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“Null-E” Magnetic Bearings

A Dissertation

Presented to

the Faculty of the School of Engineering and Applied Science

University of Virginia

In Partial Fulfillment

of the Requirements for the Degree

Doctor of Philosophy

(Mechanical and Aerospace Engineering)

by

Alexei Filatov

August 2002

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Abstract

Using electromagnetic forces to suspend rotating objects (rotors) without mechanical contact is of-

ten an appealing technical solution. However, in real life magnetic suspensions have to satisfy many

engineering performance requirements beyond the simple compensation for the rotor weight. These

typically include adequate load capacity and stiffness, low rotational loss, low price, high reliability

and manufacturability. With recent advances in permanent-magnet materials, the magnitudes of the

required forces can often be obtained by simply using the interaction between permanent magnets.

While a magnetic bearing based entirely on permanent magnets could be expected to be inexpensive,reliable and easy to manufacture, a fundamental physical principle known as Earnshaw’s theorem

maintains that this type of suspension cannot be statically stable. Therefore, some other physical

mechanisms must be included.

One such mechanism employs the interaction between a conductor and a non-uniform mag-

netic field in relative motion. Its advantages include simplicity, reliability, wide range of operating

temperature and system autonomy (no external wiring and power supplies are required). The dis-

advantages of the earlier embodiments were high rotational loss, low stiffness and load capacity.

It was realized, however, that rotational loss, load capacity and stiffness depend strongly on the

topology of the conductors and the magnetic fields.

In theory, the rotational loss in the equilibrium position in the absence of external loading can

be made zero by designing a system such that no electric field develops in the conductor during therotor rotation. In this dissertation, we introduce the term ”Null-E” to describe this condition. In the

earlier embodiments, the Null-E condition could not be satisfied exactly.

Load capacity and stiffness can be also maximized through choosing shapes of the conductors

and the fields. From this point of view the field and the conductor shapes used in the so called

Null-Flux Bearings were found to be advantageous: the conductors were shaped as planar loops

with central openings and the fields were orthogonal to the loop planes and periodic in the direction

of the conductor motion.

To reduce rotational losses an additional restriction was imposed on the field shape (“Null-Flux”

condition): the flux variation within each loop had to be zero in the equilibrium. This condition

lowered the average value of the electric field in the conductor, but the topology did not allow

making it zero everywhere. The satisfaction of the “Null-Flux” condition was extremely sensitive

to manufacturing inaccuracies.

This dissertation proposes a novel type of magnetic bearing stabilized by the field-conductor

interaction. In contrast to the other bearings based on this principle, the proposed design allows

exact satisfaction of the Null-E condition in the equilibrium regardless of the conductor shapes and

even in the presence of an axial loading. Because of this we refer to it as the Null-E Bearing. Null-E

Bearings also have potential for higher load capacity and stiffness than Null-Flux Bearings. Finally,

their performance is highly insensitive to the manufacturing inaccuracies.

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ii

The Null-E Bearing in its basic form can be augmented with supplementary electronics to im-

prove its performance. The power rating of the electronics in this case can be much smaller than

in conventional active magnetic bearings. Depending on the degree of the electronics involvement,

a variety of magnetic bearings can be developed ranging from a completely passive to an active

magnetic bearing of a novel type.

The dissertation contains theoretical analysis of the Null-E Bearing operation, including deriva-

tion of the stability conditions and estimation of some of the rotational losses. The validity of the

theoretical conclusions has been demonstrated by building and testing a prototype in which non-

contact suspension of a 3.2-kg rotor is achieved at spin speeds above 18 Hz.

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Acknowledgments

This work would not have been possible without the help of many people and many strokes of

good fortune. Therefore, in my acknowledgment section, I would like to look back on the way that

brought me to the idea of the magnetic bearing presented in this dissertation, and say thanks to God

and to all of the people who helped me in this long journey.

I first heard about magnetic bearings while studying navigational devices at Moscow N. E.

Bauman State Technical University (MSTU) in 1987. I consider myself fortunate that I have learned

the fundamentals of engineering from very talented professors teaching at MSTU. I am especiallygrateful to Dr. Konovalov - an excellent engineer, manager, and a man of principles - who was a

real mentor for me.

The year I joined MSTU was also the year when High-Temperature Superconductivity was

discovered. Soon after, I joined a team working on the development of magnetic bearings for

navigational devices based on this effect. I greatly appreciate the help of Dr. Poluschenko, Dr.

Nizhel’skii, Dr. Matveev and other members of this team who provided me with the support to

do my experiments on superconducting levitation. I am even more grateful to them for personal

support during difficult times.

In the following years, I worked on many other projects, which were more rewarding finan-

cially; however, I continued to work with Dr. Poluschenko and Dr. Nizhel’skii on superconducting

magnetic bearings. I am thankful for their persistence and devotion to the science.During my studies of superconducting bearings I understood much more about the physics of

magnetic levitation. This finally led me to a novel magnetic levitation principle, which does not rely

on superconductors, and which laid the foundation for this dissertation.

Because of the economical situation in Russia I could not continue my research, and in 1997 I

joined a British consulting company “Ove ARUP and Partners”. Here I met a very good friend of

mine, Adrian Salter, who became very interested in my principle of magnetic levitation. In 1999,

I left “Ove ARUP and Partners”. At that time Adrian and I filed an application for the US patent

on the magnetic bearing design. Adrian actively supported all my subsequent work on this project,

which would have been absolutely impossible without his help.

The same year I entered the University of Virginia, where I was very lucky to have Dr. Maslen as

my advisor. Not only did he allow me to pursue this idea as my Ph.D. project, but he also provided

intellectual and financial support. His broad expertise enabled him to understand my problems and

guide me in the right direction. Because of his continuous prodding I had to generalize the theory

of the bearing operation far beyond my original intent. This gave me many new insights.

The help of many other people was also essential for the success of my project. Dr. Gillies

helped me with my experiments and did a heroic work on proof-reading my papers. Lewis Steva

helped me with manufacturing the prototype. John Gray helped me with various practical issues.

I would also like to thank Carlos and Esther Farrar for awarding me their fellowship, and all my

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iv

advisory committee for their time, good words and valuable suggestions.

I am very grateful to everybody who gave me feedback on my work. In particular, I would like

to thank David Meeker for expressing his points of view on various aspects of my dissertation, good

insights, suggestions, and healthy criticism.

Further, I need to thank my friends John Gray, Michael Baloh and many others, for moral

support, strong drinks and good parties.

Finally, I must to say Thank You to my mom for raising me, and then supporting and encourag-

ing me all the time. I would like to dedicate this work to her.

Alexei Filatov

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Contents

1 Introduction 1

1.1 Problem definition and research motivation . . . . . . . . . . . . . . . . . . . . . 1

1.2 Summary of the dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Dissertation outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Literature review 82.1 Active Magnetic Bearings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Magnetic Bearings using tuned LC circuits . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Diamagnetic and Superconducting Bearings . . . . . . . . . . . . . . . . . . . . . 9

2.4 Bearings utilizing conductor/magnet interaction . . . . . . . . . . . . . . . . . . . 9

2.4.1 Eddy-current bearings using AC field . . . . . . . . . . . . . . . . . . . . 9

2.4.2 Eddy-current bearings involving relative motion . . . . . . . . . . . . . . . 10

2.4.3 Null-flux bearings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.5 Bearings using a combination of magnet/magnet andconductor/magnet interactions 12

2.6 Bearings stabilized by a gyroscopic torque . . . . . . . . . . . . . . . . . . . . . . 12

2.7 Position of the proposed electromagnetic bearing in the general classification . . . 12

2.8 Other relevant developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.9 Prior publications of the material presented in the dissertation . . . . . . . . . . . . 17

3 Theoretical Analysis 18

3.1 Conductor rotating about a fixed axis in an axisymmetric magnetic field . . . . . . 18

3.2 Dynamics of a rotating conductor with slow lateral motions . . . . . . . . . . . . . 29

3.3 Dynamics of a rotating conductor with damped lateral motion . . . . . . . . . . . 34

3.4 An inverse system with a stationary conductor and a rotating magnet. . . . . . . . . 35

3.5 5-DOF non-contact suspension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.6 Effects of the stationary conductor inductance on system stability . . . . . . . . . . 40

3.6.1 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.6.2 Stability conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.6.3 Analysis of the stability conditions . . . . . . . . . . . . . . . . . . . . . . 463.7 Rotational loss due to the rotor unbalance . . . . . . . . . . . . . . . . . . . . . . 53

3.7.1 Losses neglecting the damper inductance . . . . . . . . . . . . . . . . . . 54

3.7.2 Losses including the damper inductance . . . . . . . . . . . . . . . . . . . 57

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CONTENTS vi

4 Technical and Practical Considerations 59

4.1 Angular stability of the suspension . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.2 Designing the rotating conductors and stationary magnetic field . . . . . . . . . . . 60

4.3 Generation of the stationary magnetic field . . . . . . . . . . . . . . . . . . . . . . 72

4.4 Designing the stationary conductors and rotating magnetic field . . . . . . . . . . . 73

5 Experiments 78

5.1 Observation of stable non-contact levitation . . . . . . . . . . . . . . . . . . . . . 78

5.1.1 Experimental arrangement . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.1.2 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.2 Measurement of the radial suspension stiffness . . . . . . . . . . . . . . . . . . . 81

5.3 Measurement of the rotational losses . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.4 Effects of the stationary coil inductance on the suspension properties . . . . . . . . 89

5.4.1 Influence of damper inductance on lift–off speed . . . . . . . . . . . . . . 89

5.4.2 Experimental arrangement . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.4.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6 Conclusions 95

A Exact Stability Bounds 105

B Force Calculations 107

B.1 Direct force calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

B.2 Force calculation using energy methods (Section 3.1) . . . . . . . . . . . . . . . . 111

C Magnetic Field Calculations 113

C.1 Magnetic system without pole shoes. . . . . . . . . . . . . . . . . . . . . . . . . . 114

C.2 Magnetic system with pole shoes. . . . . . . . . . . . . . . . . . . . . . . . . . . 116

C.3 Choosing parameters of the magnetic system. . . . . . . . . . . . . . . . . . . . . 119

D Loading Characteristics 122

E Experimental details 128

E.1 Measurement of the rotating coil inductances and resistances . . . . . . . . . . . . 128

E.2 Measurement of the radial position of the rotor. . . . . . . . . . . . . . . . . . . . 128

F Measured Loading Characteristics 133

G The high rotational speed assumption. 150

G.1 Avoiding the high spin rate assumption . . . . . . . . . . . . . . . . . . . . . . . . 151

G.2 Bode plot analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155G.2.1 Open loop model in the frequency domain. . . . . . . . . . . . . . . . . . 155

G.2.2 Effects of τ on the system stability. . . . . . . . . . . . . . . . . . . . . . 158

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List of Figures

1.1 An example clarifying application of Earnshaw’s theorem . . . . . . . . . . . . . . 2

1.2 The structure of the proposed magnetic suspension. . . . . . . . . . . . . . . . . . 4

2.1 Explanation of the Null-E principle in eddy current bearings. . . . . . . . . . . . . 10

2.2 The operational principle of a rotational null-flux suspension. . . . . . . . . . . . . 13

2.3 The operational principle of an embodiment of the proposed suspension. . . . . . . 13

2.4 Comparison to Null-Flux bearings: the rotor rotates about its symmetry axis. . . . . 14

2.5 Comparison to Null-Flux bearings: the rotor rotates about an arbitrary axis. . . . . 14

2.6 The inverse system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.7 Further generalization of the scheme shown in Figure 2.3. . . . . . . . . . . . . . . 16

3.1 Rotating conductor G exposed to a circumferentially uniform magnetic field Bz. . 183.2 Coordinate frames. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.3 An example clarifying why the force always passes through the axis Z 0. . . . . . . 233.4 Average electromagnetic force acting on body G. Rotation axis Z is fixed. . . . . . 273.5 Dependencies of the in-plane stiffness K and the angle θ on the rotational speed ω. 283.6 Orientations of the force components. . . . . . . . . . . . . . . . . . . . . . . . . 33

3.7 Diagram of the velocity dependent forces. . . . . . . . . . . . . . . . . . . . . . . 353.8 Inverse system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.9 Explanation of the stabilizing effect due to the inductance of the stationary coils. . 41

3.10 Arrangement of stationary coils and a movable magnet. . . . . . . . . . . . . . . . 42

3.11 Possible mutual orientations of the graphs W 2(x) and W 1(x). . . . . . . . . . . . 50

3.12 Calculated stability regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.13 Analysis of the stability region . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.14 Force diagram with rotor unbalance. Stationary coil inductances are neglected. . . 56

3.15 Force diagram with rotor unbalance. Stationary coil inductances included. . . . . . 57

4.1 An explanation of static angular stabilization . . . . . . . . . . . . . . . . . . . . 59

4.2 An explanation of stabilizing mechanism due to the gyroscopic effect . . . . . . . 60

4.3 A motivation for the choice of the conducting loop and magnetic field shapes. . . . 61

4.4 A particular case of the general structure shown in Figure 3.1. . . . . . . . . . . . 61

4.5 Cross-section of the loop and magnetic field as in Figure 4.3 in the XZ plane. . . . 62

4.6 XZ-plane cross-section with conducting slab instead of a loop . . . . . . . . . . . 62

4.7 Field distribution approximated by a third order polynomial. . . . . . . . . . . . . 65

4.8 Field distribution approximated by a third order polynomial plus point currents. . . 66

4.9 A conducting loop formed by two infinitely long round wires. . . . . . . . . . . . 67

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LIST OF FIGURES ix

D.6 Predictions of the load capacity vs. bearing weight ratios. . . . . . . . . . . . . . . 126

D.7 Predictions of the stiffness vs. rotor weight ratios. . . . . . . . . . . . . . . . . . . 127

D.8 Predictions of the load capacity vs. rotor weight ratios. . . . . . . . . . . . . . . . 127

E.1 AC voltage and current measured in a coil and an additional inductor. . . . . . . . 129

E.2 Sensor output voltage vs. distance dependence measured in the x-direction. . . . . 130E.3 Sensor output voltage vs. distance dependence measured in the y-direction. . . . . 131

E.4 Sensor output voltage vs. distance dependence in the x-direction . . . . . . . . . . 131

E.5 Sensor output voltage vs. distance dependence in the y-direction . . . . . . . . . . 132

F.1 Cross–coupled force-displacement characteristic at 1320 RPM. . . . . . . . . . . . 134

F.2 Direct force-displacement characteristic in the X-direction at 1600 RPM. . . . . . . 136

F.3 Cross–coupled Force-displacement characteristic in the Y-direction at 1600 RPM. . 136

F.4 Force vs total rotor displacement, R =√

X 2 + Y 2, characteristic at 1600 RPM. . . 137F.5 The angle between displacement and force vs rotor displacements at 1600 RPM. . 137


F.7 Cross–coupled force-displacement characteristic in the Y-direction at 1800 RPM. . 139F.8 Force vs total rotor displacement, R = √ X 2 + Y 2, characteristic at 1800 RPM. . . 140F.9 Angle between displacement and force vs rotor displacement at 1800 RPM. . . . . 140




X 2 + Y 2, characteristic at 2000 RPM. . . 143F.13 Angle between displacement and force vs rotor displacement at 2000 RPM. . . . . 143

F.14 Direct force-displacement characteristic at 2200 RPM. . . . . . . . . . . . . . . . 145



X 2 + Y 2, characteristic at 2200 RPM. . . 146F.17 Angle between displacement and force vs displacement at 2200 RPM. . . . . . . . 146

F.18 Direct force-displacement characteristic at 2400 RPM. . . . . . . . . . . . . . . . 148



X 2 + Y 2, characteristic at 2400 RPM. . . 149F.21 Angle between displacement and force vs displacement at 2400 RPM. . . . . . . . 149

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List of Tables

3.1 Summary of the stability conditions. . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.2 Important axes of the system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.1 Parameters used to predict the prototype’s radial loading characteristics. . . . . . . 84

C.1 Suggested geometry of the radial suspension magnetic system with pole shoes. . . 121

D.1 Parameters of the rotating coils in the bearing prototype. . . . . . . . . . . . . . . 123

D.2 Parameters of the stationary magnetic system used in the radial suspension. . . . . 123

D.3 Densities of the major materials used in the radial bearing. . . . . . . . . . . . . . 124

E.1 Measured parameters of the rotating coils used in the bearing prototype. . . . . . . 129

F.1 Raw data for calculating the radial force-displacement characteristic at 1320 RPM. 133

F.2 Summary of the radial suspension parameters measured at 1320 RPM. . . . . . . . 134




F.6 Summary of the radial suspension parameters measured at 1800 RPM. . . . . . . . 139F.7 Raw data for calculating the radial force-displacement characteristic at 2000 RPM. 141






x

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Nomenclature

A AmplitudeA Work B Magnetic flux densityC Damping coefficientc Mass normalized damping coefficientD Determinant of a quadratic equationE Electrical field strength

E Electromotive forceF Mechanical forcef L Volume density of Lorenz forcesf Reciprocal of the inductive coil time constant (units of frequency)G Designation of a rotating conductorG Designation of a stationary conductorH Magnetic field strengthI Electrical currenti =

√ −1J Current density per unit lengthJ z Polar moment of inertia about axis Z

j Current density per unit surface areaK Stiffness

k Mass normalized stiffnessL Inductance (or inductance matrix)l LengthM Mutual inductance (or mutual inductance matrix)M Magnetizationm MassN Number of wire turns in a coiln Circular periodicity of an objectP PressureP PowerQ Designation of a magnetic field source mounted on the statorQ Designation of a magnetic field source mounted on the rotorR Resistance (or resistance matrix)S Surface area

s = c/√

k (dimensionless)T Time period

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NOMENCLATURE xii

T Mechanical torqueU Voltageu = kτ 2 (dimensionless)v VelocityW m Electromagnetic energy

Z Rotor rotation axisZ Stationary conductor symmetry axisZ 0 Stator magnetic field symmetry axisZ 0

Rotor magnetic field symmetry axis

Z ar Axial suspension axis on the rotorZ as Axial suspension axis on the statorZ m Axis on the rotor passing through its center of massα Polar angle of a displacement vectorβ Polar angle of a velocity vector∆r0 Displacement of the Z axis from the axis Z 0∆u Distance between the rotor center of mass center and the axis Z

0

(unbalance)Γ Gain of a transfer functionγ Phase of a transfer function

µ Magnetic permeabilityµ0 Magnetic permeability of vacuumµr Relative magnetic permeabilityΛ Current vs displacement ratio (or a matrix composed of such ratios)λ Root of a characteristic equationΦ Magnetic fluxθ Angle complimentary to the angle between a force and a displacementτ Inductive coil time constantσ Conductivityω Circular rotational frequency Circular frequency as a variable in the frequency domain analysisΩ Rotor whirl speedΩ p Rotor precession speed

x∗ Complex conjugate of x∂x∂t

Time derivative taken in the rotor coordinate frame

x† x is a parameter of the radial suspension affected by the axial suspensionx̆ x is a normalized length Time averaging

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Chapter 1

Introduction

1.1 Problem definition and research motivation

Using electromagnetic forces to support rotating objects (rotors) without mechanical contact is an

appealing technical solution in many situations since it allows rotating systems to operate at much

higher speeds than obtainable using conventional mechanical bearings, and without risk of over-

heating and wear of the suspension components. Such suspensions must meet specific engineering

performance requirements including (typically): adequate load capacity and stiffness in given size,

low levels of rotational loss and external energy consumption. Sufficient load capacity and stiffness

can often be obtained simply using the interaction between permanent magnets and/or between per-

manent magnets and permeable materials. Thus, if two permanent magnets having magnetization

M are brought in contact, the magnetic pressure P on their surfaces is [1]

P = (µ0M)2

2µ0. (1.1)

Recently developed relatively inexpensive and very powerful rare-earth magnets such as NdFeB [2]

have µ0M close to 1.2 T essentially independent of demagnetizing fields in a wide field range.According to Eq. (1.1), this results in a magnetic pressure of 0.57 MPa. This value is high enough

to satisfy load capacity requirements for many applications. (For comparison, the typical maximum

operating pressures for fluid film bearings is about 2 MPa and for rolling element bearings it is

about 6 MPa). High values of stiffness can also be obtained through proper design of the magnet

arrangement [3]. Importantly, no external energy is required to produce a force using permanent

magnets, as long as no net work is done. In Figure 1.1, the repulsive force between two magnets is

used to compensate for the weight of the top magnet. Note that, if the magnetic field generated by

the top magnet is uniform circumferentially about its axis Z , there will be no drag torque exertedabout this axis.

Unfortunately, the physical limitation imposed by Earnshaw’s theorem [4] rules out the pos-

sibility of realization of stable static non-contact suspension using only permanent magnets and

permeable materials. Strictly speaking, this theorem was originally formulated for constant charges

(which could be electric, magnetic or gravitational charges) interacting with fields (electric, mag-

netic or gravitational), which could be described through a potential function obeying Laplace’s

equation. In this case, the conclusion of the theorem is a direct consequence of the maximum prin-

ciple for Laplace’s equation [5] maintaining that the maximum or minimum of the potential function

1

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CHAPTER 1. INTRODUCTION 2

Figure 1.1: This magnet arrangement could be used to compensate the weight of the top magnet,

but it is unstable radially.

always occurs on the boundary and never in free space. This theorem was later expanded by Braun-

bek [6, 7] to include materials with magnetic permeability and electrical permittivity greater than 1.

Both results are usually collectively referred to as Earnshaw’s theorem. This theorem can be neatly

formulated in terms of suspension stiffnesses K in three dimensions:

K x + K y + K z ≤ 0 (1.2)or, in linear algebraic terms, [8]: the trace of the lateral suspension stiffness matrix is zero or less

than zero. Recognizing that a necessary condition of the suspension stability is that each of the

diagonal elements of this matrix is positive, this implies that stability is not possible. Note that

Earnshaw’s theorem does not impose any restrictions on angular stiffnesses. In fact, suspension

stability about all angular degrees of freedom can be achieved in a passive magnetic system by

proper design (see a simple argument in Section 4.1). The main concern is, therefore, the lateral

stability of the levitated body.

For the axially symmetric suspension shown in Figure 1.1, Eq. (1.2) simplifies to

K ax

+ 2K rad ≤

0 (1.3)

where K ax is the axial suspension stiffness and K rad is the radial suspension stiffness. From thisequation, it is apparent that if stable suspension is achieved in the axial direction (K ax > 0), it willbe unstable in the radial direction (K rad < 0) and vice versa. Thus the top magnet in Figure 1.1will tend to slip to the side from the central equilibrium position.

There are, however, several well-known methods of achieving non-contact suspension using

electromagnetic forces in spite of the restrictions imposed by Earnshaw’s theorem. To summarize,

such methods make use of either diamagnetic or superconducting materials, conducting objects

interacting with time-varying magnetic fields, gyroscopic torques, or feedback control systems.

In rotational systems, only the last method has found application so far, because in contrast to

the other it can provide a high level of load capacity and stiffness, a low level of rotational losses and

a wide range of operating temperatures. The suspension systems utilizing this method are referredto as Active Magnetic Bearings (AMB). However, they also have significant drawbacks including

requirement of rather complicated control systems, external power sources and connecting wires.

This often complicates integration of a bearing into a final device, for example if it has to operate

in a closed vessel or in vacuum. More discussion of the electromagnetic suspension methods is

provided below in the Literature Review.

In the present dissertation we investigate the potential of magnetic bearings utilizing the inter-

action of conducting objects with time-varying magnetic fields, or, more specifically, interaction

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of conducting objects with non-homogeneous magnetic fields in relative motion. The known ad-

vantages of this type of magnetic bearing include simplicity, reliability, a wide range of operating

temperatures and system autonomy (no external wiring and power supplies are required). The dis-

advantages of the earlier embodiments were high rotational loss, low stiffness and load capacity.

It was realized, however, that rotational loss, load capacity and stiffness depend strongly on the

topology of the conductors and the magnetic fields.

In theory, the rotational loss in the equilibrium position in the absence of external loading can

be made zero by designing a system so that no electrical field E is induced in the conductor duringthe rotor rotation. This is referred to as the “Null-E” condition in the dissertation. In the earlier

embodiments, the Null-E condition could not be satisfied exactly unless infinitely thin conductors

were used.

Load capacity and stiffness can be also maximized through choosing shapes of the conductors

and the fields. From this point of view the field and the conductor shapes used in the so called Null-

Flux Bearings were found to be advantageous: the conductors were shaped as planar loops with

central openings and the fields were orthogonal to the loop planes and periodical in the direction of

the conductor motion.

To reduce rotational losses at equilibrium, an additional restriction was imposed on the fieldshape (“Null-Flux” condition): the flux variation within each loop had to be zero. This condition

lowered the average value of the electrical field in the conductor, but the topology did not allow

making it zero everywhere. The satisfaction of the “Null-Flux” condition was extremely sensitive

to manufacturing inaccuracies.

After analysis of the existing designs, we propose a novel topology of the conductor/field sys-

tem. This topology takes advantage of the technical solutions leading to high load capacities and

stiffnesses in Null-Flux Bearings, but at the same time allows exact satisfaction of the Null-E con-

dition regardless of the shapes of the conductors used. In fact the Null-E condition is satisfied not

only in the absence of external loading but also when a purely axial loading is applied. Because of

this, the bearing potentially has very low rotational losses, which would be essentially zero if the

loading is axial - a condition which is often easy to satisfy in stationary applications.Considering all the above we named the proposed bearing a Null-E Bearing by analogy with

Null-Flux Bearings. Other important advantages of the Null-E Bearings include potentially higher

load capacity and stiffness than can be attained by Null-Flux Bearings and robustness to manufac-

turing inaccuracies.

The Null-E Bearing in its basic form can be augmented with supplementary electronics to im-

prove its performance. The power rating of the electronics in this case can be much smaller than

that in Active Magnetic Bearings. Depending on the degree of the electronics involvement, a vari-

ety of magnetic bearings can be developed ranging from a completely passive to an active magnetic

bearing of a novel type.

1.2 Summary of the dissertation

The subjects of this dissertation are a novel method of non-contact electromagnetic suspension and

a novel class of magnetic bearings.

The proposed class of magnetic bearing features a combination of properties which are not

collectively provided by any of the prior technologies. These include:

1. Very high reliability due to intrinsic stability;

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Figure 1.2: The structure of the proposed magnetic suspension.

2. Wide temperature operation range including room temperature;

3. High efficiency;

4. Load capacity and stiffness sufficient for many applications;

5. Low rotational loss, virtually zero if only axial loading is applied - a condition which is easy

to satisfy in stationary applications;

6. No connecting wires and power supplies, unless supplementary electronics are used to aug-

ment system performance;

7. If supplementary electronics are used, their power rating will be much lower than those in

active magnetic bearings;

8. Low price;

9. Low requirements for manufacturing accuracy.

10. Easy introduction of active components, permitting a continuum of design from fully passive

to fully active.

Further, the proposed suspension principle allows easy integration with external control systems

resulting in variety of magnetic bearing designs featuring different degrees of the active component

involvement.

The method is based on the combination of the static interaction between permanent magnets

and the dynamic interaction between conductors and permanent magnets. The former is used to

achieve axial suspension, while the latter is used to achieve radial stabilization. The moving con-

ductors interact with an axially directed magnetic field which is also required to be circumferentially

uniform. The latter requirement results in rotational losses being virtually zero under any purely

axial loading provided that it does not exceed the maximum axial load capacity of the system.

Moreover, together with some other measures, it minimizes rotational losses under radial loading.These properties exhibit very little sensitivity to variations of many system parameters including

dimensioning inaccuracies. The suspension is stable for any rotational speed above a certain critical

value.

Its main idea is schematically explained in Figure 1.2, showing the arrangement of two magnets

Q and Q as in Figure 1.1, but further including two conducting disks G and G attached to themagnet surfaces. Static interaction between magnets Q and Q provides axial suspension, whiledynamic interaction between conducting disk G and magnet Q together with dynamic interaction

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between conducting disk G and magnet Q stabilizes the system in the radial direction and providespositive radial stiffness. Note, that both conductors G and G are necessary; the suspension isunstable if only one of the conductors is used. It is easy to see that in the equilibrium position when

the axis of the top magnet coincides with the axis of the bottom magnet, no magnetic field variation

occurs at any point of the conducting disks and no electrical field is induced. This is referred to

as the “Null-E” condition throughout the dissertation. Apparently because of the Null-E operation

no currents are induced and no energy dissipation takes place in the conducting volumes when the

bearing is at equilibrium.

The dissertation includes a general analysis of the interaction in this type of system, considering

arbitrary magnetic means Q and Q, generating circumferentially uniform magnetic fields, and arbi-trary conductors G and G. For simplicity, it is assumed that the magnetic fields are directed axiallywithin the conductors. This implies that Ampere forces acting on the conductors are purely radial

and do not influence the operation of the axial suspension. The analysis includes the following

steps:

1. Analysis of the radial electromagnetic forces acting on an arbitrary conducting body G ex-

posed to a magnetic field Bz(r) generated by sources Q. The field is circumferentially uni-form about some axis Z 0 and is directed along this axis. The conductor rotates about someaxis Z parallel to Z 0. The sources Q are stationary.

The equations for the forces acting on the body G are derived.

2. Analysis of the dynamics of slow radial motions of body G superimposed on high speedrotation about axis Z .

It is shown that if the slow motion assumptions are satisfied, the equilibrium that occurs when

axis Z coincides with axis Z 0 is always unstable.

3. Analysis of the inverse system, including magnetic sources Q rotating about some axis Z 0

and interacting with a stationary conductor G

. The field generated by the sources Q

iscircumferentially uniform about axis Z 0.

It is rather surprising, but these two structurally identical systems behave completely differ-

ently. The source of this difference is clarified. The equations for the forces acting on Q areobtained assuming that the inductive properties of the body G can be neglected.

4. Analysis of the radial motion of a combined body Q + G comprising bodies Q and Garranged so that axes Z 0 and Z coincide, interacting with a combined body Q+G

comprising

bodies Q and G arranged so that their axes Z 0 and Z coincide. This analysis does not

consider the axial suspension, which can be provided using the static interaction between the

magnetic field sources Q and Q (as in Figure 1.2), or between separate sets of magnets. It isassumed that the center of mass of the body Q + G lies on the Z axis.

This time it is shown that there is an in-plane equilibrium position for which axis Z (Z 0)coincides with axis Z (Z 0) and this equilibrium can be made stable. The stability conditionis derived.

5. An analysis that includes the axial suspension.

Here, a stability condition for the whole assembly is derived.

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6. Modified stability analysis including inductive component of the impedance of the conductor

G. The analysis is carried out for a practically important case when the conductor G isformed by a set of individual current loops with negligible mutual inductance.

It is shown that the inductive component may have a stabilizing effect and that the mini-

mal stable levitation speed can be reduced through appropriate choice of the stationary coilinductance.

While Figure 1.2 indicates the major components of our design, to fully realize its potential, the

shapes of the conductors and the fields need to be optimized. In particular, the ideas which were used

earlier to increase load capacity, stiffness and minimize energy dissipation in the out-of-equilibrium

operation in Null-Flux and Superconducting Bearings were found to be useful here. These and other

design considerations as well as some practical modifications to the device structure are discussed

farther in the dissertation. One of the significant modifications is using external circuitry to control

currents in the stationary conductors G.The dissertation also includes an estimation of the rotational losses caused by the non-coincidence

of the rotor center of mass and the Z 0 axis – equivalent of unbalance in this bearing. This loss modeldoes not include the effects caused by the axial suspension.

A significant part of the dissertation is devoted to the analysis of systems involving either con-

ductors rotating in stationary circumferentially uniform magnetic fields, or sources of such fields

rotating about the field symmetry axis in the presence of stationary conductors. This analysis has a

value of its own, since systems of this type can be encountered in other electromechanical devices

such as electromechanical dampers.

To verify theoretical results, a suspension system has been built and tested, in which a 3.2-kg

rotor is suspended without mechanical contact when it rotates above approximately 18 Hz. The

dissertation describes the actual hardware and the tests conducted to establish its conformance with

the theory.

1.3 Dissertation outline

Chapter 1 gives the motivation and objectives of the dissertation research followed by a brief dis-

sertation summary.

Chapter 2 is a comprehensive survey of literature on various types of magnetic bearings along

with discussion of advantages and disadvantages of each particular type. The chapter indi-

cates the position of the principle developed in this dissertation within the context of general

classifications of magnetic suspension technologies.

Chapter 3 contains a theoretical analysis of the electromagnetic interactions encountered in the

proposed system, system dynamics and stability.Chapter 4 suggests some practical solutions which could be useful in designing magnetic bearings

based on the proposed principle.

Chapter 5 describes the experimental part of the work aimed to verify the theoretical results.

Chapter 6 concludes the dissertation with a summary of the obtained results and suggestions for

future research.

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Appendix A includes the MathCAD script used to calculate exact stability bounds taking into ac-

count the inductance of the stationary coils.

Appendix B includes an example of using the general theory developed in this dissertation to cal-

culate parameters of a particular embodiment of the Null-E Bearing, in which the conductor

G is represented by a set of conducting coils.

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Chapter 2

Literature review

There are several well-known methods of achieving non-contact suspension using electromagnetic

forces in spite of the restrictions imposed by Earnshaw’s theorem. Each method has its advantages

and disadvantages, which are discussed in this chapter.

2.1 Active Magnetic Bearings

Active Magnetic Bearings (AMB) employ control systems to vary the supporting magnetic field

in response to rotor displacements in order to produce restoring forces. For this field to be easily

variable, it has to be generated by current-carrying coils [9, 10, 11, 12, 13, 14,15,16,17,18,19,20,21,

22, 23, 24, 25, 26]. Active Magnetic Bearings can produce high load capacity and stiffness, but they

also have significant drawbacks, some of which originate from the very principle of their operation.

Indeed, compensation of high loads requires strong magnetic fields associated with large amounts

of energy. If the loads are dynamic, these fields have to be varied quickly. This requires powerful

electronics. Partially because of the high power rating, AMB controllers are relatively expensive,

bulky and can be unreliable. Another practical disadvantage of AMB’s is that they require external

power sources and connecting wires. This often complicates their integration into a final device, for

example if the bearing has to operate in a closed vessel or in vacuum.

2.2 Magnetic Bearings using tuned LC circuits

This type of magnetic bearing ( [27], [28]) is, in fact, very close to AMBs in the sense that it also

relies on coils with currents which vary depending on the rotor position. Consequently, it also

requires external wiring and power supplies. The advantages, as compared to most AMBs, are that

“LC” bearings are simple and do not have position sensors. The disadvantages are principally much

lower efficiency, load capacity and stiffness. Furthermore, in contrast to an AMB, the parameters of this type of bearing cannot be tuned easily. Finally, Active Magnetic Bearings can be also designed

so as to operate without an explicit position sensor (these are so called “Self-Sensing” Bearings,

see, for example, [16]). Because of these reasons, LC bearings can hardly compete with AMBs in

most cases.

8

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CHAPTER 2. LITERATURE REVIEW 9

2.3 Diamagnetic and Superconducting Bearings

Another approach to achieve stable non-contact suspension is to utilize magnetic substances to

which Earnshaw’s theorem is not applicable: diamagnetic materials or superconductors. The super-

conductors in some cases can also be viewed as perfectly diamagnetic materials almost completely

expelling magnetic fields from their interior [29, 30]. This phenomenon is known as the Meissner

effect and can be observed in Type I superconductors and Type II superconductors below the first

critical field value. Type II superconductors above the first critical field value allow the field to enter,

but if they belong to the category of so called Hard Type II superconductors, further variations of

the field are suppressed.

Forces acting on ponderable masses of any known diamagnetic materials such as graphite or

bismuth are negligible unless magnetic fields on the order of tens of teslas are used [ 31, 32, 33, 34,

35, 36] which cannot be obtained with permanent magnets. This makes such bearings impractical

for most commercial applications.

The interaction of a superconductor with a permanent magnet may result in significant forces

[37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58]. However, the require-

ment of cooling superconductors to cryogenic temperatures prevents the use of this type of bearingin most applications. Moreover, if the Hard Type II superconductors are used and if they interact

with magnetic fields exceeding the first critical value, the bearing force-displacement characteris-

tics exhibit strong force-displacement hysteresis. The latter makes the rotor equilibrium position

unpredictable and may cause bearing failure under vibrations [52]. At the same time, use of Hard

Type II superconductors working above the first critical field value results in much higher load ca-

pacity and stiffness than obtainable with Type I superconductors. This is because Hard Type II

superconductors can carry much higher currents in much stronger magnetic fields. Moreover, re-

cently discovered High-Temperature superconductors [59, 60] such as YBaCuO (which are typical

Hard Type II superconductors) remain superconducting at much higher temperatures (well above

the boiling point of liquid nitrogen) than known Type I Superconductors. Because of the above,

significant efforts have been made to design magnetic bearings using Hard Type II and, particu-larly, High-Temperature superconductors. Thus, a method was found to reduce force-displacement

hysteresis significantly [55, 56, 57, 58]. The same method allowed obtaining relatively high load ca-

pacity and stiffness per unit surface area of the bearing due to optimal usage of the superconductor

current-carrying capacity and the energy of the permanent magnets. Some results of this work are

applied in this dissertation to the design of a bearing which does not use superconductors.

2.4 Bearings utilizing conductor/magnet interaction

2.4.1 Eddy-current bearings using AC field

Similar to diamagnetic and superconducting materials, normal conductors such as copper suppress

AC magnetic fields within their volume. Thus a non-contact suspension can be obtained using

the interaction of a conductor with an AC magnetic field. This method has found application in

containerless melting, where a conductor suspended in AC fields melts due to heating by eddy

currents [61,62,63,64,65,66,67,68,69,70]. The advantage of non-contact suspension in this case is

that the crucible material does not contaminate the melt. While, in this particular application, high

losses are beneficial (they are needed to melt the conductor), in other applications the suspension,

in contrast, has to be highly efficient.

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Figure 2.1: Explanation of the Null-E principle in eddy current bearings.

2.4.2 Eddy-current bearings involving relative motion of a conductor and

non–homogeneous field

Note that energy in the previous type of bearing dissipates in both the suspended conductor and the

coils generating the supporting AC field. The latter are eliminated in another embodiment, in which

the supporting non-homogeneous magnetic field is generated by permanent magnets. This field is

time-invariant in the magnet coordinate frame, but seen as time-varying by a conductor moving with

respect to it. The early designs of such bearings exhibited low stiffness and load capacity and highrotational loss [71, 72].

It was realized, however, that the rotational loss in the no-load equilibrium can be made essen-

tially zero by designing a system so that no magnetic field variation occurs in the conductor during

the rotor rotation in this position [73]. Consequently, no electrical fields and no currents will be

developed. Figure 2.1 clarifies this point using a linear bearing example. It is easy to see that the

magnetic field on the middle line of this structure is zero and when the conductor is thin, the field is

close to zero within the conductor volume. Motion of the conductor along the middle line, therefore,

does not cause significant electrical fields, currents and losses.

In this dissertation we refer to the requirement of zero electrical field within the conductor

volume during its motion in the no-load equilibrium position as the “Null-E” condition even though

this term, to our knowledge, is not used in the literature. Note that the Null-E condition cannot be

satisfied exactly within the conductor shown in Figure 2.1 unless the conductor is infinitely thin.

Even an approximate satisfaction of this condition requires high geometrical accuracy, especially

in rotational systems. In contrast, the bearing structure proposed in this dissertation allows exact

satisfaction of the Null-E condition regardless of the conductor shape.

2.4.3 Null-flux bearings

Other disadvantages of the early eddy-current bearings were low load capacity and stiffness. Again,

it was realized that this is not an inherent problem of the method: both load capacity and stiffness

could be increased dramatically through careful choosing the topologies of the conductors and the

fields. Low values of these parameters in the eddy-current bearings were caused by an inefficient us-

age of the conductor current-carrying capacities and the magnet energies. The eddy currents withinthe conducting volumes were distributed and oriented with respect to the magnetic fields rather ar-

bitrarily, resulting in arbitrarily oriented Lorenz forces. This caused unwanted resistive losses and

stresses in the conductors. In the ideal scenario, the current at each point of the conducting volume

should be orthogonal to the field and directed so that the resulting Lorenz force is directed oppo-

sitely to the disturbing force acting on the rotor. Besides, it is desirable to concentrate the magnetic

fields generated by the magnets in compact areas and force the currents to flow in these areas. All

these measures would lead to increases of the load capacity, stiffness and their ratios with respect to

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the energy dissipating in the conductor when it is displaced from the equilibrium. They were major

guidelines in developing so called Null-Flux Bearings.

The Null-flux principle was first proposed for application to linear suspension (high-speed

trains). In the late 1960’s, Danby and Powell [74] suggested using a high-speed train suspension

scheme comprising an array of magnets of alternating polarity installed on a train and generating

flux trough a set of stationary, vertically oriented coils. The flux was required to vary monotonically

with the levitation height, passing through zero at some point. Consequently, this type of suspension

is generally referred to as “Null-Flux” suspension. It is easy to see that if the train moved constantly

at the height corresponding to the zero flux, there would be no macroscopic currents induced in the

coils. However, if the levitation height varied, currents would appear because the magnet polarity

alternates in the direction of motion. This would result in lifting and drag forces. This scheme has

been proven to deliver much higher lift/drag ratios (≈ 60/1) than the scheme in which simple coilsor aluminium slabs laid on the ground interact with a high-gradient magnetic field generated on the

train (lift/drag≈ 25/1) [75].The principle of null-flux suspension is implemented in Japanese MagLev trains [76], [77]. Sig-

nificant theoretical work in this direction has been done in Argonne National Laboratory [78], [79]

and by Davey [75, 80, 81, 82]. The null-flux suspension concept is also being developed by Foster-Miller (USA) with application to both high-speed trains [83] and space vehicle launch systems [84].

It is easy to see that the system topology in the Null-Flux Bearing does not allow satisfaction of

the Null-E condition in the equilibrium: the parts of the conducting loops orthogonal to the motion

will experience time-varying magnetic fields. The Null-Flux condition is weaker than the Null-E.

It is rather surprising, but the null-flux principle has been extended to rotational systems only

recently [85, 86, 87, 88]. One of the significant problems with direct application of this technology

to the rotational systems is that in this case the null-flux condition is especially difficult to satisfy

because it requires very high geometrical accuracy of the bearing components. (The author earlier

has successfully applied the methods of increasing suspension stiffness and load capacity developed

for the Null-Flux Bearings to the design of a superconducting rotational bearing [55, 56, 57, 58].

In this dissertation we point out that there is a way to extend the null-flux principle to rotationalsystems other than doing it directly, as in [85,86,87,88]. As a result we obtained a novel non-contact

suspension principle, which has much in common with null-flux suspensions, but also has significant

differences. This principle is applicable only for rotational systems and cannot be adapted to linear

ones. It allows exact satisfaction of the Null-E condition (which is a stricter condition than the

Null-Flux one) and because of this we refer to it as a ”Null-E” suspension. Importantly, the Null-E

condition is satisfied not only in the absence of any loadings, but also when the loading is purely

axial - a condition that is often easy to satisfy in stationary applications.

One of the other advantages of the bearing proposed in this dissertation is potentially higher

load capacity and stiffness than in Null-Flux Bearings. The reason for this is that in the Null-Flux

Bearings only the time-averaged Ampere force acting on each conducting coil is directed oppositely

to the rotor displacement, while there is a time period when the instantaneous force is directed in thedirection of the displacement [85]. Moreover, since the field in a Null-Flux Bearing is required to

be circumferentially periodical, its magnitude and, consequently, the Ampere force are not always

of the maximal magnitude.

In contrast, in the proposed bearing, the field is circumferentially uniform, and when the bearing

is exposed to a constant loading it responds with a time-invariant force opposite to the loading, as

will be shown later in the dissertation.

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2.5 Bearings using a combination of magnet/magnet and

conductor/magnet interactions

A passive magnetic bearing structure can be developed combining the static interaction between

permanent magnets and the dynamic interaction of a conductor with a magnetic field. The first typeof interaction provides compensation at constant loading, while the second stabilizes an otherwise

unstable system [89, 73]. If a structure like that shown in Figure 2.1 is used as a stabilizer, then the

overall system can be designed so that there would be no energy dissipation not only in the absence

of any loading, but also when the loading is purely axial [ 73]. For example, this loading can be the

rotor weight if the bearing axis is vertical. Unfortunately, this feature is very sensitive to bearing

parameter variations, especially dimensioning inaccuracies. Also, the design of the subsystem using

a conductor interacting with a magnetic field as suggested in [89, 73] is essentially an eddy current

bearing with all the subsequent disadvantages: high rotational loss when the rotor is displaced from

the equilibrium and low stiffness and load capacity.

One of the major contributions of this work is the development of an electromagnetic stabilizer

which offers significant advantages compared to both Eddy-Current and conventional Null-Flux

Bearings, including lower rotational losses and higher stiffness and load capacity.

2.6 Bearings stabilized by a gyroscopic torque

Note that the suspensions utilizing interaction of conductors with permanent magnets work only in

the dynamic mode, i.e., when the rotor rotates. It was shown recently, that a dynamically stable

suspension can be realized using only permanent magnets [90, 91, 92, 93, 94, 95]. This does not

contradict Earnshaw’s theorem which is valid only for stationary systems and does not consider

dynamic effects caused by rotation. Unfortunately, this method was found difficult to use in practice

because of low stiffness and load capacity and because the stability is very sensitive to variations of

many parameters including rotor weight, residual magnetization of the magnets and, importantly,

rotation speed: the suspension is stable only within a narrow speed range limited from both below

and above.

2.7 Position of the proposed electromagnetic bearing in the general

classification

The approach most similar to the one proposed in this dissertation is the null-flux suspension. In

principle, the proposed scheme can be viewed as a special zero-order case of a rotational null-flux

suspension, however, there are significant differences between the two schemes which should be

kept in mind. The following discussion briefly examines the major similarities and differences.

The operational principle of a rotational null-flux suspension is indicated in Figure 2.2. Whenthe rotor axis of symmetry is in the central position, the flux through the conducting coil is required

to be zero. It is easy to see that when this axis is displaced from the central position, the flux through

the coil Φ can be described in terms of a Fourier cosine series expansion, as follows:

Φ = Φ01 cos(ωt)

∞k=1

Ak cos(kn0ωt + ϕk) (2.1)

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Figure 2.2: The operational principle of a rotational null-flux suspension.

Figure 2.3: The operational principle of an embodiment of the proposed suspension.

where n0 is half of the number of magnet blocks around the circumference (n0 = 2 in Figure 2.2);

Ak and ϕk are the amplitude and phase of the harmonic with frequency kn0ω. The parts of thebearing are interchangeable: the magnets can be mounted on the rotor while coils can be made

stationary.

An embodiment of a part of the suspension proposed here can be obtained from the suspension

Figure 2.2 if only one magnet of each polarity is used as shown in Figure 2.3. (We specify that this

is a part of the suspension here because by itself it does not define an equilibrium position of the

rotor as discussed below.) The flux through the coil in this case can be represented as

Φ = Φ02 cos(ωt) (2.2)

This equation can be obtained from (2.1) by taking n0 = 0.

While the schemes shown in Figure 2.2 and Figure 2.3 look very similar, there is a significantdifference between them. In Figure 2.2, there is a unique point of equilibrium, when the rotor sym-

metry axis coincides with the stator symmetry axis. In contrast, in the system shown in Figure 2.3,

regardless of what is the axis about which the rotor spins, there will be no currents and no forces

induced as long as this rotation axis coincides with the field symmetry axis.

Figure 2.4 and Figure 2.5 clarify this difference. Figure 2.4 shows a rotor with four conducting

loops exposed to a circumferentially uniform magnetic field. This structure is a generalization of

Figure 2.3. The density of the dashed circles represents the magnetic flux density: the field has

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Figure 2.4: Rotor axis of symmetry = Rotation axis = Field axis of symmetry: no currents and no

forces.

Figure 2.5: Rotor axis of symmetry = Rotation axis = Field axis of symmetry: again no currentsand no forces.

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Figure 2.6: The inverse system: a source of the field is mounted on the rotor, a conductor is station-

ary.

to be radially non-uniform. It is easy to see that when the rotor symmetry axis coincides with the

magnetic field symmetry axis and the rotor rotates about this axis there will be no currents and,

subsequently, no forces induced. The same would be observed in the null-flux bearing (Figure 2.2)if the geometry of the coils is chosen to meet the null-flux requirement.

Assume, however, that in the system shown in Figure 2.4 the rotor rotates as a whole about

another axis, which is not its axis of symmetry. This can be absolutely any axis which, in particular,

can be located outside of the rotor as shown in Figure 2.5. Note, that in this case there still will be no

changes of the magnetic fluxes through the coils and, correspondingly, no currents and forces. The

system shown in Figure 2.3 and Figure 2.5 does not define an equilibrium as a particular position

of the rotor with respect to the stator . Instead, it defines a kind of a “pseudoequilibrium” position

where the rotor rotation axis coincides with the magnetic field symmetry axis. The rotor rotation

axis is defined arbitrarily. Apparently, small perturbations of the rotor motions accumulated over

time will redefine the rotor rotation axis: the system shown in Figures 2.3 and 2.5 by itself does not

prevent the rotor from moving around.

In contrast, it is easy to see that if in the system shown in (Figure 2.2) the rotor rotated about

some axis different from its symmetry axis, then forces would develop tending to align the rotor

symmetry axis (moving in circles) with the stator symmetry axis.

The situation with the system shown in Figure 2.3 and 2.5 becomes even worse when rotor

inertia is taken into account. In this case, there is a unique equilibrium when the rotor center of

mass coincides with the rotation axis which, in its turn, coincides with the magnetic field symmetry

axis. However, as one can anticipate and as is shown in this dissertation, this equilibrium is unstable

and the rotor center of mass would tend to spiral away from it. Therefore, the system shown in

Figures 2.3 and 2.5 is not functional on its own.

It becomes functional, however, when augmented with a second subsystem, which is structurally

identical to the first but has the source of the circumferentially uniform magnetic field mounted on

the rotor so that the field symmetry axis is located in proximity of the rotor center of mass, while theconductor is mounted stationary. We refer to this system as the “inverse” one in order to distinguish

it from the “direct” system including a conductor rotating in a stationary field. An example of the

inverse system is shown schematically in Figure 2.6.

It is easy to see that rotation of the magnet about its symmetry axis Z does not cause any currentsin the stationary coils regardless of the position of the magnet. At the same time lateral motions of

the magnet do cause currents and, consequently, forces. If the inductive properties of the stationary

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Figure 2.7: Further generalization of the scheme shown in Figure 2.3.

coils are neglected, then the force is dissipative in nature and the system acts as a “selective damper”

suppressing all the motions of the rotor but the rotation about the Z axis.Summing up, the system shown in Figures 2.3 and 2.5 dictates that in equilibrium the rotation

axis of the rotor coincides with the symmetry axis of the stationary magnetic field, but does notdetermine where this axis passes through the rotor. The system shown in Figure 2.6 dictates that

the rotation of the rotor would take place about the symmetry axis of the magnet mounted on the

rotor but does not tell how this axis has to be located with respect to the stator. Only both systems

together define a unique equilibrium. If the symmetry axis of the magnet shown in Fig. 2.6 passes

through the rotor center of mass, then the point of equilibrium is defined as being where this axis

coincides with the stationary magnetic field symmetry axis. Otherwise, the dynamic balance of

the centrifugal and damping forces acting on the rotor will define (uniquely) some other axis of

the rotor which has to coincide with the stationary magnetic field symmetry axis for equilibrium to

occur. Thus, the proposed system requires magnets and conductors mounted on both the rotor and

the stator to define the equilibrium position of the rotor with respect to the stator.

In contrast, conventional Null-Flux Bearings need only one part of the suspension to define theequilibrium, which includes either conductors mounted on the rotor and magnets mounted on the

stator or, oppositely, the magnets mounted on the rotor and conductors mounted on the stator.

(It is interesting to note that the rotational null-flux suspension (Figure 2.2) when unrolled pro-

duces a linear suspension. It is easy to see, however, that unrolled suspension shown in Figure 2.3

is not functional: there is no force acting on a coil moving along a magnetic field that is uniform in

the direction of motion.)

Note that when the proposed bearing is in equilibrium, no change of the magnetic field occurs

in either rotating or stationary conductors and, consequently, no electrical field is induced (Null-

E condition is satisfied) regardless of the shapes of the conductors. The only requirement is that

the magnetic fields have to be uniform circumferentially. Thus Figure 2.7 shows an arbitrarily

shaped conductor G instead of the conducting loop shown in Figure 2.3 and Figure 2.4. This a hugepractical advantage since it minimizes the requirements for manufacturing accuracy.The major components of the bearing in assembly are indicated in Figure 1.2 with the stationary

and rotating conductors represented by conducting discs attached to the magnet surfaces.

As shown in the dissertation, the proposed magnetic bearing has several important practical

advantages compared to Eddy-Current and conventional Null-Flux Bearings.

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2.8 Other relevant developments

Rotation of a conductor in a circumferentially uniform magnetic field, which is the basis of the

suspension principle described here, has already been studied to some extent with application to

electromagnetic eddy-current dampers. Such a damper typically includes either a conducting disc

mounted on the rotor and interacting with a stationary circumferentially uniform magnetic field or,

inversely, a source of such a field mounted on the rotor and interacting with a stationary conductor

[96, 97, 98, 99, 100, 101].

Several authors [97, 98] have indicated that electromagnetic eddy-current dampers are very ef-

fective under certain operating conditions. However, conditions (generally at high speed) in which

a rotating system becomes unstable have also been documented [99, 100].

It was noticed that system behavior at high speeds is very different depending on whether the

conductor rotates while the field source is stationary, or inversely, the field source rotates while the

conductor is stationary [100]. One of the explanations [100] was based on the well known general

fact that the effect of damping on the whirl motion of a rotor is always stabilizing if the energy

dissipates in the stationary part of the damper; however, if the energy dissipates in the rotor, the

effect is stabilizing for all whirl motions at a frequency above the rotational speed and destabilizingin other cases [102].

A detailed explanation of this phenomenon has been given in [101], where it was shown that the

destabilizing effect in the case of a rotating conductor occurs due to cross-coupling stiffness terms.

The analysis was carried out for a thin, electrically conductive, but non-ferromagnetic disk attached

to the rotor and exposed to a circumferentially uniform magnetic field normal to the disc surface. It

was assumed that the disc magnetic Reynolds number is low: Rm

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Chapter 3

Theoretical Analysis

3.1 Conductor rotating about a fixed axis in an axisymmetric

magnetic field

Consider a solid conductor G of arbitrary shape exposed to an external magnetic field. This mag-netic field is required to be circumferentially uniform about some axis Z 0 and non-uniform in theradial direction. For simplicity, the flux density is also required to be parallel to the axis Z 0, i.e. itcan be expressed as Bext = Bz = Bz(r)ez , where ez is a unit vector directed along the axis Z 0.Further, it is assumed that the conductor rotates about some axis Z parallel to Z 0 with angular speedω (Figure 3.1).

Our first goal is to find the forces acting on the rotating conductor assuming that axis Z is fixedand displaced from axis Z 0 by some very small distance ∆r0 = {∆x0, ∆y0}. Clearly, if ∆r0 = 0,there are no currents and no forces due to the system rotational symmetry.

Consider the case when ∆r0 = 0.The following coordinate frames will be used (see Fig. 3.2):

1. A stationary coordinate frame X 0Y 0Z 0;

2. A coordinate frame X 1Y 1Z moving laterally (but not rotating) together with the rotor;

3. A coordinate frame X RY RZ linked firmly with the rotor.

The equations of mechanical motion of body G will be analyzed in an inertial coordinate frameX 0Y 0Z 0 while equations for the electromagnetic field will be analyzed in a rotating coordinate

Figure 3.1: Rotating conductor G exposed to a circumferentially uniform magnetic field Bz.

18

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CHAPTER 3. THEORETICAL ANALYSIS 19

Figure 3.2: Coordinate frames.

frame X RY RZ related to conductor G. This can be done conveniently since the influence of accel-eration on electromagnetic processes is known to be negligible for most practical electromechanical

systems. [108].

Let us consider some point A within the conducting volume of the rotor, which location is givenby polar coordinates ρ and ψ in the rotor coordinate frame X RY RZ , and by ρ and φ in the X 1Y 1Z coordinate frame.

The distance from the point A to the axis Z 0 is

r = {(∆x0 + ρ cos φ)2 + (∆y0 + ρ sin φ)2}1/2 (3.1)

Since ∆r0 is small, we replace (3.1) with a Taylor’s series expansion, accurate to the second orderterms:

r = ρ + ∆x0 cos φ + ∆y0 sin φ (3.2)

Change of r:

∆r = r − ρ = ∆x0 cos φ + ∆y0 sin φ (3.3)

The magnetic flux density at point A is

Bz(ρ,φ,∆r0) = Bz(ρ) + dBz

dr ∆r

or, using (3.3)

Bz(ρ,φ,∆r0) = Bz(ρ) + dBz

dr (∆x0 cos φ + ∆y0 sin φ) (3.4)

If the rotor rotates with an angular speed ω, then φ = ψ + ωt, and Bz at each point of theconductor G varies in time.

Differentiating (3.4) with respect to time, we get

∂ Bz∂t

= dBz

dr ω(−∆x0 sin ωt + ∆y0 cos ωt) (3.5)

where ˜ indicates that the derivative is taken in the rotor coordinate frame.

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Using polar coordinate representation of the displacement vector

∆x0 = ∆r0 cos α; ∆y0 = ∆r0 sin α

we rewrite (3.5) as

∂ Bz∂t

= −dBzdr

∆r0ω sin(ωt + ψ − α(t)) (3.6)

The distributions of current density, j, and the strength of the magnetic field produced by this

current, H j , within the conductor G are given by the following quasi-stationary approximations of Maxwell’s equations:

∇ ×H j = j (3.7)

∇ ×E =

−µ

∂ H j

∂t − ∂

Bz

∂t

(3.8)

∇ · (µH j ) = 0 (3.9)where E is the electric field strength, µ = µrµ0, µ0 is the magnetic permeability of vacuum and µris the relative magnetic permeability of a given material.

In addition we need to consider a constitutive equation linking j and E (Ohm’s law):

j = σE (3.10)

in which σ is the electrical conductivity. In the most general case, the properties σ and µ in equations(3.8), (3.9) and (3.10) can be functions of spatial coordinates.

Note that all the vectors j , H j, E and ∂ Bz∂t in Eq. (3.7) through Eq. (3.7) are resolved in therotor coordinate frame X RY RZ .

Combining (3.7), (3.8) and (3.10) we get an expression for the strength of the magnetic field

induced by the current in conductor G (field diffusion equation):

∇ ×

1

σ (∇ ×H j)

+ µ

∂ H j∂t

= − ∂ ∂tBz(r, ∆r0, t)ez (3.11)

This equation together with (3.9), subject to the boundary conditions, gives us H j . Knowing

H j , we can further find j using equation (3.7).

We can be more specific about the boundary conditions here. Since equation (3.11) allows for

arbitrary spatial distributions of σ and µ, this equation, in principle, can be used to describe field

and current distributions in any rotor. (This would not be the most practical way to attack theproblem in most cases, however. In practice, the rotor can be divided in several areas with constant

σ and µ, so that the solutions can be obtained for each area separately, and then coupled togetherthrough boundary conditions. The analysis is even easier if the rotor consists of several insulated

bulk conducting coils and iron cores.) Since the bearing is supposed to be non-contact, the solution

inside the rotor should be coupled to the solution in free-space through boundary conditions on the

rotor surface: normal component of the current, j, is zero, tangential component of H j and normal

component of B j are continuous. The magnetic field in free space can be described through a scalar

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potential given by Laplace’s equation subject to the boundary conditions at infinity (the field should

be zero there).

The only information about the boundary conditions which we will really need for further anal-

ysis, however, is that they are linear and time-invariant. Then, the operators acting on H j on the

left-hand sides of equations (3.9) and (3.11) are linear and time-invariant. Further, the current den-

sity j can be obtained from H j using a linear transformation given by (3.7). We summarize allthese observations by writing in a general form

Lρ,ψ,z,t( j) = ∂ Bz

∂t ez (3.12)

where Lρ,ψ,z,t is a linear operator over variables ρ, ψ, z ,t.For our analysis, we represent two-dimensional vectors by complex numbers with real parts

corresponding to the x-components and imaginary parts corresponding to y-components. We mark

the complex representations of the vectors with hats:

∆r0 = ∆x0 + i∆y0 = ∆r0eiα

Using this form, we rewrite (3.6) as

∂ Bz∂t

= dBz

dr Re

∆r0∗ωei(ωt+ψ)eiπ/2 (3.13)Considering that the real part of a complex vector a can be represented as Re(a) = 12(a + a

∗),where * means complex conjugation, we rewrite (3.13) as

∂ Bz∂t

= 1

2

dBzdr

∆r0ωe−i(ωt+ψ)e−iπ/2 + ∆r0∗ωei(ωt+ψ)eiπ/2 (3.14)We know that when a linear system described by (3.12) is subjected to a harmonic excitation

∂ Bz∂t

= Aeiφ0eiωt (3.15)

the resulting current density is also a harmonic function: j = Γ eiγ Aeiφ0eiωt (3.16)where A is a constant amplitude of the input signal, φ0 is a constant signal phase, Γ is an amplifi-cation factor and γ is a phase shift. Thus, the current density in the conductor is

j = Γ eiγ

∂

Bz∂t

Knowing the current density, one can easily find the volume density of the Lorenz force:

f L = j × BzBecause Bz is directed along the Z axis, the f L vector is in the radial plane and it is normal to

the current density vector j . This can be easily reflected using complex vector representations bymultiplying the current density by e−iπ/2: f L = jBze−iπ/2

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or

f L = Γ Bze

−iπ/2eiγ ∂ Bz

∂t

f L = 12

Γ Bze−iπ/2eiγ dBzdr

{∆r0ωe−i(ωt+ψ)e−iπ/2 + ∆r0∗ωei(ωt+ψ)eiπ/2} (3.17)The equation (3.17) gives the projections of Lorenz force on the axes of the rotating coordinate

frame X RY RZ . To obtain the projections on the axis of the stationary coordinate frame X 0Y 0Z 0(represented by a complex number f L0), the complex representation of the force in the rotatingframe f L (3.17) has to be multiplied by eiωt:f L0 = f LeiωtThus,

f L0 = 1

2

Γ Bze−iπ/2eiγ

dBz

dr {∆r0ωe−iψe−iπ/2 + ∆r0

∗ωe2iωteiψeiπ/2

}or

f L0 = −12

Γ Bzeiγ dBz

dr {∆r0ωe−iψ − ∆r0∗ωeiψe2iωt} (3.18)

The total force acting on the conductor can be found by integration of (3.18) over the conductor

volume:

F 0 = G

f L0dv (3.19)We can make a few interesting observations about this system, which we shall call lemmas.

Lemma 1:

The electromagnetic force acting on body G is always directed along a line passingthrough the field symmetry axis Z 0.

In other words, for any current distribution in conductor G, there is never a torque aboutaxis Z 0. Also, considering that we can move a vector of a force along its line of action,we can say that the force acting on body G is always applied at some point on axis Z 0.

Proof:

If there is some current with density j flowing in body G, a torque T acting on body Gabout the axis Z 0 due to this current can be found as

T = V r× j×Bdv = ez Rmax

0Bz(r)r

S (r) j · n ds dr,

where S (r) is a cylindrical surface concentric with axis Z 0, n is the unit vector normalto the surface S (r), and Rmax is a radius of a cylindrical surface concentric with axisZ 0 and surrounding all of the conductor G. The first integral is taken over body G and

S (r) j · n ds is the current flow through a cylindrical surface S (r). Because ∇ j = 0 , S (r) j · n ds = 0, and therefore T = 0.

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Figure 3.3: An example clarifying why the force always passes through the axis Z 0.

The proof of this Lemma can be clarified by a simple example. First, assume that the conductingbody G is just a conducting loop of arbitrary shape carrying current I (see Figure 3.3). Consideran annulus with inner radius r and a differential thickness dr concentric with the Z 0 axis. We referto the inner surface of this annulus as S (r). It is easy to see that the differential tangential forcesacting on the conducting parts located within the annulus are |F τ 1| = |F τ 2| = IBz(r)dr and theyproduce opposite torques about the axis Z 0. Thus, the net torque is zero, and this is because thefield is uniform circumferentially and also because the current which crosses the surface S (r) in theoutward direction is exactly equal to the return current crossing this surface in the inward direction.

This is, in fact, true for any current distribution, as implied by the integral form of ∇ j = 0.

Lemma 2:

The time averaged electromagnetic forceFav is proportional in magnitude to therotor displacement with some proportionality coefficient K and directed at someangle 0 ≤ θ ≤ �

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