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UBC Theses and Dissertations

Long stroke magnetic levitation planar stages Usman, Irfan-ur-rab 2015

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LONG STROKE MAGNETIC LEVITATION PLANAR STAGES by  Irfan-ur-rab Usman B.A., The University of British Columbia, 2008 B.A.Sc., The University of British Columbia, 2008 M.A.Sc., The University of British Columbia, 2010  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES (Mechanical Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver)  December 2015 © Irfan-ur-rab Usman, 2015 ii  Abstract  Modern positioning applications often require long strokes in multiple degrees-of-freedom (DOF). One solution to such requirements is a planar stage capable of simultaneous large strokes in X- and Y-translation. An ideal stage concept is based on a planar motor which uses non-contact multi-axis forces to directly actuate a single moving body, without any connecting elements or bearings structures that would induce additional structural modes as well as excess inertia. This thesis presents the design, analysis and experimental results of two planar motors: 1) a permanent magnet synchronous planar motor, and 2) a permanent magnet asynchronous planar motor.  The permanent magnet synchronous planar motor is comprised of multiple one-dimensional (1-D) magnet arrays attached to a moving stage and multiple stationary 1-D coils built as a printed circuit board (https://www.youtube.com/watch?v=-r4Tv7GbB8o). This motor addresses many issues with existing synchronous planar levitation motors, including scalability of the XY stroke with minimal increase in controller or drive complexity, and simplified commutation and control due to natural force/torque decoupling and no coil/magnet array edge effects.  Modeling, analysis and design of a prototype are shown with motion control results. To use this levitation motor for high accuracy positioning applications, force and torque characteristics of the 6-DOF stage must be highly linear in all axes in order to minimize controller effort and reduce intrinsic motor disturbances. Novel magnet array designs are presented which self-attenuate both force and torque ripple in all 6-DOF, without additional controller or drive complexity. These array designs are tested via simulation and experiment.   The permanent magnet asynchronous planar motor is based on the asynchronous induction effect to simultaneously levitate and propel the motion stage. The advantages of this type of planar iii  levitation motor are: 1) passive stability in all axes; and 2) simple slab-type homogenous stator comprised of a basic conducting material such as aluminum or copper. This thesis presents novel modeling and analysis and provides a new analytical expression for the generated levitation force and torque with geometric and material property inputs. Experimental force and torque measurements are carried out and compared to the analytical model.  iv  Preface  The research presented in this thesis has been carried out at the University of British Columbia Precision Mechatronics Laboratory (UBC PML), Department of Mechanical Engineering, under the supervision of Dr. Xiaodong Lu.    Chapter 1 is a brief literature review of existing multi-degree of freedom direct drive stages using magnetic field interactions.   Chapter 2 is based in part on the work published in   Xiaodong Lu, Irfan Usman, "6D Direct-Drive Technology for Planar Motion Stages", Annals of the CIRP-Manufacturing Technology, Vol. 61, No. 1, pp. 359-362, 2012. In this work we present the novel 6-DOF synchronous levitation motor including field, force and torque modeling as well as commutation laws. Experimental motion tracking results from a prototype are also presented. The author modeled, simulated, designed, built and assembled the 6-DOF levitation motor. The 6-DOF position sensor was based on the work of Niankun Rao [1]. The real-time control computer was based on the work of Kristofer Smeds [2]. The 6-DOF motion controller and the multi-channel PWM current driver were implemented by Dr. Xiaodong Lu.   Chapter 2 is also based in part on the work published in    Irfan-ur-rab Usman, Xiaodong Lu, "Force Ripple Attenuation of 6-DOF Direct Drive Permanent Magnet Planar Levitating Synchronous Motors", IEEE Transactions on Magnetics. Vol. 51, No. 12, pp. 1-8, Dec. 2015. v  In this work a novel magnet array design is presented for force ripple attenuation. The author modelled, simulated, designed, built and measured the load characteristic of the novel split magnet array. Portions of the work have also been presented in conference proceedings as follows:  I. Usman; X. Lu; "Force ripple attenuation in 6DOF planar levitated motor"; In: 2nd International Conference on Virtual Machining Process Technology, 2013   I. Usman; X. Lu; "Torque and force ripple attenuation of 6DOF levitating motor"; In: 3rd International Conference on Virtual Machining Process Technology, 2014    Chapter 3 includes work on asynchronous levitation motors that is currently part of a manuscript that has been submitted to IEEE Transactions on Magnetics. In this work, a new analytical model for levitation force and drag torque on a rotating magnet disk over a finite thickness homogeneous conducting slab is presented, and experimental load characterization is carried out. This novel analytical model is useful for informing physical understanding and design intuition, especially when compared to existing numerical methods. The rotating magnet disk can be used to make a 6-DOF asynchronous type planar levitation motor [3]. The author carried out the modeling, analysis, simulation and experimental load characterization presented.  vi  Table of Contents  Abstract .......................................................................................................................................... ii Preface ........................................................................................................................................... iv Table of Contents ......................................................................................................................... vi List of Tables ..................................................................................................................................x List of Figures ............................................................................................................................... xi Notation ....................................................................................................................................... xix Acknowledgements ......................................................................................................................xx Dedication ................................................................................................................................... xxi Chapter 1: Introduction ................................................................................................................1 1.1 Thesis Overview ............................................................................................................. 4 1.2 Thesis Contributions ....................................................................................................... 5 1.3 Machine Architectures for Planar Positioning ................................................................ 6 1.3.1 Serial Machine Architecture ....................................................................................... 6 1.3.2 Parallel Kinematic Architecture .................................................................................. 7 1.3.3 Direct Drive Architecture ........................................................................................... 8 1.4 Multi-DOF Direct Drive Machines ................................................................................. 9 1.4.1 Long Stroke Synchronous Magnetic Planar Levitation Motors ............................... 12 1.4.2 Long Stroke Asynchronous Magnetic Planar Levitation .......................................... 22 Chapter 2: Synchronous Planar Levitation Stage ....................................................................29 2.1 Synchronous Levitation Concept and Working Principle ............................................ 30 2.2 Magnetic Field, Force and Torque Generation Modeling for 2-DOF Motor Element . 34 vii  2.2.1 3D Magnetic Field Distribution for Cuboidal Magnet ............................................. 35 2.2.2 Force and Torque Generation on Magnet Array ....................................................... 43 2.2.3 Principle of Equivalent Force Generation ................................................................. 45 2.2.4 Magnetic Field Modeling for Infinitely Extended Halbach Array ........................... 54 2.2.5 Force Due to Single-Phase Repeating Coil Group ................................................... 63 2.2.6 Force Due to 3-Phase Repeating Coil Group with Commutated Current ................. 68 2.3 Force and Torque Ripple Self-attenuation Using Magnet Array Design ..................... 72 2.3.1 Force Vector Rotation from Offset Commutation Law ............................................ 73 2.3.2 Split Magnet Array Design for Self-attenuation of Force Ripple ............................. 77 2.3.3 Experimental Load Verification for Force Ripple Self-attenuation ......................... 82 2.3.4 Quad-Split Array Design for Self-attenuation of Force and Torque Ripple ............. 87 2.4 6-DOF Motion Stage Prototype Design ........................................................................ 99 2.4.1 Prototype Architecture .............................................................................................. 99 2.4.2 Mover Design.......................................................................................................... 103 2.4.2.1 Magnet Array Geometric and Material Parameters ........................................ 103 2.4.2.2 Mover Magnet Array Configuration ............................................................... 105 2.4.2.3 Mover Mechanical Design .............................................................................. 109 2.4.3 Stator Coil Array PCB Board Design and Wiring Configuration........................... 111 2.4.4 Stator Design ........................................................................................................... 123 2.5 6-DOF Prototype Manufacturing and Assembly ........................................................ 127 2.5.1 Mover ...................................................................................................................... 127 2.5.2 Stator ....................................................................................................................... 133 2.5.3 Metrology Reference Frame ................................................................................... 135 viii  2.5.4 Complete System .................................................................................................... 136 2.6 Motion Tracking Results with 6-DOF Prototype........................................................ 138 2.6.1 Plant Modeling and Control Architecture ............................................................... 138 2.6.2 6-DOF Tracking Results ......................................................................................... 139 Chapter 3: Asynchronous Planar Levitation Stage ................................................................143 3.1 Asynchronous Planar Levitation Stage Concept and Working Principle ................... 144 3.2 Asynchronous Levitation Machine Force and Torque Modeling ............................... 148 3.2.1 Infinitely Wide 2D Moving Magnet Array over Homogeneous Conducting Slab . 148 3.2.2 Analytical Shells Model for Force and Torque of Rotating 3D Magnet Disk over Homogeneous Conducting Slab .......................................................................................... 161 3.3 Experimental Force, Torque and Power for Rotating Magnet Disk ........................... 169 Chapter 4: Conclusion ...............................................................................................................180 4.1 Synchronous Planar Levitation Motor ........................................................................ 182 4.1.1 Contribution I: Motor topology for New Type of 6-DOF Synchronous Planar Levitation Motor ................................................................................................................. 182 4.1.2 Contribution II: Analytical Model of Field, Force, Torque and Commutation Laws for New Type of 6-DOF Synchronous Planar Levitation Motor ........................................ 183 4.1.3 Contribution III: Prototype and Experimental Demonstration of New Type of 6-DOF Synchronous Planar Levitation Motor ................................................................................ 183 4.1.4 Contribution IV: Novel Split and Quad-Split Magnet Array Designs for Force and Torque Ripple Self-attenuation ........................................................................................... 184 4.1.5 Contribution V: Experimental Verification of Novel Split Magnet Array Design for Force Ripple Attenuation .................................................................................................... 185 ix  4.2 Asynchronous Planar Levitation Motor ...................................................................... 185 4.2.1 Contribution I: Novel Analytical Model of Field, Force and Torque of Magnet Disk Rotating Above a Finite Thickness Homogenous Conductor............................................. 185 4.2.2 Contribution II: Evaluation of Novel Analytical Model with Finite Element Simulation and Characterization with Experimental Load Measurements ......................... 186 4.3 Future Work ................................................................................................................ 186 4.3.1 Future Work for 6-DOF Synchronous Planar Levitation Motor ............................ 187 4.3.2 Future Work for 6-DOF Asynchronous Planar Levitation Motor Modeling and Analysis............................................................................................................................... 188 Bibliography ...............................................................................................................................189  x  List of Tables Table 2.1 Spacing for Split Array Targeting Different Force Harmonics .................................... 79 Table 2.2 Force Harmonic Coefficients in Z ................................................................................ 87 Table 2.3 Analytical Mean and 6 cpλ Ripple Forces for Sub-arrays. ........................................... 93 Table 2.4 Analytical Mean and 6 cpλ Ripple Torques for Sub-arrays. ........................................ 94 Table 2.5  Force and Torque Amplitudes. .................................................................................... 97 Table 2.6  Alpha Prototype Motor Parameters. .......................................................................... 122 Table 2.7  Predicted Alpha Prototype Motor Performance at flying height 0.5 mm. ................. 137 Table 3.1  2D Asynchronous Levitation Machine Parameters. .................................................. 161 Table 3.2  3D Rotating Disk Asynchronous Levitation Machine Parameters. ........................... 169 Table 3.3  3D Experimental rotating disk asynchronous levitation machine parameters. .......... 173  xi  List of Figures Figure 1.1 Novel 6-DOF synchronous planar levitation motor. ..................................................... 3 Figure 1.2 Asynchronous planar levitation motor mover (onboard battery pack, rotary motor controllers, and stator conducting slab not shown). ........................................................................ 3 Figure 1.3 Serial machine architectures. (a) XY H-drive style planar stage. (b) Dual stage. ......... 7 Figure 1.4 Parallel machine architecture for 6-DOF motion. ......................................................... 8 Figure 1.5 Direct drive machine architecture for 6-DOF positioning stage. .................................. 9 Figure 1.6 Planar motor for very small scale movers (such as 2mm by 2mm by 0.4mm) based on diamagnetic levitation and Lorentz force translation, figure adapted from [29]. (a) Top view of mover magnets. (b) Electromagnetic configuration of diamagnetic planar levitation motor. ...... 11 Figure 1.7 Kim’s 6-DOF synchronous levitation motor, figure adapted from [36]. .................... 13 Figure 1.8 Scaling up planar stroke of Kim and Trumper’s synchronous levitation motor. ........ 14 Figure 1.9 Moving coil synchronous planar levitation motor (umbilical cable to mover not shown), figure adapted from [39]. Red magnet blocks are magnetized in +z, and blue magnet blocks are magnetized in –z. ......................................................................................................... 15 Figure 1.10 Shifting center of pressure of example 3-phase moving coil levitation motor, given a constant levitation force command (umbilical cable not shown). ................................................ 16 Figure 1.11 Moving magnet 2D chessboard type synchronous planar levitation motor with multiple alternating layers of racetrack style coils; figure adapted from [43]. Red magnet blocks are magnetized in +z, and blue magnet blocks are magnetized in –z. .......................................... 18 Figure 1.12 Moving magnet 2D chessboard type synchronous planar levitation motors. (a) Alternating zones of racetrack coils, figure adapted from [45]. (b) Herringbone coil pattern, figure adapted from [46]. .............................................................................................................. 18 xii  Figure 1.13 Scaling up XY stroke of moving magnet 2D chessboard type synchronous planar levitation motor (example using motor type similar to [43]). ....................................................... 19 Figure 1.14 Moving magnet 2D chessboard type synchronous planar levitation motor with alternating layers of straight coils; figure adapted from [54]. ...................................................... 20 Figure 1.15 6-DOF moving magnet stage with axially magnetized cylindrical magnets over array of circular coils; figure adapted from [55]. ................................................................................... 21 Figure 1.16 Circular type 2-DOF induction planar motor, figure adapted from [57]. .................. 23 Figure 1.17 Asynchronous planar motor using electromagnets and back-iron, figures adapted from [4] (out-of-plane bearings not shown). (a) Linear 1-DOF induction motor. (b) 3-DOF configuration of asynchronous planar motor. ............................................................................... 23 Figure 1.18 Asynchronous levitation motor using permanent magnet Halbach array, adapted from [58]. ...................................................................................................................................... 24 Figure 1.19 Permanent magnet asynchronous levitation motor elements showing example magnetization patterns. (a) Circumferential magnet disk. (b) Radial magnet wheel. ................... 25 Figure 2.1 Synchronous planar levitation 6-DOF motor electromagnetic topology. ................... 31 Figure 2.2 Y2 magnetic array and coils cross-section .................................................................. 32 Figure 2.3 Cuboidal permanent magnet with vertical (z-directed) uniform magnetization M shown in red. ................................................................................................................................. 37 Figure 2.4 Example 3D finite extension Halbach-patterned magnet array (magnetization directions shown in red). (a) 3D isometric view of finite Halbach array. (b) 2D view of finite Halbach array showing segment dimensions. ............................................................................... 40 Figure 2.5 Analytical field model, flux density vector components of BFEA for example magnet array at zm = -λ/5. ....................................................................................................................... 41 xiii  Figure 2.6 Error in flux density vector components for zm = -λ/5. Error defined as BFEA-BANALYTICAL. ........................................................................................................................ 42 Figure 2.7 General 3D magnet array interacting with differential conductor element. ................ 44 Figure 2.8 Finite x-directed extension Halbach-patterned magnet array. ..................................... 46 Figure 2.9 Infinite x-directed extension Halbach-patterned magnet array ................................... 46 Figure 2.10 Bz from 3D analytical model along test line (zm = -λ5, ym = 0). ............................. 47 Figure 2.11 Equivalent force cases. (a) λ-width magnet array interacting with λ-repeating coil array. (b) repeating λ-width magnet arrays with single coil. (c) ∞-width magnet array with single coil................................................................................................................................................. 52 Figure 2.12 (a) Wm-width magnet array interacting with ηc-repeating coil array. (b) Superposition showing overlap of each identical magnet array. .................................................. 53 Figure 2.13 2D form of the infinite extension magnet array. ....................................................... 56 Figure 2.14 Magnetization distribution within magnet volume for ∞-width Halbach array with 4 magnet segments/λ separated by gap g. ........................................................................................ 58 Figure 2.15 Single phase coil array below finite width Halbach array. ........................................ 63 Figure 2.16 Partial harmonic model versus 3D analytical model at different zm (a) Bz, partial harmonic model (PHD), 3D analytical model (3DAM) evaluated at ym = 0 with Dm = 20λ, and error (PHD-3DAM), for ∞-width magnet arrays. (b) Fourier coefficients for error at different zm........................................................................................................................................................ 66 Figure 2.17 Modified 3-phase coil array below finite width Halbach array. ................................ 70 Figure 2.18 Multiple stationary coil array stack-up dimensions, N-layers of identically driven coil arrays. ..................................................................................................................................... 72 xiv  Figure 2.19 Offset magnet array from commutation center. ........................................................ 74 Figure 2.20 Force vector rotation due to commutation center offset. (a) Position independent force vector F0. (b) Force ripple vector F6. (c) Force ripple vector Fk ± 1 resulting from field harmonic Bk. ................................................................................................................................. 76 Figure 2.21 Split array solution for force ripple attenuation of the 6th harmonic of force ripple due to the 5th field harmonic. ........................................................................................................ 77 Figure 2.22 FEA models for ‘infinite’ coil arrays (extended in the x-direction over 18λ width). (a) Non-split array. (b) Split array. ............................................................................................... 80 Figure 2.23 Finite element analysis of non-split array and split array over ‘infinite’ coil arrays( extended over 18λ width). Flying height zf = 0.5 mm, Ixr = 0, Izr = 10 [A], λ = 30 mm, Br =1.325 T, Dm = 2λ,Wc = 0.1583λ, tc = 0.213 [mm] and 8 layers of active coils spaced 0.643 [mm] apart in the -zm direction. ........................................................................................ 81 Figure 2.24 Experimental arrays for load measurement, showing dimensions and magnetization directions. ...................................................................................................................................... 83 Figure 2.25 Experimental setup for load measurement. ............................................................... 83 Figure 2.26 Experimental test cases with finite coil array of 4λ width. (a) Non-split array. (b) Split array. ..................................................................................................................................... 84 Figure 2.27 Comparison experimental non-split array and split array, uncompensated for zf =0.0247λ , Ixr = 0, Izr = 9.5 A, λ = 30 mm, Dm = 2λ,Wc = 0.1583λ, Hm = λ/4, tc = 0.213 mm and N = 8 layers of active coils spaced tg =  0.643 mm apart in the -zm direction. ................... 86 Figure 2.28 Magnet array designs showing force ripple vectors acting at magnetic centers of pressure. (a) Non-split magnet array. (b) Split magnet array. ...................................................... 89 xv  Figure 2.29 Quad-split array for force and torque attenuation. (a) Embodiment 1. (b) Embodiment 2. (c) Sub-array example Halbach magnetization pattern. ...................................... 90 Figure 2.30 Quad –split force ripple vectors acting at equivalent moment arms of sub-arrays (LI, LII, RI, RII) . ............................................................................................................................. 91 Figure 2.31 Finite element model schematic showing generalized magnet array in relation to stationary coil array with 8 coil layers spaced apart by 0.643 mm in zm. .................................... 95 Figure 2.32 Simulated magnet arrays showing magnetization patterns. (a) Non-split array. (b) Split array with τ = λ/10. (c) Quad-split array with τ = λ/10. .................................................. 95 Figure 2.33 FEA forces and torques for non-split array, split array, and quad-split array; zf =λ/60 = 0.5mm, Ixr = 0, Izr = 10 A, with 8 coil layers spaced apart by 0.643 mm in zm. ........ 96 Figure 2.34 Spatial harmonic amplitudes for force and torque ripples from FEA; flying height zf = λ/60 = 0.5 mm, Ixr = 0, Izr = 10 A, with 8 coil layers spaced apart by 0.643 mm in zm. (a) Translation force, Fx. (b) Levitation force, Fz. (c) Torque in ym-axis, Ty ........................ 98 Figure 2.35 Prototype architecture schematic showing 6-DOF camera metrology, power amplifier and real-time controller. .............................................................................................. 100 Figure 2.36 Schematic cross-section of alpha prototype. ........................................................... 101 Figure 2.37 Alpha prototype solid model showing 6-DOF camera metrology. ......................... 102 Figure 2.38 Alternate mover magnet array layouts (identical λ = 30mm). ............................... 106 Figure 2.39 Alpha prototype mover design, exploded view. ...................................................... 110 Figure 2.40 Isometric cut-view of mover . ................................................................................. 110 Figure 2.41 Alpha prototype mover design, detailed planform. ................................................. 111 Figure 2.42 Coil structure isometric view, single λ-group Y-elongated coil traces. .................. 113 Figure 2.43 Eight layer coil structure, showing alternating X- and Y-elongated coil traces. ..... 114 xvi  Figure 2.44 Single phase multilayer windings, shown for three layers of Y-elongated traces. .. 115 Figure 2.45 Three phase wiring, shown for a single λ-coil group. ............................................. 116 Figure 2.46 Serial connections between two single λ-coil groups with 3-phase wiring. ............ 116 Figure 2.47 Extendable coil array PCBs. Inset shows connector PCB with solid pin connectors...................................................................................................................................................... 118 Figure 2.48 Coil structure isometric view, X- and Y-elongated coils. (a) Layer X1: x-elongated traces. (b) Layer Y1: y-elongated traces. .................................................................................... 119 Figure 2.49 Coil array PCB showing coil groupings, X- and Y-elongated coils. ....................... 120 Figure 2.50 Coil array PCB stack-up details. ............................................................................. 121 Figure 2.51 Alpha prototype stator exploded view. .................................................................... 124 Figure 2.52 Alpha prototype stator isometric cut-view. ............................................................. 125 Figure 2.53 Isometric cut-view of metrology reference frame. .................................................. 126 Figure 2.54 Magnet sub-array assembly jig. ............................................................................... 128 Figure 2.55 Magnet sub-array assembly jig with magnet array installed. .................................. 129 Figure 2.56 Magnet array assembly jig....................................................................................... 130 Figure 2.57 Magnet assembly. .................................................................................................... 132 Figure 2.58 Stator assembly steps. .............................................................................................. 134 Figure 2.59 Metrology reference frame assembly. ..................................................................... 135 Figure 2.60 Synchronous planar levitation motor system. ......................................................... 136 Figure 2.61 Controller block diagram. ........................................................................................ 139 Figure 2.62 Elliptical path of mover in XY, at zf = 1 mm. ....................................................... 141 Figure 2.63 Tracking error for elliptical path, zf = 1 mm. ........................................................ 142 Figure 2.64 Mover floating with zf = 2.5 mm. .......................................................................... 142 xvii  Figure 3.1 Asynchronous levitation planar motor concept using four levitation disks, magnetization shown in red for magnet disk 1 (central actuated pivot mechanism and onboard battery pack/motor controllers not shown). ................................................................................ 146 Figure 3.2 Levitation and propulsion principle (details of actuated pivot mechanism not shown)...................................................................................................................................................... 147 Figure 3.3 Stationary Halbach array of infinite extent over a translating conducting slab. ....... 150 Figure 3.4 Magnetic pressure on magnet array, 2D analytical model versus 2D FEA. .............. 160 Figure 3.5 Rotating permanent magnet disk asynchronous levitation machine. (a) Motor topology (with magnetization pattern for example similar to that shown in Figure 1.19a). (b) Cylindrical shell element of magnet disk. ` ................................................................................................... 165 Figure 3.6 3D FEA model, with Nm = 8. ................................................................................... 167 Figure 3.7 Circumferential versus straight magnetization of magnet segments in XY plane. ... 168 Figure 3.8 Levitation force and drag torque, 3D FEA versus analytical shells method for zf =2 mm. .......................................................................................................................................... 168 Figure 3.9 Experimental rotating disk asynchronous levitation machine. .................................. 171 Figure 3.10 Load test setup. ........................................................................................................ 172 Figure 3.11 6-DOF load characteristics for rotating disk asynchronous machine (copper). ...... 174 Figure 3.12 6-DOF load characteristics for rotating disk asynchronous machine (aluminum). . 175 Figure 3.13 Force, torque and mechanical power iso-lines at different flying heights (copper). 177 Figure 3.14 Force, torque and mechanical power iso-lines at different flying heights (aluminum)...................................................................................................................................................... 178 Figure 3.15 Error between analytical shells model and experimental load characteristic (copper)...................................................................................................................................................... 179 xviii  Figure 3.16 Error between analytical shells model and experimental load characteristic (aluminum). ................................................................................................................................. 179    xix  Notation Multiple forms of vector notation are used throughout this thesis. A general vector field 𝑾 is denoted by bold italics, while a scalar is always un-bolded. The vector field notations are defined as 𝑾 ≡ 𝑊𝑥 ?̂? + 𝑊𝑦𝒋̂ + 𝑊𝑧?̂? ≡ (𝑊𝑥,𝑊𝑦,𝑊𝑧) ≡ [𝑊𝑥𝑊𝑦𝑊𝑧] where (?̂?, 𝒋̂, ?̂?) are unit vectors in the principal directions of a Cartesian frame (𝑥, 𝑦, 𝑧), and the scalar values 𝑊𝑥,𝑊𝑦,𝑊𝑧 denote the magnitudes of the vector at a location (𝑥, 𝑦, 𝑧)  along each unit direction.   xx  Acknowledgements  Special thanks are owed to my parents, who have supported and sacrificed for me throughout my life. I wish to thank in particular my research advisor, Dr. Xiaodong Lu, without whom none of this would be possible. His incredible work ethic and creative prowess were and are a constant source of inspiration.  I thank my friends of old (in alphabetical order): Yoyo Au, Daniel Fritter, Howie Wong, and Adrian Yu.  I thank Erina Okuda Nesbit. I thank my somewhat ludicrous number of cousins who made me feel like I had an infinitely extended and close knit family array. My colleagues and friends at UBC who throughout my many, many years in the laboratory have influenced me as an engineer and as a person: Elizabeth Jean Hu, Kristofer Smeds, Darya Amin-Shahidi, Richard Graetz, Arash Jamalian, Bill Kengli Lin, Fan Chen, Alexander Yuen, Navio Kwok, Niankun Rao, Eric Buckley, Mark Dyck, Rui Chen, Jian Gao and Keir Maguire.  The machine shop and instrumentation technicians who let me “borrow” all the tools and knowledge I needed to get this done: Roland Genshorek, Erik Wilson, Bernhard Nimmervol, Markus Fengler, Sean Buxton, and Glen Jolly. And of course thanks to Samuel Earnshaw who set us a most interesting problem to solve, and James Clerk Maxwell for giving us the fundamental tools to solve it. xxi  Dedication  To my parents.   1  Chapter 1: Introduction  Non-trivial modern industrial positioning machines typically require controlled motion in multiple degrees of freedom (DOF). For example, a planar stage (X-Y table) generates simultaneous X and Y translation, producing one of the most fundamental machine elements in manufacturing. High speed, high acceleration, precision planar motion with some out of plane Z-stroke is greatly sought after in many high performance industrial applications, such as lithography wafer steppers [4]. Further, even lower precision applications may benefit from ultra-long XY planar strokes as an enabling technology for reconfigurable manufacturing, such as for agile manufacturing in modular assembly systems [5] [6]. An ideal stage concept that simultaneously satisfies the need for precision motion, large strokes and high scanning rates and accelerations is the application of non-contact multi-axis forces to a single end effector, without any intervening connecting structures or bearings which would induce structural modes and decrease overall stiffness, accuracy, and achievable acceleration. This thesis presents two methods for applying such non-contact forces to an end effector over large strokes using magnetic field interactions.  First, a novel 6-DOF long stroke synchronous planar levitation motor (Figure 1.1) is presented which uses driven current through stationary 1D coil arrays to actuate sets of 1D Halbach magnet arrays attached to a mover chassis. The motor topology naturally generates linear and decoupled 6-DOF actuation of the mover, which allows minimal controller and drive complexity. The coil structure is very well suited to be manufactured as a printed circuit board (PCB) using commercially available technology without further effort. The passive mover (unpowered and untethered) combined with low heat generation due to good power efficiency from the high copper fill factor of the coil design mean this motor is well suited to applications that require 2  simultaneously high accelerations, precision and very large planar strokes. The coil design and novel driving principles allow for ease of scaling up the planar stroke, relative to other synchronous levitation machines. Second, a novel analysis of an asynchronous planar levitation motor (Figure 1.2) is presented which utilizes sets of mechanically rotating Halbach permanent magnet disks over a conducting stator to produce an induced field in the stator which generates both lift and propulsion on the mover. A major advantage of this class of levitation motor is the simple stator: at a minimum, only a homogenous slab of conducting material without further structure is required, making this stator more cost effective than the synchronous levitation motor stator for ultra-long strokes. Another advantage is that the mover is passively stable in all 6 axes under gravity preload, meaning position feedback is not a requirement. However, the power efficiency of the asynchronous levitation motor is lower than that of the synchronous levitation motor especially at higher loads because of the Ohmic loss in the conducting stator plus power losses incurred by the mechanical rotation mechanism of each magnet disk. The levitation force must be opposed by gravity for stable operation, so the work volume of the motor is limited in orientation. And, while all axes are passively stable, they are also highly coupled, complicating 6-DOF control. Further, the mover requires some powered mechanisms to mechanically rotate the magnet disks relative to the stator and thus needs either onboard power (such as a battery) or an umbilical cable to the mover. This class of levitation motor is thus suitable for applications with lower precision requirements but very large planar strokes. 3   Figure 1.1 Novel 6-DOF synchronous planar levitation motor.    Figure 1.2 Asynchronous planar levitation motor mover (onboard battery pack, rotary motor controllers, and stator conducting slab not shown). 4  1.1 Thesis Overview This thesis is divided into four main parts:  Chapter 1: Introduction This chapter consists of the overall structure of the thesis and a literature review of both classes of synchronous and asynchronous planar levitation techniques.    Chapter 2: Synchronous planar levitation motor This chapter comprises the bulk of the research work and includes: the novel concept and working principles of the 6-DOF synchronous levitation motor; modeling and analysis of the magnetic field and force generation; presentation, analysis and simulation of magnet array designs and design methodology to attenuate force and torque ripple in synchronous machines; design and manufacture of a prototype 6-DOF planar levitation stage; experimental motion tracking results for the prototype 6-DOF stage; and experimental load testing of the force ripple-attenuating magnet array designs.    Chapter 3: Asynchronous planar levitation motor This chapter presents the conceptual design of a 6-DOF asynchronous planar levitation motor based on rotating permanent magnet disks similar to that first presented in [7] and more recently as a commercial venture by [8], [9]. Modeling and analysis of the field distribution is presented which yields a closed form analytical solution for levitation and drag forces for a 2D infinitely long linear motor geometry, which is then extended to a new model for levitation force and drag torque for a 3D magnet array disk and conductor geometry. This novel 3D analytical model has 5  the potential to give detailed insight into the effect of various geometric and material parameters on the performance of the levitation motor without the need for extensive 3D finite element analysis. An experimental asynchronous levitation motor element is built to verify the model and levitation force and drag torque are measured.  Chapter 4: Conclusion This chapter summarized the contributions of this thesis to the state of the art and discusses future work.  1.2 Thesis Contributions The original contributions of this work are: Synchronous planar levitation motor  A novel electromagnetic configuration for a new type of 6-DOF synchronous planar levitation motor.  Modeling and analysis for magnetic field, force and torque generation, and commutation laws for this new type of planar levitation stage. The novel motor topology allows the minimal complexity analytical force model, which greatly simplifies the commutation law for the motor. A simpler commutation law provides benefits in terms of computational cost in a practical implementation of the 6-DOF motion stage.  Magnet array design methodology for force and torque ripple self-attenuation in this synchronous planar levitation motor, which can be generalized to other synchronous machines. 6   Experimental load testing validation of force and torque ripple attenuation using novel magnet array designs.  Prototype and motion tracking results for 6-DOF synchronous planar levitation motor.   Asynchronous planar levitation motor  New analytical model for levitation force and drag torque of asynchronous planar levitation motor.  Experimental force, torque and power measurement and characterization of asynchronous planar levitation motor element.  1.3 Machine Architectures for Planar Positioning Three types of machine architecture can be used to solve multi-DOF positioning problems: 1) serial, 2) parallel, and 3) direct drive architectures.   1.3.1 Serial Machine Architecture Serial architectures employ multiple 1-DOF motion elements and stack them in series to achieve either multi-DOF motion or enhanced performance (precision and stroke) on the end effector. Figure 1.3a shows a widely adopted approach to multi-axis machines: a Y-axis linear stage transfers its Y-direction motion to an X-axis linear stage, producing combined XY motion on the end effector (the X-axis stage) [10]. The dual stage concept shown in Figure 1.3b is often used to achieve high precision over large strokes by using a long stroke, low precision coarse stage to carry a high precision, short stroke fine stage, achieving high precision over a long stroke [11]. The serial machine architecture is the most ubiquitous because single-DOF stages are the simplest and 7  most mature positioning technology. However, as each 1-DOF stage has only a single actively controlled axis and five passively constrained axes, the final moving stage will suffer from decreased stiffness, accumulated error motion, and usually very large moving inertia caused by the serial connection of multiple moving masses.    Figure 1.3 Serial machine architectures. (a) XY H-drive style planar stage. (b) Dual stage.  1.3.2 Parallel Kinematic Architecture An alternative approach to multi-DOF positioning is a parallel architecture (Figure 1.4), where 1-DOF motion elements are connected in parallel between inertial ground and the end effector. This approach is a popular research area due to its versatile motion potential [12] but suffers from limited motion range, varying force/velocity transmission ratio, and position-dependent stiffness within the work volume [13]. Serial and parallel approaches can also be combined together, such as a planar stage with large yaw motion [14].   8   Figure 1.4 Parallel machine architecture for 6-DOF motion.   1.3.3 Direct Drive Architecture The ideal stage architecture is the direct drive approach (Figure 1.5), where there is only one moving stage/end effector without any intermediate motion elements and all actuating forces are directly applied to the mover. In linear stages, the advantages of direct drive over a lead-screw feed drive are well understood, including improved speed, accuracy and acceleration [15]. These benefits are still true in multi-DOF direct drive stages, but such drives have not been widely utilized. Conceptually, 6-DOF direct drive systems do not need any mechanical guiding bearings, and consequently have additional benefits including zero friction and natural isolation from ambient vibration. In this thesis, we examine in particular methods using magnetic field interactions to produce non-contact forces on the moving body.  9   Figure 1.5 Direct drive machine architecture for 6-DOF positioning stage.  1.4 Multi-DOF Direct Drive Machines Many efforts have been made towards the ideal multi-DOF direct drive concept. 2-DOF direct drives have been widely used in optical disk players as fine motion stages for simultaneous focus control and radial tracking of the optical pickup unit, such as in [16], but such drives only have sub-mm strokes. Combined rotary and linear 2-DOF direct drives can be advantageously used in high precision spindles with in-feed capability [17], [18]. One well-known example of 3-DOF direct drive stages is a Sawyer stepper motor [19], which can generate XY and small θ in-plane motion on a single stage and relies on air bearings to constrain out-of-plane motion. The disadvantages of this motor include inherent cyclic errors as well as constraints imposed by the umbilical cord which powers the moving stage. For smoother planar motion, voice coil motors [20] and linear motors [21] have been used in high precision 3-DOF direct drive stages, but their strokes are very limited. There are also 3-DOF rotary direct drives for versatile joints or robotic ‘wrists’, such as variable reluctance spherical motors [22] and brushless spherical motors [23]. A magnetic field-gradient based 5-DOF actuator for micro-needle manipulation has been developed 10  that controls three translation and two rotational degrees of freedom of a 0.5mm long needle-like soft magnetic mover using externally applied and controlled magnetic fields over millimeter level work volumes [24]. Intended for micro-surgery, the work volume and size of the mover needle is severely limited.  Fully 6-DOF direct drive stages using magnetic field interactions present particular challenges. Earnshaw in [25] showed that 6-DOF passive magnetic levitation of a permanent magnet without dynamic stabilisation was impossible with only ferromagnetic materials (i.e. with materials with permeability 𝜇 ≥ 𝜇𝑜).  An exception are systems that include diamagnetic materials where 𝜇 < 𝜇𝑜 ( [26], [27]). Recently, planar actuators utilizing stationary sheets of diamagnetic material (high purity graphite) interacting with permanent magnet movers to generate levitation, in combination with stationary coil sets to provide actuation in the XY plane, have been presented for very small scale movers (such as 2mm by 2mm by 0.4mm) in [28] [29] [30] (Figure 1.6). This planar motion system has the potential to be run entirely without position feedback as all degrees of freedom are passively stable, and can be actuated over very large planar strokes potentially meters-long. However, the low force density of diamagnetic levitation makes scaling up to large payload masses problematic [27]. 11   Figure 1.6 Planar motor for very small scale movers (such as 2mm by 2mm by 0.4mm) based on diamagnetic levitation and Lorentz force translation, figure adapted from [28]. (a) Top view of mover magnets. (b) Electromagnetic configuration of diamagnetic planar levitation motor.   For 6-DOF magnetic actuation using materials with 𝜇 ≥ 𝜇𝑜, dynamic feedback stabilization is required in at least one axis. 6-DOF direct drive stages based on single-axis voice coil type Lorentz force drives have been presented in [31], [32], [33]. Verma et. al has demonstrated a 6-DOF stage using multiple 2-DOF Lorentz force motors [34]. While highly linear in terms of force characteristics and thus suitable for very high precision applications, these stage topologies are limited to small positioning ranges.  In this thesis we study two categories of long stroke 6-DOF planar levitation direct drive stages: i) synchronous machines where the magnetic field of the stator travels synchronously with the magnetic field of the mover; and ii) asynchronous machines where a changing magnetic field 12  interacts with a conducting body to produce an induction field in the conductor that resists and repels the applied dynamically changing magnetic field producing force on the conductor and dynamic field source. The use of permanent magnets in both classes of motor allows for greater force densities compared to pure current-based electromagnets.  The advantage of permanent magnet 6-DOF synchronous levitation motors are the potential for good power efficiency, making this class of motor well suited to high precision applications where thermal variation is a significant issue, as well as vacuum-environment compatibility. The disadvantages are i) position feedback is required for stability and proper synchronous commutation; and ii) coil topologies can become complex or difficult to manufacture and drive, making scaling to ultra-long planar strokes problematic.  The advantages of 6-DOF asynchronous planar levitation motors are i) that the conducting body has a very simple homogenous structure (in the case where the conducting body is the stator, the planar stroke of the stage can be extended with minimal cost and complexity); and ii) all axes are passively stable, and therefore position feedback is not a requirement for stable levitation and actuation. The biggest disadvantage to asynchronous levitation is power efficiency even with the use of permanent magnets, as there are significant Ohmic power losses incurred by the induced eddy currents in the conducting body. This makes this class of levitation motor suitable for applications requiring ultra-long strokes with lower precision positioning requirements.   1.4.1 Long Stroke Synchronous Magnetic Planar Levitation Motors Kim and Trumper demonstrated a large stroke synchronous 6-DOF direct drive planar stage demonstrated 50×50×0.4 mm XYZ stroke and small rotations with a 300×300 mm mover in [35]. This synchronous planar levitation motor used four sets of 2-DOF motor elements comprised of 13  Halbach array linear motors and stationary coils based on the Lorentz force for both levitation and translation (shown in Figure 1.7). This design is sufficient for scanning microscopy applications [36]; however, most industry applications require a motion stage with planar strokes on the order of several hundred millimetres or more. Comparing the motor topology in Figure 1.8a and Figure 1.8b, it’s clear that in order to scale the planar stroke (𝑆𝑥 × 𝑆𝑦), the mover size (𝐿 × 𝐿) must increase in proportion. Simultaneously, the magnet fill factor as a proportion of the mover area decreases meaning lower force densities relative to mover weight and thus lower maximum accelerations for a given current density limit. Achieving >100 mm stroke requires either a mover on the order of meters [37] or this drive system can only be used as a fine motion stage on top of another long-stroke coarse stage.    Figure 1.7 Kim’s 6-DOF synchronous levitation motor, figure adapted from [35]. 14   Figure 1.8 Scaling up planar stroke of Kim and Trumper’s synchronous levitation motor.  A moving-coil long-stroke planar motor for photolithography has been presented in [38] and [39]. The basic design is similar to that shown in Figure 1.9, with a set of moving 2D coils actuated above a stationary 2D magnet array. The planar stroke only depends on the size of the stationary magnet array, and therefore very high mover area to working area ratios can be achieved. It is capable of meters-long strokes in X and Y, and can achieve accelerations of several g’s with appropriate forced cooling of the moving coils. Such a design can be successfully industrialized and used in advanced lithography machines [4] [40] [41]. However, the stage requires cooling fluid and electrical power lines attached to the mover which causes significant position dependent disturbance forces. In addition, the stator is a large exposed permanent magnet array, which limits the stage to a few niche applications. In addition, the centre of magnetic pressure on each coil set shifts with the mover XY position, creating a position dependent torque acting on the mover. This 15  is the result of synchronous locking of the magnetic field of the moving coils to the magnetic field of the stationary bed of magnets as illustrated in Figure 1.10 with an example 3-phase moving coil levitation motor. Individual force vectors (green arrows) act at the centre of each coil, producing a net levitation force (black arrow) acting at the centre of magnetic pressure for each planar position. As the coils traverse in the XY plane, the centre of magnetic pressure shifts, imparting a position dependent torque to the mover.   Figure 1.9 Moving coil synchronous planar levitation motor (umbilical cable to mover not shown), figure adapted from [38]. Red magnet blocks are magnetized in +z, and blue magnet blocks are magnetized in –z.  16   Figure 1.10 Shifting center of pressure of example 3-phase moving coil levitation motor, given a constant levitation force command (umbilical cable not shown).  Cable-free long-stroke 6-DOF direct drive planar motion stages have been presented in [42] [43] and [44]. These existing solutions are essentially based on a 2D chessboard-type magnet array on the mover, actuated by a magnetic field generated by a stationary array of 2D patterned coils. Binnard [42] uses multiple layers of racetrack type coils with alternating orthogonal orientations interacting with a 2D magnet array, similar to that shown in Figure 1.11. Compter et. al [44] uses a single layer of racetrack coils with zones of alternating orientation as seen in Figure 17  1.12a. Jansen et. al [45] uses  a single layer of herringbone pattern coil arrays (Figure 1.12b), and explicitly excites coil arrays inside the mover area and away from the edges of the magnet array in order to minimize magnet array edge effects. These moving magnet synchronous levitation motors have several significant advantages over moving coil topologies [38] and Kim and Trumper’s synchronous levitation motor [35]: i) the passive mover doesn’t require power or a cooling umbilical cable; ii) forced cooling architecture for the stationary coils is easier to implement as the heat source is stationary; iii) the planar stroke can be extended much larger than the size of the mover itself, meaning a greater utilization of the machine volume. However, these 2D moving magnet/ 2D coil array designs have several disadvantages as well: high system complexity, poor coil fill factor, coil end effects, and actuating force and torque variation with stage position. Extensive research effort has been expended on modelling and compensation schemes for these intrinsic force and torque disturbances, for example as presented in [43] [46] [47] [48] [49] [50] [51]. In addition, the number of coils and drives required scales quadratically with the planar stroke (or linearly with the work area): the motion stage in Figure 1.13b has twice the XY stroke of the motion stage in Figure 1.13a but four times the number of individual coils. This significantly and in some cases prohibitively raises the manufacturing, drive and control complexity of the stage as the planar stroke increases.    18   Figure 1.11 Moving magnet 2D chessboard type synchronous planar levitation motor with multiple alternating layers of racetrack style coils; figure adapted from [42]. Red magnet blocks are magnetized in +z, and blue magnet blocks are magnetized in –z.   Figure 1.12 Moving magnet 2D chessboard type synchronous planar levitation motors. (a) Alternating zones of racetrack coils, figure adapted from [44]. (b) Herringbone coil pattern, figure adapted from [45]. 19   Figure 1.13 Scaling up XY stroke of moving magnet 2D chessboard type synchronous planar levitation motor (example using motor type similar to [42]).  Ueda and Ohsaki presented the design, analysis and simulation of a 6-DOF levitation stage using a 2D moving magnet array and stationary sets of straight coils arranged in alternating layers oriented in X- and Y-axes (similar to a proposal by Kim in appendix A, page 319 of [52]), and also at a skew angle to both planar axes [53]. The X-oriented straight conductors actuate the mover in the Y- and Z-directions, and the Y-oriented conductors actuate the mover in the X- and Z-directions. The skewed coil layer is used to help mitigate the torque coupling with other axes. This stage has the potential for high coil fill factor due to the straight 1D coil array patterns, and for large strokes and ease of scalability since the number of individual coil drives does not scale 20  quadratically with planar stroke. However, the 2D magnet array itself is quite complex to manufacture, and there is significant torque coupling between axes. In addition, in the presented design coils that do not produce force are necessarily active meaning the power efficiency of the motor is compromised. No experimental prototype demonstrating full 6-DOF actuation has been disclosed yet.      Figure 1.14 Moving magnet 2D chessboard type synchronous planar levitation motor with alternating layers of straight coils; figure adapted from [53].  A moving magnet long stroke 6-DOF stage has recently been presented in [54], using an array of stationary circular type coils interacting with cylindrical magnets attached to a mover 21  (Figure 1.15). The prototype stage has achieved 80×80×25 mm XYZ stroke with ± 40 degrees and ± 15 degrees rotations about the in-plane XY axes, and infinite rotation about the Z-axis. A significant advantage of this stage compared to other synchronous levitation stages is the very large rotation range combined with large planar strokes achievable. However, the force characteristic over the work volume is highly position dependent and non-linear, making control and commutation complex. Additionally, since the number of coils is linear with the planar area of the stage, scaling up the planar stroke presents similar issues to the moving magnet synchronous levitation stages using arrays of 2D racetrack or circular type coil arrays.    Figure 1.15 6-DOF moving magnet stage with axially magnetized cylindrical magnets over array of circular coils; figure adapted from [54].  22  In Chapter 2, we present a new long-stroke 6-DOF direct drive synchronous planar levitation motor solution comprised of sets of 1D magnet arrays and stationary 1D arrays of straight coils. This novel stage has the following advantages over existing long stroke 6-DOF asynchronous planar levitation motors:  (1) the mover can be much smaller than the planar stroke (ratio of working planar area to mover area is very high) ;  (2) the moving magnet topology requires no cables to the moving body;  (3) the number of coils increases linearly with stroke (versus quadratically in existing moving magnet 6-DOF stage solutions) and thus can scale easily to meters-long XY strokes with cm range Z-stroke;  (4) the coil terminals exit at the perimeter of the stator, greatly simplifying the system;  (5) coil fill factor can be very high and thus large force density is achievable; and  (6) it is simple to control/commutate and has superior force linearity due to freedom from coil end effects and permanent magnet field edge effects.  1.4.2 Long Stroke Asynchronous Magnetic Planar Levitation  Linear induction motors without permanent magnets have long been studied in the context of generating single axis forces alone [55] but can also be used to generate multi-DOF planar motion [56]. A 2-DOF circular type planar motor using electromagnetic coils and a back-iron was presented in [57] (Figure 1.16) that was shown to be extendable to a long stroke planar motor using linear induction motor elements. A 3-DOF asynchronous XY planar motor using linear electromagnets and a back-iron has been presented in [56], similar to that shown in Figure 1.17. These planar motors still require an out-of-plane bearing due to the field interactions with the back-23  iron, which generates an attractive force that overcomes the repulsive levitation force developed by the dynamic field in the conductor for reasonable driving current densities. The back-iron is required in these types of planar motors to increase magnetic field flux densities and thus increase thrust force densities in the XY plane.  Figure 1.16 Circular type 2-DOF induction planar motor, figure adapted from [57].   Figure 1.17 Asynchronous planar motor using electromagnets and back-iron, figures adapted from [56] (out-of-plane bearings not shown). (a) Linear 1-DOF induction motor. (b) 3-DOF configuration of asynchronous planar motor. 24  To generate levitation forces, permanent magnets can be used as the primary dynamic field in the conductor, eliminating the requirement for a back-iron to increase field densities. The Inductrack concept was proposed in [58] that used a linear permanent magnet Halbach array attached to a mover translating over a set of close-packed shorted coils to generate a repulsive levitation force counterbalanced by gravity (Figure 1.18). By using permanent magnets as the primary dynamic field source instead of coils, they were able to demonstrate very high force densities but still require an additional mechanism for propulsion to lift-off speeds.    Figure 1.18 Asynchronous levitation motor using permanent magnet Halbach array, adapted from [58].  Previously, [7] presented the concept of a permanent magnet disk rotated mechanically over a homogenous conducting body (Figure 1.19a), producing both levitation force and drag torque that are functions of the rotation speed of the disk. In this thesis we refer to this type of magnetization pattern as circumferential. The drag torque was advantageously used to simultaneously produce a propulsion force by tilting the magnet disk rotation axis or by partial 25  overlap of the magnet disk and the conducting body. This 3-DOF motor element can be used as the basis of a 6-DOF asynchronous planar levitation motor as demonstrated in [3].  A commercial venture based on this concept was recently started as seen in [8] [9]. Another permanent magnet based induction motor element has been proposed that uses a radial pattern magnet wheel rotating over a conducting body (Figure 1.19b) which is also capable of simultaneous levitation and propulsion, and is self-stable when combined with a split-sheet conducting guideway [59] [60]. However, the radial magnet wheel has a lower force density compared to the circumferentially-patterned magnet disk in [7] because only a portion of the strong side of the permanent magnet field is close to the conductor. In this thesis we therefore confine ourselves to the rotating circumferentially-magnetized magnet disk similar to that found in [7].   Figure 1.19 Permanent magnet asynchronous levitation motor elements showing example magnetization patterns. (a) Circumferential magnet disk. (b) Radial magnet wheel. 26   Much of the electrodynamic force modeling of induction effects for both magnet disks and magnet wheels is carried out numerically using finite element type methods [59] [61] [62] [63] [64] [60]. While such numerical methods have been shown to be accurate to within a few percent, a closed form analytical solution for the forces and torques produced on the magnet disk is a very useful tool for informing the optimal design of the magnet disk and conducting slab, and establishing physical understanding and intuition as to motor working principles.  Many analytical models for linear induction motors have been given in the literature. The levitation and drag force expressions for an infinitely thick and infinitely wide conducting slab with a time- and space-varying surface current as the primary field source over an infinitely permeable back iron was given in [65]. The analytical model for a linear permanent magnet Halbach array of finite horizontal extent over a uniform conductor was presented in [66], though we note that an expression for the field penetration depth into the conductor (a critical parameter for this model) is not given. A semi-analytical model was presented for eddy current damping generated by cuboidal magnets over a finite conducting plate [67]; this model assumes that the conducting plate thickness is small relative to the skin depth effect and therefore the field fully penetrates the conductor for all speeds. The analytical expression for the field in the air gap of a linear induction motor with a back iron was given in [68], but only applies to current-based dynamic field sources.  Lubin and Rezzoug [69] have presented a 3D analytical field, force and torque model for an axial flux eddy current torque coupling. The coupling is comprised of a circumferential permanent magnet disk separated by an air gap from a conducting plate, with some relative rotational velocity between the two bodies. The permanent magnet disk has an iron yoke backing, as does the conducting plate. The purpose of the iron yokes is to maximize the flux density acting 27  through the conductor. Lubin and Rezzoug extrapolate from [70] where it was shown that the curvature of the magnet disk generates second order effects and can therefore be neglected as a first approximation. They accordingly unwrap the magnet disk into an equivalent linear asynchronous machine, and solve for the field distribution using the 3D magnetic vector potential in Cartesian coordinates. By assuming infinite relative permeability of the yoke material, they are able to set the Dirichlet boundary conditions at the edge of the air gap as perfect magnetic boundaries (i.e. no field extends beyond the planar extents of the iron yokes) and solve the boundary value problem. Similarly, Koo et. al. [71] used the presence of back iron yokes in an axial flux permanent magnet disk synchronous rotary machine to set the magnetic boundary conditions in the radial direction, and used the 3D magnetic vector potential in cylindrical coordinates to solve for the no load magnetic flux distribution. The iron yokes in both the asynchronous torque coupler and the synchronous rotary motor are well designed for the particular application: induced drag torque in the eddy current coupler is maximized at reasonable slip speeds [69], and the synchronously generated torque is similarly amplified in [71]. However, iron yokes always produce an attractive force between magnet disk and yoke acting against the direction of the induced levitation force. Therefore for applications where we desire to levitate the magnet disk using asynchronously generated fields, the presence of any iron would be detrimental to generation of lifting forces at reasonable power levels and rotational velocities. Therefore neither of the solutions from [69] or [71] can be used for a practical magnet disk asynchronous levitation motor. A closed form analytical solution for Fujii et. al’s 2-DOF circular type planar motor was presented in [57], but is not applicable to permanent magnet-based primary field sources, has an iron yoke and also requires that the area covered by the dynamic field source (the eddy current region) be much smaller than the entire disk. This is non-ideal as it limits the achievable force 28  density and acceleration since the ratio of force generation area to total mover area is small. Park et al. in [72] presented an analytical model for force and drag torque on a rotating permanent magnet disk over a homogeneous conducting stator without iron yokes. This model applies only to conductors of infinite thickness, and uses experimentally derived correction factors on the motor pole pitch and stator conductivity in order to fit the analytical model to experimental results. As a design tool, this empirical approach is limited as extensive mapping of correction factors is required by either finite element analysis or (in Park et al.’s approach) building and testing different magnet array disks.   In Chapter 3, we derive a novel analytical expression for the levitation force and drag torque acting on a rotating circumferentially-patterned permanent magnet disk over a stationary homogeneous conducting slab that takes as input only the geometric and material parameters of the permanent magnet disk and conducting slab. Conceptually this method can be extended to modeling propulsion as a function of tilt angle and rotation speed similar to that presented in [7].  29  Chapter 2: Synchronous Planar Levitation Stage  We present in this chapter a novel long-stroke 6-DOF synchronous planar levitation motor which has the following advantages over existing synchronous levitation motors: (1) the number of coils increases linearly with stroke (versus quadratically in existing solutions) and thus can scale easily to meters-long X-Y strokes with cm range Z stroke; (2) the coil terminals exit at the perimeter of the stator, greatly simplifying the system mechanical and electrical connection design; (3) coil fill factor can be very high and thus large force densities are achievable; (4) it is easy to control/commutate and has superior force linearity due to freedom from coil end effects and field edge effects; (5) a minimal machine footprint relative to its working volume. It utilizes multiple 1D magnet arrays (where the permanent magnet field varies along a single axis) and multiple 1D coil arrays (where the current distribution varies along a single axis) to produce 6-DOF motion, making the design, analysis, modeling, and control much simpler than the synchronous planar levitation motors based on 2D magnet arrays and 2D coil arrays. The prototype presented here demonstrates a range of 260 mm by 60 mm, with a mover of 185 mm per side, as shown in the video [73]. This chapter consists of the following parts:  Synchronous planar levitation motor concept, topology and working principle.  Detailed force and torque modeling for 2-DOF motor element.  Magnet array designs for attenuation of intrinsic force and torque ripples.  Design and fabrication details for alpha prototype of 6-DOF synchronous planar levitation motor.  Experimental motion tracking results from prototype. 30   Experimental load test results for magnet array design for force ripple attenuation.  2.1 Synchronous Levitation Concept and Working Principle Figure 2.1 shows the concept of a novel 6-DOF direct drive motor, which is composed of a stator coil assembly and a mover. The mover travels on top of the stator coil assembly, with a controllable air gap between the two parts. Instead of a 2D magnet array, four 1D magnet arrays (X1, Y2, X3, and Y4) are installed on the bottom surface of the mover. The force actuation principle of the planar levitation stage is the interaction between the magnetic fields of the four magnet arrays bonded to the mover chassis and zones of excitation current in the stationary 1-D coil array to produce a Lorentz force acting on the mover. Each magnet array is a Halbach pattern with a fundamental spatial wavelength of 𝜆. The coil array is made of stacks of alternating layers of X-oriented and Y-oriented conductors, with end turns and coil connections made on the outside perimeter of the working planar range of the motor. Each conductor can be excited individually. The active coils (shown in colours) carry currents, while inactive coils (in grey) have no current. The zones of active coils will change dynamically based on the X and Y location of the mover.  Figure 2.1 shows the set of conductors that act on each of the four magnet arrays and the generated forces. As the mover traverses the planar range, each excitation zone (X1, X3, Y2 and Y4 active coils) moves synchronously with the corresponding magnet array. This method of force actuation means that the coils can generate a magnetic field wave that can continuously travel in the XY plane.    31   Figure 2.1 Synchronous planar levitation 6-DOF motor electromagnetic topology.  Figure 2.2 shows a cross-section of the Y2 magnet array. The Y2 magnet array has the basic pattern of a traditional 4-segment-per- 𝜆 Halbach array, composed of both vertically and horizontally magnetized elements. Generally, the advantages of a Halbach array over an array of only vertically magnetized elements include 41% higher magnetic field strength and fewer higher order harmonic fields (no 3rd, 7th, or even harmonics) [74]. The active Y-coils are immersed in the magnetic field below the magnet array, and a Lorentz force is generated on the coils from the interaction between the excitation current and the magnetic field. A force acts on the magnet array at the location of the excitation current, leading to both a force and a torque generated on the Y2 magnet array. The active Y-coils thus generate a force vector acting on the Y2 magnet array in the XZ plane and a torque in the Y-direction. The excitation current through each conductor is 32  commutated to generate two independent forces that comprise a single force vector (𝐹𝑋2 ?̂? + 𝐹𝑍2?̂?) in the XZ plane, where ?̂? and ?̂? are unit vectors in the X and Z directions, respectively. We note here that the active coils extend beyond the edges of the finite width magnet array in the X-direction further than shown in the schematic drawing in Figure 2.2. In section 2.2, this drive methodology will be shown to be a key aspect of the motor linearity.   Figure 2.2 Y2 magnetic array and coils cross-section  This force vector (𝐹𝑋2 ?̂? + 𝐹𝑍2?̂?) on magnet array Y2 is independent of the mover’s position in the Y-direction, because the active Y-coils have a uniform structure in the Y direction. As the magnet array width is designed as an integer number of 𝜆, the active coils in the orthogonal direction (X direction) have both zero interacting force on Y2 magnet array and moreover no net torque is developed by the orthogonal coils (X-coils) on the magnet array Y2 because the Halbach array pattern is symmetric with respect to the geometric center of the array as shown in [75].  It is made mirror-symmetric about its vertical middle axis by setting the width of the end magnet elements to /8 instead of the /4 used in conventional Halbach array designs. In addition, the Y-elongated magnet arrays Y2 and Y4 are spaced apart in the X direction to significantly reduce the 33  coupling between the Y2 magnet array and Y4 active coils. These features enable Y2 to mainly interact with Y2 active coils, minimizing the coupling with active coils Y4, X1, and X2 that are commutated for other magnet arrays. Due to the natural force decoupling between the magnet arrays, motion control design can be greatly simplified. Thus the magnet array Y2 and its corresponding active coil zone combine to form a 2-DOF motor element.   Similarly, magnet arrays X1 and X3 work with the X coils to generate forces (𝐹𝑌1, 𝐹𝑍1) and (𝐹𝑌3, 𝐹𝑍3) respectively; and the magnet array Y4 produces forces (𝐹𝑋4 , 𝐹𝑍4). The four Z-forces combined can be used to adjust the air gap between the motor and the stator and control the mover’s pitch and tilt. The X and Y forces combined control 2D translation and yaw. Thus the 8 independently controlled forces enable 6-DOF motion control of the mover, allowing over-actuation of the mover in translation (𝑥, 𝑦, 𝑧) and Euler angles (𝛼, 𝛽, 𝛾) (rotation around 𝑥, 𝑦, 𝑧, respectively).   Figure 2.2 also shows the cross-section of the stator coil assembly with multiple layers of X and Y coils interlaced. This coil array can be made cost-effectively with well-established printed-circuit board (PCB) manufacturing technology. Insulator layers (FR4 core or prepreg) are inserted between coils. The coils in the same direction on different layers can be connected in parallel or series, depending on the via design and end termination. Multiple PCBs can be laterally connected side by side in both X and Y directions (similar to floor tiles) to form a larger planar coil array. The board-to-board lateral connections (in X and Y) are made at the edges only by connecting pads or through-holes of adjacent boards. As a result, all the end terminals of the entire planar coil array are at the outermost perimeter for ease of wiring to the drive electronics. Therefore, this design allows the planar motor to be easily extended in both X and Y directions for various applications while the number of required coils increases linearly with the in-plane stroke, 34  instead of quadratically as in 2D-chessboard type prior art designs. The coil end effects are eliminated, because the current on each layer only changes orientation to return at the outermost perimeter, outside of the active positioning range of the mover.  2.2 Magnetic Field, Force and Torque Generation Modeling for 2-DOF Motor Element The goal of this section is to obtain an analytical model for force and torque generation on the 2-DOF motor element comprised of the magnet array Y2 and its corresponding active coil zone. This force and torque model is applied to all four 2-DOF motor elements and a general plant model is developed in the prototype design section 2.4.  We are not necessarily interested in describing the exact distribution of electromagnetic fields at every point in space but rather the net force and torque generated on each 2-DOF motor element from the interaction of the permanent magnet field and the current in the stator. A harmonic field model was presented in [52]. This harmonic field model applies to periodic magnet arrays of infinite horizontal extent and sufficient depth as to render the problem 2D; that is, without considering edge effects or 3D leakage. We desire to use this harmonic model in our case where we have finite width magnet arrays and cannot neglect edge effects out of hand, without losing accuracy or generality. We prove the equivalence between force generated between an infinitely extended repeating magnet array and a single coil, and the force generated between a finite magnet array and a repeating coil pattern, which we term the principle of equivalent force generation. We further show that the extent of the active coil zone for the Y2 magnet array allows us to approximate the infinite repeating coil array case. We thereafter use a partial harmonic field model to derive the force on a Halbach array from its interaction with the current distribution in the active coil zone.  35   This section is comprised of the following parts: 1) General 3D magnetic field distribution for a cuboidal magnet, showing how the exact spatial field distribution, while accurate, gives us little insight into the operating principles of the motor. 2) Force and torque generated by interaction of a permanent magnet field and current distribution. 3) Principle of Equivalent Force Generation, which allows us to use the simplified harmonic model of field distribution for an infinitely extended repeating magnet array under certain driving conditions. 4) The harmonic field model derivation for a Halbach array of infinite horizontal extent. 5) Force and torque generation expressions which we will use to generate commutation algorithms for 6-DOF motor control.  2.2.1 3D Magnetic Field Distribution for Cuboidal Magnet Several analytical models for permanent magnets have been presented in [43] including the magnetic surface charge model and harmonic model. Here we use the analysis for cuboidal permanent magnets based on the fictitious magnetic charge method shown in [76] and [77]. All electromagnetic field quantities for the synchronous planar levitation motor can be modelled under the magnetoquasistatic (MQS) assumption as we can assume that for synchronous driving conditions all such fields will propagate very quickly throughout the volume of the planar stage relative to the time rate of change of driving current and magnet array position for practically sized stages [78]. Under the MQS assumption, the differential form of Maxwell’s equations can be written as: 36   𝛻 × 𝑬 = −𝜕𝑩𝜕𝑡 (2.1)   𝛻 × 𝑯 = 𝑱 (2.2)   𝛻 ∙ 𝜖𝑜𝑬 = 𝜌  (2.3)   𝛻 ∙ 𝑩 = 0  (2.4)  and  𝑩 = 𝜇𝑜(𝑯 + 𝑴)  (2.5)  Here 𝑬,𝑯,𝑩 denote the electric field [𝑉/𝑚], magnetic field intensity [𝐴/𝑚] and magnetic field density [𝑇] vectors in Cartesian coordinates respectively; 𝑱 is the current density vector; 𝜌 is the volumetric charge density; 𝜇𝑜 is the permeability of free space and is equal to 4𝜋 × 10−7 [𝐴/𝑚]; and 𝜖𝑜 is the permittivity of free space and is equal to 8.854 × 10−12 [𝐹/𝑚]. The magnetization 𝑴 [𝐴/𝑚] represents the macroscopic effects of many microscopic magnetic dipoles. Further, the current induced in a body is described by    𝑱 = 𝜎(𝑬 + 𝒗 × 𝑩) (2.6)  where 𝜎 is the conductivity of the material, 𝒗 is the relative velocity of the medium with respect to the 𝑩 distribution. A cuboidal 3D magnet segment is shown in Figure 2.3 with a coordinate system (𝑥, 𝑦, 𝑧) centered inside the body of the permanent magnet which has dimensions (2𝑎, 2𝑏, 2𝑐). For this 37  analysis the relative permeability of the entire volume (both inside and outside the magnet) is equal to one, which is a reasonable estimation of the relative permeability of rare earth magnets with 𝜇𝑟 ≈ 1.01 − 1.05. A uniform z-directed magnetization 𝑴 = 𝑀𝑜?̂? is given where the magnetization amplitude is 𝑀𝑜 = 𝐵𝑟/𝜇𝑜 and 𝐵𝑟 is the magnet remanence. Using the magnetic surface charge method [76], [78], a fictitious magnetic surface charge everywhere can be defined as:  𝝈 = −∇ ∙ 𝜇𝑜𝑴 (2.7)   Figure 2.3 Cuboidal permanent magnet with vertical (z-directed) uniform magnetization M shown in red.   Since the magnetization is uniform inside the permanent magnet, it follows that it is divergence free within the magnet volume and only the top and bottom surface of the cuboidal magnet will have magnetic charge. The magnetic surface charge at the top surface is 𝜎𝑡𝑜𝑝 = 𝜇𝑜𝑀𝑜 and the 38  magnetic surface charge at the bottom surface is 𝜎𝑏𝑜𝑡𝑡𝑜𝑚 = −𝜇𝑜𝑀𝑜. In the absence of currents, the curl of the field intensity 𝑯 is zero and can be solved using the scalar magnetic potential:  𝑯 = −∇𝜓 (2.8)  From [78], with only the magnetic surface charges as sources, the scalar potential can be formulated as   𝜓(𝒓) = ∫𝜎(𝒓)𝑑𝑣4𝜋𝜇𝑜|𝒓|𝑉 (2.9)  where 𝒓 is a point in the (𝑥, 𝑦, 𝑧) frame centered on the magnet body. Thus from (2.7), (2.8) and (2.9) the field distribution can be analytically solved as:  𝐵𝑥 =𝐵𝑟4𝜋∑∑∑(−1)𝑖+𝑗+𝑘𝐿𝑛(𝑅 − 𝑇)1𝑘=01𝑗=01𝑖=0 (2.10)   𝐵𝑦 =𝐵𝑟4𝜋∑∑∑(−1)𝑖+𝑗+𝑘𝐿𝑛(𝑅 − 𝑆)1𝑘=01𝑗=01𝑖=0 (2.11)   𝐵𝑧 =𝐵𝑟4𝜋∑∑∑(−1)𝑖+𝑗+𝑘𝑎𝑡𝑎𝑛2(𝑆𝑇𝑅𝑈)1𝑘=01𝑗=01𝑖=0 (2.12)  where the magnetic flux density vector is  𝑩𝒂𝒏𝒂𝒍𝒚𝒕𝒊𝒄𝒂𝒍 = 𝐵𝑥 ?̂? + 𝐵𝑦𝒋̂ + 𝐵𝑧?̂?, and  𝑅 = √𝑆2 + 𝑇2 + 𝑈2 (2.13)   𝑆 = 𝑥 − (−1)𝑖𝑎 (2.14)   𝑇 = 𝑦 − (−1)𝑗𝑏 (2.15)  39   𝑈 = 𝑧 − (−1)𝑘𝑐 (2.16)  and atan2 is a four quadrant arctangent operator [77]. Since the problem is stated as linear with uniform relative permeability everywhere, we can obtain the full field distribution for a cuboidal magnet pattern by simply translating and rotating the field for each cuboidal magnet segment and using superposition to obtain the net magnetic field. Thus we can solve for the exact spatial distribution of any pattern of cuboidal magnet segments.  As an example, we take a Halbach-patterned magnet array as shown in Figure 2.4a made up of five cuboidal magnet segments with X- and Z-directed magnetizations only and finite extensions in each direction. The Halbach pattern has a repeating pattern every 𝜆, the fundamental spatial wavelength for the magnet array. Figure 2.4b shows the magnetization pattern with segment dimensions. Each cuboidal magnet segment is assumed to be uniformly magnetized in the direction shown within the magnet volume. This is a close approximation of the real case, however, in reality the local magnetization will be influenced by the magnetic fields generated by the neighbouring magnet segments (the so-called ‘bucking field’). Setting magnet remanence 𝐵𝑟 = 1.325 [𝑇], array width 𝑊𝑚 =  𝜆 and array depth 𝐷𝑚 = 2𝜆, the analytical field 𝑩𝒂𝒏𝒂𝒍𝒚𝒕𝒊𝒄𝒂𝒍 is calculated as per (2.10)-(2.12). Figure 2.5 shows the vector component of flux density 𝑩𝒂𝒏𝒂𝒍𝒚𝒕𝒊𝒄𝒂𝒍 at 𝑧𝑚 = −𝜆/5. As a partial validation of the analytical 3D field model, commercial physics modeling software COMSOL in 3D mode [79] is used to determine the 3D magnetic field for the same magnet array via finite element analysis (FEA) yielding 𝑩𝑭𝑬𝑨. The difference between the analytical model and FEA is shown in Figure 2.6, with error defined as  𝑩𝑭𝑬𝑨 − 𝑩𝒂𝒏𝒂𝒍𝒚𝒕𝒊𝒄𝒂𝒍. The worst case error as a percentage of the maximum amplitude of the analytical field for this  𝑧𝑚-level is 1.1% for  𝐵𝑥, 0.48% for 𝐵𝑦, and 1.37% for 𝐵𝑧. The solutions converge with finer meshing in the finite element 40  model; however due to computational hardware limits the FEA mesh was not further refined. The analytical model coded in the MATLAB environment gave a result in 7.8 seconds on an Intel Core i7-3770 CPU at 3.4 GHz, 12 GB of DDR2 RAM, running Windows 7 64bit. The FEA model took approximately 3 hours to yield a solution on the same computer.   Figure 2.4 Example 3D finite extension Halbach-patterned magnet array (magnetization directions shown in red). (a) 3D isometric view of finite Halbach array. (b) 2D view of finite Halbach array showing segment dimensions. 41   Figure 2.5 Analytical field model, flux density vector components of 𝑩𝑭𝑬𝑨 for example magnet array at 𝒛𝒎 =−𝝀/𝟓.  42   Figure 2.6 Error in flux density vector components for 𝒛𝒎 = −𝝀/𝟓. Error defined as 𝑩𝑭𝑬𝑨 − 𝑩𝒂𝒏𝒂𝒍𝒚𝒕𝒊𝒄𝒂𝒍.  43   We therefore have a 3D analytical field model that captures essentially all characteristics of the spatial distribution of any cuboidal magnet pattern given uniform permeability through the solution space and a known magnetization distribution. Here we note that this analytical model was determined for magnetizations that are parallel with two faces of the cuboidal magnet geometry.  However, despite having an exact analytical expression for the field distribution for a finite Halbach array with a very significant improvement in solution time compared to 3D finite element models, we gain little design insight into the operation of the motor. From the summed analytical expression for the total field it is non-trivial to determine a commutation law for multi-DOF force and torque actuation of the Halbach array using only this form of the field distribution [77]. Sensitivity analyses can be carried out on each parameter to determine an optimized motor design [80]. We will show in later sections that the topology and driving method of this synchronous levitation motor lends itself to a simplified analysis for force and torque generation.  In the next section we determine the force and torque generated from interaction between a current distribution and the magnetic field from a general magnet array.    2.2.2 Force and Torque Generation on Magnet Array Once again assuming the relative permeability of all materials in the motor is close to unity, the force generated on the magnet array from the interaction of the current distribution in the stator 𝑱 and the magnetic field from the Halbach array can be calculated from the Lorentz force expression alone [81]. As illustrated in Figure 2.7, the force acting on the magnet array due to a differential conductor volume with current density 𝑱 is  44   𝒅𝑭𝒎𝒂𝒈𝒏𝒆𝒕 = −(𝑱 × 𝑩𝒇)𝑑𝑣 (2.17)  Here 𝑩𝒇 is the magnetic field due to the magnet array alone, 𝑱 is the applied current density in the stator and 𝑑𝑣 is the volume of the conductor element. The total force on the magnet array is therefore the integration of (2.17) over the entire conductor volume 𝑉𝑐:  𝑭𝒎𝒂𝒈𝒏𝒆𝒕 = −∭ (𝑱 × 𝑩𝒇𝑉𝑐)𝑑𝑣 (2.18)  The total torque on magnet array is similarly  𝑻𝒎𝒂𝒈𝒏𝒆𝒕 = −∭ 𝒑 × (𝑱 × 𝑩𝒇𝑉𝑐)𝑑𝑣 (2.19)  where 𝒑 is the moment arm vector between the magnet array center of gravity and the location of the differential conductor volume.  Figure 2.7 General 3D magnet array interacting with differential conductor element.  45  2.2.3 Principle of Equivalent Force Generation From section 2.2.1 and the force and torque expressions (2.18) and (2.19) it can be seen that solving for the magnetic field distribution for a 3D Halbach array (or any repeating pattern magnet array) with finite extension in each dimension yields a complex and non-intuitive analytical expression for the magnetic field spatial distribution, which must then be integrated across the volume of the conductor. While this integration is at least numerically solvable, obtaining the exact spatial distribution for such a magnet array is not the primary goal of this analysis nor is it the most useful magnetic field model for design insight into motor operation. Rather, we are interested in the net force and torque generated on the array via the interaction of a repeating pattern current distribution and the permanent magnet field from the Halbach array. In this sub-section we show that the force generated on a λ-wide Halbach magnet array by a coil array repeating with the same period of λ is equivalent to the force generated on an infinitely wide Halbach magnet array interacting with a single coil driven by the same current. We later extend this analysis to examine the effect of different spatial wavelengths for the repeating coil pattern versus the width of the magnet array.   An example Halbach magnet array is shown in Figure 2.8, with equivalent dimensions to those shown in Figure 2.4 and x-directed width 𝑊𝑚 = 𝜆 (hereafter referred to as the 𝜆-width magnet array). Note the magnetization pattern is flipped compared to the array in Figure 2.4 though the strong field side remains below the magnet array (𝑧𝑚 < 0). Figure 2.9 shows a Halbach magnet array with the same spatial wavelength and repeating magnetization pattern, with infinite x-directed extension 𝑊𝑚 = ∞ (hereafter referred to as the ∞-width magnet array); other dimensions are equal to the 𝜆-width array in Figure 2.8.  46   Figure 2.8 Finite x-directed extension Halbach-patterned magnet array.    Figure 2.9 Infinite x-directed extension Halbach-patterned magnet array  The z-directed field component 𝐵𝑧 at the test line shown in Figure 2.8 (𝑧𝑚 = −𝜆/5, 𝑦𝑚 = 0) is calculated for the 𝜆-width magnet array using (2.12) for a 𝑥𝑚 ∈ [−3𝜆, 3𝜆] (with 𝐵𝑧 falling to 47  0.0152% of the max|𝐵𝑧| at these boundaries). In order to calculate 𝐵𝑧 over the same x-range for the ∞-width magnet array, we note that magnetic field values converge far away from the edges of the magnet array as 𝑊𝑚 increases. That is, 𝐵𝑧 for 𝑊𝑚 = 500𝜆 is that same as that for 𝑊𝑚 =1000𝜆 over 𝑥𝑚 ∈ [−3𝜆, 3𝜆] to within 2.1 × 10−5 ppm (relative to the maximum amplitude of |𝐵𝑧|). Therefore for the x-range of interest, we consider 𝑊𝑚 = 1000𝜆 equivalent to a true ∞-width magnet array.  Figure 2.10 shows 𝐵𝑧 for both the 𝜆-width and ∞-width magnet arrays. The 𝜆-width magnet array clearly shows edge/leakage effects and no strong periodicity. The ∞-width magnet array has a repeating pattern with a fundamental wavelength of 𝜆. At 𝑧𝑚 = −𝜆/5, the field from the ∞-width magnet array is almost entirely a single fundamental. This in no way reduces the generality of the force analysis which applies to force generated by coils at any 𝑧𝑚.  Figure 2.10 𝑩𝒛 from 3D analytical model along test line (𝒛𝒎 = −𝝀/𝟓, 𝒚𝒎 = 𝟎).  48  For the synchronous planar levitation motor 2D forcer, the force generation case is a repeating pattern of y-directed coils immersed in the magnetic field of a finite width magnet array. In Figure 2.11a, a single λ-wide Halbach magnet array M0 interacts with a group of y-directed coil traces Ck (k = −∞,… ,−2,−1,0,1,2,… ,∞) which repeat every λ and extend from 𝑦𝑚 ∈ (−∞,∞). The coils are simple line elements and have zero size in the XZ plane, reducing the force expression (2.18) to a single spatial integration along the y-axis. The x-locations of the coil traces in the magnet frame are 𝑥𝑚 = 𝛿 for coil C0, 𝑥𝑚 = 𝛿 + 𝜆 for coil C1, 𝑥𝑚 = 𝛿 − 𝜆 for coil C−1, and so forth. Correspondingly, assuming that the z-directed flux density distribution from the magnet array M0 is 𝐵𝑧(𝑥𝑚) in Figure 2.11a, the field at each coil location is 𝐵𝑧(𝛿) for coil C0, 𝐵𝑧(𝛿 + 𝜆) for coil C1, 𝐵𝑧(𝛿 − 𝜆) for coil C−1, and so forth. As a result, if each coil Ck is driven by the same current 𝑖 [𝐴] the total x-directed force as determined from the x-component of (2.18) on the magnet array is   𝐹𝑥,0 = −∫ ∑ 𝐵𝑧(𝛿 + 𝑘𝜆) ∞𝑘=−∞𝑖 𝑑𝑦∞−∞ (2.20)  The order of operations for the discrete summation and continuous integration are interchangeable. Figure 2.11b shows an alternate case, with a single line-element coil C0 interacting with an infinitely extended magnet array which is the combination of the magnet array M0 and its duplicates Mk (k = −∞,… ,−2,−1,0,1,2, … ,∞) which repeat every 𝜆. The magnet array M0 and the coil  C0 in Figure 2.11b are at identical locations to those in Figure 2.11a. The magnetic field from magnet array M0 at the location of  C0 is 𝐵𝑧(𝛿), from magnet array M1 at the location of C0 is 𝐵𝑧(𝛿 + 𝜆), from magnet array M−1 at the location of C0 is 𝐵𝑧(𝛿 − 𝜆), and so forth. Therefore the total field at the location of C0 is the superposition of the fields from each magnet 49  array, 𝐵𝑧(𝛿 − ∞) + ⋯+ 𝐵𝑧(𝛿 − 𝜆) + 𝐵𝑧(𝛿) + 𝐵𝑧(𝛿 + 𝜆) + ⋯+ 𝐵𝑧(𝛿 + ∞). The corresponding x-directed force on the magnet array is therefore   𝐹𝑥,∞ = −∫ 𝑖 ∑ 𝐵𝑧(𝛿 + 𝑘𝜆) ∞𝑘=−∞𝑑𝑦∞−∞ (2.21)  Comparing the forces (2.20) and (2.21), the x-directed force between a group of coil traces spaced 𝜆 apart in 𝑥𝑚 and a single 𝜆-wide magnet array is equivalent to the x-directed force between a single coil trace and an infinitely wide magnet array with a magnetization pattern that repeats every 𝜆. It can be similarly shown that the z-directed force is also equivalent between the two cases. It is clear that the repeating pattern magnet array in Figure 2.11b is exactly equivalent to the ∞ -width magnet array as shown in Figure 2.11c. Since the field from the 𝑊𝑚-wide magnet array goes to zero very rapidly as 𝑥𝑚 → ±∞, the contribution from coils far away from the edges of the magnet array in Figure 2.11a don’t contribute much to the total force on the magnet array. The field is significantly attenuated (below 1% of its maximum value as shown in Fig. 11 for the λ-wide array) beyond λ/2 from the edges of the magnet array. Practically, only coils immediately under the magnet array or extending beyond the edge of magnet array less than λ/2 need to be excited in order to consider the magnetic field as purely periodic from the point of view of net force generation. We therefore make the approximation that so long as the extent of the active coil zone for each magnet array is at least λ/2 from the edges of the magnet array in 𝑥𝑚, the  principle of equivalent force generation applies. The force equivalence analysis thus far has been for the special case of a repeating coil pattern with a spatial periodicity equal to the width of the magnet array, which is in turn an integer value of the Halbach fundamental wavelength 𝜆. A more general case is shown in Figure 2.12a, where the magnet array width is 𝑊𝑚 and the coil array spatial periodicity is 𝜂𝑐. For 𝜂𝑐 ≤ 𝑊𝑚, the 50  superposition method implies that each identical magnet array will have some overlap 𝜀 where 𝜀 =𝑊𝑚 − 𝜂𝑐 as shown schematically in Figure 2.12b. As determined in the motor topology section 2.1, the width 𝑊𝑚 of the magnet array should be constrained to an integer value of the Halbach spatial period, and the coil pattern should repeat with a spatial wavelength of 𝜆 for maximum force generation. Therefore, a well-designed coil and magnet array will be constrained to 𝑊𝑚 = 𝑛𝜆 and 𝜂𝑐 = 𝜆 where 𝑛 is preferred to be a real positive integer. From these constraints a general force equivalence can be obtained where a Halbach magnet array of width 𝑊𝑚 = 𝑛𝜆 interacting with a repeating coil array with spatial wavelength 𝜂𝑐 = 𝜆 has the same force as a single coil interacting with 𝑛- ∞-width magnet arrays superimposed in the same location. Thus the force expression (2.21) becomes  𝐹𝑥,∞ = −𝑛∫ 𝑖 ∑ 𝐵𝑧(𝛿 + 𝑘𝜆) ∞𝑘=−∞𝑑𝑦∞−∞ (2.22)  and  𝐹𝑧,∞ = 𝑛 ∫ 𝑖 ∑ 𝐵𝑥(𝛿 + 𝑘𝜆) ∞𝑘=−∞𝑑𝑦∞−∞ (2.23)  where 𝑛 = 𝑊𝑚/𝜂𝑐. Therefore the force on the 2D motor element can be determined by analyzing the magnetic field of an ∞-width magnet array interacting with a single coil and multiplying by the magnet array width factor 𝑛 = 𝑊𝑚/𝜂𝑐. We note here that for non-integer 𝑛, while the mean force generated from the interaction with a repeating coil array will still be proportional to 𝑛, additional force harmonics will be introduced which may have to be addressed depending on the performance required. For the general case, the practical excitation zone remains one the order of 𝜆/2 away from the edges as the decay rate in 𝑥𝑚 is dependent on the Halbach period and not the width of the magnet array. 51  We conclude that the net force generated on the 2-DOF motor element can be appropriately analyzed by collapsing a group of coil traces interacting with a finite width magnet array to a single coil interacting with an infinitely extended magnet array. It remains to determine an appropriate and useful analytical model of the field from an ∞-width magnet array. 52   Figure 2.11 Equivalent force cases. (a) 𝝀-width magnet array interacting with 𝝀-repeating coil array. (b) repeating 𝝀-width magnet arrays with single coil. (c) ∞-width magnet array with single coil.  53   Figure 2.12 (a) 𝑾𝒎-width magnet array interacting with 𝜼𝒄-repeating coil array. (b) Superposition showing overlap of each identical magnet array.  54  2.2.4 Magnetic Field Modeling for Infinitely Extended Halbach Array Sub-section 2.2.3 established that modeling the net force on the 2D motor element as a series of single coils interacting with an infinitely extended array is appropriate so long as the excitation zone extends sufficiently beyond the edges of the finite magnet array. In this sub-section we develop a simple analytical model to analyze the strong-side field of an infinitely extended Halbach array. This analysis is based on the original work carried out in [52] where the harmonic model for a general magnet array was presented. Kim formed the problem as 2D in the XZ plane, assuming that the depth of the magnet array was sufficient to neglect leakage effects. Here we begin with the 2D assumption and compare the resulting harmonic model to the exact 3D analytical model integrated over 𝑦𝑚 ∈ (−∞,∞) for an infinitely extended array as presented in section 2.2.3.  Though we assume no current in any region, in Chapter 3 we will use this analysis as a starting point to determine the force from the induction levitation effect on the asynchronous planar levitation motor, with induced currents in the stator region. We therefore solve field distributions using the vector magnet potential rather than with the scalar magnet potential because unlike the scalar magnet potential the vector magnet potential applies to each region regardless of the presence of currents, i.e. where the curl of the field intensity 𝑯 is not zero. The field density distribution must satisfy Gauss’ law for magnetic field (2.4) everywhere, i.e. the field density 𝑩 is everywhere solenoidal (divergence free). One solution is   𝑩 = [𝐵𝑥𝐵𝑦𝐵𝑧] = ∇ × 𝑨 (2.24)  where 𝑨 is the vector magnet potential in Cartesian coordinates. This automatically satisfies Gauss’ law (2.4) since the divergence of the curl of a vector field is always zero. Plugging into Ampere’s law (2.2), and substituting field intensity using (2.24), we obtain 55   ∇ × ∇ × 𝑨 = 𝜇𝑜(𝑱 + ∇ × 𝑴) (2.25)  Using the vector identity ∇ × ∇ × 𝑨 = ∇(∇ ∙ 𝑨) − ∇2𝑨 and setting the Coulomb gauge ∇ ∙ 𝑨 = 0 (i.e. the vector potential is divergence free, which is correct for the MQS assumption), we obtain the vector Poisson’s equation:  ∇2𝑨 = −𝜇𝑜(𝑱 + ∇ × 𝑴) (2.26)  This vector Poisson’s equation is equivalent to three scalar Poisson’s equations. From the Biot-Savart law and the definition of the vector magnet potential (2.24), it is possible to write the magnetic vector potential in integral form in the absence of free surface currents as [52] :  𝑨 =𝜇𝑜4𝜋∫𝑱 + ∇ × 𝑴𝒓 − 𝒓′𝑑𝑣 +𝜇𝑜4𝜋∫𝑴 × 𝒏𝒓 − 𝒓′𝑑𝑠𝑆𝑉 (2.27)  Here the volume 𝑉 is enclosed by the surface 𝑆, the source location is 𝒓′ and the observer location is 𝒓. The second term in (2.27) insures continuity of the vector potential across the boundary between permanent magnet and non-permanent magnet. Boundary conditions are given by (2.2), (2.3) and (2.4), and the continuity of the vector magnet potential across each boundary. Figure 2.13 shows the solution volume broken up into three regions: region I-the volume of air on the top (weak side for a Halbach array) of the magnet array, region II-the permanent magnet array with magnetization 𝑴 = 𝑀𝑥 ?̂? + 𝑀𝑧?̂? which has only 𝑥𝑚-dependence within the magnet volume (this is assumed to be rare earth magnetic material and therefore 𝜇𝑟 ≅ 1), and region III- the air gap between magnet array and current distribution. In the 2D limiting case the vector potential 𝑨 has only a y-directed component, 𝑨 = 𝐴𝑦𝒋.̂ For the general magnetization 𝑴 and in the absence of currents, the vector Poisson equation (2.26) reduces to the scalar equation 56   (∂2∂2𝑥+∂2∂2𝑧)𝐴𝑦 = 𝜇𝑜(−∂𝑀𝑥∂𝑧+∂𝑀𝑧∂𝑥) (2.28)    Figure 2.13 2D form of the infinite extension magnet array.   The magnetization distribution 𝑴 can be written as a sum of Fourier coefficients as described by [52]. The field due to each magnetization harmonic can be solved separately and summed for the total field solution, and therefore the vector potential can be treated similarly. Therefore in each region we can assume that the form of the vector potential will satisfy the usual Laplace function criterion that the vector potential will be the multiplication of functions that depend only on a single coordinate:  𝐴𝑦 = 𝐹(𝑥𝑚)𝐺(𝑦𝑚)𝐻(𝑧𝑚) (2.29)  57  Given a sinusoidal variation in two of the functions, the third must be dependent on hyperbolic trigonometric functions.  For a four-segment per spatial wavelength Halbach array of infinite width in the x-direction, the magnetization distribution is shown in Figure 2.14 [74]. The gap 𝑔 is introduced between magnet elements to maintain generality and allow for later analysis of the effects of practical arrays. The magnetization amplitude is 𝑀𝑜 = 𝐵𝑟/𝜇𝑜 where 𝐵𝑟 is the magnet remanence. We define the characteristic harmonic wavelength 𝜆𝑐𝑘 = 𝜆/𝑘2𝜋 where 𝑘 is the harmonic wave number with 𝑘 > 1, and fundamental characteristic wavelength 𝜆𝑐 = 𝜆/2𝜋. The Fourier coefficients are calculated as   𝑀𝑥𝑘 =1𝜆∫ 𝑀𝑥𝑒−𝑗𝑥𝑚/𝜆𝑐𝑘𝑑𝑥𝑚𝜆/2−𝜆/2 (2.30)   𝑀𝑧𝑘 =1𝜆∫ 𝑀𝑧𝑒−𝑗𝑥𝑚/𝜆𝑐𝑘𝑑𝑥𝑚𝜆/2−𝜆/2 (2.31)  In the remaining analysis we solve the field for the fundamental component of magnetization. All field components can be similarly solved by substituting harmonic magnetization components calculated as per (2.30), (2.31). The total field is the summation of all field harmonics and the fundamental. The magnetization fundamental harmonic in region II is   𝑴𝟏 = [𝑀𝑥10𝑀𝑧1] = 𝑅𝑒 [𝑗𝑀1𝑒−𝑗𝑥𝑚/𝜆𝑐0𝑀1𝑒−𝑗𝑥𝑚/𝜆𝑐] (2.32)  with the fundamental spatial wavelength 𝜆 and the magnetization amplitude  𝑀1 =4𝑀𝑜𝜋cos (𝜋4+𝑔2𝜆𝑐) (2.33)  58   Figure 2.14 Magnetization distribution within magnet volume for ∞-width Halbach array with 4 magnet segments/𝝀 separated by gap 𝒈.   Since the Poisson equation (2.28) reduces to a pure Laplace equation in region I, the fundamental component of vector potential in region I can have the form:  𝐴𝑦1 = 𝑅𝑒([𝑓𝐼𝑒𝑧𝑚𝜆𝑐 + 𝑔𝐼𝑒−𝑧𝑚𝜆𝑐 ]𝑒−𝑗𝑥𝑚/𝜆𝑐)  (2.34)  Similarly in region III, the vector potential has the form: 59   𝐴𝑦1 = 𝑅𝑒([𝑓𝐼𝐼𝐼𝑒𝑧𝑚𝜆𝑐 + 𝑔𝐼𝐼𝐼𝑒−𝑧𝑚𝜆𝑐 ]𝑒−𝑗𝑥𝑚/𝜆𝑐)  (2.35)  Since the potential must go to zero at  𝑧𝑚 = −∞,∞, 𝑓𝐼 = 𝑔𝐼𝐼𝐼 = 0. In region II, the Poisson equation (2.28) is driven by the magnetization term. Since the magnetization is given as uniform in 𝑧𝑚 within the magnet volume, (2.28) further reduces to   (∂2∂2𝑥+∂2∂2𝑧)𝐴𝑦1 = 𝜇𝑜∂𝑀𝑧1∂𝑥 (2.36)  The solution will have a homogenous part which solves (∂2∂2𝑥+∂2∂2𝑧)𝐴𝑦1ℎ = 0 and allow us to fit boundary conditions and a particular solution driven by (∂2∂2𝑥+∂2∂2𝑧)𝐴𝑦1𝑝 = 𝜇𝑜∂𝑀𝑧1∂𝑥. Since 𝑀𝑧1 has the x-dependence 𝑒−𝑗𝑥𝑚/𝜆𝑐 , the particular solution must satisfy  (∂2∂2𝑥+∂2∂2𝑧)𝐴𝑦1𝑝 =−𝑗𝜇𝑜𝜆𝑐𝑀𝑧1 (2.37)  From [52] the vector potential along each boundary in region II has the form    𝐴𝑦1 = 𝐴𝑦1𝑝 + (𝐴𝑦1,𝑏 − 𝐴𝑦𝑝)𝑠𝑖𝑛ℎ (𝑧𝑚𝜆𝑐)𝑠𝑖𝑛ℎ (𝐻𝑚𝜆𝑐)   − (𝐴𝑦1,𝑐 − 𝐴𝑦1𝑝)𝑠𝑖𝑛ℎ (𝑧𝑚 − 𝐻𝑚𝜆𝑐 )𝑠𝑖𝑛ℎ(𝐻𝑚𝜆𝑐)    (2.38)  where we assume that 𝐴𝑦1 has the x-dependence 𝑒−𝑗𝑥𝑚/𝜆𝑐, and 𝐴𝑦1,𝑏 and 𝐴𝑦1,𝑐 are the potentials along the 𝑏- and 𝑐- boundaries respectively. The particular solution 𝐴𝑦1𝑝 is obtained by solving (2.37) and noting that since 𝐴𝑦1𝑝 must also have the x-dependence 𝑒−𝑗𝑥𝑚/𝜆𝑐, then ∂2𝐴𝑦1𝑝∂2𝑥=−1𝜆𝑐𝑘2 𝐴𝑦1𝑝. Further, since the magnetization is uniform in z within the magnet volume, the particular solution is  60   𝐴𝑦1𝑝 = 𝑗𝜇𝑜𝜆𝑐𝑀𝑧1 (2.39)  From (2.24), the field components as a function of the vector potential are  𝑩 = [𝐵𝑥𝐵𝑦𝐵𝑧] = 𝑅𝑒[    −∂𝐴𝑦∂𝑧0∂𝐴𝑦∂𝑥 ]     (2.40)  Using the definition of 𝐴𝑦1 in each region, and the definition of flux density (2.43), the value of field densities at each boundary are obtained:   𝑩𝒂 = 𝑅𝑒[    1𝜆𝑐𝑔𝐼𝑒−𝐻𝑚/𝜆𝑐𝑒−𝑗𝑥𝑚/𝜆𝑐0−𝑗𝜆𝑐𝑔𝐼𝑒−𝐻𝑚/𝜆𝑐𝑒−𝑗𝑥𝑚/𝜆𝑐]     (2.41)    𝑩𝒃 = 𝑅𝑒[    1𝜆𝑐{−(𝐴𝑦1,𝑏 − 𝐴𝑦1𝑝) 𝑐𝑜𝑡ℎ (𝐻𝑚𝜆𝑐) + (𝐴𝑦1,𝑐 − 𝐴𝑦1𝑝)𝑐𝑠𝑐ℎ (𝐻𝑚/𝜆𝑐)}0−𝑗𝜆𝑐𝐴𝑦1,𝑏 ]     (2.42)    𝑩𝒄 = 𝑅𝑒[    1𝜆𝑐{−(𝐴𝑦1,𝑏 − 𝐴𝑦1𝑝) 𝑐𝑠𝑐ℎ (𝐻𝑚𝜆𝑐) + (𝐴𝑦1,𝑐 − 𝐴𝑦1𝑝)𝑐𝑜𝑡ℎ (𝐻𝑚/𝜆𝑐)}0−𝑗𝜆𝑐𝐴𝑦1,𝑐 ]     (2.43)    𝑩𝒅 = 𝑅𝑒[    −1𝜆𝑐𝑓𝐼𝐼𝐼𝑒−𝑗𝑥𝑚/𝜆𝑐0−𝑗𝜆𝑐𝑓𝐼𝐼𝐼𝑒−𝑗𝑥𝑚/𝜆𝑐]     (2.44)  61  Boundary continuity conditions are derived from the continuity of vector potential, field continuity due to Gauss’ law (2.4), and from (2.2) and (2.5) which in the absence of currents yields  ∇ × 𝑩 = 𝜇𝑜∇ × 𝑴. (2.45)  Boundary conditions are summarized as  𝐵𝑥,𝑎 − 𝐵𝑥,𝑏 = −𝜇𝑜𝑀𝑥1,𝑏 (2.46)   𝐵𝑧,𝑎 − 𝐵𝑧,𝑏 = 0 (2.47)   𝐵𝑥,𝑐 − 𝐵𝑥,𝑑 = 𝜇𝑜𝑀𝑥1,𝑐 (2.48)   𝐵𝑧,𝑐 − 𝐵𝑧,𝑑 = 0 (2.49)   𝐴𝑦,𝑎 = 𝐴𝑦,𝑏 (2.50)  and  𝐴𝑦,𝑐 = 𝐴𝑦,𝑑 (2.51)  where the subscript denotes vector amplitude direction and the boundary location. In addition, the field 𝑩 as well as the vector potential 𝑨 go to zero at 𝑧𝑚 = −∞,∞. 𝑀𝑥1,𝑏 and 𝑀𝑥1,𝑐 are the fundamental components of magnetization in the 𝑥𝑚-direction along the 𝑏- and 𝑐- boundaries respectively. We are particularly interested in region III below the magnet array, as this is where the stator coils will be located. Solving the system of equations (2.41)-(2.44) using the boundary 62  conditions and relations (2.46)-(2.51), the fundamental component of magnetic field in region III is found to be    𝑩𝑰𝑰𝑰,𝟏 = 𝑅𝑒[     −𝑗𝜇𝑜𝑀1 (1 − 𝑒−𝐻𝑚𝜆𝑐 ) 𝑒𝑧𝑚/𝜆𝑐𝑒−𝑗𝑥𝑚/𝜆𝑐0𝜇𝑜𝑀1 (1 − 𝑒−𝐻𝑚𝜆𝑐 ) 𝑒𝑧𝑚/𝜆𝑐𝑒−𝑗𝑥𝑚/𝜆𝑐]      (2.52)  For the following sections, the primary harmonics of interest are those present in 𝑩𝑰𝑰𝑰, in particular the fundamental harmonic with characteristic wavelength 𝜆𝑐 =𝜆2𝜋 and the fifth harmonic with characteristic wavelength 𝜆𝑐5 =15𝜆2𝜋  As discussed in [52], for a four-segment-per- 𝜆 magnet array with an infinite number of periods, the magnetic field can be modelled as the sum of a base component at 1 cycle-per-λ (cpλ) and harmonic components at 5, 9, and 13 cpλ. Further harmonics can be added depending on the accuracy desired. We note that at 𝑧𝑚 = −𝜆/60, the 13th field harmonic amplitude is less than 0.1% of the fundamental component, and that the harmonics decay faster as a function of the harmonic number. From (2.52) the fundamental flux density is  𝑩𝑰𝑰𝑰,𝟏 = [−𝐵1𝑒𝑧𝑚/𝜆𝑐1𝑠𝑖𝑛 (𝑥𝑚/𝜆𝑐)0𝐵1𝑒𝑧𝑚/𝜆𝑐1𝑐𝑜𝑠 (𝑥𝑚/𝜆𝑐)] (2.53)  where 𝐵1 =4𝐵𝑟cos (𝜋4+𝑔2𝜆𝑐)𝜋(1 − 𝑒−𝐻𝑚𝜆𝑐 ). Repeating the above analysis for the fifth harmonic of magnetization, the 5 cpλ component of flux density is  𝑩𝑰𝑰𝑰,𝟓 = [𝐵5𝑒𝑧𝑚/𝜆𝑐5𝑠𝑖𝑛 (𝑥𝑚/𝜆𝑐5)0−𝐵5𝑒𝑧𝑚/𝜆𝑐5𝑐𝑜𝑠 (𝑥𝑚/𝜆𝑐5)] (2.54)  where 𝐵5 =4𝐵𝑟cos (𝜋4+𝑔2𝜆𝑐5)5𝜋(1 − 𝑒−𝐻𝑚𝜆𝑐5). We note that the full field is the summation of all harmonic contributions plus the fundamental 63   𝑩𝑰𝑰𝑰 = 𝑩𝑰𝑰𝑰,𝟏 + ∑ 𝑩𝑰𝑰𝑰,𝒌∞𝑘=2 (2.55)  We use this result in the following sections to determine both the force coefficient and force ripple due to the fundamental field component and the fifth field harmonic. We first derive the force ripple due to a single field harmonic and later generalize to force ripple due to all other harmonics.   2.2.5 Force Due to Single-Phase Repeating Coil Group  Now that we have determined an appropriate analytical model for the field from an ∞-width Halbach magnet array, we can determine the force on the 𝑊𝑚-magnet array shown in Figure 2.15 by the group of coils identically driven by the current 𝐼𝑎 and spaced apart by 𝜂𝑐 = 𝜆 (implying that the current distribution in the coil array has a spatially periodic current distribution). Each coil trace has width 𝑊𝑐 and thickness 𝑡𝑐, with the top surface located at 𝑧𝑚 = −𝑧𝑓. For this analysis we assume the depth of the magnet array is sufficient to ignore leakage flux and the integral expression ∫ 𝐼𝑎𝑑𝑦𝑚∞−∞≅ 𝐷𝑚𝐼𝑎.   Figure 2.15 Single phase coil array below finite width Halbach array.   64  From the general Lorentz force (2.18) and the equivalent force expressions (2.22) and (2.23), combined with the field expression for the ∞-width magnet array (2.55), the total force from the total field on the magnet array due to the group of coils is   𝑭𝒂 =𝑊𝑚𝜂𝑐𝐷𝑚 ∫ ∫ (𝐽𝑎𝒋̂ × 𝑩𝑰𝑰𝑰)𝑑𝑥𝑚𝑑𝑧𝑚𝑊𝑐/2−𝑊𝑐/2−𝑧𝑓−(𝑧𝑓+𝑡𝑐) (2.56)  where the current density is 𝐽𝑎 =𝐼𝑎𝑊𝑐𝑡𝑐[𝐴𝑚2]. Note that for the case shown in Figure 2.15,  𝑊𝑚𝜂𝑐= 2.  As we will show section 2.2.6, the fundamental component of 𝑩𝑰𝑰𝑰 contributes to the desired mean force generation on the mover and all other field harmonics interact with stator currents to generate undesired force and torque ripples. In particular, the 5-cpλ field harmonic combined with a 3-phase coil pattern produces a 6-cpλ force ripple. Theoretically, the higher harmonic field components can be eliminated by using continuously varying Halbach magnet arrays as described in [74]; however, these are not practical to manufacture. To develop a means to minimize the effect from harmonic field components, here we use a partial harmonic model which includes both the fundamental component and a 5th harmonic component. Thus, given a repeating coil pattern, for the purpose of deriving the net force on the magnet array the spatial variation of the magnetic field below the magnet array can be approximated as the partial harmonic model 𝑩𝑷𝑯𝑴 = 𝑩𝑰𝑰𝑰,𝟏+𝑩𝑰𝑰𝑰,𝟓 for 𝑧𝑚 < 0 for the 2D limiting case:  𝑩𝑷𝑯𝑴 =[   −𝐵1𝑒𝑧𝑚𝜆𝑐 𝑠𝑖𝑛𝜃+ 𝐵5𝑒5𝑧𝑚𝜆𝑐 𝑠𝑖𝑛(5𝜃) 0𝐵1𝑒𝑧𝑚𝜆𝑐 𝑐𝑜𝑠𝜃 −𝐵5𝑒5𝑧𝑚𝜆𝑐 𝑐𝑜𝑠(5𝜃) ) ]   . (2.57)  Here we introduce the electrical angle 𝜃 =𝑥𝑚𝜆𝑐1[𝑟𝑎𝑑] and 𝐵1, 𝐵5 are defined as in (2.53), (2.54). Higher order spatial harmonics decay faster than the fundamental harmonic as 𝑧𝑚 decreases (the decay rate being proportional to 𝑘). For a change in 𝑧𝑚 from 𝑧𝑚 = −𝜆/60   to 𝑧𝑚 = −𝜆/5, the 65  fundamental field harmonic decreases by 68%, whereas the 5th field harmonic decreases by over 99%. Figure 2.16a shows the comparison between this PHD and the exact 3D analytical model (3DAM) solution for an ∞-width magnet array with depth 𝐷𝑚 = 20𝜆 evaluated at 𝑦𝑚 = 0 (middle of the array in 𝑦𝑚). The error is defined as PHD-3DAM. The field at 𝑧𝑚 = −𝜆/60  is highly non-sinusoidal, while at 𝑧𝑚 = −𝜆/5  the fundamental harmonic dominates. The maximum modelling error at 𝑧𝑚 = −𝜆/60 is 11.2% relative of the fundamental component, and this error is dominated by the 9th, 13th and 17th harmonics as shown in Figure 2.16b, with 6% due to the 9th harmonic alone. At 𝑧𝑚 = −𝜆/5, the maximum modelling error falls to 0.0006% relative to the fundamental component, with the dominant error harmonic at 9- cpλ making up essentially all of it. This is the result of the higher harmonics decaying faster than the 9th harmonic. If required, at lower flying heights, 9th, 13th and 17th field harmonics can be added into the PHM to reduce modeling error. However, as shown in the following analysis, with a 3-phase coil arrangement the dominant force components are due to the fundamental and 5th field harmonics for practical stator coil geometries with the effects of the 9th field harmonic entirely eliminated, and therefore the analytical field model is usefully simplified to only these harmonics.  66   Figure 2.16 Partial harmonic model versus 3D analytical model at different 𝒛𝒎 (a) 𝑩𝒛, partial harmonic model (PHD), 3D analytical model (3DAM) evaluated at 𝒚𝒎 = 𝟎 with 𝑫𝒎 = 𝟐𝟎𝝀, and error (PHD-3DAM), for ∞-width magnet arrays. (b) Fourier coefficients for error at different 𝒛𝒎.  67  The exact force expression (2.56) can be approximated using the PHM field as  𝑭𝒂,𝑷𝑯𝑴 =𝑊𝑚𝜂𝑐𝐷𝑚 ∫ ∫ (𝐽𝑎𝒋̂ × 𝑩𝑷𝑯𝑴)𝑑𝑥𝑚𝑑𝑧𝑚𝑊𝑐/2−𝑊𝑐/2−𝑧𝑓−(𝑧𝑓+𝑡𝑐) (2.58)  Evaluating the spatial integrations across the coil geometry, the field on the magnet array is therefore   𝑭𝒂,𝑷𝑯𝑴 = 𝐼𝑎𝑊𝑚𝜂𝑐[−𝛾1𝑐𝑜𝑠𝜃𝑒𝑧𝑚𝜆𝑐 + 𝛾5 𝑐𝑜𝑠(5𝜃) 𝑒5𝑧𝑚𝜆𝑐 }0−𝛾1 𝑠𝑖𝑛 𝜃 𝑒𝑧𝑚𝜆𝑐 + 𝛾5 𝑠𝑖𝑛(5𝜃) 𝑒5𝑧𝑚𝜆𝑐] (2.59)  where magnet and coil geometry are combined in the factors  𝛾1 = 𝐵1𝐷𝑚 𝑠𝑖𝑛𝑐 (𝑊𝑐2𝜆𝑐) (𝜆𝑐𝑡𝑐) (1 − 𝑒−𝑡𝑐/𝜆𝑐) (2.60)   𝛾5 = 𝐵5𝐿𝑦𝑠𝑖𝑛𝑐 (5𝑊𝑐2𝜆𝑐1) (𝜆𝑐15𝑡𝑐) (1 − 𝑒−5𝑡𝑐/𝜆𝑐1) (2.61)  The torque developed on the finite magnet array is more complex as there is no analogue to the equivalent force generation principle for torque; rather the exact field for the finite width array as calculated by the 3DAM must be used to determine the torque from each coil individually and summed to find total torque:   𝑇𝑦(𝑥𝑚, 𝑧𝑚) = ∑ 𝑟𝑎,𝑝⃑⃑ ⃑⃑ ⃑⃑  × 𝐹𝑎,𝑝 ⃑⃑ ⃑⃑ ⃑⃑  ⃑  ∞𝑝=−∞ (2.62)  where 𝑟𝑎,𝑝⃑⃑ ⃑⃑ ⃑⃑  is the individual moment arm from the center of gravity of the magnet array to the center of the current distribution of coil trace 𝑝 (assumed to be at the center of the coil trace for all practical driving conditions)  as shown in Figure 2.15, and the force 𝐹𝑎,𝑝 ⃑⃑ ⃑⃑ ⃑⃑  ⃑  is the individual force from coil trace 𝑝 on the magnet array. 68  2.2.6 Force Due to 3-Phase Repeating Coil Group with Commutated Current From (2.59), a constant current 𝐼𝑎 with the magnet array traversing the X-direction produces a rotating force vector in the XZ plane with fundamental and 5th spatial harmonics. Noting that the force expression is sinusoidal with a fundamental wavelength 𝜆 (a result that was impossible to determine using only the exact field distribution model 3DAM for the finite width array), a commutation law for the coil group can be determined. Since the force on the magnet array is modulated by the sinusoidally varying field, in order to generate a specified force as the mover traverses the XZ plane, the current 𝐼𝑎 is modulated to have the form  𝐼𝑎 = −𝐼𝑥𝑟 𝑒−𝑧𝑚𝜆𝑐 𝑐𝑜𝑠 𝜃 − 𝐼𝑧𝑟𝑒−𝑧𝑚𝜆𝑐 𝑠𝑖𝑛 𝜃 (2.63)  where 𝐼𝑥𝑟 and 𝐼𝑧𝑟 represent the force commands on the magnet array for the 𝑥  and 𝑧  directions respectively. This commutation law is function of the coil’s location (𝑥𝑚, 𝑧𝑚) in the magnet frame; accordingly, another active coil trace located an integer number of 𝜆 away in the x-direction will be excited with the same current. By changing the current 𝐼𝑎 sinusoidally with electrical angle  𝜃 such that the fundamental harmonic of current is synchronous with the mover’s permanent magnet field at the location of the conductor, 𝐼𝑎 will demodulate the permanent magnet field to produce a useful position independent or mean force, as well as 2, 4 and 6 cpλ force ripples when we consider only the fundamental and 5th harmonics of field. To obtain the maximum mean force, it is clear that the coil array spatial period of 𝜂𝑐 must match the magnet array spatial period 𝜆. The mean force and the 2 cpλ force ripple are from the modulation of the fundamental harmonic of the magnetic field with the fundamental spatial harmonic of excitation current. The 4 and 6 cpλ force ripples in force are due to the modulation of the 5th field harmonic with the fundamental harmonic in the excitation current. If the force ripple due to the 9th field harmonic were calculated, it would 69  produce 8 and 10 cpλ force ripples. The majority of the force ripples are eliminated using a modified 3-phase coil configuration as shown in Figure 2.17 with currents through additional conductors commutated as  𝐼𝑏 = −𝐼𝑥𝑟𝑒−𝑧𝑚𝜆𝑐 𝑐𝑜𝑠 (𝜃 −𝜋3) − 𝐼𝑧𝑟𝑒−𝑧𝑚𝜆𝑐 𝑠𝑖 𝑛 (𝜃 −𝜋3) (2.64)   𝐼𝑐 = −𝐼𝑥𝑟𝑒−𝑧𝑚𝜆𝑐 𝑐𝑜𝑠 (𝜃 −2𝜋3) − 𝐼𝑧𝑟𝑒−𝑧𝑚𝜆𝑐 𝑠𝑖 𝑛 (𝜃 −2𝜋3) (2.65)  and 𝐼𝑎′ = −𝐼𝑎, 𝐼𝑏′ = −𝐼𝑏 , 𝐼𝑐′ = −𝐼𝑐. This multi-phase coil pattern approximates a sinusoidal current distribution with six steps perλ. In particular, the 2, 4, 8 and 10 cpλ force ripples are canceled out in this configuration. In each magnet spatial period 𝜆, six individual coil traces contribute to force generation as shown in Figure 2.17. Because the magnetic field components in x- and z-directions are electrically orthogonal (90 degrees apart in electrical angle), this commutation law allows for the fully independent actuation of a translation (x-directed) and levitation (z-directed) force. The mean position independent force generated is therefore  𝑭𝒐 = 𝐾0 [𝐼𝑥𝑟0𝐼𝑧𝑟] (2.66)  where the motor force constant is 𝐾0 = 3𝑊𝑚𝜂𝑐𝛾1. Thus the vector force in (2.66) is the desired force used for 6-DOF motion actuation. However, due to the existence of the 5th harmonic field from the permanent magnet array, an undesirable force ripple of 6 cpλ is also produced. By applying the commutation method (2.63)-(2.65), this force ripple can be calculated as  𝑭𝟔 = 𝐾6𝑒4𝑧𝑚𝜆𝑐 [−𝐼𝑥𝑟 𝑐𝑜𝑠(6𝜃) − 𝐼𝑧𝑟 𝑠𝑖𝑛(6𝜃)0−𝐼𝑥𝑟 𝑠𝑖𝑛(6𝜃) + 𝐼𝑧𝑟 𝑐𝑜𝑠(6𝜃)]] (2.67)  70  where 𝐾6 = 3𝑊𝑚𝜂𝑐𝛾5. Other force ripple vectors due to field harmonics 𝑩𝒌 can be determined similarly.    Figure 2.17 Modified 3-phase coil array below finite width Halbach array.    The practical coil stator array is a set of coil traces stacked in 𝑧𝑚. Figure 2.18 shows the general case with the trace spacing in the z-direction 𝑡𝑔. As the ‘gap factor’ (the field decay term from 𝑩𝟏), is of the form 𝑒𝑧𝑚/𝜆𝑐1, the total mean force for 𝑁 coil arrays spaced 𝑡𝑔 apart is simply   𝑭𝒐𝑵 = 𝐾01 − 𝑒−𝑁(𝑡𝑐+𝑡𝑔)/𝜆𝑐1 − 𝑒−(𝑡𝑐+𝑡𝑔)/𝜆𝑐 [𝐼𝑥𝑟0𝐼𝑧𝑟] (2.68)  Similarly the total 6 cp𝜆 force ripple is   𝑭𝟔𝑵 = 𝐾6𝑒4𝑧𝑚𝜆𝑐1 − 𝑒−4𝑁(𝑡𝑐+𝑡𝑔)/𝜆𝑐1 − 𝑒−4(𝑡𝑐+𝑡𝑔)/𝜆𝑐 [−𝐼𝑥𝑟 𝑐𝑜𝑠(6𝜃) − 𝐼𝑧𝑟 𝑠𝑖𝑛(6𝜃)0−𝐼𝑥𝑟 𝑠𝑖𝑛(6𝜃) + 𝐼𝑧𝑟 𝑐𝑜𝑠(6𝜃)]] (2.69)  We note that given a continuously varying current sheet, rather than the discretely distributed 3-phase coil arrangement analyzed above, the magnetic field produced by the current distribution 71  would be perfectly locked to the magnet array position and therefore the magnet array would see no variation in magnetic pressure with motion in 𝑥𝑚. A continuously varying current distribution is impractical to manufacture, but increasing the number of phases will decrease the position dependent force component at the expense of current driver complexity. The net torque depends on the net force combined with the location of the center of magnetic pressure. This center depends partly on the discrete nature of the coil distribution, which causes a spatial variation of the center of pressure dependent on the coil pitch (for the 3-phase arrangement, 𝜆/6). As a first approximation, the center of magnetic pressure can be assumed to be stationary in the magnet frame and close to the center of the array at the location of the middle of the magnet array and in the center of the current distribution in in 𝑧𝑚 for thin 𝑡𝑐; i.e. 𝑥𝑚 = 0, 𝑧𝑚 = −(𝑧𝑓 +𝑡𝑐2) for a single layer of coils. Therefore the mean torque is 𝑻𝟎 = 𝐾𝑜(𝐻𝑚+𝑡𝑐2+ 𝑧𝑓)𝐼𝑥𝑟𝒋̂.  In general, the force and torque model may differ from the real motor in several ways. First, the geometric variation of the motor from the model will introduce modeling error. Second, and more significantly depending on the scale of the motor, variations in magnetization strength and magnetization direction (up to 10% and +/- 10 degrees for some typical commercial magnets) will contribute to overall modeling error. One advantage of this motor is that since there is no ferritic or highly permeable material in the system, variations between motors can be limited by careful attention to geometric tolerancing and uniform magnet selection alone. 72   Figure 2.18 Multiple stationary coil array stack-up dimensions, N-layers of identically driven coil arrays.   2.3 Force and Torque Ripple Self-attenuation Using Magnet Array Design The force ripple (2.67) at low flying heights can become a performance limiting factor, especially for high precision applications. As shown in (2.53) and (2.54), higher order spatial field harmonics decay with 𝑧𝑚 at a higher rate compared to the fundamental field harmonic, so increased flying height reduces force ripple. Flying higher, however, also decreases the achievable acceleration of the mover since the fundamental field harmonic which produces the mean force is reduced, and there is a maximum current achievable by the current driver as well as a maximum current carrying capacity of the coils. Lee et al. [82] used a higher-order Halbach array with 8 magnetization segments per spatial period to reduce higher harmonics of field. Other strategies to minimize linear motor force ripples include non-rectangular-shaped magnet elements for lower force ripples as in Zhang et al. [83] and Okamoto et al. [84]. Here we ignore such software based controller methods for addressing intrinsic motor disturbances since we assume that it is always better to develop a more linear motor than to attempt correction using software/control methods. Such methods can always be applied in addition if computation cost is not a factor. 73  In this section we present two magnet array designs and a design methodology to reduce force and torque ripple in the 2D motor element. We term these designs self-attenuating because they rely only on magnet array configuration and do not change either commutation laws or controller laws. A whole magnet array is split into sections and each section is offset relative to the commutation center of the excitation current zone. This method of splitting magnet arrays to reduce intrinsic motor disturbances can be generalized to other synchronous machines, in particular other planar levitation synchronous motors, ironless linear motors where the primary source of force ripple is due to field and current interactions, and can also be extended to rotary synchronous machines.  2.3.1 Force Vector Rotation from Offset Commutation Law Since different spatial force harmonics are caused by modulation of different field harmonics, it is useful to examine the effect on each spatial force ripple vector of offsetting the magnet center (𝑥𝑚, 𝑧𝑚) in the 𝑥𝑚 direction without compensating for this offset in the electrical angle 𝜃 used in the commutated currents.  Figure 2.19 shows the true magnetic center of the array at (𝑥𝑚, 𝑧𝑚) and the offset magnet coordinate system (𝑥𝑚′, 𝑧𝑚′) used as the virtual magnet center in the commutation law of (2.63)-(2.65), offset by 𝑥𝑜 from the true magnet center at (𝑥𝑚, 𝑧𝑚). As 𝑥𝑚′ = 𝑥𝑚 + 𝑥𝑜, the magnetic field PHM from (2.57) can be re-written as  𝑩𝑷𝑯𝑴′ =[    −𝐵1𝑒𝑧𝑚𝜆𝑐 𝑠𝑖𝑛(𝜃′ −𝜃𝑜) + 𝐵5𝑒5𝑧𝑚𝜆𝑐 𝑠𝑖𝑛 (5(𝜃′ −𝜃𝑜)) 0𝐵1𝑒𝑧𝑚𝜆𝑐 𝑐𝑜𝑠(𝜃′ −𝜃𝑜) −𝐵5𝑒5𝑧𝑚𝜆𝑐 𝑐𝑜𝑠 (5(𝜃′ −𝜃𝑜)) ]    . (2.70)  Here we define the virtual electrical angle 𝜃′ =𝑥𝑚′𝜆𝑐1 and offset electrical angle as 𝜃𝑜 =𝑥𝑜𝜆𝑐1.  74   Figure 2.19 Offset magnet array from commutation center.  The commutation law (2.63)-(2.65) is modified to be synchronous with the virtual electrical angle 𝜃′ and thus  𝐼𝑎 = −𝐼𝑥𝑟 𝑒−𝑧𝑚𝜆𝑐 𝑐𝑜𝑠 𝜃′ − 𝐼𝑧𝑟𝑒−𝑧𝑚𝜆𝑐 𝑠𝑖𝑛 𝜃′ (2.71)   𝐼𝑏 = −𝐼𝑥𝑟𝑒−𝑧𝑚𝜆𝑐 𝑐𝑜𝑠 (𝜃′ −𝜋3) − 𝐼𝑧𝑟𝑒−𝑧𝑚𝜆𝑐 𝑠𝑖 𝑛 (𝜃′ −𝜋3) (2.72)   𝐼𝑐 = −𝐼𝑥𝑟𝑒−𝑧𝑚𝜆𝑐 𝑐𝑜𝑠 (𝜃′ −2𝜋3) − 𝐼𝑧𝑟𝑒−𝑧𝑚𝜆𝑐 𝑠𝑖 𝑛 (𝜃′ −2𝜋3) (2.73)  Carrying out the same force analysis from 2.2.6 using the shifted field (2.70), the position independent force is   𝑭𝒐′ = 𝐾0 [𝐼𝑥𝑟 cos 𝜃𝑜 + 𝐼𝑧𝑟sin 𝜃𝑜0𝐼𝑥𝑟 𝑠𝑖𝑛 𝜃𝑜 + 𝐼𝑧𝑟𝑐𝑜𝑠 𝜃𝑜] (2.74)  Similarly, the 6-cp𝜆 force ripple vector becomes in the true magnet frame (𝑥𝑚, 𝑧𝑚)  𝑭𝟔′ = 𝐾6𝑒4𝑧𝑚𝜆𝑐 [−𝐼𝑥𝑟 𝑐𝑜𝑠(6𝜃 − 5𝜃𝑜) − 𝐼𝑧𝑟 𝑠𝑖𝑛(6𝜃 − 5𝜃𝑜)0−𝐼𝑥𝑟 𝑠𝑖𝑛(6𝜃 − 5𝜃𝑜) + 𝐼𝑧𝑟 𝑐𝑜𝑠(6𝜃 − 5𝜃𝑜)]] (2.75)  75  It can be seen from (2.74) that the position independent mean force vector is rotated through the angle 𝜃𝑜, while (2.75) shows that the 6-cp𝜆 force ripple vector is rotated by 5 times that same angle. From a similar analysis for a general force ripple vector due to the 𝑘𝑡ℎ field harmonic, the rotation of each of the force ripple vectors can be determined whereby the 𝑘𝑡ℎ field harmonic produces a (𝑘 ± 1) harmonic force ripple, rotated through the angle 𝑘𝜃𝑜, for 𝑘 > 1 as shown in Figure 2.20.  76   Figure 2.20 Force vector rotation due to commutation center offset. (a) Position independent force vector 𝑭𝒐. (b) Force ripple vector 𝑭𝟔. (c) Force ripple vector 𝑭𝒌±𝟏 resulting from field harmonic 𝑩𝒌.   77  2.3.2 Split Magnet Array Design for Self-attenuation of Force Ripple  As shown in Figure 2.21, the magnet array of Figure 2.17 is now split into two sub-arrays in the Y-direction: the array half 𝑣 is offset in the +𝑥𝑚 direction by 𝜆/20, and the array half 𝑢 is offset in the −𝑥𝑚 direction by 𝜆/20. The total spacing between array halves is thus 𝜏 = 𝜆/10, which is half the spatial wavelength of the 5th field harmonic responsible for the dominant and undesired 6 cp𝜆 force ripple. The coil current is commutated based on the relative position of each coil trace with respect to the center of the whole magnet array according to (2.63)-(2.65). The magnet array width 𝑊𝑚 used to calculate force coefficients is the total width covered by the magnet volume in the 𝑥𝑚-direction (i.e. for the array shown in Figure 2.21 the magnet array width used to calculate 𝐾0, 𝐾6  is 𝑊𝑚 = 2𝜆, not the total physical extent of the array 2𝜆 + 𝜏).   Figure 2.21 Split array solution for force ripple attenuation of the 6th harmonic of force ripple due to the 5th field harmonic.   With the split in the magnet array, the 6 cp𝜆 force ripple vector in (2.75) becomes two parts, one acting on each array half. The 6 cp𝜆 force ripple on array half 𝑢 is 78   𝑭𝟔,𝒖 =𝐾62𝑒4𝑧𝑚𝜆𝑐[    −𝐼𝑥𝑟 𝑐𝑜𝑠 (6𝜃 +𝜋2) − 𝐼𝑧𝑟 𝑠𝑖𝑛 (6𝜃 +𝜋2)0−𝐼𝑥𝑟 𝑠𝑖𝑛 (6𝜃 +𝜋2) + 𝐼𝑧𝑟 𝑐𝑜𝑠 (6𝜃 +𝜋2)]]     (2.76)  and the 6-cp𝜆 force ripple array half  𝑣 is  𝑭𝟔,𝒗 =𝐾62𝑒4𝑧𝑚𝜆𝑐[    −𝐼𝑥𝑟 𝑐𝑜𝑠 (6𝜃 −𝜋2) − 𝐼𝑧𝑟 𝑠𝑖𝑛 (6𝜃 −𝜋2)0−𝐼𝑥𝑟 𝑠𝑖𝑛 (6𝜃 −𝜋2) + 𝐼𝑧𝑟 𝑐𝑜𝑠 (6𝜃 −𝜋2)]]     (2.77)  The 6-cp𝜆 force ripples generated on two array halves have a full 180∘ angle and equal magnitude. The resulting sum of (2.76) and (2.77) is 𝑭𝟔 = 𝑭𝟔,𝒖 + 𝑭𝟔,𝒗 = 0, independent of lateral position 𝑥𝑚. The position independent force for the split array design is  𝑭𝒐 = 𝐾0 cos (𝜋10) [𝐼𝑥𝑟0𝐼𝑧𝑟] = 0.95𝐾0 [𝐼𝑥𝑟0𝐼𝑧𝑟] (2.78)  Thus the 6 cp𝜆 force ripple is eliminated. In comparison with (2.68), the position independent force from the split array design decreases by 5%. The splitting gap 𝜏 = 𝜆/10 between array halves also increases the XY footprint area of the whole mover by 3.3%.  This magnet array splitting method can be generalized to target different force ripple harmonics by spacing the array halves apart by half the spatial wavelength of the field harmonic responsible for the targeted force ripple. Table 2.1 shows the designed gap 𝜏 for a split array targeting different force ripples present in the 4-segment Halbach array with a 3-phase coil arrangement. In reality, for 4-segment Halbach pattern the 5th harmonic of the permanent magnet array field is much larger than the others, and thus is the main design consideration for the split array.  79  SPACING FOR SPLIT ARRAY TARGETING DIFFERENT FORCE HARMONICS Field Harmonic k Field Harmonic Wavelength Force Ripple Separation Gap 𝜏 5 𝜆/5 6 𝑐𝑝𝜆 𝜆/10 13 𝜆/13 12 𝑐𝑝𝜆 𝜆/26 17 𝜆/17 18 𝑐𝑝𝜆 𝜆/34 33 𝜆/33 32 𝑐𝑝𝜆 𝜆/66 Field harmonics for a 4-segment Halbach array, force harmonics from actuation by a 3-phase coil array. Table 2.1 Spacing for Split Array Targeting Different Force Harmonics   To verify the analytical model and split array force analysis, an FEA was carried out in COMSOL Multiphysics using the AC/DC module in 2D planar mode [79]. The 2D planar mode is used to minimize computation time and to more closely follow the assumptions of the analysis. As before, the leakage effects from the finite extension in 𝑦𝑚 will cause a small reduction in the overall force generated. Figure 2.22 shows the finite element models in schematic form for each case. In order to separate the effects of a finite coil array from the effects of the modulation of higher spatial harmonics of field with the commutated current, an ‘infinite’ or extended array was used; the coil array was 18𝜆 wide compared to 2𝜆 magnet arrays. The coil array width was verified to be wide enough to emulate an infinite repeating current pattern by examining the 1 cp𝜆 force ripple: the 1 cp𝜆 force ripple amplitude should go to zero as the current sheet goes to an infinite width based on the force equivalence principle. Different coil array widths were attempted until the 1 cp𝜆 force ripple amplitude was below the numerical precision of the FEA. 80   Figure 2.22 FEA models for ‘infinite’ coil arrays (extended in the x-direction over 18𝛌 width). (a) Non-split array. (b) Split array.  Figure 2.23 shows the results of a pure levitation (Z) command on the non-split and split arrays. The mean force for the non-split array is 14.57 N compared to the mean force for the split array at 13.85 N for a command of 𝐼𝑧𝑟 = 10 𝐴 at a flying height of 𝑧𝑓 = 𝜆/60 above the top conductor layer. This shows a 5% reduction in the force coefficient as predicted by (2.78). The 6 cp𝜆 force ripple amplitude in the non-split array is 0.215 N, and in the split array is 3.65 × 10−6 N. The targeted 6 cp𝜆 force ripple is reduced to 0.26 parts per million (ppm) (the numerical 81  precision of the FEA) of the mean force in the split array, from 14756 ppm of the mean force in the non-split array: a reduction of more than four orders of magnitude. The remaining force ripple in the split array is at 12 and 18 cp𝜆, due to the 13th and 17th field harmonics respectively. The amplitude of the remaining force ripple is 260 ppm of the mean force and can be targeted by further coil averaging techniques and combinations of different array splitting as first described in [75].  Figure 2.23 Finite element analysis of non-split array and split array over ‘infinite’ coil arrays( extended over 18𝛌 width). Flying height 𝒛𝒇 =  𝟎. 𝟓 [𝐦𝐦], 𝑰𝒙𝒓 = 𝟎, 𝑰𝒛𝒓 = 𝟏𝟎 [𝐀], 𝝀 = 𝟑𝟎 [𝐦𝐦], 𝑩𝒓 = 𝟏. 𝟑𝟐𝟓 [𝐓], 𝑫𝒎 =𝟐𝛌,𝑾𝒄 = 𝟎. 𝟏𝟓𝟖𝟑𝛌, 𝒕𝒄 = 𝟎. 𝟐𝟏𝟑 [𝐦𝐦] and 8 layers of active coils spaced 𝟎. 𝟔𝟒𝟑 [𝐦𝐦] apart in the −𝐳𝐦 direction.    82  2.3.3 Experimental Load Verification for Force Ripple Self-attenuation  In order to demonstrate experimentally the effectiveness of the magnet array splitting method presented in 2.3.2, two magnet arrays were built as shown in Figure 2.24: a non-split array of (2𝜆 ×𝐻𝑚 × 2𝜆) dimensions and a split array of ((2𝜆 +𝜆10) × 𝐻𝑚 × 2𝜆)  dimensions designed to attenuate the 6 cp𝜆 force ripple. Figure 2.25 shows the experimental setup designed to measure the load characteristic of two 1-D magnet arrays actuated by a 1-D coil array. The magnet array is attached to an ATI Industrial Automation MINI-45 6DOF load cell that measures three forces and three torques via an adapter plate. The load cell is attached to the spindle housing of a CNC. The CNC spindle is used to translate the magnet array relative to the stationary coils, while a set of linear power amplifiers is used to drive the excitation current through the coils. We did not use the switching amplifier used for the alpha prototype in order to get a cleaner load characterization unsullied by electromagnetic switching noise. The drive capacity of the linear amplifiers is limited to 10 A continuous, so all experimental measurements are done near this limit at 9.5 A to maximize the signal-to-noise ratio of the load cell. 83   Figure 2.24 Experimental arrays for load measurement, showing dimensions and magnetization directions.   Figure 2.25 Experimental setup for load measurement.  Figure 2.26 shows the two test cases measured, with a flying height 𝑧𝑓 = 0.0247𝜆 =0.74 mm and 𝐼𝑧𝑟 = 9.5 A (a pure levitation force is commanded). Since the amplifiers were near their maximum continuous current limit, the flying height was not compensated in the commutation command (i.e. 𝑧𝑚 in (2.63)-(2.65) was set to zero). The excitation zone was kept to 84  4𝜆 wide (four coil groups) due to experimental limitations of the test coil board, and this was deemed sufficient to meet the requirement to consider the magnetic field purely periodic (based on the principle of equivalent force generation in 2.2.3) and is the practical excitation zone used in the alpha prototype.    Figure 2.26 Experimental test cases with finite coil array of 4𝛌 width. (a) Non-split array. (b) Split array.  Figure 2.27 shows the comparison of the experimental non-split array to the split array. Table 2.2 shows the individual force components of interest, comparing the analytical predictions of each force component using (2.84) to experiment. Due to the variation of actual magnet 85  remanence (±5%), the 𝐵𝑟 value in the analytical model is calibrated by matching the analytical levitation force 𝐹0𝑧 for the non-split array with the experimentally measured levitating force. All other ratios were then compared without further adjustment of any kind.  The key result is that the experimental 6 cp𝜆 force ripple for the split array is 10 times smaller than that of the non-split array, from an amplitude of 1.1% of the levitation force for the non-split array to 0.12% of the levitation force for the split array. This indicates that the split array design solution is an effective and practical method of force ripple attenuation in synchronous planar levitation motors. The analytical model predicts that the 6 cp𝜆 force ripple should be entirely eliminated. In reality, several practical limitations can cause non-perfect cancelation between the array halves: (a) each magnet element is not necessarily magnetized to an identical magnet remanence; (b) the gap spacing 𝜏 on the experimental split array could not be manufactured to be exactly 𝜆/10; and (c) each magnet segment has a geometric tolerance. There exists a small amount of 1 cp𝜆 force ripple, due to the use of 4 𝜆 wide coils excited in this experimental setup (as opposed to the infinitely wide repeating coil array assumed by the analytical model) which is predicted by numerical simulation (both FEA and numerical integration using the exact 3D field model from 2.2.1). We note also that the orthogonal force 𝐹𝑦 for both magnet array types is essentially zero, at the level of the noise floor of the sensor. This confirms that the motor has minimal coupling to the orthogonal axis.  86   Figure 2.27 Comparison experimental non-split array and split array, uncompensated for 𝐳𝐟 = 𝟎. 𝟎𝟐𝟒𝟕𝝀 ,𝐈𝐱𝐫 = 𝟎, 𝐈𝐳𝐫 = 𝟗. 𝟓 𝐀, 𝛌 = 𝟑𝟎 𝐦𝐦,𝐃𝐦 = 𝟐𝛌,𝐖𝐜 = 𝟎. 𝟏𝟓𝟖𝟑𝛌,𝐇𝐦 =𝝀𝟒, 𝐭𝐜 = 𝟎. 𝟐𝟏𝟑 𝐦𝐦 and N = 8 layers of active coils spaced 𝒕𝒈 =  𝟎. 𝟔𝟒𝟑 𝐦𝐦 apart in the −𝐳𝐦 direction.      87  FORCE HARMONIC COEFFICIENTS IN Z    F0z % F0z F6z % F0z Non-split Array Analytical 12.33 N 100% 0.153 N 1.24% Experimental 12.33 N 100% 0.137 N 1.11%             Split Array Analytical 11.72 N 95% 0 N 0 % Experimental 11.79 N 95.6% 0.014 N 0.11% Br = 1.2 T , adjusted  to match experimental and analytical non-split array F0z. All harmonics are compared to the analytical non-split array F0z. Table 2.2 Force Harmonic Coefficients in Z  2.3.4 Quad-Split Array Design for Self-attenuation of Force and Torque Ripple  The split magnet array method in 2.3.2 demonstrated the elimination of a targeted spatial harmonic of force ripple to within FEA simulation accuracy; further, experimental characterization in 2.3.3 showed an order of magnitude attenuation of force ripple. However, the issue of torque ripple was not addressed. To meet the positioning and control requirements for high precision stages both torque and force ripple must be attenuated as much as possible. For example, force and torque ripple are discussed in the context of magnetically levitated stages in [45]. In this section, the split array method is extended to a magnet array design that simultaneously self-attenuates targeted spatial harmonics of torque and force ripple without inducing any coupling between any axes or additional distortion at the targeted spatial frequency. We term this new magnet array topology a quad-split array. The quad-split array presented is designed to attenuate both 6 cp𝜆 force and torque ripple simultaneously. Finite element analysis (FEA) based simulations are carried out to validate the new magnet array design. 88  Figure 2.28a shows the 6 cp𝜆 force ripple vectors (𝑭𝟔,𝒖, 𝑭𝟔,𝒗) acting on each half of the non-split magnet array at their respective centers of magnetic pressure for a given 𝑥𝑚  position. This force ripple cancels out almost cancels out all net 6 cp𝜆 torque on the array due to the modulation of the 5th field harmonic and the current distribution for flying heights, notably 𝑧𝑓 =𝜆/60. We note, however, that there still always exists a net 6 cp𝜆 torque due to the pitch of the coil array (𝜆/6) and the discrete nature of the coil distribution (approximately a stepwise function in 𝑥𝑚).  Figure 2.28b shows the split magnet array with 6 cp𝜆 force ripple vectors (𝑭𝟔,𝒖, 𝑭𝟔,𝒗) acting on each magnet array half at moment arms (𝒅𝒖, 𝒅𝒗). The summation of these force vectors cancels out any net 6 cp𝜆 force on the array, as described in 2.3.2. However, an additional torque ripple is induced by the opposition of the 6 cp𝜆 force ripple vectors (𝑭𝟔,𝒖, 𝑭𝟔,𝒗) as they have a relative angle of 180∘ at all positions and are on opposing sides of the magnet array CG. We note that only the z-direction components of force ripple contribute to this torque as the x-directed force components act at the same z-height. If the center of magnetic pressure for each array half is modelled at 𝒅𝒖 =−𝜆−𝜏2?̂? − (𝐻𝑚+𝑡𝑐2+ 𝑧𝑓)?̂? and 𝒅𝒗 =𝜆+𝜏2?̂? − (𝐻𝑚+𝑡𝑐2+ 𝑧𝑓)?̂? respectively, the additional torque ripple resulting from the 6 cp𝜆 force ripple is  𝑻𝟔 = −𝜆 + 𝜏2𝐾6𝑒4𝑧𝑚𝜆𝑐 [𝐼𝑥𝑟 𝑐𝑜𝑠(6𝜃) + 𝐼𝑧𝑟 𝑠𝑖𝑛(6𝜃)] 𝒋̂ (2.79)  where 𝒋̂ is a unit vector in the 𝑦𝑚–direction. This torque ripple will reduce the positioning accuracy of the moving stage and must be addressed for high performance positioning stage applications such as lithography wafer steppers. It is desired to attenuate this 6 cp𝜆 torque ripple while simultaneously attenuating force ripple without inducing any additional coupling between axes. 89  Figure 2.29 shows two example embodiments of the quad-split array.  Each is comprised of eight identical sub-arrays (𝐿𝐼 , 𝐿𝐼𝐼 , 𝐿𝐼𝐼𝐼 , 𝐿𝐼𝑉, 𝑅𝐼 , 𝑅𝐼𝐼 , 𝑅𝐼𝐼𝐼 , 𝑅𝐼𝑉). Each sub-array is magnetized in a Halbach-pattern in the XZ-plane and is uniform in the Y-axis. Figure 2.29c shows an example magnetization pattern for the sub-array with horizontally and vertically magnetized elements, with a sub-array width in X and depth in Y of 𝜆 and magnet height in Z of 𝐻𝑚. In embodiment 1 shown in Figure 2.29a, sub-arrays (𝑅𝐼𝐼 , 𝑅𝐼𝐼𝐼) are shifted by 𝜏 = 𝜆/10 in 𝑥𝑚 and sub-arrays (𝐿𝐼𝐼 , 𝐿𝐼𝐼𝐼) are shifted by – 𝜏 = −𝜆/10 in 𝑥𝑚; the remaining sub-arrays (𝐿𝐼 , 𝐿𝐼𝑉 , 𝑅𝐼 , 𝑅𝐼𝑉) are not shifted. In embodiment 2 shown in Figure 2.29b, sub-arrays (𝑅𝐼 , 𝑅𝐼𝑉) are shifted by 𝜏 = 𝜆/10 in 𝑥𝑚 and sub-arrays (𝐿𝐼 , 𝐿𝐼𝑉) are shifted by – 𝜏 = −𝜆/10  in 𝑥𝑚; the remaining sub-arrays (𝐿𝐼𝐼 , 𝐿𝐼𝐼𝐼 , 𝑅𝐼𝐼 , 𝑅𝐼𝐼𝐼) are not shifted. As the force and torque analysis for both embodiments are similar, the remaining analysis will be for embodiment 1.  Figure 2.28 Magnet array designs showing force ripple vectors acting at magnetic centers of pressure. (a) Non-split magnet array. (b) Split magnet array. 90   Figure 2.29 Quad-split array for force and torque attenuation. (a) Embodiment 1. (b) Embodiment 2. (c) Sub-array example Halbach magnetization pattern. Figure 2.30 shows 6 cp𝜆 force ripple vectors for sub-arrays (𝐿𝐼 , 𝐿𝐼𝐼 , 𝑅𝐼 , 𝑅𝐼𝐼). From (2.75), the 6 cp𝜆 force ripple vector on sub-array 𝐿𝐼 is  𝑭𝟔,𝑳𝑰 =𝐾68𝑒4𝑧𝑚𝜆𝑐 [−𝐼𝑥𝑟 𝑐𝑜𝑠(6𝜃) − 𝐼𝑧𝑟 𝑠𝑖𝑛(6𝜃)0−𝐼𝑥𝑟 𝑠𝑖𝑛(6𝜃) + 𝐼𝑧𝑟 𝑐𝑜𝑠(6𝜃)}] (2.80)  acting on the moment arm 𝑑𝐿𝐼⃑⃑ ⃑⃑  ⃑, and the 6 cp𝜆 force ripple vector on sub-array 𝐿𝐼𝐼 is  𝑭𝟔,𝑳𝑰𝑰 =𝐾68𝑒4𝑧𝑚𝜆𝑐 [−𝐼𝑥𝑟 𝑐𝑜𝑠(6𝜃 − 𝜋) − 𝐼𝑧𝑟 𝑠𝑖𝑛(6𝜃 − 𝜋)0−𝐼𝑥𝑟 𝑠𝑖𝑛(6𝜃 − 𝜋) + 𝐼𝑧𝑟 𝑐𝑜𝑠(6𝜃 − 𝜋)}] (2.81)  acting on the moment arm 𝑑𝐿𝐼𝐼⃑⃑ ⃑⃑ ⃑⃑  ⃑ . The total 6 cp𝜆  force ripple is cancelled out between these two sub-arrays. Similarly (𝑅𝐼 , 𝑅𝐼𝐼) internally cancels out 6 cp𝜆 force ripple. In addition, as shown schematically in Figure 2.29a, sections (𝐿𝐼 , 𝐿𝐼𝐼) and (𝑅𝐼 , 𝑅𝐼𝐼) are mirrored across the 𝑦𝑚𝑧𝑚-plane. Thus, the 6 cp𝜆 torque ripple induced in the 𝑦𝑚-axis for section 𝐿𝐼 is cancelled out by opposing 91  torque ripple induced by section 𝑅𝐼. Similarly the 6 cp𝜆 torque ripple induced by section 𝐿𝐼𝐼 is cancelled by the torque induced by section 𝑅𝐼𝐼. The entire array is symmetric with respect to both 𝑥𝑚𝑧𝑚- and 𝑦𝑚𝑧𝑚-planes. This ensures that there is no 6 cp𝜆 torque ripple generated on any of the 𝑥𝑚-,  𝑦𝑚- or 𝑧𝑚-axes on the quad-split array.   Figure 2.30 Quad –split force ripple vectors acting at equivalent moment arms of sub-arrays (𝑳𝑰, 𝑳𝑰𝑰, 𝑹𝑰, 𝑹𝑰𝑰).  Table 2.3 summarizes the mean and 6 cp𝜆 ripple forces acting on each sub-array as well as the mean and 6 cp𝜆 net forces acting on the quad-split array. Similarly, Table 2.4 summarizes the mean and 6 cp𝜆 ripple torques acting on each sub-array as well as the mean and 6 cp𝜆 net torques acting on the quad-split array. From this analysis, all net 6 cp𝜆 force and torque ripples are shown to cancel out in every axis. The net mean force that remains on the quad-split magnet array is  92   𝑭𝟎 = 0.9𝐾0 [𝐼𝑥𝑟0𝐼𝑧𝑟] (2.82)  From Table 2.4, the remaining mean torque is approximately  𝑻𝟎 = 0.9𝐾𝑜(𝐻𝑚 + 𝑡𝑐2+ 𝑧𝑓)𝐼𝑥𝑟𝒋̂ (2.83)  which can be compensated easily as it is a constant torque across all positions and dependent only on the X-direction force command 𝐼𝑥𝑟. If other force and torque ripple harmonics are to be targeted for attenuation, the spacing 𝜏 should be adjusted as per Table 2.1, where 𝜏 is determined by the half-wavelength of the targeted field harmonic generating the force and torque ripple. To validate the quad-split array design and examine the remaining torque and force ripple, FEA based simulations are carried out using COMSOL Multiphysics [79] in 2D planar mode using motor parameters similar to the final prototype (described in later sections). Figure 2.31 shows a finite element model schematic, with an 18𝜆-wide active coil zone used to minimize edge effects from the finite width of the magnet arrays as described in 0. Figure 2.32 shows the three magnet array types modelled. Figure 2.33 shows the force and torque results of the three FEA simulations, with a flying height 𝑧𝑓 = 𝜆/60 = 0.5 mm and 𝐼𝑧𝑟 = 10 A (a pure levitation force command) over an X-direction traverse of 𝜆. The flying height component in the commutation law (2.63)-(2.65) is not compensated (𝑧𝑚 = 0) to mimic a maximum current capacity of 10 A on each drive channel.     93   Table 2.3 Analytical Mean and 6 cp𝝀 Ripple Forces for Sub-arrays. 94   Table 2.4 Analytical Mean and 6 cp𝝀 Ripple Torques for Sub-arrays. 95   Figure 2.31 Finite element model schematic showing generalized magnet array in relation to stationary coil array with 8 coil layers spaced apart by 0.643 mm in 𝒛𝒎.   Figure 2.32 Simulated magnet arrays showing magnetization patterns. (a) Non-split array. (b) Split array with 𝝉 = 𝝀/𝟏𝟎. (c) Quad-split array with 𝝉 = 𝝀/𝟏𝟎. 96   Figure 2.33 FEA forces and torques for non-split array, split array, and quad-split array; 𝒛𝒇 = 𝛌/𝟔𝟎 =𝟎. 𝟓𝐦𝐦, 𝑰𝒙𝒓 = 𝟎, 𝑰𝒛𝒓 = 𝟏𝟎 𝐀, with 8 coil layers spaced apart by 0.643 mm in 𝒛𝒎.  Figure 2.34 shows the harmonic amplitudes of the forces and torques from each FEA. The 6 cp𝜆 force and torque ripples are summarized in Table 2.5 along with the mean levitation force values. From Figure 2.34a and Figure 2.34b, both the split array and quad-split array show an almost complete attenuation of 6 cp𝜆 force ripple compared to the non-split array (approximately four orders of magnitude). This is on the order of the numerical precision of the finite element analysis. Figure 2.34c shows that the split array induces a torque ripple at 6 cp𝜆 which is almost 97  six times larger than the non-split array, while the quad split array has attenuated the 6 cp𝜆 torque ripple to 5.1% of the non-split array and 0.88% of the split array. From symmetry, 6 cp𝜆 torque ripple on the other axes are zero. Thus the quad-split magnet array is able to attenuate both force and torque ripple at a targeted spatial frequency without additional coupling between axes. This is at the cost of a 10% loss in mean force generation and corresponding increase in mover area In addition, this attenuation of torque is partial validation for the approximation of the magnetic center of pressure for each sub-array, as discussed at the end of section 2.2.6.  FORCE AND TORQUE AMPLITUDES   F0z (N) % F6z (N) % T6y (Nm) % NON-SPLIT 30.1  100 0.453  100 0.00127  100        SPLIT 28.6  95.0 0.000135  0.003 0.007327  577        QUAD-SPLIT 27.2  90.4 0.0000636  0.0014 0.0000652  5.1 𝑧𝑓 =λ60= 0.5mm, 𝐼𝑥𝑟 = 0, 𝐼𝑧𝑟 = 10 A, with 8 coil layers spaced apart by 0.643 mm in 𝑧𝑚. Table 2.5  Force and Torque Amplitudes.          98   Figure 2.34 Spatial harmonic amplitudes for force and torque ripples from FEA; flying height 𝒛𝒇 = 𝛌/𝟔𝟎 =𝟎. 𝟓 𝐦𝐦, 𝑰𝒙𝒓 = 𝟎, 𝑰𝒛𝒓 = 𝟏𝟎 𝐀, with 8 coil layers spaced apart by 0.643 mm in 𝒛𝒎. (a) Translation force, 𝑭𝒙. (b) Levitation force, 𝑭𝒛. (c) Torque in 𝒚𝒎-axis, 𝑻𝒚  99  2.4 6-DOF Motion Stage Prototype Design  A 6-DOF alpha prototype (i.e. basic proof of concept prototype) is designed to test the synchronous planar levitation principle. The mover under gravity preload is passively stable in three axes (𝑧, 𝛼, 𝛽) and inherently unstable in three axes (𝑥, 𝑦, 𝛾). We therefore require position feedback and control in at least three axes. In order to test the prototype, three additional technologies were developed by colleagues in the UBC Precision Mechatronics Laboratory: a 6-axis position sensing stereo-vision camera was developed by Dr. Xiaodong Lu and Niankun Rao [85]; a 48-channel PWM 3-phase power amplifier to drive the stator currents; and a real time computer Tsunami developed by Kristofer Smeds and presented in [2] to accept the position feedback, execute the digital controller, output current commands to the power amplifier and interface with a host user interface on a non-real time PC.   2.4.1 Prototype Architecture  The prototype basic architecture is shown in schematic form in Figure 2.35. The system components are a mover with magnet arrays attached, a coil array (built as a printed circuit board) attached to a stationary base frame, a stationary metrology reference frame attached to the same base frame, and a stereo-vision based position sensor. There are four infrared light-emitting diodes (IR-LEDS) attached to the corners of the mover, and another four IR-LEDS attached to the stationary metrology reference frame with known relative positions. The stereo-vision camera outputs the six-axis relative position (𝑥, 𝑦, 𝑧, 𝛼, 𝛽, 𝛾) (where (𝛼, 𝛽, 𝛾) are rotations about (𝑥, 𝑦, 𝑧) respectively) between the metrology frame and the mover based on the 3-DOF (𝑥, 𝑦, 𝑧) position of 100  each LED in the position sensor coordinate frame. This six-axis position is fed back to the real time computer and processed by the digital controller which outputs current commands to the power amplifier. The power amplifier drives current through each coil trace in the coil array, actuating the mover in 6-DOF. Not shown is the host computer which runs the user interface to the real time computer.  As shown in the schematic cross section of the alpha prototype in Figure 2.36, a stiffening layer is used to support the coil array PCB. This stiffening layer should be electrically insulating to minimize eddy current losses. The mover chassis should be able to support both the actuating forces applied by the stator current to the magnet arrays as well as the forces between the magnet arrays themselves (these forces are analyzed in [86]).    Figure 2.35 Prototype architecture schematic showing 6-DOF camera metrology, power amplifier and real-time controller. 101   Figure 2.36 Schematic cross-section of alpha prototype.   Figure 2.37 shows the solid model of the alpha prototype. To obtain the necessary field of view using the 6-DOF position sensor, the stereo-vision cameras must be placed approximately a meter away from the plane of the LEDs (i.e. the top of the mover) in the orientation shown. The camera frame is extruded aluminum. This necessarily means that the camera will be subject to vibration as the camera frame is relatively un-stiff. Hence the stationary metrology reference allows a differential measurement in all six axes, meaning the position feedback is decoupled from camera frame vibration.  102   Figure 2.37 Alpha prototype solid model showing 6-DOF camera metrology.  103  2.4.2 Mover Design 2.4.2.1 Magnet Array Geometric and Material Parameters Many different optimization cost functions can be assembled from the force coefficient expression. Here we choose to maximize acceleration based on a prescribed current density maximum, assuming the mass of the mover chassis is negligible relative to the magnet arrays. From the force expression (2.68), the net force coefficient for each magnet array is  𝐾 =12𝐵𝑟cos (𝜋4 +𝑔2𝜆𝑐)𝜋(1 − 𝑒−𝐻𝑚𝜆𝑐 )𝐷𝑚𝑊𝑚𝜂𝑐 𝑠𝑖𝑛𝑐 (𝑊𝑐2𝜆𝑐) (𝜆𝑐𝑡𝑐) (1 − 𝑒−𝑡𝑐𝜆𝑐)1 − 𝑒−𝑁(𝑡𝑐+𝑡𝑔)𝜆𝑐1 − 𝑒−𝑡𝑐+𝑡𝑔𝜆𝑐   [𝑁𝐼𝑟] (2.84)  where 𝐼𝑟 is the amplitude of the force commands 𝐼𝑥𝑟 , 𝐼𝑧𝑟. These force commands can be interpreted as current commands given a constant flying height. The mass for each 2-DOF motor element is 𝑚 = 𝜌𝑚𝑎𝑔𝑛𝑒𝑡𝐻𝑚𝐷𝑚𝑊𝑚. Therefore acceleration per force command unit is simply 𝑎 = 𝐾/𝑚.  Halbach Magnet Array Geometry We choose a four segment per wavelength Halbach array to minimize cost and manufacturing complexity. Force is proportional to (1 − 𝑒−𝐻𝑚𝜆𝑐 ) and acceleration is proportional to 1𝐻𝑚(1 − 𝑒−𝐻𝑚𝜆𝑐 ). To maximize both we define a cost function 𝜂𝐻 =1𝐻𝑚(1 − 𝑒−𝐻𝑚𝜆𝑐 )2. This cost function is maximum at 𝐻𝑚 = 𝜆/5, implying a rectangular cross-section since each magnet segment width will be 𝜆/4. However, non-square cross-section magnet arrays would require three different types of magnets for a Halbach array mirror-symmetric about the magnet array 𝑦𝑚𝑧𝑚 plane; therefore we design the magnet arrays as 𝐻𝑚 = 𝜆/4. This configuration requires only two types of arrays for symmetric Halbach-pattern arrays and is relatively close to 𝜆/5 𝜆/4 × 𝜆/4 ×𝐷𝑚 (width and height) and 𝜆/8 × 𝜆/4 × 𝐷𝑚 segments magnetized in the 𝐻𝑚 direction. The gap 104  between segments 𝑔 should be kept as close to zero as possible. Adhesive two part epoxy will be used to bond magnet segments together and this necessitates a 50-100 micron gap. We examine coil parameters as they relate to acceleration and power in 2.4.3. The choice of magnet spatial period is driven by cost and maximum load capacity required, as well as the relative achievable geometric tolerances of each magnet segment. The cost of magnet segments scales more than linearly by volume; hence we have a push to minimize the overall size of the magnet array and the spatial wavelength 𝜆. We also want some reasonably large force capacity as well as insensitivity to machining and fabrication tolerances on the order of 50-100 micron. We therefore choose 𝜆 = 30 𝑚𝑚 as a compromise between the objectives of cost, manufacturability and actuator performance. Further, a ~2:1 ratio between 𝐷𝑚:𝑊𝑚 yields the largest magnet area per mover (highest magnetic fill factor) and therefore the highest acceleration.   Permanent Magnet Material  Rare earth magnets offer very high remanence; however, the higher the remanence the lower the overall operating temperature of the magnetic material before permanent losses in magnetization occur (at significantly lower than the Curie temperature). In addition, in the Halbach configuration the operating temperature may be less than an isolated magnet segment as the bucking fields from neighboring magnet segments will exert a de-magnetizing field, meaning permanent magnetization losses occur at an even lower temperature.  An N44SH grade magnet material is chosen which yields a remanence 𝐵𝑟 = 1.325 [𝑇] (from a magnetic energy product of 44 MGOe and a relative permeability 𝜇𝑟 of 1.027). 105  2.4.2.2 Mover Magnet Array Configuration The analysis from 2.3 allows us to design the magnet array configuration of the mover based on the specific needs of our application. As the alpha prototype is a basic proof of concept, we prioritize simple manufacturability and minimal cost. Figure 2.38 shows several mover topologies (note the view is from the bottom of the mover, or the strong side of the Halbach arrays). All array force coefficients are normalized to the high force basic large mover shown in Figure 2.38a.   a. High Force Basic Large Mover The high force basic large mover is a set of four 2𝜆 × 𝐻𝑚 × 4𝜆 Halbach arrays. Each magnet array requires only 2 types of magnet segments. The 6 cp𝜆 force ripple on the mover is at a maximum. This basic mover gives the largest mean force and acceleration of all of the mover topologies considered here.  This topology has the highest acceleration and simplest structure and is therefore suitable for applications that are not high precision but benefit from high acceleration.  b. High Force, Low Force Ripple Large Mover This mover topology uses four 2𝜆 × 𝐻𝑚 × 4𝜆 Halbach arrays with the centers of each pair of (X1, X3) and (Y2, Y4) arrays separated by 4𝜆 +𝜆12. For operating conditions where each pair of   106   Figure 2.38 Alternate mover magnet array layouts (identical 𝝀 = 𝟑𝟎𝒎𝒎). 107  magnet arrays is driven by similar force commands with near zero tip/tilt (𝛼, 𝛽), the additional 𝜆12 offset insures that the 6 cp𝜆 force ripple cancels out on the CG of the mover without compromising mean force generation. However, this will also induce a large 6 cp𝜆 torque ripple on the mover.  Intrinsic force disturbance is minimized in this topology while maintaining high acceleration and simplicity of manufacture. The induced torque ripple is still an issue.  c. Low Force and Torque Ripple Large Mover Using four split arrays as discussed in 2.3.2, this mover uses four (2𝜆 +𝜆10) × 𝐻𝑚 × 4𝜆 split Halbach arrays which generate internal 6 cp𝜆 torque ripples. Note the magnitude of the torque ripple induced from each magnet array pair is smaller than the induced torque ripple in Figure 2.38b by a factor of 4𝜆+𝜆12𝜆+𝜆10= 3.717. To further cancel this torque ripple, the magnet array pairs (X1, X3) and (Y2, Y4) arrays are separated by 4𝜆 +𝜆12 to make the torque ripple on each array out of phase by 180∘ under similar force commands for each magnet array pair. This mover topology requires twice the number of non-square magnet segments compared to either Figure 2.38a or b and has a larger planform area. The maximum force on the mover is 95% of that generated by Figure 2.38a or b.  This mover topology has no 6 cp𝜆 intrinsic force or torque ripple at the cost of 5% maximum acceleration and a marginal increase in manufacturing complexity and planform area.     108  d. High Accuracy Low Internal Stress Large Mover Four quad-split arrays  of (2𝜆 +𝜆5) × 𝐻𝑚 × 4𝜆 are arranged as discussed in 2.3.3 to eliminate 6 cp𝜆 torque and force ripples at the magnet array level. The advantages of this mover topology are: i) magnet array force commands can be very different and internal cancellation of 6 cp𝜆  force and torque ripple will still occur; ii) because the internal force and torque cancellations on each magnet array occur over much smaller moment arms compared to Figure 2.38b or c, cyclic internal stresses acting on the mover chassis will be significantly reduced. This renders this topology very suitable for high precision applications. The cost of this force and torque linearity and low internal stresses leading to vibrations is a 10% reduction in maximum force generated compared to Figure 2.38a and a much more complex assembly requiring four different magnet segments as shown.    e. High Force Basic Small Mover This mover topology is shown to illustrate that multiple sizes of mover using the same magnetic wavelength 𝜆 can be actuated on the same motor stator. With 25% of the mover area, the total maximum force on the mover is ¼ of that generated on the high force basic mover in Figure 2.38a, but with ¼ of the magnet weight the acceleration is the same.  The high force, low force ripple mover topology in Figure 2.38b is chosen for the alpha prototype as a compromise between maximum acceleration, cost and intrinsic force disturbance. The intrinsic torque disturbance was not initially considered to be an issue until later analysis.   109  2.4.2.3 Mover Mechanical Design Figure 2.39 shows an exploded view of the mover design. The mover chassis is a lightweight aluminum honeycomb structure. The magnet arrays are attached to the bottom of the mover chassis using adhesive epoxy. A non-magnetic central spacer is used as part of the assembly process (described in the manufacturing section 2.5.1) and left in place after assembly. The plastic side plates are to protect the honeycomb core which, while very strong in sheer, is yet very thin aluminum foil and easily damaged by side loads.  Figure 2.40 shows an isometric cut view of the mover with internal details. The mover chassis is a commercially available aluminum honeycomb sandwich panel with solid aluminum top and bottom skins of 1.59 mm thickness and a honeycomb core of 1/8” hex cell density and 0.002” aluminum foil thickness. The core thickness is 50.8 mm. This was chosen as a compromise between stiffness and weight, with the aluminum skins thicker than optimal (as calculated in [52]) to allow tapping with M2 threads. The IR-LEDs are powered by a lithium-ion battery and driven by a constant current circuit as shown and attached directly to the top of the mover chassis using M2 socket head cap screws [1]; a battery powered solution is chosen to negate the need for power cables to the mover.  The central spacer block is aluminum.  Magnet array and mover topology dimensions are shown in Figure 2.41, with 𝜆 = 30 𝑚𝑚. Each magnet array is made up of two subarrays with dimensions 2𝜆 × 𝐻𝑚 × 2𝜆, with each sub-array made of magnet segments with a depth of 𝐷𝑚2= 60 𝑚𝑚 to minimize cost and risk of damaging the magnet segments due to too large an aspect ratio. The four magnet arrays combined weigh 1.68 kg and the overall weight of the mover is 𝑀 = 2.3 kg.  110   Figure 2.39 Alpha prototype mover design, exploded view.   Figure 2.40 Isometric cut-view of mover . 111   Figure 2.41 Alpha prototype mover design, detailed planform.  2.4.3 Stator Coil Array PCB Board Design and Wiring Configuration  The coil array PCB must implement the coil structure presented in 2.1. A single set of 3-phase coil traces elongated in the Y-direction is shown in Figure 2.42 with zoomed inset showing coil geometry. Ideally 𝑊𝑐 = 𝜆/6 for the 3-phase coil arrangement, but a practical insulation gap is required and therefore 𝑊𝑐 =𝜆6− 𝑔𝑐 where 𝑔𝑐 is the insulating gap dimension. Figure 2.43 shows an example of an eight layer coil array with both X- and Y-elongated coil trace layers alternating 112  in Z, with four layers each. Note that the top layer is X1. From (2.68) we define array force coefficients for each magnet array pair as   𝐾𝑋 =12𝐵𝑟cos (𝜋4 +𝑔2𝜆𝑐)𝜋(1 − 𝑒−𝐻𝑚𝜆𝑐 )𝐷𝑚𝑊𝑚𝜂𝑐 𝑠𝑖𝑛𝑐 (𝑊𝑐2𝜆𝑐) (𝜆𝑐𝑡𝑐) (1 − 𝑒−𝑡𝑐𝜆𝑐)1 − 𝑒−𝑁(𝑡𝑐+𝑡𝑔)𝜆𝑐1 − 𝑒−𝑡𝑐+𝑡𝑔𝜆𝑐   [𝑁𝐼𝑟] (2.85)   𝐾𝑌 = 𝑒−(𝑡𝑐+𝑡𝑖𝑛𝑠)/𝜆𝑐𝐾𝑋   [𝑁𝐼𝑟] (2.86)  where 𝐾𝑋 is the force coefficient for the (X1, X3) and 𝐾𝑌 is the force coefficient for (Y2, Y4) and we can define an insulation thickness in the z-direction as 𝑡𝑖𝑛𝑠. From (2.85), minimal separation thickness 𝑡𝑔 provides the highest acceleration. The separation thickness between layers of coil traces acting in the same direction is 𝑡𝑔 = 2𝑡𝑖𝑛𝑠 + 𝑡𝑐. For the ideal case of zero insulation thickness, the minimum separation thickness is still 𝑡𝑔 = 𝑡𝑐. From this structural limitation we see that simply maximizing the copper thickness 𝑡𝑐 is non-optimal as the difference between the force coefficients will be too large (with 𝐾𝑌𝐾𝑋= 𝑒−(𝑡𝑐+𝑡𝑖𝑛𝑠)/𝜆𝑐).  The Ohmic power loss in a single 𝜆-coil group is   𝑃 = 𝐼𝑎2𝑅 + 𝐼𝑏2𝑅 + 𝐼𝑐2𝑅   [𝑊] (2.87)  with the resistance 𝑅 = 𝜌𝑅𝑊𝑐𝑡𝑐𝑁𝐿, 𝜌𝑅the resistivity of the coil material, and 𝐿 the total length of the coil trace in each phase for each layer. Therefore the Ohmic loss is directly proportional to the number of coil layers 𝑁. From (2.86) and (2.88) we can define a force/power ratio that is related to the motor electrical efficiency, 𝜂𝑒 = 𝐾𝑥/𝑁. We note that the total effective board thickness is 𝑁(2𝑡𝑐 + 𝑡𝑖𝑛𝑠). We can see that there are diminishing returns for additional coil layers; this is 113  intuitively true because each additional coil trace layer is immersed in a field decaying in 𝑧𝑚 as 𝑒𝑧𝑚/𝜆𝑐 .    Figure 2.42 Coil structure isometric view, single 𝝀-group Y-elongated coil traces. 114   Figure 2.43 Eight layer coil structure, showing alternating X- and Y-elongated coil traces.  Coil trace end turns are implemented outside the active region of the coil array as shown for a single phase in Figure 2.44 (three layers only are shown). The three phase schematic is shown in Figure 2.45 for a single 𝜆-coil group. Note that as per standard three phase motors, only two currents are actively controlled, with the third current automatically generated by the wye- or star-center configuration. Because the commutation law and the physical placement of the coil traces mean that 𝐼𝑎, 𝐼𝑏 and 𝐼𝑐 have a 60∘ electrical angle between them, the standard three phase wiring 115  is modified as shown and the star center insures that 𝐼𝑎 − 𝐼𝑏 + 𝐼𝑐 = 0, i.e. the three phase currents balance. The current controller actively commands only two currents, 𝐼𝑎 and −𝐼𝑏 with 𝐼𝑐 automatically generated.   Figure 2.44 Single phase multilayer windings, shown for three layers of Y-elongated traces.   To minimize the number of drivers required for the alpha prototype, series connections between different 𝜆-coil groups are desired. Figure 2.46 shows an example of two 𝜆-coil groups connected in series with three-phase wiring implemented.  116   Figure 2.45 Three phase wiring, shown for a single 𝝀-coil group.  Figure 2.46 Serial connections between two single 𝝀-coil groups with 3-phase wiring. 117  In addition to proving the basic synchronous levitation motor concept, it was desired that the coil array PCB be able to demonstrate at least one method of scaling up the planar stroke. Figure 2.47 shows a multiple coil array PCB tile arrangement. Y-elongated coil traces on each board (I, II, III) are attached in series using connector PCBs on the bottom side of the coil array, with individually soldered pins. This allows the use of multiple coil array PCBs without interfering with the working area of the mover and minimal increase in the number of current drivers. We mention this additional functional requirement to explain the difference in layout geometry for the connection points for the X-elongated and Y-elongated coil trace layers, as well as the choice of the X-elongated traces as the first layer (making the Y-stroke have a higher maximum acceleration; the Y-direction is the long stroke axis in the multi-tile configuration). Figure 2.48 shows the coil layer structure. As shown, the via connections for the Y-elongated traces are optimized for minimum Y-width (to minimize the non-uniform coil trace region) and minimal interference with the X-elongated traces. The via connections for the X-elongated traces are outside the working region of the multi-tile coil array stator and are therefore laid out more simply, in straight lines. Figure 2.49 shows the overall coil array PCB layout, with 16 Y-elongated 𝜆-coil groups and 9 X-elongated 𝜆-coil groups. Groups of the same colour are in series (e.g. 𝐺𝑌1 is in series with 𝐺𝑌9). Series groups are chosen based on the stroke of the mover and the need to dynamically shift the active coil zones for each magnet array. The active coil zones for each array are four 𝜆 -coil groups wide. The Y-elongated coil traces provide actuation in the longer stroke for the alpha prototype, the X-direction. A minimum of eight individually addressable Y-elongated coil groups allows the independent actuation of (Y2, Y4); with the series connection shown it is possible to traverse the entire X-stroke of the stator with only the drivers necessary for these eight coil groups. The X-elongated coil traces provide actuation in the short axis (Y-direction) which is less than the 118  length of the mover and therefore coil groups (𝐺𝑋1, 𝐺𝑋2, 𝐺𝑋3) actuate only magnet array X1 and coil groups (𝐺𝑋7, 𝐺𝑋8, 𝐺𝑋9) actuate only magnet array X3 across the whole Y-stroke of the mover. These groups are therefore wired in series to save on the number of drives.  Figure 2.47 Extendable coil array PCBs. Inset shows connector PCB with solid pin connectors. 119   Figure 2.48 Coil structure isometric view, X- and Y-elongated coils. (a) Layer X1: x-elongated traces. (b) Layer Y1: y-elongated traces. 120   Figure 2.49 Coil array PCB showing coil groupings, X- and Y-elongated coils.  A 16 layer PCB is designed with 8 layers of X-elongated coil traces and 8 layers of Y-elongated coil traces. Copper thickness of 6 oz corresponding to 210 𝜇𝑚 chosen based on available 121  heavy copper PCB manufacturing capability and the trade-off between different force coefficients between planar axes and maximum acceleration in the strong planar axis. The minimum insulation thickness available for this weight of copper was 𝑡𝑖𝑛𝑠 = 127 𝜇𝑚. The minimum insulation gap for this copper thickness based on a standard copper etching process is 𝑔𝑐 = 254 𝜇𝑚, making the coil width 𝑊𝑐 = 4.746 𝑚𝑚. Figure 2.50 shows the final board stack-up, with corresponding separation thickness 𝑡𝑔 = 464 𝜇𝑚.   Figure 2.50 Coil array PCB stack-up details. 122  Motor parameters are summarized in Table 2.6. From (2.85), (2.86) the force coefficients for each magnet array are 𝐾𝑋 = 3.28 [𝑁𝐼𝑟], 𝐾𝑌 = 3.06 [𝑁𝐼𝑟]. The maximum force achievable for equal current is 7% lower for the weak stroke direction. We note that the stack-up fill factor for the coil trace array is 63%, and the area fill factor for each coil trace layer is 94%, significantly higher than achievable through typical motor coil windings using round wire. In addition, the PCB feature creation technology automatically allows for alignment tolerances on the order of 50 micron without further effort.  ALPHA PROTOTYPE MOTOR PARAMETERS Parameter Symbol Value Units Halbach wavelength 𝜆   30  mm Magnet remanence 𝐵𝑟   1.325  T Magnet height 𝐻𝑚   7.5  mm Magnet depth 𝐷𝑚   120  mm Magnet array width 𝑊𝑚   60  mm Halbach segment gap 𝑔   50 𝜇m Coil trace width 𝑊𝑐   4.746  mm Coil trace thickness 𝑡𝑐   0.210 mm Coil separation thickness 𝑡𝑔   0.464  mm Coil layer number in each direction 𝑁   8  -- Table 2.6  Alpha Prototype Motor Parameters.   123  2.4.4 Stator Design Figure 2.51 shows the exploded view of the stator assembly. The coil array PCB is supported by a ceramic stiffener made from an electrically insulating quartz-based ceramic. The stator base frame is a lightweight and stiff honeycomb of the same stack-up dimensions as the mover chassis in 2.4.2.3. The coil array PCB is bonded to the ceramic stiffener which in turn is bonded to the top surface of the base frame. The metrology reference frame is a removable part that is installed on the base frame using a kinematic mount. The kinematic mount divorces the metrology frame from internal stresses that may be induced on the stator base frame and coil array by (for example) Ohmic heating. The mounting blocks for the kinematic mount as well as the electrical terminal blocks used to strain relieve the coil array wiring (not shown) are attached to the top surface of the base frame using M2 and M3 socket head cap screws.   Figure 2.52 shows detailed dimensions of the stator components. The thickness of the ceramic stiffener insures that the field strength at the top surface of the aluminum skinned base frame is 0.6% of the field strength at the top surface of the coil array, providing low eddy current losses without unduly increasing the overall stator height.  124   Figure 2.51 Alpha prototype stator exploded view.   125   Figure 2.52 Alpha prototype stator isometric cut-view.  Figure 2.53 shows the details of the metrology reference frame. The monolithic block is solid aluminum. Three 12mm diameter silicon nitride ceramic balls are bonded to cones machined into the bottom surface of the monolithic block. These ceramic balls mate with commercially available v-groove blocks (VB-375-SM from Bal-tec™ [87]) mounted to the base frame to form a minimally constrained kinematic mount. The weight of the monolithic block is sufficient to provide a preload for the mount. Kinematic constraints allow for repeatable and stress free installation of the metrology reference frame. The height of the reference frame is designed so that 126  the reference block’s IR-LEDs are in the same focal plane as the IR-LEDs on the mover for a flying height of 1 mm. This minimizes position feedback error due to lens distortion. The IR-LEDs and constant current circuit PCB are identical to those used on the mover, with the exception of the lithium-ion battery which is unnecessary since the metrology frame is stationary and can be powered by a benchtop power supply.   Figure 2.53 Isometric cut-view of metrology reference frame.   127  2.5 6-DOF Prototype Manufacturing and Assembly 2.5.1 Mover Sub-arrays are built using the assembly jig shown in Figure 2.54. To partially stabilize the magnet segments in the Halbach array, a non-magnetic spacer/datum plate is placed on top of a magnetic steel base, with datum plane blocks attached to the spacer plate. The attraction force between the magnet segments and the magnetic base help stabilize the magnet segments as they are brought close together. Three planes constrain the sub-array in XYZ, and lateral and vertical clamping screws are used to push the magnet segments against the datum planes. Loctite E-120HP two part epoxy [88] was used to bond each magnet segment together, and plastic spacing sheets (not shown) were used to isolate the epoxy from the assembly jig. Figure 2.55 shows the actual assembly jig with magnet sub-array installed. 128   Figure 2.54 Magnet sub-array assembly jig.  129   Figure 2.55 Magnet sub-array assembly jig with magnet array installed.   To assemble the entire mover, a magnet array assembly jig was designed (Figure 2.56). A non-magnetic stainless steel spacer/datum plate is used as the datum surface on which the mover is built and is sandwiched between the sub-arrays and the magnetic steel base plate. The magnetic steel base plate helps stabilize the sub-arrays when they are brought close together in the XY plane, and provides more than gravity preload to push the magnet sub-arrays onto the spacer/datum plate. The magnet arrays are placed strong side facing this plate. Screw clamp devices are used to laterally constrain the magnet sub-arrays while the epoxy cures. The same epoxy [88] is used. The magnetic steel base plate also has M12 tapped holes through which screws can be threaded and used to separate the non-magnetic spacer/datum plate from the magnetic base plate after the epoxy cures. 130   Figure 2.56 Magnet array assembly jig.   Figure 2.57 shows the mover assembly process. In Figure 2.57a, the sub-arrays are brought close together in a stable fashion, and the lateral screw clamps begin to push them together. In Figure 2.57b the lateral screw clamps are fully tightened and the magnet sub-arrays locations are 131  checked. The mover chassis is epoxied to the weak side of the magnet arrays in Figure 2.57c. Once the epoxy has cured, the magnet arrays and the mover chassis are an integral unit; however, now the force between the mover and the magnetic steel base plate is very large. Separation screws threaded into the base plate are used to force the spacer/datum plate apart from the base plate against the large normal force between mover and base plate until the mover can be detached from the spacer/datum plate (Figure 2.57d). After the mover is removed from the magnet array assembly jig, the IR-LED and battery circuit PBCs are test fit (Figure 2.57e). Figure 2.57f shows the bottom (strong side) of the mover with flux sensitive paper showing |𝐵𝑧|. 132   Figure 2.57 Magnet assembly.    133  2.5.2 Stator The coil array PCB was manufactured by a commercial vendor using a standard heavy copper PCB process. Figure 2.58 shows the coil array PCB as it arrived from the vendor. Figure 2.58b shows the end turn wiring partially installed using 12AWG solid core wire. Using solder paste and conventional solder, care was taken to insure that solder flowed all the way through the via connections to create minimal electrical resistance between the solid core wire connection and the coil traces. Figure 2.58c shows the ceramic stiffener epoxied to the coil array PCB after the end turn wiring is installed using [88]. Figure 2.58d shows the now integral PCB/stiffener unit being epoxied to the honeycomb base frame. The motor is to be installed on the cement floor, so a rubber damping layer is added to the bottom of the base frame to both provide grip and some vibration isolation (Figure 2.58e). The complete stator with the kinematic mount v-groove blocks installed using M2 screws and three-phase wiring complete is shown in Figure 2.58f. The coil array structure is visible from the edge of the PCB (Figure 2.58f). 134   Figure 2.58 Stator assembly steps.  135  2.5.3 Metrology Reference Frame The metrology reference frame monolithic block is machined from a single block of 7071-T6 aluminum. Figure 2.59a shows the top side of the monolithic block with features for attachment of the IR-LED PCBs and the constant current circuit PCB. Figure 2.59b shows the bottom side of the monolithic block with cone features to accept the kinematic mount silicon nitride balls. Figure 2.59c shows the bottom side of the monolithic block after the silicon nitride balls have been bonded to the block using [88]. Figure 2.59d shows the complete metrology reference frame with IR-LEDs and constant current circuit PCB installed.    Figure 2.59 Metrology reference frame assembly.   136  2.5.4 Complete System The complete motor system is shown in Figure 2.60, with metrology reference frame installed. The position sensor frame is shown in the background. Not pictured are the 48-channel switching amplifier and real time computer. The maximum force and acceleration parameters for a flying height of 0.5 mm and maximum current density of 20 A/mm2 is shown in Table 2.7. Due to amplifier limitations, these values were not experimentally tested beyond 1g accelerations.   Figure 2.60 Synchronous planar levitation motor system.     137  PREDICTED PERFORMANCE OF ALPHA PROTOTYPE MOTOR Parameter Symbol Value Units Mover mass 𝑚  2.3  kg Maximum current density 𝐽𝑚    20  A/mm2 Maximum levitation height 𝑧𝑓𝑚𝑎𝑥   11.6  mm Maximum levitation force 𝐹𝑧𝑚𝑎𝑥   228  N Maximum x-translation force 𝐹𝑥𝑚𝑎𝑠    118  N Maximum y-translation force 𝐹𝑦𝑚𝑎𝑠    110  N Maximum levitation acceleration 𝑎𝑧𝑚𝑎𝑥   9.9  g Maximum x-translation acceleration 𝑎𝑥𝑚𝑎𝑠   5.2  g Maximum y-translation acceleration 𝑎𝑦𝑚𝑎𝑠   4.9  g Table 2.7  Predicted Alpha Prototype Motor Performance at flying height 0.5 mm.        138  2.6 Motion Tracking Results with 6-DOF Prototype 2.6.1 Plant Modeling and Control Architecture The eight actuating forces shown in Figure 2.1 are related to the mover translation and Euler angles via the following dynamic equation:  [     𝑀?̈?𝑀?̈?𝑀?̈?𝐽𝑥?̈?𝐽𝑦?̈?𝐽𝑧?̈? ]     =[     0 0 1 0 0 0 1 01 0 0 0 1 0 0 00 1 0 1 0 1 0 1ℎ(𝑧) 𝑎 0 𝑏 ℎ(𝑧) −𝑎 0 −𝑏0 −𝑏 −ℎ(𝑧) 𝑎 0 𝑏 −ℎ(𝑧) −𝑎𝑏 0 −𝑏 0 −𝑏 0 𝑏 0 ]     [        𝐹𝑦1𝐹𝑧1𝐹𝑥2𝐹𝑧2𝐹𝑦3𝐹𝑧3𝐹𝑥4𝐹𝑧4]          (2.88)  where 𝑀 is the mover mass, (𝐽𝑥, 𝐽𝑦, 𝐽𝑧) are the moments of inertia around three principal axes, (𝑎, 𝑏) are the X-distance and Y-distance between the magnet array Y2 centre and the mover center of gravity (CG) as defined in Figure 2.41, and ℎ(𝑧) is the Z-distance between the mover CG and the stator coil. The ℎ(𝑧) term represents the dynamic coupling between X-Y translation and pitch/roll motion. For example, acceleration in Y will also induce the mover to rotate around the X axis, because the motor forces 𝐹𝑋1 and 𝐹𝑋3 act laterally below the mover, instead of ideally through the mover CG. In addition, this coupling coefficient varies with the mover Z position. To eliminate motion coupling and control the mover in 6-DOF, a multi-input-multi-output (MIMO) motion controller is implemented in the digital controller (Figure 2.61). The MIMO controller output (𝐼𝑥𝑟, 𝐼𝑧𝑟) for each magnet array is demodulated into 3-phase currents (𝐼𝑎, 𝐼𝑏, 𝐼𝑐) for each array, and these current commands are dynamically mapped to the stator X and Y coils in the coil switching block based on the mover position. The coordinate transform block converts the position sensor output into the mover CG translation and rotation coordinates. 139  The MIMO controller allows the implementation of 6 independent SISO control loops. Due to sensor feedback limitations which are beyond the scope of this work, a loop transmission cross-over frequency of 50 Hz was implemented for each axis, with approximately 100Hz closed loop -3dB bandwidth. The limitation is purely due to the feedback sensor and not the drive electronics or the levitation motor itself. The approximate minimum dynamic stiffness within the closed loop bandwidth is calculated as 𝑘𝑑𝑦𝑛𝑎𝑚𝑖𝑐,𝑚𝑖𝑛 = (2𝜋𝑓𝑏𝑤)2𝑚, with the closed loop bandwidth 𝑓𝑏𝑤 = 100 𝐻𝑧 and the mass of the mover 𝑚 = 2.3 𝑘𝑔, giving 𝑘𝑑𝑦𝑛𝑎𝑚𝑖𝑐,𝑚𝑖𝑛 =0.9 𝑁/𝜇𝑚. Within the bandwidth of the system the dynamic stiffness will be higher, with infinite DC stiffness due to integrator action within the load capacity of the motor.   Figure 2.61 Controller block diagram.  2.6.2 6-DOF Tracking Results  The sensor has a position update rate of 8 kHz and a root-mean-square (RMS) error in the three translation axes of 2.0 𝜇𝑚, 2.1 𝜇𝑚, 5.1 𝜇𝑚 in 𝑥, 𝑦, 𝑧 respectively, and regulation RMS error in the three rotational axes of  0.0014∘, 0.0013∘, 0.0004∘ in 𝛼, 𝛽, 𝛾 as discussed in [1]. For a stationary 140  position command in all 6 axes, the mover has position regulation RMS errors equal to this sensor noise floor. Motion tracking results for a reference elliptical path are shown in Figure 2.62; the reference path is for 260 mm X-stroke and 60 mm Y-stroke at a flying height of 1mm above the coil array PCB surface. Figure 2.63 shows the 6-DOF tracking error for this trajectory. Tracking RMS error in the three translation axes is 7.49 𝜇𝑚, 5.02 𝜇𝑚, 9.20 𝜇𝑚 in 𝑥, 𝑦, 𝑧 respectively, and RMS error in the three rotational axes of  0.0066∘, 0.0082∘, 0.0029∘ in 𝛼, 𝛽, 𝛾 respectively.  The alpha prototype succeeds in its purpose as a proof of concept; therefore we leave the minimization of this tracking error using this position sensor to future work. In theory, the achievable precision of the synchronous levitation motor is largely limited by the sensor noise floor. If nanometer-level precision is desired, laser interferometers with a target mirror block can be used as in a lithography machine.  The achievable translation stroke for the alpha prototype is 260 𝑚𝑚 × 60 𝑚𝑚 × 2.5 𝑚𝑚 (Figure 2.64 shows the mover floating at 𝑧𝑓 = 2.5 𝑚𝑚). Small rotational strokes are achievable, on the order of a ±5 degrees in 𝛾, with rotation in tip/tilt only limited by the flying height. 141   Figure 2.62 Elliptical path of mover in XY, at 𝒛𝒇 = 𝟏 𝒎𝒎. 142   Figure 2.63 Tracking error for elliptical path, 𝒛𝒇 = 𝟏 𝒎𝒎.   Figure 2.64 Mover floating with 𝒛𝒇 = 𝟐. 𝟓 𝒎𝒎.  143  Chapter 3: Asynchronous Planar Levitation Stage  We present in this chapter an analysis of levitation and drag torque of an asynchronous planar levitation motor element using rotating permanent magnet arrays. The conceptual design of a levitation and propulsion motor based on rotating permanent magnets was first presented in [7] with a stable 6-DOF planar levitation stage described in [3] and utilized in [9] [8]. The advantage of asynchronous levitation methods over synchronous magnetic levitation are: i) very simple and passive stator structure requiring at a minimum only an electrically conducting medium; ii) passively stable in all axes under gravity preload, therefore requiring no feedback for stable levitation and actuation. This work focusses on modeling and analysis of the field distribution and generation of levitation and drag forces for a linear 2D motor geometry of infinite horizontal extent, which is extended to an analytical expression for force and torque for rotating 3D magnet disk geometries. The novel analytical expression for force and torque can yield design insight into the motor topology without the need for sensitivity studies using 3D FEA or experimentation. This chapter consists of:  6-DOF asynchronous levitation motor basic concept and working principle.  Modeling and novel analysis for a limiting 2D case.  Extension of the analytical expression for force, drag and power from 2D geometry to 3D geometry using superposition of fields of cylindrical shell elements.   Experimental load test of single magnet disk over a conducting stator and comparison to the analytical shells model.   144  3.1 Asynchronous Planar Levitation Stage Concept and Working Principle The 6-DOF asynchronous planar levitation stage is comprised of a mover with multiple magnet disks with independently controllable rotation speeds and tilt/tip angles attached to a single mover chassis, levitating over a stationary conducting slab. Figure 3.1 shows a particular embodiment of the planar stage with four rotating magnet disks attached to a mover chassis (note that details of the tilt/tip mechanism are not shown, nor are the onboard power source or controllers for the rotary motors). Each magnet disk is a circumferential Halbach array with the strong field side on the –z face of the disk, and is driven by a rotary motor. In typical operation, pairs of magnet disks are driven in counter-rotation. That is, magnet disk 1 is typically driven in an opposing rotation to magnet disk 3; and similarly magnet disk 2 is driven in opposition to the rotation of magnet disk 4. Each rotating magnet disk generates an asynchronous induction field in the conducting body which acts to repel and counteract the motion of the disk; thus when the disk plane is parallel with the conducting slab, each magnet disk is capable of generating a z-directed levitating force and a drag torque in the axis of rotation. When the disk plane is not parallel with the face of the conducting slab, each disk will produce a z-directed levitation force, a drag torque around the axis of rotation, and a propulsion force in the XY plane.  Figure 3.2a shows the principle of propulsion and levitation in the zero tilt/tip case. Magnet disk 1 rotates at 𝜔1 in the +z-direction and generates a drag torque 𝑇𝑧1 in the –z-direction, and a levitation force 𝐹𝑧1. Similarly magnet disk 3 rotates at 𝜔3 in the –z-direction and magnet disk 3 generates a drag torque 𝑇𝑧3 in the +z-direction and a levitation force 𝐹𝑧3. When 𝜔1 = 𝜔3, the drag torques on each disk have equal magnitudes and the torque contribution on the mover due to the pair of magnet disks is 𝑇𝑧1 + 𝑇𝑧3 = 0, and the levitation force contribution is 𝐹𝑧1 + 𝐹𝑧3 and is always in the +z-direction (because the induced field in the conductor always acts to repel the 145  permanent magnet field from the magnet disk). For 𝜔1 < 𝜔3, |𝑇𝑧1| < |𝑇𝑧3| and a +z-directed torque is generated on the mover as well as a +y-directed torque. Similarly, for 𝜔1 > 𝜔3, |𝑇𝑧1| >|𝑇𝑧3| and a net –z-directed torque is produced on the mover and a -y-directed torque. In each driving case, the levitation forces on each magnet disk are in the +z-direction and will always produce a levitation force contribution in the +z-direction.  Figure 3.2b shows the principle of propulsion and levitation in the non-zero tilt/tip case. Magnet disk 1 is rotated by 𝛽 in the -y-direction while spinning about its own axis at 𝜔1 in the +z-direction, producing drag torque drag torque 𝑇𝑧1, levitation force 𝐹𝑧1 and translation force 𝐹𝑦1. The translation force is produced because the left-hand side of the disk is at a lower flying height over the conducing slab, producing a larger induction field acting on that half of the disk; thus a differential in-plane force is produced. Similarly, magnet disk 3 is rotated by 𝛽 in the +y-direction while spinning about its own axis at 𝜔3 in the -z-direction, producing drag torque drag torque 𝑇𝑧3, levitation force 𝐹𝑧3 and translation force 𝐹𝑦3. For positive values of 𝛽, a +y-directed force is generated on the mover. Thus by controlling the tilt angle 𝛽 in the y-axis and the rotational velocities 𝜔1, 𝜔3, the pair of magnet disks 1,3 is capable of generating controllable levitation force, translation force in y-direction, and torques in the y-axis and z-axis. Similarly by controlling the tip angle 𝛼 in the x-axis and the rotational velocities 𝜔2, 𝜔4, the magnet disks 2,4 are capable of generating a levitation force, a translation force in the x-direction, and torques in the x-axis and z-axis. Combining all four magnet disks allows 6-DOF actuation of the mover. We note that the axes are coupled. Under gravity preload, the mover is passively stable in all axes.  146   Figure 3.1 Asynchronous levitation planar motor concept using four levitation disks, magnetization shown in red for magnet disk 1 (central actuated pivot mechanism and onboard battery pack/motor controllers not shown).    147    Figure 3.2 Levitation and propulsion principle (details of actuated pivot mechanism not shown).         148  3.2 Asynchronous Levitation Machine Force and Torque Modeling We desire to develop a purely physics based analytical model that can be used as a design tool to develop optimized asynchronous levitation planar machines, without having to rely on experimentally derived correction factors. Here we focus on levitation and drag torque of the magnet disk over a widely extended stationary conducting slab, and leave propulsion modeling to future work. A 2D asynchronous levitation machine comprised of an infinitely extended linear Halbach array and an infinitely extended conducting slab with finite thickness is analyzed. Analytical expressions for the levitation force and drag force are derived that are functions only of material parameters and magnet array and stator geometry. The 2D analytical model is then extended to the case of a rotating 3D circumferential Halbach array over a conducting slab of finite thickness by first splitting the problem into cylindrical shell elements with infinitesimally thin radial thickness, solving for the fields generated by each shell element using the 2D model assuming no radial flux leakage, and superimposing the generated fields from each cylindrical shell element; we term this the analytical shells method. This analytical shells method provides a useful design tool that gives levitation force and drag torque for any geometry of magnet array and stator conductor.   3.2.1 Infinitely Wide 2D Moving Magnet Array over Homogeneous Conducting Slab Linear induction motors have been well analyzed in the literature [65], [55]. The asynchronous machine shown in Figure 3.3 has a permanent magnet array with periodic magnetization distribution 𝑴, relative permeability 𝜇𝑟 = 1 and thickness 𝐻𝑚 over a homogeneous conducting slab of conductivity 𝜎, relative permeability 𝜇𝑟 = 1 and thickness 𝐷𝑐. Both magnet array and conducting slab are of infinite horizontal extent, and depth in 𝑦𝑚 is considered sufficient such that 149  the problem is 2D. We note that only the relative motion between the permanent magnet source and the conducting body is relevant to induced forces, therefore we are free to choose to make either one or the other body stationary for ease of analysis (this relative velocity is the slip speed as discussed in the literature). Here we define the relative motion between the magnet array and the conducting slab using the magnet frame of reference 𝑥𝑚, 𝑦𝑚, 𝑧𝑚 with a stationary magnet array and the conductor translating in the 𝑥𝑚-direction at 𝑣𝑥  [𝑚𝑠]. We are interested in the field solutions after steady state conditions have been established and therefore MQS assumptions can be applied. Five regions are defined with boundaries as shown in Figure 3.3: region I is the air above the magnet array (𝑧𝑚 ∈ (𝐻𝑚, ∞)), region II is the permanent magnet region (𝑧𝑚 ∈ [0, 𝐻𝑚]), region III is the air gap between the magnet array and the conducting slab (𝑧𝑚 ∈ (−𝑧𝑓 − 𝐷𝑐 , −𝑧𝑓)), region IV is the conducting slab, and region V is the air below the conducting slab (𝑧𝑚 ∈ (−∞,−𝑧𝑓 −𝐷𝑐]).  As in 2.2.4, we define a magnetic vector potential 𝑨 such that the magnetic flux density everywhere is 𝑩 = ∇ × 𝑨 (2.24). The vector potential expression in the absence of surface currents is once again (2.27). The magnetization distribution is defined as in 2.2.4. Since the problem is 2D, the vector potential has only a single component in the 𝑦𝑚-direction so 𝑨 = 𝐴𝑦𝑗,̂ everywhere. From Gauss’ law (2.4), the definition of the flux density (2.24) and Ampere’s law (2.2) and the absence of currents in the air regions (I, III, V), we obtain the Laplace equations for regions I, III and V as  ∇2𝑨 = 0 (3.1)  150  For region II, assuming the permanent magnetic material is electrically insulating (relative to the conducting slab this is a good assumption), from (2.26) we obtain  Figure 3.3 Stationary Halbach array of infinite extent over a translating conducting slab.   151     ∇2𝑨 = −𝜇𝑜(∇ × 𝑴) (3.2)  just as in 2.2.4. The conducting slab region will have induced currents because every part of the conducting slab will experience a changing magnetic field as the slab moves relative to the permanent magnet array. Re-arranging (2.6) and plugging in (2.2), we get the electric field within the conducting slab purely as a function of the total field distribution and the velocity of the conducting slab:  𝑬 =1𝜇𝑜𝜎(∇ × 𝑩) − 𝒗 × 𝑩 (3.3)  Substituting (3.3) into Faraday’s induction law (2.1), we get ∇ × {1𝜇𝑜𝜎(∇ × 𝑩) − 𝒗 × 𝑩} = −𝜕𝑩𝜕𝑡 and rearranging,  1𝜇𝑜𝜎∇ × ∇ × 𝑩 = −𝜕𝑩𝜕𝑡+ 𝒗 × 𝑩 Applying the vector identity (∇ × (∇ × 𝑩) =  ∇(∇ ∙ 𝑩) − ∇2𝑩) and Gauss’ law (2.4), we obtain  −1𝜇𝑜𝜎∇2𝑩 = −𝜕𝑩𝜕𝑡+ 𝒗 × 𝑩 (3.4)  We have assumed steady state conditions and a stationary magnet array, with only the conductor moving with some velocity 𝒗 = 𝑣𝑥 ?̂?. Therefore 𝜕𝑩𝜕𝑡= 0, and (3.4) reduces  to the vector Poisson equation   −1𝜇𝑜𝜎∇2𝑩 = 𝒗 × 𝑩 (3.5)  From (3.5) two scalar Poisson’ equations arise: 152   𝜕2𝐵𝑥𝜕𝑥𝑚2   +𝜕2𝐵𝑥𝜕𝑧𝑚2   = −𝜇𝑜𝜎𝑣𝑥𝜕𝐵𝑧𝜕𝑧𝑚 (3.6)  and   𝜕2𝐵𝑧𝜕𝑥𝑚2   +𝜕2𝐵𝑧𝜕𝑧𝑚2   = 𝜇𝑜𝜎𝑣𝑥𝜕𝐵𝑧𝜕𝑥𝑚 (3.7)   The magnetization distribution for a general four segment Halbach array is periodic in 𝑥𝑚 and uniform within the magnet volume of each segment as depicted in Figure 2.14. We use the fundamental component of a four segment Halbach array as defined in (2.32) for the following analysis. This analysis can be identically repeated for any spatial harmonic of magnetization by substituting the magnetization fundamental component for any other magnetization harmonic component. The full field solution will be the superposition of all solutions for each harmonic. As will be shown in later sections, the fundamental component of magnetization is responsible for the vast majority of the force generation on asynchronous levitation machines, so we focus on the solution using just this Fourier component.   Within the conductor volume, we can guess the form of the magnetic field density as   𝑩𝐼𝑉 = [𝐵𝑥𝐼𝑉𝐵𝑦𝐼𝑉𝐵𝑧𝐼𝑉] = 𝑅𝑒 [{𝑚𝐼𝑉𝑒𝛼𝑐𝑧𝑚 + 𝑛𝐼𝑉𝑒−𝛼𝑐𝑧𝑚}𝑒−𝑗𝑥𝑚/𝜆𝑐0{𝑓𝐼𝑉𝑒𝛼𝑐𝑧𝑚 + 𝑔𝐼𝑉𝑒−𝛼𝑐𝑧𝑚}𝑒−𝑗𝑥𝑚/𝜆𝑐] (3.8)  The field everywhere must obey (2.4) and therefore we can immediately find that since ∇ ∙ 𝑩𝐼𝑉 =0, −𝜕𝐵𝑥𝐼𝑉𝜕𝑥𝑚=𝜕𝐵𝑧𝐼𝑉𝜕𝑧𝑚 and therefore   𝑚𝐼𝑉 =𝛼𝑐𝜆𝑐𝑗𝑓𝐼𝑉 (3.9)  and 153   𝑛𝐼𝑉 = −𝛼𝑐𝜆𝑐𝑗𝑔𝐼𝑉 (3.10)  So (3.9) becomes  𝑩𝐼𝑉 = 𝑅𝑒[   {𝛼𝑐𝜆𝑐𝑗𝑓𝐼𝑉𝑒𝛼𝑐𝑧𝑚 −𝛼𝑐𝜆𝑐𝑗𝑔𝐼𝑉𝑒−𝛼𝑐𝑧𝑚}𝑒−𝑗𝑥𝑚/𝜆𝑐0{𝑓𝐼𝑉𝑒𝛼𝑐𝑧𝑚 + 𝑔𝐼𝑉𝑒−𝛼𝑐𝑧𝑚}𝑒−𝑗𝑥𝑚/𝜆𝑐 ]    (3.11)  Evaluating the scalar Poisson’s equation (3.6) using (3.12), we obtain the relation  (−1𝜆𝑐2+ 𝛼𝑐2)𝐵𝑥𝐼𝑉 = −𝜇𝑜𝜎𝑣𝑥𝜕𝐵𝑧𝐼𝑉𝜕𝑧𝑚 (3.12)  Evaluating (3.13) using (3.12), we find that the 𝑧𝑚-direction field decay term 𝛼𝑐 depends only on the material properties, the magnetization harmonic characteristic wavelength 𝜆𝑐and the velocity of the conducting slab:  𝛼𝑐 =1𝜆𝑐√1 − 𝑗𝜇𝑜𝜎𝑣𝑥𝜆𝑐   (3.13)  Just as in 2.2.4, boundary conditions are obtained from the continuity of vector potential, field continuity based on Gauss’ law (2.4), Ampere’s law (2.2) and (2.5). Boundary conditions are summarized below:   𝐵𝑥,𝑎 − 𝐵𝑥,𝑏 = −𝜇𝑜𝑀𝑥,𝑏 (3.14)   𝐵𝑧,𝑎 − 𝐵𝑧,𝑏 = 0 (3.15)   𝐵𝑥,𝑐 − 𝐵𝑥,𝑑 = 𝜇𝑜𝑀𝑥,𝑐 (3.16)  154   𝐵𝑧,𝑐 − 𝐵𝑧,𝑑 = 0 (3.17)   𝐵𝑥,𝑒 − 𝐵𝑥,𝑓 = 0 (3.18)   𝐵𝑧,𝑒 − 𝐵𝑧,𝑓 = 0 (3.19)   𝐵𝑥,𝑔 − 𝐵𝑥,ℎ = 0 (3.20)   𝐵𝑧,𝑔 − 𝐵𝑧,ℎ = 0 (3.21)    𝐴𝑦,𝑎 = 𝐴𝑦,𝑏 (3.22)   𝐴𝑦,𝑐 = 𝐴𝑦,𝑑 (3.23)  where the subscript denotes both the direction of the field component and the boundary location.  As in 2.2.4, the field 𝑩 as well as the vector potential 𝑨 go to zero at 𝑧𝑚 = −∞,∞. We know the vector potential must be continuous everywhere and can confirm this separately, but only use this property to evaluate the boundaries (𝑎, 𝑏) and (𝑐, 𝑑).  We guess that 𝐴𝑦 in region I has the form   𝐴𝑦 = 𝑅𝑒([𝑓𝐼𝑒𝑧𝑚𝜆𝑐 + 𝑔𝐼𝑒−𝑧𝑚𝜆𝑐 ]𝑒−𝑗𝑥𝑚/𝜆𝑐)  (3.24)  Similarly in regions III and V, we guess that 𝐴𝑦 has the form: 155   𝐴𝑦 = 𝑅𝑒([𝑓𝐼𝐼𝐼𝑒𝑧𝑚𝜆𝑐 + 𝑔𝐼𝐼𝐼𝑒−𝑧𝑚𝜆𝑐 ]𝑒−𝑗𝑥𝑚/𝜆𝑐)  (3.25)   𝐴𝑦 = 𝑅𝑒([𝑓𝑉𝑒𝑧𝑚𝜆𝑐 + 𝑔𝑉𝑒−𝑧𝑚𝜆𝑐 ]𝑒−𝑗𝑥𝑚/𝜆𝑐)  (3.26)  The vector potential in region II, the permanent magnet region, is defined as per (2.38) for the fundamental spatial component as   𝐴𝑦 = 𝐴𝑦𝑝 + (𝐴𝑦,𝑏 − 𝐴𝑦𝑝)𝑠𝑖𝑛ℎ (𝑧𝑚𝜆𝑐)𝑠𝑖𝑛ℎ (𝐻𝑚𝜆𝑐)   − (𝐴𝑦,𝑐 − 𝐴𝑦𝑝)𝑠𝑖𝑛ℎ (𝑧𝑚 − 𝐻𝑚𝜆𝑐 )𝑠𝑖𝑛ℎ(𝐻𝑚𝜆𝑐)    (3.27)  where 𝐴𝑦𝑝 is the particular solution to the Poisson equation (3.2):  𝐴𝑦𝑝 = 𝑗𝜇𝑜𝜆𝑐𝑀𝑧1 (3.28)  From (2.24) and the definition of the potential amplitude 𝐴𝑦 in each region other than the conductor, the field along each boundary can be calculated as below:  𝑩𝑎 = 𝑅𝑒[    1𝜆𝑐{−𝑓𝐼𝑒𝐻𝑚𝜆𝑐 + 𝑔𝐼𝑒−𝐻𝑚𝜆𝑐 }𝑒−𝑗𝑥𝑚/𝜆𝑐0−𝑗𝜆𝑐{𝑓𝐼𝑒𝐻𝑚𝜆𝑐 + 𝑔𝐼𝑒−𝐻𝑚𝜆𝑐 }𝑒−𝑗𝑥𝑚/𝜆𝑐]     (3.29)    𝑩𝑏 = 𝑅𝑒[    1𝜆𝑐{−(𝐴𝑦,𝑏 − 𝐴𝑦𝑝) 𝑐𝑜𝑡ℎ (𝐻𝑚𝜆𝑐) + (𝐴𝑦,𝑐 − 𝐴𝑦𝑝)𝑐𝑠𝑐ℎ (𝐻𝑚/𝜆𝑐)}0−𝑗𝜆𝑐𝐴𝑦,𝑏 ]     (3.30)   156   𝑩𝑐 = 𝑅𝑒[    1𝜆𝑐{−(𝐴𝑦,𝑏 − 𝐴𝑦𝑝) 𝑐𝑠𝑐ℎ (𝐻𝑚𝜆𝑐) + (𝐴𝑦,𝑐 − 𝐴𝑦𝑝)𝑐𝑜𝑡ℎ (𝐻𝑚/𝜆𝑐)}0−𝑗𝜆𝑐𝐴𝑦,𝑐 ]     (3.31)    𝑩𝑑 = 𝑅𝑒[    1𝜆𝑐{−𝑓𝐼𝐼𝐼 + 𝑔𝐼𝐼𝐼}𝑒−𝑗𝑥𝑚/𝜆𝑐0−𝑗𝜆𝑐{𝑓𝐼𝐼𝐼 + 𝑔𝐼𝐼𝐼}𝑒−𝑗𝑥𝑚/𝜆𝑐]     (3.32)    𝑩𝑒 = 𝑅𝑒[     1𝜆𝑐{−𝑓𝐼𝐼𝐼𝑒−𝑧𝑓𝜆𝑐 + 𝑔𝐼𝐼𝐼𝑒𝑧𝑓𝜆𝑐}𝑒−𝑗𝑥𝑚/𝜆𝑐0−𝑗𝜆𝑐{𝑓𝐼𝐼𝐼𝑒−𝑧𝑓𝜆𝑐 + 𝑔𝐼𝐼𝐼𝑒𝑧𝑓𝜆𝑐}𝑒−𝑗𝑥𝑚/𝜆𝑐]      (3.33)    𝑩𝑓 = 𝑅𝑒 [−𝑗𝛼𝑐𝜆𝑐{𝑓𝐼𝑉𝑒−𝛼𝑐𝑧𝑓 − 𝑔𝐼𝑉𝑒𝛼𝑐𝑧𝑓}𝑒−𝑗𝑥𝑚/𝜆𝑐0{𝑓𝐼𝑉𝑒−𝛼𝑐𝑧𝑓 + 𝑔𝐼𝑉𝑒𝛼𝑐𝑧𝑓}𝑒−𝑗𝑥𝑚/𝜆𝑐] (3.34)    𝑩𝑔 = 𝑅𝑒 [−𝑗𝛼𝑐𝜆𝑐{𝑓𝐼𝑉𝑒−𝛼𝑐(𝑧𝑓+𝐷𝑐) − 𝑔𝐼𝑉𝑒𝛼𝑐(𝑧𝑓+𝐷𝑐)}𝑒−𝑗𝑥𝑚/𝜆𝑐0{𝑓𝐼𝑉𝑒−𝛼𝑐(𝑧𝑓+𝐷𝑐) + 𝑔𝐼𝑉𝑒𝛼𝑐(𝑧𝑓+𝐷𝑐)}𝑒−𝑗𝑥𝑚/𝜆𝑐] (3.35)    𝑩ℎ = 𝑅𝑒[     1𝜆𝑐{−𝑓𝑉𝑒−(𝑧𝑓+𝐷𝑐)𝜆𝑐 + 𝑔𝑉𝑒(𝑧𝑓+𝐷𝑐)𝜆𝑐 }𝑒−𝑗𝑥𝑚/𝜆𝑐0−𝑗𝜆𝑐{𝑓𝐼𝑒−(𝑧𝑓+𝐷𝑐)𝜆𝑐 + 𝑔𝐼𝑒(𝑧𝑓+𝐷𝑐)𝜆𝑐 }𝑒−𝑗𝑥𝑚/𝜆𝑐]      (3.36)  157  Solving the system of equations from the boundary conditions (3.15)-(3.24) using the field definitions (3.30)-(3.37) and the additional boundary conditions 𝑩 = 0, 𝑨 = 0 at 𝑧𝑚 = −∞,∞, we find that the complex valued coefficient 𝑓𝐼𝑉 is  𝑓𝐼𝑉 = 2𝐵𝑜 (1 − 𝑒−𝐻𝑚𝜆𝑐 )1(𝛼𝑐𝜆𝑐 + 1) + (−𝛼𝑐𝜆𝑐 + 1)(𝛼𝑐𝜆𝑐 − 1𝛼𝑐𝜆𝑐 + 1)𝑒−2𝛼𝑐𝐷𝑐  (3.37)  where 𝐵𝑜 =4𝐵𝑟𝜋cos (𝜋4+𝑔2𝜆𝑐). The field at the conducting slab boundary is  𝑩𝑓 = 𝑅𝑒[    {    −𝑗𝛼𝑐𝜆𝑐[1 −𝛼𝑐𝜆𝑐 − 1𝛼𝑐𝜆𝑐 + 1𝑒−2𝐷𝑐𝛼𝑐]0[1 +𝛼𝑐𝜆𝑐 − 1𝛼𝑐𝜆𝑐 + 1𝑒−2𝐷𝑐𝛼𝑐]}    𝑓𝐼𝑉𝑒−𝑧𝑓/𝜆𝑐𝑒−𝑗𝑥𝑚/𝜆𝑐]     (3.38)  For a conducting slab of infinite thickness (𝐷𝑐 → ∞) the field along the top of the conducting slab simplifies to  𝑩𝑓 = 𝑅𝑒 [{−𝑗𝛼𝑐𝜆𝑐01} 2𝐵𝑜1 − 𝑒−𝐻𝑚𝜆𝑐𝛼𝑐𝜆𝑐 + 1𝑒−𝑧𝑓/𝜆𝑐𝑒−𝑗𝑥𝑚/𝜆𝑐] (3.39)  We can partially validate the field solution by setting the conducting slab velocity to zero (𝑣𝑥 = 0) and comparing the field distribution to that of a static Halbach array (since the conducting slab has a permeability 𝜇𝑜, under static conditions it is considered air). It is easily shown that (3.39) becomes the field solution (2.52) for the stationary infinitely extended Halbach array with 𝑧𝑚 =−𝑧𝑓 for 𝑣𝑥 = 0. To evaluate the force on the conducting slab, we use Maxwell’s stress tensor on the boundary shown in Figure 3.3. The total force on any volume is the integration of the Maxwell stress tensor over the surface of the volume enclosed (Chapter 8 of [65]): 158   𝑭 = ∯ 𝐓 ∙ 𝑑𝒔𝑆 (3.40)  with the stress tensor under MQS conditions defined as   𝐓 = [𝑇𝑥𝑥 𝑇𝑥𝑦 𝑇𝑥𝑧𝑇𝑦𝑥 𝑇𝑦𝑦 𝑇𝑦𝑧𝑇𝑧𝑥 𝑇𝑧𝑦 𝑇𝑧𝑧] (3.41)  Each term within the stress tensor (for a system with uniform permeability 𝜇𝑜) is  𝑇𝑚𝑛 = 𝜇𝑜𝐻𝑚𝐻𝑛 −𝜇𝑜2𝐻𝑘𝐻𝑘𝛿𝑚𝑛 (3.42)  where 𝑚 is the direction of the tensor pressure and 𝑛 is the normal to the surface, and the Kronecker delta 𝛿𝑚𝑛 is   𝛿𝑚𝑛 = [1 𝑚 = 𝑛0 𝑚 ≠ 𝑛] (3.43)  and   𝐻𝑘𝐻𝑘 = 𝐻𝑥2 + 𝐻𝑦2 + 𝐻𝑧2 (3.44)  Examining the boundary drawn around the conducting slab, we note that the bottom surface of the boundary is at 𝑧𝑚 → −∞ with 𝑩 → 0 and will contribute nothing to the force on the slab. The boundaries at 𝑥𝑚 = −∞,∞ will cancel out due to symmetry in the limit. Conveniently, this allows us to evaluate the total force on the conducting slab by evaluating only the field at the surface of the conductor, 𝑩𝒇. The total force on the conductor is therefore   𝑭𝒔𝒍𝒂𝒃 =12𝜇𝑜[𝐵𝑥𝑓𝐵𝑧𝑓012(𝐵𝑧𝑓2 − 𝐵𝑥𝑓2 )] 𝑆𝑙𝐷𝑚 (3.45)  159  where 𝑆𝑙 is the width of the boundary and 𝐷𝑚 is the depth of the array and conductor in the 𝑦𝑚 direction. Because the field is periodic in 𝑥𝑚, the mean force can be evaluated using the identity for spatial averaging of two periodic functions [65]: the spatial average of the product of two periodic functions 𝐺 = 𝑅𝑒(?̂?𝑒−𝑗𝑥𝑚𝜆𝑐 ), 𝐾 = 𝑅𝑒(?̂?𝑒−𝑗𝑥𝑚𝜆𝑐 ) where ?̂?, ?̂? are the complex valued amplitudes of each function is  < 𝐺𝐾 >=12𝑅𝑒[?̂??̂?∗] (3.46)  Here we use the notation <> to mean a spatial averaging operation, and ?̂?∗ is the complex conjugate of ?̂?. Therefore (3.46) becomes   𝑭𝒔𝒍𝒂𝒃 =12𝜇𝑜𝑅𝑒 [?̂?𝑥𝑓?̂?𝑧𝑓∗012(?̂?𝑧𝑓?̂?𝑧𝑓∗ − ?̂?𝑥𝑓?̂?𝑥𝑓∗ )] 𝑆𝑙𝐷𝑚 [𝑁] (3.47)  The magnetic pressure developed on the magnet array is therefore   𝑷𝒎𝒂𝒈𝒏𝒆𝒕 = [𝑃𝑥0𝑃𝑧] = −12𝜇𝑜𝑅𝑒 [?̂?𝑥𝑓?̂?𝑧𝑓∗012(?̂?𝑧𝑓?̂?𝑧𝑓∗ − ?̂?𝑥𝑓?̂?𝑥𝑓∗ )] [Pa] (3.48)  The field amplitudes used to evaluate (3.48) are simply the complex valued amplitudes of the vector components in (3.38) with 𝑓𝐼𝑉 defined as in (3.37):  ?̂?𝑥𝑓 = 2𝐵𝑜 (1 − 𝑒−𝐻𝑚𝜆𝑐 ) 𝑒−𝑧𝑓/𝜆𝑐−𝑗𝛼𝑐𝜆𝑐[1 −𝛼𝑐𝜆𝑐 − 1𝛼𝑐𝜆𝑐 + 1𝑒−2𝐷𝑐𝛼𝑐](𝛼𝑐𝜆𝑐 + 1) + (−𝛼𝑐𝜆𝑐 + 1)(𝛼𝑐𝜆𝑐 − 1𝛼𝑐𝜆𝑐 + 1)𝑒−2𝛼𝑐𝐷𝑐 (3.49)     ?̂?𝑧𝑓 = 2𝐵𝑜 (1 − 𝑒−𝐻𝑚𝜆𝑐 ) 𝑒−𝑧𝑓/𝜆𝑐1 +𝛼𝑐𝜆𝑐 − 1𝛼𝑐𝜆𝑐 + 1𝑒−2𝐷𝑐𝛼𝑐(𝛼𝑐𝜆𝑐 + 1) + (−𝛼𝑐𝜆𝑐 + 1)(𝛼𝑐𝜆𝑐 − 1𝛼𝑐𝜆𝑐 + 1)𝑒−2𝛼𝑐𝐷𝑐 (3.50)  160  We can evaluate the closed form analytical solution for levitation and drag forces (3.48) by comparing it to the results from a finite element simulation in 2D. We use [79] in 2D mode, using an analytical array of 15𝜆 width, and a copper conducting slab with a thickness of 𝐷𝑐 =0.055𝜆 = 2 mm (well below the limit where we would start considering the conductor thickness infinite). Model parameters are listed in Table 3.1. We determine the force on the slab via the Lorentz force integrated through the volume of the conducting slab. Figure 3.4 shows the comparison between the analytical magnetic pressure and average magnetic pressure from the 2D FEA. The error decreases as the mesh for the 2D FEA is refined; for the case shown the maximum error is 1.4% between the 2D analytical model and the FEA. The error decreases as velocity 𝑣𝑥 increases. The effects of field harmonics other than the fundamental should be less than this error.  Figure 3.4 Magnetic pressure on magnet array, 2D analytical model versus 2D FEA.    161  2D ASYNCHRONOUS LEVITATION MACHINE PARAMETERS Parameter Symbol Value Units Halbach wavelength 𝜆   36.13  mm Magnet remanence 𝐵𝑟   1.325  T Magnet height 𝐻𝑚   7.5  mm Halbach segment gap 𝑔   2.682 mm Conductor slab thickness  𝐷𝑐    2 mm Conductor slab conductivity 𝜎   5.998× 107  S/m Flying height 𝑧𝑓   1.7  mm Table 3.1  2D Asynchronous Levitation Machine Parameters.   3.2.2 Analytical Shells Model for Force and Torque of Rotating 3D Magnet Disk over Homogeneous Conducting Slab In this section we extend the linear 2D asynchronous machine model to a 3D rotating permanent magnet asynchronous motor element. Figure 3.5a shows the motor element topology with a circumferential Halbach patterned magnet disk at some height over a stationary conducting slab, with some relative rotational velocity between the two bodies. The magnet disk has a magnet height of 𝐻𝑚, inner radius 𝑅𝑖, and outer radius 𝑅𝑜. The conducting slab is below the magnet disk at a flying height of 𝑧𝑓, and the conductor has a thickness in the z-direction of 𝐷𝑐. The conductor dimensions in the XY plane are large compared to the outer diameter of the magnet disk. The magnet array coordinate frame is fixed to the magnet disk and expressed in cylindrical coordinates (𝑟𝑚, 𝜃𝑚, 𝑧𝑚). The magnetization distribution 𝑴 is now a periodic function of 𝜃𝑚 only and uniform in the other axes. The disk rotates at 𝜔 around the z-axis relative to the stator coordinate frame 162  expressed in Cartesian coordinates (𝑥, 𝑦, 𝑧). We define the Halbach pole number 𝑁𝑚 as the number of full wavelengths of the Halbach array. Figure 3.5b shows a differential cylindrical shell element of the magnet disk at radius 𝑟𝑚. If we imaging cutting the circumferential shell element at some point and unwrapping it to form a linear magnet array, we can form a conceptual equivalence between the 2D linear asynchronous levitation machine discussed in 3.2.1 and the cylindrical shell element, with coordinate transforms:  𝑥𝑚 = 𝜃𝑚3𝐷𝑟𝑚3𝐷 (3.51)   𝑦𝑚 = 𝑟𝑚3𝐷 (3.52)   𝑧𝑚 = 𝑧𝑚3𝐷 (3.53)  where the coordinates (𝑥𝑚, 𝑦𝑚, 𝑧𝑚) are identical to the magnet frame used in 3.2.1 for the 2D infinitely extended case. We note that this shell element thus unwrapped has no end effects in 𝑥𝑚 due to circular closure of the magnet array pattern in 3D. We can thus model the field distribution from each shell as the result of an infinitely extended 2D linear asynchronous machine with infinitesimal depth 𝑑𝑟 using (3.49) and (3.50) determine the field distribution at the top surface of the conductor assuming for the moment no flux leakage in the radial direction. Minimal leakage in the radial direction is a reasonable assumption away from the inner and outer radius of the magnet disk.   The total field is obtained by superimposing the field contributions of each shell element. To obtain the total force on the conductor we can use Maxwell’s stress tensor, and define a closed surface with the top surface at the conductor face 𝑧𝑚3𝐷 = −𝑧𝑓 from 𝑟𝑚3𝐷 ∈ (𝑅𝑖, 𝑅𝑜) and 𝜃𝑚3𝐷 ∈163  (0, 2𝜋); bottom surface at 𝑧𝑚3𝐷 = −∞ with the same radial and angular extents; and cylindrical faces at 𝑟𝑚3𝐷 = 𝑅𝑖 , 𝑅𝑜 from 𝑧𝑚3𝐷 ∈ (−𝑧𝑓 , −∞). As per the analysis in 3.2.1, the surface at 𝑧𝑚3𝐷 =−∞ will contribute zero force, and assuming no radial flux leakage in the first approximation, the cylindrical faces will also contribute no force. Therefore we can determine the differential force and torque generated on each shell element from the average magnetic pressures calculated in (3.48).  By integrating the differential force and torque contributions from all shells along the radius of the magnet array, an approximation of the total force and drag torque on the magnet disk can be calculated, and is equivalent to evaluating the total Maxwell stress tensor.   The Halbach spatial wavelength is a function of the radius 𝑟𝑚 and the Halbach pole number 𝑁𝑚 as  𝜆 =2𝜋𝑟𝑚3𝐷𝑁𝑚 (3.54)  The shell element has a tangential speed at every point 𝑟𝑚3𝐷𝜔; the 2D analytical model in 3.2.1 is defined based on a stationary magnet array and traveling conductor, therefore   𝑣𝑥 = −𝑟𝑚3𝐷𝜔 (3.55)  The differential levitation force in the z-direction on the shell element is obtained by plugging in the now radially dependent wavelength (3.55) and relative conductor velocity (3.56) into (3.48):   𝑑𝐹𝑧 = 2𝜋𝑃𝑧𝑟𝑚3𝐷𝑑𝑟 (3.56)  The total force on the magnet disk is the integration   𝐹𝑧 = ∫ 𝑑𝐹𝑧𝑅𝑜𝑅𝑖= 2𝜋 ∫ 𝑃𝑧𝑟𝑚3𝐷𝑑𝑟𝑅𝑜𝑅𝑖 (3.57)  164  Similarly the differential torque around the z-axis is    𝑑𝑇𝑧 = 2𝜋𝑃𝑥𝑟𝑚3𝐷2 𝑑𝑟 (3.58)  and the total torque is   𝑇𝑧 = ∫ 𝑑𝑇𝑧𝑅𝑜𝑅𝑖= 2𝜋 ∫ 𝑃𝑥𝑟𝑚3𝐷2 𝑑𝑟𝑅𝑜𝑅𝑖 (3.59)  The integrations in (3.57) and (3.59) can be carried out numerically. Since each is only an integration along a single axis, compared to the 3D numerical integration required by FEA, this model still has the advantage in terms of computation time. In addition, even without solving the integrations analytically, the relative shapes of the force and torque curves can be determined by the closed form solutions to the 2D case (3.48) for parameter optimization.  165   Figure 3.5 Rotating permanent magnet disk asynchronous levitation machine. (a) Motor topology (with magnetization pattern for example similar to that shown in Figure 1.19a). (b) Cylindrical shell element of magnet disk. `   We compare this analytical shells model to a 3D finite element study. An example magnet disk is designed with parameters listed in Table 3.2. We note that the total force producing area is much larger than the inner diameter of the magnet disk, so no assumptions about magnet geometry are being used as was the case for the analytical model in [57]. Figure 3.6 shows the FEA model 166  created in [79], with a conductor diameter 10% wider than the outer diameter of the magnet array disk (increasing the conductor diameter made the FEA simulation time excessively long). This array is designed as a practically buildable magnet disk, and therefore each of the magnet segments magnetized in the XY plane (all others are magnetized in the Z-direction) are straight magnetized and not circumferentially magnetized as shown in Figure 3.7. We wish to evaluate the model in the context of a practical design and not as a descriptive model for an impractical case.  Figure 3.8 compares the levitation force and torque from the analytical shells model (with integrations in (3.57) and (3.59) carried out numerically with 104 points) to the 3D FEA at different rotational speeds (where rotational speed in revolutions-per-minute (rpm) is 60|𝜔|2𝜋). The maximum error is under 15%, and is due in part to the assumption of no leakage flux at the inner and outer radii of the magnet array, the relatively small XY extent of the conductor slab, and the straight magnetization of the magnet segments. For engineering design applications, this level of error in the magnitude of the force and torque is acceptable, especially as the shape of the load characteristic (e.g. the saturation rate of the levitation force) is accurate to less than 1%. This is predicted from the form of the expressions (3.58), (3.60) because the rotational velocity dependent terms are easily separable from the magnitude terms. We conclude that the analytical shells model can be used to design practical magnet array disks.  167   Figure 3.6 3D FEA model, with 𝑵𝒎 = 𝟖.  168   Figure 3.7 Circumferential versus straight magnetization of magnet segments in XY plane.    Figure 3.8 Levitation force and drag torque, 3D FEA versus analytical shells method for 𝒛𝒇 = 𝟐 𝒎𝒎.      169  3D ROTATING DISK ASYNCHRONOUS LEVITATION MACHINE PARAMETERS Parameter Symbol Value Units Outer magnet array radius 𝑅𝑜   100  mm Inner magnet array radius 𝑅𝑖   20  mm Magnet height 𝐻𝑚   6.35  mm Magnet remanence 𝐵𝑟   1.2347  T Halbach segment gap 𝑔   0 mm Magnet array pole number 𝑁𝑚   8 -- Conductor slab thickness  𝐷𝑐    25.4 mm Conductor slab conductivity 𝜎   5.998× 107  S/m Flying height 𝑧𝑓   2  mm Table 3.2  3D Rotating Disk Asynchronous Levitation Machine Parameters.   3.3 Experimental Force, Torque and Power for Rotating Magnet Disk  An experimental magnet array disk (Figure 3.9) is designed and built to evaluate the analytical shells model. We note that the magnet segments are not true arc geometries, but rather cube magnet segments which were more easily available. We approximate the gap between magnet segments for each shell element as a linear function of the radius 𝑟𝑚 and the minimum and maximum gaps (𝑔𝑖, 𝑔𝑜):   𝑔 =𝑔𝑜 − 𝑔𝑖𝑅𝑜 − 𝑅𝑖(𝑟𝑚 − 𝑅𝑖) + 𝑔𝑖 (3.60)  170  In addition the magnet segments are coated with a thin layer of conductive nickel. This will affect the asynchronous field as nickel has a conductivity of 14.3 MS/m (compared to 60 MS/m for copper) and the nickel coating is approximately 100 micron thick. The experimental load characterization setup is shown in Figure 3.10, with a CNC spindle used to locate the magnet disk in XYZ and rotate the magnet disk. A conducting slab is attached with adhesive to the top of an electrically insulated spacer which is bonded to an adapter plate and an ATI Industrial Automation MINI-45 6-DOF load cell as shown. The thickness of the insulating layer was determined by spinning the magnet array at the maximum test speed condition of 7700 rpm over the adapter plate and load cell alone and increasing the height of the disk until the measured forces and torques were below the noise floor of the sensor, then increasing the height a further 25%. This insures that only the effects of the conducting slab will be measured and there is no extraneous coupling with the load sensor and other apparatus. Two different conducting slab materials with different thicknesses were tested: copper and aluminum. The test setup parameters and magnet disk design parameters are listed in Table 3.3.    171   Figure 3.9 Experimental rotating disk asynchronous levitation machine.  172   Figure 3.10 Load test setup.       173  EXPERIMENTAL ROTATING DISK ASYNCHRONOUS LEVITATION MACHINE PARAMETERS Parameter Symbol Value Units Outer magnet array radius 𝑅𝑜   26.30  mm Inner magnet array radius 𝑅𝑖   16.70  mm Magnet height 𝐻𝑚   6.35  mm Magnet remanence 𝐵𝑟   1.2347  T Halbach segment gap, maximum 𝑔𝑜   3.79 mm Halbach segment gap, minimum 𝑔𝑖   1.21 mm Magnet array pole number 𝑁𝑚   4 -- Conductor slab thickness  𝐷𝑐   (Copper) 25.4 mm  𝐷𝑐   (Aluminum) 21.5 mm Conductor slab conductivity 𝜎  (Copper) 5.998× 107  S/m  𝜎  (Aluminum) 3.5× 107  S/m Table 3.3  3D Experimental rotating disk asynchronous levitation machine parameters.   Figure 3.11 and Figure 3.12 shows the 6-DOF load characteristic of the magnet disk over copper and aluminum respectively. The disk generates large levitation force 𝐹𝑧 and drag torque 𝑇𝑧 as predicted. The torques 𝑇𝑥,  𝑇𝑦 are non-zero and detectable at higher rotational speeds, and some very small in-plane XY forces on the order of the load sensor noise floor. This is because the magnet array disk rotation axis is out of perpendicularity to the surface of the conducting slab for each case to within 250 micron, and therefore there will be a net torque generated on an axis in the 174  XY-plane as well as some net translation force. The torque is at least an order of magnitude less than the drag torque 𝑇𝑧. This is in line with our conceptual understanding of the asynchronous levitation motor element as discussed in 3.1. The levitation force generated on the copper slab versus the aluminum slab for a given height and rotation speed is larger by approximately the ratio of the conductivity of copper to aluminum; this agrees with the findings from the experimental work in [62].  Figure 3.11 6-DOF load characteristics for rotating disk asynchronous machine (copper).  175   Figure 3.12 6-DOF load characteristics for rotating disk asynchronous machine (aluminum).   Figure 3.13 and Figure 3.14 shows a comparison of the experimental load characteristic with derived mechanical power and the analytical shells model at iso-lines of constant flying height (with integrations in (3.57) and (3.59) carried out numerically with 104 points) for the copper and aluminum cases respectively. The levitation force shows good agreement at lower flying heights, while the drag torque shows good agreement at higher flying heights. Figure 3.15 shows the error 176  at all measured points between the analytical model and the experimental measurement. The worst case error in levitation force prediction is 17.82% for the copper slab; in drag torque -29.8%; the power, since it is derived from the spindle speed and measured drag torque, 𝑝𝑜𝑤𝑒𝑟 = |𝜔|𝑇𝑧, has the same worst case error at -29.8%. Similarly, the worst case error for levitation force prediction is 16.4% for the aluminum slab and 26.3% for the drag torque and power. This is relatively good agreement with the experimental case, given the assumptions about circumferential magnetization, arc geometry, leakage effects and the presence of the nickel coating. In addition, the magnet disk chassis is aluminum with conductivity of 36.9 MS/m, which will affect the induced field. The magnet disk field may also have been affected by the magnetic spindle chuck. We conclude that the analytical shells method is a useful design tool for real magnet array disks, without any need for empirical data fitting.   177   Figure 3.13 Force, torque and mechanical power iso-lines at different flying heights (copper).  178   Figure 3.14 Force, torque and mechanical power iso-lines at different flying heights (aluminum).   179   Figure 3.15 Error between analytical shells model and experimental load characteristic (copper).    Figure 3.16 Error between analytical shells model and experimental load characteristic (aluminum). 180  Chapter 4: Conclusion  In this concluding chapter, I outline the original contributions of this thesis to the state of the art and suggest future work based on the presented research.    Non-contact 6-DOF direct drives have the potential to greatly improve motion stage performance by eliminating friction, bearing/guiding errors and the inertia of intermediate motion elements that are used in traditional stage designs. Magnetic field interactions can be used to apply these non-contact forces to a single mover body. In this thesis I have explored two classes of magnetic non-contact multi-DOF planar motion stages: i) synchronous planar levitation stages and ii) asynchronous planar levitation stages. Permanent magnet-based synchronous levitation stages have the potential to be highly power efficient if the motor topology allows for high copper fill factors and utilization of the permanent magnet field. The novel 6-DOF motor topology presented in this thesis achieves these design goals in addition to minimizing controller and commutation complexity through the novel design which allows for the highly decoupled, linear 6-DOF actuation of a single mover body with very simple control and commutation architecture. Compared to other synchronous planar levitation motors, this novel 6-DOF actuator can be scaled up in planar stroke with minimal increase in the controller complexity or number of current drives required. The high copper fill factor of the coil design, in addition to increasing the overall power efficiency of the motor, also produces very high force densities and thus accelerations.  One limitation to scaling up this synchronous motor technology is that this technology relies on high performance 6-DOF metrology solutions (such as planar encoders and multi-axis laser interferometers) for large planar work volumes. The nature of magnetic levitation systems 181  means that the final positioning performance is in large part determined by the performance of the sensor itself. Force generation capabilities are also limited by the cooling solution of the motor, due to excessive heat dissipation at high current working conditions. However, the heat source in the described synchronous moving magnet planar motor is stationary and thus the required active cooling method is easier to incorporate into existing stages than moving coil motors with a necessarily moving heat source. The development of the stationary cooling architecture is beyond the scope of the work in this thesis, but should be relatively simple.  While synchronous planar levitation motors have intrinsically better power efficiency compared to asynchronous induction type planar levitation motors, even the minimal complexity of driving alternating 1D coil structures can become prohibitive for ultra-long stroke planar applications. Asynchronous planar levitation motors have several advantages over synchronous levitation techniques: simple homogeneous stator structures (essentially just slabs of conducting material) allow for cost effective ultra-long planar work areas, and the passively stable nature of the 6-DOF actuation means feedback sensors are not required for stable motion. The disadvantages of this type of levitation motor are: i) the power efficiency, especially at higher loads, is limited because of the Ohmic losses induced in the conducting stator plus the power loss in the rotation mechanism for each magnet disk; ii) the coupled nature of all axes means the 6-DOF motion control is not simple; iii) the mover itself must have some active means of mechanically rotating the permanent magnet disks relative to the motor slab, as well as tilting them in tip/tilt axes. These latter requirements mean that either the mover uses some battery to power onboard rotary motors, or an umbilical cable must be attached to the mover to provide power. Nevertheless it may be beneficial to choose asynchronous levitation techniques for certain applications where the complexity of synchronous driving and/or the cost of the stator may be the limiting factor. 182  4.1 Synchronous Planar Levitation Motor Contributions I, II and III have been in part published in [89]. Parts of contributions IV and V have been published as [90].  4.1.1 Contribution I: Motor topology for New Type of 6-DOF Synchronous Planar Levitation Motor I presented a novel long stroke moving magnet 6-DOF synchronous planar levitation motor that uses symmetric linear 1D magnet arrays interacting with coil distributions in alternating layers of straight 1D coil patterns manufactured as printed circuit boards. This motion stage has the following advantages compared to existing 6-DOF synchronous levitation planar stages: the mover can be much smaller than the planar stroke and therefore the utilization of the work volume can be much higher; the number of coils increases linearly with planar stroke, compared to quadratically for existing 6-DOF synchronous levitation motors; the coil fill factor can be very high making this motor very power efficient with high force density (critical requirements for high precision applications); the driving method (with coil excitation zones deliberately wider than the magnet arrays) minimizes magnet array edge effects; the coil topology (where all end turns and coil return paths are on the perimeter of the working zone) minimizes coil edge effects; and this natural force linearity combined with the natural force decoupling of each 2-DOF motor element makes control and commutation of the stage very simple.  183  4.1.2 Contribution II: Analytical Model of Field, Force, Torque and Commutation Laws for New Type of 6-DOF Synchronous Planar Levitation Motor In this thesis I have presented new analysis and modeling of the relevant magnetic field, force, torque, and commutation laws of this new 6-DOF motion stage. The novel motor topology and driving method allows the application of an analytical force model of minimal complexity, which further allows us to generate quite simple commutation laws and feedback motion controllers based on a rigid body plant model. Simple commutation laws with low computational cost for 6-DOF actuation are highly desired for practical implementations of the motion stage.   4.1.3 Contribution III: Prototype and Experimental Demonstration of New Type of 6-DOF Synchronous Planar Levitation Motor An experimental alpha prototype has been built and successfully demonstrated 6-DOF planar actuation over a 60mm by 260 mm planar stroke with 11.6 mm maximum out of plane stroke and small rotations around all axes. Position feedback control has been implemented with 50 Hz cross-over frequencies for all axes with ~100 Hz closed loop bandwidth and 0.9 N/𝜇𝑚 minimum dynamic stiffness. This control bandwidth is limited by the position sensor performance. Regulation positioning results (i.e. with a stationary reference position command in all axes) show RMS error in the three translation axes of 2.0 𝜇𝑚, 2.1 𝜇𝑚, 5.1 𝜇𝑚 in 𝑥, 𝑦, 𝑧 respectively, and RMS regulation error in the three rotational axes of  0.0014∘, 0.0013∘, 0.0004∘ in 𝛼, 𝛽, 𝛾. These results are consistent with the noise floor of the position sensor.  For an XY elliptical reference path of 260 mm X-stroke and 60 mm Y-stroke at a flying height of 1mm above the coil array PCB surface, the tracking RMS error in the three translation 184  axes is 7.49 𝜇𝑚, 5.02 𝜇𝑚, 9.20 𝜇𝑚 in 𝑥, 𝑦, 𝑧 respectively, and RMS tracking error in the three rotational axes of  0.0066∘, 0.0082∘, 0.0029∘ in 𝛼, 𝛽, 𝛾 respectively.  The prototype presented in this thesis was designed for in-plane accelerations of 5.2g in the strong axis and 4.9g in the weak axis, and out of plane (z-directed) accelerations of 9.9g for a maximum current density of 20A/mm2 and a 0.5 mm flying height, with a total mover mass of 2.3 kg. The maximum flying height of 11.6 mm is determined by the ability to exert 1g levitation acceleration at this maximum current density. This current density limit may be exceeded through the use of forced cooling solutions.  4.1.4 Contribution IV: Novel Split and Quad-Split Magnet Array Designs for Force and Torque Ripple Self-attenuation I modeled sources of force and torque ripple arising from the interaction of higher order field harmonics of the permanent magnet field of each array and the commutated current in the discrete 1D coil pattern. I presented novel magnet array designs to minimize targeted force and torque ripples by splitting and offsetting sections of the magnet array relative to a common commutation center. These designs attenuate force and torque ripple without changing either commutation or control algorithms, and without increasing the complexity of the coil design. Finite element simulations of the novel quad-split magnet array showed that both force and torque ripples at the targeted spatial frequency were attenuated to within the numerical accuracy of the simulation for force and by four orders of magnitude for torque; this is at the cost of 10% mean force generation. This method of splitting magnet arrays can be generalized to other synchronous machines, in particular other synchronous planar levitation motors, ironless linear motors where 185  the primary source of force ripple is due to field and current interactions, and can also be extended to rotary synchronous machines.  4.1.5 Contribution V: Experimental Verification of Novel Split Magnet Array Design for Force Ripple Attenuation Experimental load testing of the novel split array was carried out and showed an order of magnitude reduction in the dominant 6-cp𝜆 force ripple at the expense of 5% mean force generation. This verifies the practicality of the split array methodology for targeted force ripple attenuation.  4.2 Asynchronous Planar Levitation Motor These contributions have been submitted as a paper to IEEE Transactions on Magnetics and is currently under review.   4.2.1 Contribution I: Novel Analytical Model of Field, Force and Torque of Magnet Disk Rotating Above a Finite Thickness Homogenous Conductor I presented a novel analytical model of a rotating permanent magnet disks over a stationary homogenous conducting slab to produce actuation forces and torques; this magnet disk can be used in a 6-DOF asynchronous planar levitation motor. While there is significant force and torque coupling between axes, the advantages of this asynchronous topology are a combined levitation and propulsion system and a very simple stator structure, as well as passive stability of all axes. The novel analytical expression for levitation force and drag torque is based on an integration of differential shell elements equivalent to multiple 2D infinitely extended asynchronous levitation 186  motors. This force and torque model allows for quick optimization of magnet disk and conducting slab design, and can offer physical and design insight into motor operation based on the form of the expression. This is advantageous compared to existing force and torque models in literature, which are primarily based on numerical finite element methods. Of the two relevant analytical models presented in the literature, one was restricted to non-permanent magnet sources [57] and limited in terms of applicable geometry, and the other used parametric empirical curve fitting to fit the model to experiment with two parameters making it less useful for design [72].   4.2.2 Contribution II: Evaluation of Novel Analytical Model with Finite Element Simulation and Characterization with Experimental Load Measurements I presented comparisons of this analytical model to a 3D finite element model showing a worst case error of 15% in both levitation force and drag torque. I built and measured a real experimental magnet disk with cuboidal magnet segments and showed a worst case force and torque modeling error of under 18% in levitation force and under 30% in drag torque for a copper conducting slab, and under 17% for levitation force and under 27% for drag torque for an aluminum conducting slab. This analytical model is therefore useful in the design of an asynchronous planar levitation motor using rotating circumferential magnet disks.  4.3 Future Work Common future work for both classes of planar levitation motor should include work on multiple simultaneous movers in the same work volume.   187  4.3.1 Future Work for 6-DOF Synchronous Planar Levitation Motor An important part of the future work on this class of levitation motor is the study of different motor topologies using the same 2-DOF motor elements. Since the 2-DOF motor is a combination of 1D magnet arrays and 1D coils, it is easy to conceptualize the transformation of these 1D patterns on the surface of flat planes onto curved surfaces e.g. spherical surfaces, or curved in plane (non-straight coils). This transformability of the 2-DOF motor has already been implemented for a rotary version of this synchronous levitation motor by Mark Dyck in [91] as the rotary table for a micro-machining application (a topology first disclosed in [75]). Where the synchronous planar levitation motor presented in this thesis has large translation strokes and small rotation range, the rotary version has infinite rotation range in 𝜃 and small translations in the XY plane. We can conceptualize other such versions of this synchronous levitation motor, possibly for applications where large rotations around multiple axes with small translations are required, similar to applications for the spherical motor presented in [22]. Other possible future work on this synchronous levitation motor include a study of the dynamic switching algorithm of the active coil zones and the effect on mover actuation force; minimal constraint actuation where only three of the quadrants are working (which would have applications with multiple movers partially overlapping in one of the in-plane axes); over-actuation of a large mover with more than four 2-DOF motor elements attached, allowing the control of large structural modes (as outlined in [75]); and topology, control and commutation methods for a synchronous planar motor with simultaneous large rotations and translations.  Force and torque ripple self-attenuation of multiple simultaneous harmonics should be analyzed in the context of magnet array designs presented by this author in [75]. An analysis of the effects of skewed and patterned coil shapes on force and torque ripple attenuation has yet to be 188  carried out. Multiple such coil designs have been presented by Dr. Xiaodong Lu and the author in [75].  4.3.2 Future Work for 6-DOF Asynchronous Planar Levitation Motor Modeling and Analysis  Future modeling work for the asynchronous 3-DOF motor element should include expanding the novel analytical shells model to include the effects of tilt/tip in order to model propulsion in the XY plane. In particular, the in-plane translation velocity should be included, similar to the work done numerically on rotating and translating magnet wheels in [61]. Combining a full 3-DOF analytical model with combinations of multiple magnet disks, a 6-DOF dynamic model can be produced. The 6-DOF characterization of this planar motor can be used to linearize and uncouple the axes of the stage for simplified control and actuation. 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