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Natural Variable Modeling and Performance of Interior Permanent Magnet Motor with Concentrated and Distributed Windings
By
Aliyu, Nasiru
PG/M.ENG/12/62492
DEPARTMENT OF ELECTRICAL ENGINEERING
FACULTY OF ENGINEERING
UNIVERSITY OF NIGERIA, NSUKKA
NOV, 2014
2
UNIVERSITY OF NIGERIA, NSUKKA
DEPARTMENT OF ELECTRICAL ENGINEERING
NATURAL VARIABLE MODELING AND PERFORMANCE OF INTERIOR PERMANENT MAGNET MOTOR WITH
CONCENTRATED AND DISTRIBUTED WINDINGS
A PROJECT REPORT SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENT FOR THE AWARD OF MASTERS OF
ENGINEERING DEGREE (M.ENG) IN ELECTRICAL ENGINEERING
BY
ALIYU, NASIRU
PG/M.ENG/12/62492
NOVEMBER, 2014
3
UNIVERSITY OF NIGERIA, NSUKKA
DEPARTMENT OF ELECTRICAL ENGINEERING
NATURAL VARIABLE MODELING AND PERFORMANCE OF INTERIOR PERMANENT MAGNET MOTOR WITH
CONCENTRATED AND DISTRIBUTED WINDINGS
BY
ALIYU, NASIRU
PG/M.ENG/12/62492
NOV, 2014
AUTHOR: ……………………… ENGR. NASIRU ALIYU SUPERVISOR: ………………………. ENGR. PROF. E. S. OBE EXTERNAL EXAMINER: ………………………. ENGR. PROF. S. N. NDUBISI HEAD OF DEPARTMENT: ………………………. ENGR. PROF. E. C. EJIOGU
4
TITLE PAGE
NATURAL VARIABLE MODELING AND PERFORMANCE OF INTERIOR PERMANENT MAGNET MOTOR WITH
CONCENTRATED AND DISTRIBUTED WINDINGS
NOVEMBER, 2014
5
CERTIFICATION
Engr. Nasiru Aliyu, a postgraduate student in the Department of Electrical Engineering
with Registration Number, PG/M.ENG/12/62492, has satisfactorily completed the
requirements for the course work and project report for the Degree of Masters of
Engineering in the Department of Electrical Engineering, University of Nigeria Nsukka.
The work contained in this thesis is original and has not been submitted in part or full for
any other diploma or degree of this or any other university to the best of my knowledge.
…………………………… ……………….……………
ENGR. PROF. E. S. OBE ENGR. PROF. E. C. EJIOGU
Supervisor Head of Department
6
DEDICATION
This work is dedicated to the unprecedented support of my parents Sheikh Alhaji Aliyu
Babando and Hajiya Aishatu Garba Muri (Asabe) who have been very supportive to me
to fulfill my academic goals.
7
ACKNOWLEDGEMENT
I hereby express my sincere gratitude to my supervisor, Engr. Prof. E. S. Obe for giving
me the privilege to benefit from his great intellectual resources, the time devoted to this
work, the material made available and advice he always dished out to me, I appreciate all.
I also use the opportunity to thank the Head of Department, Electrical Engineering. Engr.
Prof. E. C. Ejiogu for his support and pioneering the affairs of this great department well
and Engr. Prof. L. U. Anih for his moral support and encouragement.
My appreciation also goes to Engr. Prof. M. U. Agu, former Head of Department, Engr.
Dr. B. O. Anyaka Engr. Dr. C. I. Ode, and Engr. B. A. Ugochukwu for their support and
encouragement.
Naturally, I thank my family for bringing so much happiness into my life and my loving
wife Rashida and daughter Gumaisa’u for there understanding during this period of my
programme, and my brother Dr. Abubakar Aliyu Babando for downloading some
materials for this project report from Kwazulu Natal University South Africa.
Finally, I wish to appreciate my colleagues for assisting me in one way or another.
8
ABSTRACT
Interior Permanent Magnet (IPM) motor is widely used for many industrial applications
and has relatively high torque ripple generated by reluctance torque. Since the
configuration of the stator has great influence on reluctance torque, different stator
configuration is necessary to improve the torque performance of IPM motor. Natural
variable modeling and performance comparison of Interior Permanent Magnet Motor
with Concentrated winding (CW), Short pitched and Full pitched distributed winding
(DW) is presented in this project report. Three phase Interior Permanent Magnet Motor
with identical rotor dimensions, air gap length, series turn number, stator outer radius,
and axial length was studied with different stator winding configuration. Basic
parameters and machine performance, such as inductances, copper losses, power density,
efficiency at high and low speed, torque ripple, rotor speed with load torque, phase
currents, electromagnetic torque, controllability and demagnetization tolerance are
compared. As a means of supplementing analysis of the IPM motor, winding function
theory (WFT) is used to analyze the motor. Winding function theory has enjoyed success
with induction, synchronous, and even switched reluctance machines in the past. It is
shown that this method is capable of analyzing IPM motor with different stator
configuration and the simulations were carried out by using Embedded MATLAB
function. It was observed that, the concentrated winding IPM motor has a lower copper
loss of 0.3 kw and 3.7 kw at low and high speed respectively and 133 Nm high peak
torque developed, pull out power of 58 kw, torque ripple of 96 Nm, average torque of
142 Nm, demagnetization tolerance of 60%, amplitude of the fundamental winding is
26.45 and efficiency of 89. the short pitched distributed winding IPM motor has a lower
copper loss of 0.35 kw and 3.6 kw at low and high speed respectively and 116 Nm high
peak torque developed, pull out power of 57 kw, torque ripple of 71 Nm, average torque
of 185 Nm, demagnetization tolerance of 78%, amplitude of the fundamental winding is
27.53 and efficiency of 87. As for full pitched distributed winding IPM motor has a lower
copper loss of 0.35 kw and 3.6 kw at low and high speed respectively and 116 Nm high
peak torque developed, pull out power of 56 kw, torque ripple of 71 Nm, average torque
of 185 Nm, demagnetization tolerance of 78%, amplitude of the fundamental winding is
29.3 and efficiency of 88.
9
TABLE OF CONTENTS
TITLE PAGE ……………………………………………………………………………..i
CERTIFICATION ………………………………………………………………………..ii
DEDICATION …………………………………………………………………………...iii
ACKNOWLEDGEMENT …………………….…………………………………………iv
ABSTRACT …………………………………………………………….………………..v
TABLE OF CONTENTS ……………………………………………………..………..vii
LIST OF FIGURES & DIAGRAMS ……………………………………………………x
LIST OF TABLES …………………………………………………………….……….viii
CHAPTER ONE
1.0 Introduction………………….……………………………………………….1
1.1 Overview ………….…………………………………………………………1
1.2 Research Objectives …………………………………………………………6
1.3 Thesis Outline ………………………………………………………………..6
1.4 Study limitation ……………………………………………………………...7
CHAPTER TWO
2.0 Literature Review …. ………………………………………………………..8
2.1 Introduction ………………………………………………………………….8
2.2 Permanent Magnet Materials ………………………………………………...8
2.3 IPM Machine Technology ………………………………………………...... 9
2.4 Winding Function Theory ………………………………………….…….…15
2.5 Why Winding Function Theory ………………………………………… ..17
10
CHAPTER THREE
3.0 Analysis of IPM Motor with Winding Function Theory..………………….18
3.1 Introduction ………………………………………………………………..18
3.2 Winding Function Theory and its Modifications …………………….…....18
3.2.1 Basic Winding Function Theory …………………………………18
3.2.2 WFT for machines with salient air gaps …………………………22
3.2.3 WFT Applied to magnetic devices ………………………………28
3.2.4 Verification of a single phase per rotor ………………………….30
3.2.5 Matlab Program for Solving Machine Equations ………………..33
3.2.6 Torque calculated from inductance ………………………………39
3.3 Clock diagram of IPM motor ………………………………………………41
3.4 Total Harmonic Distortion (THD)………………………………………….41
3.5 Winding factor (kw)……..………………….………………………….……42
3.6 Slot-fill factor ………………………………………………………….…...45
3.7 The Voltage Equations ……………………………….…………………….45
3.8 Solution of Equation (3.52) ……………………………………………….48
3.9 Torque Ripple ………………………………………………………………49
3.10 Losses in IPM motor ………………………………………………………49
3.10.1 Core Loss ……………………………………………………….50
3.10.2 Magnet Loss ………………………………………………….....50
3.10.3 Stator Winding Loss …………………………………………….51
3.10.4 Mechanical Losses ………………………………………………51
11
CHAPTER FOUR
4.0 Dynamic Simulation in MATLAB Simulink ………………………………52
4.1 Simulation Tools …………………………………………………………...52
4.2 Simulink Simulation of IPM with Short pitched, Full pitched & Concentration
Winding using Embedded MATLAB Function Blocks …………………...52
4.3 Simulation Results …………………………………………………………52
4.4 Discussions ………………………………………………………………...66
CHAPTER FIVE
5.0 Conclusion and Recommendation …………………………………………69
5.1 References …………………………………………………………………70
Appendix 1……………………………………………………………………………77
Appendix 2……………………………………………………………………………78
Appendix 3……………………………………………………………………………80
Appendix 4……………………………………………………………………………82
Appendix 5……………………………………………………………………………83
Appendix 6 …………………………………………………………………………...84
12
LIST OF FIGURES AND DIAGRAMS
Figure 1.1: Classification of AC machine types ………………………………………….1
Figure 1.2: Various IPM rotor geometries ………………………………………………..3
Figure 1.3: Various stator winding layouts ……………………………………………….5
Figure 2.1: Flux Density vs Magnetizing Field of Permanent Magnetic Materials …........9
Figure 2.2: Rotor structures of interior IPM type………………………………………..11
Figure 3.1: An idealized machine with conductors in the air gap……………………….19
Figure 3.2: An idealized machine with multiple coils in the air gap…………………….21
Figure 3.3: An idealized machine with a salient rotor and conductors in the air gap……22
Figure 3.4: Fundamental MMF Diagram for Short pitched phase A ………………...….24
Figure 3.5: Fundamental MMF Diagram for Full-Pitched phase A ………………….…24
Figure 3.6: Fundamental MMF Diagram for Concentrated phase A ……………………25
Figure 3.7: Fundamental MMF Diagram for Short pitched Distributed Winding ………25
Figure 3.8: Fundamental MMF Diagram for Full-Pitched Distributed Winding ……….26
Figure 3.9: Fundamental MMF Diagram for Concentrated Winding …………………..26
Figure 3.10: Four Poles magnet flux density ……………………………………………27
Figure 3.11: Airgap ……………………………………………………………………..27
Figure 3.12: Model of a PM which produces an equivalent magnetization vector ……..29
Figure 3.13: IPM with alternating vectors form the basis for a single turns function over
the circumference of the rotor …………………………………………………………..30
Figure 3.14: The turns function modeled with the Fourier series ………………………30
Figure 3.15: An ideal four pole machine ………………………………………………..31
Figure 3.16a: Turns and winding functions for stator phases ……………………….…..32
13
Figure 3.16b: WF, from top figure to bottom: winding arrangements in the slots turns
function of winding A, WF of windings A, B and C……………………………………33
Figure 3.17: Self inductance of the stator winding for Short pitched ………………….34
Figure 3.18: Self inductance of the stator winding for Full-Pitched …………………...34
Figure 3.19: Self inductance of the stator winding for Concentrated …………………..34
Figure 3.20: Mutual inductance between stator winding for Short pitched …………….35
Figure 3.21: Mutual inductance between stator winding for Full-pitched ………………35
Figure 3.22: Mutual inductance between stator winding for Concentrated ……………..35
Figure 3.23a: Flowchart for calculated inductances using MWFT in Matlab/Simulink ..36
Figure 3.23b: Flowchart of the iterative procedure for solving machine equations …….37
Figure 3.24: Inverse gap function used for ideal machines in inductance calculations …38
Figure 3.25: Harmonic Order for phase A ………………………………………………38
Figure 3.26: Clock Diagram for Short pitched Distributed Stator winding..……..……..42
Figure 3.27: Clock Diagram for Full pitched Distributed Stator Winding ………….…..43
Figure 3.28: Clock Diagram for Concentrated Stator Winding ………………………....44
Figure 4.1: Stator phase A’s currents …………………………………………………...55
Figure 4.2: d-axis rotor currents …………………………………………………………55
Figure 4.3: q-axis currents ………………………………………………………………56
Figure 4.4: Electromagnetic torques ……………………………………………………56
Figure 4.5: Electromagnetic torques at synchronization ………………………………..57
Figure 4.6: Electromagnetic torques when the load was applied …………………….....57
Figure 4.7: Short pitched Rotor Speeds and its response to a load torque ……………..58
Figure 4.8: Full pitched Speeds during starting and response to a load torque ………..58
14
Figure 4.9: Concentrated Speeds during starting and its response to a load torque …..58
Figure 4.10: Rotor Speeds during synchronization for Short pitched …………………59
Figure 4.11: Rotor Speeds during synchronization for full pitched. …………………..59
Figure 4.12: Rotor Speeds during synchronization for Concentrated …………………59
Figure 4.13: Rotor Speeds response to a load torque for Short pitched ……………….60
Figure 4.14: Rotor Speeds response to a load torque for Full-pitched..………….…….60
Figure 4.15: Rotor Speeds response to a load torque for Concentrated ………………..60
Figure 4.16: Rotor Speeds with the application of ramp for Short pitched …………….61
Figure 4.17: Rotor Speeds with the application of ramp for Full-pitched ……………...61
Figure 4.18: Rotor Speeds with the application of ramp for Concentrated ……………..61
Figure 4.19: Chorded Torque-Speed characteristic ……………………………………..62
Figure 4.20: Non-Chorded Torque-Speed characteristic ………………………………..62
Figure 4.21: Concentrated Torque-Speed characteristic ………………………………..62
Figure 4.22: Output power against stator current for Short pitched ……………………63
Figure 4.23: Output power against stator current for Full-pitched ……………………..63
Figure 4.24: Output power against stator current for Concentrated …………………….63
Figure 4.25: Stator losses against stator current for Short pitched …………………..…64
Figure 4.26: Stator losses against stator current for Full-pitched ……………….………64
Figure 4.27: Stator losses against stator current for Concentrated ……………….…….64
Figure 4.28: Efficiency against stator current for Short pitched ……………………….65
Figure 4.29: Efficiency against stator current for Full-pitched ………………………...65
Figure 4.30: Efficiency against stator current for Concentrated ………………………..65
15
LIST OF TABLES
Table 2.1: Advantages and Disadvantages of different magnet type ………………..…10
Table 4.1: IPM with Short pitched, Full-pitched and Concentrated winding ………..…53
Table 4.2: Summary of the Performance comparison of the each IPM Motor …………54
16
CHAPTER ONE
1.0 Introduction
1.1 Overview
Over the years, the application of electric motors has replaced vast numbers of
mechanical rotating devices. From tiny motors used in wristwatches, to very large motors
used for ship propulsion and wind turbines. There are numerous types of electric motors
available for present-day applications, of which the AC types are most commonly used in
high performance applications due to its increased efficiency and excellent dynamic
performance [1]. The classifications of common types of AC motors are shown in Figure
1.1[2].
Figure 1.1: Classification of AC machine types
The Induction, Surface Permanent Magnet (SPM), Inset Permanent Magnet Machine, and
Interior Permanent Magnet (IPM) machine types have already been applied to present
day drive systems. Induction, SPM and inset PM machines usually have a lower power
rating compared to the IPM machine and are most commonly applied as an Integrated
17
Motor Assist (IMA) system, where the main driver of the vehicle is the internal
combustion engine while the electric motor assists. On the other hand, the IPM machine
itself produces up to 73kW or more of power and can be driven in full electric mode,
producing zero emissions. [2]
This project report will focus on the IPM machine type, which is generally preferred due
to three main reasons: Firstly, the buried magnets make the rotor structurally stronger,
which make it more capable of withstanding higher speeds. Secondly, the additional
useful reluctance torque, resulting from the salient pole structure, thus giving the motor
greater field-weakening capabilities. Additionally, this saliency allows sensorless control,
properties which the SPM does not offer [3]. Lastly, the possibility of changing the
geometry of buried magnets in the rotor makes it possible to employ flux concentration,
and provides the possibility of saliency ratio optimisation [4].
With the availability of high energy permanent magnet materials and advanced power
electronics, the fields in which IPM machines can be applied to are rapidly broadening.
They include aerospace, nautical, automobile, rail transportation, medical, generation and
industrial process automation [5]. Common magnet geometries include single-piece/pole,
rectangular shaped magnet design (Figure 1.2a), segmented magnet design (Figure 1.2b),
v-shaped magnet design (Figure 1.2c), and the multi-barrier design (Figure 1.2d) [6].
Each of these designs has its advantages and disadvantages: The single-piece/pole
magnet design, for example, is the easiest to manufacture, but has larger magnet losses
compared to the other designs due to the larger magnet pole surface [7]. The segmented
magnet design has lower magnet losses and better field-weakening capability but requires
more magnet pieces. It also results in decreased magnet flux density due to the leakage
18
flux in the iron bridges [8]. The v-shaped magnet design provides flux concentration but
also requires more magnet pieces compared to the single-piece/pole design.
(a) rectangular single-piece/pole magnets (b) segmented magnets
(c) v-shaped magnets (d) multi-barrier magnets
Figure 1.2: Various IPM rotor geometries
The multi-barrier magnet design creates a very high saliency ratio, but consequently
results in an increased amount of structural stress on the rotor steel [6]. In practice, there
is no single rotor that can satisfy all applications. The pros and cons of each design as
well as more specific magnet type and shape have to be altered to meet desired
specifications. As magnets are very brittle, there are also practical limits to the
manufacturability of the magnets.
19
Before the 21st century, the majority of IPM machines were designed with distributed
stator windings (DW). The use of concentrated winding (CW) was not popular due to a
poor torque to magnetomotive force (MMF) ratio. However in the early 21st century,
Cros and Viarouge [9], Magnussen and Sadarangani [10] proved that by an appropriate
choice of slot and pole combination, the winding factor can be significantly increased,
thus increasing output torque. Additionally it was also shown that with appropriate slot
and pole combination, cogging torque can also be reduced.
Stator windings can either be single-layer or double-layer. The choice depends on the
desired machine performance characteristics. Single-layer CW creates high self-
inductance and low mutual-inductance which leads to better fault-tolerant capability. On
the other hand, double-layer CW has lower airgap MMF harmonic components, thereby
resulting in smaller torque ripples and lower magnet eddy current losses [11]. The
winding layouts for single-layer and double-layer DW are shown in (Figure1.3a) and
(Figure 1.3b), while the layouts for single-layer and double-layer CW are shown in
(Figure1.3c) and (Figure 1.3d) respectively [12].
Finally, the study is primarily concerned with the natural variable modeling and
performance comparison of Interior permanent magnet (IPM) motors with a short pitched
distributed winding, full-pitched distributed winding and concentrated winding. An
Interior permanent magnet (IPM) motor has many advantages such as high power
density, efficiency and wide speed operation. These merits make it particularly suitable
for automotive and other applications where space and energy savings are critical.
However, IPM motor has relatively high torque ripple generated by reluctance torque
which results in noise and vibration [13].
20
(a) Single-layer distributed windings (b) Double-layer distributed windings
(c) Single-layer concentrated windings (d) Double-layer concentrated windings
Figure 1.3: Various stator winding layouts
Torque ripple in IPM motor is often a major concern in applications where speed and
position accuracy are great important. Since the component configuration such as a stator
has great impact on reluctance torque, different stator configuration is necessary to
improve the performance of IPM motor [14]. Most previous works to obtain optimal
design for torque ripple reduction have been restricted to size optimization in which
design parameters are known in priori and fixed throughout the optimization process
[15]. The size design variables include slot opening, depth of rotor yoke, angle of one
pole magnet, thickness of permanent magnet and so on.
21
1.2 Research Objectives
The general objective of this project report is firstly to present a feasibility study on the
IPM motor with different stator configuration. Subsequently to carryout a natural variable
modeling and performance comparison on a three phase IPM motor with short pitched
distributed winding, full-pitched distributed winding and concentrated winding. Despite
the fact that the machines are now widely in use, there has been considerable interest in a
three phase Interior Permanent Magnet Motor. Finally, the study will help clarify the
advantages and disadvantages of implementing the different stator configuration in IPM
motor, as well as its prospects in industrial applications requiring high efficiency.
1.3. Thesis Outline
In order to conduct the stated project objectives, this project report is outlined as
following:
Chapter I covers overview and some backgrounds on interior permanent magnet motor.
In this chapter the main objectives and motivations of this project, thesis outline and
study limitation are introduced.
Chapter II, this chapter gives the literature reviews on Interior Permanent Magnet
machine technology, permanent magnet materials, winding function theory and why
winding function theory.
Chapter III gives the analysis of Interior Permanent Magnet Motor with winding function
theory and its modifications which include basic winding function theory, winding
function theory for machines with salient air gaps, winding function theory applied to
magnetic devices, verification of a single phase per rotor and Torque calculated from
22
inductance. It also include the clock diagram of IPM motor, Total Harmonic Distortion
(THD), winding factor (kw), Slot-fill Factor, the Voltage Equations, Torque Ripple and
Losses in IPM motor which includes the core loss, magnet loss, stator winding loss and
mechanical losses
Chapter IV is a set performance, simulation and results.
Chapter V is a Conclusions and Recommendations.
1.4 Study Limitations
A major limitation encountered in this project report thesis occurred during the modeling
of the different stator configuration of the machines and only the first harmonics was
used since third harmonic would be eliminated if the motor is connected in star.
Eventually, certain assumptions deemed necessary.
23
CHAPTER TWO
2.0 Literature Review
2.1 Introduction
Distributed Winding IPM machines have been widely used in high performance
industrial applications over decades. Concentrated Winding IPM machines on the other
hand have only more recently found their way into industrial markets. This section
reviews permanent magnet materials and well established work in IPM machine
technology as well as recent research in winding function theory for AC machines.
2.2 Permanent Magnet Materials
The properties of the permanent magnet material will affect directly the performance of
the motor and proper knowledge is required for the selection of the materials and for
understanding PM motors.
The earliest manufactured magnet materials were hardened steel. Magnets made from
steel were easily magnetized. However, they could hold very low energy and it was easy
to demagnetize. In recent years other magnet materials such as Aluminum Nickel and
Cobalt alloys (ALNICO), Strontium Ferrite or Barium Ferrite (Ferrite), Samarium Cobalt
(First generation rare earth magnet) (SmCo) and Neodymium Iron-Boron (Second
generation rare earth magnet) (NdFeB) have been developed and used for making
permanent magnets. The rare earth magnets are categorized into two classes: Samarium
Cobalt (SmCo) magnets and Neodymium Iron Boride (NdFeB) magnets. SmCo magnets
have higher flux density levels but they are very expensive. NdFeB magnets are the most
24
common rare earth magnets used in motors these days. A flux density versus magnetizing
field for these magnets is illustrated in Figure 2.1 [16].
Figure 2.1 Flux Density versus Magnetizing Field of Permanent Magnetic Materials
NdFeB magnets are the preferred choice in present day applications due to its reasonable
cost, high coercivity and high remanent flux density. However, NdFeB magnets have
comparatively low operating temperatures. Table: 2.1 shows the advantages and
disadvantages of different magnet types [17].
2.3 IPM Machine Technology
A permanent magnet synchronous motor (PMSM) is a motor that uses permanent
magnets to produce the air gap magnetic field rather than using electromagnets. These
motors have significant advantages, attracting the interest of researchers and industry for
use in many applications.
25
Table: 2.1
Advantages and disadvantages of different magnet types
Magnet type Advantages Disadvantages
Ferrite Least expensive magnet
material
High operating
temperature up to 300oC
Hard and brittle
Lowest remanent flux
density of up to 0.42T
Alnico High operating
temperature up to 520 ºC
Lowest temperature
coefficient (0.02%/ºC)
Extremely hard and
brittle
Can be easily
demagnetised
Samarium-cobalt High remanent flux
density of up to 1.16T
Extremely resistant to
corrosion
High resistance to
demagnetisation
High operating
temperature of up to
350ºC
Low temperature
coefficient (0.04%/ºC)
High coercivity
Extremely hard and
brittle
Most expensive
magnetic material
Neodymiumiron-boron Highest remanent flux
density of up to 1.48T
High resistance to
demagnetisation
Least brittle
Lower cost compared to
SmCo
High coercivity
Low operating
temperature up to 200ºC
High temperature
coefficient (0.12%/ºC)
Susceptible to corrosion
26
The recent development of improved ferrites and rare earth cobalt magnet materials has
caused a renewed interest in permanent magnet machine design. These materials have
straight-line BH characteristics in the second quadrant so that demagnetization is of a
lesser concern than with the earlier Alnicos. Such machines have their permanent
magnets inserted below a rotor squirrel cage winding and are known as Interior
Permanent Magnet Synchronous Motors (as shown in Figure 2.2) which as in the case of
an induction motor lies in slots near the air gap. The squirrel cage currents provide
starting torque while the magnet torque pulls the rotor into synchronism so that under
steady state operation there are no rotor I2R losses. Interior Permanent Magnet Motors
have come to be replacement for high efficient induction machines in small power
applications of two to 25 horsepower [18]. By enhancing this motor parameter it was
suggested that improved performance can be obtained over a wider speed range. These
claims were contested in a latter publication by Richter and Neumann [19] in which it
was shown that for some IPM machines, the cross coupling reactance is saturation
dependent and therefore disappears at low flux high speed operating conditions.
Figure 2.2: Rotor structures of interior IPM type.
27
Global research on the IPM machine dates back to the late 1970s and early 1980s when
the first few papers were published; there is still very significant research interest in this
area today. The popularity of the IPM machine is due to the embedded structure of its
magnets, which lowers the risk of demagnetisation, increases the mechanical robustness
of the rotor and provides additional reluctance torque. The small airgap design in most
IPM machines makes it excellent for flux weakening, as the negative armature reaction
can effectively reduce airgap flux. The IPM machine also gives the machine designer the
freedom to vary the magnet pole geometry, thereby broadening the machine’s area of
application.
Leading studies in IPM machine technology has included patents and several papers
setting the basis for research in this area. Steen [20] filed a patent on synchronous motors
with buried permanent magnets having several geometrical configurations. He stated that
the buried magnets produced additional direct-axis (d-axis) flux in aid of the flux
generated by the inductive copper bars during no-load operation. Honsinger [21]
illustrated a detailed mathematical representation of the IPM machine which included its
magnetic fields and parameters. Rahman et al. [22] presented the equivalent circuit model
to determine the d-axis and quadrature-axis (q-axis) reactance. Consoli and Renna [23]
illustrated a detailed representation of the IPM machine in the rotor reference frame, and
demonstrated an equivalent circuit model to determine iron losses. Chalmers et al. [24]
presented a study of the IPM machine through extensive experiments with frequency
variations.
Jahns et al. [25] first addressed the IPM machine’s characteristics when used as a high-
performance variable speed drive. Jahns [26] then performed a novel study on the flux
28
weakening of the IPM machine, thereby successfully extending its constant power speed
range. Since then, the use of IPM machines has soared. Honda et al. [27] did a study on
the effects of various winding types and rotor configuration on the performance of the
IPM machine. The IPM machine’s characteristics were constantly compared to those of
other AC machines. Fratta et al. [28] compared the torque density ability of the IPM
machine with the induction machine, and highlighted that the IPM would have better
electromagnetic performance if mechanical issues were resolved.
A controllability comparison between the IPM and SPM machines under various
operating requirements of the current vector control scheme was done by Morimoto et al.
[29]. Zhu et al. [30] compared the iron loss between the IPM and SPM machines. They
indicated that the iron loss of the IPM machine would be lower under open-circuit
conditions but significantly higher in the field-weakening region compared to the SPM
machine, due to the increased harmonic content in the armature field. Kyung-Tae et al.
[31] compared the effects of rotor eccentricity on the IPM and SPM rotor, in which they
concluded that the IPM is more prone to the effects of rotor eccentricity. Jung Ho et al.
[32] studied the inductance variation of a hybrid synchronous reluctance/IPM motor and
found out that the addition of buried magnets increased the saliency ratio, thus increasing
output torque and power factor.
The end of the 20th century saw drastic improvements in computational resources and
techniques. This allowed machine designers to efficiently determine parameters and
perform optimisation strategies. Yamazaki [33] illustrated a method to calculate IPM
machine parameters, including rotor and stator iron losses. Efficiency optimisation by
geometric variation was carried out by Sim et al. [34]. Ki-Chan et al. [35] studied the
29
effects on machine parameters and torque performance by varying the shape of rotor link
sections. They showed that small variations of the link sections could significantly
saliency ratio. Parsa and Lei [36] studied the effects of torque ripple and performance
characteristics when key machine parameters were varied. Kioumarsi et al. [37]
attempted to reduce torque ripple and by drilling additional holes in the rotor. Han et al.
[38] attempted to reduce torque ripple by varying the number of slots and number of rotor
barriers. They showed that multi-barrier rotors with an odd number of slots per pole pair
resulted in low torque ripple. Sanada et al. [39] experimented with several designs and
proved that the use of asymmetric flux barriers was beneficial in reducing torque ripple in
multi-barrier IPM machines. Fang at al. [40] showed that torque ripple can be reduced
with a double-layer rotor. Kim at al. [41] studied the effects of geometric variations of
magnets in the IPM machine to reduce torque ripple.
Computational methods also made the calculation of losses more accurate and realisable.
Loss minimisation was also made less costly and more effective. Kawase [42] analysed
Permanent magnet eddy current losses in the IPM machine with three-dimensional finite
element (FE) analysis, and showed the effectiveness of reducing eddy current losses by
axially dividing the magnets. Zivotic-Kukolj [43] proposed geometric variations to the
multi-barrier IPM machine to reduce iron losses. Ionel et al. [44] improved the accuracy
of modeling core losses by allowing hysteresis loss to vary with both flux density and
frequency, but leaving the eddy current and excess losses to vary only with flux density.
Wang et al. [45] studied the effects of temperature on the torque performance and
machine losses in an IPM machine. Seo et al. [46] studied iron loss on machines with
integral and fractional slot DW. They showed that with fractional-slot configuration, iron
30
loss in the stator is reduced slightly, but iron loss in the rotor is increased significantly,
especially at high speeds. Yamazaki and Abe [47] further investigated the effects of
magnet segmentation in IPM motors, and concluded that the axial length of each magnet
segment should not be more than twice the skin depth of the eddy currents produced by
the dominant harmonics. Han et al. [48] attempted to minimize eddy current loss in the
stator teeth of IPM rotors and highlighted that double-layer rotor magnets resulted in
lower losses compared to single-layer rotor magnets. Tseng and Wee [49] investigated
various methods to determine core loss in the IPM machine, in which they stated the
relationship between flux and core loss as well as appropriate core loss calculation
methods to use at different stages of the machine design to save resources. Stumberger et
al. [50] studied the iron losses under different stator configuration and stated that the
rotor iron losses is substantial, despite there being a very small portion of iron is present
above the magnets. Ma et al. [51] proposed a method to increase the accuracy of
calculating iron loss using rotational fields and flux density harmonics. Barcaro et al. [52]
also studied how the design of rotor flux barriers and the amount of PM material affected
the losses in an IPM machine.
2.4 Winding Function Theory
Winding function theory has been used for years on induction and synchronous machines
[53]. Little has been done however to apply winding function theory to machines with
permanent magnets, just two examples of which are found in [54] and [55]. Its usefulness
lies in the fact that it can take into account geometrical variations of the machines and be
used to simulate transient performance.
31
The basis for applying winding function theory to interior permanent magnet motor is to
treat the magnets as coils. A permanent magnet which is magnetized in a parallel manner
can be modeled as a coil wrapped around material with the same permeability as that of
the magnet. The coils, with sufficient current, can provide the same magnetomotive force
(MMF) as the magnet itself. The MMF imposed by the magnetic layout, along with the
gap function imposed by the stator pole pieces will be used to calculate the inductance
variation. These inductances are evaluated using winding function and other equations
within the theory. The changing inductance will then be used to calculate the torque of
the rotors.
Winding function theory is based on the basic geometry and winding layout of machine
[56]. The only information required in winding function approach is the winding layout
and machine geometry. By this approach, it is possible to analyze performance of any
faulty machine with any type of winding distribution and air gap distribution around
rotor, while taking into account all spatial and time harmonics. Hence this method has
found application in the analysis of asymmetrical and fault conditions in machines, Such
as broken rotor bars [57] and fault condition in stator windings [58]. The modified
winding function approach (MWFA) for asymmetrical air-gap in an IPM machine has
been proposed in [59]. It was later more developed for non-uniform air gap machines
[60]. This theory has been applied to analyze static, dynamic and mixed eccentricity in
induction and synchronous machines [61]. Some techniques were also developed to
include rotor bar skewing, slotting, saturation effect and inter-bar currents [62].
In the previous works, based on winding function theory, the analysis of IPM machines
performance under eccentricity conditions is carried out assuming uniformity down the
32
axial length of the machine. In the other hand, the rotor axis is considered to be parallel to
that of the stator.
2.5 Why Winding Function Theory?
The d-q model of an ac machine is based on the assumption that stator windings
are sinusoidally distributed and is used to represent healthy motors and motors
under rotor faults.
The d-q model reduces the number of equations required for simulation. However
it can't give any information about rotor bars and end rings currents, and require a
modification in model structure for each fault case.
The analysis of machines as a magnetic field problem needs large resources of
digital computers to solve for complicated equations.
Winding function method is based on basic geometry and windings layout of ac
machine.
Winding function method with coupled magnetic circuits approach is a suitable
model for analyzing machines including faults.
The advantages of this method is that it is possible to predict transient and steady-
state performance of any machine with any type of winding distribution and air-
gap length, while taking into account the effect of all spatial and time harmonics.
This means that all faults occurring in the stator windings, rotor turns and air-gap
eccentricity can be included in the model obtained using this theory.
33
CHAPTER THREE
3.0 Analysis of Interior Permanent Magnet Motor with Winding Function Theory
3.1 Introduction
Winding function theory (WFT) has been used primarily on machines without permanent
magnets since its early development. It has seen great success when applied to induction
machines [63], switched reluctance machines [64], and synchronous reluctance machines
[65]. One of the only examples of WFT applied to a permanent magnet machine is found
in [66]. However, the radial flux density component due to the magnets is used only to
analyze the back-electromotive force (EMF) of the brushless DC (BLDC) machine. The
study undertaken here is to apply WFT theory to interior permanent magnet motor with
different stator configuration.
3.2 Winding Function Theory and its Modifications
3.2.1 Basic winding function theory
The derivation of winding function theory begins with a perfectly cylindrical and
concentric machine with conductors in the air gap. Later in the analysis, salient air gaps
will be handled. Figure 3.1 shows the machine with a two-pole winding residing within
the air gap. By Ampere’s law in (3.1), the line integral is taken around the conductor in
the path 1-2-3-4-1.
Following the path 1-2-3-4-1 around the conductor in Figure 3.1, the total MMF is calculated
in (3.2). The resultant turns function, )(n is directly proportional to the MMF in the air gap.
The MMF drop in paths 2-3 and 4-1 can be neglected however.
34
Fig. 3.1. An idealized machine with conductors in the air gap.
dsJdlH (3.1)
This is due to the assumption of the use of linear iron with high relative permeability. With a
relative permeability much greater than air, i.e. 10,000 the MMF drop through the iron is then
neglected, leaving the two terms in (3.3). The magnetic field strength around the air gap,
)(H is thus a function of the turns function, in (3.4).
in )(41342312 (3.2)
s
r
r
s
r
rrs
r
rsr
gHHrrdrrH
gHHrrdrrH
)()()(),(
)0()0()()0,(
34
12
(3.3)
)0()()( HginH (3.4)
Using Gauss’ law in (3.5), the value for the magnetic field strength around the air gap can be
found. This assumes a fictitious cylinder encompassing only the rotor of the ideal machine.
35
2
00
2
00
1
0
0)(
)(0
dHlr
dzrdHdsB
(3.5)
One of the key assumptions of normal winding function theory follows from (3.5), in that the
value of the MMF taken around the circumference of the machine must have no average
component, shown in (3.6).
0)(2
034
d (3.6)
Through the process shown in (3.7), the value of )(H is found by using the turns function.
)()()(
)(21)()0(
)(21)0(
0)()0(2
0)0()(
0)(
2
0
2
0
2
0
2
00
2
00
nngiH
dnngiH
dngiH
dngiH
dHginl
dHl
r
r
(3.7)
From the end result of (3.7), it is apparent that the magnetic field strength around the air gap
is a function of the turns function, )(n minus its average value, which is the winding
function, )(N shown in brackets. The differential flux is found in (3.8). In order to calculate
the inductances among windings in the machine, Figure 3.2 is used to make sure that
orientation is accounted for.
36
diNgrl
BrlddABd )(0 (3.8)
Figure 3.2: An idealized machine with multiple coils in the air gap. Care is taken in (3.9) to distinguish between the values of flux which can be calculated
depending upon the orientation of the winding shown in Figure 3.2. The value of (3.9) could
be positive or negative depending upon which coil side of winding A1 is encountered first.
dNigrl A
A
AAAA )('1
1
11'11
0
(3.9)
The problem with accounting for different winding orientations can be handled by including
the turns function in the calculation of the flux in (3.10). Assuming that each winding
constitutes its own phase, the mutual flux linkage is shown in (3.11).
dNingrl
dNingrl
AAAAAAAA
A
A
)()()()(2
0
00111
'1
1
111'11
(3.10)
dNingrl
AAAAA )()(2
0
011212 (3.11)
37
Now that the flux linkage has been found, the mutual inductance of the windings easily
follows in (3.12). The derivation is also valid if the windings are the same, thus giving the
magnetizing inductance in (3.13) [67] [68].
dNngrl
iL AA
A
AAAA )()(
2
0
021
21
21 (3.12)
dNngrl
iL AA
A
AAAA )()(
2
0
011
11
11 (3.13)
3.2.2 Winding function theory for machines with salient air gaps
In basic winding function theory, the inverse gap function is assumed to be composed of
only even harmonics. Figure 3.3 shows the machine with a two-pole winding residing
within the air gap. By Ampere’s law, the line integral is taken around the conductor in the
path 1-2-3-4-1 in (3.14). The MMF drops, are equivalent to the turns function of the
winding, )(n multiplied by the current, i in the winding. As with regular WFT and the
assumption of linear iron with a high relative permeability, the MMF drops through the
back iron and rotor are negligible and can thus be ignored.
Figure 3.3: An idealized machine with a salient rotor and conductors in the air gap.
38
in )(41342312 (3.14)
Winding function theory is amended as done in [67] to include a salient air gap for the
geometries studied here. An inverse gap function will be used to account for the changing
gap thickness, in the same manner that the turns function accounts for the changing MMF
throughout the machine. In (3.15), it is assumed that the average value of the MMF over the
circumference of the machine is zero, when multiplied by an inverse gap function )(1 g :
0)()( 12
0
dg (3.15)
Following the assumptions associated with (3.16), the value for the MMF can be derived as
follows:
2
0
2
0
113412 )()()()()0( digndg (3.16)
In [67], it was assumed that the salient air gap was composed solely of even harmonics,
allowing for the simplification which resulted in regular WFT. However, in the case of the
permanent magnet machine, the salient air gaps will not be limited to even harmonics. The
salient air gap could be composed of first harmonics only. In [59] and [69] which led to the
derivation and use of modified winding function theory (MWFT). However, eccentricity is
not the only factor that necessitates the use of MWFT; the simple fact of using a salient air
gap that is not built from even harmonics necessitates MWFT as well. The derivation
continues with (3.17) to find the right MMF term in (3.16), assuming that the left MMF term
is zero. Figure 3.4 to 3.9 shows Fundamental MMF Diagram for phase A and for the three
phases. The term in brackets is the modified winding function, )(M . In (3.18) and (3.19),
the modified winding function is shown as a new average value which must be subtracted
from the turns function. Figure 3.10 to 3.11 shows the Four Poles magnet flux density and airgap.
39
0 50 100 150 200 250 300 350 400-30
-20
-10
0
10
20
30
Stator Circumferential angle, [deg]
MM
F pe
r uni
t cur
rent
, Am
pere
-turn
s
Short-Pitched phase A
Figure 3.4: Fundamental MMF Diagram for Short pitched phase A
0 50 100 150 200 250 300 350 400-30
-20
-10
0
10
20
30
Stator Circumferential angle, [deg]
MM
F pe
r uni
t cur
rent
, Am
pere
-turn
s
Full-Pitched phase A
Figure 3.5: Fundamental MMF Diagram for Full-Pitched phase A
40
0 50 100 150 200 250 300 350 400-30
-20
-10
0
10
20
30
Stator Circumferential angle, [deg]
MM
F pe
r uni
t cur
rent
, Am
pere
-turn
s
Concentrated phase A
Figure 3.6: Fundamental MMF Diagram for Concentrated phase A
0 50 100 150 200 250 300 350 400-30
-20
-10
0
10
20
30
Stator Circumferential angle, [deg]
MM
F pe
r uni
t cur
rent
, Am
pere
-turn
s
Fundamenral MMF Diagram for Short-Pitched Distributed Winding
phase Aphase Bphase C
Figure 3.7: Fundamental MMF Diagram for Short pitched Distributed Winding
41
0 50 100 150 200 250 300 350 400-30
-20
-10
0
10
20
30
Stator Circumferential angle, [deg]
MM
F pe
r uni
t cur
rent
, Am
pere
-turn
s
Fundamental MMF Diagram for Full-Pitched Distributed Winding
phase Aphase Bphase C
Figure 3.8: Fundamental MMF Diagram for Full-Pitched Distributed Winding
0 50 100 150 200 250 300 350 400-30
-20
-10
0
10
20
30
Stator Circumferential angle, [deg]
MM
F pe
r uni
t cur
rent
, Am
pere
-turn
s
Fundamenral MMF Diagram for Concentrated Winding
phase Aphase Bphase C
Figure 3.9: Fundamental MMF Diagram for Concentrated Winding
42
0 1 2 3 4 5 6 7-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
rotor position [rad]
[T]
Four Poles MMF Diagram
Figure 3.10: Four Poles magnet flux density
0 1 2 3 4 5 6 7-1
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
rotor position [rad]
Sta
tor t
ooth
leng
th
AIRGAP
Figure 3.11: Airgap
43
2
0
1134
3412
2
0
1112
2
0
1112
)()()(2
1)()(
)()()0(
)()()(2
1)0(
)()()()0(2
dgng
ni
in
digng
digng
(3.17)
2
0
11
)()()(2
1)( dgng
M (3.18)
)()()( MnM (3.19)
Assuming two windings A and B, a radius r, and a stack length l, the new inductance values
using the modified winding function are given in (3.20) for the mutual inductance and in
(3.21) for the magnetizing or self inductance. The flux linkage AB is the flux in winding A
due to the current in winding B, while the flux linkage AA is the flux in winding A due to its
own current.
2
0
10 )()()( dgMnrl
iL BA
B
ABAB (3.20)
2
0
10 )()()( dgMnrl
iL AA
A
AAAA (3.21)
3.2.3 Winding Function Theory Applied to Magnetic Devices
Now that the need for MWFT has been presented, it will be applied to interior permanent
magnet motor. The modeling must begin with the most fundamental component of the
IPM motor, the individual magnets themselves, first looked at in [70] and [71]. In Figure
3.12 a single magnet is shown with a magnetization vector which is parallel to its
thickness. A coil wrapped around a material of equal permeability to that of the magnet
44
will provide an equivalent magnetization vector, assuming sufficient current. The
magnets used in this analysis have a thickness, lm, of 2.5 mm and a remanent magnetism,
Br, of 1.21 T. The magnets on the interior are stacked doubly. Hence, the required MMF
for one stack of magnets is:
AtlB
lHr
mrmm 2288
052.1104)0025.0(21.1
70
(3.22)
Figure 3.12: Model of a permanent magnet which produces an equivalent magnetization vector.
In the simplest case, the magnets lie adjacent to each other and form the basis for a single
turns function, as visualized in Figure 3.13 the adjacent magnets are simply modeled as coils
which are wrapped in opposing directions adjacent to each other. These assumptions allow us
to treat the adjacent magnets as though they are part of a single phase. The turns function
reaches a magnitude of 2N if each magnet has N windings. This is due to the fact that
adjacent coil sides lie next to each other to create alternating magnetic flux vectors.
The turns functions for the windings in this study are built from Fourier series, given in
(3.23) and shown in Figure 3.14 which can be phase-shifted easily in simulations.
xh
xdh
hxdn
h
h
cossin)1(2)(1
(3.23)
45
Figure 3.13: IPM with alternating vectors form the basis for a single turns function over the circumference of the rotor.
Figure 3.14: The turns function modeled with the Fourier series.
3.2.4. Verification of a single phase per rotor
The first assumption must be verified that the magnets can be assumed to be a single phase,
rather than as multiple phases. It will be done by looking at a simple case, shown in Figure
3.15, in which an ideal four-pole machine is shown. The inductances will be calculated for
the case in which the two coils, A and B, are treated as distinct coils, and ones which are
connected in series. The stator winding arrangements in 36 slots along with the turns
function of winding A and the winding functions of windings A, B and C are shown in Figure
3.16a. The turns function shows the number of turns as a function of , and the winding
function is the turns function minus its average. The phases are assumed to have Nx windings.
The average value is used to turn the turns function into the winding function. The average
number is found in (3.24).
46
Figure 3.15: An ideal four pole machine.
4)(
21)(
2
0
Ndnn
(3.24)
The self or magnetizing inductance is found in (3.25) for phase A.
8
344
3)( 202
0
2
2
220
2
0
20 Ngrl
dNdNgrl
dNgrl
L AAA
(3.25)
Likewise, the mutual inductance is calculated in (3.26) between phases A and B.
8
14
)( 202
3
02
0
0 Ng
rlNdN
grl
dnNgrl
L BAAB
(3.26)
The total flux treating A and B as separate phases is given by (3.27). This assumes that the
currents in the phases are equal, and that the mutual inductances are equivalent as well. Then
the total inductance is computed in (3.28).
AABAAABBAAABBBBAAAs iLiLiLiLiLiL 22 (3.27)
21
81
81
83
83 2020 N
grl
Ngrl
iL
s
ss
(3.28)
Now the inductance is calculated by treating the stator phases as set of series connected coils,
47
done in (3.29). It is apparent that the inductance is the same as that found in (3.28).
2
14
4 202
0
20 N
grl
dNg
rlLAB
(3.29)
The flux linkages are calculated for each of the h magnets in the four models, and a full
rotation. The inductance is then calculated from the flux linkage for each magnet, k
produced from (3.30). The self and mutual inductances calculated for each of the different
stator configurations is shown in Figure 3.17 to Figure 3.22.
ii
Lh
kk
tot /1
(3.30)
Figure 3.16a WF, from top figure to bottom: winding arrangements in the slots turns function of winding
A, WF of windings A, B and C.
48
3.2.5 Matlab Program for Solving Machine Equations
The winding function method discussed above for calculation of the machine inductances
along with the machine equations are used to develop a Matlab program for simulating
the IPM motor. The flowchart for using MWFT to calculate the inductance is shown in
Figure 3. 23a and the flowchart showing the iterative process of solving the machine
equations is presented in Figure 3.23b. In this program, all winding functions
corresponding to the stator windings are defined as a function of stator angle according
to their winding layouts. The magnetizing and mutual inductances are calculated over one
complete revolution of the rotor.
3.2.6 MMF rise across slots
As can be seen, mmf rises linearly across slots and not in form of steps. This is for every
slots, the WF is defined as in Figure 3.16b
Figure 3.16b A slot filled with conductors and WF diagram
49
0 2000 4000 6000 8000 10000 120000.012
0.014
0.016
0.018
0.02
0.022
0.024
0.026
0.028
0.03
0.032
2 [rad]
[H]
Stator self inductance
Short-Pitched (Laa)
Figure 3.17: Self inductance of the stator winding for Short pitched
0 2000 4000 6000 8000 10000 120000.01
0.015
0.02
0.025
0.03
0.035
2 [rad]
[H]
Stator self inductance
Full-Pitched (Laa)
Figure 3.18: Self inductance of the stator winding for Full-Pitched
0 2000 4000 6000 8000 10000 120000.012
0.014
0.016
0.018
0.02
0.022
0.024
0.026
0.028
0.03
2 [rad]
[H]
Stator self inductance
Concentrated (Laa)
Figure 3.19: Self inductance of the stator winding for Concentrated
50
0 2000 4000 6000 8000 10000 12000-16
-14
-12
-10
-8
-6
-4
-2
0
2x 10
-3
2 [rad]
[H]
Stator mutual inductance
Short-Pitched (Lab)
Figure 3.20: Mutual inductance between stator winding for Short pitched
0 2000 4000 6000 8000 10000 12000-20
-15
-10
-5
0
5x 10
-3
2 [rad]
[H]
Stator mutual inductance
Full-Pitched (Lab)
Figure 3.21: Mutual inductance between stator winding for Full-pitched
0 2000 4000 6000 8000 10000 12000-16
-14
-12
-10
-8
-6
-4
-2
0
2x 10-3
2 [rad]
[H]
Stator mutual inductance
Concentrated (Lab)
Figure 3.22: Mutual inductance between stator winding for Concentrated
51
Each inductance as a function of the angular rotor position is saved in a matrix to be used
when solving the machine equations. The inductance calculations and machine equations are
run in Matlab/Simulink using a fixed time step solver. While the turns functions are built
from Fourier series, the inverse gap functions are built from piecewise-linear functions in
their exact form since they do not have to be shifted. The inverse gap function is shown in
Figure 3.24. While Figure 3.25 shows the Harmonic Order for phase A. These equations are
solved iteratively and in each iteration the rotor position is found by taking the derivative of
the rotor speed ω. Based on the new rotor position, the inductance matrices are updated to be
used in the next iteration.
Figure 3.23a Flowchart for calculated inductances using MWFT in Matlab/Simulink.
52
Figure 3.23b: Flowchart of the iterative procedure for solving machine equations.
53
Figure 3.24: Inverse gap function used for ideal machines in inductance calculations
0 5 10 15 20 250
5
10
15
20
25
30
Short-PitchedFull-PitchedConcentrated
Figure 3.25: Harmonic Order for phase A
54
3.2.7. Torque calculated from inductance
In finding the torque for IPM motor, will once again be used as a basis for comparison. In
this section, the operating point for conventional interior permanent magnet machines is
below the remanent magnetization. This is due to the demagnetization from the stator
field, as well as the excitation requirement of the air gap [72]. This reduction however
comes only due to the excitation requirement for the air gap in the IPM motor. Adjusting
for the operating point, the MMF for the magnets used in the inductance calculations is
2151 ampere-turns. The principal cause of developed torque in IPM machines is the
interaction of the fields from the permanent magnet and rotating stator electromagnetic
field. Therefore the quality of torque produced is largely dependent on the winding
design and the configuration of the magnet. This changing reluctance, is represented
in (3.31) and (3.32) as either unaligned, u, or aligned, a.
u
uu A
l
0 (3.31)
a
aa A
l
0 (3.32)
The changing reluctance translates into a change in inductance, and thus a change in energy,
as shown by the distinct energy values in (3.33) and (3.34). This is also known as the
coenergy, Wco, shown in matrix form in (3.35) as a function of the four inductances for the
IPM motor.
2
21 iLE uu (3.33)
2
21 iLE aa (3.34)
o
i
oo
io
oi
iioico i
iLL
LL
iiW21 (3.35)
55
The expression for the electromagnetic torque developed by the machine can be obtained
from the component of the input power that is transferred across the air gap. The total
input power into the machine is given by:
cscsbsbsasasin iViViVP (3.36)
And the relationship between the electromechanical torque and the load torque is given as
lr
mr
mrmlr
mem Tdt
dB
dtd
JBTdt
dJT
2
2 (3.37)
where mB is the friction coefficient, lT is the load torque and mJ is the moment of inertia.
For simulation of the dynamic characteristics of the drive, we can rewrite equation in two
first order equations as:
mrmlemr JBTTP /)( (3.38)
rrP (3.39)
The mechanical dynamic equation, ignoring friction, is:
dtJTT
p lerr 2
(3.40)
where lT is the load torque, rp is the number of pairs of rotor poles and J is the total
inertia of the rotating mass. For simulation the input power is given as:
abcsabcscscsbsbsasasin iViViViVP 3 (3.41)
lossesPP inout (3.42)
where,
sabcs rilosses 23 (3.43)
And efficiency is given by:
%100Input
OutputEfficiency (3.44)
56
3.3 Clock diagram of IPM motor
The clock diagram of the machines stator winding as shown in Figure 3.26 to 3.28
indicates that,
A+ is the coil carrying current into the paper from phase A,
A- is the coil carrying current out of the paper from phase A,
B+ is the coil carrying current into the paper from phase B,
B- is the coil carrying current out of the paper from phase B,
C+ is the coil carrying current into the paper from phase C,
C- is the coil carrying current out of the paper from phase C,
3.4 Total Harmonic Distortion (THD)
The total harmonic distortion (THD) can be represented by several different methods. In
one of the most common, the THD is defined as the root mean square (RMS) value of the
total harmonics of the signal, divided by the RMS value of its fundamental signal. For
example, for signal X, the THD or harmonic factor is defined as [73]:
(3.45)
Where 22
322 ... nH XXXX
nX RMS value of the harmonic n
FX =RMS value of the fundamental signal
F
H
XX
THD
57
Figure 3.26: Clock Diagram for Short pitched (7/9) Distributed Stator Winding
3.5 The winding factor (kw)
The winding factor ( wk ) affects the shape and magnitude of flux linkage across the air
gap, and affects how harmonics of individual coil back EMF phasor are summed together
to form the overall phase EMF. A low winding factor means that the harmonic
components of the EMF are relatively high as compared to the fundamental component
resulting in lower magnitude of useful EMF. A lower winding factor can be treated as a
58
reduction in the effective number of turns per phase in stator windings [73]. The EMF in
turn affects the efficiency and torque density of the machine. It is therefore important to
obtain a high winding factor in the initial design stages with an appropriate choice of
Slots per pole per phase (Spp). This is the main contributor to the useful torque produced
by the machine. Te is related to the variation of mutual inductances between the stator
and rotor. In particular, it is the interaction between the back EMF and stator current.
Generally, the winding factor is made up of three parts [20]:
Figure 3.27: Clock Diagram for Full-pitched Distributed Stator Winding
59
Figure 3.28: Clock Diagram for Concentrated Stator Winding
sdpw kkkk (3.46)
where,
pk = Pitch factor,
dk = Distribution factor
sk = Skew factor
60
Concentrated windings are usually not skewed, as it will significantly increase the
complexity and cost of constructing the machine. Therefore the third term can be omitted.
Although the pitch factor can be easily calculated, calculating the distribution factor can
be quite complex. An alternative method that can be used to determine both the pitch and
distribution factor together is the EMF phasor method. [20]
3.6 Slot-fill Factor
The slot-fill factor otherwise known as the packing factor is inversely proportional to the
copper loss in the machine. Therefore, having the highest possible slot-fill factor is
essential for achieving optimal efficiency in the machine. The advantage that
Concentrated Winding has over Distributed Winding is that coils are wound around
individual stator teeth. This allows the use of more advanced winding methods such as
the joint lapped core method and pre-pressed windings in separable tooth pieces. Typical
slot-fill factors of up to 35% can be achieved for Distributed Winding and up to 45% can
be achieved for Concentrated Winding [74].
3.7 The Voltage Equations
The stator of a 220 V, 36-slot, 4-pole, 3 phase chorded and non-chorded distributed
winding and 220 V, 12 slot 4-pole 3 phase concentrated winding interior permanent
magnet motor was used for this study. The main dimensions of the machine are as given
in Table 4.1
In the stator reference frame, the first-order differential equations describing the electrical
circuit of a three-phase IPM motor using conventional notations are given below.
61
dtd
irV asasasas
(3.47)
dtd
irV bsbsbsbs
(3.48)
dtd
irV cscscscs
(3.49)
dtd
irV qrqrqrqr
(3.50)
dtd
irV drdrdrdr
(3.51)
Equations (3.47-3.51) can be written in matrix form as
dr
qr
cs
bs
as
drdrdrqrdrcsdrbsdras
qrdrqrqrqrcsqrbsqras
csdrcsqrscsbscsas
bsdrbsqrbscsbsbsbsas
asdrasqrascsasbsasas
dr
qr
cs
bs
as
dr
qr
cs
bs
as
dr
qr
cs
bs
as
i
iiii
LLLLL
LLLLL
LLLLL
LLLLL
LLLLL
dtd
i
iiii
r
rr
rr
V
VVVV
csc
0000
0000000000000000
(3.52)
Where,
rBAlsasas LLLL 2cos (3.53)
32cos
21
rBAasbs LLL (3.54)
32cos
21 rBAascs LLL (3.55)
32cos
21 rBAbsas LLL (3.56)
322cos rBAlsbsbs LLLL (3.57)
rBAbscs LLL 2cos21 (3.58)
32cos
21 rBAcsas LLL (3.59)
62
rBAcsbs LLL 2cos21 (3.60)
322coscsc rBAlss LLLL (3.61)
10
2
2 rl
NL r
sA
(3.62)
20
2
221
rlN
L rs
B
(3.63)
rmqasqr LL cos (3.64)
)sin( rmdasdr LL (3.65)
32cos
rmqbsqr LL (3.66)
32sin
rmdbsdr LL (3.67)
32cos
rmqcsqr LL (3.68)
32sin
rmdcsdr LL (3.69)
BAmq LLL 23 (3.70)
BAmd LLL 23 (3.71)
rmcsascsbsasbsasasasas iLiLiL sin (3.72)
32sin
rmcsbscsbsbsbsasbsasbs iLiLiL (3.73)
32sincsc
rmcssbscsbsascsascs iLiLiL (3.74)
63
3.8 Solution of Equation (3.52)
We attempt to solve the first order differential equation (3.52) to illustrate its complexity.
Now we may rewrite (3.52) as:
LIdtdIRV (3.75)
L is a function of rotor position r which is also a function of time and I, is only a time
varying function so we may write:
dtdIL
dtdL
IIRV rr )()(
(3.76)
This now implies a differential equation with time-varying coefficient. But equation
(3.95) can be rewritten as:
dtdIL
dtd
ddL
IIRV rr
r
r )()(
(3.77)
But we know that dt
d r is the derivative of position which means “speed”, or r so
(3.96) now becomes:
dtdIL
ddL
IIRV rr
rr )(
)(
(3.78)
Equation (3.89) may finally be written as:
1))(()(
r
r
rr L
ddLRIV
dtdI
(3.79)
Where dt
d rr
(3.80)
and L( r ) is the 5 5 inductance matrix appearing in (3.52).
64
Equation (3.79) shows that the solution of current at every time step involves the
computation of the derivative of the inductance matrix, r
r
ddL
)( for the particular rotor
position and the computation of the inverse of the inductance matrix 1))(( rL [75].
3.9 Torque Ripple Pulsating torque is harmonic torque which is caused by the interaction between induced
EMF harmonic and the stator current harmonic. In order to reduce the ripple torque, the
harmonic are minimum as much as possible, while induced EMF harmonic is related with
the spatial distribution and winding design of excitation magnetic field produced by
magnets. The ripple torque calculation is defined as [20]:
AvegRipp TTTT /)( minmax (3.81)
3.10 Losses in IPM Motor
A key aim in almost all high performance PM machine design is to minimise losses. In
machines used for field weakening applications, frequency related losses in particular
have to be carefully considered. Losses in PM machines are separated into two main
areas – electrical losses and mechanical losses.
Electrical Losses:
Core losses – Eddy current and hysteresis losses
Permanent magnet losses
Copper loss
Mechanical losses:
Bearing losses and Windage losses
65
3.10.1 Core Loss
In IPM motor frequency-related losses have to be carefully considered and minimised
due to constant operation at high frequencies. Furthermore the increase in leakage
harmonic terms as a result of applying CW makes the machine more susceptible to
increased core and rotor losses. One method of separating the hysteresis and eddy current
losses is derived from the relationship mentioned by Yeadon [76];
Thus, this approximation is valid for the breakdown of hysteresis and eddy current loss at
60Hz and the total core loss can be expressed below and this estimation gives us an eddy
current loss and hysteresis loss.
hyseddycore PPP (3.82)
Where
coreeddy PP31
(3.83)
corehys PP32
(3.84)
3.10.2 Magnet Loss
With the core loss substantially reduced by the choice of thin silicon steel laminations,
the time varying fields may still create substantial losses in the magnets, especially in
SPM machines. A great deal of research interest is focused on the study and minimisation
of magnet losses in Concentrated Winding PM machines [77]. Commonly used strategies
to reduce magnet losses are by the use of bonded instead of sintered magnets at the
expense of lower magnet strength, or the use of sintered-segmented magnets [78].
66
3.10.3 Stator Winding Loss
Stator winding loss – also known as I2R, copper or joule loss – occurs when the armature
windings are excited by an external source. Of the total loss in PM machines, the largest
portion is usually due to I2R loss [79]. I2R is not frequency dependent and is constant
throughout the speed range, so the CPSR is not affected by this loss. This is due to the
copper conductors being thin enough to have 100% skin-depth throughout the operating
region. I2R loss is described in the following formula:
(3.85)
where,
coilN = Number of turns per coil
= Conductor resistivity (1.68X10-8 for copper)
wA = Cross-sectional area of wire
3.10.4 Mechanical Losses
Mechanical losses consist of mainly bearing and windage losses [80]. Bearing loss is
dependent on factors such as the bearing type, bearing diameter, rotor speed, load and
lubricant used. Windage loss occurs when friction is created with the rotating parts of the
machine and the surrounding air. Bearing losses can be calculated with the following
formula [81]:
bbmbearing FDkP 5.0 (3.86)
where,
m = Mechanical speed of the rotor, F = Force acting on the bearing
bD = Bearing inner diameter, bk = Bearing loss constant
w
effcoil A
lINRI 22 2
67
CHAPTER FOUR
4.0 Dynamic Simulation in MATLAB/SIMULINK®
4.1 Simulation Tools
Computer-based analysis of electrical machines requires that appropriate measures are
made towards the proper selection of a simulation tool. The complex models of some
electrical machines need computing tools capable of performing dynamic simulations
with greater efficiency and accuracy. [82]. SIMULINK® has the advantages of being
capable of complex dynamic simulations, graphical environment with visual real time
programming and broad selection of toolboxes [83].
4.2 Simulation of IPM motor with Short pitched, Full-pitched Distributed Winding
and Concentrated winding using Embedded MATLAB Function Blocks
The interior permanent motor models constructed in this research work were achieved
using the Embedded MATLAB Function blocks of the MATLAB/SIMULINK® toolbox.
The machine is simulated using the actual abc phase variables model as developed.
4.3 Simulation Results
The system built in SIMULINK for the test interior permanent magnet motor is operated
with a load torque to enable notice the maximum load it can carry as shown in Figure 4.1
to 4.30 below.
68
Table 4.1: IPM motor with Short pitched, Full-pitched and concentrated winding
Voltage: 220V Rating: 110 kW Frequency: 50Hz
Rotor inertia: 0.22 kg.m2
Poles: 4 & Speed: 2400 rpm Rated Current (RMS): 208.33A
Magnetic Steel: N28UH & Remanent flux density: 1.21T Magnet coating: Ni-Cu-Ni, Core material: Non-oriented silicon steel & Magnet thickness:2.5mm
Parameter Value
Configuration Short pitched Full-pitched Concentrated Stator winding resistance, sr 0.62 0.62 0.62
d-axis leakage inductance, ldrL 0.0055 0.0055 0.0055
q-axis leakage inductance, lqrL 0.0062 0.0062 0.0062
Resistance, Rm 0.03 0.03 0.03
Stator winding leakage inductance, lsL 0.0083 0.0083 0.0083
Rotor d-axis winding resistance, drr 0.12 0.12 0.12
Rotor q-axis winding resistance, qrr 0.25 0.25 0.25
Inductance, Lm 0.09 0.09 0.09
1g 0.0003 0.0003 0.0003
2g 0.012 0.012 0.012
Amplitude of the 1st order harmonics 27.53 29.30 26.45
Stator outer diameter 130mm 130mm 130mm
Rotor outer diameter 80mm 80mm 80mm
Airgap length 1.2mm 1.2mm 1.2mm
Stator inner diameter 82.4mm 82.4mm 82.4mm
Shaft outer diameter 24mm 24mm 24mm
Stack length (Stator) 80mm 80mm 80mm
Stack length (rotor) 79mm 79mm 79mm
Area per half slot 86.3mm2 86.3mm2 86.3mm2
Stator and Rotor lamination thickness 0.35mm 0.35mm 0.35mm
Slot-opening width 1.2mm 1.2mm 1.2mm
Number of turns per coil (around each tooth) 115turns 115turns 115turns
Slot-fill factors (%) 35 35 45
69
Table 4.2: Summary of the Performance comparison of the each IPM Motor
Variables Configuration
Short pitched Full-pitched Concentrated Fundamental winding factor 0.933 0.933 0.866
Amplitude of the fundamental winding 27.53 29.30 26.45
Copper loss (kw) Low speed High speed
0.35 3.6
0.35 3.6
0.30 3.7
Pull out power (kw) 57 56 58
Quantity of copper (kg) 20 20 10
THD of mmf (%) 3.07 3.07 3.51
Torque ripple (Nm) 71 71 96
Average torque (Nm) 185 185 142
Peak torque develop (Nm) 116 116 133
Controllability Better Best Worsened
Demagnetization tolerance (%) 78 78 60
Efficiency 87 88 89
70
0 1 2 3 4 5 6 7 8-200
0
200
Ia, A
0 1 2 3 4 5 6 7 8-200
0
200
Ia, A
0 1 2 3 4 5 6 7 8-200
0
200
Time, sec
Ia, A
Chorded
Non-Chorded
Concentrated
Figure 4.1: Stator phase A’s currents
0 1 2 3 4 5 6 7 8-200
0
200
Idr,
A
0 1 2 3 4 5 6 7 8-200
0
200
Idr,
A
0 1 2 3 4 5 6 7 8-200
0
200
Time, sec
Idr,
A
Chorded
Non-Chorded
Concentrated
Figure 4.2: d-axis rotor currents
--------Short-Pitched
---------- Full-Pitched
--------Short-Pitched
--------Full-Pitched
71
0 1 2 3 4 5 6 7 8-200
0
200
Iqr,
A
0 1 2 3 4 5 6 7 8-200
0
200
Iqr,
A
0 1 2 3 4 5 6 7 8-200
0
200
Time, sec
Iqr,
A
Chorded
Non-Chorded
Concentrated
Figure 4.3: q-axis currents
0 1 2 3 4 5 6 7 8-200
0
200
0 1 2 3 4 5 6 7 8-100
0
100
200
Ele
ctro
mag
netic
Tor
que,
Te,
N-m
0 1 2 3 4 5 6 7 8-100
0
100
200
Time, sec
Chorded
Non-Chorded
Concentrated
Figure 4.4: Electromagnetic torques
--------Short-Pitched
-------- Full-Pitched
--------Short-Pitched
------- Full-Pitched
72
1 1.5 2 2.5 3-200
-100
0
100
1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8
-500
50100150
Ele
ctro
mag
netic
Tor
que,
Te
N-m
1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8
0
50
100
Time, sec
Non-Chorded
Chorded
Concentrated
Figure 4.5: Electromagnetic torques at synchronization
3.8 4 4.2 4.4 4.6 4.8 5-150
-100-50
0
50
4 4.2 4.4 4.6 4.8 5 5.2
-50
0
50
Ele
ctro
mag
netic
Tor
que,
Te
N-m
4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9
-200
2040
Time, sec
Non-Chorded
Chorded
Concentrated
Figure 4.6: Electromagnetic torques when the load was applied
-------Short-Pitched
------- Full-Pitched
------- Short-Pitched
------- Full-Pitched
73
0 1 2 3 4 5 6 7 8-200
0
200
400
600
800
1000
1200
1400
1600
1800
Time, sec
Roto
r Spe
ed, r
.p.m
Chorded
Figure 4.7: Rotor Speeds during starting and its response to a load torque for Short pitched
0 1 2 3 4 5 6 7 8-200
0
200
400
600
800
1000
1200
1400
1600
Time, sec
Rotor S
peed
, r.p.m
Non-Chorded
Figure 4.8: Rotor Speeds during starting and its response to a load torque for Full-pitched
0 1 2 3 4 5 6 7 8-200
0
200
400
600
800
1000
1200
1400
1600
Time, sec
Rotor
Spe
ed, r
.p.m
Concentrated
Figure 4.9: Rotor Speeds during starting and its response to a load torque for Concentrated
--------Short-Pitched
---------Full-Pitched
74
0.8 1 1.2 1.4 1.6 1.8 2 2.2
1200
1250
1300
1350
1400
1450
1500
1550
1600
1650
Time, sec
Rotor S
peed
, r.p.m
Chorded
Figure 4.10: Rotor Speeds during synchronization for Short pitched
0.8 1 1.2 1.4 1.6 1.8 2 2.2
1300
1350
1400
1450
1500
1550
1600
Time, sec
Rotor S
peed
, r.p.m
Non-Chorded
Figure 4.11: Rotor Speeds during synchronization for Full-pitched
0.8 1 1.2 1.4 1.6 1.8 2 2.2
1300
1350
1400
1450
1500
Time, sec
Rotor S
peed
, r.p.m
Concentrated
Figure 4.12: Rotor Speeds during synchronization for Concentrated
------- Short-Pitched
------- Full-Pitched
75
3.9 3.95 4 4.05 4.1 4.15 4.2
1465
1470
1475
1480
1485
1490
1495
1500
1505
1510
1515
Time, sec
Rotor S
peed
, r.p.m
Chorded
Figure 4.13: Rotor Speeds response to a load torque for Short pitched
4 4.05 4.1 4.15 4.2 4.25 4.3 4.35 4.4 4.45 4.51440
1460
1480
1500
1520
1540
Time, sec
Rotor S
peed
, r.p.m
Non-Chorded
Figure 4.14: Rotor Speeds response to a load torque for Full-pitched
3.9 4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8
1460
1470
1480
1490
1500
1510
1520
1530
Time, sec
Rot
or S
peed
, r.p
.m
Concentrated
Figure 4.15: Rotor Speeds response to a load torque for Concentrated
------- Short-Pitched
------- Full-Pitched
76
0 0.5 1 1.5 2 2.5 3 3.5 4-200
0
200
400
600
800
1000
1200
1400
1600
Time, sec
Rot
or S
peed
, r.p
.m
Chorded
Figure 4.16: Rotor Speeds with the application of ramp for Short pitched
0 0.5 1 1.5 2 2.5 3 3.5 4-200
0
200
400
600
800
1000
1200
1400
1600
Time, sec
Rot
or S
peed
, r.p
.m
Non-Chorded
Figure 4.17: Rotor Speeds with the application of ramp for Full-pitched
0 0.5 1 1.5 2 2.5 3 3.5 4-200
0
200
400
600
800
1000
1200
1400
1600
Time, sec
Rot
or S
peed
, r.p
.m
Concentrated
Figure 4.18: Rotor Speeds with the application of ramp for Concentrated
------- Short-Pitched
------- Full-Pitched
77
-200 0 200 400 600 800 1000 1200 1400 1600 1800-200
-150
-100
-50
0
50
100
150
200
Rotor Speed, r.p.m
Elect
rom
agne
tic T
orqu
e, T
e, N
-m
Chorded
Figure 4.19: Torque-Speed characteristic for Short pitched
-200 0 200 400 600 800 1000 1200 1400 1600-100
-50
0
50
100
150
Elect
rom
agne
tic T
orqu
e, T
e, N
-m
Rotor Speed, r.p.m
Non-Chorded
Figure 4.20: Torque-Speed characteristic for Full-pitched
-200 0 200 400 600 800 1000 1200 1400 1600-100
-50
0
50
100
150
Elect
rom
agne
tic T
orqu
e, T
e, N
-m
Rotor Speed, r.p.m
Concentrated
Figure 4.21: Torque-Speed characteristic for Concentrated
------- Short-Pitched
------- Full-Pitched
78
20 40 60 80 100 120 140 1602
2.5
3
3.5
4
4.5
5
5.5
6x 104
Stator current, A
Outpu
t pow
er, w
atts
Chorded
Figure 4.22: Output power against stator current for Short pitched
20 40 60 80 100 120 1402
2.5
3
3.5
4
4.5
5
5.5
6x 104
Stator current, A
Outpu
t pow
er, w
atts
Non-Chorded
Figure 4.23: Output power against stator current for Full-pitched
40 50 60 70 80 90 100 110 120 130 1402
2.5
3
3.5
4
4.5
5
5.5
6x 10
4
Stator current, A
Out
put p
ower
, wat
ts
Concentrated
Figure 4.24: Output power against stator current for Concentrated
------- Short-Pitched
-------- Full-Pitched
79
20 40 60 80 100 120 140 1600
0.5
1
1.5
2
2.5
3
3.5
4x 104
Stator current, A
Sta
tor l
osse
s, w
atts
Chorded
Figure 4.25: Stator losses against stator current for Short pitched
20 40 60 80 100 120 1400
0.5
1
1.5
2
2.5
3
3.5
4x 104
Stator current, A
Sta
tor l
osse
s, w
atts
Non-Chorded
Figure 4.26: Stator losses against stator current for Full-pitched
40 50 60 70 80 90 100 110 120 130 1400
0.5
1
1.5
2
2.5
3
3.5
4x 10
4
Stator current, A
Sta
tor l
osse
s, w
atts
Concentrated
Figure 4.27: Stator losses against stator current for Concentrated
------- Short-Pitched
------- Full-Pitched
80
20 40 60 80 100 120 140 16060
65
70
75
80
85
90
Stator current, A
Efficien
cy, 1
00%
Chorded
Figure 4.28: Efficiency against stator current for Short pitched
20 40 60 80 100 120 14060
65
70
75
80
85
90
Stator current, A
Effi
cien
cy, 1
00%
Non-Chorded
Figure 4.29: Efficiency against stator current for Full-pitched
40 50 60 70 80 90 100 110 120 130 14060
65
70
75
80
85
90
Stator current, A
Efficien
cy, 1
00%
Concentrated
Figure 4.30: Efficiency against stator current for Concentrated
------- Short-Pitched
------- Full-Pitched
81
4.4 Discussions
In this work, various performance characteristics of both concentrated winding IPM
motor short pitched and full-pitched distributed winding IPM motor are investigated and
compared in detail. All time-stepping simulations were carried out at a constant supply
voltage of 220 V, 50 Hz using Matlab Simulink. All the results are shown for
comparison. A load torque of 40 Nm (60% of rated torque) at t = 4.0s followed by a ramp
at the same time were introduced into the simulations this leads to small speed departures
from interior permanent magnet motor. The harmonics originating from the slots and the
winding are responsible for the distortions in the waveforms of different variables shown.
Table 4.1 show different stator configuration since the Embedded MATLAB function
block was employed; the abc phase variables were written and coded inside the block
(see Appendix 1 and 2).
As it is well known, the distributed winding motor has lower torque ripple because of the
distributed magnetic flux through the teeth. Analysis results show that it has lower torque
ripple of 71 Nm compared with the 96 Nm of concentrated winding motor. And the
average torque of the distributed winding motor at the maximum speed is higher than that
of the concentrated winding motor. Hence, it results in that the distributed winding motor
is better for the torque performance.
Losses and efficiency as the concentrated winding IPM motor has shorter end coil as
mentioned, it has lower phase resistance and lower copper loss at lower speed than the
distributed winding IPM motor. In addition, the distributed winding IPM motor has more
magnetic saturation parts at the stator teeth and core loss is bigger than that of the
concentrated winding IPM motor. However, the concentrated winding IPM motor causes
82
huge amount of magnet eddy current loss by the slot harmonics at high speed operation
because permanent magnet has conductivity inherently.
Harmonics and controllability are also considered. The 3rd and 15th harmonics of the
concentrated winding IPM motor are removed inherently, but the magnitudes of
harmonics are bigger than those of the distributed winding IPM motor. As the
concentrated winding IPM motor has more harmonics than the distributed winding IPM
motor, torque ripple would be bigger and the controllability would be worsened. It can be
said that the distributed winding is superior to the concentrated winding IPM motor.
However, this torque ripple can be ignored in case the moment of inertia of the rotor and
load is enough big.
Demagnetization is generally occurred by high temperature, airgap variation and locked-
rotor high current. For both concentrated winding and distributed winding IPM motors,
demagnetization from high temperature and airgap variation would be similar and
because permanent magnets of the IPM motor are inserted in the rotor. For the
concentrated winding IPM motor, the intensive flux from the stator core affects directly
to the permanent magnet and the minimum magnetic flux density of the permanent
magnet is 0.2515 T. On the other hand, the minimum magnetic flux density of the
permanent magnet in case of the distributed winding IPM motor is 0.6422 T. This
analysis result elicits that the distributed winding IPM motor has much higher
demagnetization tolerance than the concentrated winding IPM motor.
In general, IPM motor with concentrated winding is superior to that with distributed
winding in the power density point of view because of less end coils. The approximate
weight of the IPM motor with concentrated winding is 162.6 kg and power density of that
83
is 0.670 kW/kg. On the other hand, the approximate weight of the IPM motor with
distributed winding is 182.9 kg and power density of that is 0.596 kW/kg. Major source
of the difference is the weight of stator and coil. From the calculation, it is confirmed that
the concentrated winding IPM motor is better in the power density point of view.
Table 4.2 shows the summary of the performance comparison of the each IPM motor.
Also observed that there is a slightly faster damping of oscillations in the full-pitched
distributed winding and concentrated winding than the short pitched distributed winding
and current drawn is also less for the full-pitched distributed winding and concentrated
winding. These are due to the increased impedances from the additional harmonic terms.
Another observation is that the current plots of dri and qri appears to oscillate before
arriving at steady-state when a load torque was applied for full-pitched distributed
winding and concentrated winding. This is not unconnected to the fact that there is
significant deviation from synchronous speed mode for these configurations. The farther
the speed deviates from synchronous, the longer it takes the system to attain steady-state.
84
CHAPTER FIVE
5.0 Conclusion and Recommendation
An Interior Permanent Magnet motor has been successfully simulated using the
Embedded MATLAB function. Various performance characteristics of concentrated
winding IPM, short pitched DW IPM and full-pitched DW IPM are investigated and
compared in detail using the winding function theory, harmonic contents analysis and
inductance calculations. The DW motor has lower torque ripple and high average torque
at maximum speed. Hence, DW motor is better for the torque performance. The CW IPM
motor has shorter end coil, lower phase resistance and lower copper loss at lower speed.
In addition, the DW IPM motor has more magnetic saturation parts at the stator teeth and
core loss is bigger. However, the CW IPM motor causes huge amount of magnet eddy
current loss by the slot harmonics at high speed operation and has more harmonics,
torque ripple would be bigger and the controllability would be worsened. It can be said
that the DW is superior. Demagnetization from high temperature and airgap variation
would be similar for both windings and because permanent magnets of the IPM motor are
inserted in the rotor, the DW IPM motor has much higher demagnetization tolerance than
the CW IPM motor. In general, IPM motor with CW is superior to that with short pitched
and full-pitched DW in the power density point of view because of less end coils. Major
source of the difference is the weight of stator and coil. The continued interest in IPM
motor is very promising. The number of institutions performing research into IPM motor is
increasing and continued research should yield industrial products down the road.
85
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92
Appendices APPENDIX 1: Simulink Model of the IPM with Ramp for Short pitched, Full-pitched and Concentrated Winding
wr
f(u)
Vcs
Vbs
Vas
To Workspace9
IAAcon
To Workspace8
Laacon
To Workspace7
Ic
To Workspace6
Ib
To Workspace5
Iacon
To Workspace4
Idrcon
To Workspace3
Iqrcon
To Workspace2
Time
To Workspace13
Labcon
To Workspace12
Effcon
To Workspace11
Losscon
To Workspace10
Poutcon
To Workspace 1
TorqueconTo Workspace
Speedcon
Te
Speed
Ramp
Pout
Loss
Lab
Laa
Iqdr
Int 6
1s
Int 5
1s
Int 4
1s
Int 3
1s
Int 2
1s
Int 1
1s
Int
1s
Ic
Ib
Ia
IAA
EmbeddedMATLAB Function
Ia
Ib
Ic
wr
tr
Vas
Vbs
Vcs
TL
Iqr
Idr
pIa
pIb
pIc
pwr
Te
IAA
Pout
Laa
Lab
Loss
Eff
LA
LB
pIqr
pIdr
fcn
Eff
Display 1
Display
Clock
93
APPENDIX 2: Embedded MATLAB code of IPM Motor for Short pitched, Full-pitched and concentrated winding in per unit. function [pIa, pIb, pIc, pwr, Te, LA, LB, pIqr, pIdr]... = fcn(Ia, Ib, Ic, wr, tr, Vas, Vbs, Vcs, TL, Iqr, Idr) P=4; J=0.22; Lls = 8.3e-3; Llqr = 6.2e-3; Lldr = 5.5e-3; rs = 0.62; rqr = 0.25; rdr = 0.12; Rm=30/1000; Lm=90/1000; mu=4*pi*10^-7; g1=0.3/1000; g2=g0*40; Na=27.53; Nb=29.30; Nc=26.45; LA = 1/2*mu*Rm*Lm*Na^2*pi*(g0+g1)/g0/g1; LB = 3183/10000*mu*Rm*Lm*Na^2*pi*(g0+g1)/g0/g1; al=2*pi/3; Lmq=3/2*(LA-LB); Lmd=3/2*(LA+LB); Laa=Lls+LA-LB*cos(2*tr); Lab=-.5*LA-LB*cos(2*(tr-pi/3)); Lac=-.5*LA-LB*cos(2*(tr+pi/3)); Lba=-.5*LA-LB*cos(2*(tr-pi/3)); Lbb=Lls+LA-LB*cos(2*(tr-2*pi/3)); Lbc=-.5*LA-LB*cos(2*(tr+pi)); Lca=-.5*LA-LB*cos(2*(tr+pi/3)); Lcb=-.5*LA-LB*cos(2*(tr+pi)); Lcc=Lls+LA-LB*cos(2*(tr+2*pi/3)); LSS=[Laa Lab Lac; Lba Lbb Lbc; Lca Lcb Lcc]; LS2= -0*LB*[cos(2*2*tr) cos(2*2*(tr-pi/3)) cos(2*2*(tr+pi/3));... cos(2*2*(tr-pi/3)) cos(2*2*(tr-2*pi/3)) cos(2*2*(tr+pi));... cos(2*2*(tr+pi/3)) cos(2*2*(tr+pi)) cos(2*(tr+2*pi/3))]; dLS2= 0*LB*4*[sin(2*2*tr) sin(2*2*(tr-pi/3)) sin(2*2*(tr+pi/3));... sin(2*2*(tr-pi/3)) sin(2*2*(tr-2*pi/3)) sin(2*2*(tr+pi));... sin(2*2*(tr+pi/3)) sin(2*2*(tr+pi)) sin(2*(tr+2*pi/3))]; LS=LSS+LS2; LSRa=[Lmq*cos(tr) Lmd*sin(tr);... Lmq*cos(tr-al) Lmd*sin(tr-al);... Lmq*cos(tr+al) Lmd*sin(tr+al)]; LSR1=.0*[Lmq*cos(3*tr) Lmd*sin(3*tr);... Lmq*cos(3*(tr-al)) Lmd*sin(3*(tr-al));... Lmq*cos(3*(tr+al)) Lmd*sin(3*(tr+al))]; dLSR1=.0*3*[-Lmq*sin(3*tr) Lmd*cos(3*tr);... -Lmq*sin(3*(tr-al)) Lmd*cos(3*(tr-al));... -Lmq*sin(3*(tr+al)) Lmd*cos(3*(tr+al))]; dLSS=[2*LB*sin(2*tr) -2*LB*sin(2*tr+1/3*pi) 2*LB*cos(2*tr+1/6*pi);... -2*LB*sin(2*tr+1/3*pi) 2*LB*cos(2*tr+1/6*pi) 2*LB*sin(2*tr);... 2*LB*cos(2*tr+1/6*pi) 2*LB*sin(2*tr) -2*LB*sin(2*tr+1/3*pi)];
94
dLS=dLSS+dLS2; dLSRa=[-Lmq*sin(tr) Lmd*cos(tr);... -Lmq*sin(tr-al) Lmd*cos(tr-al);... -Lmq*sin(tr+al) Lmd*cos(tr+al)]; LSR=LSRa+LSR1; dLSR=dLSRa+dLSR1; LRR=[Llqr+Lmq 0;... 0 Lldr+Lmd]; L = [LS LSR; 2/3*(LSR)' LRR]; dL = [dLS dLSR; 2/3*(dLSR)' zeros(2,2)]; V=[Vas; Vbs; Vcs; 0; 0]; R=diag([rs rs rs rqr rdr]); II=[Ia; Ib; Ic; Iqr; Idr]; pII=inv(L)*(V-(R+wr*dL)*II); pIa=pII(1); pIb=pII(2); pIc=pII(3); pIqr=pII(4); pIdr=pII(5); Ias=[Ia; Ib; Ic]; Iqdr=[Iqr; Idr]; Te = (P/2)*(.5*Ias'*dLSS*Ias + Ias'*dLSR*Iqdr); pwr=(Te-TL)*P/(2*J); TT = 2/3*[cos(tr) cos(tr-al) cos(tr+al); sin(tr) sin(tr-al) sin(tr+al); 1/2 1/2 1/2]; IQD = TT*Ias; Iq = IQD(1); Id = IQD(2); IAA = (Iq.^2+Id.^2).^.5; Loss = 3*IAA.^2*rs; Pout = 3*220*IAA - Loss; Eff=(Pout/(3*220*IAA))*100;
95
APPENDIX 3: M-file code of IPM Motor for Short pitched, Full-pitched and concentrated winding for Fundamental MMF Diagram for phase A’s and the Harmonic Order clear;clc; s1=[zeros(1,25) linspace(0,2,50) 2*ones(1,25)]; s2=[2*ones(1,25) linspace(2,3,50) 3*ones(1,25)]; s3=[3*ones(1,25) linspace(3,4,50) 4*ones(1,25)]; s4=4*ones(1,400); s5=[4*ones(1,25) linspace(4,3,50) 3*ones(1,25)]; s6=[3*ones(1,25) linspace(3,2,50) 2*ones(1,25)]; s7=[2*ones(1,25) linspace(2,0,50) 0*ones(1,25)]; s8=[0*ones(1,25) linspace(0,-1,50) -1*ones(1,25)]; s9=[-1*ones(1,25) linspace(-1,-2,50) -2*ones(1,25)]; s10=-2*ones(1,400); s11=[-2*ones(1,25) linspace(-2,-1,50) -1*ones(1,25)]; s12=[-1*ones(1,25) linspace(-1,0,50) 0*ones(1,25)]; s13=[zeros(1,25) linspace(0,2,50) 2*ones(1,25)]; s14=[2*ones(1,25) linspace(2,3,50) 3*ones(1,25)]; s15=[3*ones(1,25) linspace(3,4,50) 4*ones(1,25)]; s16=4*ones(1,400); s17=[4*ones(1,25) linspace(4,3,50) 3*ones(1,25)]; s18=[3*ones(1,25) linspace(3,2,50) 2*ones(1,25)]; s19=[2*ones(1,25) linspace(2,0,50) 0*ones(1,25)]; s20=[0*ones(1,25) linspace(0,-1,50) -1*ones(1,25)]; s21=[-1*ones(1,25) linspace(-1,-2,50) -2*ones(1,25)]; s22=-2*ones(1,400); s23=[-2*ones(1,25) linspace(-2,-1,50) -1*ones(1,25)]; s24=[-1*ones(1,25) linspace(-1,0,50) 0*ones(1,25)]; s25=[zeros(1,25) linspace(0,2,50) 2*ones(1,25)]; s26=[2*ones(1,25) linspace(2,4,50) 4*ones(1,25)]; s27=[4*ones(1,25) linspace(4,6,50) 6*ones(1,25)]; s28=6*ones(1,600); s29=[6*ones(1,25) linspace(6,4,50) 4*ones(1,25)]; s30=[4*ones(1,25) linspace(4,2,50) 2*ones(1,25)]; s31=[2*ones(1,25) linspace(2,0,50) 0*ones(1,25)]; s32=0*ones(1,600); s33=[zeros(1,25) linspace(0,2,50) 2*ones(1,25)]; s34=[2*ones(1,25) linspace(2,4,50) 4*ones(1,25)]; s35=[4*ones(1,25) linspace(4,6,50) 6*ones(1,25)]; s36=6*ones(1,600); s37=[6*ones(1,25) linspace(6,4,50) 4*ones(1,25)]; s38=[4*ones(1,25) linspace(4,2,50) 2*ones(1,25)]; s39=[2*ones(1,25) linspace(2,0,50) 0*ones(1,25)]; s40=0*ones(1,600); s41=[zeros(1,25) linspace(0,2,50) 2*ones(1,25)]; s42=2*ones(1,200); s43=[2*ones(1,25) linspace(2,0,50) 0*ones(1,25)]; s44=0*ones(1,200); s45=[zeros(1,25) linspace(0,2,50) 2*ones(1,25)]; s46=2*ones(1,200); s47=[2*ones(1,25) linspace(2,0,50) 0*ones(1,25)]; s48=0*ones(1,200);
96
NCS=16; NPS=42; A1=NCS/2*[s1 s2 s3 s4 s5 s6 s7 s8 s9 s10 s11 s12 s13 s14 s15 s16 s17 s18 s19 s20 s21 s22 s23 s24]; A2=NCS/2*[s25 s26 s27 s28 s29 s30 s31 s32 s33 s34 s35 s36 s37 s38 s39 s40]; A3=NPS/2*[s41 s42 s43 s44 s45 s46 s47 s48]; NN=3600; NQ=1200; Aw=A1-(sum(A1)/NN); Ax=A2-(sum(A2)/NN); Az=A3-(sum(A3)/NQ); tt=linspace(0,2*pi,NN); tt1=linspace(0,2*pi,NQ); td=tt*180/pi; td1=tt1*180/pi; AA=fft(Aw);AB=fft(Ax);AC=fft(Az); MA=2/NN*abs(AA); MB=2/NN*abs(AB); MC=2/NQ*abs(AC); ta=tan(atan2(real(AA), imag(AA)));tb=tan(atan2(real(AB), imag(AB)));tc=tan(atan2(real(AC), imag(AC))); figure(3); %%subplot(311); bar(MA(3:80)); subplot(312); bar(MB(3:80)); subplot(313); bar(MC(3:80)); MM=[MA(3:80);MB(3:80);MC(3:80)]; MMM=MM'; bar(MMM,1.0,'group'); legend('Chorded','Non-Chorded','Concentrated'); figure(2); subplot(311); plot(td,Aw,'k-',td, MA(3)*sin(2*tt-ta(3)),'k--','linewidth',2);grid minor; legend('Chorded phase A'); subplot(312); plot(td,Ax,'k-',td,MB(3)*sin(2*tt-tb(3)),'k--','linewidth',2);grid minor; ylabel('MMF per unit current, Ampere-turns'); legend('Non-Chorded phase A'); subplot(313); plot(td1,Az,'k-',td1,MC(3)*sin(2*tt1-tc(3)),'k--','linewidth',2); grid minor; xlabel(' Stator Circumferential angle, \phi [deg]'); legend('Concentrated phase A');
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APPENDIX 4: M-file code of IPM Motor for Short pitched, Full-pitched and concentrated winding for Inductance calculation. clear;clc; syms ph Na Nb Nc tt g1 g0 mu R L wbb=Na*cos(2*ph); waa=Na*cos(2*ph-2*pi/3); wcc=Na*cos(2*ph-4*pi/3); k=mu*R*L; giv=1/2*(1/g0+1/g1)-2/pi*(1/g0-1/g1)*cos(4*(ph-tt)); Laa=k*int(waa*waa*giv,ph,0,2*pi); Laa=simple(Laa) Lab=k*int(waa*wbb*giv,ph,0,2*pi); Lab=simple(Lab) Lac=k*int(waa*wcc*giv,ph,0,2*pi); Lac=simple(Lac) Lba=k*int(wbb*waa*giv,ph,0,2*pi); Lba=simple(Lba) Lbb=k*int(wbb*wbb*giv,ph,0,2*pi); Lbb=simple(Lbb) Lbc=k*int(wbb*wcc*giv,ph,0,2*pi); Lbc=simple(Lbc) Lca=k*int(wcc*waa*giv,ph,0,2*pi); Lca=simple(Lca) Lcb=k*int(wcc*wbb*giv,ph,0,2*pi); Lcb=simple(Lcb) Lcc=k*int(wcc*wcc*giv,ph,0,2*pi); Lcc=simple(Lcc) R=30/1000; L=90/1000; mu=4*pi*10^-7; g1=0.3/1000; g2=g1*40; Na=27.53; Nb=29.30; Nc=26.45; tt=linspace(0,2*pi,3600); Lasas=eval(Laa); Lbsbs=eval(Lbb); Lcscs=eval(Lcc); figure(1);plot(tt,Lasas,'k',tt,Lbsbs,'r',tt,Lcscs,'b','linewidth',4);
98
APPENDIX 5: M-file code of IPM Motor for Short pitched, Full-pitched and concentrated winding for four
poles MMF diagram
clear; clc; B=0.8; C=0.6; D=pi*(1-C)/2; E=pi*C; s1=[zeros(1,180) ones(0,0.8)]; s2=[0.8*ones(1,540) ones(0.8,0)]; s3=[zeros(1,180)]; s4=[zeros(1,180) ones(0,-0.8)]; s5=[-0.8*ones(1,540) ones(-0.8,0)]; s6=[zeros(1,180)]; s7=[zeros(1,180) ones(0,0.8)]; s8=[0.8*ones(1,540) ones(0.8,0)]; s9=[zeros(1,180)]; s10=[zeros(1,180) ones(0,-0.8)]; s11=[-0.8*ones(1,540) ones(-0.8,0)]; s12=[zeros(1,180)]; A=[s1 s2 s3 s4 s5 s6 s7 s8 s9 s10 s11 s12]; tt1=linspace(0,2*pi,3600); plot(tt1,A,'k-','linewidth',2); grid minor; xlabel('4\pi [rad]'); ylabel('[B]'); title('Four Poles MMF Diagram'); ylim([-1 1]);
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APPENDIX 6: M-file code of IPM Motor for Short pitched, Full-pitched and concentrated winding for
Airgap
clear; clc; g1=0.3; g2=50; s1=[ones(1,25)*(1/g1) ones(1,50)*(1/g2)]; s2=[ones(1,50)*(1/g1) ones(1,50)*(1/g2)]; s3=[ones(1,50)*(1/g1) ones(1,50)*(1/g2)]; s4=[ones(1,50)*(1/g1) ones(1,50)*(1/g2)]; s5=[ones(1,25)*(1/g1)]; A=[s1 s2 s3 s4 s5]; tt1=linspace(0,2*pi,400); plot(tt1,A,'k-','linewidth',2);grid minor; xlabel('\pi [rad]'); ylabel('Stator tooth length'); title('AIRGAP'); ylim([-1 4]);
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