of simplicity the position at anytime will be considered known by a position sensor,
but will not be assumed to be in one of the cogging locations. The reluctance torque of the
PMSRM is significantly larger than the torque generated from the PM allowing for startup in the
same fashion of the SRM.
3.2 Region of Operation
In the most basic sense, the operation of the PMSRM is determined by when, how long,
and how much current is applied to each phase and the resulting torque created by the motor.
Torque of a PMSRM is a nonlinear function of current and rotor position which can be
represented as three distinct control inputs. The first is the advance angle, 𝜃𝑎 , which designates
how many mechanical degrees prior to a set point the current is excited. The second control
variable is the dwell angle, 𝜃𝑑 , which indicates how long excitation lasts before the current is
commutated. Together, the advance and dwell angle are referred to as the machines firing
0 5 10 15 20 25 30 35 -1
-0.5
0
0.5
1
1.5
2
Rotor Position (deg)
To
rqu
e (N
m)
19
angles. The third input is the current reference, 𝑖∗, which must be controlled with an added
current regulator.
Figure 3.2: Torque Profile of 4ecore PMSRM
All three control inputs have significant contributions to the operation of the motor and
often under-constrain operating points. An under-constrained system means that multiple
combinations of inputs can result in the same torque and speed output. In order to select the best
control inputs additional constraints must be considered; including the efficiency and acoustic
noise produced. In certain cases the current controller may be omitted to allow for single pulsing
operation, and controlled solely by the firing angles. However for a speed regulator the current
command is usually used as the single control output variable. This allows for the firing angles
to be determined offline based on the maximum torque production. Another option, which is
shown later in this thesis, is to use a real time algorithm to select the firing angles to maximize
efficiency.
3.2.1 Effect of the Advance Angle
0 5 10 15 20 25 30 35 -20
-15
-10
-5
0
5
10
15
20
Rotor Position (deg)
To
rqu
e (N
m)
a
d
22A
0A
20
The advance angle determines when the phase is excited relative to the rotor position,
which in turn affects the current rise as well as the torque produced by motor. The ratio between
torque produced and amp seconds of excitation is directly related to the efficiency of the
machine. The resulting tradeoff between torque produced and efficiency must be maximized for
ideal operation. The advance angle should be chosen to produce the necessary torque while
minimizing the total time of excitation however as the advance angle is minimized the current
reference must be increased
Figure 3.3: Simulated Current and Torque with Selected Advance Angles (𝜃𝑑 = 10°, 𝑖∗ =
6𝐴, 𝜔𝑟 = 3600𝑟𝑝𝑚)
Figure 3.3 shows the simulated effect of different advance angles on the torque
production of the PMSRM. For this simulation the speed, dwell angle and current reference
were kept constant. The current was regulated by an ideal current controller and the torque was
measured as the average over one mechanical revolution of the rotor.
A small advance angle prohibits the current from reaching the current reference. The
larger advance results in excitation through a region of increased inductance which impedes the
current rise. With the increased advanced angle the current rise is accelerated in addition to the
time in the positive torque region increased. However with too large of an advance, as in 3.3(c),
negative torque occurs at the beginning of excitation which negates any additional torque
produced by a faster rise and decreases the average torque production.
0
5
10
0 10 20 30 40 -10
-5
0
5
10
θa = 10
θ (deg)
(c)
0
5
10 θa = 5
0 10 20 30 40 -5
0
5
10
θ (deg)
(b)
0
5
10
ia
0 10 20 30 40 -5
0
5
10
θ (deg)
Ta
θa = 2
(a)
21
Figure 3.4: Simulated Average Torque as a Function of Advance Angle and Reference
Current(𝜃𝑑 = 10°, 𝜔𝑟 = 3600𝑟𝑝𝑚)
The effects of an increasing advance angle with different reference currents are shown in
figure 3.4. The peak torque production happens with an advance angle similar to figure 3.3(b) in
which the excitation begins prior to any negative torquing. In addition, the advance angle must
increase to maintain maximum average torque as the current reference rises or else the reference
is not achieved indicated in figure 3.4 when the same average torque is produced regardless of
the current references until the advance angle is increased.
3.2.2 Effect of the Dwell Angle
The dwell angle controls how long the current is maintained at the reference level. As
the rotor leaves the primary torque production region the current needs to be commutated in
order to prevent excitation in the negative torque region. As the dwell angle increases the
inductance also increases. In all cases, it is important to have the phase current completely
commutated at a position no further than the peak inductance. If this is not done, then the motor
will create negative torque.
0 1 2 3 4 5 6 7 8 9 10 1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
Avera
ge T
orq
ue
(N
m)
Advance Angle, a , (deg)
12A
9A
6A
5A
11A
8A
7A
10A
22
Figure 3.5: Simulated Current and Torque with Selected Dwell Angles(𝜃𝑎 = 5°, 𝑖∗ = 6𝐴, 𝜔𝑟 =
3600𝑟𝑝𝑚)
Simulated results of increasing dwell angles are shown in figure 3.5. For small dwell
angles, negative torque is produced from the magnet as the current comes to zero, then it rises
back as the magnet provides its positive torque. Keeping the dwell on for too long will enter the
negative torquing region as in figure 3.5 (d). Therefore, the dwell should be long enough to
prevent negative torque contribution from the PM but not so long as to produce negative
reluctance torque. In addition, the positive contribution of the PM is at the end of the
commutation period. Leaving current during this time increases the number of amp seconds into
the windings while getting only a minimal increase in torque production.
0 5
10 θd =2
ia
10 20 30 40 -5
0
5
10
10 20 30 40 10 20 30 40
θd =5 θd =10 θd =15
10 20 30 40
Ta
θ (deg) θ (deg) θ (deg) θ (deg)
(a) (b) (c) (d)
23
Figure 3.6: Simulated Average Torque as a Function of Dwell Angle and Reference
Current(𝜃𝑎 = 5°, 𝜔𝑟 = 3600𝑟𝑝𝑚)
As the dwell angle is increased there is a noticeable peak in torque production which
varies with current reference. For larger current references maximum torque output is achieved
with smaller dwell angles because it will take a longer amount of time to decrease the current.
On the other hand, small current references maintain maximum output from larger dwell angles
which allow for longer positive torque production. The relationship between the dwell and the
average torque appears to be linear for a constant speed.
3.2.3 Effect of Speed on the Firing Angles
The speed of the motor is not an implicit control variable; however, like the firing angles,
it has an impact on the rate of change of the current which in turn has an effect on the torque
produced each phase. Since the dynamic operation of the motor will occur with a functioning
closed loop speed controller, it will be assumed that the reference speed is the same as the actual
speed. Generally, higher speeds limit the rise of current and require a larger advance angle to
produce torque equivalent to that of lower speeds. Figure 3.7 shows the simulated current and
torque of the PMSRM for set firing angles and a constant current reference.
5 6 7 8 9 10 11 12 13 14 15 1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
Dwell Angle, d , (deg)
12A
10A
9A
11A
8A
7A
6A
5A
Av
erag
e T
orq
ue
(Nm
)
24
Figure 3.7: Simulated Current and Torque for Varying Speeds (𝜃𝑎 = 5°, 𝜃𝑑 = 10°, 𝑖∗ = 6𝐴, )
At low speeds the current waveform has a sharp rise and is maintained at the reference
level for the positive torque region. As the speed increases, the rise is not as sharp, indicating
that the advance angle must be increased for higher speeds. The decline of current after
commutation in the 1200rpm simulation is so rapid that it begins to produce negative torque.
Thus a longer dwell is required at lower speeds to ensure that no negative cogging torque is
produced.
0 2 4 6 8 10 12 14 16 18 20 0
0.5
1
1.5
2
2.5
3
3.5
1200rpm 1600rpm 2400rpm
3000rpm 3600rpm
Dwell Angle (deg)
Av
erag
e T
orq
ue
(Nm
) 0
5
10
10 20 30 40 -5
0
5
10
10 20 30 40 10 20 30 40
𝜔𝑟 = 1200 𝑟𝑝𝑚 𝜔𝑟 = 2400 𝑟𝑝𝑚 𝜔𝑟 = 3600 𝑟𝑝𝑚
θ (deg) θ (deg) θ (deg)
ia
Ta
25
Figure 3.8: Simulated Average Torque as a Function of Dwell Angle and Speed(𝜃𝑎 = 5°, 𝑖∗ =
6𝐴,
When the current reference and the advance angle are held constant the maximum
average output torque is nearly constant for all speeds with a decreasing dwell angle. The trend
of increasing speeds correlates to figure 3.7 where it was noted that at lower speeds the sharp
decrease in current created negative torque before the positive torque contribution from the PM.
Therefore at lower speeds the dwell angle must be increased to avoid negative torque. If the
dwell is increased past a certain point there is a sharp drop-off in average torque for all speeds.
Figure 3.8: Simulated Average Torque as a Function of Advance Angle and Speed(𝜃𝑑 = 10°,
𝑖∗ = 6𝐴,
Holding the current reference and dwell angle constant shows the maximum torque for
increasing speed with an increased advance angle. In this case, a dwell angle of 10 degrees was
used which, as seen in figure 3.7, indicates that there will be larger torque production for higher
speeds. The larger values of average torque for the higher speeds are slightly misleading since
different combinations of dwell angles could create more torque for any speed. What can be
taken from this simulation is that larger advance angles need to be used to maximize torque at
higher speeds. At higher speeds the excitation period is shorter in time, which means that by
0 1 2 3 4 5 6 7 8 9 10 1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3
Advance Angle (deg)
Av
erag
e T
orq
ue
(Nm
)
1200rpm
1800rpm
2400rpm
3000rpm
3600rpm
26
increasing the advance a longer amount of time is allowed for the current to rise and reach its
maximum.
3.2.4 Selection of Firing Angles
Analysis of the effects of the firing angles gives a clear picture on how these variables
affect steady state operation of the motor. With current control in mind, choosing firing angles
that produce the maximum torque at the highest current will allow for the greatest range of
torque control. The selection of firing angles can neglect the maximum torque level at smaller
current levels since the current level can be raised for additional torque. For instance, choosing a
dwell angle of 8.5 will give a current controller the ability to generate anywhere from 0 to 5.75
Nm of torque while a dwell of 14 would only allow for 0 to 4.75 Nm.
Offline selection of the firing angles are possible and can be done so as to maximize the
torque output, however the angles will only be optimum for one operating condition. For more
precise control of the PMSRM the performance of the motor must be characterized based on its
operating conditions as well as its control variables. In addition, the control structure must
change in order to allow the firing angles to be dynamically updated.
0
5
10
5
10
150
1
2
3
4
a (deg)
d (deg)
Torq
ue (
Nm
)
27
Figure 3.9: Simulated Average Torque with Set Speed and Current Command
Each current reference and speed combination has a map similar to the one shown in
figure 3.9, with increasing torque output for increased current reference and decreased torque
output for increased speed. Likewise, the firing angle combination that yields the peak will vary
as a function of speed and current. Visualizing all four dimensions of this performance map is
difficult, however it can be thought of in a numeric method. The complete map of average
torque can be expanded to a four dimensional matrix with each element referring to a specific
advance and dwell angle with a set speed and current reference. This torque matrix is
represented as:
𝑇 𝜔, 𝑖∗, 𝜃𝑎 , 𝜃𝑑 =
𝑇 𝜔0, 𝑖0∗, 𝜃𝑎0
, 𝜃𝑑0 ⋯ 𝑇 𝜔0, 𝑖0
∗, 𝜃𝑎0, 𝜃𝑑𝑚
⋮ ⋱ ⋮𝑇 𝜔0, 𝑖0
∗, 𝜃𝑎𝑛, 𝜃𝑑0
⋯ 𝑇 𝜔0, 𝑖0∗, 𝜃𝑎𝑛
, 𝜃𝑑𝑚 ⋯
𝑇 𝜔0, 𝑖𝑗∗, 𝜃𝑎0
, 𝜃𝑑0 ⋯ 𝑇 𝜔𝑘 , 𝑖𝑗
∗, 𝜃𝑎0, 𝜃𝑑𝑚
⋮ ⋱ ⋮𝑇 𝜔0, 𝑖𝑗
∗, 𝜃𝑎𝑛, 𝜃𝑑0
⋯ 𝑇 𝜔𝑘 , 𝑖𝑗∗, 𝜃𝑎𝑛
, 𝜃𝑑𝑚
⋮ ⋱ ⋮
𝑇 𝜔𝑘 , 𝑖0∗, 𝜃𝑎0
, 𝜃𝑑0 ⋯ 𝑇 𝜔𝑘 , 𝑖0
∗, 𝜃𝑎0, 𝜃𝑑𝑚
⋮ ⋱ ⋮𝑇 𝜔𝑘 , 𝑖0
∗, 𝜃𝑎𝑛, 𝜃𝑑0
⋯ 𝑇 𝜔𝑘 , 𝑖0∗, 𝜃𝑎𝑛
, 𝜃𝑑𝑚 ⋯
𝑇 𝜔𝑘 , 𝑖𝑗∗, 𝜃𝑎0
, 𝜃𝑑0 ⋯ 𝑇 𝜔𝑘 , 𝑖𝑗
∗, 𝜃𝑎0, 𝜃𝑑𝑚
⋮ ⋱ ⋮𝑇 𝜔𝑘 , 𝑖𝑗
∗, 𝜃𝑎𝑛, 𝜃𝑑0
⋯ 𝑇 𝜔𝑘 , 𝑖𝑗∗, 𝜃𝑎𝑛
, 𝜃𝑑𝑚
(3.1)
In order to obtain a complete performance map of a motor the elements of the matrix in
3.1 can be obtained through an extensive simulation. For simplicity, the simulations used in this
thesis were performance using an ideal current regulator to remove additional disturbances from
overshoot or slow rise time which may come with actual control implementation. In addition,
the simulations need to be run at the different intervals of speed which requires the speed to be
held constant. The simulation of the current is dynamic but the mechanical model is considered
constant, which means that the speed is not affected by the torque production.
When the simulation is completed, the resulting performance map has figures similar to
3.9 for each of the reference current and speed combination. For a given operating point
maximum efficiency will occur when the largest amount of torque is produced with the smallest
amount of current, or at the peak of the graph. Hence, the torque can be maximized for each
unique reference current and speed combination. The point of maximum torque also corresponds
to a pair of firing angles, which can be thought of as two separate sets of dependent variables
28
matched to each point of maximum torque. The set of maximum torque and the corresponding
firing angles as a function of speed and reference current are represented as:
𝑇𝑚𝑎𝑥 , 𝜃𝑎𝑚𝑎𝑥 , 𝜃𝑑𝑚𝑎𝑥 =
𝑇𝜔0 ,𝑖0
∗ , 𝜃𝑎𝜔0 ,𝑖0∗ , 𝜃𝑑𝜔0 ,𝑖0
∗ ⋯ 𝑇𝜔0 ,𝑖𝑗∗ , 𝜃𝑎𝜔0 ,𝑖𝑗
∗ , 𝜃𝑑𝜔0 ,𝑖𝑗∗
⋮ ⋱ ⋮
𝑇𝜔𝑘 ,𝑖0∗ ,𝜃𝑎𝜔𝑘 ,𝑖0
∗ ,𝜃𝑑𝜔𝑘 ,𝑖0∗ ⋯ 𝑇𝜔𝑘 ,𝑖𝑗
∗ , 𝜃𝑎𝜔𝑘 ,𝑖𝑗∗ , 𝜃𝑑𝜔𝑘 ,𝑖𝑗
∗
(3.2)
Figure 3.10: Maximum Average Torque
The maximum average torque output of the motor is linear for low speeds and smaller
current reference levels. As the speed and current reference increase, the maximum torque
declines. When simulating or running the motor the torque is an output variable determined by
the speed, current and firing angles. However, in closed loop control design it is necessary to
have reference current as a function of torque since the output of the system is torque, and the
input is reference current. Current can be represented as a function of torque performing a
transformation of the maximum torque matrix.
𝑇𝑚𝑎𝑥 𝑖∗, 𝜔𝑟 → 𝑖∗ 𝑇, 𝑤𝑟
(3.3)
This transformation is accomplished through a relatively simple linear interpolation of
the simulated torque matrix. The first step is to determine the range of the torque inputs. On the
1000 2000
3000 4000 0 2 4 6 8 10 12
0
1
2
3
4
5
6
7
Current Reference (A)
Speed (rpm)
Max
imu
m T
orq
ue
(Nm
)
29
upper side range of torque is limited by the average torque output of the motor and on the lower
side it is limited by the smallest average torque of the lowest current reference. The step size of
the torque lookup is user defined. A larger step size will give a poorer resolution but a smaller
table that is easier to implement in a microcontroller. For each speed, the corresponding torque
is interpolated for every point in the range. The fraction of the each torque relative to a known
torque is proportional to the current. The following figure shows the transform process with a
step size of 1Nm. A horizontal line is brought from the torque command to the line. This
projection onto the current axis provides the corresponding current.
Figure 3.11: Transformation of the Torque Table
The advance and dwell angle matricides do not need to undergo the same transformation
because they can be looked up based on the speed, which is the control input, and the current
reference, which is found with the previous current lookup table. When the maximization of the
torque map occurs, each index from for the dwell and advance angle need to be preserved so that
there is no offset.
2 4 6 8 10 12 1000
2000
3000
4000 0
1
2
3
4
5
6
𝜔𝑟 (rpm)
Torq
ue
Com
man
d (
Nm
)
Reference Current (A)
30
Figure 3.12: Lookup Tables for, (a) the Dwell Angle and, (b) the Advance Angle
The data in figure 3.12 shows the firing angle lookup tables which was acquired through
simulation. The dwell angle is at a maximum when the current and the speed are at their lowest
which may seem counterintuitive at first. However with slower speeds and current levels the
current will drop to zero very quickly. If the current reaches zero too soon then negative torque
1000
2000
3000
4000
0 2
4 6
8 10
12
8
10
12
14
16
18
𝜔𝑟 (rpm)
Reference Current (A)
𝜃𝑑
(a)
1000
2000
3000
4000
0 2 4 6 8 10 12 4
4.5
5
5.5
6
6.5
𝜔𝑟 (rpm)
Reference Current (A)
𝜃𝑎
(b)
31
will occur before the magnet torque comes into effect. At higher current levels the dwell needs
to be shorter so that there is enough time for the current to reach zero before the negative torque
region. Generally, the dwell angle seems to have a negative linear relationship between both the
current and the speed.
The advance angle lookup table shows a peak advance angle when the speed and the
current are at a maximum. The lowest advance angle comes at lower speeds and high current
reference due to the fact that the current will have the additional time to rise at lower speeds,
rendering additional advance angle unnecessary. The advance is constant for the lowest current
level; however it drops off for increased current.
3.2.5 Sensitivity Analysis
The previous simulations have all been performed assuming a constant dc link voltage.
While this is acceptable to demonstrate trends and the general operation of the motor it fails to
account for the voltage ripple caused by rectification. The converters presented in chapter two
are dc converters, which in almost all cases will come from a rectified ac signal which will have
an oscillating ac component, or ripple. The size of the ripple will be determined mainly by the
size of the capacitors. In most applications total system cost is a priority, meaning that expensive
electronics with higher performance will be substituted by the lowest cost working replacement.
Finding the limits of the system’s electronics are important to determining the most cost effective
solution.
0 5 10 15 20 25 30 35 40 0
1
2
3
4
5
6
7
8
9
Cu
rren
t (A
)
Rotor Position (deg)
340V
310V
280V
32
Figure 3. 13: Current Waveform with Varying Dc-link Voltage
The magnitude of the voltage on the dc link has its primary impact on the rise and fall
rate of the current. This in turn affects how much torque is produced for a given set of firing
angles. For example, a smaller dc link voltage will account for a slower rise time which would
require a larger advance angle to match the performance of a larger voltage. The accompanying
slower turn-off time will require a shorter dwell angle so as to avoid negative torque production.
Figure 3.14: Average Torque as a Function Dwell Angle with Variable Dc-link Voltage
The simulation above shows the average torque as a function of dwell and current with a
dc-link voltage that is 310V ±10%. As the current reference increases, the discrepancy between
the maximum and minimum voltages becomes larger. In addition, the location of peak torque
occurs at increasing dwell angles for increasing bus voltages since with a quicker turn-off time
the dwell can be larger without producing negative torque. With an increased dwell angle the
overall average torque increases.
6 8 10 12 14 16 1
1.5
2
2.5
3
3.5
4
4.5
5
Av
erag
e T
orq
ue
(Nm
)
Dwell Angle (deg)
8A
6A
4A
33
Figure 3.15: Average Torque as a Function Advance Angle with Variable Dc-link Voltage
3.3 Control Overview
Given the performance benefits coupled with relative low cost, microcontrollers and
digital signal processors (DSP) are the most effective and efficient way to implement the control
scheme. The absolute position feedback can come from an encoder wheel, a magnetic Hall
Effect sensor, or potentially a sensor-less position observer. For closed loop current regulation
the controller also needs current feedback from analog current sensors discretized using an
analog to digital converter (ADC). In addition, dc-link voltage of the converter is useful for
control and can be scaled down with a voltage divider and then fed through the ADC. The PWM
output of the controller can be directly connected to the gate driver on the converter.
0 1 2 3 4 5 6 7 8 9 10 1
1.5
2
2.5
3
3.5
4
4.5
Av
erag
e T
orq
ue
(Nm
)
Advance Angle (deg)
8A
6A
4A
34
Figure 3.16: Hardware Overview
For variable speed operation of the PMSRM an outer-loop speed regulator will be
implemented in software to control the current regulator. With this type of operation the dwell
and advance angles will remain constant in order to reduce torque ripple; one of the primary
contributors to acoustic noise in a motor. In turn, the software defined current regulator will
produce a reference voltage output. For PWM current control is used the reference will be a duty
cycle with 1 representing the dc link voltage and 0 indicating the negative dc link voltage. The
PWM will use a DTA converter to send the duty cycle to each phase. In addition, the PWM will
control when each phase should be on or off based on the rotor position and the firing angles.
2-Phase
Asymmetric
Converter
PM-
SRM
Control
Logic PWM
Timer
ADC ADC ADC
Voltage Sensor
Current Sensor
Position Sensor
𝑖𝑎 ,𝑏 𝜃
𝑣𝑑𝑐
DSP
M
35
Figure 3.17: General Two Phase PMSRM Control Block Diagram
The self-tuning efficiency based algorithm presented in chapter 5 will retain the same
structure as the above control diagram as the above controller with the only difference being the
switching signals come from an additional controller. Once the algorithm has completed then
operation of the motor will continue with fixed angles.
The previous control method uses set firing angles (computed either online or off) with a
variable current. However, as seen in section 3.2.4, the average torque output of the PMSRM
can only be controlled for efficiency by using all three variables. The following control scheme
can be implemented to maximize the efficiency by maximizing the average torque.
Figure 3.18: Average Torque Control Block Diagram
𝑖∗ 𝐺𝑇
𝜃𝑎
+
-
𝜔𝑟∗
𝜔𝑟
𝑖∗ 𝑇∗, 𝜔𝑟 𝑇𝑒
∗
𝜃𝑎 𝑖∗,𝜔𝑟
𝜃𝑑 𝑖∗, 𝜔𝑟
𝜃𝑑
Phase A
𝑖∗
𝜔𝑚
𝜔𝑓
𝜔∗
𝐺𝑐−𝑎
𝐺𝑐−𝑏
PWM
𝐻𝑐
𝐻𝑐
Asymmetric
Converter
𝐺𝜔
𝐻𝜔 Phase B
PMSRM
Switching
Signals
𝑖𝑏
𝑖𝑎
𝜃
+
+
+ -
-
-
𝑑𝑎
𝑑𝑏
36
The speed error is placed through a torque controller which generates a torque command.
Given the torque command and the speed reference a current command value is determined with
a two dimensional interpolation from a lookup table generated from the simulation in section
3.2.4. This value will then be used to calculate the firing angles that match the current and speed
reference based on another lookup table. For any combination of current command and speed
reference there is one unique set of firing angles which guarantees that torque will be the
maximal possible for the current, and therefore the most efficient combination. In this control
scheme the current regulator remains the same; and is provided a reference current from the
lookup table. The PWM functionality remains the same even though the firing angles are time
varying.
37
4 Current Control Design and Simulation
As stated in [2], “the heart of any motor drive’s control system is current control”. The
electromagnetic nonlinearities of both the SRM and the PMSRM make current control a non-
trivial task. Although they are slightly different, both machines have inductances that vary with
position and current with similar values. In addition, both machines use the same electronics and
have similar operational regions; therefore the approach to controlling the PMSRM will closely
follow the control of the SRM.
Hysteresis controller has been proven to be a simple and effective approach; however, its
lack of sophistication is evident in its large current ripple and significant switching losses. Pulse
width modulation is a more efficient means of current control which can also provide a large
performance increase. Of the available types of linear control, proportional plus integral (PI) is
the most common controller for SRM current control, and is considered the benchmark of all
controllers.
A drawback for the PI controller is that the controller must be specifically designed for
set operating point. During variable speed operation the speed and current reference will change
considerably, rendering the gains of the controller ineffective. In addition, based on
manufacturing methods and materials, the actual motor may vary up to 30 percent from the
model. Gain scheduling uses gains selected for linearized points within the operating region to
improve performance at variable speeds and loads. With a hysteresis band to avoid excessive
changing of gains the speed and current feedback are used to select gains from a lookup table of
gains designed from either linear or experimental design.
Another approach for current control design is to use a nonlinear adaptive control
algorithm that recognizes both parameter uncertainty and model nonlinearity. The controller is
designed so that the known linear model is separated from the parameter uncertainties and
nonlinearities. From this form, the controller is designed to drive the system to a reference
current while adapting for the unknown. Thus, the more that is known about the plant model the
better the performance. If the nonlinearities are only partially known and there is significant
uncertainty stable control can still be achieved. The downside of adaptive control is that it can be
difficult to implement the sophisticated adaptive algorithm on a DSP or MCU that has significant
computational and numerical limitations.
38
4.1 Hysteresis Current Control
Due to its simplicity and ease of implementation, hysteresis current control is a viable
option for low performance applications and general operation. The most basic hysteresis
control strategy is implemented by applying full positive voltage to the phase whenever the
current feedback is less than the reference current value. Likewise, whenever the current
feedback is greater than the reference then full negative voltage is applied. The resulting current
ripple is directly related to the controller frequency.
When using the asymmetric converter the hysteresis controller can also output zero volts
by turning on the switch T1 and turning off T2. With the asymmetric converter positive voltage
is applied when both switches are on, and negative voltage is applied when both switches are off.
A slightly higher performance control strategy can be used given this extra degree of freedom by
adding a boundary around the current command. When the current feedback is within the
boundary then zero volts are applied. When the feedback is out of bounds, then the controller
behaves like a typical hysteresis controller.
Figure 4.1: Hysteresis Controller with Asymmetric Converter
The switching behavior for the controller is summed up as:
𝑇1 = 𝑂𝑁, 𝑖 ≥ 𝑖∗ + ∆𝑖∗ 𝑂𝐹𝐹, 𝑒𝑙𝑠𝑒
𝑇2 = 𝑂𝑁, 𝑖 ≤ 𝑖∗ − ∆𝑖∗ 𝑂𝐹𝐹, 𝑒𝑙𝑠𝑒
39
(4.1)
Correlating to figure 2.5 𝑇1 is the switch connected to the positive dc link and 𝑇2 is the
switch connected to the negative terminal of the dc link. The error boundary, ∆𝑖∗, can either be a
set value or, for more accuracy, a percentage of the current command. The controller was
simulated for the PMSRM as shown in figure 4.2. In this simulation the boundary was chosen to
be 10% of the current command.
Figure 4.2: Simulation of the Hysteresis Current Controller
From the simulation it is clear that the hysteresis controller can accurately track the
current reference. As the reference is increased the magnitude of the current ripple becomes
significant. The simplicity of implementing the hysteresis controller makes it an attractive
option in certain low performance settings; however the large current ripple is a cause for
concern in higher performance applications.
4.2 PI Current Control
PI control is a proven method of current control in SRM’s as well as in the control of
other motors. The integral term is vital to eliminating the steady state error that will
0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045
0
5
10
15
Time (s)
Curr
ent
(A)
i a i b
i *
40
undoubtedly be present due to the systems nonlinearities combined with general model
uncertainties. The linear transfer function will give a transfer function with current as the output
signal and voltage as the input. To apply this, a PWM chopping scheme described in chapter 2
must be implemented. While the PI controller may be a simple solution it is a proven and robust
method for current control which is very easily implemented.
4.2.1 Linearization of the PMSRM Current Model
The relationship between voltage and current for the PMSRM is:
𝑣 𝑡 = 𝑅𝑠𝑖 𝑡 +𝑑𝜆(𝜃, 𝑖)
𝑑𝑡
(4.2)
The voltage is considered to be the control input to the system and the current is the state
and the output of the system. When calculating the effect of the current for the PMSRM the flux
is given by 𝜆 = 𝐿 𝜃, 𝑖 𝑖 + 𝜆𝑝𝑚 𝜃 . Substituting the flux equation and taking the partial
derivative results in:
𝑣 𝑡 = 𝑅𝑠𝑖 𝑡 + 𝐿 𝜃, 𝑖 𝑑𝑖 𝑡
𝑑𝑡+ 𝑖 𝑡
𝑑𝐿 𝜃, 𝑖
𝑑𝑡+
𝑑𝜆𝑝𝑚 𝜃
𝑑𝑡
(4.3)
The derivative of the cogging flux is only a function of speed and can be assumed to be a
constant value since the change in flux will be near constant for the excitation range. When
considering the linear system, this term will be absorbed by the large signal voltage term since it
is a time invariant constant. The linear system can be found by substituting the following small-
signal perturbations at the following operating points:
𝑖 = 𝑖0 + 𝛿𝑖
𝑣 = 𝑣0 + 𝛿𝑣
𝜔𝑚 = 𝜔𝑚0 + 𝛿𝜔𝑚
(4.4)
For current control it is desired to have current as the output signal and voltage as the input
signal. Combining the perturbations with the voltage equation results in:
𝑑𝛿𝑖
𝑑𝑡𝐿 𝜃, 𝑖 = − 𝑅𝑠 +
𝑑𝐿(𝜃, 𝑖)
𝑑𝑡𝜔0 𝛿𝑖 + 𝛿𝑣
(4.5)
41
Taking the Laplace transform of the above equation the transfer function is found to be:
𝐺𝑝 𝑠 = 𝛿𝑖 𝑠
𝛿𝑣 𝑠 =
1
𝑠𝐿0 + 𝑅𝑒𝑞
(4.6)
The equivalent resistance has been substituted into the equation as:
𝑅𝑒𝑞 = 𝑅𝑠 + Δ𝐿0𝜔𝑚
(4.7)
Also the nominal inductance, 𝐿0 , and the nominal change in inductance, Δ𝐿0 , have
been introduced . To complete the linearization these nominal points must be calculated for each
operating point of a particular speed and current. The speed can be chosen based upon whatever
the desired operations of the motor. However the current will be varying over the operating
region, an operating current is chosen for based on what the average current will be at each
specific operating point.
Figure 4.3: Self Inductance of the PMSRM
The inductance of the PMSRM is a nonlinear function of position and current. When the
rotor pole is aligned with the stator pole the inductance is at a maximum. Likewise, when the
rotor and stator are unaligned the inductance is at a minimum. As the current increases the
change in inductance becomes smaller. The nominal inductance is found by taking the arithmetic
mean of inductance for the current specified by the operating region.
0 5 10 15 20 25 30 35 0.005
0.01
0.015
0.02
0.025
0.03
Rotor Position (deg)
Ind
uct
ance
(H
)
22A 20A 18A 16A
14A
12A
10A
8A
6A 4A 2A
42
𝐿0 = 𝐿𝑎 + 𝐿𝑢
2 𝑖=𝑖0
(4.8)
Where 𝐿𝑎 is the inductance at the aligned position (the maximum inductance), and 𝐿𝑢 is
the inductance at the unaligned position (the minimum inductance). Likewise, the nominal
change in inductance is found by taking the difference in inductance divided by the difference in
position.
Δ𝐿0 = 𝑑𝐿0
𝑑𝜃 𝑖=𝑖0
= 𝐿𝑎 − 𝐿𝑢
𝜃𝑎 − 𝜃𝑢 𝑖=𝑖0
(4.9)
The aligned position is 𝜃𝑎 , and 𝜃𝑢 is the unaligned position. Variable speed operation will
require a range of operating currents, all of which will have unique nominal inductance values,
as seen in table 4.1.
Table 4.1: Nominal Inductance values for the 2Hp PMSRM
The nominal values for the equivalent resistance are a function of both current and the
speed of the motor.
Req
𝜔0 𝑟𝑝𝑚 𝜔0(rad/sec) 2A 4A 6A 8A 10A 12A
3600 377.0 6.25 6.25 5.87 5.12 3.99 3.61
3300 345.6 5.78 5.78 5.43 4.74 3.710 3.36
3000 314.1 5.31 5.31 4.99 4.36 3.42 3.11
2700 282.7 4.84 4.84 4.55 3.99 3.14 2.86
2400 251.3 4.36 4.36 4.11 3.61 2.86 2.61
2100 219.9 3.89 3.89 3.67 3.23 2.57 2.35
1800 188.5 3.42 3.42 3.23 2.86 2.29 2.10
𝑖0 2A 4A 6A 8A 10A 12A
𝐿0 0.0194 0.0187 0.0182 0.0175 0.0171 0.0161
Δ𝐿0 0.015 0.015 0.014 0.012 0.009 0.008
43
1500 157.1 2.95 2.95 2.79 2.48 2.01 1.85
1200 125.7 2.48 2.48 2.35 2.10 1.73 1.60
Table 4.2: Nominal Req Values for the 2Hp PMSRM
4.2.2 PI Control Design
The next step in linear control design is the addition of a PI controller. The following
block diagram shows the closed loop current control strategy. Each phase will need a spate
controller since the error signals will be different, although the model, controller, and reference
will be the same
Figure 4.4: Block Diagram of the Linear 2-phase PMSRM Current Controller
The plant transfer function is represented by, 𝐺𝑝 𝑠 and the PI controller is represented by:
𝐺𝑐 𝑠 =𝐾𝑃 𝑠 + 𝐾𝑖
𝑠
(4.10)
It is desirable to place the controller zero as close to the systems real pole as possible in order to
cancel it out. This means that the integral gain should be selected to have the same value which is the
equivalent resistance divided by the nominal inductance. That is:
𝐾𝑖 =𝑅𝑒𝑞
𝐿0
(4.11)
In the real system the controller’s zero will not cancel the system’s pole out since the plant model
will not be identical to the actual system. If the system pole is smaller than the controller pole then the
root locus will have a part on the real axis from the pole at the origin to the controller zero and a second
𝐾𝑃 𝑠 + 𝐾𝑖
𝑠
1
𝑠𝐿0 + 𝑅𝑒𝑞
𝑖 ∗
𝑖𝑏 𝑒𝑖𝑏
−
𝐾𝑃 𝑠 + 𝐾𝑖
𝑠
1
𝑠𝐿0 + 𝑅𝑒𝑞
𝑒𝑖𝑎 𝑖𝑎
−
+
+
44
part moving into the left hand plane away from the system pole. If the system pole is larger, then the
locus will split before the zero, and then rejoin the real axis somewhere past the zero. In both cases one
of the poles will go towards the zero and the other will go to negative infinity. Selecting a large
proportional gain will have the same effect in either case.
Figure 4.5: Root Locus of the Linear Electrical Model of the PMSRM
The proportional gain must be large enough to ensure that the full dc voltage is
commanded as soon as the initial excitation command is seen. If full voltage is not seen
immediately, the rise time of the system will be limited. Inaccuracies in the linear model can be
seen by simulating the linear and the nonlinear models side by side as shown in figure 4.6.
-350 -300 -250 -200 -150 -100 -50 0
-20
-15
-10
-5
0
5
10
15
20
Root Locus
Real Axis
Imagin
ary
Axis
45
Figure 4.6: Small Signal Step Response of the PI Controller for Current Reference of (a) 2A and
(b) 6A for the linear and nonlinear model
The accuracy of the linear design can be evaluated by simultaneous simulation of the
linear and nonlinear models with the same PI controller. For the case in figure 4.5(a), when the
current reference is 2A, the two models behave slightly different. First of all, the rise time of the
0
2
4
6
8
Ph
ase
Cu
rren
t (A
)
(A)
i a
i *
i a lin
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
-400
-200
0
200
400
Ph
ase
Volt
age
(V)
(V)
Time (ms)
v a v a lin
(b)
v a
v a lin
(a)
0
0.5
1
1.5
2
2.5
Ph
ase
Cu
rren
t (A
)
(A)
i a
i *
i a lin
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
-400
-200
0
200
400
Ph
ase
Volt
age
(V)
(V)
Time (ms)
46
linear model is slower, meaning that the actual system pole is further in the left half plane. The
slower rise time is also a result of the nonlinear inductance being lower than the nominal
inductance at the time of excitation. The command of the current regulator is limited in the
nonlinear model since the voltage cannot rise above the dc-link voltage, yet in the linear model
the voltage does rise above the dc-link threshold. For the 6A current command, the transient
response of the two models are very similar, however the nonlinear model looks more like a 2nd
order response and has a faster response despite having a limited voltage command. In both
cases the linear approximation does not exactly match the nonlinear system response although it
does provide a good general guideline for control design.
4.2.3 Anti-windup PI Control
For any converter topology there will always be a limited dc bus voltage which limits the
control signal which means that the control signal must be limited to – 𝑉𝑑𝑐 ≤ 𝑉∗ ≤ 𝑉𝑑𝑐 . Adding
a saturation function to the control signal is a necessary step in implementation of the controller
however it leads to a large build up of integral term (windup) as error continues to compound.
As the current approaches the reference the large magnitude of the integral term produces an
excessively large contribution resulting in overshoot.
A solution to avoid the negative effects of saturation is to introduce an anti-windup PI
controller. The anti-windup can be implemented in different ways, but the goal is to limit when
the integration occurs and at what rate. One way to do this is to stop or reset the integration
process once the controller is saturated. Another way to avoid windup is to initially apply a full
command signal. Then when the output is within a certain threshold of the reference begin the
control algorithm.
Within typical operating conditions the PMSRM’s PI current controller will saturate as
soon as excitation begins. In addition, the inductance is increasing through the excitation range,
resulting in a further increase from integrator windup. Therefore the anti-windup action can be
chosen to set the integral value to zero every time the output of the controller reaches saturation.
The block diagram for such a controller is shown in figure 4.7.
47
Figure 4.7: Anti-windup PI Current Controller
The addition of the integral limitation from the anti-windup adds additional non-linearity
to the system. When the controller is in saturation the controller behaves strictly as a
proportional controller. As the current waveform reaches the reference the control comes out of
saturation and the integral term begins to function. After each phase excitation is over, the error
signal for the integral term must be reset so as to not affect the next excitation period.
Figure 4.8: Simulated Current Response with Anti-Windup PI Current Control on Phase
A and PI Current Control of Phase B
0
2
4
6
8
10
12
Phas
e C
urr
ent
(A)
(A)
i a i b i*
1 2 3 4 5 6 7 8 9
x 10 -3
-400
-200
0
200
400
Aver
age
Volt
age
(V)
(V)
Time (sec)
v a
v b
PI w/Antiwindup
PI
48
Simulation shoes the increased performance of anti-windup current control and verifies
that the overshoot resulting from saturation is eliminated. Without anti-windup the full voltage is
applied more than two times longer that with anti-windup, all after the reference has been
reached. Since the PMSRM’s current controller will always be operating in saturation, anti-
windup is a necessity for even average performance.
4.2.4 PI Controller Simulation
In actual implementation the current control will be limited by whatever frequency the
control law can be operated at. The control frequency is selected according to the limitations on
the DSPs clock frequency, the chopping frequency the IGBTs or MOSFETs and the sampling
frequency of the ADC channels that the feedback signals are brought through. A typical
frequency is between 12 and 25 kHz. Larger chopping frequencies increase the performance of
the control since the controllers bandwidth is increased; however the efficiency of the system is
decreased as the power devices are switched more frequently. For each simulation the firing
angles are selected from the lookup table created in chapter 3 based on the speed and the
reference current. The current controller gains are Ki = 322. Kp =310, which are selected for the
operating point of 3600rpm with a current reference of 6A.
0
2
4
6
8
Ph
ase
Cu
rren
t
Cu
rren
t
i a i b i*
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x 10 -3
-400
-200
0
200
400
Ph
ase
Vo
ltag
e
Vo
ltag
e
v a
v b
0
2
4
6
8
i a i b i*
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x 10 -3
-400
-200
0
200
400
v a
v b
(a) 3600rpm (b) 3000rpm
49
Figure 4.9: Simulated phase current and voltage with 6A command
Simulation shows that as the operating point changes so do the performance of the PI
controller. As the speed moves away from the operating point the overshoot of the controller
becomes larger as does the number of oscillations. For all of the different speeds the rise time is
consistently good and there is no steady state error.
0
2
4
6
8
Ph
ase
Cu
rren
t (A
)
(A)
i a i b i*
0.5 1 1.5 2 2.5 3 3.5 4 4. 5 x 10 -
-400
-200
0
200
400
Ph
ase
Vo
ltag
e (V
)
(V)
v
0
2
4
6
8
i a i b i*
0. 1 1. 2 2.5 3 3.5 4 4.5 5 x 10 -
-400
-200
0
200
400
Time (sec)
(c) 2400rpm (d) 1600rpm
Time (sec)
50
Figure 4.10: Simulated Phase Current at 3600rpm
Similar to when the operating point is varied with speed, as the current reference
becomes smaller the overshoot and oscillations become more prominent. However, in when the
operating point is varied the controller performs with marginal performance in the worst case,
and good performance in the best case. Although it may not have the highest performance
control, PI control is a reliable and robust choice that is easy to implement in PMSRM current
control.
4.3 Adaptive Current Control
Adaptive control is a well suited approach to addressing current control in the PMSRM
from a non-linear perspective. The general structure of adaptive control allows for matched non-
linearities, such as the inductance of the PMSRM, in addition to system uncertainties which will
0 1 2 3 4 5
i a i b i*
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-400
-200
0
200
400
v a
v b
0
5
10
0
15
5
i a i b i*
*
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time (ms)
v a
v b
0
1
2
3
Ph
ase
Cu
rren
t (A
)
i a i b i*
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x 10
10
-3 -400
-200
0
200
400
v a
v b
0
4
8
i a i b i*
v a
v b
Ph
ase
Vo
ltag
e (V
) P
has
e C
urr
ent
(A)
(a) i*=2A (b) i*=4A
(c) i*=8A (d) i*=10A
Time (ms)
-400
-200
200
0
400
Ph
ase
Vo
ltag
e (V
)
51
arise in the difference between FEA data and the actual motor. Model reference adaptive control
(MRAC) uses dynamic gains to asymptotically track an ideal model of the system.
Figure 4.11: Block Diagram of One Phase of the PMSRM Model Reference Adaptive Controller
The adaptive laws are generated using the error between the system and the reference
model as they dynamically update the control law. There is a significant increase in
computational power needed to implement the MRAC controller as compared to the PI controller
since the MRAC controller needs the additional dynamic laws as well as the reference model
observer, both of which require a significant increase in the number of computations. In
addition, the observer needs to run at a greater frequency than the controller, requiring an
additional timing loop in an embedded controller.
4.3.1 PMSRM System Model with Structured Non-Linearites
The governing electrical equation for the PMSRM has nonlinearities as well as unknown
and time varying parameters. This can be approached by separating the equation to have the
known linear terms separate from the unknown linear and nonlinear terms. The voltage equation
for the PMSRM derived in section 4.2 can be rearranged as:
𝑑𝑖
𝑑𝑡= −
𝑅𝑠
𝐿𝑛𝑜𝑚𝑖 +
1
𝐿𝑛𝑜𝑚𝑣 +
1
𝐿𝑛𝑜𝑚 −Δ𝑅𝑖 − 𝜔 𝑖
𝑑𝐿 𝜃, 𝑖
𝑑𝜃
1
𝐿 𝜃, 𝑖 − 𝐿𝑛𝑜𝑚
𝑑𝜆𝑝𝑚 𝜃
𝑑𝜃
(4.12)
The electrical equation is represented in terms of a nominal linear plant together with
bounded nonlinearities and disturbances. The nominal plant is represented by replacing the
PMSRM
Reference Model
Adaptive
Gains
Adaptive
Controller
+
-
𝑣∗
𝑖∗
𝑘𝑥 ,𝑘𝑟 , 𝑘𝑊
𝑖
𝑖𝑚
𝑒
52
nominal inductance by 𝐿𝑛𝑜𝑚 , and placing the time varying inductance along with other
uncertainties in the last term. An additional term, Δ𝑅, has been added to compensate for
uncertainties as well as the time varying change of resistance with temperature change. The
speed, current, inverse inductance, as well as the position derivatives of the inductance and
magnetic flux are all known to be bounded. Given an arbitrary 𝜖∗ > 0 and an arbitrary compact
set 𝐵𝑟 ∈ ℝ𝑛 , there exists a positive integer 𝑚 such that for any arbitrary continuous
function 𝑑 𝑥 : 𝐵𝑟 → ℝ,
𝑑 𝑥 = 𝑊Φ 𝑥 + 𝜖 𝑥 , 𝜖 𝑥 < 𝜖∗
(4.13)
Where W is an 𝑚 × 1 vector of unknown constants and Φ 𝑥 is an 𝑚 × 1 vector and
is a radial bias function such that Φ 𝑥 = Φ 𝑥 [17]. Appling this to the bounded
nonlinearities and uncertainties:
𝑊Φ 𝜃, 𝑖 + 𝜖 𝑥 = −Δ𝑅𝑖 − 𝜔 𝑖𝑑𝐿 𝜃, 𝑖
𝑑𝜃
1
𝐿 𝜃, 𝑖 − 𝐿𝑛𝑜𝑚
𝑑𝜆𝑝𝑚 𝜃
𝑑𝜃
(4.14)
Substituting (4.14) into the nonlinear plant model results in:
𝑑𝑖
𝑑𝑡= −
𝑅𝑠
𝐿𝑛𝑜𝑚𝑖 +
1
𝐿𝑛𝑜𝑚𝑣 +
1
𝐿𝑛𝑜𝑚 𝑊Φ 𝜃, 𝑖 + 𝜖 𝑥
(4.15)
4.3.2 MRAC Current Control
The performance of any adaptive control is limited by the reference system to which it
adapts. The closer the model is to the actual system the more accurately the system will track it.
For the PMSRM a first order approximation is appropriate for the observer design; therefore the
reference model can be defined as:
𝑖𝑚 = 𝐴𝑚 𝑖𝑚 𝑡 + 𝑏𝑚𝑟 𝑡
(4.16)
The linear reference plant parameters are represented by 𝐴𝑚 < 0 and 𝑏𝑚 > 0 and are
calculated based on the linear system parameters found in section 4.2 and 𝑟 𝑡 = 𝑖∗ is the
reference current input. The feedback control law for model reference adaptive control is given
by:
53
𝑣 𝑡 = 𝑘𝑥 𝑡 𝑖 𝑡 + 𝑘𝑟 𝑡 𝑟 𝑡 − 𝑘𝑊 𝑡 Φ(𝜃, 𝑖)
(4.17)
The state, reference and uncertainty adaptive gains are represented by 𝑘𝑥 , 𝑘𝑟 , and
𝑘𝑊 respectively. Substituting the adaptive control law into (4.15) yields:
𝑑𝑖
𝑑𝑡= −
𝑅𝑠
𝐿𝑛𝑜𝑚+
𝑘𝑥
𝐿𝑛𝑜𝑚 𝑖 𝑡 +
𝑘𝑟
𝐿𝑛𝑜𝑚𝑟 𝑡 −
1
𝐿𝑛𝑜𝑚 Δ𝑊 𝑡 Φ 𝜃, 𝑖 + 𝜖 𝑥
(4.18)
The difference in unknown parameters and its corresponding adaptive law is represented
by Δ𝑊 = 𝑘𝑊 𝑡 − 𝑊. The tracking error is the difference between the reference system and the
actual system and is represented by 𝑒 𝑡 = 𝑖 𝑡 − 𝑖𝑚 𝑡 . The error dynamics are found by
substituting (4.18) and (4.16) into the error equation as:
𝑒 𝑡 = 𝐴𝑚𝑒 𝑡 + Δ𝑘𝑟𝑟 𝑡 + Δ𝑘𝑥 𝑖 𝑡 +1
𝐿𝑛𝑜𝑚 −Δ𝑊 𝑡 Φ 𝜃, 𝑖 + 𝜖 𝑥
(4.19)
Where Δ𝑘𝑟 =𝑘𝑟
𝐿𝑛𝑜𝑚− 𝑏𝑚and Δ𝑘𝑥 =
𝑘𝑥−𝑅𝑠
𝐿𝑛𝑜𝑚− 𝐴𝑚 . The adaptive laws are chosen to be:
𝑘 𝑥 𝑡 = −Γ𝑥 𝑖 𝑡 𝑒 𝑡 𝑃
𝐿𝑛𝑜𝑚+ 𝜎𝑥𝑘𝑥 𝑡
(4.20)
𝑘 𝑟 𝑡 = −γ𝑟 𝑟 𝑡 𝑒 𝑡 𝑃
𝐿𝑛𝑜𝑚+ 𝜎𝑟𝑘𝑟 𝑡
(4.21)
𝑘 𝑊 𝑡 = −Γ𝑊 Φ 𝑡 𝑒 𝑡 𝑃
𝐿𝑛𝑜𝑚+ 𝜎𝑊𝑘𝑊 𝑡
(4.22)
Γ𝑥 > 0, 𝛾𝑟 > 0, Γ𝑊 > 0 are the adaptive gains which can be tuned to improve the
controllers performance, 𝜎𝑥 , 𝜎𝑟 ,𝜎𝑊 are control parameters and P solves the algebraic Lyapunov
equation:
𝐴𝑚𝑇 𝑃 + 𝑃𝐴𝑚 = 𝑄 > 0
(4.23)
In order to determine the stability of the closed loop system, the following positive
definite Lyapunov candidate is suggested:
54
𝑉 𝑒 𝑡 , Δ𝑘𝑥 𝑡 ,Δ𝑘𝑟 𝑡 , Δ𝑊 𝑡
= 𝑃 𝑒 𝑡 2
+ Γ𝑥−1 Δ𝑘𝑥 𝑡
2+ 𝛾𝑟
−1 Δ𝑘𝑟 𝑡 2
+ Γ𝑊−1 Δ𝑊 𝑡
2≥ 0
(4.24)
The rate of the Lyapunov function is evaluated in [17] as:
𝑉 = 2𝑒𝑃𝑒 + 2Δ𝑘𝑥Γ𝑥−1Δ𝑘𝑥
+ 2Δ𝑘𝑟γ𝑟−1Δ𝑘𝑟
+ 2Δ𝑘𝑊Γ𝑊−1Δ𝑘𝑊
= 2𝑃𝐴𝑚𝑒2 + 2𝑒𝑃Δ𝑘𝑟𝑟 𝑡 + 2𝑒𝑃Δ𝑘𝑥 𝑖 𝑡 + 2𝑒𝑃1
𝐿𝑛𝑜𝑚 −Δ𝑊 𝑡 Φ 𝜃, 𝑖 + 𝜖 𝑥
− 2Δ𝑘𝑥 𝑖 𝑡 𝑒 𝑡 𝑃
𝐿𝑛𝑜𝑚+ 𝜎𝑥𝑘𝑥 𝑡 − 2Δ𝑘𝑟
𝑟 𝑡 𝑒 𝑡 𝑃
𝐿𝑛𝑜𝑚
𝑘𝑟
𝐿𝑛𝑜𝑚− 𝑏𝑚 + 𝜎𝑟𝑘𝑟 𝑡
− 2Δ𝑘𝑊 Φ 𝑡 𝑒 𝑡 𝑃
𝐿𝑛𝑜𝑚+ 𝜎𝑊𝑘𝑊 𝑡
≤ − 𝑒 𝜆𝑚𝑖𝑛 𝑄 𝑒 − 2 𝑃
𝐿𝑛𝑜𝑚 𝜖∗
(4.25)
Therefore 𝑉 ≤ 0 as long as:
𝑒 ≥ 2 𝑃𝐿𝑛𝑜𝑚
−1 𝜖∗
𝜆𝑚𝑖𝑛 𝑄
(4.26)
Thus the controller is asymptotically stable as long as the error remains outside of a
region determined by the values of P, Q and the nominal inductance. When the error is within
that region, the stability cannot be guaranteed, however as long as epsilon is sufficiently small
the region will remain small.
4.3.3 Adaptive Current Control Simulation
Implementing the closed loop MRAC current control requires the addition of the model
reference observer and the dynamic gain equations. Prior to simulating the adaptive control the
reference model must be designed, the structured uncertainties must be determined and the
model parameters and gains must be defined. The observer must be significantly faster than
controller so as to have an accurate system. The linear gains for the reference system do not
have to be based on the actual system response and should be selected to have the fastest rise
time obtainable by the system and no steady state error. However, ensuring a realistic rise time
55
is limited by the operating conditions of the plant. A reference model which is much faster than
the actual system will guarantee tracking error and will result in overshoot. If the model is not
fast enough then the system will track it without error, however the performance will be
marginal.
The modeled nonlinearities need to be selected so as to closely match the actual system
parameters. In the case of the DSPMSRM no specific radial basis function can exactly match the
nonlinearities of the inductance, the derivative of the inductance and the derivative of flux.
While any radial basis function will work, the performance of the controller will be limited by
the accuracy of the approximation. The function Φ = 𝑒−𝑖2/50 is a relatively simple function to
calculate in the controller and follows the general trends of the system nonlinearities.
Figure 4.12: Comparison of Actual Nonlinearities to Matched Nonlinearities
For a simulation at 1600 rpm a linear model with 𝐴𝑚 = −15000 and 𝑏𝑚 = 15000
ensures an adequate time response and no steady state error. Since the system model is of first
order, the value for P is a constant and not a matrix. This can be trivially assigned the value of 1,
which means that 𝑄 = 2𝐴𝑚 . The gains for the controller cannot be easily selected with the same
ease as in linear control design. In addition to the difficulty is the fact that there are six control
parameters, with three gains and three scaling values. Increased the value of a gain makes that
adaptive gain more dominant in the overall control law. For instance, increasing the gain 𝛾𝑟 will
make the reference value more of an impact in the overall control law. The control parameters
𝜎𝑥 , 𝜎𝑟 and 𝜎𝑊 determine the amount that a parameter can adapt. By increasing the control
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 -0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
𝑒−𝑖2/2
Φ(𝜃, 𝑖)
Rotor Position (rad)
PM
SR
M N
on
lin
eari
ties
56
parameter the net effect of the gain will be decreased which will reduce the overshoot but limit
the amount the controller can adapt to the model.
Parameter Value
𝐴𝑚 -15000
𝑏𝑚 15000
Γ𝑥 5000
γ𝑟 3000
Γ𝑊 100
𝜎𝑥 0.1
𝜎𝑟 0.1
𝜎𝑊 0.01
Table 4.3 Adaptive Parameters for simulation at 1600rpm with 𝑖∗ = 7𝐴
Figure 4.13: Adaptive Current Control Simulations at 1600rpm with 𝑖∗ = 7𝐴
0
2
4
6
8
10
Ph
ase
Cu
rren
t (A
)
(A)
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 -500
0
500
Vo
ltag
e C
om
man
d (
V)
ia
im
Time (s)
57
The simulation in figure 4.13 shows a good steady state tracking response of the adaptive
controller with a high initial overshoot. The effect of the adaptation can be seen as the number
of oscillations and size of the overshoot are decreased over time. This simulation was done at a
slower speed than the reference in order to show the desired performance of the adaptive
controller which requires time to converge to the steady state value. For current control of a
PMSRM each excitation period is a new step response with a new transient response. The
adaptive controller is very good at tracking the steady state value; however gain tuning is
required to improve the transient response. Over time the adaptive gains will settle, as shown in
figure 4.14, and although there is still significant overshoot the change in gains does make a
nominal improvement in transient performance.
Figure 4.14: Simulated Adaptive Gain Convergence at 1600rpm
Although adaptive control can guarantee stability, it cannot guarantee performance at
different operating points. Therefore, the reference model as well as the control parameters and
gains must be adjusted for the best performance at different operating points. Another option is
to increase the control frequency of the MRAC in order to compensate for the reduced total time
the controller has to adapt at higher speeds. Yet after a certain point the frequencies are not
hardware implementable.
Parameter Value
𝐴𝑚 -11500
𝑏𝑚 11500
Γ𝑥 18000
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 -600
-400
-200
0
200
400
600
800
k x
k r
k W
Time (s)
Ad
apti
ve
Gai
ns
58
γ𝑟 3000
Γ𝑊 8000
𝜎𝑥 3
𝜎𝑟 2.5
𝜎𝑊 0.5
Table 4.4: Control Parameters for Adaptive Control at 3600rpm with 𝑖∗ = 7𝐴
Figure 4.15: Adaptive Current Control Simulations at 3600rpm with 𝑖∗ = 7𝐴
0
1
2
3
4
5
6
7
8
9
10
Ph
ase
Cu
rren
t (A
)
(A)
-400
-200
0
200
400
Vo
ltag
e C
om
man
d (
V)
(V)
0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 -100
-50
0
50
100
150
Ad
apti
ve
Gai
ns
Gai
ns
ia
im
k x
k r
k W
Time (s)
59
The control parameters were selected for the simulation in figure 4.15 so as to minimize
the overshoot and steady state error of the control. To decrease the overshoot 𝜎𝑥 and 𝜎𝑟 were
increased to limit the overall change of the adaptation laws. In addition, the adaptive gains were
increased to make the control more responsive to error. From the simulation it is clear that the
overall change in the adaptive gains is much smaller than the previous simulation and although
the performance is better it is only because the gains are specifically tuned for this operating
point and performance cannot be guaranteed at other operating points.
60
5 Speed Control Design and Simulation
Overall system efficiency is an increasingly important selling point for commercial and
industrial motor drives. Induction motors or any other fixed speed motors will waste a large
amount of energy when running at their set speeds, regardless of the immediate system need.
The addition of a closed-loop, feedback speed regulator will allow the PMSRM to have variable
speed operation. For general applications, the speed controller will use a current reference as its
control variable while keeping the firing angles set. The current controller will be running at a
much greater frequency than the speed regulator allowing for the assumption that any delay to
the current regulator can be neglected and the gain will be considered unity. In addition to
allowing variable speed access, the speed regulator will provide disturbance rejection in the
presence of changes in load.
Although the advance and dwell angles are powerful control inputs, they will be kept
constant in the initial design of the speed regulator. However, in cases when the current
reference is larger than the nominal estimation, it is necessary to increase the advance angle to
allow the current to reach its reference level. The current reference output of the linear speed
regulator can also be augmented with a proportional gain to change the current reference into a
torque reference. The resulting torque command can be used to lookup a current, and the
appropriate firing angles from the lookup tables in chapter 3.
Considering the nonlinearities of the PMSRM coupled with the model uncertainty and
large range of control inputs, attempting to maximize the efficiency through simulation would be
largely inaccurate. However, using an embedded self-tuning efficiency algorithm a motor can
find its maximum operating point in a short period of time. The algorithm searches for
efficiency while varying the advance and dwell angles while maintaining a constant speed via the
speed regulator. The algorithm will run once at startup, then only as needed for changes in the
speed or load.
5.1 Speed Loop Linearization
The back EMF of the DSPMPSRM is the time derivative of its flux linkage, which is the
sum of the magnetic flux and the motor’s self inductance times current.
61
𝑒 =𝑑𝜆 𝜃, 𝑖
𝑑𝑡=
𝑑 𝐿 𝜃, 𝑖 𝑖 + 𝜆𝑝𝑚 𝜃
𝑑𝑡
= 𝑖𝑑𝐿
𝑑𝑡+ 𝐿
𝑑𝑖
𝑑𝑡+
𝑑𝜆𝑝𝑚
𝑑𝑡
(5.1)
Taking partial derivatives of its terms, the back EMF can be rewritten as:
𝑒 = 𝑖𝑑𝐿
𝑑𝜃𝜔 + 𝐿
𝑑𝑖
𝑑𝑡+
𝑑𝜆𝑝𝑚
𝑑𝜃 𝜔
(5.2)
The speed, 𝜔 , has replaced the time derivate of position. The electrical power produced by one
phase of the motor is found by multiplying the EMF by the phase current.
𝑃 = 𝑒𝑖 = 𝐿𝑖𝑑𝑖
𝑑𝑡+
1
2𝑖2
𝑑𝐿
𝑑𝜃𝜔 +
𝑑𝜆𝑝𝑚
𝑑𝜃 𝑖𝜔
(5.3)
The first term in this equation is the change in stored magnetic energy. The second and
third terms are the power generated by the switched reluctance and the permanent magnet,
respectively. The torque produced by the PMSRM is:
𝑇𝑚 =1
2𝑖2
𝑑𝐿
𝑑𝜃+
𝑑𝜆𝑝𝑚
𝑑𝜃 𝑖
(5.4)
The general linear mechanical model for an electric motor is:
𝑇𝑚 − Tl = Jdωm
𝑑𝑡+ 𝐵𝜔𝑚
(5.5)
Where J represents the inertia constant of the rotor, B represents the friction constant of
the motor’s bearings, and 𝑇𝑙 is the applied load torque. The complete equation for the torque of
the PMSRM is found by combining 5.4 and 5.5. As previously shown, the torque of the
PMSRM is a nonlinear function of current, inductance and magnetic flux.
i2dL θ, i
𝑑𝜃+
𝑑𝜆𝑝𝑚
𝑑𝜃 𝑖 = J
dωm
𝑑𝑡+ 𝐵𝜔𝑚 + Tl
(5.6)
62
In the desired linear model for a speed regulator, the output signal will be the rotor speed,
the input signal will be current, and the load torque will be considered a disturbance input. The
small signal perturbations are:
𝑖 = 𝑖0 + 𝛿𝑖
𝜔𝑚 = 𝜔𝑚𝑜 + 𝛿𝜔𝑚
𝑇𝑙 = 𝑇𝑙0 + 𝛿𝑇𝑙
(5.7)
In this representation the input and output signals are equal to the sum of their nominal
value at the operating point and the small signal perturbation. The perturbation signals can then
be substituted into the mechanical system equation resulting in:
𝐽𝛿𝜔𝑚
𝑑𝑡= −(𝐾𝑏 + Kpm ) δi − B𝛿ωm + 𝛿𝑇𝑙
(5.8)
The EMF constant, 𝐾𝑏 , and the PMF constant, 𝐾𝑝𝑚 , have been substituted into 5.8 to
show the contribution of the PMSRM’s torque production. The EMF and PMF constants are:
𝐾𝑏 =𝑑𝐿
𝑑𝜃𝑖0 𝐾𝑝𝑚 =
𝑑𝜆𝑝𝑚
𝑑𝜃 𝑖0
(5.9)
The mechanical model of the system can be represented in the Laplace domain as the
following transfer function:
𝐺𝑝𝜔 =1
𝐵 + 𝑠𝐽
(5.10)
The open loop linear system can be represented in the s-domain as a block diagram with
disturbance input as.
Figure 5.1: Open-loop Small Signal PMSRM Mechanical Model
𝛿𝜔𝑚 𝑠
- + 𝛿𝑖 𝑠 𝛿𝑇𝑒 𝑠
𝐾𝑏 + 𝐾𝑝𝑚 𝐺𝑝𝜔
𝛿𝑇𝑙 𝑠
63
To implement the linear model of the mechanical system the values of the EMF and PMF
constants must be calculated for specific operating points. The EMF constant calculation is the
product of the nominal current while the input of the system is the time varying current
command. Thus, as the input changes, so does the linear model of the system.
𝑖0 (𝐴) 4 6 8 10 12
𝐾𝑏 3.3𝑒−5 4.8𝑒−5 5.4𝑒−5 5.5𝑒−5 5.4𝑒−5
Table 5.1: Selected Values of the EMF Constant
With currents larger than 4 amps, values of the EMF constant tend to be fairly similar.
There is a peak at 10 amps, which is when the change in inductance decreases. The PMF
constant is derived from average magnetic flux and can be calculated by averaging the flux of the
motor with no current applied which is called the cogging flux.
Table 5.2: Average Cogging Flux
5.2 Design of a Speed Feedback Filter
The pulsing nature of the PMSRM’s torque production creates a ripple in the rotor speed.
A digital filter can be used to remove the ripple and give a clear signal to the speed controller.
Since this controller will be implemented digitally using a microprocessor using digital filter
design is an appropriate approach. The transfer function a second order digital filter is given by:
𝐻𝜔 𝑧 = 𝜔𝑓 𝑧
𝜔𝑚 𝑧 =
𝑎0 + 𝑎1𝑧−1 + 𝑎2𝑧
−2
1 + 𝑏1𝑧−1 + 𝑏2𝑧−2
(5.11)
The infinite impulse response filter can be represented as a block diagram as shown below.
Kpm (phase A) 9.1𝑒−6
Kpm (phase B) 8.6𝑒−6
64
Figure 5.2: IIR Speed Feedback Filter
Considering that the inertia of the rotor is large enough to prevent instantaneous change
in speed the cutoff frequency for this filter can be chosen to be 250 Hz. For simplicity the filter
can be implemented in the current regulator which will avoid the burden of an additional control
loop, and will be guaranteed to be running at least ten times, if not more, the speed of the speed
regulator. The coefficients are calculated using a low pass Butterworth filter design in matlab.
Filter Frequency, 𝑓𝜔 20kHz
Cut off Frequency, 𝑓𝑐𝑜𝜔 500Hz
Normalized Frequency 0.025
Poles 0.9445 ± 𝑗0.0526
Zeros -1,-1
Table 5.3: Speed Filter Parameters
The frequency response of the speed feedback filter is shown in the following figure.
𝑏1[𝑛]
𝑏2[𝑛] 𝑎2[𝑛]
𝑎1[𝑛]
𝑎0[𝑛] 𝜔𝑓[𝑛] 𝜔𝑚 [𝑛] +
+
+
+
+
+
𝑧−1
𝑧−1
𝑧−1
𝑧−1
65
Figure 5.3: Speed Filter Frequency Response
This bode plot confirms the filter’s 500Hz low pass cut-off frequency, and the -180
degree phase shift shows that the filter is second order. The bandwidth of the filter is 10 kHz,
which is half of its sampling frequency.
Although this filter has been designed, and will be implemented, as a discrete function,
for the rest the analog control design it will be transformed to a continuous transfer function,
𝐻𝜔 (𝑠). The transformation to the Laplace domain is necessary for control design. In all
actuality, the filter has large enough bandwidth that it will not affect the linear mechanical
system used in following control design.
5.3 PI Speed Control
To achieve closed loop speed control a PI controller will be added to the forward path of
the linear speed model. This controller type is chosen because the integral term will eliminate
steady state error while the proportional term can be tuned to increase the controller’s
performance. The form of the PI controller is:
-100
-80
-60
-40
-20
0
Magnitu
de (
dB
)
101
102
103
104
-180
-90
0
Phase (
deg)
Frequency (Hz)
66
𝐺𝑐𝜔 𝑠 = 𝐾𝑝𝜔 𝑠 + 𝐾𝑖𝜔
𝑠
(5.12)
Although the controller and speed filter will be implemented digitally, the control design
will take place in the s-domain for simplicity and assuming that the bandwidth of the regulator
will be significantly larger than the bandwidth of the controller. The PI controlled linear system
is represented in the following block diagram.
Figure 5.4: Closed-loop Speed Control Block Diagram
The closed-loop transfer function for the system is given by:
𝐺𝜔 𝑠 =𝜔∗
𝜔𝑚=
𝐾𝑏𝐺𝑐𝜔𝐺𝑝(𝑠)
1 + 𝐾𝑏𝐺𝑐𝜔𝐺𝑝𝜔𝐻𝜔(𝑠)
(5.13)
Root locus techniques are a powerful method of controller design, and can be used to
select the gains for the speed controller. The transfer function of the uncompensated linear
system has a pole on the real axis at -2.25e-4. By design, the integral term from the PI controller
adds a pole at the origin. The selection of the zeros is a critical choice in determining to the
transient response of the compensator.
𝜔𝑚 𝜔∗
-
+ 𝑖∗ 𝑇𝑒 𝐾𝑏 + 𝐾𝑝𝑚 𝐺𝑝𝜔
𝑇𝑙
𝐻𝜔
𝐺𝑐𝜔 + +
67
Figure 5.5: Step Response and Root Locus of the
PI Compensated Mechanical System
The integral gain determines where the compensator zero is. By making it much larger
than the system pole, the zero pulls the locus further into the left half plane. With such a small
plant frequency the system naturally has a slow rise time. Theoretically the zero could be placed
far into the left hand plane and a large gain could be given in order to get a fast first order
response. However, the current command is limited and to avoid saturation the gains must be
limited. In order to get the fastest possible rise time without saturation or oscillations the system
will have some overshoot. A dynamic simulation of the speed regulator with modeled load
disturbances is shown in the following figure.
0 50 100 1500
0.5
1
1.5
Step Response
Time (sec)
Am
plit
ude
-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1-0.1
-0.05
0
0.05
0.1
Root Locus
Real Axis
Imagin
ary
Axis
68
Figure 5.6: Dynamic Closed Loop Speed Control Simulation with Disturbance Inputs
The simulation results indicate that the speed regulator can successfully maintain the
speed when the load is changed with a quick rise time and minimal overshoot. In addition, the
controller does not become saturated and the current command stays well within its typical
range. There is a noticeable oscillation when the load is reduced; however the magnitude of
ripple in the actual speed is less than one rpm so this should not be a problem.
Although the speed regulator can maintain load disturbance without saturation when the
speed reference is changed saturation will occur because of the time it takes to accelerate or
decelerate the rotor with a current range. Specifically, when a smaller speed reference is
commanded the controller can only take the current to zero and then wait as the rotor slows.
Likewise, when the speed command is larger than the current speed the controller can only take
the current up to the rated level, or even single pulse, and wait for the speed to rise. In either
case, the response of the system is limited by nonlinear limitations not included in the linear
model.
0
5
10
15
Ph
ase
Cu
rren
t (A
)
(A)
3590
3595
3600
3605
Sp
eed
(rp
m)
(rp
m)
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0
5
10
15
Time (sec)
Cu
rren
t C
om
man
d (
A)
(A)
i a
i b
i *
m
f
2.8Nm 3.8Nm 4.5Nm
69
As with the current control, the problem related to controller saturation can be reduced
with the addition of an anti-windup PI control. The controller implementation will be identical
to that of section 4. As the controller will move towards the commanded speed at either
minimum or maximum the integral term will remain zero until the current command becomes
unsaturated. At the point the controller will return to normal operation. The results of this
controller are seen in figure 5.7. When the speed command is initially incremented the controller
reaches saturation until it reaches the new speed with almost no overshoot. When the speed is
increased again the rise time is increased, but there is no overshoot. When the reference is
dropped the current command is zero for approximately 0.12 seconds then it reaches the desired
speed. For the decreasing speed there is again no overshoot in the actual speed; however, there
is overshoot in the control signal.
Figure 5.7: Dynamic Closed Loop Speed Control
Minor modifications can be made to increase the speed controller’s performance during
both negative and positive saturation. When the current reference is saturated at the maximum
level, the current regulator can be turned off to allow single pulsing operation. In addition, the
advance and dwell angles can be increased in order to allow more torque production. For high
1000
2000
3000
4000
Sp
eed
(rp
m)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0
5
10
15
20
Time (sec)
Cu
rren
t C
om
man
d (
A)
(A)
70
performance deceleration in servo motors the opposite phase can be fired to apply negative
torque and to slow the motor even more rapidly. While this is certainly an option for the
PMSRM, it is not necessary.
5.4 PI Torque Control
The average torque control strategy presented in chapter 3 can be implemented as a
closed loop torque controller with both current and firing angle lookup tables. The PI speed
control designed in section 3 of this chapter is essentially a torque controller since the
mechanical model is a function of torque. Therefore, all that is needed to change the controller
to torque control is to scale the current reference to a torque reference with an additional
gain, 𝐾𝑇 , which is the ratio of the maximum torque to the maximum current reference.
Figure 5.8: Closed-loop Torque Control Block Diagram with Firing Angle Lookup
The timing structure of the controller remains the exact same with the firing angle lookup
added to the outer speed control loop which requires three two dimensional interpolations in
addition to computation required for the PI control. Since it occurs in the speed loop, which will
occur at a frequency between 2 and 5 kHz, the additional computation should not be a problem to
any DSP or MCU.
𝜔𝑚 𝜔∗
-
+ 𝑇𝑒 𝐺𝑝 𝐺𝑝𝜔
𝑇𝑙
𝐻𝜔
𝐺𝑐𝜔 𝐾𝑇 + +
𝜃𝑎
𝑖∗ 𝑇∗, 𝜔𝑟
𝜃𝑎 𝑖∗,𝜔𝑟
𝜃𝑑 𝑖∗, 𝜔𝑟
𝜃𝑑
𝑇𝑒∗ +
𝑖∗
71
Figure 5.9: Dynamic Closed Loop Torque Control Simulation with Disturbance Inputs
Dynamic simulation of the torque controller shows very similar performance to the speed
controller which is expected since the PI controller is the same for both approaches. The time
varying firings angles converge at the same rate that the current reference and the torque
command converges to a value very close to the load which indicates the gains are scaled
properly. In the case of the larger load there appears to be some oscillation of the advance angle
which comes from the interpolation in the lookup table occurring on the boundary of two values.
Increasing the number of samples of the firing angles during simulation would increase the
accuracy of the lookup tables and improve the torque controller’s performance.
5.5 Efficiency Searching Algorithm
Considering that energy efficiency and green technology have become important selling
points in commercial motor markets coupled with recent consumer electronic energy regulations,
maximizing the efficiency of the PMSRM is a critical task for its development and success. The
3580
3590
3600
3610
0
5
0
5
10
3
4
5
6
𝜃𝑎
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
10
14
18
Time (s)
𝜃𝑑
𝑖 ∗
𝑇 ∗
𝜔𝑟
3.8Nm
4.8Nm 2.8Nm
72
three areas of efficiency for a variable speed motor are converter efficiency, motor efficiency and
system efficiency, where converter efficiency is given by:
𝜂𝑐 =Σ𝑃𝑝𝑖
𝑃𝑖𝑛× 100%
(5.14)
The sum of the electrical power applied to each phase, the phase power, is represented
by Σ𝑃𝑖 , and 𝑃𝑖𝑛 is the input electrical power. The converter efficiency is in large part determined
by the design of the power electronics and the specific components used. Also, PWM chopping
The motor efficiency is represented as:
𝜂𝑚 =𝑃𝑜𝑢𝑡
Σ𝑃𝑝𝑖× 100%
(5.15)
The mechanical output of the system, 𝑃𝑜𝑢𝑡 , is the product of speed and torque. The
motor design, including copper and core losses are the primary factors in its efficiency, however
the control scheme is critical to maximizing this efficiency. The system efficiency is:
𝜂𝑠 = 𝜂𝑐 ∗ 𝜂𝑚 =𝑃𝑜𝑢𝑡
Pin× 100%
(5.16)
Thus the system efficiency is the product of the converter and motor efficiencies. When
marketing a motor’s efficiency, the system efficiency is the only number that truly matters. Yet
knowing the location of the losses is an important part in maximizing efficiency.
In order to measure and control the efficiency, the input and output power levels must be
measured. In typical laboratory testing the input and the phase power is measured with a power
meter or oscilloscope and the output power is measured by a torque meter connected to a
dynamometer or brake. However, for real time control, the power signal must be measured using
only sensors available on the controller which requires simplifications to be made for the input
power.
For the sake of simplicity it will be assumed that load torque will remain constant while
the optimization algorithm is running. If the controller notices a change in load, then the
algorithm will be run again. It is fair to say that the speed will be properly regulated by a well
designed speed regulator. Therefore, the output power is assumed to be constant. Another
73
assumption that must be made is that the converter efficiency will remain more or less constant,
and is based on the design of the converter as well as its devices.
With the output power assumed to be constant finding the most efficient operating point
is reduced to finding the combination of control signals resulting in the lowest phase power.
With a PMSRM there are three control variables, the current, the advance angle and the dwell
angle. With the current required by the speed regulator the advance and dwell angles can be
varied in order to characterize the efficiency of the motor. The advance angle can be held
constant while the dwell is slowly incremented. When either the dwell has reached its
predefined limit or the current is at a point of saturation then the cycle is done. The advance
angle is increased and the cycle begins again.
Measure efficiency
Advance In range?
Dwell In range?
Yes
Change dwell increment direction
No
Start
End
Increment advance
Increment dwell
Yes
Set advance and dwell to maximum
efficiencyNo
Figure 5.10: Efficiency Searching Algorithm Flow Chart
The searching algorithm requires a maximum and minimum range for dwell and advance
angles. The selection of these values must be limited so as to not allow the motor to enter an
74
operating region where it comes to a stop, or if it begins to have negative torque and spins in
reverse. The angles also must have enough breadth so that all potential operating points can be
tested to give an accurate performance map. In addition to the angular limits, the decision to
increment the dwell should also take into consideration the current reference signal from the
speed regulator. The current reference should not exceed any maximum levels which could
potentially cause harm to the motor.
The operating condition with the maximum efficiency is recognized as the point with the
smallest phase power which is calculated by multiplying the current feedback signal and the dc
bus voltage, both of which are readily accessible feedback signals on the converter. In order to
maintain accurate instantaneous calculations, the sampling frequency must be large with a large
filtering period. At the nominal speed the time of each electrical cycled can be calculated as:
3600𝑟𝑝𝑚
2𝜋60
180𝜋
10
−1
=1
2160°/𝑠 = 463𝜇𝑠
To get an accurate reading of each phase current the current and voltage need to be read
at least 20 times per cycle which would require a sampling frequency of 40 kHz. In addition, the
ADC on the DSP can be set to read multiple readings at a higher frequency. These readings are
then averaged to increase the resolution of the sampling. The algorithm to measure the
instantaneous power is:
Every 25𝜇𝑠:
1. Read 𝑖𝑎 , 𝑖𝑏 , 𝑉 𝑑𝑐 from the ADC
2. Average each of the inputs.
3. Calculate each phase power: 𝑃𝑝𝑎 = 𝑖𝑎 ∗ 𝑑𝑎 ∗ 𝑉𝑑𝑐 and 𝑃𝑝𝑏 = 𝑖𝑏 ∗ 𝑑𝑏 ∗ 𝑉𝑑𝑐
4. Calculate total phase power: 𝑃𝑝 = 𝑃𝑝𝑎 + 𝑃𝑝𝑏
5. Filter phase power
The filter for the power measurements will be an I.I.R. digital low-pass filter with a
cutoff frequency of approximately 10 Hz. The sampling frequency will be the speed of the
power calculations, which is 40 kHz, resulting in a normalized filter frequency of 0.0005.
75
Figure 5.11: Frequency Response of the Power Averaging Filter
This efficiency optimization loop needs to run at least half the speed of the filter’s cutoff
frequency, most likely around one second. This will allow for the advance and dwell angles
commands to be registered by the commutator and the speed regulator reach steady state.
However if more accuracy is desired the time can be increased. In general, there will be about
10 mechanical degrees of advance and 20 degrees of dwell to be tested. With the magnetic
sensor having a resolution of 0.5 degrees there are 1600 possible combinations to be tested. At a
1 Hz frequency the optimization will be made in approximately 30 seconds.
10 -1
10 0
10 1
10 2
10 3
-180
-135
-90
-45
0
Ph
ase
(deg
)
(deg
)
Frequency (Hz)
-100
-80
-60
-40
-20
0
Frequency (Hz): 10.1
Magnitude (dB): -3.15 M
agn
itu
de
(dB
)
(dB
)
76
Figure 5.12: Simulation of the Efficiency Searching Algorithm
Dynamic simulation of the firing angle searching algorithm shows the controller cycle
through the dwell and advance angle as the speed regulator maintains the speed. At the rated
speed of 3600 rpm the algorithm returned a dwell angle of 9 degrees and an advance angle of 5
degrees which produced a motor efficiency of 95%. From the simulation, it is clear that the
angle selection has a large effect on the ripple in the speed, although this does not necessarily
affect the efficiency. In addition, the speed ripple is at most ±5rpm which is less than one
percent of the actual speed.
Areas of possible problems in implementation of the efficiency searching algorithm are
accurately measuring power and determining the range of firing angles. If the power
measurement is inaccurate then the resulting angles could be any arbitrary efficiency yet as long
as the general trend is accurate then the algorithm will work. That is, the actual value of the
power is unimportant; the only thing that matters is the relative value of power at one firing
angle combination to another. Selection of the general range of firing angles is easy, but in
certain cases, such as when the dwell is at a maximum and the advance is at a minimum, the
3590
3600
3610
1000
2000
3000
5
10 15
0 1 2 3 4 5 6 7 8 0 5
10 15
𝜃𝑎
Time (s)
𝜃𝑑
𝑖 ∗
𝑃𝑖𝑛
𝜔𝑟 F
irin
g
Angle
s
77
regulator may not be able to maintain speed. If the range is too selective then the best efficiency
is not guaranteed. By augmenting the algorithm to recognize when the controller is saturated and
adjusting the angle this problem can be avoided.
5.6 Comparison of Speed Control Designs
This section has presented two general strategies for speed control of the PMSRM. The
first being a closed loop PI based speed control that uses set firing angles and a current control
input. This controller was modified with an efficiency searching algorithm to find the best set of
firing angles to minimize input power, thus finding the most efficient operating point. The
second strategy was to use a torque controller to determine reference current and firing angles
from a lookup table that found the maximum average torque. Simulation has shown both
methods to function properly for variable speed operation and disturbance rejection; however the
most important aspect of the control is the overall efficiency. The results from dynamic
simulations of the different efficiency control strategies are shown in table 5.4. The firing angles
reported for the torque control as well as the reference current for both control schemes are
averaged over the sample period to give a basis for comparison. The load is a constant 3.8Nm for
all speeds.
Table 5.4: Motor Efficiency of the Different Control Schemes
The results of simulation indicate that the efficiency searching algorithm can find higher
efficiency points that the average torque method. Although the average torque simulations find
1200rpm 1600rpm 2400rpm 3000rpm 3600rpm
Efficiency
Searching
Algorithm (Fixed
Firing Angles)
𝑖∗ = 5.85𝐴
𝜃𝑎 = 4°
𝜃𝑑 = 16°
𝜂 = 85.4%
𝑖∗ = 6.22𝐴
𝜃𝑎 = 4°
𝜃𝑑 = 15°
𝜂 = 86.8%
𝑖∗ = 6.2𝐴
𝜃𝑎 = 4°
𝜃𝑑 = 12°
𝜂 = 92.5%
𝑖∗ = 6.33𝐴
𝜃𝑎 = 4.5°
𝜃𝑑 = 11°
𝜂 = 95.1%
𝑖∗ = 6.74𝐴
𝜃𝑎 = 5°
𝜃𝑑 = 9°
𝜂 = 94.5%
Torque Control
(Dynamic Firing
Angles)
𝑖∗ = 5.85𝐴
𝜃𝑎 = 4.0°
𝜃𝑑 = 16.1°
𝜂 = 82.5%
𝑖∗ = 5.99𝐴
𝜃𝑎 = 4°
𝜃𝑑 = 15.3°
𝜂 = 82.8%
𝑖∗ = 6.15𝐴
𝜃𝑎 = 4.0°
𝜃𝑑 = 13.4°
𝜂 = 87.0%
𝑖∗ = 6.31𝐴
𝜃𝑎 = 4.5°
𝜃𝑑 = 12.2°
𝜂 = 88.0%
𝑖∗ = 6.54𝐴
𝜃𝑎 = 4.5°
𝜃𝑑 = 11.2°
𝜂 = 89.8%
78
the maximum torque production for the current input, the points of highest torque are not
necessarily the most efficient, and in these cases are not. The torque is being maximized for the
current reference, not the power being put in the motor, hence the efficiency and torque are not
directly related. The current reference is minimized for all speeds with torque control, which is
expected since the lookup tables were created to find the firing angles to maximize torque for
each current which requires a larger dwell angle to compensate for any torque lost to having a
smaller reference. In turn there is a decline in efficiency. As long as the power measurements
are accurate, which they are in simulation, the searching algorithm is guaranteed to find the most
efficient firing angles; therefore it can be used as the baseline for the motors performance.
79
6 Conclusions
6.1 Summary
This thesis has presented a control strategy for the unknown PMSRM. The approach
started by looking at the operation of the motor in terms of its control inputs, firing angles and
current reference and its outputs, speed and torque. The firing angles and reference current were
selected so as to maximize the average torque for any given current reference. Extensive
simulations were performed on the motor to produce an extensive four dimensional map of the
motors average torque output. Three lookup tables were created from this performance data to
later be used in the closed loop torque control of the motor.
Three different approaches for current control were presented including: hysteresis
control, anti-windup PI and MRAC. The addition of the anti-integrator windup in the PI
controller greatly improved its performance by avoiding the impending saturation every
excitation. Of the three types, the PI control seems to be the best choice for implementation due
to its above average performance and relative simplicity to program to a DSP. While MRAC
control can provide stability amidst nonlinearities and uncertainties its performance cannot be
guaranteed. The periodic turn-on turn-off process of PMSRM current control makes the benefits
of adaption negligent. In addition, the adaptation algorithm calls for more bandwidth than is
available on an imbedded digital controller and once implemented the performance would be
severely limited.
The speed controller design for the DSPMSRM started with a linear model of the motors
mechanical properties to design an anti-windup PI controller. This same controller was used for
both speed control, where the output of the controller is a current reference, and a torque
controller, where the output is a torque reference. The two designs showed comparable
performance results, however the average torque control is a more robust design because it can
more easily handle changes in operating point with the ability to varying all three control
parameters in a unified fashion. Speed control on the other hand has set firing angles which must
be adjusted for significant changes in load or speed. One way to solve this is to schedule the
firing angles based on the speed, or adaptively based on saturation of the current controller.
In chapter 5 simulations showed that the firing angles corresponding to maximum torque
do not relate to the points of maximum efficiency. Previous research ([9], [10]), has used
maximum torque as a general way to optimize efficiency, yet the results of this thesis prove
80
otherwise. The average torque controller had good performance and robust control, yet the
efficiency was up to 6% less than found by the efficiency searching algorithm. The discrepancy
has to do with the tradeoff of power used for longer excitation versus the magnitude of the
current reference.
6.2 Future Research
While this paper provided a complete overview of control design for the PMSRM there is
still a great amount of work to be done in both expanding the findings and the implementation of
the controllers. Firstly, the specific data found in this thesis was found through brute-force style
simulations that are specific to this design alone. The trends found between average torque,
firing angles and efficiency need to be verified not only on other PMSRMs but also applied to
the SRM. If possible, finding general relationships between the firing angles, current reference,
rotor speed and average torque would be immensely useful to control design and implementation
of the PMSRM. In addition, the average torque simulations done in this thesis focused on
minimizing the current reference while maximizing torque, however the relationship between the
average torque and the efficiency, if there is one, would be more useful. While much more
difficult to simulate, this could prove to be of higher significance than average torque control.
The implementation of this control must be verified experimentally.
81
References
[1] Nasar. S.A., D.C.-switched reluctance motor, in Proceedings of the Institution of
Electrical Engineers (London), Vol. 116, No. 6, June 1969, pp. 1048
[2] R. Krishnan, Switched Reluctance Motor Drives: Modeling, Simulation, Analysis,
Design and Applications. Boca Raton, Fla.: CRC Press, 2001.
[3] Cheewoo Lee; Krishnan, R.; Lobo, N.S., "Novel Two-Phase Switched Reluctance
Machine Using Common-Pole E-Core Structure: Concept, Analysis, and Experimental
Verification," Industry Applications, IEEE Transactions on , vol.45, no.2, pp.703-711,
March-april 2009
[4] Cheewoo Lee; Krishnan, R., "New Designs of a Two-Phase E-Core Switched Reluctance
Machine by Optimizing the Magnetic Structure for a Specific Application: Concept,
Design, and Analysis," Industry Applications, IEEE Transactions on , vol.45, no.3,
pp.1804-1814, Sept.-oct. 2009
[5] Xiaogang Luo; Dinyu Qin; Lipo, T.A., "A novel two phase doubly salient permanent
magnet motor," Industry Applications Conference, 1996. Thirty-First IAS Annual
Meeting, IAS '96., Conference Record of the 1996 IEEE , vol.2, no., pp.808-815 vol.2, 6-
10 Oct 1996
[6] R. Krishnan. Permanent Magnet Synchronous and Brushless DC Motor Drives. CRC,
2009.
[7] K. Ramu, “High Power Disunity Switched Reluctance Machines with Hybrid
Excitation”, Provisional US Patent Application, April 2010.
[8] Lobo, N. "Doubly-Salient Permanent-Magnet Flux-Reversal-Free Switched Reluctance
Machine,” Blacksburg, Va.: Virginia Polytechnic & State University, 2010.
[9] Fisch, J.H.; Yun Li; Kjaer, P.C.; Gribble, J.J.; Miller, T.J.E.; , "Pareto-optimal firing
angles for switched reluctance motor control ," Genetic Algorithms in Engineering
Systems: Innovations and Applications, 1997. GALESIA 97. Second International
Conference On (Conf. Publ. No. 446) , vol., no., pp.90-96, 2-4 Sep 1997
[10] Gribble, J.J.; Kjaer, P.C.; Miller, T.J.E.; , "Optimal commutation in average torque
control of switched reluctance motors," Electric Power Applications, IEE Proceedings - ,
vol.146, no.1, pp.2-10, Jan 1999
82
[11] Blanque, B.; Perat, J.I.; Andrada, P.; Torrent, M.; , "Improving efficiency in switched
reluctance motor drives with online control of turn-on and turn-off angles," Power
Electronics and Applications, 2005 European Conference on , vol., no., pp.9 pp.-P.9, 0-0
0
[12] Sozer, Y.; Torrey, D.A.; , "Optimal turn-off angle control in the face of automatic turn-on
angle control for switched-reluctance motors," Electric Power Applications, IET , vol.1,
no.3, pp.395-401, May 2007
[13] Schulz, S.E.; Rahman, K.M.; , "High-performance digital PI current regulator for EV
switched reluctance motor drives," Industry Applications, IEEE Transactions on , vol.39,
no.4, pp. 1118- 1126, July-Aug. 2003
[14] Szamel, L.; , "Convergence test of model reference parameter adaptive SRM drives,"
Power Electronics and Applications, 2005 European Conference on , vol., no., pp.10 pp.-
P.10, 0-0 0
[15] Bortoff, S.A.; Kohan, R.R.; Milman, R., "Adaptive control of variable reluctance motors:
a spline function approach," Industrial Electronics, IEEE Transactions on , vol.45, no.3,
pp.433-444, Jun 1998
[16] Le-Huy, H.; Dessaint, L.A.; , "An adaptive current control scheme for PWM synchronous
motor drives: analysis and simulation," Power Electronics, IEEE Transactions on , vol.4,
no.4, pp.486-495, Oct 1989
[17] N. Hovakimyan, “ME6984 Adaptive Control Course Notes”, Blacksburg, VA: Virginia
Polytechnic and State University, 2009.
83
Appendix A: 8/10 4ecore PMSRM Specifications
Maximum gir gap radius, mm 0.3
Maximum air gap radius, mm 0.6
Stator outer radius, mm 75
Shaft radius, mm 11.26
Shared pole arc, deg 36
Stator pole arc, deg 12
Rotor pole arc, deg 19
Stack length, mm 55
Winding turns per pole 103
Winding resistance, ohm 0.527
Magnet type NdFeB N42
Magnet thickness, mm 1
Table A.1: Dimensions of the 4ecore PMSRM
84
Appendix B: Deadbeat Current Controller Design
The simulations used to find the average torque based on varied firing angles and
reference currents assumed an ideal current controller to get results that are unbiased by the
controllers performance. The digital deadbeat current controller solves the inverse of the
PMSRM’s electrical equation to find the ideal voltage command. This type of control is an
inherently discrete way to solve a difference equation; however it requires knowledge of the
future inductance and PM flux. In simulation, the same data that is used for the dynamic
electrical model simulation can also be used for the future current.
Numerically, the solution to ordinary differential can be found by forward rectangular
approximation as:
𝑥 𝑘 =𝑥 𝑘 + 1 − 𝑥[𝑘]
𝑇
(A2.1)
The future state of the PMSRM can be found by using definition of (A2.1) in the
electromagnetic equation of (2.8). That is:
𝑣 𝑘 = 𝑖 𝑘 𝑅𝑠 +𝐿 𝑘 + 1 − 𝐿 𝑘
𝑇 + 𝐿 𝑘
𝑖 𝑘 + 1 − 𝑖 𝑘
𝑇+
𝜆 𝑘 + 1 − 𝜆 𝑘
𝑇
(A2.2)
For deadbeat control the assumption must be made that the current will reach the
reference in the next time period. That is, 𝑖 𝑘 + 1 = 𝑖∗[𝑘 + 1]. Therefore, the future voltage
can be calculated as:
𝑣∗ 𝑘 = 𝑖 𝑘 𝑅𝑠 +𝐿 𝑘 + 1
𝑇− 2
𝐿 𝑘
𝑇 + 𝑖∗ 𝑘 + 1
𝐿 𝑘
𝑇+
𝜆 𝑘 + 1 − 𝜆 𝑘
𝑇
(A2.3)
The lookup of inductance and flux can be interpolated in the same fashion that they are
for the dynamic simulation. Although these look up terms will most likely be unknown in a real
situation, their exact values are known during the simulation. In order to make the current rise
realistic it is necessary to limit the voltage command to the dc link voltage so as to not command
an unrealistic voltage. While the actual control used through simulation is not going to be this
ideal, the use of deadbeat current control for the simulations in section 3 allows for the most
accurate relationships between firing angles, reference current and the average torque.
