CITY UNIVERSITY OF HONG KONG 香港城市大學

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1 CITY UNIVERSITY OF HONG KONG 香港城市大學 Experimental study on stabilizing range extension of diamagnetic levitation under modulated magnetic field 磁場調控對擴大反磁性穩定懸浮範圍的實驗研究 Submitted to Department of Manufacturing Engineering and Engineering Management 製造工程及工程管理學系 in Partial Fulfillment of the Requirements for the Degree of Master of Philosophy 哲學碩士學位 by Chow Tsz Chun, Samuel 周子雋 March 2010 二零一零年三月

2 Abstract ii Abstract The real energy-free levitation can be achieved with the help of diamagnetic materials. The floater for diamagnetic levitation can be a magnet or made of diamagnetic material. The stabilized levitation range of this floater is defined by two parameters: Levitation Stabilizing Range, R S, and Local Range, R L. R S signifies the range (both vertical and horizontal) that floaters of different masses can be stably levitated under a given magnetic field while R L is the maximum range that a floater of a given mass can be deviated from its stabilized levitation position. Regarding to engineering applications, the two levitation parameters, R S and R L, are significant and crucial to the loading and self-rotating performance of a diamagnetic levitation system. For example, the larger the R L, the less the energy dissipated of a self-rotating levitated floater due to the eddy current effect and the larger the R S, the greater the load range of the system can be, i.e. higher application potential. A recently published paper of Cazacu and Stanciulescu carried out a theoretical study of the stabilizing range, R S and R L. It found that both of the ranges could be increased by manipulating the magnetic field gradient, B and the magnetic field curvature, B using a stepped shape coil as a field source. The numerical simulation proved that a stepped coil shape with a larger outer diameter at the bottom and a smaller outer diameter at the top could extend the stabilized levitation range. This project furthered the work of Cazacu and Stanciulescu to carry out an experimental study of the shape and geometric effects of the coil or ring magnet stacking arrays on the Local Range, R L. Furthermore, the effects of using electromagnetic coils and permanent magnets were verified by comparing the results to the corresponding standard cylindrical shaped coil or ring magnet. The aim of the project was to find out

3 Abstract iii how the stacked coils or permanent ring magnets of different shapes, layers and forming angles would affect the maximum performance of the R L. Coil stacking arrays of different shapes were constructed with four types of element coils of constant length, constant inner diameter, different numbers of turns and outer diameters. Experiments were carried out with three different overall shapes, namely Ascending, Descending and Centre-Diverging. The shape effects on R L were studied with different numbers of coil layers and applied currents. An additional shape Centre-Converging was evaluated when permanent ring magnets were used as the source. A list of ring magnets of different outer diameters with constant thickness, inner diameter and magnetization were chosen as the elements for the magnet stacking array experiment. The experiments from this project produced original data for validating the theory. Besides, it was found that the local stabilizing range, R L, of a given diamagnetic levitation system can only be extended if the coil of uniform shape is replaced by stacked coils of Ascending shape stacked coils with the same total number of coil turns N and overall length L. Other shape types of stacked coil arrays were found reducing the R L comparing to their corresponding standard coils. Similarly, the comparison of the effect of different shapes of permanent ring magnets was made based on a constant volume of the magnets. The results of permanent magnet experiments were depicted that no positive effect on R L could be made by using different forming shapes, use of layers and forming angles.

4 Table of Contents iv Table of Contents Acknowledgement... i Abstract.. ii Table of Contents... iv List of Figures... x List of Tables... xxi Chapter 1 Introduction Introduction Project Motivation, Aim and Objectives Chapter 2 Literature Review Magnetostatics and Diamagnetism Advanced basics of magnetostatics Electrostatic and magnetostatic dipole, magnetic dipole moment and magnetic dipole field equation Magnetic flux density, strength, magnetic permeability and susceptibility Biot-Savart law and a single loop on axis field equation Maxwell s equations Divergence and Curl Material properties to magnetic field Atomic theory of magnetism

5 Table of Contents v Atomic response classifying diamagnetic, para-magnetic, ferro-magnetic and ferri-magnetic material under external magnetic field Magnetic and Diamagnetic Levitation Implication of Earnshaw s theorem to magnetic levitation Common magnetic levitation system Short history of diamagnetism and diamagnetic levitation General diamagnetic levitation system Levitating magnet Levitating diamagnetic material Benefits and limitations of diamagnetic levitation.. 45 Chapter 3 Theory for Stabilized Diamagnetic Levitation Derivations of finite element magnetic field equations For coils For disc and ring magnets Stabilized diamagnetic levitation theory Stabilized diamagnetic levitation conditions for levitating a magnet Theoretical derivations of diamagnetic levitation stabilized maximum local range, D max Theoretical Stabilizing Range (conditional range) and Local Range (practical levitation stabilized moving range) The Stabilizing Range given by source of coils and permanent ring magnets as the levitation source... 69

6 Table of Contents vi Chapter 4 Research Methodology Introduction Magnetic Field Source Unit Coil design, stacking method and list of stacking coils Permanent magnet design, stacking method and list of stacking magnets The LABVIEW 7.1 platform for finite element equations of stacking coils and ring magnets Magnetic Levitation Floater and Diamagnetic Plate Unit Cubic neodymium-iron-boron (NdFeB) magnet Diamagnetic graphite plate Hardware and Power Supply Design Coils and permanent magnets supporting structure and descriptions Measurement system supporting structure Permanent ring magnets stacking and separating jig Electrical power supply for coils Measurement System Design Vibration isolation breadboard Global vertical translation stage with optical measurement unit and local vertical micrometer Gap sizes translation micrometer with the graphite holder and optical laser unit Design of Experiments Preparation procedures Promising data criteria and precautions

7 Table of Contents vii 4.7 Other Equipment Accessories Mitutoyo digital caliper and leveling indicator Mitutoyo optical measurement display unit. 128 Chapter 5 Geometric effect of stacking coils on stabilizing local range of diamagnetic levitation Introduction Experimental Procedures Experimental Results Calibration test on each coil Effect of stacking shapes on D max for different current I under a constant differential layer in NI configurations Effect of stacking shapes on D max for different current I under a constant differential layer in NNI configurations Effect of stacking shapes on D max for different number of differential layers under a constant current I in NI configurations Effect of stacking shapes on D max for different number of differential layers under a constant current I in NNI configurations Discussions Preliminary experimental questions Results verification The relationship between D max and current I The relationship between D max and shape effect The relationship between D max and number of differential layers Comparison between NI and NNI configurations

8 Table of Contents viii The summarized points for improving the local range of diamagnetic levitation under stacking coils source Conclusions Chapter 6 Geometric effect of stacking permanent ring magnets on stabilizing local range of diamagnetic levitation Introduction Experimental Procedures Experimental Results Calibration test on each permanent ring magnet Effect of stacking shapes on D max for different numbers of layers under a constant geometric shaping angle θ Effect of stacking shapes on D max for different geometric shaping angle θ under a constant number of layers Discussions Results verification The relationship between D max and number of layers The relationship between D max and shape effect The relationship between D max and number of differential layers Comparison between stacking coils and stacking magnets Conclusions. 227

9 Table of Contents ix Chapter 7 Conclusions Conclusive Remark Summary of Conclusions Suggestions on Future Work Coils and permanent magnet combinations and utilizations Stiffness test for micro/nano force sensor Novel Diamagnetic Levitated Atomic Force Microscopy (Diamaglev AFM) concept Novel Diamagnetic Levitated Solar Energy Cell concept References Appendix Appendix (I) Referral equations deviations and proofs Appendix (II) Optical linear scale specification Appendix (III) Optical linear scale display unit specification Appendix (IV) Optical laser sensor and amplifier specification

10 List of Figures x List of Figures Figure 1-1: The three shapes performed in stacking coils 7 Figure 1-2: Figure 1-3: Figure 2-1: Figure 2-2: Examples of (i) NI and (ii) NNI configurations with Ascending coil array illustrating the meaning of differential ( diff. ) layer 7 The four shapes constructed with stacking permanent ring magnets and the geometric shaping angle, θ for each shape... 8 Electric field direction of a positively charge (left) and an electron (right). 10 (a) Magnetic field direction in a closed loop of a dipole (north and south poles) and (b) Electric field between two different charges. 10 Figure 2-3: Mono-pole for a magnetic being impossible.. 11 Figure 2-4: m dipole analogous to the concept of a current carrying circular loop Figure 2-5: Figure 2-6: Figure 2-7: Figure 2-8: Magnetic field db generated a field loop at P (out of paper) and P (into the paper) by a current element I dl Schematic representation of the volume current density J (left) and the surface current density J s (right).. 15 Schematic of a single loop current wire on axis field contribution to point A (0,0,z) 15 Schematic of a single loop current wire on axis field cancellation in db ρ direction Figure 2-9: Graphical expression of the Ampere s law. 19 Figure 2-10: Surface area, S(t) integrated by element da s bounded by a loop l s moving with velocity, v defined for Faraday s law of induction and the vector magnetic field B(r, t) [35]. 21 Figure 2-11: Graphical illustration of the three cases in field divergence Figure 2-12: Graphical illustration of the Cylindrical (left) and Spherical (right) coordinates system. 23 Figure 2-13: Graphical illustration of four cases in curl behavior.. 24 Figure 2-14: Graphical illustration of the atomic magnetic moment generated by the electron orbit motion and the minor magnetic moment due to the electron spins 26 Figure 2-15: Graphical illustration of the zero resultant magnetic moment of the material at normal state due to different orbit orientation.. 27 Figure 2-16: Graphical illustration of diamagnetic, paramagnetic, ferromagnetic and ferromagnetic materials atomic moment response to applied external magnetic field... 30

11 List of Figures xi Figure 2-17: Graphical illustration of a magnetic levitation system by permanent magnets interactions with guiding fixtures [1].. 34 Figure 2-18: Graphical illustration of an example of a magnetic levitation system by active controls Figure 2-19: Graphical illustration of flux exclusion property of a Type I superconductor [1]. 36 Figure 2-20: Graphical illustration of flux trapping property of a Type II superconductor Figure 2-21: Graphical illustration of magnetic levitation example by eddy current generated in alternative current magnetic field [1] Figure 2-22: Graphical illustration of hybrid system example by active control with permanent magnet embedded core [1].. 38 Figure 2-23: Graphical illustration of the vertical diamagnetic levitation in two diamagnetic plates (left) and single diamagnetic plate (right) with their corresponding force balancing expression. 42 Figure 2-24: Graphical illustration of the horizontal diamagnetic levitation with their corresponding force balancing expression. 43 Figure 2-25: Graphical illustration of levitating diamagnetic material in comparative heavy (top) and light (bottom) objects.. 44 Figure 3-1: Schematic drawing of a one layer thin solenoid with defined parameters for on axis field derivation Figure 3-2: Schematic drawing of a one layer thin solenoid with defined parameters for on axis field derivation outside the solenoid.. 49 Figure 3-3: Schematic drawing of a finite solenoid with inner radius r 1, outer radius r 2, length L and its corresponding element with thickness dr 50 Figure 3-4: Graphical illustration of the internal current loop cancellation of a disc magnet with magnetization M. 52 Figure 3-5: Schematic illustration of a coordinate system and related parameters for the central axis field equation of a disc magnet for equation (3-12).. 54 Figure 3-6: Schematic illustration of a coordinate system and related parameters for the central axis field equation of a ring magnet for equation (3-15).. 55 Figure 3-7: Graphical illustration of the three conditions on the existence of a diamagnetic levitation 60 Figure 3-8: Schematic illustration of the dipole magnet and the dipole current explanation. 62

12 List of Figures xii Figure 3-9: Schematic illustration of the θ and K parameter in the cylindrical coordinate for equation (3-34).. 64 Figure 3-10: Schematic illustration of R S and R L of a diamagnetic levitation in a vertical field case (diamagnetic plates not shown) 69 Figure 3-11: Schematic illustration of K v and K h curves behavior of using coil (top) and permanent ring magnet (below) as the field source in a vertical diamagnetic levitation configuration. 70 Figure 4-1: Bobbin design with different parts made of POM and Al.. 74 Figure 4-2: The fabricated bobbin (left) and the wounded coil by 1600 turns (right). 75 Figure 4-3: Example of different sizes of designed ring magnets. 82 Figure 4-4: Schematic illustration of the field magnitude superposition from a 4 element coils example on the same position at z Figure 4-5: The finite solenoid interface for coils Base coil parameters input and range input Figure 4-6: The finite solenoid interface for coils 2 nd coil parameters input, diamagnetic and floater specifications, diamagnetic plates control and D value input Figure 4-7: The finite solenoid interface for coils The superposed B vs. z profile with individuals coil field indication and the real-time illustration of the stacking coil Figure 4-8: The finite solenoid interface for coils The field gradient B vs. z profile with exporting ability and schematic referred to the coil input parameters. 92 Figure 4-9: The finite solenoid interface for coils The field gradient B vs. z profile with exporting ability.. 93 Figure 4-10: The finite solenoid interface for coils The diamagnetic levitation conditions curves and the results in levitation position, levitation achievement, stabilizing range R s and other values Figure 4-11: The finite solenoid interface for ring magnets The input parameters of ring magnets 95 Figure 4-12: The performance deviation under different length L to radius R factor between dipole approx. and surface current approx. in Force and Force gradient against the gap size d to floater radius R, captured in paper [14] Figure 4-13: The experiment used cubic NdFeB magnet floater 97 Figure 4-14: The diamagnetic plate response in magnetic moment and applied field relationship. 98

13 List of Figures xiii Figure 4-15: The diamagnetic plate response in magnetization and applied field strength relationship Figure 4-16: The 1000 times magnified surface of the specimen from the diamagnetic plate (top) and its corresponding spectrum (below) 100 Figure 4-17: Whole test rig with different structures indication. 102 Figure 4-18: Schematic structure of the supporting bar Figure 4-19: Schematic structure of supporting plate. 103 Figure 4-20: Schematic structure of one slot frame work 104 Figure 4-21: An example of 5 slots frame structure Figure 4-22: An example of coil inserting guide by the coil screw head and the U hole of the plate 104 Figure 4-23: The pin fixture (top) and the supporting table mounting hole for the pin fixture (bottom) Figure 4-24: Graphical illustration of measurement system supporting structure 106 Figure 4-25: Ring magnet stacking and separating jig 107 Figure 4-26: Insert the first ring magnet (left) and move close to the separating plate (view from back) (right) Figure 4-27: Insert the second ring magnet to the short pin Figure 4-28: Gradually move the front arm and finally two magnets attracted together with the plate in between Figure 4-29: Insert the alignment rod to connect the two arms Figure 4-30: Front arm movement causing the attachment of the magnets (the alignment rod not shown here for a better illustration). 110 Figure 4-31: The stacked magnet are pushed to the back side pin Figure 4-32: The separating model and attachment of a thick plate behind the separating plate Figure 4-33: Insert the stacked magnet to the back side pin Figure 4-34: Stacked magnet blocked by the thick plate 112 Figure 4-35: Insert the outmost ring magnet to the short pin Figure 4-36: Separate the ring magnet by pushing down the front arm to shear the stacked magnet. 113 Figure 4-37: A dual channel power supply. 114 Figure 4-38: 15 coils connection to power supply channel. 114 Figure 4-39: Measurement system with breadboard, micrometers and optical units 115 Figure 4-40: Graphical illustration of the vibration isolation breadboard (left) and its sectional view (right). 116 Figure 4-41: Global vertical translation stage. 117

14 List of Figures xiv Figure 4-42: Schematic structure of optical linear scale with guided platform Figure 4-43: Local vertical micrometer Figure 4-44: Schematic structure of the connection between local vertical micrometer, global vertical translation stage and the optical linear scale through the guiding platform. 118 Figure 4-45: Schematic structure of the gap size micrometer with the holder attached to the local vertical micrometer Figure 4-46: Schematic structure of the optical laser micrometer placement with the horizontal movable platform 121 Figure 4-47: The preparation tools Figure 4-48: The four positions leveling check Figure 4-49: The clear view between transmitting head and receiving head (top) and calibrating control at the amplifier (bottom) Figure 4-50: The vertical perturbation (left) and radial perturbation (right). 126 Figure 4-51: The cross spirit level indicator (left) and digital caliper (right) Figure 4-52: The optical linear scale display unit Figure 5-1: Hirst gaussmeter with axial probe (left) and axial probe head (right). 130 Figure 5-2: Coil calibration setup. 130 Figure 5-3: Graphical illustration of the inchworm movement of the vertical translation ruler Figure 5-4: Axial probe touching the 0.05 mm shim 131 Figure 5-5: Magnetic flux profile B vs. z of a single coil with N = 1600 under constant I = 3A Figure 5-6: Magnetic flux profile B vs. z of 2 individual coils with N = 800 under constant I = 3A. 136 Figure 5-7: Magnetic flux profile B vs. z of 4 individual coils with N = 400 under constant I = 3A. 136 Figure 5-8: Magnetic flux profile B vs. z of 8 individual coils with N =200 under constant I = 3A. 137 Figure 5-9: Graphical illustration of the three stacking designs in 2 differential layers with each differential layer having N = Figure 5-10: Ascending shape effects on the maximum gap size, D max at different applied coil currents I (Two differential layers coil array, each layer N = 1600, constant NI for each layer). 139 Figure 5-11: Descending shape effects on the maximum gap size, D max at different applied coil currents I (Two differential layers coil array, each layer N = 1600, constant NI for each layer). 139

15 List of Figures xv Figure 5-12: Centre-Diverging shape effects on the maximum gap size, D max at different applied coil currents I (Two differential layers coil array, each layer N = 1600, constant NI for each layer). 140 Figure 5-13: Stacking shape effects on the maximum gap size, D max at different applied coil currents I (Two differential layers coil array, each layer N = 1600, constant NI for each layer) Figure 5-14: Graphical illustration between the NI and NNI configurations of the Centre-Diverging shape in 2 and 3 differential layers Figure 5-15: Graphical illustration of the NNI stacking design in 2 differential layers for the Ascending and Descending shapes with 3 differential layers for the Centre-Diverging shape 143 Figure 5-16: Ascending shape effects on the maximum gap size, D max at different applied coil currents I (Two differential layers coil array in NNI configurations, with one N=1600 and one N=800) Figure 5-17: Descending shape effects on the maximum gap size, D max at different applied coil currents I (Two differential layers coil array in NNI configurations, with one N = 1600 and one N = 800) Figure 5-18: Centre-Diverging shape effects on the maximum gap size, D max at different applied coil currents I (Three differential layers coil array in NNI configurations, with one N=1600, two N=800 and two N = 400 ) Figure 5-19: Ascending and Descending shapes effect on the maximum gap size, D max at different applied coil currents I for NNI configurations (Centre-Diverging shape not shown) 145 Figure 5-20: Graphical illustration of the outer radius calculation of the corresponding single cylindrical coil to a given total turns, length and inner radius from the shaped stacking coils 148 Figure 5-21: Theoretical results of the relationship between D max and the number of differential layers in NI configurations under a constant I = 2A for Ascending, Descending and Centre-Diverging shapes stacking coils with a comparison to the referred cylindrical coils Figure 5-22: Theoretical results of the relationship between D max and the number of differential layers in NNI configurations under a constant I = 2A for Ascending and Descending shapes stacking coils with a comparison to the referred cylindrical coils

16 List of Figures xvi Figure 5-23: Theoretical results of the relationship between D max and the number of differential layers in NNI configurations under a constant I = 2A for Centre-Diverging shape stacking coils with a comparison to the referred cylindrical coil Figure 5-24: (top) B and (bottom) B against the displacement z of different applied current I in Ascending shape of a 2 differential NI configured layers with levitation position indication. 158 Figure 5-25: (top) B and (bottom) B against the displacement z of different shapes under a constant I = 2A in 2 differential NI configured layers and the referred cylindrical coil with levitation position indication 160 Figure 5-26: Graphical illustration of the superposed field profile started at z = 0 with each individual superposing coils profiles allocation of the Ascending shape in 2 differential layers under NI configuration Figure 5-27: Graphical illustration of the superposed field profile started at z = 0 with each individual superposing coils profiles allocation of the Descending shape in 2 differential layers under NI configuration. 162 Figure 5-28: Graphical illustration of the superposed field profile started at z = 0 with each individual superposing coils profiles allocation of the Centre-Diverging shape in 2 differential layers under NI configuration Figure 5-29: (top) B' and (bottom) B vs. z graph of the Ascending shape in 2 differential layers NI configured coils and the referred cylindrical coil with I = 2A, Outer radius = 79.2mm, Total N = 3200 and L = 120 mm Figure 5-30: (top) B' and (bottom) B vs. z graph of the Ascending shape in 3 differential layers NI configured coils and the referred cylindrical coil with I = 2A, Outer radius = 59.6 mm, Total N = 4800 and L = 280 mm Figure 5-31: (top) B' and (bottom) B vs. z graph of the Ascending shape in 4 differential layers NI configured coils and the referred cylindrical coil with I = 2A, Outer radius = 47 mm, Total N = 6400 and L = 600 mm Figure 5-32: B vs. z displacement of the 2, 3 and 4 differential layer referred standard coils to the NI configured Ascending shape under a constant I = 2 A Figure 5-33: B vs. z displacement of the 2, 3 and 4 differential layer referred standard coils to the NI configured Ascending shape 168

17 List of Figures xvii Figure 5-34: (top) B' and (bottom) B vs. z graph of the Ascending shape in 2 differential layers NNI configured coils and the referred cylindrical coil with I = 2A, Outer radius = 84.8 mm, Total N = 2400 and L = 80 mm. 170 Figure 6-1: The clamping fixture of the transverse field probe 174 Figure 6-2: Fine adjustment of the probe head to align the centre-axis 174 Figure 6-3: Moving the global vertical micrometer upward. 175 Figure 6-4: Pin fixture for ring magnet placement 175 Figure 6-5: An example on monitoring the vertical movement in 2 mm Figure 6-6: Magnetic flux profile B vs. z of two permanent ring magnets in Figure 6-7: Magnetic flux profile B vs. z of two permanent ring magnets in Figure 6-8: Magnetic flux profile B vs. z of two permanent ring magnets in Figure 6-9: Magnetic flux profile B vs. z of two permanent ring magnets in Figure 6-10: Magnetic flux profile B vs. z of two permanent ring magnets in Figure 6-11: Magnetic flux profile B vs. z of two permanent ring magnets in Figure 6-12: Magnetic flux profile B vs. z of two permanent ring magnets in Figure 6-13: Magnetic flux profile B vs. z of a permanent ring magnets in Figure 6-14: Magnetic flux profile B vs. z of a permanent ring magnets in

18 List of Figures xviii Figure 6-15: Magnetic flux profile B vs. z of two permanent ring magnets in Figure 6-16: Magnetic flux profile B vs. z of two permanent ring magnets in Figure 6-17: Magnetic flux profile B vs. z of two permanent ring magnets in Figure 6-18: Ascending shape effects on the maximum gap size, D max at different number of layers at a constant geometric shaping angle θ = 63 o of a stacking ring magnet Figure 6-19: Descending shape effects on the maximum gap size, D max at different number of layers at a constant geometric shaping angle θ = 63 o of a stacking ring magnet Figure 6-20: Centre-Diverging shape effects on the maximum gap size, D max at different number of layers at a constant geometric shaping angle θ = 63 o of a stacking ring magnet Figure 6-21: Centre-Converging shape effects on the maximum gap size, D max at different number of layers used at a constant geometric shaping angle θ = 63 o of a stacking ring magnet. 189 Figure 6-22: Theoretical results of the relationship between D max and the number of layers in Ascending and Descending shapes stacking ring magnets with the comparison to the results of the referred single ring magnets Figure 6-23: Theoretical results of the relationship between D max and the number of layers in Centre-Diverging shape stacking ring magnets with the comparison to the results of the referred single ring magnets Figure 6-24: Theoretical results of the relationship between D max and the number of layers in Centre-Converging shape stacking ring magnets with the comparison to the results of the referred single ring magnets Figure 6-25: Theoretical results of the relationship between D max and the geometric shaping angle θ in Ascending and Descending shapes stacking ring magnets with comparison to the results of the referred single ring magnets

19 List of Figures xix Figure 6-26: Theoretical results of the relationship between D max and the geometric shaping angle θ in Centre-Diverging shape stacking ring magnets with comparison to the results of the referred single ring magnets Figure 6-27: Theoretical results of the relationship between D max and the geometric shaping angle θ in Centre-Converging shape stacking ring magnets with comparison to the results of the referred single ring magnets Figure 6-28: (top) B and (bottom) B against the displacement z in different number of layers of Ascending shape stacking ring magnets at a constant geometric shaping angle θ = 63 o Figure 6-29: (top) B and (bottom) B against the displacement z in different number of layers of Descending shape stacking ring magnets at a constant geometric shaping angle θ = 63 o. 205 Figure 6-30: (top) B and (bottom) B against the displacement z in different number of layers of Centre-Diverging shape stacking ring magnets at a constant geometric shaping angle θ = 63 o Figure 6-31: (top) B and (bottom) B against the displacement z of different number of layers in Centre-Converging shape stacking ring magnets at a constant geometric shaping angle θ = 63 o. 207 Figure 6-32: Graphical illustration of the superposed field profile of a 3 layers Descending shape started at z = 0 with the field profiles allocation of each elemental ring magnet Figure 6-33: (top) B and (bottom) B against the displacement z of Ascending, Descending shapes in a 3 layers stacking configurations at a constant geometric shaping angle θ = 63 o and the referred single ring magnet 209 Figure 6-34: (top) B and (bottom) B against the displacement z of Centre-Diverging shape in a 3 layers stacking configurations at a constant geometric shaping angle θ = 63 o and the referred single ring magnet. 211 Figure 6-35: (top) B and (bottom) B against the displacement z of Centre-Converging shape in a 3 layers stacking configurations at a constant geometric shaping angle θ = 63 o and the referred single ring magnet. 213 Figure 6-36: (top) B and (bottom) B against the displacement z of Ascending shape in a geometric shaping angle θ = 27 o under a 5 layers stacking configurations and the referred single ring magnet. 215

20 List of Figures xx Figure 6-37: (top) B and (bottom) B against the displacement z of Ascending shape in a geometric shaping angle θ = 45 o under a 5 layers stacking configurations and the referred single ring magnet. 216 Figure 6-38: (top) B and (bottom) B against the displacement z of Ascending shape in a geometric shaping angle θ = 56 o under a 5 layers stacking configurations and the referred single ring magnet. 217 Figure 6-39: (top) B and (bottom) B against the displacement z of Ascending shape in a geometric shaping angle θ = 63 o under a 5 layers stacking configurations and the referred single ring magnet. 218 Figure 6-40: (top) B and (bottom) B against the displacement z of Centre-Diverging shape in a geometric shaping angle θ = 27 o under a 5 layers stacking configurations and the referred single ring magnet 220 Figure 6-41: (top) B and (bottom) B against the displacement z of Centre-Diverging shape in a geometric shaping angle θ = 45 o under a 5 layers stacking configurations and the referred single ring magnet 221 Figure 6-42: (top) B and (bottom) B against the displacement z of Centre-Diverging shape in a geometric shaping angle θ = 56 o under a 5 layers stacking configurations and the referred single ring magnet 222 Figure 6-43: (top) B and (bottom) B against the displacement z of Centre-Diverging shape in a geometric shaping angle θ = 63 o under a 5 layers stacking configurations and the referred single ring magnet 223

21 List of Tables xxi List of Tables Table 2-1: Overview of four Maxwell equations in total charge and current with differential and integral form Table 3-1: A summary for the R S and R L comparison between using coil and permanent ring magnet as the lifter field source Table 4-1: Copper wire diameter selection provided by the engineering company Table 4-2: Dimensions of the four types experimental coils. 76 Table 4-3: Stacking coils models in Ascending, Descending and Centre-Diverging shapes under NI configurations for 2, 3 and 4 differential layers.. 77 Table 4-4: Stacking coils models in Ascending, Descending and Centre-Diverging shapes under NNI configurations for 2, 3 and 4 differential layers.. 79 Table 4-5: Specifications of the elemental ring magnets with the diameter type and quantities 81 Table 4-6: Stacking ring magnets models in Ascending, Descending, Centre-Diverging and Centre-Converging shapes under constant geometric shaping angle θ = 63 o in different layers. 82 Table 4-7: Stacking ring magnets models in Ascending, Descending, Centre-Diverging and Centre-Converging shapes under constant stacking layers = 5 in different geometric shaping angle θ.. 85 Table 4-8: The floater magnet specifications 97 Table 4-9: The diamagnetic graphite plate composition table Table 4-10: The diamagnetic plate specifications Table 5-1: Dimension table of Ascending, Descending and Centre-Diverging shapes in 2 differential layers stacking coils under NI configurations and the outer radius value of the corresponding single cylindrical coil 146 Table 5-2: Dimension table of Ascending, Descending and Centre-Diverging shapes in 3 differential layers stacking coils under NI configurations and the outer radius value of the corresponding single cylindrical coil 147

22 List of Tables xxii Table 5-3: Dimension table of Ascending, Descending and Centre-Diverging shapes in 4 differential layers stacking coils under NI configurations and the outer radius value of the corresponding single cylindrical coil 147 Table 5-4: Theoretical results of the relationship between D max and the number of differential layers in NI configurations under a constant I = 2A for Ascending, Descending and Centre-Diverging shapes stacking coils with a comparison to the referred cylindrical coils Table 5-5: Dimension table of Ascending and Descending shapes in 2 differential layers stacking coils under NNI configurations and the outer radius value of the corresponding single cylindrical coil Table 5-6: Dimension table of Ascending and Descending shapes in 3 differential layers stacking coils under NNI configurations and the outer radius value of the corresponding single cylindrical coil Table 5-7: Dimension table of Ascending and Descending shapes in 4 differential layers stacking coils under NNI configurations and the outer radius value of the corresponding single cylindrical coil Table 5-8: Dimension table of Centre-Diverging shape in 2 differential layers stacking coils under NNI configurations and the outer radius value of the corresponding single cylindrical coil. 153 Table 5-9: Dimension table of Centre-Diverging shape in 3 differential layers stacking coils under NNI configurations and the outer radius value of the corresponding single cylindrical coil. 153 Table 5-10: Theoretical results of the relationship between D max and the number of differential layers in NNI configurations under a constant I = 2A for Ascending and Descending shapes stacking coils with a comparison to the referred cylindrical coils Table 5-11: Theoretical results of the relationship between D max and the number of differential layers in NNI configurations under a constant I = 2A for Centre-Diverging shape stacking coils with a comparison to the referred cylindrical coils Table 6-1: Specifications of the elemental permanent ring magnets with the diameter type, quantities and the name 179 Table 6-2: Outer radius calculation of the referred single ring magnet from the total volume of the stacking permanent ring magnets in different number of layers of the Ascending and Descending shapes 190

23 List of Tables xxiii Table 6-3: Outer radius calculation of the referred single ring magnet from the total volume of the stacking permanent ring magnets in different number of layers of the Centre-Diverging shape. 191 Table 6-4: Outer radius calculation of the referred single ring magnet from the total volume of the stacking permanent ring magnets in different number of layers of the Centre-Converging shape Table 6-5: Theoretical results of the relationship between D max and the number of layers in Ascending and Descending shapes stacking ring magnets with the comparison to the results of the referred single ring magnets 192 Table 6-6: Theoretical results of the relationship between D max and the number of layers in Centre-Diverging shape stacking ring magnets with the comparison to the results of the referred single ring magnets Table 6-7: Theoretical results of the relationship between D max and the number of layers in Centre-Converging shape stacking ring magnets with the comparison to the results of the referred single ring magnets Table 6-8: Outer radius calculation of the referred single ring magnet from the total volume of the stacking permanent ring magnets in different geometric shaping angle θ of the Ascending and Descending shapes 196 Table 6-9: Outer radius calculation of the referred single ring magnet from the total volume of the stacking permanent ring magnets in different geometric shaping angle θ of the Centre-Diverging shape Table 6-10: Outer radius calculation of the referred single ring magnet from the total volume of the stacking permanent ring magnets in different geometric shaping angle θ of the Centre-Converging shape 197 Table 6-11: Theoretical results of the relationship between D max and the geometric shaping angle θ in Ascending and Descending shapes stacking ring magnets with comparison to the results of the referred single ring magnets Table 6-12: Theoretical results of the relationship between D max and the geometric shaping angle θ in Centre-Diverging shape stacking ring magnets with comparison to the results of the referred single ring magnets Table 6-13: Theoretical results of the relationship between D max and the geometric shaping angle θ in Centre-Converging shape stacking ring magnets with comparison to the results of the referred single ring magnets Table 7-1: Summary of stacking coil results Table 7-2: Summary of stacking ring magnet results 230