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The University of Sheffield The University of Sheffield Novel Permanent Magnet Brushless Machines Having Segmented Halbach Array Yang Shen A thesis submitted for the degree of Doctor of Philosophy Department of Electronic and Electrical Engineering The University of Sheffield Mappin Street, Sheffield, S1 3JD, UK 15 November 2012 1 ABSTRACT Permanent magnet brushless machines having Halbach array exhibit a number of attractive features. Therefore, they have been increasingly applied to different market sectors, including aerospace, industrial, domestic, renewable, and healthcare, etc. The need of fast global optimization, cost-effective design, and physical understanding of the relationship between parameters and performance requires a powerful analytical model. This thesis develops a general analytical model which is capable of predicting the electromagnetic performance of slotted/slotless permanent magnet brushless machines with both even- and odd-segment Halbach array, having different magnet remanence, magnetization angle and arc for each single magnet segment. The validity of proposed analytical model is examined by finite- element analyses. The price of Neodymium Iron Baron magnet has been raised rapidly over the last few years, which has increased the awareness of cost-effective design and leads the magnet usage efficiency, viz. the ratio of average output torque over permanent magnet volume, to be an important design concern for industry applications. Meanwhile, the needs of high electromagnetic performance including lower torque ripple and sinusoidal air-gap flux density are also critically required. In order to meet such demands, the magnet poles having unequal-magnet height, modular high cost and low cost magnets, together with Halbach magnetization are proposed in this thesis. Based on the developed analytical models, extensive investigation has been performed. Furthermore, in order to fast optimize design parameters, e.g. optimal magnet arc ratio for 2-segment Halbach array and split ratio for external rotor PM machine, two analytical models are developed and validated by finite- element analysis and measurements. With the aid of developed analytical models and finite element analyses, the findings provide useful guidelines for design and analysis of permanent magnet brushless machine having conventional and proposed Halbach arrays. 2 ACKNOWLEDGEMENTS Many thanks are due to my supervisor, Prof. Zi-Qiang Zhu, for his continued encouragement, advice and support throughout this PhD study, which has been invaluable in the completion of this thesis. Thanks are also due to many other current and former members of the Electrical Machines and Drives Group at the University of Sheffield, who have assisted in the research process, particularly to Dr Jintao Chen and Dr Lijian Wu for their many helpful and productive discussions. Finally, I thank my parents for their love, encouragement and constant support for the strength to bring this task to completion. 3 Contents CHAPTER 1 GENERAL INTRODUCTION ....................................................................................... 10 1.1 Permanent Magnet Machines ...................................................................................................... 10 1.2 Halbach Magnetization ............................................................................................................... 15 1.2.1 PMLM .................................................................................................................................. 16 1.2.2 PMTM .................................................................................................................................. 18 1.2.3 SPMRM ............................................................................................................................... 19 1.3 Magnetic Field Models ............................................................................................................... 20 1.3.1 Numerical Models ................................................................................................................ 20 1.3.2 Analytical Models ................................................................................................................ 20 A. 1-D Field Model ....................................................................................................................... 21 B. 2-D Field Slotless Model .......................................................................................................... 21 C. 2-D Field Slotted Model ........................................................................................................... 21 D. LPC Model ............................................................................................................................... 22 1.3.3 Analytical Models for Halbach Array .................................................................................. 22 1.4 Outline of Thesis ......................................................................................................................... 24 1.5 Major Contributions of Thesis .................................................................................................... 25 CHAPTER 2 DETERMINATION OF THE MAXIMUM ELECTROMAGNETIC TORQUE IN PM BRUSHLESS MACHINES Having 2-SEGMENT HALBACH ARRAY ........................................... 26 2.1 Introduction ................................................................................................................................. 26 2.2 Development of Analytical Model .............................................................................................. 27 2.2.1 Magnetization Vectors ......................................................................................................... 27 A. Radially Magnetized 2-Segment Halbach Array ..................................................................... 27 B. Parallel Magnetized 2-Segment Halbach Array ....................................................................... 29 2.2.2 Field Produced by Magnets .................................................................................................. 31 2.2.3 Flux-Linkage, Back-EMF, and Electromagnetic Torque ..................................................... 34 A. Phase Flux-Linkage, Back-EMF, and Electromagnetic Torque............................................... 34 B. Electromagnetic Torque ........................................................................................................... 36 2.3 Simple Equation for determining Optimal Magnet Ratio ........................................................... 36 2.4 Finite-Element Validation and Investigations ............................................................................. 39 2.5 Measurements ............................................................................................................................. 45 2.5 Summary ..................................................................................................................................... 51 CHAPTER 3 GENERAL ANALYTICAL MODEL FOR CALCULATING ELECTROMAGNETIC PERFORMANCE OF PM BRUSHLESS MACHINES HAVING SEGMENTED HALBACH ARRAY ................................................................................................................................................ 52 4 3.1 Introduction ................................................................................................................................. 52 3.2 Development of General Analytical Model ................................................................................ 53 3.3 Finite-Element Validation ........................................................................................................... 64 3.3.1 2-Segment Halbach Array .................................................................................................... 67 3.3.2 3-Segment Halbach Array ........................................................................................................ 70 3.3.3 4-Segment Halbach Array ........................................................................................................ 71 3.3.4 5-Segment Halbach Array ........................................................................................................ 72 3.4 Investigation Based on General Analytical Model ..................................................................... 73 3.4.1 Curvature Effect in 2- and 3-Segment Halbach Array ............................................................. 73 3.4.2 Performance Analysis for 3-Phase 12-Slot Machines Having 2-Segment Halbach Array ...... 75 3.5 Summary ..................................................................................................................................... 77 3.6 Appendix ..................................................................................................................................... 77 CHAPTER 4 INVESTIGATION OF PM BRUSHLESS MACHINES HAVING UNEQUAL- MAGNET HEIGHT POLE ................................................................................................................... 87 4.1 Introduction ................................................................................................................................. 87 4.2 Analytical Sub-Domain Models for Hat- and T-Type PM Machines ......................................... 88 4.3 Finite-Element Validation ........................................................................................................... 93 4.4 Comparison
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