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Title Design, analysis and application of low-speed permanentmagnet linear machines

Advisor(s) Chau, KT

Author(s) Li, Wenlong; g ‡Ÿ™

Citation

Issued Date 2012

URL http://hdl.handle.net/10722/173931

Rights The author retains all proprietary rights, (such as patent rights)and the right to use in future works.

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Design, Analysis and Application of Low-speed

Permanent Magnet Linear Machines

by

LI, Wenlong

B.Sc.(Eng), M.Sc.(Eng.)

A thesis submitted in partial fulfillment

of the requirements for the degree of

Doctor of Philosophy

at the

Department of Electrical and Electronic Engineering

The University of Hong Kong

in

September 2012

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DECLARATION

I hereby declare that this thesis represents my own work, except where due

acknowledge is made, and that it has not been previously included in a thesis,

dissertation or report submitted to this University or to any other institution for a

degree, diploma or other qualifications.

Signed

LI, Wenlong

September 2012

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To my parents

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Abstract of thesis entitled

Design, Analysis and Application of Low-speed

Permanent Magnet Linear Machines

Submitted by

LI Wenlong

for the degree of Doctor of Philosophy

at The University of Hong Kong in September 2012

With the growing interests and high requirements in low-speed linear drives, the

linear machines possessing high force density, high power density and high efficiency

feature become in great demands for the linear direct-drive applications. There are

many available linear machine topologies, but their performances for exhibiting the

high-force density capability dissatisfy the industrial requirements. In order to solve

this problem, the new machine topologies emphasizing on high force density are

explored and studied. The objective of this thesis is to present the design, analysis,

and application of permanent magnet (PM) linear machines which can offer a higher

force density at the same magnetic loading and electric loading than the conventional

machines.

Although in recent years there are many emerging advanced PM rotational

machines for direct-drive rotational drives, the development of advanced PM linear

machines for direct-drive linear drives is sparse. In spite of the motion type of electric

I

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II

machines, the inherent operating principle is the same. By studying and borrowing

concepts of the high torque density rotational electric machines, the linear machine

morphologies of the promising candidates are designed and analyzed. The problems

and side effects resulting from the linearization are discussed and suppressed.

Two main approaches for machine design and analysis are developed and applied,

namely the analytical calculation and the finite element method (FEM). By

analytically solving the magnetic field problem, the relationships between the field

quantities and the machine geometry are unveiled. With the use of analytical

calculation, the machine design and dimension optimization are conveniently

achieved. With the use of FEM, the machine design objective and its electromagnetic

performance are verified and evaluated.

Finally, the proposed low-speed PM linear machine is applied for direct-drive

wave power generation. By mathematically modeling the wave power, generation

system and the generator, the conditions for maximum power harvesting are

determined. By using the vector control, the generator output power is maximized

which is verified by the simulation results.

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ACKNOWLEDGEMENTS

Firstly and foremost, I greatly appreciate and express my deepest gratitude to my

supervisor Professor K.T. Chau, for his generous support and guidance on my

academic and professional career. His profound knowledge and extensive

professional experience and invaluable discussion lead me into the science world and

make me understand what the in depth research is. He helps me grasp the research

skills and enrich my study in the academic ocean. His academic knowledge and life

attitude benefit me all my life.

I also would like to express my thanks to Prof. C.C. Chan, Prof. J.Z. Jiang and

Prof. M. Cheng. Prof. C.C. Chan is always full of energy in his career and is an

amiable person. Prof. J.Z Jiang is an extremely nice teacher. His experience and

knowledge are the treasure for me. Prof. M. Cheng provides me greatly convenience

in fabrications for machine prototype and its test-bed. Here, I am very grateful to

them again.

My sincere thanks also owns to Mr. Raymond S.C. Ho, who always give his

selfless help to me. Whatever my research or daily life, he always supports me

quickly.

Many thanks are also given to my research group, my teachers and my friends.

Their help, advice, guidance, encouragement and support are very helpful during my

study, most notably Dr. Y.B. Li, Dr. Y. Fan, Dr. S. Ye, Dr. Z. Wang, Dr. X.Y. Zhu,

III

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IV

Dr. W.X. Zhao, Dr. C. Liu, Dr. S. Niu, Dr. C. Yu, Dr. L. Jian, Dr. X. Zhang, Miss J.

Li, Miss. S. Gao, Miss. D. Wu, Mr. Z. Zhang, Mr. F. Li, Mr. M. Chen, Mr.

Christopher H.T. Lee, Mr. D. Yi, and Miss R.Y. Ma.

I would like to express my deepest appreciation to my parents and my sisters.

Their love gives me the power and strength. With their understanding and

encouragement, my life is always energized and full-hearted.

This work was supported in part by a grant (Project No. HKU 710711E) from the

Research Grants Council, Hong Kong Special Administrative Region, China.

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CONTENTS

DECLARATION

ABSTRACT I

ACKNOWLEDGMENTS III

CONTENTS V

CHAPTER 1 INTRODUCTION

1.1 Background .........................................................................1

1.2 Objective and Contribution.................................................4

1.3 Overview of Emerging Advanced PM Machines ...............5

1.3.1 Stator-PM Machines ..................................................... 6

1.3.2 Variable Reluctance Machines.................................... 11

1.3.3 Magnetic Gear and its Integrated Machines ............... 13

1.4 Thesis Outlines..................................................................15

CHAPTER 2 ANALYSIS APPROACHES FOR

PERMANENT MAGNET LINEAR

MACHINES

2.1 Introduction .......................................................................17

2.2 Maxwell’s Equations.........................................................18

2.2.1 Integral Form .............................................................. 18

2.2.2 Differential Form ........................................................ 18

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2.3 Analytical Calculation.......................................................19

2.3.1 Magnetic Scalar Potential ........................................... 20

2.3.2 Magnetic Vector Potential........................................... 20

2.3.3 Boundary Conditions .................................................. 21

2.4 Finite Element Method......................................................22

2.5 Parameter Calculation.......................................................26

2.5.1 Induced Voltage Calculation....................................... 26

2.5.2 Inductance Calculation ............................................... 26

2.5.3 Force Calculation........................................................ 27

2.6 Summary ...........................................................................28

CHAPTER 3 TRANSVERSE-FLUX PERMANENT

MAGNET LINEAR MACHINES

3.1 Introduction .......................................................................29

3.2 Linear Morphology of Transverse-flux Machines ............30

3.3 Cogging Force Migration..................................................35

3.4 Proposed TFPM Linear Machine and its Improvement....38

3.4.1 Proposed Machine Structure....................................... 38

3.4.2 Thrust Force Generation Principle.............................. 40

3.4.3 Analytical Results....................................................... 41

3.5 Summary ...........................................................................46

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CHAPTER 4 LINEAR MAGNETIC GEARS AND THE

INTEGRATED MACHINES

4.1 Introduction .......................................................................47

4.2 Linear Magnetic Gears......................................................50

4.2.1 Operating Principle..................................................... 50

4.2.2 Transmission Capacity Improvement ......................... 55

4.3 Analytical Computation ....................................................59

4.3.1 Analytical Model ........................................................ 59

4.3.2 Magnetic Field Solution ............................................. 64

4.3.2.1 Field Solution in Regions without PMs ............ 64

4.3.2.2 Field Solution in the Region with PMs ............. 64

4.3.2.3 Boundary Conditions......................................... 66

4.3.3 Calculation Results and Verification .......................... 68

4.4 Linear Magnetic-geared Machines ...................................74

4.4.1 Linear Machine Selection........................................... 76

4.4.2 Performance Analysis ................................................. 79

4.5 Quantitative Comparison ..................................................86

4.6 Summary ...........................................................................88

CHAPTER 5 PERMANENT MAGNET LINEAR VERNIER

MACHINES

5.1 Introduction .......................................................................89

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5.2 Vernier Structure ...............................................................91

5.2.1 Configurations ............................................................ 91

5.2.2 Operating Principle..................................................... 94

5.3 Design Procedure ............................................................100

5.4 Mathematical Modeling ..................................................103

5.5 Analysis...........................................................................105

5.6 Discussion ....................................................................... 112

5.7 Summary .........................................................................114

CHAPTER 6 INDUSTRIAL APPLICATION FOR

DIRECT-DRIVE WAVE ENERGY

HARVESTING

6.1 Introduction.....................................................................115

6.2 Overview of Wave Energy Harvesting Techniques ........116

6.2.1 Rotational Type......................................................... 116

6.2.2 Linear Type ............................................................... 121

6.3 Modeling of the Oceanic Waves .....................................123

6.4 Modeling of Direct-drive Wave Energy Converter.........125

6.5 Modeling of PMLV Machine ..........................................126

6.6 Power Conditioning System............................................128

6.7 Summary .........................................................................135

VIII

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IX

CHAPTER 7 CONCLUSIONS AND

RECOMMENDATIONS

7.1 Conclusions .....................................................................136

7.2 Recommendations...........................................................138

LIST OF FIGURES 140

LIST OF TABLES 147

REFERENCES 148

APPENDICES 162

PUBLICATIONS 167

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CHAPTER 1

INTRODUCTION

1.1 BACKGROUND

Linear motion is a fundamental motion type that an object travels in a straight

line. It is quite universal in industrial field, such as transportation and factory

automation system, etc. For conventional industrial application, the linear motion is

usually converted from the rotational motion by a rotational electric motor with theintermediate mechanical components such as ball screw, lead screw and rack and

pinion, etc. As shown in Figure 1.1, the motion type conversion is commonly realized

by teeth of the different mechanical devices meshing with each other. The meshing

engagement of the mechanical devices for motion type conversion inevitably incurs

loss, noise, vibration, regular maintenance, and degrades the precious positioning

capability. Therefore, the direct-drive electric machines are highly expected.

The linear machine operation principle can described as the following model. As

show in Figure 1.2, when the switch is closed on, the DC current flows anticlockwise

in the circuit. Since the sliding bar is exposed into a magnetic field directed out of the

page, a Lorentz force is exerted on the sliding bar which drives the sliding bar

straightly forward to the left hand side. When the battery is short circuited, the sliding

bar is driven by man hand, a current can also be drawn in the circuit. At this situation,

the linear machine operates as a generator. The linear machine was firstly invented by

Sir Charles Wheatstone in 1840s [1]. This prototype has the same structure as modern

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Introduction

linear machines which can be considered as slitting the rotational one longitudinally

and unrolling it into a flat one. Due to the low efficiency and difficulty in control, the

linear machine in its early ages was not applied widely. Since 1960s, with

advancement of the material industry, computer technology and control theory,

development and application of linear electric machines are in an accelerated pace.

Particularly with the widespread applications of the high energy product permanent

magnet (PM) material for providing the excitation magnetic field, the research and

development of PM machines attracts more and more attention. Compared to the

electrically excited machine, the PM machines possessing features of simplestructure, robust, high energy density, and high efficiency, etc., are widely used in

industrials and household appliances. As its rotational counterpart, the linear machine

topologies ranges in induction, synchronous, stepping, reluctance, etc. Its application

spreads in various fields, such as industrial automation, robotics, power generation,

and transportation, etc [2]-[6].

Low-speed drives attract more and more attention in recent years with the active

demands for renewable energy related industrials, such as wind power generation and

electric vehicle motor drives, etc [7]-[10]. For the conventional electric machine

which usually operates at a high speed compared to the speed of wind turbine or

vehicle wheels, the low-speed gearless drives usually render a large physical volume

and relative low efficiency at the same power rating. In order to solve these problems,

mechanical gearboxes for speed reduction and torque transmission are applied which

can improve the efficiency of the whole driving system. However, the mechanical

transmission units inevitably incur the system complexity, increased cost and further

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Introduction

deteriorate the control performance and reliability. Therefore, direct and gearless

driven approaches are put on the agenda. In order to satisfy the above requirement,

the electric machine possessing high thrust density, high power density and high

efficiency features is high expected.

Figure 1.1 Rack and pinion for linear-rotational motion conversion.

Figure 1.2 Idealized linear DC machine model.

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Introduction

1.2 OBJECTIVE AND CONTRIBUTION

Although various direct-drive rotational machines are proposed and studied,

there is not much literature focusing on the direct-drive and low-speed linear drives.

The objective of this thesis is to develop PM linear machines exhibiting high force

density, high power density and high efficiency for low-speed, direct-drive and linear

motion applications. This thesis deals with the following aspects:

Extending the promising rotational PM machine morphologies into linear

morphologies. Study and discuss the problems raised by the morphology

extension. Propose a design methodology for PM linear machine design.

Analyzing electromagnetic performances of the proposed PM linear machine

with both numerical method and analytical method. The two analysis

methods have their own pros and cons which can be complementary to each

other to some extent. The numerical method gives a detailed and precious

evaluation, but lack of physical insight and time consuming. The analytical

method describes the relationship between the machine performances with its

geometry which can guide the machine design. In addition, the analytical

method gives fast and relatively precious results.

Based on the two analysis methods, optimal design for machine structure can

be carried out. According to the analytical expressions of the machine

performance, the needed field quantities are optimized under several

limitations. Finally, the results are verified by the numerical methods.

Application for oceanic wave power generation is assessed. Wave energy has

an abundant storage with low-frequency and time-varying feature. In order to

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Introduction

maximize the harvesting efficiency, the vector control of the PM linear

machine is applied and evaluated.

1.3 OVERVIEW OF EMERGING ADVANCED PM MACHINES

The electric machine what ever it operates in linear or rotational motion has the

same the operating principle that is to engage for the mechanical energy and electric

energy inter-convention. The thrust force/torque in electrical machines can be

developed by two traveling/rotating magnetic field interactions or by switching

magnetic field with variable reluctance mover/rotor. For PM linear machines, the

force generation can be deduced by derivative of the magnetic field co-energy [11].

The stored magnetic field co-energy can be expressed as:

PM PM co W i LiW 2

21

(1)

where L is the synchronous inductance, i is the armature current, Y PM is the armature

flux linkage produced by PMs, W PM is the magnetic field co-energy only produced by

PMs.

Consequently, the thrust force can be easily obtained by derivative of the

magnetic co-energy field when the current is kept unchanged:

cogPM co

em F idxd

idxdL

dxdW

F 221

(2)

The thrust force consists of three force components: reluctance force component,

PM force component, and cogging force component. For non-saliency machines, the

synchronous inductance is space-invariant, and the reluctance force component can

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Introduction

be ignored. The cogging force component is a parasitic component caused by slot-

effect and end-effect.

The research and development of low-speed and direct-drive rotational machines

for renewable energy application becomes a hot topic in recent years. However, the

machine topologies of PM linear machines are not diversiform as that of PM

rotational machines. In order to fulfill the research objective — — research and

development of low-speed PM linear machines, it is necessary to draw on experience

from that of the PM rotational machines. Due to the booming development of wind

power generation and electric vehicles, advanced PM machine topologies emerge in

an endless stream. The following overview reviews representatives of these PM

machines.

1.3.1 Stator-PM Machines

Switched reluctance machines (SRMs) utilizing a double salient structure for

torque production, has many distinct merits: simple structure, inherent fault tolerance

and high reliability, etc. Therefore, they are widely applied in wind power generation,

and wave power generation. However, due to only one excitation source, they suffer

some major drawbacks: excitation penalty, acoustic noise, torque jerk, and relative

low torque density. In order to solve the above problems, a new class machine which

incorporates PMs into the stator of SRM was proposed to overcome the shortcomings

[12]. Thanks to the PMs, the torque production of this machine can be greatly

improved. According to the PM location in the stator, they are classified as following

categories.

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Introduction

A. Doubly Salient PM (DSPM) Machines [13]-[15]

In DSPM machines, the PMs usually located in the stator yoke as shown in

Figure 1.3. Due to the doubly saliency of the stator and the rotor, the flux links the

armature winding in a variation mode along with the rotating of the rotor. Although it

has salient poles in the stator and rotor, the PM torque significantly dominates the

reluctance torque, hence exhibiting low cogging torque. Thus, the torque density of

DSPM machine is higher than that of the SR machine. Since the variation of flux

linkage with each coil as the rotor rotates is unipolar, it is very suitable for the BLDC

operation.

Figure 1.3 DSPM machine.

B. Flux-reversal PM (FRPM) Machines [16], [17]

PMs in the FRPM machine are placed on surface of stator teeth, as shown in

Figure 1.4. Each stator tooth has a pair of magnets of different polarity mounted at its

surface. When a coil is excited, the field under one magnet reduced while another one

is increased, and the salient rotor pole rotates towards the stronger magnetic field.

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Introduction

The flux-linkage with each coil reverse polarity as the rotor rotates. Thus, the phase

flux-linkage variation is bipolar, while the phase back-EMF waveform is trapezoidal.

Such a machine topology exhibits a low winding inductance, while the magnets are

more vulnerable to partial irreversible demagnetization.

Figure 1.4 FRPM machine.

C. Flux-switching PM (FSPM) Machines [18], [19]

PMs in the FSPM machines are located in the stator teeth. As shown in Figure

1.5 , the stator consists of U-shaped segments with PMs sandwiched between them.

The PMs are circumferentially magnetized, thus they possess the flux-focusing and

low energy density PMs can be employed. In addition, the PMs are immune to the

armature reaction, thus the electric loading can be set very high which results in a

high per-unit winding inductance. Therefore, they are very suitable for constant

power operation over a wide speed range. The phase flux-linkage waveform is

bipolar. The back-EMF waveform of this kind of machines is sinusoidal.

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Introduction

Figure 1.5 FSPM machine.

D. Flux-controllable PM (FCPM) Machines

The field excitation of the stator-PM machine introduced above is provided by

the PMs. Due to the unchangeable work point, the air-gap flux density can not be

achieve flexible adjust which may not satisfy the high demand drives. In order to

online tune the air-gap flux, DC field windings are invited to online regulate the air-

gap flux density which results in a new class machine named FCPM machine. With

the DC field windings for flexible flux control, the constant power operation range of

FCPM machines can further extended. The two types of FCPM are introduced as

following:

1) PM Hybrid Brushless (PMHB) Machines [20]-[22]

As shown in Figure 1.6, this machine has a similar structure with DSPM machine,

but it has a DC field winding located in the inner stator. With this field winding, the

hybrid excitation of this machine can enable an online flux controllable ability. Thus,

the flux strengthening can be used in the starting and acceleration stage and the flux

weakening function can be applied in the high-speed operation range which can

enhance the machine performances.

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Figure 1.6 PMHB machine.

Figure 1.7 Memory machines. (a) Single-magnet arrangement. (b)Dual-magnetarrangement.

2) Memory Machines [23]-[25]

The PMHB machine adopts a field winding for flux control, but due to a

continuous dc current fed for hybrid excitation, it suffers an extra copper loss. In

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Introduction

order to avoid this loss, the AlNiCo PM alloy is adopted for flux control. The high

effectiveness is due to its direct magnetization of PMs by magnetizing windings,

whilst the high efficiency is due to the use of temporary current pulse for PM

magnetization. The memory machine can be designed with only AlNiCo PM for

excitation [9] or AlNiCo PM and NdFeB PM for hybrid excitation [10], as shown in

Figure 1.7.

1.3.2 Variable Reluctance PM Machines

The variable reluctance PM (VRPM) machine is a class of PM brushlessmachines dedicated to low-speed high-torque direct-drive applications. The essential

of this machine family is that the interaction of multi-pole PMs with a group of teeth

which results in the variation of flux linkage in the stator windings [26].

A. Transverse-flux PM (TFPM) Machines

The TFPM machine featuring as high-force density is very suitable for direct-

drive applications. Since their magnetic flux paths are orthogonal to the current flow

plane of the armature winding, the magnetic loading is totally decoupled from the

electric loading, as shown in Figure 1.8 [26]-[27]. The corresponding electric loading

can be much higher than that of conventional one which can achieve a higher

electromagnetic force. As shown in Figure 1.8, another merit of the TFPM machine is

that the phases are decoupled and have little influence on each other which may have

a good capability of fault-tolerance applications. However, due to the 3-D flux path,

the complicated machine structure is often criticized by the users.

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Figure 1.8 3-phase TFPM machine.

B. PM Vernier (PMV) Machines

The PMV machine is another key member of the VRPM machine family. The

PMV machine has a conventional flux path and its magnetic circuit is featured as the

slotted structure and multi-pole PM configuration [28]-[31]. As shown in Figure 1.9,

it can be designed as a toothed-pole stator with PMs mounted on its rotor, and a stator

with PM mounted on its tooth surface and a slotted rotor. The first one operates due

to the two rotating magnetic field, and the second one works as the FSPM machine. A

small movement of the rotor can cause a large flux-linkage variation in the armature

winding which further results in a high torque. This is also known as the magnetic

gearing effect which results from the interaction between the PMs and toothed-pole

structure. Due to features of the high torque/force density and the compact structure,

it is very suitable for the direct-drive applications.

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Figure 1.9 PMV machine. (a) Rotational morphology. (b) Linear morphology.

1.3.3 Magnetic Gear and its Integrated Machines

The Magnetic gear is reported as a high torque density and high power density

device for torque transmission and speed reduction. Compared to the mechanical one,

the torque transmission is realized by the interaction between two rotating magnetic

fields, and no physical contact is needed. Therefore, they have many distinct merits

such as high efficiency, reduced acoustic noise, and maintenance free, etc. By

integrating the magnetic gear with a conventional PM brushless machine, the

integrated machine can retain the merits of magnetic gears, and adopt a high-speed

machine design to improve its efficiency [32], [33]. As shown in Figure 1.10, it is

proposed to replace the conventional power train system where combination of the

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Introduction

mechanical gear and the electric motor is often used. Due to the prominent

advantages, the integrated machines are very suitable for direct-drive application,

such as electric vehicle drive system and wind power generation.

Figure 1.10 A magnetic-geared machine.

The literature review covers the emerging PM machines of the near decades

which gives us enough knowledge of various machine topologies and their

performances. Three promising machine topologies fall into our research and

development candidates, namely transverse-flux permanent magnet (TFPM) machine,

magnetic-geared machine (MGM), permanent magnet vernier (PMV) machine. The

research will be carried out based on the above three machine topologies.

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Introduction

1.4 THESIS OUTLINES

This thesis consists of seven chapters. The content of each chapter is briefly

introduced as follows:

In Chapter 2, the analysis approaches for PM linear machine which provides

theoretical background for this thesis are presented. This chapter focuses on the

analytical approach and numerical approach for magnetic field calculation and

electromagnetic performance assessment. The cogging force minimization for PM

linear machine is discussed.

Chapter 3 devotes the design and analysis for linear TFPM machine. Due to the

end-effect of linear machines, the design consideration and faced problem is

discussed which is different from the design of a rotational machine. For further

improve the force density, the high temperature superconductor (HTS) bulks are

utilized for field shielding which contributes much for force improvement.

In Chapter 4, the linear magnetic gear operating principle and its mathematical

modeling is intensively studied. The analytical computation modeling for the linear

magnetic gear in cylindrical coordinates is developed. Thereafter, the linear

magnetic-geared machine is proposed. Its static performance and dynamic

performance are assessed.

In Chapter 5, a new machine structure named vernier machine is proposed which

can be regarded as evolution from the linear magnetic-geared machine. This machine

attains the magnetic field modulation effect of the magnetic gear but has only one air-

gap and one moving parts. Based on the analytical expression, the toothed-pole

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Introduction

16

structure for field modulation is optimized. A prototype is also fabricated to testify

the analytical results which show good agreement.

The application of PM linear vernier machine for wave power generation is

discussed in Chapter 6. Firstly, the wave power generation techniques are reviewed.

Then, the direct-drive wave energy conversion is selected. In order to maximize the

harvesting power, vector control of PMLV machine is applied.

Chapter 7 is the last chapter and gives the conclusion of the whole thesis and

recommendations for future work.

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CHAPTER 2

ANALYSIS APPROACHES FOR PERMANENT

MAGNET LINEAR MACHINES

2.1 INTRODUCTION

Permanent magnet (PM) electric machines apply PMs for providing excitation

field without external excitation circuit, therefore the machine structure can be

simplified and efficiency can be improved.

The PM in the PM machines not only serves as a magnetomotive force (MMF)

source, but also composes part of the magnetic circuit. Due to special features of the

PMs, design and analysis approach of PM machines can not totally refer to that of the

electrically-excited machines. In general, there are two main approaches for PM

machine analysis [34]. One is based on the equivalent magnetic circuit method, and

the other is based on the magnetic field. The first approach simplifies the magnetic

field problem into magnetic circuit with PM considered as MMF source or flux

source. The computation complexity is low but accuracy is not high. Although the

magnetic circuit method can satisfy the industrials at some situation, it can not

preciously predict the flux distribution, some nonlinear characteristics and the

saturation problems which are common in the real cases. The magnetic field approach

can give preciously assessment of the PM machines, since the saturation of

ferromagnetic materials, motion of the mover, tooth-slot effect and the skin effect etc.

The magnetic field problems describe by a set of Maxwell’s equations. By solving

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Analysis Approaches for Permanent Magnet Linear Machines

these equations, the field quantities can be obtained, and the electromagnetic

performance of PM machines can be predicted. Two popular methods for solving the

Maxwell’s equations are analytical method and numerical method. They are

introduced in the following section.

2.2 MAXWELL’S EQUATIONS [35]

Maxwell’s equations are organized and improved from the Ampere’s law,

Faraday’s law and Gauss’s law by James Clerk Maxwell. These equations express the

sources, field quantities and the interaction between them.

2.2.1 INTEGRAL FORM

Ad t

D I l d H

s sc (1)

Ad t

Bl d E

sc (2)

0 s

Ad B (3)

Q Ad E s

(4)

where H is the magnetic field intensity, I s is the free current within the surface s,

D is the electric displacement, E is the electric field intensity, B is the magnetic

flux density, Q is the net electric charge within the surface s, and c is the closed

boundary of the surface s.

2.2.2 DIFFERENTIAL FORM

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By using the Stokes’ theorem and Gauss theorem, the integral form of Maxwell’s

equation can be converted into the differential form which is most used for solving

the field problems.

t D

J H (5)

t B

E (6)

0 B (7)

D (8)

where J is the free current density and ρ is the free charge density.

The above quantities obey following conditions:

E J (9)

E D (10)

H B 0 (11)

H B B r 0 (12)

where σ is the electric conductivity, ε is the electric permittivity, μ is the magnetic

permeability of the free space, and r B is the remanence of the magnetic material. Eq.

(11) is applicable for the electromagnetic field in the free space, whereas (12) is

applicable for the electromagnetic filed in the magnetic materials.

2.3 ANALYTICAL CALCULATION

The field density B and intensity H can not be easily obtained from the above

differential equations in the most cases. In order to simplify the problem and reduce

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the variables, the potential functions are usually used as the assistant quantities [35].

According to the Curl of intensity H , the vector field can be classified as the

irrotational field and the solenoidal field.

2.3.1 MAGNETIC SCALAR POTENTIAL

In vector calculus, the curl of a gradient of a scalar field always gains the zero

vector. Therefore, in the irrotational field, the field vector can be expressed as a

gradient of a function in terms of the magnetic scalar potential φ :

0)( H (13)

)( k z

j y

i x

H z y x

(14)

In the PM region, Eq. (13) can be re-organized as:

M 2 (15)

where is the magnetization vector of PM materials.

In other region, Eq. (13) can be re-organized as:

0 (16)2

2.3.2 MAGNETIC VECTOR POTENTIAL

In the solenoidal field, the field vector can be expressed as a curl of a function in

terms of the magnetic vector potential A [36]. In Cartesian coordinates, the magnetic

field density can be expressed as:

k y

A x

A j

x A

z A

i z

A

y A

A B x y z x y z )()()( (17)

The equation in current region:

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J A H H )()( (18)

In the PM region:

)()( r B J A (19)

In other region:

0)( A (20)

2.3.3 BOUNDARY CONDITIONS [35]

The analytical calculation of the magnetic field can be conducted by the above

two approaches which are finally deduced into a set of partial differential equations.

To solve these equations, a set of expression called general solution can be achieved.

In order to gain the unique solution for the partial differential equations, the

conditions for describing the field boundaries and initial values can make the problem

solvable. In most cases, only the boundary conditions can be listed out. There are

three kinds of conditions which are elaborated as follows.

(1) First type boundary condition

It is also called the Dirichlet condition. For this situation, the potential u along

the boundary s can be expressed by a function.

)(1 s f u (21)

When the value is zero, the boundary condition is also called homogenous

Dirichlet condition.(2) Second type boundary condition

It is also called the Neumann condition. For this situation, the normal derivative

of the potential u along the boundary s can be expressed by a function.

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)(2 s f nu

(22)

When the value is zero, the boundary condition is also called homogenous

Neumann condition.

(3) Third type boundary condition

It is also called the Robin condition which is the linear combination of the first

type and second type boundary conditions.

)(321 u f nu

k uk (23)

where k 1 and k 2 are constants.

In electrical machine analysis, the first type and second type boundary conditions

are applied in most cases. For the boundary between two different media, the normal

component of the flux density and the tangential component of the field intensity are

kept unchanged which indicate that:

nn B B 21 (24)

t t H H 21 (25)

2.4 FINITE ELEMENT METHOD [37]-[39]

The analytical calculation can give sufficient insight of machine performance and

its dimension. However, for the complicated structure and nonlinear materials, the

analytical calculation may have no closed solutions. In order to simplify the

calculation process, some assumptions are made such as reluctivity, saturation effect,

and core losses for ferromagnetic material which are usually ignored.

In order to consider the nonlinear feature for PMs and ferromagnetic materials,

the numerical analysis including the finite element method (FEM), the boundary

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element method (BEM) and the finite difference method (FDM) can have a precious

result and a general application. Especially the finite element method (FEM) is

applied widely for electrical machine design and analysis.

FEM uses discrete method to solve the partial differential equations raised by

Maxwell’s equation. The triangles are often adopted for space variable discretization

as show in Figure 2.1 where the target surface is split into 5 regions with 6 nodes.

2 512

3

4

5

1

3 4

6

Figure 2.1 FEM using triangles.

In each triangle, the potential can be expressed by its geometry and the potential

at the three vertexes.

n

iii u y x N u

1

),( (26)

where N i(x, y) is the shape function and u i is the potential at each vertex of the

triangle.

As shown in Figure 2.2, the shape function can be expressed as:

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2

2

2

yc xba N

yc xba N

yc xba N

mmmm

j j j j

iiii

(27)

where the coefficients a i, b i, c i, a j , b j , c j , a m, b m and cm can be determined by:

i jm jimi j jim

mi jim jmimi j

jmimii jmm ji

x xc y yb y x y xa

x xc y yb y x x xa

x xc y yb y x x xa

,,

,,

,,

,

and D is the triangle area, D = ( b ic j - b jc i)/2.

Figure 2.2 Vector potential presentation using a triangle.

Therefore, the regions to be solved in the electrical machine can be discretized

according to the above approaches.

In the stator winding region, according to (18), it yields:

dxdyS i

Nidxdy A N y y

N A N

x x N

v j

j ji

j j j

i

3

1

3

1

(28)

In the PM region, according to (19), it yields:

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dxdy N x

B N y

Bv

dxdyt

A N dxdy A N

y y N

A N x x

N v

j jry

j jrx

ii PM

j j j

i

j j j

i

3

1

3

1

3

1

3

1

(29)

In the airspace region including air-gap, according to (20), it yields:

03

1

3

1

dxdy A N y y

N A N

x x N

v j

j ji

j j j

i (30)

In the iron core, if the eddy current effect is taken into account, it yields:

03

1

3

1

dxdyt

A N dxdy A N

y y N

A N x x

N v ii Fe

j

j ji

j

j ji

(31)

The above discretized equation in each region can be reformed into the following

matrix:

][][][ P

t it

A

Di

AC (32)

where [A] is the vector potential matrix, [i] is the current matrix, [C] and [D] are the

coefficient matrix, and [P] is the matrix related to the output voltage and PM material.

The variables in (32) all relate to time, thus time discretization of these variables

should be carried out for solving the matrix. By applying the implicit Euler method, ()

can be expressed as:

][][][ t t

t t

t t t t

t t

t t t t t t

P i

At

Di

At

DC

(33)

Due to the use of the ferromagnetic material, coefficients in (33) contain the

electric resistivity which depends on the electric field intensity. For solving the

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nonlinear problem, the Newton-Raphson method is commonly used. After equation

linearization, the ICCG method is adopted for solving these linear equations.

2.5 PARAMETER CALCULATION

2.5.1 INDUCED VOLTAGE CALCULATION

According to (2), the induced voltage can be calculated by derivative of the flux

linkage in the coil. With the knowledge of magnetic vector potential, it is easy to find

out the flux linking one coil by the following equation:

ef l A A 21 (34)

where A1 and A2 is the magnetic vector potential at the two sides of one coil and l ef is

the effective length of the coil. When the 2-D analysis is applied, the magnetic vector

potential is degraded to a scalar value.

Therefore, the induced voltage in one coil can be deduced by:

dt dx

dxd

N e

(35)

where N is the number of turns of the coil.

When the induced voltage of one coil is obtained, the voltage of one phase can

be determined by summing of the each coil of that phase.

2.5.2 INDUCTANCE CALCULATION

The phase inductance can also be determined by the flux linkage method. Due to

the PM excitation, the total flux linkage of the winding sum of the flux linkage

produced by current and PMs. The inductance of one winding is determined by:

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ii L PM tot i

(36)

where Y i and Y PM is the flux linkage produced by current i and PMs respectively, and

Y tot is the sum of the two items.

When the flux density in the iron core goes to saturation, the winding inductance is

different from the calculation by (). The actual inductance should be calculated by the

incremental inductance:

i L i (37)

2.5.3 FORCE CALCULATION

With the information of magnetic field, the thrust force of the linear machine can

be determined by Maxwell stress tensor which expresses the force pre unit area on a

surface produced by the magnetic field.

The tangential force of a point which is parallel to the surface can be calculated

by:

0

t nt nt

B B H B f (38)

where Bn and B t is the normal and tangential component of flux density at one point

in the air-gap respectively, and H t is the field intensity at that point.

The normal force of a point which is perpendicular to the surface can be

calculated by:

0

2

nnnn

B H B f (39)

Therefore, the thrust force acting on the surface is:

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28

ds f F s t

(40)

2.6 SUMMARY

In this chapter, the analysis approaches for PM linear machine are discussed. The

two approaches are focused, namely analytical calculation and the FEM analysis. The

first one deals with a set of partial differential equation derived from the Maxwell’s

equation. Since the analytical expression shows the relationship of field quantities

with the machine structure, it is helpful for machine design and parameter

optimization. The latter one is a numerical approach to find the approximate solution

of the partial differential equation using the discretization. The FEM method can

gives accurate solution of a particular machine structure with considering the

nonlinear characteristics. With the assist of the two approaches, the design and

analysis of PM linear machines are carried out in the following chapters.

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CHAPTER 3

TRANSVERSE-FLUX PERMANENT MAGNET LINEAR

MACHINES

3.1 INTRODUCTION

In the conventional machines, the developed torque/thrust is determined by the

magnetic loading and the electrical loading. Thus, the average thrust force developed

in a linear electrical machine can be estimated in terms of Lorentz force equation:

l JS Bb

p Il Bb

p IlB F st t

t t

t

t ag em

(1)

where l is the stack length of the flat linear machines and the circumferential length

of the tubular linear machines, Bag is the air-gap flux density, p is the mmf pole-pair

numbers of the field excitation, b t is the tooth width, τ t is the tooth pitch, B t is the

flux density in stator tooth, J is the current density in one slot, and S s is the slot area.

The thrust density per unit area can be obtained by:

st t m

t st

t

t

ag

emd JS B

bl JS B

b p

S F

F

(2)

where S ag is the total area of the air-gap and τ m is the mmf pole-pitch of the field

excitation.

According to (2), for improving the thrust density, the flux density in the tooth,

current density, slot area and tooth width should be increased. The flux density in the

tooth and the current density depends on the ferromagnetic permeability and the

cooling method respectively. Therefore, it is quite straightforward to increase thrust

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Transverse-flux Permanent Magnet Linear Machines

density by enlarging the tooth width b t and slot area S s . However, in the radial-flux or

longitudinal-flux electrical machines, the product of two variables b t and S s which

has the inverse relationship can not be increased.

To solve this problem, a new class of electric machine named transverse-flux

permanent magnet (TFPM) machine was proposed by H. Weh [40]. In this kind of

electric machines, the flux path plane is orthogonal to the rotor movement plane, the

magnetic loading and electrical loading which related to b t and S s can be adjusted

independently. Therefore, the torque density is higher compared to their radial-flux

counterparts.

3.2 LINEAR MORPHOLOGY OF TRANSVERSE-FLUX

MACHINES

Figure 3.1 TFPM machine arrangements. (a) U-shaped core stator. (b) C-shaped corestator.

Figure 3.1 (a) shows the principle model of a typical TFPM motor [40]. It adopts

the double-stator arrangement with the rotor/mover sandwiched between the two

stators. Its stator consists of U-shaped cores and windings on both sides of the

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translator. The U-shaped cores of the upper stator and the lower stator have a

separation of a PM pole-pitch to form the flux path. Its mover consists of two rows of

PMs and flux concentrators with nonmagnetic material in between. The stator has

two sets of windings placed in the upper and lower stator core respectively. Since

their magnetic flux paths via the upper and lower U-shaped stator cores are

orthogonal to the current flow of the armature winding, the magnetic loading is

totally decoupled from the electric loading. Hence, the corresponding electric loading

can be much higher than that of its longitudinal-flux counterpart. Figure 3.1 (b)

shows the TFPM machine model with C-shaped cores [41]. Compared to the U-shaped ones, the stator configuration is simpler. It consists of outer stator core, stator

joint core and inner stator-core to form a stator core unit. Then several of the same

units are assembled to compose a single phase. These two topologies suffer from the

drawbacks that it involves too many components which make the structure

complicated and cause manufacturing difficulty. Thus, several other shaped stators

are invited for TFPM design to ease the fabrication, such as E-shaped core [42] and

soft material composite (SMC) stator core [43]. In this chapter, a new C-shaped core

is adopted for the linear TFPM machine design.

Figure 3.2 depicts the proposed C-shaped stator core of the linear TFPM

machine, in which the PM mover lies between the core teeth. The dimensions of the

C-shaped stator core are w1 = 52 mm, w2 = 20 mm, w3 = 30 mm, h 1 = 34 mm and

h2 = 18 mm. It can be observed that it retains the orthogonal feature between the

magnetic loading and electric loading, which enables the motor to achieve high force

density. Compared with the two U-shaped stator cores, the proposed C-shaped core

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takes the definite advantage of simple structure and hence easy manufacturing. Also,

it can provide a larger cross-sectional area for armature windings, leading to further

increase the electric loading and hence force density.

Figure 3.2 Cross-section of proposed linear TFPM machine.

Figure 3.3 Linear TFPM machine with C-shaped cores.

This linear TFPM machine is depicted in Figure 3.3, in which the stator contains

three segments of C-shaped iron cores as embraced by armature windings, while the

translator consists of 7 PM poles moving in between the C-shaped iron cores with the

length of each air-gap equal to 1.5mm. It can be seen that plane of the magnetic

loading is perpendicular to the plane of electric loading, and this characteristic

ensures a high power density because of no competition between magnetic circuit and

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electric circuit, however, in those longitude flux linear machines these two planes are

parallel.

Figure 3.4 Performance analyses. (a) Back-EMF waveforms. (b) Cogging force. (c) Normal force.

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For assessing its performance, the finite element method (FEM) analysis is

adopted for evaluating its static features. Figure 3.4 shows the back-electromagnetic

force (EMF) waveforms, cogging force and normal force features. It can be found

that phase B of the back-EMF waveform is distorted which shown that the magnetic

circuit of the 3-phase TFPM linear machine is not symmetrical. The cogging force is

very large which may cause large force ripples at operation mode. Due to the double-

sided design, the normal force is appropriate for application which does not require a

high strength linear bearing.

The distorted EMF waveform and large cogging force are resulted from the end-effect of the PM linear machine. In linear machines, there are two kinds of end-effect.

One is the transverse end-effect which also exists in the rotational electric machine.

When the stack length of the linear machine is far more than its air-gap length, the

influence of the transverse end-effect can be ignored. The other is the longitudinal

end-effect which is due to the finite length of the stator or mover. Compared to the

rotational one, the field distribution in the linear machine is distorted at the two ends

in its traveling direction. The unsymmetrical field distribution causes unbalanced

magnetic structure and thrust ripple. This is the appearance of the longitudinal end-

effect which can also contribute a cogging force component and further deteriorate

the linear machine performance.

Because of the finite length of the stator, its unbalanced magnetic circuit is not

symmetrical, thus the flux linkages in three phases are asymmetrical. In order to solve

the above problem, the 3-phase machine is modified into 2-phase machine, as show

in Figure 3.5. Figure 3.6 gives the electromagnetic performance of the 2-phase

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machine. It can be observed that the back-EMF waveforms are symmetrical, but its

cogging force still keeps a high value. Therefore, the cogging force should be

minimized for industrial applications.

Besides above method for solving the magnetic asymmetry, in design practice,

when stator poles are increased to 6 or above, the phenomenon of magnetic

asymmetry also can be alleviated [44], [45].

Figure 3.5 2-phase linear TFPM machine.

3.3 COGGING FORCE MIGIRATION

Cogging force is an important parameter in PM linear machines, which is caused

by two effects: (i) the PM segments on the mover prefer to align with the teeth of the

stator core (so-called the slot-effect); (ii) there are attractive forces between the ends

of the stator core and the PM mover (so-called the end-effect). This cogging force

causes force ripples superimposed on the thrust force, thus causing annoying jerk and

vibration of the mover.

The cogging torque due to slot-effect is intensively studied in design of rotational

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PM electric machines. It can be suppressed by the various approaches, namely

increasing the least common multiplier (LCM) of the slot number and PM pole

number [46], [47], skewing the PMs or stator stack [48], optimizing the PM width or

shape [49], and asymmetrically arranging PMs [50], etc.

Figure 3.6 Performance analyses. (a) Back-EMF waveforms. (b) Cogging forcewaveform.

Firstly, the cogging force of the proposed TFPM linear machine due to the

interaction between stator teeth and PM segments can be reduced by using the

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technique adopted by rotational PM motors. Namely, the cogging force is governed

by the LCM of the number of stator slots Q and the number of PM poles p. The larger

the LCM value, the smaller the cogging force is resulted. For this design, Q = 11 and

p = 19 are selected.

Secondly, the cogging force due to the end-effect of stator core is modeled as a

slotless PM linear machine as illustrated in Figure 3.7, where F Lx and F Rx are the

attractive forces at the left and right ends of the stator core exerted on the PM mover,

respectively [51]-[53]. These two forces can be expressed as the summation of a real

Fourier series:

(3)1

00 sink

k x Lx xk F F F

(4))sin(1

00 k

k x Rx xk F F F

where F x0 is the DC component, F k is the coefficient of the k -th harmonic

omponent, w c

orce is given by:

0 = 2 π / τ is the fundamental frequency, τ is the pole pitch, d is the

phase difference between F Lx and F Rx. Thus, the resultant cogging f

2

cos)2

sin(21

0

k k Lx Rx x xk F F F F (5)

It can be found that F x will become zero if q = (2n − 1) π , where n is an integer. Since q

is governed by the magnetic length of the stator L sm and τ , the condition for F x = 0

can be rewritten as:

)12( n L sm , n is the natural number (6)

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Practically, the magnetic length is not exactly equal to the physical length of the

stator L s . So, after obtaining L sm from (6), the optimal value of L s needs to be further

tuned. Figure 3.8 shows the relationship between F x and the L s / τ ratio.

Figure 3.7 Cogging force component due to end-effect of stator core.

Figure 3.8 Variation of cogging force with respect to physical stator length.

3.4 PROPOSED TFPM LINEAR MACHINE AND ITS

IMPROVEMENT

3.4.1 PROPOSED MACHINE STRUCTURE

Figure 3.9 shows the detailed structure of the proposed motor. The stator consists

of 12 C-shaped iron cores with a stack length of 10 mm. The cores alternate with

phases A, B and C, while every four of them are grouped together to form a phase.

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The machine specifications are listed in Table 3.1. For this design, Q = 11 and p = 19

are selected. It can be seen that the minimal cogging force is about 5 N occurred at

the ratio of 19.25. Therefore, when the PM pole-pitch is sized as 12 mm, the physical

length of the stator is selected as 231 mm

Figure 3.9 Proposed machine structure.

TABLE 3.1 Specifications of Proposed Machine

Rated power 300 W

Phase number 3

Rated phase voltage (RMS) 30 V

Rated phase current (RMS) 3.3 A

Rated speed 1 m/s

No. of turns per armature coil 50

Stator length 231 mm

Air-gap length 1 mm

Stack length 52 mm

PM dimension 4 mm × 12 mm × 30 mm

PM material NdFeB

PM coercivity 940 kA/m

PM remanence 1.05 T

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3.4.2 THRUST FORCE GENERATION PRINCIPLE

The principle of thrust force generation of the proposed motor can be illustrated

by Figure 3.10. There are two stator teeth and an effective air-gap (including two

actual air-gaps and the PM) between them. The magnetic flux generated by the

armature winding flows through the air-gap from one stator tooth to another. Because

of the fringing effect, there is a portion of flux passing through the air-gap beside the

stator teeth. So, the thrust force F exerted on the PM can be expressed as [30]:

(7)1 2 1 2( ) pm pm F F F B B I l

where F 1 and F 2 are the magnetic forces developed at the left and right hand sides of

the PM, respectively, B1 is the magnetic flux density under the stator teeth, B2 is the

magnetic flux in the fringing areas, I pm is the equivalent current sheet of the PM, and

l pm is the length of the PM. Also, I pm can be written as:

pm c pm I H h (8)

where H c and h pm are the coercive force and thickness of the PM, respectively. From

(7), it is obvious that the thrust force can be maximized by increasing the difference

between B1 and B2 . In order to achieve this goal, HTS bulks are inserted into the slot

between the stator teeth so as to provide magnetic shielding of the fringing flux.

Thus, B2 is suppressed to almost zero while B1 is improved, hence maximizing the

difference between them.

In order to enlarge the difference of B1 and B2 , the high temperature

superconductor (HTS) bulks are placed between two teeth for field shielding. Due to

the Meissner effect of HTS materials, the use of HTS bulks can force all PM flux

passing through the stator teeth [54], thus significantly decreasing the flux leakage in

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the slot area. The concept machine is shown in Figure 3.11. In this design, we focus

on proposing a new machine structure. So, the analysis is based on the standard FEM

and the HTS bulk is considered as an ideal superconductor. The property of HTS is

only considered as a material with ultra-low permeability which shielding the

fringing magnetic field. Practically, for using HTS inside the machine, the

refrigerator is engaged which provides cooling liquid for avoiding the so-called

quench effect.

Figure 3.10 Principle of thrust force generation.

Figure 3.11 Improved machine structure with HTS bulks.

3.4.3 ANALYTICAL RESULTS

As the proposed motor has a simple magnetic circuit in which the yoke of each

stator core is equivalent to the tooth with periodic boundary, the two-dimensional (2-

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D) FEM is adopted for analysis. For simplification of analysis, since the HTS bulks

serve as flux barriers, they are considered as an ideal superconductor where the

induced magnetization always opposes the field attempting to cross it. When the

magnetic flux is solely excited by the armature winding with 200 A-turn, the

magnetic flux distributions with and without using HTS bulks are shown in Figure

3.12. It can be observed that the use of HTS bulks can effectively shield the fringing

flux. The corresponding air-gap flux density is shown in Figure 3.13. It can be found

that the air-gap flux density under the slots is nearly zero, thus confirming the

effectiveness of the HTS bulks. As shown in Figure 3.13, the use of HTS bulks canshield the fringing flux which then reduces the magnitude of flux density at the

positions causing force retardation. So, even though their fundamental components

are essentially unchanged, the thrust force can be significantly improved. Actually,

the reduction of force retardation due to the use of HTS bulks can be interpreted as

the force contribution by the harmonic components of the flux density distribution.

When the magnetic flux is solely excited by the PM, the air-gap flux density is

shown in Figure 3.14. It further confirms that the HTS bulks can effectively shield the

fringing flux, and hence improve the thrust force. Then, the no-load electromotive

(EMF) waveform is deduced when the mover travels at 1 m/s. As shown in Figure

3.15, this EMF waveform is trapezoidal which enables the motor to perform brushless

DC operation, hence offering higher force density than that at brushless AC

operation. Consequently, when both of the phase A and phase B windings are excited

by 0 A-turn, 200 A-turn and 400 A-turn, the thrust force waveforms with and without

using the HTS bulks are plotted in Figure 3.16. It confirms that the peak thrust force

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can be improved by 175% at 200 A-turn and 183% at 400 A-turn due to the use of

HTS bulks.

Under no excitation, the thrust force is simply due to the cogging force. Figure

3.17 shows the cogging force normalized by the rated thrust force under 400 A-turn.

Although the cogging force also increases with the use of HTS bulks, the

corresponding peak value is less than 6% which is actually due to the improvement of

the thrust force.

Finally, in order to verify the design using the 2-D FEM analysis, the thrust force

at the rated armature current excitation of 400 A-turn is also calculated by using the3-D FEM analysis as shown in Figure 3.18. It can be found that the maximum error

and root-mean-square error between them are 8.7% and 4.8%, respectively. However,

based on a standard PC with Intel Core 2 Duo Processor 2.66 GHz and 2 GB

SDRAM, the computational time of the thrust force waveform using the 2-D FEM is

52 min whereas that using the 3-D FEM is 644 min (over 12 times longer time).

Therefore, the 2-D FEM is preferred to the 3-D FEM for the analysis of the proposed

motor design, since the corresponding errors are acceptable.

Figure 3.12 Magnetic flux distributions with and without HTS bulks underarmature winding excitation.

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Figure 3.13 Air-gap flux density waveforms with and without HTS bulks underarmature winding excitation.

Figure 3.14 Air-gap flux density waveforms with and without HTS bulks underPM excitation.

Figure 3.15 No-load EMF waveforms.

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Figure 3.16 Thrust force waveforms with and without HTS bulks under differentarmature winding excitations.

Figure 3.17 Normalized cogging force waveform with HTS bulks.

Figure 3.18 Comparison of thrust force waveforms with HTS bulks using 2-DFEM and 3-D FEM.

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46

3.5 SUMMARY

In this chapter, a novel linear TFPM machine has been designed and analyzed.

Firstly, with the introduction of C-shaped stator cores, the motor possesses a simple

structure which is easy to fabricate. Secondly, by properly selecting the numbers of

stator slots and PM poles as well as tuning the physical stator length, the cogging

force can be significantly suppressed to less than 6%. Thirdly, by using the HTS

bulks to perform magnetic shielding, the rated thrust force can be significantly

improved by 183%. Therefore, the proposed motor is very promising for those

applications desiring high thrust force, low cogging force and easy to manufacture

such as industrial linear actuators and vehicular linear drives.

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CHAPTER 4

LINEAR MAGNETIC GEARS AND THE INTEGRATED

MACHINES

4.1 INTRODUCTION

Mechanical gears are widely used in industry as the tools for transmission of

torque/thrust, speed scaling up/down and direction conversion. Especially for low-

speed applications, such as wind power generation, electrical vehicle power train

system, the electrical machines can operate at a high efficiency working condition via

mechanical gear transmission system. However, the drawbacks of mechanical gears,

namely noise, vibration, regular maintenance, mechanical loss and wear and tear,

may degrade the performance and efficiency of the whole system accordingly.

In order to solve the above problems, the magnetic gears which imitate the

operation of mechanical ones were proposed and developed. These magnetic gears

employ magnetic field interaction for torque transmission without physical contact,

hence eliminating the transmission loss and wear-and-tear problem [55]. In the early

stage, the magnetic gear adopts the topology resembles the mechanical gears [56]. As

shown in Figure 4.1, only parts of PMs are engaged for torque transmission, thus it

exhibits a low torque density. In order to fully utilize PMs, the coaxial magnetic gear

was proposed. Coaxial magnetic gears consist of three main parts: the outer-rotor, the

stationary ferromagnetic segments and the inner-rotor, as shown in Figure 4.2. The

key of coaxial magnetic gears is the ferromagnetic segments which locate between

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Linear Magnetic Gears and the Integrated Machines

the inner rotor and the outer rotor. When the sum of the outer-rotor PM pole-pair

number N 1 and the inner-rotor PM pole-pair number N 2 equals the number of

ferromagnetic segments N s , torque transmission between the inner rotor and the outer

rotor can be achieved without any mechanical assistance [57]-[61].

Figure 4.1 Gears. (a) Mechanical spur gear. (b) Magnetic spur gear.

Figure 4.2 Coaxial magnetic gear.

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Since they take some distinct advantages over the mechanical ones and magnetic

spur gears, such as higher efficiency, higher reliability, lower acoustic noise, inherent

overload protection, and free from maintenance, coaxial magnetic gears are becoming

attractive in some niche areas such as electric vehicle propulsion and wind power

generation [62], [32]. By readily integrating the magnetic gears into various electric

machines, hence the so-called geared machines which exhibit some distinct merits are

created. As for the electric vehicle application, they can enable high-speed rotating-

field design to increase the torque density while offering low-speed output rotation

for in-wheel direct-drive electric vehicles [62]. Also, they can perform online powersplitting of the engine power for electric variable transmission, hence offering the

optimal operation line for hybrid electric vehicles [63].

For satisfying the low-speed application in linear motion, the concept of coaxial

magnetic gears has been extended to the linear morphology so as to improve the force

capability of a linear motor [58], [64]. The linear magnetic gear, flat or tubular, has a

similar structure as its rotational counterpart. As shown in Figure 4.3, the tubular

linear magnetic gear consists of three parts: the low-speed mover, the high-speed

mover and the stationary field-modulation ferromagnetic rings. Due to principle of

field modulation, the two movers with different PM pole-pair numbers interact with

one another to achieve force transmission.

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Figure 4.3 Structure of a tubular linear magnetic gear.

Figure 4.4 Model of conventional tubular linear magnetic gear in cylindricalcoordinates.

4.2 LINEAR MAGNETIC GEARS

4.2.1 OPERATING PRINCIPLE

For unveiling the operating principle, magnetic circuit approach is adopted which

gives a visual and understandable expression. In order to derive the analytical model

of the magnetic circuit, some assumptions are made: the permeability of the back

irons of two movers and the ferromagnetic rings is assumed to be infinite, the

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permeability of the PMs is assumed to be equal to that of air, and the magnetic field

only varies in the longitudinal direction. In this modeling, the one-dimensional (1-D)

path is adopted so that the flux go straightly up and down and close at the infinite

distance [65]. Figure 4.4 shows the model of the tubular linear magnetic gear in the

two-dimensional (2-D) cylindrical polar coordinates. Based on the aforementioned

assumptions, the magnetic circuit can be considered as linear so that the resultant

magnetic field can be treated as the superposition of the fields separately excited by

PMs on the two movers. Figure 4.5 shows the equivalent magnetic circuit when

excited by PMs on the high-speed mover only. Thus, the equivalent total magnetic permeance in the longitudinal direction can be expressed as:

1 1 1 1 1 1( ) ( )hpm oag fm iag lpm z z

(1)

where Λ hpm = μ0 /hhpm , Λ oag = μ0/h oag , Λ iag = μ0 /h iag , and Λ lpm = μ0 /h lpm are the

magnetic permeances in the longitudinal direction of the PMs on the high-speed

mover, outer air-gap, inner air-gap and PMs on the low-speed mover, respectively;

Λ fm( z ) is the magnetic permeance in the longitudinal direction of the field modulation

segment area which is a function of the axial position z ; and h hpm , h oag , h iag , h lpm and

h fm are the longitudinal lengths of PMs on the high-speed mover, outer air-gap, inner

air-gap, PMs on the low-speed mover and ferromagnetic ring, respectively. When the

segment area is the ferromagnetic ring, the corresponding Λ fm( z ) is infinite. On the

contrary, when the segment area is air space, Λ fm( z ) = μ0/h fm; and when it is the HTS

bulk, Λ fm( z ) = 0.

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Figure 4.5 Magnetic circuit excited by PMs on high-speed mover.

Figure 4.6 Magnetic permeance waveform.

Figure 4.6 shows the magnetic permeance waveform of the equivalent magnetic

circuit of a conventional linear magnetic gear. It can be resolved into a Fourier series:

01

2( ) cos( )m s

m

mN z L

(2)

where λ0 is the DC offset of the total equivalent magnetic permeance, λm is the

amplitude of the mth harmonic magnetic permeance, N s is the number of

ferromagnetic rings, and L is the active length of the linear magnetic gear which is

also equal to the total length of the high-speed mover.

The magnetomotive force (MMF) of PMs on the high-speed mover can also be

expressed in a Fourier series:

)](2

cos[4

)2

cos(4

)( 0,

1,

z z L

nN h H n

z L

nN h H n

z F hpmhpmhcodd n

hpmhpmhcodd n

hpm

(3)

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where H hc is the coercive force of PMs on the high-speed mover, h hpm is the magnet

thickness, N hpm is the number of PM pole-pairs on the high-speed mover, z 1 is the

axial position of PMs on the high-speed mover with respect to z 0 , and z 0 is the

corresponding initial position as shown in Figure 4.4. Thus, the magnetic flux density

excited by PMs on the high-speed mover can be calculated:

321

01

0100

100

22)(cos

2

22)(cos

2)(

2cos

4

)2

cos()](2

cos[4

)()(

hpmhpmhpm

hpmhpm shpmhc

hpm shpmhpmhchpmhpmhc

shpmhpmhchpmhpm

B B B

z L

N z L

N N h H

z L

N z L

N N h H z z L

N h H

z L

N z z L

N h H z z F B

(4)

where B1 hpm has the same pole-pair number with that of PMs on the high-speed

mover, and B3 hpm has the same pole-pair number with that of PMs on the low-speed

mover. Thus, a thrust force can be produced by B3 hpm and PMs on the low-speed

mover.

In order to obtain the expression of the thrust force, an equivalent current sheet is

used to substitute the MMF of PMs on the low-speed mover. The fundamental MMF

component of PMs on the low-speed mover is given by:

14 2

( ) cos[ ( )]lpm lc lpm lpm F z H h N z z L

2

(5)

where H lc , h lpm and N lpm are the coercive force, thickness and pole-pair number of

PMs on the low-speed mover, and z 2 is its initial position as shown in Figure 4.4.

Thus, the corresponding equivalent current sheet is given by:

24 2

( ) sin[ ( )]lpm lc lpm lpm I z H h N z z L

(6)

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Consequently, by using Lorentz force law, the thrust force exerted on the low-

speed mover can be obtained by:

cos8

)( 12/

2/

3lpmhpmlchcl lpmlpm

L

Lhpml lm hh H H D N dz I B D F (7)

where θ is the angular displacement between the centers of PMs

of the two movers, hence written as θ = 2 N π z /L+ 2 N π z /L, and D is the

diameter of the low-speed mover. The maximum thrust force occurs at θ equal to

zero:

hpm 0 lpm 2 l

_ 18

lm Max lpm l hc lc hpm lpm F N D H H h

h (8)

By using the same derivation, the magnetic flux density due to PMs on the low-

speed mover can be expressed as:

31

2 2cos( )lpm l c lpm hpm hpm B H h N z N z

L L2

2

(9)

Then, the thrust force exerted on the high-speed mover can be obtained as:

cos8)( 12/

2/

3lpmhpmlchchhpmhpm

L

Llpmhlm hh H H D N dz I B D F (10)

where D is the diameter of the high-speed mover.h

From (8), it can be found that the developed thrust force is directly proportional

to λ1 , namely the fundamental harmonic of the magnetic permeance of the equivalent

magnetic circuit, which is governed by the field-modulation segment area. As shown

in Figure 4.6, the longitudinal magnetic permeance waveform of a conventional

linear magnetic gear is a symmetrical square wave. By using Fourier analysis, the

analytical formula λ1 can be expressed as:

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12( )

sinh l hm

(11)

where λh = μ0 /(h hpm + hoag + h iag + h lpm ), λl = μ0 /(h fm + h hpm + hoag + h iag + h lpm ), τ h

is the length of ferromagnetic ring in z direction, and τ fm is the pole-pitch of

ferromagnetic ring. When τ h equals one half of τ fm, λ1 achieves the peak value which

is 2( λh - λl ) / π .

4.2.2 TRANSMISSION CAPACITY IMPROVEMENT

If the thrust force of the linear magnetic gear can be improved, its transmission

capability will be enhanced as a result. According to equations (7) and (10), the thrust

force can be improved by increasing the PM thickness, PM pole-pair numbers, the

mover diameter and the first harmonic component of the magnetic permeance λ1 . In

this section, only the last item is improved. According to (11), in order to increase λ1 ,

the difference between λh and λl should be enlarged. Therefore, the magnetic material

which presents the lower permeability than the airspace is favorable. The high

temperature super conductor (HTS) material is adopted.

In this analytical model, the HTS bulks are considered to be an ideal

superconductor, in which the magnetic field is totally ejected. Thus, the

corresponding permeability is zero so that the value of λl becomes zero. Consequently,

it yields λ1 = 2 λh / π , which physically means that the developed thrust forces of both

movers can be improved from λ1 = 2( λh - λl ) / π to λ1 = 2 λh / π . This theoretical

improvement is then verified by applying finite element analysis to simulate the

thrust forces of the proposed tubular linear magnetic gear with and without using

HTS bulks.

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The finite element method (FEM) is employed for field calculation. In order to

take into account magnetic saturation during analysis, the permeability of back irons

and ferromagnetic rings is based on practical data of iron materials. On the other

hand, the permeability of PMs is a constant based on the NdFeB material, while the

permeability of HTS bulks is set to zero.

The field-modulation segments of the proposed gear adopts the zebra-striped

design, namely the HTS bulks are inserted between the ferromagnetic rings while

they have the same pole-pitch. The HTS bulks are located in airspaces of the

stationary rings, thus facilitating the cooling arrangement. Also, it can achieve thethrust force density of 3.2 MN/m 3.

Firstly, the inner air-gap flux densities of the two magnetic gears are analyzed

when only PMs on the high-speed mover serve as field excitation whereas PMs on

the low-speed mover are set as air space. Figure 4.7 shows the corresponding

waveforms and harmonic spectra. It can be found that by using the HTS bulks, the

amplitude of air-gap flux density can be improved greatly. By using spectrum

analysis, it can also be found that the 6th and 15th harmonic components improve

dramatically by using the HTS bulks. The largest asynchronous space harmonic

which has 15 pole-pairs interacts with the 15 pole-pair number PMs on the low-speed

mover, hence developing the desired steady thrust force.

Secondly, the outer air-gap densities are analyzed when only PMs on the low-

speed mover serve as excitation whereas PMs on the high-speed mover are set as air

space. Figure 4.8 shows their waveforms and spectra. It can be also found that the 6th

and 15th harmonic components improve greatly. The corresponding largest

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asynchronous space harmonic which has 6 pole-pairs interacts with the 6 pole-pair

number PMs on the high-speed mover, hence developing the desired thrust force.

Thirdly, when their low-speed movers travel at 1 m/s while their high-speed

movers are fixed, the static thrust force characteristics of the low-speed mover are

analyzed. As shown in Figure 4.9, the maximum thrust force of the proposed

magnetic gear is improved by 1.8 times than that of the conventional one, which

agrees with the theoretical 2.1 times as predicted by (8).

(a)

(b)

Figure 4.7 Comparison of inner air-gap flux densities excited by PMs on high-speedmover. (a) Waveforms. (b) Spectra.

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(a)

(b)

Figure 4.8 Comparison of outer air-gap flux densities excited by PMs on low-speedmover. (a) Waveforms. (b) Spectra.

Figure 4.9 Comparison of static thrust force characteristic of low-speed mover.

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4.3 ANALYTICAL COMPUTATION

The finite element method is an excellent tool for numerical field calculation, but

it provides little information on the relationship of the machine geometry and its

performance, and usually needs lengthy computation [66]-[68]. To complement the

FEM, the analytical calculation for field analysis of machines including magnetic

gears is highly desirable.

4.3.1 ANALYTICAL MODEL

In the linear tubular magnetic gear, the magnetic fields are only produced by

PMs and no current source is involved. Thus the magnetic scalar potential is

adopted for the magnetic field calculation. In order to facilitate the analytical

modeling, the following assumptions are made:

(1) The permeability of back irons of two movers is assumed to be infinite.

(2) The relative recoil permeability of PMs is assumed to be linear.

(3) The axial length is infinite so that the field distribution is axially symmetric

and periodic.

(4) The field-modulation region is considered to be

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